0$ (positive bias voltages) the first term of (5) contributes to the N(E) because of\n$\\delta(E_k-E)$.\nIn this case, as can see from Fig. 7 and Fig. 9c and Fig. 14a $|T_k|^2$ selects only a\nrelatively short region of states in k-space in which $u_k^2>0$.\nThese are the states with $\\xi_k>0$ (above the FS).\nFor the majority of states integrated over ( see again Fig.7\nand Fig.9c).\nAt $E< 0$ (negative bias voltages) the second term of (5) contributes to the\nDOS because of $\\delta(E_k+E)$. In this case, as can see from Fig.7 and\nFig. 9c and Fig. 14b, $|T_k|^2$ selects out a large region of k states where $v_k^2 >0$,\nin fact, equal to one. These states are below the Fermi surface, where $\\xi_k<0$.\nThe overall effect then is to have a large negative bias conductance compared\nto the positive one. This is true as for s- and d-wave symmetry. Hence, the\nunderlying asymmetry of the conductance peaks is primarily due to the band\nstructure $\\xi_k$ and d-wave symmetry simply enhances the degree of asymmetry\nof the peaks. So, the peaks' asymmetry existing as for s-wave and d-wave symmetry\nis sensitive to\nband structure $\\xi_k$.\n\\end{multicols}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{13as.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{13bs.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-1.78cm}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{13ad.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{13bd.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.13} Numerical calculation of the quasiparticle DOS with a s-wave (a,b)\nand d-wave (c,d)\ngap symmetry at $\\Gamma$=3 meV (a,c)\n and $\\Gamma$=9 meV (b,d)\nfor different spread $\\Theta_0$.\n%\\vspace{-1cm}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=2.0in\n\\epsffile{15a.eps}\n\\hspace{1cm}\n\\epsfxsize=2.0in\n\\epsffile{15b.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.14} ARPES energy spectrum along $\\Theta=0.25$. The values of\ncoherent factors correspond to $E>0$ (a) and $E<0$ (b).\n\\begin{multicols}{2}\nIn summary, by changing of the energy gap $\\Delta_0$ in HTSC one may\nmodel the influence of\nnonmagnetic impurities on the DOS. We consider that the asymmetry of the\nquasiparticle peaks is due to the\nspecific features of the energy spectrum of HTSC and that the d-wave gap symmetry\nonly enhances\n the peaks' asymmetry.\\par\nThe absence of the VHS peak on the experimental $dI/dV$-charactristics\nmeans the large enough lifetime\nbroadening factor $\\Gamma$ in HTSC.\n\\section{Acknowledgement}\nWe thank Professor Y. Sobouti and Professor M.R. H. Khajehpour for useful discussions.\n\\end{multicols}\n\\begin{multicols}{2}\n\\begin{references}\n\n\\bibitem{} Yusof Z., Zasadzinski J.F., Coffey L., Miyakawa N., Modeling of tunneling\nspectroscopy in the high-Tc superconductors incorporating band structure,\ngap symmetry, group velocity, and tunneling directionality, {\\it Phys.Rev.} {\\bf B 58}\n(1998) pp.514-521.\n\\bibitem{} Schlenga K., Kleiner R., Hechtfischer G., Moessle M., Schmitt S., Mueller\nPaul, Helm Ch., Preis Ch.,Forsthofer F., Keller J., Veith M., Steinbess E.,\nTunneling Spectroscopy with intrinsic Josephson junctions in $BiSrCaCuO$ and\n$TlBaCaCuO$,{\\it Phys.Rev.} {\\bf B 57}(1998) pp.14518-14536.\n\\bibitem{}Shukrinov Yu.M., Nasrulloev Kh., Mirzoaminov Kh., Sarhadov I., Layered\nmodels and tunneling in HTS,{\\it Appl. Superconductivity.} {\\bf 2} (1994) pp.741-745.\n\\bibitem{} Shukrinov Yu.M., Stetsenko A., Nasrulloev Kh., Kohandel M., Tunneling in\nHTS, {\\it IEEE Transaction on Appl. Supercond.} {\\bf 8} (1998) pp.142-145.\n\\bibitem{} Latyshev Yu.I.,Yamashita T., Bulaevskii L.N., Graf M.J.,Balatsky A.V.,\nMaley M.P., Interlayer Transport of Quasiparticles and Cooper pairs in\n$BiSrCaCuO$ Superconductors, Cond-Mat/9903256, 17 Mar 1999, pp.1-4.\n\\bibitem{} Shukrinov Yu., Seidel P., Scherbel J., Quasiparticle current in the\nintrinsic Josephson junctions in $TBCCO$, will be published.\n\\bibitem{}Cucolo A.M., Zero bias conductance peaks in high-Tc superconductors:\nclues and ambiguities of two mutually excluding models, {\\it Physica} {\\bf C 305}\n(1998) pp.85-94.\n\\bibitem{} Ozyuzer L., Yusof Z., Zasadzinski J., Li T., Hinks T., Gray K.E.,\nTunneling Spectroscopy of $Tl_2Ba_2CuO_6$, Cond-mat/9905370, 25 May 1999,pp.1-7.\n\\bibitem{} Y.DeWilde, N.Miyakawa, P.Guptasarma, I.Iavarone, L.Ozyuzer, J.F.Zasadzinski, P.Romano, D.G.Hinks, C.Kendziora, G.W.Crabtree, and K.E.Gray, Unusual Strong-Coupling Effects in the Tunneling Spectroscopy of Optimally Doped and Overdoped $BiSrCaCuO$, {\\it Phys.Rev.Lett.} 80 (1998) pp.153-156.\n\\bibitem{} K.Kouznetsov and L.Coffey, Theory of tunneling and photoemmission spectroscopy for high-temperature superconductors, {\\it Phys.Rev.}{\\bf B, 54} (1996) pp.3617-3621.\n\\bibitem{} J.R.Kirtley and D.J.Scalapino, Inelastic-tunneling model for the linear conductance\nbackground in the high-$T_c$ superconductors, {\\it Phys.Rev.Lett.}, 65 (1990) pp.798.\n\\bibitem{} A.J.Fedro and D.D.Koelling, Interplay between band structure and gap symmetry in a two-dimmensional anisotropic superconductor, {\\it Phys.Rev.}{\\bf B, 47} (1993) pp.14342-14347.\n\\bibitem{} M. R. Norman, M. Randeria, H. Ding, and Campuzano, Phenomenological models for the gap anisotropy of $Bi_2Sr_2CaCu_2O_8$ as measured by angle-resolved photoemission spectroscopy, {\\it Phys. Rev.} {\\bf B 52 }(1995) pp.615-622.\n\\bibitem{} A. A. Abrikosov, On the nature of the order parameter in HTSC\nand influence of impurities, {\\it Physica} {\\bf C}, 244 (1995) 243-255\n\\end{references}\n\\end{multicols}\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n"}],"string":"[\n {\n \"name\": \"modeling.tex\",\n \"string\": \"%\\\\documentstyle[aps,multicol,epsf]{revtex}\\n\\\\documentstyle[multicol,aps,prb,tabularx,epsf]{revtex}\\n%\\\\documentstyle[preprint,aps]{revtex}\\n\\\\def\\\\d{{\\\\rm d}}\\n\\\\begin{document}\\n\\\\title{Modeling of Tunneling Spectroscopy in HTSC}\\n\\\\author{Yu. M. Shukrinov~$^{a,b}$, A. Namiranian~$^{a}$,A. Najafi~$^{a}$}\\n\\\\address{$^{a}$ Institute for Advanced Studies in Basic Sciences, Gava Zang,\\nZanjan 45195-159, Iran\\\\\\\\\\n$^{b}$Physical Technical Institute of Tajik Academy of\\nSciences\\\\\\\\\\n 299/1 Aini Str., Dushanbe, 734063, Tajikistan, C.I.S.}\\n\\\\date{\\\\today}\\n\\\\maketitle\\n\\\\begin{abstract}\\nThe tunneling density of states of HTSC is calculated taking\\ninto account tight binding band structure, group\\nvelocity, and tunneling directionality for s-wave and d-wave gap symmetry.\\nThe characteristic density of states has quasiparticle peaks' asymmetry, flat\\ns-wave and cusplike d-wave case subgap behavior, and asymmetric background.\\nWe consider that the underlying asymmetry of the conductance peaks\\nis primarily due to the features of quasiparticle energy spectrum, and the\\nd-wave symmetry enhances the degree of the peaks' asymmetry. Increasing of the\\nlifetime broadening factor changes the degree of tunneling conductance peaks'\\nasymmetry, and leads to the confluence of the quasiparticle and van Hove singularity\\npeaks.\\n\\n\\\\end{abstract}\\n\\\\pacs{74.50, 74.80.F}\\n%%\\n\\\\begin{multicols}{2}\\n%%\\n\\n\\n\\\\section{ INTRODUCTION}\\nTunneling measurements on HTSC have revealed a rich variety of properties\\nand characteristics [1-4]. They may be classified according to their low and high energy\\nfeatures. With the low energy features we may attribute: (i) variable subgap\\nshape of conductance, ranging from sharp, cusplike, to a flat, BCS-like\\nfeature [1]; (ii) voltage and temperature dependence of quasiparticle\\nconductivity [5,6]; (iii) subgap structure [2]; (iv) zero bias conductance\\npeak (ZBCP) [7]. The high energy features include: (i) asymmetry of conductance\\npeaks [1],(ii) van Hove singularity (VHS), (iii) conductance shape outside of the gap region (background (BG)) and\\nits asymmetry [1]; (iv) dip feature [8]; (v) hump feature [8].These features are collected in schematic Fig 1 .\\n\\n\\n\\n\\n\\\\begin{figure}\\n\\\\epsfysize=2.0in\\n\\\\centerline{\\\\epsffile{1.eps}}\\n{\\\\bf Fig.1} Schematic $\\\\frac{dI}{dV}$-characteristic of NIS-structure with the main features.\\n\\\\label{Fig1}\\n\\\\end{figure}\\nWhile the tunneling spectroscopy on\\nconventional superconductors allows directly to find the energy gap of\\nsuperconductor, the same measurements in HTSC are not as easily interpreted. Some times the same\\nexperiments on the same samples show different results [9] : cusplike or flat subgap feature, symmetric or asymmetric\\nconductance peaks. Usually the sharpest gap features are obtained when BG\\nis weakly decreasing. A quantitative measure of it is the ratio\\nof the conductance peak height (PH) to the background conductance:\\n PHB=PH/BG.\\nWhen the BG conductance is decreasing, the $PHB>2$, but when\\nBG conductance is linearly increasing ($\\\\sim$V),\\n$PHB<2$ . Kouznetsov and Coffey [10] and Kirtly and Scalapino [11]\\nsuggested that the linearly increasing BG is arising from inelastic tunneling.\\nAs was mentioned in [1], the conductance is dominated by quasiparticle tunneling and that the effect of Andreev reflection is not significant.\\nA theoretical model for tunneling spectroscopy employing tight-binding\\nband structure, $d_{x^2-y^2}$ gap symmetry, group velocity and tunneling\\ndirectionality was studied by Z. Yusof, J. F. Zasadzinski, L. Coffey and\\nN. Miyahawa [1]. An angle resolved photoemission spectroscopy (ARPES)\\nband structure specific to optimally-doped BSCCO (Bi-2212) was used to\\ncalculate the tunneling density of states for a direct comparison to the\\nexperimental tunneling conductance. This model produces an asymmetric,\\ndecreasing conductance background, asymmetric conductance peaks and\\nvariable subgap shape, ranging from sharp, cusplike to a flat, BCS-like\\nfeature.\\nA standard technique in analyzing the tunneling conductance\\nis to use a smeared BCS function\\n\\\\begin{equation}\\nN(E)=N(0)\\\\frac{E-i\\\\Gamma}{\\\\sqrt{(E-i\\\\Gamma)^2-\\\\Delta^2}}\\n\\\\end{equation}\\nin which a scattering rate parameter (lifetime broadening factor) $\\\\Gamma$ is used to take into account\\nany broadening of the gap region in the DOS. Fig. 2 shows the DOS\\ncalculated by formula (1) at $\\\\Delta=46 $ meV and $\\\\Gamma=9$ meV (a),\\n$\\\\Gamma=3$ meV (b) and $\\\\Gamma=0$ (c). The characteristic features\\nof the DOS is the flat subgap structure at small\\n$\\\\Gamma$. This method can not explain the asymmetry of the conductance peaks observed in the\\ntunneling experiments.\\n\\nIn the case of d-wave symmetry we have\\n\\\\end{multicols}\\n\\n\\\\begin{figure}\\n\\\\nonumber\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=2.4in\\n\\\\epsffile{2a.eps}\\n\\\\epsfxsize=2.3in\\n\\\\epsffile{2b.eps}\\n\\\\epsfxsize=2.3in\\n\\\\epsffile{2c.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.8cm}\\n{\\\\bf Fig.2} s-wave DOS at $\\\\Delta=46$ meV and different values of $\\\\Gamma$, calculated by formula (1).\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=2.3in\\n\\\\epsffile{3a.eps}\\n\\\\epsfxsize=2.3in\\n\\\\epsffile{3b.eps}\\n\\\\epsfxsize=2.4in\\n\\\\epsffile{3c.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.8cm}\\n{\\\\bf Fig.3} d-wave DOS at $\\\\Delta=46$ meV and different values of $\\\\Gamma$, calculated by formula (2).\\n\\\\vspace{0.3cm}\\n\\\\begin{multicols}{2}\\n\\\\begin{equation}\\nN(E)=N(0)Re\\\\int_{0}^{2\\\\pi}\\\\frac{d\\\\phi}{2\\\\pi}\\n\\\\frac{E-i\\\\Gamma}{\\\\sqrt{(E-i\\\\Gamma)^2-\\\\Delta_{0}^2cos^{2}(2\\\\phi)}}\\n\\\\end{equation}\\n\\n\\\\noindent and DOS calculated by this formula are presented in Fig. 3.\\nThe characteristic features of the DOS is the cusplike subgap structure. As was mentioned\\nin [1] this standard technique requires that the comparison be made with\\nnormalized tunneling conductance date, and since HTSC tunneling\\nconductance can exhibit a varied and complex background shape, this\\nprocedure may \\\"filter out\\\" too much information from the conductance\\ndata. An alternative is to simply normalize the data by a constant.\\\\par\\nIn [8] the tunneling data were first normalized by constructing a\\n\\\"normal state\\\" conductance obtained by fitting the high bias data to a\\nthird order polynomial. The normalized conductance date were compared\\nto a weighted momentum averaged d-wave DOS\\n\\\\begin{equation}\\nN(E)=\\\\int f(\\\\phi)\\\\frac{E-i\\\\Gamma}\\n{\\\\sqrt{(E-i\\\\Gamma)^2-\\\\Delta_{0}^2cos^{2}(2\\\\phi)}}d\\\\phi\\n\\\\end{equation}\\nHere $f(\\\\phi)$ is an angular weighting function, which allows for a better fit\\nwith the experimental date in the gap region. A weighting function\\n$f(\\\\phi)=1+0.4cos(4\\\\phi)$ was used which imposed a preferential angular\\nselection of the DOS along the absolute maximum of the d-wave gap and\\ntapers off towards the nodes of the gap. This is a rather weak directional\\nfunction since the minimum of $f(0)$ along the nodes of the d-wave\\ngap is still none-negligible [8].\\\\par\\n\\nA. J. Fedro and D. Koelling [12] have done the modeling of the normal state\\nand superconducting DOS of HTSC, using tight-binding band structure,\\nincluding the next nearest neighbors\\n\\\\begin{equation}\\n\\\\xi_k=-2t(cos(k_xa)+cos(k_ya))+4t'cos(k_xa)cos(k_ya)-\\\\mu\\n\\\\end{equation}\\n\\nThe calculation showed two singularities in DOS: a van Hove\\nsingularity in the center of energy band due to saddle point near ($\\\\pi$,0) at\\n$t'=0$ and another at the lower edge of the energy band due to extra flattening\\nout at (0,0).\\nAs extended s-wave and d-wave superconducting DOS were considered\\nin case of hole-doped situation ($\\\\mu<0$) for different hole concentration.\\nThe Fermi surface for $t'=0$ and $t'=0.45t$ at the same concentration,\\ncorresponding $\\\\mu/2t=-0.187$ which was used in [12] are presented in\\nFig. 4a. It is needed to move up from the Fermi surface (set as the zero energy)\\nto reach the point ($\\\\pi$,0) in case $t'=0$ and move down in case\\n$t'=0.45t$. So, for $t'=0$ the Fermi energy lies to the left of the van Hove\\nsingularity and will move away from it with increased hole doping while for\\n$t'=0.45t$ it lies to the right and will move towards to with increased hole\\ndoping (See Fig. 4b where the DOS for $t'=0$ and $t'=0.45$ are presented ).\\n For calculation of the superconducting DOS Fedro and Koelling used formula\\n\\\\begin{equation}\\nN(E)=\\\\frac{1}{2}\\\\sum_{k}(1+\\\\frac{\\\\xi_k}{E_k})\\\\delta(E-E_k)+\\n(1-\\\\frac{\\\\xi_k}{E_k})\\\\delta(E+E_k)\\n\\\\end{equation}\\n\\nThis formula is the limit of the expression for tunneling\\ndensity of states (6) at $\\\\Gamma=0$ and $|T_k|^2=1$, where $T_k$ is tunneling matrix element.\\n\\\\end{multicols}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{4a.eps}\\n\\\\hspace{0.5cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{4b.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n{\\\\bf Fig.4} Fermi surfaces (left) and DOS (right) for $t'$=0 (solid lines) and $t'$=0.45t (dashed lines) in formula (4) at $\\\\frac{\\\\mu}{2t}$=-0.187 which corresponds\\nhole - doped situation.\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{5s.eps}\\n\\\\hspace{0.3cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{5d.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n{\\\\bf Fig.5} DOS for $t'$=0 at different $\\\\Gamma$ for s-wave symmetry (a) and\\nd-wave symmetry (b), calculated by formula (6).\\n\\\\begin{multicols}{2}\\nThe Fig.5a shows the result of calculation\\n of the DOS for $t'$ = 0 at $\\\\Gamma=$ 0.07, 0.1 and 0.2 meV for s-wave\\n symmetry which reflect the results of Fedro and Koelling. Fig.5b shows the same DOS for d-wave symmetry. In both cases the Fermi energy\\n(set as the zero of energy) lies to the left of the van Hove singularity.\\nThere is the peaks' asymmetry which is more pronounced at large $\\\\Gamma$.\\n\\\\section{ Models and Methods}\\nIn this paper we use the method for calculation of the DOS presented in [1].\\nThe tunneling DOS of a superconductor is determined by the imaginary part\\nof the retarded single particle Green's function\\n\\\\begin{equation}\\nN(E)=-\\\\frac{1}{\\\\pi}Im\\\\sum_{k}|T_k|^2G^{R}(k,E)\\n\\\\end{equation}\\nFor the superconducting state\\n\\\\begin{equation}\\nG^{R}(k,E)=\\\\frac{u_{k}^{2}}{E-E_k+i\\\\Gamma}+\\n\\\\frac{v_{k}^{2}}{E+E_k+i\\\\Gamma}\\n\\\\end{equation}\\nwhere $u_k^2$ and $v_k^2$ are the usual coherence factors,\\n\\\\begin{eqnarray}\\nu_k^2=\\\\frac{1}{2}(1+\\\\frac{\\\\xi_k}{E_k})\\\\cr\\nv_k^2=\\\\frac{1}{2}(1-\\\\frac{\\\\xi_k}{E_k})\\n\\\\end{eqnarray}\\nand $\\\\Gamma$ is the\\nquasiparticle lifetime broadening factor. The energy spectrum of\\nquasiparticles in the superconducting state is determined by\\n\\\\begin{equation}\\nE_k=\\\\sqrt{|\\\\Delta(k)|^2+\\\\xi_k^2}\\n\\\\end{equation}\\nwith the effective band structure extracted from ARPES experiments [13]\\n\\\\begin{eqnarray}\\n\\\\xi_k=C_0+0.5C_1[cos(k_xa)+cos(k_ya)]~~~~~~~~~~~~~~~~~~~~\\\\cr\\n~~+C_2cos(k_xa)cos(k_ya)+\\n0.5C_3[cos(2k_xa)+cos(2k_ya)]\\\\cr\\n~+0.5C_4\\n[cos(2k_xa)cos(k_ya)+cos(k_xa)cos(2k_ya)]\\\\cr\\n+C_5cos(2k_xa)cos(2k_ya)\\n\\\\end{eqnarray}\\nHere $\\\\xi_k$ is measured with respect to the Fermi energy ($\\\\xi_k$=0), and\\nthe phenomenological parameters are (in units of eV) $C_0=0.1305$,\\n$C_1=-0.5951$, $C_2=0.1636$, $C_3=-0.0519$, $C_4=-0.1117$, $C_5=0.0510$.\\\\par\\n\\\\end{multicols}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=2.1in\\n\\\\epsffile{6a.eps}\\n\\\\hspace{0.5cm}\\n\\\\epsfxsize=2.1in\\n\\\\epsffile{7a.eps}\\n\\\\hspace{0.5cm}\\n\\\\epsfxsize=1.6in\\n\\\\epsffile{8.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n\\\\noindent {\\\\bf Fig.6}(left) 3D-plot of energy spectrum of normal\\nstate according to formula (9).\\\\\\\\\\n{\\\\bf Fig.7}(center) 3D-plot of coherence factor $u_k^2$ according to formula (8).\\\\\\\\\\n{\\\\bf Fig.8}(right) Fermi surface corresponding to the $\\\\xi_k=0$ in formula (10). Dark straight line shows the line of directional tunneling, the dashed lines show the angular spread $\\\\Theta_0$.\\n\\\\begin{multicols}{2}\\nFig. 6 shows the three dimensional image of function (10) .\\nThere are saddle\\npoint in ($\\\\pi$,0) and flattening out of the energy band at (0,0) which\\nlead to the van Hove singularities in the DOS. The three dimensional graph of the coherence factor $u_k^2$\\nis shown in Fig. 7.\\\\par\\nSince quasiparticles with momentum perpendicular to the barrier interface\\nhave the highest probability of tunneling, the tunneling matrix element $|T_k|^2$ reveals\\na need for factors of directionality $D(k)$ and group velocity $v_g(k)$ [1]. The group velocity\\nfactor is defined by\\n\\\\begin{equation}\\nv_g(k)=|\\\\vec \\\\nabla_k\\\\xi_k.\\\\hat n|=\\n|\\\\frac{\\\\partial\\\\xi_k}{\\\\partial k_x}cos(\\\\theta)+\\n\\\\frac{\\\\partial\\\\xi_k}{\\\\partial k_y}sin(\\\\theta)|\\n\\\\end{equation}\\nwhere the unit vector $n$ defines the tunneling direction as shown in Fig. 8,\\nwhich is perpendicular to the plane of the junction.\\\\par\\nThe directionality function $D(k)$ is defined by\\n\\\\begin{equation}\\nD(k)=exp[-\\\\frac{k^2-(\\\\bf k.\\\\hat n)^2}{( {\\\\bf k.\\\\hat n})^2\\\\Theta_0^2}]\\n\\\\end{equation}\\nHere $\\\\Theta_0$ defines the angular spread of the quasiparticle momentum with none-negligible tunneling probability\\nwith respect to n. The tunneling matrix element $|T_k|^2$ is written as\\n\\\\begin{equation}\\n|T_k|^2=v_g(k)D(k)\\n\\\\end{equation}\\nThe three dimensional graphs of group velocity $v_g(k)$, directionality $D(k)$\\nand tunneling matrix element $|T_k|^2$\\nfunctions are shown in Fig 9.\\n\\\\section{Results and discussions}\\nDifferent factors may lead to the changing of the energy gap $\\\\Delta _0$ in HTSC.\\nIn particular, strong effects are caused by nonmagnetic impurities [14].\\nIn superconductors with d-wave symmetry the nonmagnetic impurities\\ndestroy the superconductivity very efficiently. Possibility to destroy of Cooper pairs by\\nimpurities leads to their finite lifetime.\\nIf the state with the quasiparticle is not stationary state, it must attenuate with time\\ndue to transitions to other states.\\nThe corresponding wave function has the form\\n$e^{-i\\\\xi (p)t/\\\\hbar-\\\\Gamma t/\\\\hbar}$, where $\\\\Gamma$ is\\nproportional to the probability of the transitions to the other states.\\nIt may be interpreted as the energy of the quasiparticle\\nhas the imaginary addition $-i\\\\Gamma$. The relation between $\\\\Gamma$ and\\n\\\\end{multicols}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=2.1in\\n\\\\epsffile{9a.eps}\\n\\\\epsfxsize=2.3in\\n\\\\epsffile{9b.eps}\\n\\\\epsfxsize=2.0in\\n\\\\epsffile{9c.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n{\\\\bf Fig.9} 3D plot of the group velocity function (a), the directionality function\\n(b) and the tunneling matrix element according to formulas (11), (12) and (13),\\ncorrespondingly.\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{10as.eps}\\n\\\\hspace{-0.6cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{10bs.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-1.78cm}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{10ad.eps}\\n\\\\hspace{-0.6cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{10bd.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n{\\\\bf Fig.10} The changing of DOS with energy gap $\\\\Delta_0$ for s- (a,b) and d-wave (c,d) symmetry at $\\\\Gamma$=3 meV (a,c) and $\\\\Gamma$=9 meV (b,d) without of effects of directionality and group velocity.\\n\\\\begin{multicols}{2}\\n\\\\noindent lifetime of quasiparticle $\\\\tau_s$\\nis $\\\\Gamma=\\\\hbar/\\\\tau_s$. Hence, the impurities lead to changing $\\\\Delta_0$ and\\nwe may do modeling of the\\ninfluence of impurities on tunneling conductance by numerical calculation of DOS\\nN(E) considering different\\nvalues of $\\\\Delta_0$ in formula (6). Here we present results of calculation N(E) at\\n$\\\\Delta_0=\\\\alpha\\\\Delta_{00}$,\\nwhere $\\\\alpha$=0.2, 0.4, 0.6, 0.8, 1 and $\\\\Delta_{00}$= 46 meV.\\\\par\\nThe peculiarities of the quasiparticle energy spectrum (10) play an essential role in\\nexplanation of the\\nconductance features. Here, based on the numerical calculation of DOS\\nwe consider that\\nthe underlying asymmetry of the conductance peaks is primarily due to the features\\nof quasiparticles energy\\nspectrum. The d-wave gap symmetry simply enhances the degree of the peaks\\nasymmetry. The last one is also\\nchanged by changing the tunneling direction.\\\\par\\nFig. 10 shows the results of the numerical calculations of the DOS at\\n$\\\\Gamma _0 =3$ meV (a,c) and $\\\\Gamma _0 =9$ meV (b,d) without effects\\nof group velocity and directionality as for s-wave (a,b) and d-wave (c,d) gap\\nsymmetry, correspondingly for different values of energy gap $\\\\Delta_0$.\\nWe have decreased the energy gap $\\\\Delta _0$, starting from $\\\\Delta _0=46$meV.\\nFor clarity we present only three characteristic curves, which corresponds to\\n$\\\\alpha \\\\Delta _0$ with $\\\\alpha=$1, 0.6 and 0.2.\\nWe exclude the effects of group velocity and directionality to demonstrate that\\nthey are not responsible\\nfor peaks asymmetry.\\\\par\\nThere is the asymmetry of the quasiparticle peak heights as for s- and d-wave\\nsymmetry. So, the origin of\\nthe peaks asymmetry is not due to d-wave symmetry of the energy gap of HTSC.\\nThere is more flat subgap behavior of DOS in the case of s-wave symmetry in\\ncomparing with the d-wave case. The increase of lifetime broadening factor\\n$\\\\Gamma$ leads to the enhance of the peaks' asymmetry. There are van Hove\\nsingularities in the DOS at small $\\\\Gamma$. The increase of $\\\\Gamma$ leads to\\nthe confluence\\nof the quasiparticle and VHS peaks and this results to the enhance\\nof the DOS peaks asymmetry due to saddle point in energy spectrum (10)\\nat ($\\\\pi$,0).\\nAlso note to the asymmetry of the background as for s- and d-wave\\ngap symmetry.\\\\par\\nFig. 11 shows the $\\\\Delta _0$-dependence of DOS taking into account the effects\\nof group velocity and directionality at $\\\\Gamma=3$ meV (a,c) and\\n$\\\\Gamma=9$ meV (b,d) for s-wave (a,b) and d-wave (c,d) gap symmetry.\\nAs in [1] we have taken $\\\\Theta =0.25$ and $\\\\Theta _0 =0.1$.\\\\par\\nThere is also quasiparticle peaks' asymmetry similar to s- and d-wave cases.\\nBut in d-wave case the asymmetry is more stronger than in s-wave.\\nThe effects of group velocity and directionality lead to disappear of the VHS\\nin DOS.\\nThe increase of $\\\\Gamma$ enhances the quasiparticle peak asymmetry.\\nThe most strong effect of energy band structure on the DOS occurs along\\n$k_x$-axis due to van Hove singularity at $(\\\\pi,0)$.\\\\par\\n\\\\end{multicols}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{11as.eps}\\n\\\\hspace{-0.6cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{11bs.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-1.8cm}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{11ad.eps}\\n\\\\hspace{-0.6cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{11bd.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n{\\\\bf Fig.11} The changing of DOS with energy gap $\\\\Delta_0$ for s- wave\\n(a,b) and d-wave (c,d) gap symmetry at $\\\\Gamma$=3 meV (a,c) and\\n$\\\\Gamma$=9 meV (b,d) with the\\n effects of directionality and group velocity.\\n\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{12as.eps}\\n\\\\hspace{-0.6cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{12bs.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-1.78cm}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{12ad.eps}\\n\\\\hspace{-0.6cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{12bd.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n{\\\\bf Fig.12} Effects of directionality on the DOS as in s-wave (a,b) and d-wave gap symmetry at $\\\\Gamma$=3 meV (a,c) and $\\\\Gamma$=9 meV (b,d).\\n\\n\\\\begin{multicols}{2}\\nFig. 12 has demonstrated this effect. We have presented the DOS at different\\n$\\\\Theta$ at $\\\\Gamma=3$ meV (a,c) and $\\\\Gamma=9$ meV (b,d) as for s-wave\\n(a,b) and d-wave (c,d) gap symmetry.\\nIn the case of s-symmetry the position of the quasiparticle peaks is constant excluding the direction\\nalong $k_x$( $\\\\Theta=0$). We pay attention to the strong peaks' asymmetry\\nin this case.\\\\par\\nIn the case of d-wave symmetry we have practically the same behavior around\\n$k_x$ direction as for s-wave, but energy gap is changed due to the\\n$\\\\Theta$-dependence of $\\\\Delta_0$ and correspondingly, the quasiparticle peaks\\nare shifted to the zero energy.\\\\par\\n Fig.13 shows the $\\\\Theta_0$-changing of DOS at $\\\\Gamma=3$ meV (a,c)\\nand $\\\\Gamma=9$ meV (b,d) as for s-wave (a,b) and d-wave (c,d) gap symmetry.\\nThe increase of $\\\\Theta_0$ means the taking into play (inclusion) the states,\\nclose to ($\\\\pi,0 $). It is reflected as an appearance of the van Hove singularity\\nas in case of s-wave and d-wave gap symmetry at small $\\\\Gamma$. The VHS\\nis more pronounced in case of d-wave in comparing with s-wave symmetry. The increase of $\\\\Gamma$ leads to confluence of the quasiparticle and VHS peaks.\\\\par\\nWe consider that the absence VHS peak on the experimental\\n$dI/dV$-characteristics means the enough large lifetime broadening factor\\n$\\\\Gamma$ in that HTSC material.\\n\\nThe origin of the peaks asymmetry in the tunneling DOS was studied in [1]\\nby considering the role of the tunneling matrix element $|T_k|^2$ in the clean\\nlimit case ($\\\\Gamma=0$), where for the calculation N(E) was used the formula (5).\\\\par\\nWe repeat the explanation of the paper [1] because we believe that the following\\nconclusion on the origin of the peaks asymmetry must be different.\\nAt $E> 0$ (positive bias voltages) the first term of (5) contributes to the N(E) because of\\n$\\\\delta(E_k-E)$.\\nIn this case, as can see from Fig. 7 and Fig. 9c and Fig. 14a $|T_k|^2$ selects only a\\nrelatively short region of states in k-space in which $u_k^2>0$.\\nThese are the states with $\\\\xi_k>0$ (above the FS).\\nFor the majority of states integrated over ( see again Fig.7\\nand Fig.9c).\\nAt $E< 0$ (negative bias voltages) the second term of (5) contributes to the\\nDOS because of $\\\\delta(E_k+E)$. In this case, as can see from Fig.7 and\\nFig. 9c and Fig. 14b, $|T_k|^2$ selects out a large region of k states where $v_k^2 >0$,\\nin fact, equal to one. These states are below the Fermi surface, where $\\\\xi_k<0$.\\nThe overall effect then is to have a large negative bias conductance compared\\nto the positive one. This is true as for s- and d-wave symmetry. Hence, the\\nunderlying asymmetry of the conductance peaks is primarily due to the band\\nstructure $\\\\xi_k$ and d-wave symmetry simply enhances the degree of asymmetry\\nof the peaks. So, the peaks' asymmetry existing as for s-wave and d-wave symmetry\\nis sensitive to\\nband structure $\\\\xi_k$.\\n\\\\end{multicols}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{13as.eps}\\n\\\\hspace{-0.6cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{13bs.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-1.78cm}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{13ad.eps}\\n\\\\hspace{-0.6cm}\\n\\\\epsfxsize=3.0in\\n\\\\epsffile{13bd.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n{\\\\bf Fig.13} Numerical calculation of the quasiparticle DOS with a s-wave (a,b)\\nand d-wave (c,d)\\ngap symmetry at $\\\\Gamma$=3 meV (a,c)\\n and $\\\\Gamma$=9 meV (b,d)\\nfor different spread $\\\\Theta_0$.\\n%\\\\vspace{-1cm}\\n\\\\begin{figure}\\n\\\\begin{center}\\n\\\\leavevmode\\n\\\\hbox{%\\n\\\\epsfxsize=2.0in\\n\\\\epsffile{15a.eps}\\n\\\\hspace{1cm}\\n\\\\epsfxsize=2.0in\\n\\\\epsffile{15b.eps}}\\n\\\\end{center}\\n\\\\end{figure}\\n\\\\vspace{-0.6cm}\\n{\\\\bf Fig.14} ARPES energy spectrum along $\\\\Theta=0.25$. The values of\\ncoherent factors correspond to $E>0$ (a) and $E<0$ (b).\\n\\\\begin{multicols}{2}\\nIn summary, by changing of the energy gap $\\\\Delta_0$ in HTSC one may\\nmodel the influence of\\nnonmagnetic impurities on the DOS. We consider that the asymmetry of the\\nquasiparticle peaks is due to the\\nspecific features of the energy spectrum of HTSC and that the d-wave gap symmetry\\nonly enhances\\n the peaks' asymmetry.\\\\par\\nThe absence of the VHS peak on the experimental $dI/dV$-charactristics\\nmeans the large enough lifetime\\nbroadening factor $\\\\Gamma$ in HTSC.\\n\\\\section{Acknowledgement}\\nWe thank Professor Y. Sobouti and Professor M.R. H. Khajehpour for useful discussions.\\n\\\\end{multicols}\\n\\\\begin{multicols}{2}\\n\\\\begin{references}\\n\\n\\\\bibitem{} Yusof Z., Zasadzinski J.F., Coffey L., Miyakawa N., Modeling of tunneling\\nspectroscopy in the high-Tc superconductors incorporating band structure,\\ngap symmetry, group velocity, and tunneling directionality, {\\\\it Phys.Rev.} {\\\\bf B 58}\\n(1998) pp.514-521.\\n\\\\bibitem{} Schlenga K., Kleiner R., Hechtfischer G., Moessle M., Schmitt S., Mueller\\nPaul, Helm Ch., Preis Ch.,Forsthofer F., Keller J., Veith M., Steinbess E.,\\nTunneling Spectroscopy with intrinsic Josephson junctions in $BiSrCaCuO$ and\\n$TlBaCaCuO$,{\\\\it Phys.Rev.} {\\\\bf B 57}(1998) pp.14518-14536.\\n\\\\bibitem{}Shukrinov Yu.M., Nasrulloev Kh., Mirzoaminov Kh., Sarhadov I., Layered\\nmodels and tunneling in HTS,{\\\\it Appl. Superconductivity.} {\\\\bf 2} (1994) pp.741-745.\\n\\\\bibitem{} Shukrinov Yu.M., Stetsenko A., Nasrulloev Kh., Kohandel M., Tunneling in\\nHTS, {\\\\it IEEE Transaction on Appl. Supercond.} {\\\\bf 8} (1998) pp.142-145.\\n\\\\bibitem{} Latyshev Yu.I.,Yamashita T., Bulaevskii L.N., Graf M.J.,Balatsky A.V.,\\nMaley M.P., Interlayer Transport of Quasiparticles and Cooper pairs in\\n$BiSrCaCuO$ Superconductors, Cond-Mat/9903256, 17 Mar 1999, pp.1-4.\\n\\\\bibitem{} Shukrinov Yu., Seidel P., Scherbel J., Quasiparticle current in the\\nintrinsic Josephson junctions in $TBCCO$, will be published.\\n\\\\bibitem{}Cucolo A.M., Zero bias conductance peaks in high-Tc superconductors:\\nclues and ambiguities of two mutually excluding models, {\\\\it Physica} {\\\\bf C 305}\\n(1998) pp.85-94.\\n\\\\bibitem{} Ozyuzer L., Yusof Z., Zasadzinski J., Li T., Hinks T., Gray K.E.,\\nTunneling Spectroscopy of $Tl_2Ba_2CuO_6$, Cond-mat/9905370, 25 May 1999,pp.1-7.\\n\\\\bibitem{} Y.DeWilde, N.Miyakawa, P.Guptasarma, I.Iavarone, L.Ozyuzer, J.F.Zasadzinski, P.Romano, D.G.Hinks, C.Kendziora, G.W.Crabtree, and K.E.Gray, Unusual Strong-Coupling Effects in the Tunneling Spectroscopy of Optimally Doped and Overdoped $BiSrCaCuO$, {\\\\it Phys.Rev.Lett.} 80 (1998) pp.153-156.\\n\\\\bibitem{} K.Kouznetsov and L.Coffey, Theory of tunneling and photoemmission spectroscopy for high-temperature superconductors, {\\\\it Phys.Rev.}{\\\\bf B, 54} (1996) pp.3617-3621.\\n\\\\bibitem{} J.R.Kirtley and D.J.Scalapino, Inelastic-tunneling model for the linear conductance\\nbackground in the high-$T_c$ superconductors, {\\\\it Phys.Rev.Lett.}, 65 (1990) pp.798.\\n\\\\bibitem{} A.J.Fedro and D.D.Koelling, Interplay between band structure and gap symmetry in a two-dimmensional anisotropic superconductor, {\\\\it Phys.Rev.}{\\\\bf B, 47} (1993) pp.14342-14347.\\n\\\\bibitem{} M. R. Norman, M. Randeria, H. Ding, and Campuzano, Phenomenological models for the gap anisotropy of $Bi_2Sr_2CaCu_2O_8$ as measured by angle-resolved photoemission spectroscopy, {\\\\it Phys. Rev.} {\\\\bf B 52 }(1995) pp.615-622.\\n\\\\bibitem{} A. A. Abrikosov, On the nature of the order parameter in HTSC\\nand influence of impurities, {\\\\it Physica} {\\\\bf C}, 244 (1995) 243-255\\n\\\\end{references}\\n\\\\end{multicols}\\n\\\\end{document}\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\\n\"\n }\n]"},"bibtex":{"kind":"list like","value":[{"name":"cond-mat0002077.extracted_bib","string":"\\bibitem{} Yusof Z., Zasadzinski J.F., Coffey L., Miyakawa N., Modeling of tunneling\nspectroscopy in the high-Tc superconductors incorporating band structure,\ngap symmetry, group velocity, and tunneling directionality, {\\it Phys.Rev.} {\\bf B 58}\n(1998) pp.514-521.\n\n\\bibitem{} Schlenga K., Kleiner R., Hechtfischer G., Moessle M., Schmitt S., Mueller\nPaul, Helm Ch., Preis Ch.,Forsthofer F., Keller J., Veith M., Steinbess E.,\nTunneling Spectroscopy with intrinsic Josephson junctions in $BiSrCaCuO$ and\n$TlBaCaCuO$,{\\it Phys.Rev.} {\\bf B 57}(1998) pp.14518-14536.\n\n\\bibitem{}Shukrinov Yu.M., Nasrulloev Kh., Mirzoaminov Kh., Sarhadov I., Layered\nmodels and tunneling in HTS,{\\it Appl. Superconductivity.} {\\bf 2} (1994) pp.741-745.\n\n\\bibitem{} Shukrinov Yu.M., Stetsenko A., Nasrulloev Kh., Kohandel M., Tunneling in\nHTS, {\\it IEEE Transaction on Appl. Supercond.} {\\bf 8} (1998) pp.142-145.\n\n\\bibitem{} Latyshev Yu.I.,Yamashita T., Bulaevskii L.N., Graf M.J.,Balatsky A.V.,\nMaley M.P., Interlayer Transport of Quasiparticles and Cooper pairs in\n$BiSrCaCuO$ Superconductors, Cond-Mat/9903256, 17 Mar 1999, pp.1-4.\n\n\\bibitem{} Shukrinov Yu., Seidel P., Scherbel J., Quasiparticle current in the\nintrinsic Josephson junctions in $TBCCO$, will be published.\n\n\\bibitem{}Cucolo A.M., Zero bias conductance peaks in high-Tc superconductors:\nclues and ambiguities of two mutually excluding models, {\\it Physica} {\\bf C 305}\n(1998) pp.85-94.\n\n\\bibitem{} Ozyuzer L., Yusof Z., Zasadzinski J., Li T., Hinks T., Gray K.E.,\nTunneling Spectroscopy of $Tl_2Ba_2CuO_6$, Cond-mat/9905370, 25 May 1999,pp.1-7.\n\n\\bibitem{} Y.DeWilde, N.Miyakawa, P.Guptasarma, I.Iavarone, L.Ozyuzer, J.F.Zasadzinski, P.Romano, D.G.Hinks, C.Kendziora, G.W.Crabtree, and K.E.Gray, Unusual Strong-Coupling Effects in the Tunneling Spectroscopy of Optimally Doped and Overdoped $BiSrCaCuO$, {\\it Phys.Rev.Lett.} 80 (1998) pp.153-156.\n\n\\bibitem{} K.Kouznetsov and L.Coffey, Theory of tunneling and photoemmission spectroscopy for high-temperature superconductors, {\\it Phys.Rev.}{\\bf B, 54} (1996) pp.3617-3621.\n\n\\bibitem{} J.R.Kirtley and D.J.Scalapino, Inelastic-tunneling model for the linear conductance\nbackground in the high-$T_c$ superconductors, {\\it Phys.Rev.Lett.}, 65 (1990) pp.798.\n\n\\bibitem{} A.J.Fedro and D.D.Koelling, Interplay between band structure and gap symmetry in a two-dimmensional anisotropic superconductor, {\\it Phys.Rev.}{\\bf B, 47} (1993) pp.14342-14347.\n\n\\bibitem{} M. R. Norman, M. Randeria, H. Ding, and Campuzano, Phenomenological models for the gap anisotropy of $Bi_2Sr_2CaCu_2O_8$ as measured by angle-resolved photoemission spectroscopy, {\\it Phys. Rev.} {\\bf B 52 }(1995) pp.615-622.\n\n\\bibitem{} A. A. Abrikosov, On the nature of the order parameter in HTSC\nand influence of impurities, {\\it Physica} {\\bf C}, 244 (1995) 243-255\n"}],"string":"[\n {\n \"name\": \"cond-mat0002077.extracted_bib\",\n \"string\": \"\\\\bibitem{} Yusof Z., Zasadzinski J.F., Coffey L., Miyakawa N., Modeling of tunneling\\nspectroscopy in the high-Tc superconductors incorporating band structure,\\ngap symmetry, group velocity, and tunneling directionality, {\\\\it Phys.Rev.} {\\\\bf B 58}\\n(1998) pp.514-521.\\n\\n\\\\bibitem{} Schlenga K., Kleiner R., Hechtfischer G., Moessle M., Schmitt S., Mueller\\nPaul, Helm Ch., Preis Ch.,Forsthofer F., Keller J., Veith M., Steinbess E.,\\nTunneling Spectroscopy with intrinsic Josephson junctions in $BiSrCaCuO$ and\\n$TlBaCaCuO$,{\\\\it Phys.Rev.} {\\\\bf B 57}(1998) pp.14518-14536.\\n\\n\\\\bibitem{}Shukrinov Yu.M., Nasrulloev Kh., Mirzoaminov Kh., Sarhadov I., Layered\\nmodels and tunneling in HTS,{\\\\it Appl. Superconductivity.} {\\\\bf 2} (1994) pp.741-745.\\n\\n\\\\bibitem{} Shukrinov Yu.M., Stetsenko A., Nasrulloev Kh., Kohandel M., Tunneling in\\nHTS, {\\\\it IEEE Transaction on Appl. Supercond.} {\\\\bf 8} (1998) pp.142-145.\\n\\n\\\\bibitem{} Latyshev Yu.I.,Yamashita T., Bulaevskii L.N., Graf M.J.,Balatsky A.V.,\\nMaley M.P., Interlayer Transport of Quasiparticles and Cooper pairs in\\n$BiSrCaCuO$ Superconductors, Cond-Mat/9903256, 17 Mar 1999, pp.1-4.\\n\\n\\\\bibitem{} Shukrinov Yu., Seidel P., Scherbel J., Quasiparticle current in the\\nintrinsic Josephson junctions in $TBCCO$, will be published.\\n\\n\\\\bibitem{}Cucolo A.M., Zero bias conductance peaks in high-Tc superconductors:\\nclues and ambiguities of two mutually excluding models, {\\\\it Physica} {\\\\bf C 305}\\n(1998) pp.85-94.\\n\\n\\\\bibitem{} Ozyuzer L., Yusof Z., Zasadzinski J., Li T., Hinks T., Gray K.E.,\\nTunneling Spectroscopy of $Tl_2Ba_2CuO_6$, Cond-mat/9905370, 25 May 1999,pp.1-7.\\n\\n\\\\bibitem{} Y.DeWilde, N.Miyakawa, P.Guptasarma, I.Iavarone, L.Ozyuzer, J.F.Zasadzinski, P.Romano, D.G.Hinks, C.Kendziora, G.W.Crabtree, and K.E.Gray, Unusual Strong-Coupling Effects in the Tunneling Spectroscopy of Optimally Doped and Overdoped $BiSrCaCuO$, {\\\\it Phys.Rev.Lett.} 80 (1998) pp.153-156.\\n\\n\\\\bibitem{} K.Kouznetsov and L.Coffey, Theory of tunneling and photoemmission spectroscopy for high-temperature superconductors, {\\\\it Phys.Rev.}{\\\\bf B, 54} (1996) pp.3617-3621.\\n\\n\\\\bibitem{} J.R.Kirtley and D.J.Scalapino, Inelastic-tunneling model for the linear conductance\\nbackground in the high-$T_c$ superconductors, {\\\\it Phys.Rev.Lett.}, 65 (1990) pp.798.\\n\\n\\\\bibitem{} A.J.Fedro and D.D.Koelling, Interplay between band structure and gap symmetry in a two-dimmensional anisotropic superconductor, {\\\\it Phys.Rev.}{\\\\bf B, 47} (1993) pp.14342-14347.\\n\\n\\\\bibitem{} M. R. Norman, M. Randeria, H. Ding, and Campuzano, Phenomenological models for the gap anisotropy of $Bi_2Sr_2CaCu_2O_8$ as measured by angle-resolved photoemission spectroscopy, {\\\\it Phys. Rev.} {\\\\bf B 52 }(1995) pp.615-622.\\n\\n\\\\bibitem{} A. A. Abrikosov, On the nature of the order parameter in HTSC\\nand influence of impurities, {\\\\it Physica} {\\\\bf C}, 244 (1995) 243-255\\n\"\n }\n]"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":5,"numItemsPerPage":100,"numTotalItems":1318468,"offset":500,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc2NTYxNjQyNCwic3ViIjoiL2RhdGFzZXRzL3N5bnRoZXRpeC1pbnN0aXR1dGUvbGF0ZXgtZGF0YS1wdWIiLCJleHAiOjE3NjU2MjAwMjQsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.8xfcZdK6LRChnUvKBK9Wd8aqzKa5VvGFvH971K_V_5EcQWbwLCUAalTe2es4A6FqIEzIm6ghJqCLcuSVmL-FAw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":1318468,"numBytesParquet":110673004212}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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astro-ph0002509
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OBSERVATIONAL CONSTRAINTS TO THE EVOLUTION OF MASSIVE STARS
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[
{
"author": "N. PANAGIA"
}
] |
We consider some aspects of the evolution of massive stars which can only be elucidated by means of ``indirect" observations, \ie measurements of the effects of massive stars on their environments. We discuss in detail the early evolution of massive stars formed in high metallicity regions as inferred from studies of HII regions in external galaxies.
|
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"name": "vulchem.tex",
"string": "% From:\tIN%\"francesc@mizar.ap.sissa.it\" \"Francesca Matteucci\" 21-OCT-1999 12:11:02.19\n%\n\n% The CRCKAPB.STY should be in your LaTeX directory.\n\n% Begin your text file with:\n\n%\\documentstyle[editedvolume]{crckapb} \n\n% Alternatives:\n% \\documentstyle[proceedings]{crckapb} \n% \\documentstyle[monograph]{crckapb} \n \\documentstyle[nato,psfig]{crckapb} \n\n\\newcommand{\\stt}{\\small\\tt}\n\n\n% This document needs the CRCKAPB.STY file to create a \n% document with font size 12pts. \n% The title, subtitle, author's name(s) and institute(s) \n% are handled by the `opening' environment.\n\n\\begin{opening}\n\\title{OBSERVATIONAL CONSTRAINTS TO THE EVOLUTION OF MASSIVE STARS}\n\n\\vskip -.5cm\n\n% You can split the title and subtitle by putting \n% two backslashes at the appropriate place. \n\\author{N. PANAGIA} \n\\institute{ESA/Space Telescope Science Institute\\\\ \n3700 San Martin Drive, Baltimore, MD 21218, USA\\\\\nE-mail {\\it panagia@stsci.edu}}\n% If there are more authors at one institute, you should first\n% use \\author{...} for each author followed by \\institute{...}.\n\\end{opening}\n\n\\runningtitle{Massive Star Evolution Constraints}\n\n\\begin{document}\n\n\\def\\Te+{{$\\rm{T_e}(\\rm{O^+})$}}\n\\def\\Te++{{$\\rm{T_e}(\\rm{O^{++}})$}}\n\\def\\dZOdZ{{$\\Delta$Z$_O$/$\\Delta$Z~}}\n\\def\\dYdZO{{$\\Delta$Y/$\\Delta$Z$_O$~}}\n\\def\\dYdZ{{$\\Delta$Y/$\\Delta$Z~}}\n\\def\\Msun{{M$_{\\odot}$~}}\n\\def\\msun{{\\rm{M}_{\\odot}~}}\n\n\\newcommand{\\kms}{$km~s^{-1}$ }\n\\newcommand{\\eg}{{\\it e.g., }} \n\\newcommand{\\ie}{{\\it i.e.~}} \n\\newcommand{\\etal}{{\\it et al.}} \n\\newcommand{\\apj}{ApJ, } \n\\newcommand{\\aap}{A\\&A, }\n\\newcommand{\\apjl}{ApJ, } \n\\newcommand{\\aj}{AJ, } \n\\newcommand{\\mnras}{MNRAS, }\n\n% The \\begin{document} command comes after the \\end{opening}\n% command.\n\n\\begin{abstract}\nWe consider some aspects of the evolution of massive stars which can\nonly be elucidated by means of ``indirect\" observations, \\ie\nmeasurements of the effects of massive stars on their environments. We\ndiscuss in detail the early evolution of massive stars formed in high\nmetallicity regions as inferred from studies of HII regions in\nexternal galaxies. \n\n\n\\end{abstract} \n\n\\vskip -.5cm \n\n\\section{Introduction} \n\nMassive stars play a crucial role in the evolution of galaxies and the\nwhole Universe, because they are the primary sources of radiative\nionization and heating of the diffuse medium, they provide most of the\nnucleosynthesis products to boost the metal content of galaxies and the\nintergalactic medium, and they constitute a major supply of kinetic\nenergy for galaxies, both through stellar winds during their quiescent\nphases and, eventually, in the form of fast ejecta from supernova\nexplosions. \n\nTherefore, it is fundamental to reach a proper understanding of the\nformation processes, the detailed properties and the evolution of\nmassive stars. Despite the fact that their high luminosities make\nthem ``easy\" targets for detailed observational studies, many aspects\nand properties of massive star evolution are far from being fully\nunderstood. This is because in any stellar generation, massive stars\nconstitute a small fraction of the newly formed stars (say, less than\n1\\% by number), they are ``elusive\" in that their lifetimes are very\nshort (say, less than 10--20 million years), and often they are\nheavily obscured by the parent molecular clouds where they were\nformed, making even their identification rather cumbersome. As a\nconsequence it is not easy to cover all evolutionary phases with {\\it\ndirect} observations of a statistically significant sample of objects.\n\nOne can overcome these difficulties and gain additional insights by\nconsidering phenomena that {\\it indirectly} can provide hints and\nclues to the problem. In other words, besides studying individual\nmassive stars, one can look at the effects that these stars have on\ntheir environments (\\eg HII regions, circumstellar nebulae, SNRs), and\ninfer from there what the stars were doing in special phases of their\nevolution (\\eg formation, LBV and pre-SN phases, etc.) that would not\nbe accessible in other ways. \n\nThus, one can use radio observations of supernovae, which probe the\ncircumstellar material ejected by the progenitor stars several\nthousand years before explosion, to study the very last phases of\ntheir evolution. These phases represent a tiny fraction of a massive\nstar lifetime, $\\sim$0.1\\%, and, therefore, they are extremely\ndifficult to reveal and study with direct observations. Although\nthis is an interesting aspect, we are not going to review it here, but\nrather we refer the reader to recent papers (Montes \\etal~ 1998,\nPanagia \\etal~ 1999, Weiler \\etal~ 1999). \n\nHere, we consider and discuss one particular aspect of the evolution\nof massive stars, namely their formation and early evolution in high\nmetallicity environments. We will show that observations of HII\nregions in external galaxies show that the ionization of He is much\nlower than that of H when the O/H ratio in the gas is appreciably\nhigher than solar. This implies that at high metallicities either very\nmassive stars ($M>25~M_\\odot$) do not form, or they never reach their\nexpected ZAMS location. \n\n\\section{Ionized Helium in the Milky Way}\n\nThere is clear observational evidence in the H~II regions of our\nGalaxy that the fractional abundance of ionized helium\nn(He$^+$)/n(H$^+$) is not a monotonic function of the galactocentric\nradius. Moving outwards from the Galactic Center, the ionized He\nabundance is found to increase in the inner Galaxy, then it attains a\nmaximum near the solar circle, and finally drops in the outer Galaxy\n(\\eg Mezger \\& Wink 1983 and references therein). The negative\ngradient in the outer galaxy reflects a genuine decrease in the He\nabundance in the outward direction (\\eg Panagia 1980; G\\\"usten \\&\nMezger 1982). The positive gradient in the inner Galaxy instead is an\neffect of the radial metallicity gradient which produces a systematic\nvariation of the spectrum of the ionizing radiation (Panagia 1980).\n\nThe fractional ionization of helium is extremely sensitive to the most\nenergetic part of the radiation field powering an H~II complex. Hence,\nit can provide valuable information on the presence and the abundance\nof the most massive ($m \\ge$ 20 \\Msun) stars, which are responsible\nfor most of the radiation with energy in excess of 24.6~eV. Therefore,\nit is a powerful tool to study how the details of the star formation\nprocess vary in different physical environments. There are several\nmechanisms through which a higher metallicity lowers the He ionization\nin an H~II region: \\\\ \n$\\bullet~$ The relative number of He-ionizing photons in the stellar \nspectrum is reduced because of both a stronger line blanketing in the \n200--500 \\AA ~wavelength range, and a higher continuum opacity.\\\\\n%\n$\\bullet~$ The stellar radius becomes larger and the effective temperature \ndecreases for a star of given mass, because of the increased continuum \nopacity in the sub-atmospheric layers of the star.\\\\\n%\n$\\bullet~$ The upper cut-off of the Initial Mass Function (IMF, $m_U$, \nmay be shifted \nto lower masses (\\eg Kahn 1974; Shields \\& Tinsley 1976).\\\\\n%\n$\\bullet~$ A higher metallicity may induce a steeper IMF (i.e. a larger value \nof the slope $\\alpha$ of the IMF $N(m) \\propto m^{-\\alpha}$) at least \nfor $m >$ 10 \\Msun, where the bulk of the ionizing radiation is produced\n(\\eg Terlevich \\& Melnick 1983).\n\nIn the first two cases, metallicity acts ``directly'' on the radiation\nfield of the ionizing star cluster, by modifying the stellar spectra\nwithout affecting the star formation processes. In the third and\nfourth case instead, metallicity acts ``indirectly'' and the changes\nin the radiation field result from changes in the properties of the\nIMF. \n\nPanagia (1980) demonstrated that the combined effects of at least\nthe first three processes are needed to explain the He ionization in\nthe Milky Way. Moreover, these processes appear to account for the\nobserved gradient of the effective temperature of the ionizing\nradiation inferred from the fitting of theoretical models to\nobservations of low-metallicity objects (Talent 1980; Campbell 1988). \n\n\n\\section{Ionized Helium in External Galaxies } \n\nConsidering external galaxies, several authors (\\eg Pagel 1986, \nViallefond 1988, Robledo-Rella \\& Firmani 1991) have suggested that a\nsystematic change of the IMF with metallicity is required by observations.\nOthers (\\eg Fierro, Torres - Peimbert \\& Peimbert 1986, McGaugh 1991) have\ncome to the opposite conclusion, and the controversy is still open.\nA thorough assessment of this subject is now possible and necessary.\n\nWe have considered a large sample of extragalactic H~II regions which\nprovides an extensive coverage of a very wide metallicity range\n(almost a factor of 100), and includes galaxies with a variety of\nmorphological types and luminosities. Such a sample is in many\nrespects much more homogeneous than any sample of galactic H~II\nregions. All the H~II regions observed are large (diameter D $>$ 50\npc), tenuous (n$_e$ $<$ 500 cm$^{-3}$ as derived from the [S II]\nI(6717)/I(6731) ratio) and must be ionized by large OB associations.\n\nHere, we limit our analysis to data published as of February 1992. (A\nmore complete investigation, including the discussion of data\npublished as of December 1999, is in progress and will be completed\nsoon; Lenzuni and Panagia 2000, in preparation). Thus, our sample \ncurrently includes 287\nH~II regions in 46 spiral and irregular galaxies with positive\ndetections of the [O II] lines at 3726 and 3729 \\AA~ (usually\nunresolved), of the [O III] lines at 4959 and 5007 \\AA~, and of at\nleast one of the He I lines. Additional observations were also\ncollected for 87 ``Blue Compact Galaxies'' (BCG's). None of these\nobjects is resolved into individual H~II regions, the observations being\nrelative to the entire galaxy or, possibly, to its central, brightest\nparts. These galaxies appear to be undergoing a stage characterized by\na collective mode of star-formation. Their spectra are heavily\ndominated by H~II region-like emission, hence they can be treated for\nour purposes as giant, isolated, extragalactic H~II regions. \n\n\\section{Analysis and Discussion}\n% \\subsection{The Helium Enrichment Curve}\n\nIonized helium abundances are shown in Figure 1 as a function of\noxygen abundances, for all of the H~II regions in spiral and irregular \ngalaxies and the Blue Compact\nGalaxies of our sample. \n%\nThe long baseline in metallicity offers a unique opportunity to\nconstrain {\\it both } the abundance of primordial helium Y$_P$ {\\it\nand} to the \\dYdZ gradient, thus fully determining the ``helium\nenrichment curve'' (HEC) which relates the total abundance of helium\nto the abundance of oxygen. Considering that evolutionary effects\nalways decrease the He$^+$/H$^+$ ratio because the\naging of a stellar cluster results in a softening of the ionizing\nradiation field, and that young clusters are observationally favoured\nbecause they are intrinsically brighter than older clusters, the HEC\ncan be derived by determining the upper envelope of the distribution\nshown in Figure 1. Among the class of curves $Y = Y_0 + \\Delta\nY/\\Delta Z \\times Z$ the best fit to the upper envelope of the\nobservations is obtained for $Y_0$ = 0.243 and \n(\\dYdZ)$_{\\odot}$ = 3.2 (see dashed curve in Fig.~1).\n This relation is consistent with the observational results of Pagel\n\\etal~ (1992) as well as with Maeder's (1992) theoretical models. \n%\n\n%\nAn inspection to Figure 1 reveals that the observed He$^+$/H$^+$ ratio\nappears to be almost constant up to solar O abundance\n(log(O/H)$_\\odot+12\\simeq$8.8) and then it declines rather quickly for higher\nmetallicities. This is a clear sign that He is progessively less\nionized as the O abundance increases, and implies that the mean\nradiation temperature of the ionizing stars becomes lower than about\n38,000~K around log(O/H)$\\simeq8.5$. \n\n\\begin{figure}\n\\centerline{\n \\psfig{figure=fig_vulchem.ps,width=9cm,height=9cm}\n}\n%\\vskip-.3cm\n\\caption{ He$^+$/H$^+$ ratios (by number) as a function of the \nO~abundances for the 287 H~II regions in 46 Spiral and Irregular\ngalaxies (filled symbols) and for the 87 Blue Compact Galaxies (open\nsquares) of our sample. The cross in the lower left corner represents\nthe typical uncertainty of the data. The boxes correspond the\n$\\pm1\\sigma$ deviations from the averages calculated within 0.2 dex\nintervals in log(O/H). The dashed line represents the helium\nenrichment curve (HEC) and the solid line is a model calculation in\nwhich the He ionization decreases exponentially with increasing \nO~abundance.} \n\\end{figure}\n\n% \\vskip-.8cm\nWe find that mechanisms through which metallicity acts ``directly'' on\nthe radiation field of the ionizing star cluster are not enough to explain the\nobserved gradient of the He ionization fraction with O \nabundance. Most of the effect appears instead to be due to\n``indirect'' mechanisms, \\ie a marked deficiency of hot stars\nwith increasing metal abundances. There are at least three possible\nscenarios to explain this fact:\\\\ \n%\n1. The IMF slope becomes steeper for higher metallicities.\\\\\n%\n2. The IMF upper cutoff moves to lower masses for higher \nmetallicities.\\\\\n%\n3. The most massive stars become progressively unable to provide\nionizing radiation, either because at high metallicities the remnant\nof their pre-MS cocoons remains optically thick over most of a star's\nlifetime, or because pulsational instabilities prevent the most massive\nstars from reaching their expected ZAMS surface conditions. \\\\\n%\nOur model calculations show that varying only the slope of the IMF,\n\\ie point (1), does not give a satisfactory fit to the data because\nthe resulting ionization decline would be too shallow. On the other hand,\npoint (2), \\ie a systematic variation of the IMF upper mass cut-off\nwith metal abundance ($m_U \\propto$ Z$^{-\\beta}$, $\\beta > 0$) can\nreproduce the observed trend of the He ionization, with $m_{U\\odot}$ =\n48 \\Msun and $\\beta$ = 0.60. Point (3) could also account for the\nobservations provided that the invoked effects are indeed capable to\nproduce the sharp decline of He ionization as observed. \n\nFrom the observational point of view there are no direct studies to\nconclusively discriminate between points (2) and (3). Observations of\nmassive stars near the Galactic Center, such as the Pistol star, \nthe Sickle and the Quintuplet clusters (\\eg Figer \\etal~ 1999 and\nreferences therein) seem to favor the third possibility, because they\nare so bright ($log(L/L_\\odot)>6$) that they must be quite massive. \nOn the other hand, one may argue that those clusters are so close to\nthe Galactic Center that tidal forces may drastically affect the\ndynamical processes that lead to the formation of stars and, therefore,\nthey may not be representative of normal situations. The ideal\ninvestigation to clarify this issue should include nebular and stellar\nspectroscopy of a large sample of HII regions in galaxies which\ndisplay marked effects of incomplete He ionization, such as M51 or \nM83. Another discriminant between hypotheses (2) and (3) is that if\na lowering of the IMF upper cutoff is {\\it the } explanation (\\ie\npoint (2)), then the frequency of Wolf-Rayet stars relative to early\ntype stars is expected to be abnormally low in the high-Z regions\nbecause in this case the reduced ionization is entirely due to the\nlack of truly massive stars that are expected to end up as WR stars in\ntheir final stages of evolution. \n\n\\begin{thebibliography}{99} \n\n\\bibitem{} Campbell A., 1988, {\\it Ap. J.} {\\bf 335}, 644\n\\bibitem{} Fierro J., Torres-Peimbert S., Peimbert M., 1986, {\\it P.A.S.P.} \n\t{\\bf 98}, 1032\n\\bibitem{} Figer, D., McLean, I.S., Morris, M., 1999, {\\it Ap. J.} \n\t{\\bf 514}, 202\n\\bibitem{} G\\\"usten R., Mezger P., 1982, {\\it Vistas in Astronomy} \n\t{\\bf 26}, 159\n\\bibitem{} Kahn F.D., 1974, {\\it Astr. Ap.} {\\bf 37}, 149\n\\bibitem{} Maeder A., 1992, {\\it Astr. Ap.} {\\bf 264}, 105\n\\bibitem{} McGaugh S.S., 1991, {\\it Ap. J.} {\\bf 380}, 140\n\\bibitem{} Mezger P.G., Wink J.E., 1983, in {\\it Primordial Helium}, \n\teds. P.A. Shaver, D. Kunth and K. Kj\\\"ar (~ESO, Garching, Germany~)\n\\bibitem{} Montes, M.J., Van Dyk, S.D., Weiler, K.W., Sramek, R.A., \n\tPanagia, N., 1998, {\\it Ap. J.} {\\bf 506}, 874\n\\bibitem{} Pagel B.E.J., 1986, in {\\it Highlights of Astronomy} Vol. 7, \n\ted. J.P. Swings (~Reidel, Dordrecht, The Netherlands~)\n\\bibitem{} Pagel B.E.J., Simonson E.A., Terlevich R.J., Edmunds M.G.,\n\t1992, {\\it M.N.R.A.S.} {\\bf 255}, 325\n\\bibitem{} Panagia N., 1980, in {\\it Radio Recombination Lines}, ed. \n\tP. A. Shaver (~Reidel, Dordrecht, The Netherlands)\n\\bibitem{} Panagia, N., \\etal, 1999, {\\it Mem S.A.It.}, in press\n\\bibitem{} Robledo-Rella V., Firmani C., 1990, {\\it Rev. Mexicana Astron. \n\tAstrof.} {\\bf 21}, 236\n\\bibitem{} Shields G.A., Tinsley B., 1976, {\\it Ap. J.} {\\bf 203}, 66\n\\bibitem{} Terlevich R.J., Melnick J., 1983, {\\it M.N.R.A.S.} {\\bf 195}, 839\n\\bibitem{} Viallefond F., 1988, in {\\it Galactic and Extra-Galactic Star\n\tFormation}, eds. R. Pudritz and M. Fich (~Reidel, Dordrecht, \n\tThe Netherlands~)\n\\bibitem{} Weiler, K.W., \\etal, 1999, in {\\it ``The Largest Explosions since\n\tthe Big Bang: Supernovae and Gamma Ray Bursts\"}, eds. M. Livio, K.\n\tSahu \\& N. Panagia, (CUP, Cambridge, England), in press\n\\end{thebibliography}\n\n\\end{document}\n\n"
}
] |
[
{
"name": "astro-ph0002509.extracted_bib",
"string": "\\begin{thebibliography}{99} \n\n\\bibitem{} Campbell A., 1988, {\\it Ap. J.} {\\bf 335}, 644\n\\bibitem{} Fierro J., Torres-Peimbert S., Peimbert M., 1986, {\\it P.A.S.P.} \n\t{\\bf 98}, 1032\n\\bibitem{} Figer, D., McLean, I.S., Morris, M., 1999, {\\it Ap. J.} \n\t{\\bf 514}, 202\n\\bibitem{} G\\\"usten R., Mezger P., 1982, {\\it Vistas in Astronomy} \n\t{\\bf 26}, 159\n\\bibitem{} Kahn F.D., 1974, {\\it Astr. Ap.} {\\bf 37}, 149\n\\bibitem{} Maeder A., 1992, {\\it Astr. Ap.} {\\bf 264}, 105\n\\bibitem{} McGaugh S.S., 1991, {\\it Ap. J.} {\\bf 380}, 140\n\\bibitem{} Mezger P.G., Wink J.E., 1983, in {\\it Primordial Helium}, \n\teds. P.A. Shaver, D. Kunth and K. Kj\\\"ar (~ESO, Garching, Germany~)\n\\bibitem{} Montes, M.J., Van Dyk, S.D., Weiler, K.W., Sramek, R.A., \n\tPanagia, N., 1998, {\\it Ap. J.} {\\bf 506}, 874\n\\bibitem{} Pagel B.E.J., 1986, in {\\it Highlights of Astronomy} Vol. 7, \n\ted. J.P. Swings (~Reidel, Dordrecht, The Netherlands~)\n\\bibitem{} Pagel B.E.J., Simonson E.A., Terlevich R.J., Edmunds M.G.,\n\t1992, {\\it M.N.R.A.S.} {\\bf 255}, 325\n\\bibitem{} Panagia N., 1980, in {\\it Radio Recombination Lines}, ed. \n\tP. A. Shaver (~Reidel, Dordrecht, The Netherlands)\n\\bibitem{} Panagia, N., \\etal, 1999, {\\it Mem S.A.It.}, in press\n\\bibitem{} Robledo-Rella V., Firmani C., 1990, {\\it Rev. Mexicana Astron. \n\tAstrof.} {\\bf 21}, 236\n\\bibitem{} Shields G.A., Tinsley B., 1976, {\\it Ap. J.} {\\bf 203}, 66\n\\bibitem{} Terlevich R.J., Melnick J., 1983, {\\it M.N.R.A.S.} {\\bf 195}, 839\n\\bibitem{} Viallefond F., 1988, in {\\it Galactic and Extra-Galactic Star\n\tFormation}, eds. R. Pudritz and M. Fich (~Reidel, Dordrecht, \n\tThe Netherlands~)\n\\bibitem{} Weiler, K.W., \\etal, 1999, in {\\it ``The Largest Explosions since\n\tthe Big Bang: Supernovae and Gamma Ray Bursts\"}, eds. M. Livio, K.\n\tSahu \\& N. Panagia, (CUP, Cambridge, England), in press\n\\end{thebibliography}"
}
] |
astro-ph0002510
|
The MACHO Project: Microlensing Optical Depth towards the Galactic Bulge from Difference Image Analysis
|
[
{
"author": "C. Alcock\\altaffilmark{1,2}"
},
{
"author": "R.A. Allsman\\altaffilmark{3}"
},
{
"author": "D.R. Alves\\altaffilmark{4}"
},
{
"author": "T.S. Axelrod\\altaffilmark{5}"
},
{
"author": "A.C. Becker\\altaffilmark{6}"
},
{
"author": "D.P. Bennett\\altaffilmark{2,7}"
},
{
"author": "K.H. Cook\\altaffilmark{1,2}"
},
{
"author": "A.J. Drake\\altaffilmark{1,5}"
},
{
"author": "K.C. Freeman\\altaffilmark{5}"
},
{
"author": "M. Geha\\altaffilmark{1}"
},
{
"author": "K. Griest\\altaffilmark{2,8}"
},
{
"author": "%L.J. King\\altaffilmark{7}"
},
{
"author": "M.J. Lehner\\altaffilmark{9}"
},
{
"author": "S.L. Marshall\\altaffilmark{1,2}"
},
{
"author": "D. Minniti\\altaffilmark{1,10}"
},
{
"author": "C.A. Nelson\\altaffilmark{1,2}"
},
{
"author": "B.A. Peterson\\altaffilmark{5}"
},
{
"author": "P. Popowski\\altaffilmark{1}"
},
{
"author": "M.R. Pratt\\altaffilmark{11}"
},
{
"author": "P.J. Quinn\\altaffilmark{12}"
},
{
"author": "C.W. Stubbs\\altaffilmark{2,5,6}"
},
{
"author": "W. Sutherland\\altaffilmark{13}"
},
{
"author": "A.B. Tomaney\\altaffilmark{6}"
},
{
"author": "T. Vandehei\\altaffilmark{8}"
},
{
"author": "{\\sc and} D.L. Welch\\altaffilmark{14}"
}
] |
We present the microlensing optical depth towards the Galactic bulge based on the detection of 99 events found in our Difference Image Analysis (DIA) survey. This analysis encompasses three years of data, covering $\sim 17$ million stars in $\sim 4$ deg$^2$, to a source star baseline magnitude limit of $V = 23$. The DIA technique improves the quality of photometry in crowded fields, and allows us to detect more microlensing events with faint source stars. We find this method increases the number of detection events by $85\%$ compared with the standard analysis technique. DIA light curves of the events are presented and the microlensing fit parameters are given. The total microlensing optical depth is estimated to be $\tau_{total}= 2.43^{+0.39}_{-0.38}\times 10^{-6}$ averaged over 8 fields centered at $l=2\fdg68$ and $b=-3\fdg35$. %^{\!\!\circ} For the bulge component we find $\tau_{bulge}=3.23^{+0.52}_{-0.50}\times 10^{-6}$ assuming a $25\%$ stellar contribution from disk sources. These optical depths are in good agreement with the past determinations of the MACHO \shortcite{ALC97a} and OGLE \shortcite{USKK94d} groups, and are higher than predicted by contemporary Galactic models. We show that our observed event timescale distribution is consistent with the distribution expected from normal mass stars, if we adopt the stellar mass function of Scalo (1986) as our lens mass function. However, we note that as there is still disagreement about the exact form of the stellar mass function, there is uncertainty in this conclusion. Based on our event timescale distribution we find no evidence for the existence of a large population of brown dwarfs in the direction of the Galactic bulge.
|
[
{
"name": "MS.tex",
"string": "\\documentstyle[aas2pp4,lscape,astrobib]{article}\n\n%\\documentstyle[apjpt4]{article}\n%\\documentstyle[aj_pt4]{article}\n%\\documentstyle[mn]{article}\n%\\documentstyle[11pt,,aaspp4]{article} %eqsecnum\n%\\documentstyle[12pt,aasms4]{article}\n%\\documentclass{aastex}\n%\\usepackage{rotating}\n%\\rotatefoilhead\n%\\special{landscape}\n%\\usepackage{epsfig}\n%\\usepackage{rotating}\n\n% Authors are permitted to use the fonts provided by the American Mathematical\n% Society, if they are available to them on their local system. These fonts\n% are not part of the AASTeX macro package or the regular TeX distribution.\n\n%\\documentstyle[12pt,amssym,aasms4]{article}\n%\\received{1 April 1999}\n%\\accepted{1 February 2000}\n%\\journalid{337}{15 1 April 2000}\n%\\articleid{11}{14}\n\n%\\shorttitle{Microlensing Optical Depth}\n%\\shortauthors{Alcock et al.}\n%\\slugcomment{To appear in }\n\\lefthead{Alcock et al.}\n\\righthead{Microlensing Optical Depth}\n\n\\begin{document}\n\n\\title{The MACHO Project: Microlensing Optical Depth towards the Galactic Bulge from Difference Image Analysis}\n\n\\author{C. Alcock\\altaffilmark{1,2}, R.A. Allsman\\altaffilmark{3}, D.R. Alves\\altaffilmark{4},\nT.S. Axelrod\\altaffilmark{5}, A.C. Becker\\altaffilmark{6}, D.P. Bennett\\altaffilmark{2,7},\\\\\nK.H. Cook\\altaffilmark{1,2}, A.J. Drake\\altaffilmark{1,5}, K.C. Freeman\\altaffilmark{5}, \nM. Geha\\altaffilmark{1}, K. Griest\\altaffilmark{2,8}, %L.J. King\\altaffilmark{7},\nM.J. Lehner\\altaffilmark{9},\\\\ S.L. Marshall\\altaffilmark{1,2}, D. Minniti\\altaffilmark{1,10}, \nC.A. Nelson\\altaffilmark{1,2}, B.A. Peterson\\altaffilmark{5}, P. Popowski\\altaffilmark{1},\\\\\nM.R. Pratt\\altaffilmark{11},\nP.J. Quinn\\altaffilmark{12}, C.W. Stubbs\\altaffilmark{2,5,6}, W. Sutherland\\altaffilmark{13},\nA.B. Tomaney\\altaffilmark{6},\\\\\nT. Vandehei\\altaffilmark{8}, {\\sc and} D.L. Welch\\altaffilmark{14}}\n\\author{\\bf(The MACHO Collaboration)}\n\\vspace*{0.2cm}\n%\\author{\\bf Final Draft, Feb 2nd}\n\n\\altaffiltext{1}{Lawrence Livermore National Laboratory, Livermore, CA 94550} \n\\altaffiltext{2}{Center for Particle Astrophysics, University of California, Berkeley, CA 94720} \n\\altaffiltext{3}{Supercomputing Facility, Australian National University, Canberra, ACT 0200, Australia}\n\\altaffiltext{4}{Space Telescope Science Institute, 3700 San Martin Dr, Baltimore, MD 21218}\n\\altaffiltext{5}{Research School of Astronomy and Astrophysics, Weston Creek, Canberra, ACT 2611, Australia}\n\\altaffiltext{6}{Department of Astronomy and Physics, University of Washington, Seattle, WA 98195}\n\\altaffiltext{7}{Department of Physics, University of Notre Dame, Notre Dame, IN 46556}\n\\altaffiltext{8}{Department of Physics, University of California, San Diego, CA 92093}\n\\altaffiltext{9}{Department of Physics, University of Sheffield, Sheffield S3 7RH, UK}\n\\altaffiltext{10}{Departmento de Astronomia, P. Universidad Cat\\'olica, Casilla 104, Santiago 22, Chile}\n\\altaffiltext{11}{Center for Space Research, MIT, Cambridge, MA 02139}\n\\altaffiltext{12}{European Southern Observatory, Karl Schwarzchild Str.\\ 2, D-85748 G\\\"{a}rching bei M\\\"{u}nchen, Germany}\n\\altaffiltext{13}{Department of Physics, University of Oxford, Oxford OX1 3RH, UK}\n\\altaffiltext{14}{Department of Physics and Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada}\n\n\\newpage\n\\begin{abstract}\n\n We present the microlensing optical depth towards the Galactic bulge based\n on the detection of 99 events found in our Difference Image Analysis (DIA)\n survey. This analysis encompasses three years of data, covering $\\sim 17$\n million stars in $\\sim 4$ deg$^2$, to a source star baseline magnitude\n limit of $V = 23$. The DIA technique improves the quality of photometry\n in crowded fields, and allows us to detect more microlensing events with\n faint source stars. We find this method increases the number of detection\n events by $85\\%$ compared with the standard analysis technique.\n DIA light curves of the events are presented and the microlensing fit\n parameters are given. The total microlensing optical depth is estimated to\n be $\\tau_{total}= 2.43^{+0.39}_{-0.38}\\times 10^{-6}$\n averaged over 8 fields centered at $l=2\\fdg68$ and $b=-3\\fdg35$. %^{\\!\\!\\circ}\n For the bulge component we find $\\tau_{bulge}=3.23^{+0.52}_{-0.50}\\times 10^{-6}$ \n assuming a $25\\%$ stellar contribution from disk sources. These optical\n depths are in good agreement with the past determinations of the MACHO\n \\shortcite{ALC97a} and OGLE \\shortcite{USKK94d} groups, and are higher\n than predicted by contemporary Galactic models. We show that our observed\n event timescale distribution is consistent with the distribution expected\n from normal mass stars, if we adopt the stellar mass function of Scalo\n (1986) as our lens mass function. However, we note that as there is still\n disagreement about the exact form of the stellar mass function, there is\n uncertainty in this conclusion. Based on our event timescale distribution\n we find no evidence for the existence of a large population of brown\n dwarfs in the direction of the Galactic bulge.\n\n\\end{abstract}\n\n\\keywords{dark matter - Galaxy: structure - gravitational lensing - \nstars: low-mass}\n\n\\section{\\sc Introduction}\n\nOver the past seven years the MACHO group has been making observations\nof the Galactic bulge in order to determine some of the\nfundamental properties of our Galaxy. \nThe Milky Way is expected to be an\nSAB(rs)bc or SAB(r)bc type spiral galaxy \\shortcite{DEV64,Fux97} with four\nspiral arms \\shortcite{VA95}. However, very little is known about the mass\ndistributions of the various components of our Galaxy (bulge, spheroid,\ndisk, halo). Galactic microlensing surveys provide some insight into the\nstructure and dynamics of the inner Galaxy, spiral arms and the halo.\nUnlike most types of observation, the presence of lensing objects can be\ndetected independent of their luminosities. Microlensing is sensitive to\nthe mass distribution rather than light, this makes microlensing a powerful\nway of investigating the mass density within our Galaxy. Furthermore,\nmicrolensing can be used to investigate the stellar mass function to the\nhydrogen burning limit, both within our Galaxy and other nearby galaxies.\n\nThe amplification of a source star during gravitational microlensing\nis related to the projected lens-source separation $u$ normalised \nby the angular Einstein Ring radius $R_{E}$. This is given by\n\n\\begin{equation}\\label{chisw}\nA(u) = \\frac{u^{2}+2}{u \\sqrt{u^{2}+4}}.\n\\end{equation}\n\n\\noindent\nThe timescale of a microlensing event, $\\hat t$, is characterised by the time\nit takes for the Einstein ring associated with a foreground compact lensing\nobject, to transit a background source star at velocity $v_{\\perp}$. The\nsize of the Einstein ring, for a lens with mass $M$ (in $\\rm M_{\\odot}$), an\nobserver-lens distance $D_{d}$, and a source-observer distance $D_{s}$, is\ngiven by\n%(eq. \\ref{chiswolsonx})\n\n\\begin{equation}\nR_{E} = 2.85 {\\rm AU} \\sqrt{\\frac{M D_{d}(1-\\frac{D_{d}}{D_{s}})}{1\\,\\rm kpc}}.\n\\label{chiswolsonx}\n\\end{equation}\n\n\\noindent\nThe lensing timescale is $\\hat t \\equiv 2R_{E}/v_{\\perp}$.\nHence, if $R_{E}$ were known, this would enable us to constrain some of the\nphysical parameters of a microlensing event ($M$, $D_{s}$, $D_{d}$).\nHowever, $R_{E}$ is not known, generally, so it is not possible to determine\nthe lens masses from individual microlensing events. Nevertheless, under\nspecial circumstances it is possible to impose additional constraints on these\nmicrolensing event parameters when quantities such as, the physical size of\nthe source star \\cite{AAAA99} or the projected transverse velocity of the\nlens \\cite{ALC95a} are measured.\n\nPhotometry of the stars monitored by the MACHO project has previously\nbeen carried out using a fixed position PSF photometry package SoDoPhot (Son\nof DoPhot, \\shortciteNP{BEN93}). In 1996 we introduced a second reduction\nmethod, Difference Image Analysis (hereafter DIA). The DIA technique enables\nus to detect microlensing events which go undetected with the SoDoPhot\nphotometry because the events are due to stars which are too faint to be detected\nwhen unlensed. This technique follows on from the work of \\citeN{Crotts92},\n\\citeN{PHL95} and \\citeN{TC96}, and allows us to detect and perform accurate\nphotometry on these new microlensing events found in the reanalysis of bulge\nimages.\n\n\nRecently, the MACHO and OGLE groups reported that the microlensing optical\ndepth towards the Galactic bulge was a factor of 2 larger than expected from\nstellar number density. That is, the optical depth found by OGLE is\n$3.3\\pm{1.2} \\times 10^{-6}$ \\shortcite{USKK94d} and by MACHO is\n$3.9^{+1.8}_{-1.2} \\times 10^{-6}$ for 13 clump giant source star events out\nof a 41 event sample \\shortcite{ALC97a}. It was suggested that the size of\nthese measurements could be explained by the presence of a bar oriented\nalong our line-of-sight to the bulge (\\shortciteNP{PSUS94b,ZSR95}). The\ndensity profile of the proposed bar is given by\n\n\\begin{equation}\\label{pt1}\n\\rho_{b} = \\frac{M}{20.65 abc}\\; {\\rm exp}\\left (-\\frac{w^{2}}{2}\\right),\n\\end{equation}\n\n\\noindent\nwhere\n\n\\begin{equation}\\label{pt2}\nw^{4} = \\left[\\left(\\frac{x}{a}\\right)^{2} + \\left(\\frac{y}{b}\\right)^{2}\\right]^{2} \n+ \\left(\\frac{z^{\\prime}}{c}\\right)^{4}.\n\\end{equation}\n\n\\noindent\nFor the bulge galactocentric coordinates (x,y,z$^{\\prime}$):\n$x = \\cos{\\theta} - \\eta \\cos{b} \\cos{(l-\\theta)}$,\n$y= -\\sin{\\theta} - \\eta \\cos{b}\\sin{(l-\\theta)}$,\n$z^{\\prime} = \\eta \\sin{b}$.\nThe bar inclination angle $\\theta$ is oriented in the direction\nof increasing $l$, and $\\eta=D_{s}/D_{8.5}$ is the ratio of the \nsource distance relative to a galactocentric distance, taken to be\n8.5 kpc. The terms $a$, $b$ and $c$ define the bar scale lengths.\n\nThe idea that our Galaxy harbours a bar at its centre is not a new one as it\nwas first suggested by \\shortciteN{DEV64} because of the similarity of the\ngas dynamics observed in our galaxy with other barred galaxies.\n\\citeN{BGSBU91} provided further evidence for a bar from star counts.\nThe DIRBE results of \\citeN{DAH95} were also found to be consistent with\nthis prediction. The presence of such a bar is an important way of\nexplaining the interaction of the disk, halo and the spiral density waves in\nthe disk.\n\nA number of authors have adopted a bar into their Galactic models and have \nadopted various values of the bar orientation \\shortcite{PSUS94b,Peale98,ZM96} \nand bar mass \\shortcite{Peale98,ZM96}. Other authors have also proposed that\nlarge optical depth contributions could come from the disk component \\shortcite{EGTB98}, \nor the Galactic stellar mass function (M\\'era, Chabrier, \\& Schaeffer 1998; \n\\shortciteNP{HC98,ZM96}).\\nocite{MER98a} \n\nIn this paper we present a new value for the microlensing optical depth and\ninvestigate what is known about the Galactic parameters with most influence\nthe optical depth. In the next section we will detail the\nobservational setup. In \\S 3, we will outline the reduction procedure. We\nwill next review the microlensing event selection process in \\S 4. The\nresults of our analysis are presented in \\S 5, and we will discuss how the\nmicrolensing detection efficiency was calculated in \\S 6. The microlensing\noptical depth for the sample of fields presently analysed with DIA and for\neach of the individual field will be presented in \\S 7. In \\S 8, we will\nreview what is known about the most important factors affecting the observed\noptical depth and discuss the implications of our results. In the final\nsection we summarise the results of this work.\n\n\n\\section{\\sc Observations}\n\nThe MACHO observation database contains over 90000 individual observations\nof the Galactic bulge and Magellanic Clouds. The Galactic bulge\ndataset consists of $\\sim 30000$ observations of 94, $43\\arcmin$ $\\times$\n$43\\arcmin$ fields. The largest set of microlensing events reported to\ndate was given by \\citeN{ALC97e}. This consisted of 41 events\ndetected from one year of observations in 28 of those Galactic bulge fields.\nThe observations in this work were taken between March 1995 and August 1997.\nWe consider $\\sim3000$ Galactic bulge observations from eight fields.\nThe central location of the eight fields \nis\n%$\\alpha_{2000}=18^{\\rm h} 04^{\\rm s} 30^{\\rm s} $, \n%$\\delta_{2000} = -28\\arcdeg 23\\arcmin$ %$(J2000)$, or \n$l = 2\\fdg68$, $b =-3\\fdg35$.\n\nAll observations were taken with the Mount Stromlo and Siding Spring Observatories'\n1.3m Great Melbourne Telescope with the dual-colour wide-field {\\em Macho\n camera}. The Macho camera consists of two sets of four CCDs, one for red\nband ($R_{M}$) images, and the other for blue band ($B_{M}$) images. These\nobservations were taken simultaneously by employing a dichroic\nbeam splitter. Each CCD is 2k by 2k with an on-the-sky pixel size of\n$0.63\\arcsec$. All bulge observations used 150 second exposures.\n\nThe median seeing of the data subset is $\\sim2.1\\arcsec$ and the median \nsky levels are $B_{M} \\sim 1300$ ADU and $R_{M} \\sim 2200$ ADU. To improve \nthe average data quality of our light curves we have chosen to reject \n$\\sim 350$ observations where the seeing of the observation \nwas $>4\\arcsec$ or $B_{M}$ band sky level was $> 8000$ ADU.\nThe number of observations for each field varies. The smallest number of\nobservations in this subset is 204 for field 159 and the largest is \n334 for field 104. Low Galactic latitude fields such as field 104 have a \nhigher observing priority than, for example field 159, because the \nmicrolensing optical depth is expected to be higher closer to the \nGalactic center.\n\n\\section{\\sc Reduction Technique}\n\nThe DIA technique involves matching a good seeing {\\em reference} image to\nother images of the same field, so called {\\em test} images. The test\nimages are first spatially registered to the reference images and the PSFs\nof the images are matched. The test images are then photometrically\nnormalised (flux matched) to the reference image and the images are\ndifferenced to reveal variable stars, asteroids, novae, etc. These variables\nand microlensing events are searched for in each difference frame and\nphotometry is performed on the entire set of difference images (see\n\\shortciteNP{ALC99b}). Initial results of our Galactic bulge DIA were\npresented for a single field in Alcock {et~al.\\ }(\\citeyearNP{ALC99c,ALC99b}). In this analysis we combine that data with images from seven\nadditional fields to determine an accurate value for the microlensing\noptical depth towards the Galactic bulge.\n\nOur implementation of the DIA technique closely follows that given in\n\\citeN{ALC99b}. In short, this involves the initial selection of\n$200-300$ bright ``PSF'' stars from the Macho star database. The centroids\nof these stars are coordinate-matched in the reference and test images. The\ncoordinate transform between the observations is calculated and used to\nspatially register the test observation to the reference image. The PSF\nstars are ``cleaned'' of neighbouring stars and then combined to provide a\nhigh signal-to-noise ratio (hereafter S/N) PSF for each observation. Next, the image\nmatching convolution kernel is calculated from the Fourier Transforms of the\ntwo PSFs employing the IRAF task PSFmatch. The reference image is convolved\nwith the empirical kernel to seeing-match the images. The fluxes in\nthe two observations are matched to account for variances in the atmospheric\nextinction, and differences in the observed sky level of the two images. Lastly\na difference image is formed by subtracting one of these images from\nthe other. An object detection algorithm is applied to the images with a\nthree $\\sigma$ threshold, in both $R_{M}$-band and $B_{M}$-band\ndifference images. The coordinates of objects found within each passband\nare matched to sort the real sources from spurious ones, such as\ncosmic rays and noise spikes. The systematic noise is determined from\nthe residuals of the PSF stars in each difference image. The photon\nnoise is calculated from the test images and the PSF-matched reference\nimage. For further detail the reader is referred to \\shortciteN{ALC99b}\n(see also \\shortciteNP{AL98,ALA99a} for similar type of difference \nimage technique).\n\n\n\\subsection{\\it Photometry}\n\nThe MACHO project has focussed on the detection of gravitational\nmicrolensing events by repeatedly imaging millions of stars in the Galaxy.\nThe search for microlensing events has provided a consistent set of\nphotometry for these stars spanning the duration of the experiment. This\ndata is thus also extremely valuable for studying the properties of the\nstellar populations present.\n\nIn this analysis of difference images, aperture photometry is performed on\nall the objects detected in the difference images. For each field this\naperture photometry database consisted of $\\sim 40000$ objects. As all\nimages of each field were registered and matched to a single reference\nimage, the centroid coordinates for each photometry measurement are the same\nin each image. No extinction corrections are needed as image fluxes are also\nmatched in the DIA process. With DIA we are in the unique situation of being\nable to perform aperture photometry in a field which is usually very\ncrowded with non-variable stars. All these constant sources were removed,\nleaving behind only the residual noise and the variations from the\nrelatively small number of variable stars and other transient phenomena,\nsuch as microlensing events (\\shortciteNP{ALC99b},\n\\citeyearNP{ALC99c}).\n\nAt present the photometry data is put into a simple ASCII database since\nthere are relatively few variable objects in each field relative to the\ntotal number of stars ($\\sim 4e4/1e6$).\nIn Figure \\ref{figBin} we present an example of the photometry for one of\nthe microlensing events\\footnote{The asymmetry is most likely due to either\nbinary lensing or parallax. A similar microlensing event was presented in\n\\shortciteN{AAAA99}.}. The oscillations in the SoDoPhot photometry\nlight curve are due to the offset between the nearest SoDoPhot object centroid\nand the centroid of the microlensing source star. Changes in the seeing can\ncause varying amounts of flux to be sequestered into the nearest SoDoPhot PSF.\nThe SoDoPhot centroid is fixed in position, whereas DIA uses the actual event's \ncentroid.\n\n%\\placefigure{figBin}\n\\begin{figure}[ht]\n\\epsscale{1.0}\n\\plotone{./Figures/fig1.ps}\n\\figcaption{Red bandpass difference flux light curves\nfor one of the exotic microlensing events detected in this \nanalysis. Top: baseline subtracted SoDoPhot (PSF) photometry.\nBottom: DIA aperture photometry.\\label{figBin}}\n\\end{figure}\n\n\n\\section{\\sc Event Selection}\n\nThere are a number of well known theoretical properties of microlensing\nwhich can be used to select events. That is to say, the light curves of\nstars affected by microlensing should exhibit: a single, symmetric,\nachromatic excursion from a flat baseline. In reality these properties\nserve only as a guide, since there are a number of exceptions to each of\nthem (see \\citeNP{DrakeTh}). For instance, the amplification may not be\nachromatic when various types of blending are considered \\cite{ALC99b}. \nMultiple peaks can, and do occur for many types of exotic\nmicrolensing events, such as binary lensing and binary source events.\nFurthermore, all microlensing events are affected to some degree by the\nparallax induced by the Earth's orbital motion around the Sun. In most cases\nthis is negligible, but in some circumstances the magnified peak shows a\nsignificant asymmetry (see \\citeNP{ALC95a}). Therefore, a more\nrigorous set of selection criteria is necessary.\n\nIn our DIA microlensing event selection process we firstly required that the\nevents had a total S/N $> 10$, in three or more photometry measurements\nbracketing the maximum amplification, in each bandpass. We selected only\nthose light curves exhibiting a positive excursion from the baseline\nflux\\footnote{The baseline itself can be negative.}. An initial estimate of\nthe baseline flux level was determined from the median flux, $F_{med}$, of\nthe difference flux light curve. In addition to these criteria, we required\nthat each light curve passed a set of level-1.0 criteria presented in Table\n\\ref{tab1}. These level-1.0 ``cuts'' use the flux values $F_{i}$, and\nuncertainties $\\sigma_{i}$, measured at time $t_{i}$, to discern whether a\nlight curve is following a microlensing-like profile, or a more\nvariable-like curve. These initial cuts are targeted at removing particular\ntypes of variables from the event candidate list based on the general\ncharacteristics of a variable type.\n\nIn the level-1.0 selection, one cut is aimed at removing variables by the nature\nthat they repeat. A cut on the existence of a second peak is an efficient\nway of removing variables from a candidate list. However, this also can have\nthe negative effect of removing binary lensing events. For this reason we\napply only a loose cut on the occurrence of a second deviation from\nbaseline. This cut can only remove binary events with high S/N and a long\nduration between caustic crossings ($> 110$ days). The standard of the light\ncurve photometry was also accessed in this level-1.0 selection. These t\nlevel-1.0 cuts remove the high S/N variable stars from the candidate\nlensing event list, but a number of the lower S/N variables remained, so\nfurther cuts were necessary.\n\nThe DIA source light curves passing level-1.0 cuts were then fed through a\nstricter set of level-1.5 cuts. These cuts remove lower S/N variables and\nare based on the microlensing goodness-of-fit statistic $\\chi^{2}_{m}$, and\na constant baseline goodness-of-fit statistic $\\chi^{2}_{c}$ (performed in\nthe region $t_{max} \\pm 2 \\hat{t}$). Here, $\\chi^{2}_{m}$ and $\\chi^{2}_{c}$\ndenote reduced chi-squared statistics. To give weight to higher S/N events\nwe enforced what we call an $\\Omega\\chi^{2}$ cut, this was defined as\n$\\Omega\\chi^{2}$ =$1000/pf \\times (\\chi^{2}_{c} + \\chi^{2}_{m})$. The\nsymbol $pf$ refers to the flux at peak amplification. For all light\ncurves we required $\\Omega\\chi^{2} < 3.6$.\n\n\n%\\placefigure{figCN}\n\\begin{figure}[ht]\n\\epsscale{1.0}\n\\plotone{./Figures/fig2.ps}\n\\figcaption{An example of a DIA light curve for one of the dwarf novae\n removed by our colour and fit microlensing cuts. For this event \n $V-R \\sim 0$. Inset is a blow up of the outburst event.\n\\label{figDN}}\n\\end{figure}\n\n\n\nDuring our analysis we found that the DIA technique was very sensitive to\nthe detection of dwarf novae. Most of these variables can be rejected during\nevent selection based on their poor microlensing fits. An example of these\ndwarf novae is presented in Figure \\ref{figDN}. However, some faint dwarf\nnovae light curves have large uncertainties and can thus pass a microlensing\nfit $\\chi^{2}$ cut. Because of this, we found it necessary to impose a\ncolour cut $(V-R)_{108} > 0.55$ and a $V$-magnitude cut for $V > 17$.\n$(V-R)_{108}$ denotes the colour an event would have if seen at the average\nreddening of field 108 ($E(V-R) = 0.51$). To determine this we have made use\nof the reddening values given in \\shortciteN{ALC98a}.\nTransformations from our $B_{M}$ and $R_{M}$ colours to {\\it Cousins} $V$\nand $R$ are given in \\shortciteN{Alves99}. We note that, although the\ncolours of the dwarf novae companion stars can vary, the difference flux\ncolour is the hue of the outburst flux component. These outburst fluxes\nappear to generally be bluer than $V-R = 0.55$. This colour cut removes\nvery few stars potential sources of microlensing events since the stars\nobserved on the blue side of $V-R = 0.55$ are generally brighter than $V = 17$.\nIn this analysis we do not apply any colour cut on the red stars in the CMD\nwhere a large number of variables stars are known to lie.\n\nWithin the analysed observations there are a number of spurious detections\ndue to satellite trails and other transient objects. These trails generally\ncause multiple object detections within a given observation. We remove\nthese spurious objects by requiring that the time of peak flux $t_{max}$ for\na microlensing event $i$, in image section $j$, does not occur at the same time\nas another event $k$, in the same image section ($t_{max_{ij}} \\neq\nt_{max_{kj}}$). The probability that two microlensing events will occur with\nthe same value of $t_{max}$ in close proximity, is very small. %($< 0.002\\%$). \nThis cut also removes any spurious detections due to systematic effects\nwhich can occur when the seeing in an image is poor, or the telescope \nslips during an exposure.\n\nLong timescale events are subject to significant parallax effects. To\ndetermine the efficiency of detecting such events we would have to a priori\nassume a distribution of sources and lenses towards the Galactic bulge.\nTherefore, in this analysis we impose an upper-limit on $\\hat{t}$ of one\nyear. Similar restrictions are true for binary lensing events since the\ndistribution of parameters for binaries is not well known. However, we\nthink our selection should not be significantly biased against these\nevents as our final selection contains all the events we believed were\nbinaries in the loose level-1.0 cuts.\n\nTo summarise our level-1.5 cuts, all events must meet the following\ncriteria: $V-R > 0.55$, $A > 1.34$, $\\Omega\\chi^{2} < 3.6$, $\\hat{t} < 365$\ndays, $t_{max_{ij}} \\neq t_{max_{kj}}$, $\\chi^{2}_{c} < 30$, plus one\nlevel-1.5 cut which is based on the event's microlensing fit peak-flux in\nthe $R_{M}$-band\\footnote{The light curves have higher S/N in the $R_{M}$\nband since $B_{M}-R_{M} > 0$ in general.}, $F_{pr}$, from Table \\ref{tab2}.\n\nThe last set of selection cuts are the level-2.0 cuts. These cuts are\ndesigned to remove the low S/N variables which are well fitted by\nmicrolensing light curves, but are nevertheless obviously variable by eye.\nFor final selection the candidate microlensing event must pass all the\nlevel-2.0 cuts where its peak flux ($F_{pr}$) is in the range of the cut.\nThe level-2.0 cuts are also specified in Table \\ref{tab2}. To quantify the\neffect of our cuts let us mention that there were $\\sim 300000$ variable\nobjects detected in the eight fields, only 776 of these passed our level-1.0\nselection. 219 then passed the level-1.5 cuts and 99 passed the final\nevent selection cuts (level-2.0).\n%143 + 84 + 90 + 132 + 78 + 132 + 79 + 38 %level-1.5 776 %level-2 219\n\nWe note that it is possible to express these level-1.5 and level-2.0 cuts in\na more direct form. However, the procedure presented here reflects the real\nselection process which is progressive and most easily accomplished in\nstages. Furthermore, the selection of final microlensing event candidates\nthis way is necessary in order to quantify the detection efficiency of the\nanalysis system. The experimental determination of the microlensing optical\ndepth requires that this detection efficiency is known. Some microlensing\ngroups still select final candidates by eye, for instance, see \\citeN{AUB99}.\n\n\n%\\placetable{tab0}\n%\\placetable{tab3}\n\n\n\\section{\\sc Results}\n\n\nWe have discovered 99 microlensing event candidates. The light curves and\nmicrolensing fits for 83 events are presented in Figure \\ref{lc1}. The\nactual amplification of each event was obtained by fitting the light curves\nwith the baseline source flux as an additional parameter. The other 16 events\nin field 108 were shown in \\citeN{ALC99c}. In each of these figures only a\nsingle season of data for each light curves is displayed. However, all the\nDIA light curves span three observing seasons.\n\n%\\placefigure{lc1}\n\nThe numbers of events detected in each field are given in Table \\ref{tab3},\nalong with results from the SoDoPhot analysis of the same images. The\ncoordinates and labels used for identifying the events in the datasets are\ngiven in Table \\ref{tab4}. We will refer to events primarily using their\nmicrolensing event ``alert'' ID\\footnote{The details of microlensing alerts\n can be found at\\\\ {\\em http://darkstar.astro.washington.edu/}}. Each\nalert ID includes the year, observation target (BLG, LMC or SMC), and the\norder of the detection. For example, the first alert event detected in the\nbulge in 1995 is labeled 95-BLG-1. Microlensing events which were detected\nby DIA, but not in the alerts or SoDoPhot re-analysis, have a ``d'' before\nthe event number (e.g. 95-BLG-d1). Events which were not found as alerts,\nbut were detected in the subsequent analysis of SoDoPhot photometry, are\nlabeled with an ``s'' (e.g. 95-BLG-s1).\n\n%\\newpage\n\\begin{figure}\n \\vskip 19cm\n %\\figurenum{3}\n \\special{psfile=./BULGE/fig3a.ps vscale=110 hscale=110 voffset=-185 hoffset=-30} \n \\figcaption{Events from Field 101\\label{lc1}. New DIA events have Ids\n containing ``d''. New events selected with this analysis and a \n reanalysis of the SoDoPhot data have Ids containing ``s''.}\n\\end{figure}\n\n%.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3b.ps vscale=110 hscale=110 voffset=-185 hoffset=-30} \n\\figcaption{Events from\n Fields 101 \\& 104. New DIA events have Ids containing ``d''. New events\n selected with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3c.ps vscale=110 hscale=110 voffset=-185 hoffset=-30} \n\\figcaption{Events from\n Field 104. New DIA events have Ids containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3d.ps vscale=110 hscale=110 voffset=-185 hoffset=-30}\n\\figcaption{Events from Field 104 cont. New DIA events are have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3e.ps vscale=110 hscale=110 voffset=-185 hoffset=-30}\n\\figcaption{Events from Fields 104 \\& 113. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3f.ps vscale=110 hscale=110 voffset=-185 hoffset=-30}\n\\figcaption{Events from Field 113. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3g.ps vscale=110 hscale=110 voffset=-185 hoffset=-30}\n\\figcaption{Events from Field 113 cont. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3h.ps vscale=110 hscale=110 voffset=-185 hoffset=-30}\n\\figcaption{Events from Fields 113 \\& 118. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3i.ps vscale=110 hscale=110 voffset=-185 hoffset=-30}\n\\figcaption{Events from Field 118. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3j.ps vscale=90 hscale=90 voffset=-80 hoffset=-10}\n\\figcaption{Events from Fields 118 \\& 119. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3k.ps vscale=90 hscale=90 voffset=-85 hoffset=0}\n\\figcaption{Field 119 events. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3l.ps vscale=90 hscale=90 voffset=-85 hoffset=0}\n\\figcaption{Events from Fields 119 \\& 128. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3m.ps vscale=90 hscale=90 voffset=-85 hoffset=0}\n\\figcaption{Events from Field 128. New DIA events have Ids\n containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n\n.\n\\newpage\n\\begin{figure}\n\\vskip 19cm\n\\figurenum{3}\n\\special{psfile=./BULGE/fig3n.ps vscale=90 hscale=90 voffset=-85 hoffset=0}\n\\figcaption{Events from Fields 128 \\& 159\\label{lclast}. New DIA events\n have Ids containing ``d''. New events selected\n with this analysis and a reanalysis of the SoDoPhot data have Ids\n containing ``s''.}\n\\end{figure}\n.\n\\newpage\n.\n\\newpage\n\n%\\addtocounter{figure}{1}\n\\begin{figure}[ht]\n\\epsscale{1.0}\n\\plotone{./Figures/fig4.ps}\n\\figcaption{The fitted $V$ magnitudes and colours of the microlensing events \noverlaid on a CMD of neighbouring resolved stars. The solid line corresponds\nto a pixel lensing cut. Circles with dots represent classical events\nand circles with crosses are pixel lensing events.\nSome pixel lensing event sources appear above the cut because the \nare not associated with monitored SoDoPhot sources.\\label{figCMD}}\n\\end{figure}\n\nThe microlensing event's source star V-magnitudes and $V-R$ colours are\ndisplayed in Figure \\ref{figCMD}. This figure is an average CMD, which\nhas been constructed by combining the SoDoPhot photometry for the\n$\\sim 250$ stars nearest each microlensing event source star. Notice that\nthere are very few events with bright main sequence sources. This is to be\nexpected, since not only are there few of these bright stars, but most of\nthem are foreground disk stars which ``see'' a much lower microlensing\noptical depth.\nThe SoDoPhot analysis is sensitive to microlensing events occurring with\nrelatively bright source stars ( $V < 21$) or fainter stars which are blended\nwith these monitored bright stars. The DIA technique on the other hand is\nsensitive to events independent of the locations of the brighter stars.\nIn this way the DIA technique is expected to mainly detect events where\nthe source star is faint and not blended with a bright star monitored by\nSoDoPhot. Events with unresolved sources are termed pixel lensing events.\n\n\\placetable{tab1}\n\\placetable{taball}\n\n\n\\subsection{\\it Pixel Lensing Events}\n\nOur set of microlensing events contains both classical microlensing events\nand pixel lensing events. In recent years, the division between these two\ngroups of events has been unclear because of the difference between pixel\nlensing and the pixel method. We will attempt to clarify this division in\nregard to our events. Our definition of pixel lensing derives from\nthe theory of \\shortciteN{Gould96a} who defined Pixel Lensing as the\n``gravitational microlensing of unresolved stars''. Whereas, the ``Pixel\nMethod'' is a method of binning images, which is used by the AGAPE group to\ndetect microlensing events towards M31 \\shortcite{AABB97,GON97}.\n\nThroughout this work when we refer to pixel lensing, we shall mean:\n``gravitational microlensing events where the source stars are unresolved in\ninitial template/reference images of a field''. This seems the most logical\ndefinition as it is equally applicable for events seen towards extra-galactic \ntargets, such as M31, as it is for events in the line-of-sight to the \nMagellanic clouds and Galactic bulge. In contrast to this, a classical \nmicrolensing event is taken to be one where the source is resolved both \nbefore and after the lensing has taken place. \nAs a division between classical and pixel microlensing still remains somewhat\nunclear, we will further refine our concept of pixel lensing.\n\nFor all the microlensing surveys presently being carried out,\nthere is a significant amount of crowding and blending.\nSpecifically, there are many faint unresolved stars within the seeing disk\nof any star bright enough to be detected. A microlensing event detected by\nmonitoring such a {\\em group} of stars, could be due to either a bright\ndetectable star (classical lensing), or one of the many faint unresolvable\nneighbouring stars blended with it (pixel lensing).\n\nWhen events are due to the unresolved faint stars, there can be a measurable\nshift in the flux centroid position of the event as the event brightens\n\\shortcite{ALC97d}. However, the offset between the faint source's centroid\nand the group centroid is a random quantity. Such an offset will take an\narbitrarily small value in some events. Hence, this approach can not quantify\nthe degree of blending for all events. Another property of these events is\nthat they can exhibit a significant chromatic signature when the event's\nsource star colour varies from the overall colour of the group. Sources on\nthe main sequence near our detection limit ($V\\sim21$) make up the bulk of\nthe monitored stars and have a narrow range of colours. For this reason, in\nmany cases, there may be little difference between an individual star's\ncolour and that of the blended group of stars it resides amongst. The final\npiece of information about blended events comes from the shape of the light\ncurve. That is, a given amplification microlensing event has a specific\nshape. Once again there is only a slight difference between the shapes of\nevents with different amplifications if various timescales are considered\n(see \\shortciteNP{GH92}, \\shortciteNP{WP97} \\& \\shortciteNP{ALC99c}).\n\nFrom these considerations, a sensible method of differentiating between\nclassical and pixel microlensing events is to put limits on the\nallowable centroid offset, the true amount of flux (amplification), and the\nevent's colour. Beyond such hypothetical limits an event could be defined\nto be pixel lensing with significant confidence. For this reason we will\nset further limits on our definition of pixel lensing events. To quantify\nthis decision we define that, for classical lensing the centroid offset\nbetween the event centroid and the nearest photometered object centroid\nshould be less than three times the centroid uncertainty. Otherwise this is\na pixel lensing event. The average uncertainty in our event centroid\npositions is $\\sim 0.2 \\arcsec$ for the results presented here. This\nuncertainty includes the error in the transformation between DIA and\nSoDoPhot templates. We thus adopt an offset of 1 pixel ($0.63\\arcsec$) as\nour classical event limit. In addition, if the microlensing event fit gives\na source $V$ magnitude which is significantly below the detection threshold\n(for isolated stars), the event is deemed a pixel lensing event. Because\nnone of the stars in the observed fields are truly isolated, the actual\ndetection threshold will be higher. Thus, this serves as a robust lower\nlimit on classical lensing event sources. However, the uncertainty in the\nfitted baseline source flux increases as the brightness of the source\ndecreases. For the analysed fields we have set this threshold limit to be\n$V_{pix} < 20.8 + 0.3(V-R)$ (see Figure \\ref{figCMD}).\n\nMeasured variations between the microlensing event's colour, and the\nsource's baseline colour, do not provide sufficient evidence to distinguish\nwhether an event is due to pixel lensing or classical lensing. For some\nevents the chromatic signature of blending can be measured in the light\ncurve of the event, although the source is bright enough to be detected if\nit were isolated. Such colour changes can tell us the magnitude of the\ndifference in the colours, and the colour of the group. But colour\ninformation does not guarantee an event is due to an unresolvable star.\nHowever, the larger the colour differences, the more likely an event is\npixel lensing.\n\nApplying these definitions to our events we separate the classical events\nand pixel lensing events. Slightly different information is available for\nclassical and pixel lensing events. With classical events there is a\nmeasured baseline flux of the source star. This baseline flux is a blend of\nall the stars within the seeing disk. For these events one can perform blend\nfits on the photometry in order to determine the true baseline source flux\nand hence amplification and event timescale. The fit parameters for the\nindividual events are given in Table \\ref{tab5}. A blended baseline flux is\nnot applicable for pixel lensing events. However, the baseline source flux,\nthe amplification, and the timescale can still be determined from the\nmicrolensing light curve fit, these are given in Table \\ref{tab6}.\n\nTwo types of fits were performed for classical events: one which assumed the\nnearest SoDoPhot source star flux was the true source flux, and the other\nwhere the source star flux was a free parameter. The results of the first\napproach was expected to yield biased results because of the blended flux.\nHowever, this set is useful to examine timescale and other biases\ncaused by blending. We fitted with the baseline source flux as a free\nparameter for all our pixel lensing events. This enabled us to determine the\namplification of each lensing event relative to a fitted baseline flux\nvalue.\n\n%\\placetable{tab2}\n%\\placetable{tabpix}\n\n\\subsection{\\it Comparison with the Standard Analysis}\n\nDuring our normal data reduction procedure we carry out a detailed\nreanalysis of all the SoDoPhot photometry. This reanalysis is designed to\nfind new microlensing events and better characterise the known {\\em alert}\nevents. The reanalysis typically finds a few more events than the alert\nsystem as the alert scheme was designed for real time notification,\nand as such only includes the data reduced at the time the alert \nis issued.\n\n%\\placefigure{fig1}\n\\begin{figure}[ht]\n\\epsscale{1.0}\n\\plotone{./Figures/fig5.ps}\n\\figcaption{The locations of the 8 Galactic bulge fields presented in this\n paper are shown in Galactic coordinates (with their corresponding MACHO Id\n numbers). The 99 microlensing events are represented as dark spots at the\n event locations. The area of each spot is proportional to the $\\hat{t}$\n value of the event. Baade's Window is in field 119.\\label{fig1}}\n\\end{figure}\n\nImplementing the DIA technique we have found 41 more events than the\nSoDoPhot analysis using the same image data. This method thus gives us $\\sim\n71\\%$ more events than our approach based solely on PSF photometry. We\nbelieve this is not due to the failure of our PSF analysis, but instead is a\ntribute to the advantage of the new technique. To emphasise this, we point\nout that there are in fact 57 new events in the DIA which were not found in\nthe standard SoDoPhot analysis (see Table \\ref{tab3}). However, 16 events\ndetected with the SoDoPhot analysis were not found with the DIA reduction.\nThese missed events fall into a number of categories. Seven events\nwere missed because of differences between the microlensing cuts used (1 low\nS/N event, 3 poor photometry events, 1 event with $\\hat{t} > 300$ days, 2\nevents fail the $\\chi^{2}_{m}$ cut). A further 4 of these events were never\ndetected because of the slight differences between the pointing of the\nSoDoPhot and DIA reference images. Lastly, there is a dead amplifier on one\nof our CCD mosaics. Five SoDoPhot microlensing candidates fall into this\nlocation in our fields, thus photometry for these events is only available for\none colour ($B_{M}$). However, in the difference image analysis we have only\nanalysed regions where two colour photometry is possible.\n\nFor a fair comparison of the two techniques it is necessary to compare\nequivalent areas of the target fields. Small differences in the pointing\nshould not affect the overall number of events and any variation in the cuts\napplied is part of the reduction technique. Therefore, to fairly compare\nthe two sets we set aside the five single colour events in the region not\nanalysed with DIA. We thus conclude that the SoDoPhot analysis yields 53\nevents compared to 99 events from the DIA technique. This method thus\nprovides {$\\sim \\bf 87\\%$} more events than the standard SoDoPhot analysis.\n\nThe entire MACHO dataset consists of around 350 candidate microlensing\nevents. A complete DIA-based reanalysis of the Galactic bulge database\ncould therefore provide $> 600$ events. The number of events detected\ntowards the LMC, by SoDoPhot analysis, is still relatively small\n\\cite{ALC00p}. A factor of $2$ increase in the number of events detected\ntowards the LMC would be an important way of reducing the statistical\nuncertainty in the microlensing optical depth and hence the baryonic\nfraction of the Galactic halo. We believe this factor of 2 could be\nachieved in a future reanalysis of LMC data with DIA.\n\n\n\\section{\\sc Microlensing Detection Efficiency}\n\nTo determine the microlensing detection efficiency of this analysis we\nproduced a Monte-Carlo simulation for each of the eight fields reduced.\nIn this simulation we attempted to include all the known aspects of the\nanalysis, such as the seeing, sky-level, transmission, systematic noise, \netc., of each observation.\n\n%\\placefigure{figEff}\n\\begin{figure}[ht]\n\\epsscale{0.8} \n\\plotone{./Figures/fig6.eps} \n\\figcaption{Combined\n microlensing detection efficiency for source stars to a limiting\n magnitude of $V \\sim 23$ (solid line), and for clump sources (dotted\n line). Efficiencies differ from these lines for individual fields\n because of variations in sampling, reddening seeing, etc.\\label{figEff}}\n\\end{figure}\n\n\\noindent\nThe combined average detection efficiency, as a function of event timescale,\nis given in Figure \\ref{figEff}. The calculation of the optical depth\ncontribution from each microlensing event uses the efficiency for the field\nwhere the event was found.\n\n\\subsection{\\it The Combined Luminosity Function}\n\nThe observed detection efficiency is dependent on the number and colour\ndistribution of target source stars. In the case of pixel lensing events\nthese stars may lie below the detection threshold of our experiment. To\ndetermine the properties of stars below our detection threshold we have used\nthe Hubble Space Telescope (HST) Luminosity Function (LF) and\ncolour-magnitude diagrams (CMDs) for Baade's Window of \\citeN{HWBG98}. These observations were taken with the HST Wide Fields Planetary\nCamera 2 (WFPC2) which has a field-of-view of $\\sim 5\\Box\\arcmin$. %(<2.5x2.5)\\arcmin\nThere are very few bright stars ($ V < 16$) in these HST data, so the\nbright end of the luminosity function is not well defined from these data\nalone. However, the MACHO camera's field-of-view is hundreds of times larger\nthan that of WFPC2 ($43\\arcmin \\times 43\\arcmin$), so many bright stars are\nin the MACHO observations of Baade's Window. We thus combine the luminosity\nfunction and CMD of \\citeN{HWBG98} with those of our field $119$ \n(our Baade's Window field). \n\nTo combine these data we transformed the \\citeN{HWBG98}'s HST photometry\nfrom $(F555-F814)$ to Landolt's $(V-I)$ with the calibration given in \\citeN{HBCTWW95}. \nThe HST magnitudes were converted to $Kron-Cousins$ $V-R$\nwith the transformations of \\citeN{Bessell95}. For MACHO data we used the\nconversion from $R_{M}$ and $B_{M}$ to $Kron-Cousins$ $V$ and $R$\ngiven in \\citeN{ALC99c}. We note this calibration varies slightly from the\nmost recent determination given in \\shortciteN{Alves99}. The two data sets\nwere combined at $V$ $\\sim 18.5$ where the MACHO detection efficiency starts\nto decrease from unity. The uncertainty in the number of stars from the\ncombined LF is $9.5\\%$. This uncertainty carries through to our optical\ndepth estimate with the same magnitude. The number of stars in each of our 8\nfields is calculated using this combined luminosity function and the\nmeasured stellar density of clump stars relative to field 119. By taking\nthis approach we have implicitly assumed that the stellar density does not\nvary greatly within Baade's Window.\n\nA major cause of variations in the apparent stellar density towards the\nGalactic center is the patchy extinction caused by dark dust clouds.\nHowever, Baade's Window is well known as a region of low extinction.\nFurthermore, the extinction in Baade's Window appears to be relatively\nhomogeneous \\cite{ALC97b}. Therefore, we believe that the\ncombined LF should only be slightly affected by extinction inhomogeneities.\nWe have also made the assumption that the morphology of the CMD does \nnot vary between the HST field and the MACHO one. We expect that the \npresence of the Galactic bar \\cite{PSUS94b} could cause \nthe number of clump giants to vary slightly across our \nBaade's Window field.\n\n\\subsection{\\it Artificial Events}\n\nTo simulate microlensing events, firstly we weighted the $V$-magnitude axis\nof the combined (HST+MACHO) CMD to reflect the luminosity function\ndistribution. We then binned this CMD and determine the relative\nprobability of selecting a source star in any given bin. Next, we randomly\nselected a source star for the event based on the bin probability. A random\nimpact parameter $u_{min}$, timescale $\\hat{t}$, and time at maximum\n$t_{max}$ were then assigned to the artificial event. An artificial image\nwas produced for each observation and the artificial source star was added.\nEach image is only $\\sim 25\\arcsec \\times 25\\arcsec$ in size. These images\nwere further populated with neighbouring so called {\\em blend-stars}. These\nstars were selected from the CMD in the same way as the source star and\nallow us to simulate the effect of blending on light curves. The number of\nblend-stars we added to the images was based on the observed stellar density\nof the field we were simulating. Each of the artificial stars was placed at\na random location within the artificial image. An artificial image was made\nfor each observation of the event. These images were produced with seeing\nconditions that matched those observed during the data reduction.\n\nFor each simulated lensing event we performed photometry on the set of artificial\nimages making up the light curve. This photometry included the\nuncertainties produced by the photon and systematic noise of all the stars\ncontributing to the flux aperture. The photometry of each of the simulated\nevents was piped through the detection and selection processes used in the\nreal analysis to determine the overall efficiency.\n\nThe simulation was performed separately on each of the eight Galactic bulge\nfields using the seeing conditions, sky background level, etc., of the field\nselected. For each field we produced $100,000$ artificial microlensing\nevents each with $\\sim 300$ observations. The seven fields adjacent to\nBaade's Window were simulated using the combined Baade's Window luminosity\nfunction. In each case the LF and the combined CMD were reddened to the\naverage reddening of each field. Since the fields are at Galactic latitudes\nspanning $\\sim 2\\arcdeg$ we expect the contributions from the number of disk\nstars and bulge stars to vary between fields. The stellar densities in\nsimulation of each field were adjusted to reflect the observed densities.\nThese field-dependent stellar densities were employed because the amount of\nblending is dependent on the observed density.\n\n%\\placefigure{vdist}\n\\begin{figure}[ht]\n\\epsscale{1.0} \n\\plotone{./Figures/fig7.eps}\n\\figcaption{The distribution of source\n magnitudes. The solid histogram shows the distribution of the events in\n $V_{obj}$ corrected to the average reddening of field 119 (Baade's\n Window). The dashed line is the luminosity function modified by the\n efficiency and normalised to the area of the histogram.\\label{vdist}}\n\\end{figure}\n\nTo compare the results from our simulation with the observed distribution of\nevents, we have plotted the actual (histogram) and expected (dashed line)\n$V$-magnitude distributions in Figure \\ref{vdist}. In this plot we have\nmultiplied the Baade's Window LF by the detection efficiency, thus giving\nthe expected source star $V$-magnitude distribution. The microlensing event\nsource star magnitudes ($V_{obj}$) have been extinction corrected to the\naverage reddening observed in our Baade's Window field. The good agreement\nbetween these two distributions suggests that our simulated efficiency\nanalysis reproduces the actual detection sensitivity quite well. For\nfurther details of the detection efficiency simulation, see \\citeN{DrakeTh}.\n\n\n\\section{\\sc The Optical Depth}\n\nThe microlensing optical depth is defined as the probability that any given\nstar is microlensed with impact parameter $u_{min} < 1$ (i.e. $A_{max} >\n1.34$) at any given time. This optical depth is independent of the mass of\nthe lensing objects, so no assumption is required about velocity\ndistributions and mass functions. The optical depth can be estimated by,\n\n\\begin{equation}\n\\tau_{est} = \\frac{\\pi}{4N_{*}T_{*}} \\sum_{i} \\frac{\\hat t_{i}}{\\varepsilon(\\hat t_{i})}.\n\\end{equation}\n\n\\noindent\n$N_{*}$ is the number of stars and $T_{*}$ is the exposure time of\nthe experiment in years, $\\hat{t_{i}}$ is the Einstein ring crossing time for\nthe $i$th event, and $\\epsilon(\\hat{t_{i}})$ is the detection efficiency for\na given event timescale. In this analysis we use the source fit $\\hat{t}$\nvalues, given in Tables \\ref{tab5} and \\ref{tab6}, to determine the\noptical depth. For the {\\em exotic}: finite source, binary lensing and\nparallax affected events, we use the $\\hat{t}$ values given in Alcock \n{et~al.\\ }(\\citeyearNP{ALC97a}, \\citeyearNP{Becker00} \\&\n\\citeyearNP{Beck00b}). For microlensing events with timescales which our\nexperiment is sensitive to ($2 - 300$ days), we obtain\n$\\tau^{300}_{2}=2.43^{+0.31}_{-0.29}\\times 10^{-6}$. When we include the\nuncertainty in the effective number of stars monitored, we obtain\n$\\tau^{300}_{2}=2.43^{+0.39}_{-0.38}\\times 10^{-6}$.\n\n%\\placefigure{figOptUn}\n\\begin{figure}[ht]\n\\epsscale{0.9}\n\\plotone{./Figures/fig8.ps}\n\\figcaption{The distribution of statistical uncertainty in the microlensing \noptical depth from Monte-Carlo simulation for our 99 event sample.\nThe dashed lines mark one $\\sigma$ confidence limits. The solid line gives\nthe observed optical depth.\\label{figOptUn}}\n\\end{figure}\n\nThe statistical uncertainties in the optical depth have been obtained by\nusing the fact that the number of events obey Poisson statistics. We have\nsimulated ``experiments'', where the number of observed events $N$, follows\na Poisson distribution. For each event in the simulated timescale\ndistribution we randomly assign one of our measured event timescales (see\nalso \\shortciteNP{USKK94d,ALC97e}). The optical depth\nfor each of these distributions is then evaluated. Counting the fraction of\nthese experiments which yield a larger optical depth than our measured value\n$\\tau$, allows us to determine the statistical distribution of optical\ndepth. This distribution is shown in Figure \\ref{figOptUn}, from which we\ndetermine confidence limits on our measured optical depth $\\tau_{meas}$.\n\n%\\placefigure{figKS}\n\\begin{figure}[ht]\n\\epsscale{0.9} \n\\plotone{./Figures/fig9.ps} \n\\figcaption{The cumulative distribution\n of impact parameter for the 99 microlensing event sample. Here the\n straight line shows the expected $u_{min}$ distribution for microlensing\n without the effect of efficiency. The dotted histogram show the\n distribution for the efficiency simulation and the solid histogram the\n observed distribution.\\label{figKS}}\n\\end{figure}\n\nThe simple geometry of microlensing events results in the theoretical\nprediction of a linear distribution in impact parameters ($u_{min}$).\nHowever, experimentally events with faint source stars require larger\nmagnifications to be detected. This tendency skews the impact parameter\ndistribution towards smaller $u_{min}$ values. The effect of this is\nclearly seen in the cumulative distribution shown in Figure \\ref{figKS}.\n\nA Kolmogorov-Smirnov (K-S) test is a useful method for comparing whether\nsamples of events are drawn from the same distribution. A comparison\nbetween the expected distribution from our efficiency analysis and that\nobserved, gives a K-S statistic of $D = 0.086$. This corresponds to a\nprobability of $P(KS)=0.39$. In a comparison between the observed\ndistribution and a uniform distribution, we get a K-S statistic of $D=\n0.338$ corresponding to $P(KS)= 5.78e^{-10}$. The observed $u_{min}$\ndistribution is only compatible with that found using our efficiency \nsimulation, which demonstrates the bias towards high amplification \nevents.\n\n\\subsection{\\it The Optical Depth in Individual Fields}\n\nThe microlensing optical depth is a measure of the mass in lensing objects\nalong a line-of-sight. This quantity is independent of the individual masses\nof lensing objects, as long as they have characteristic timescales which lie\nwithin the region to which the experiment is sensitive. To calculate this\nstatistic from the measurements we do not need to know the velocity or mass\ndistributions of the object causing the lensing. This value is therefore\nuseful to compare the measurements with the predictions from Galactic\nmodels, where the distribution are assumed. However, any Galactic models\nwhich match the observed total optical depth must also match the observed\nevent timescale distribution. Event timescales provide a constraint on a\nmodel's velocity and mass function. By splitting the dataset of observed\nevents into sub-regions, one can also compare the optical depth distribution\nas a function of location.\n\nSince we expect the relative number of disk stars and bulge stars to vary\nbetween fields we decided to model this variation. To account for this we\nproduced a disk-LF by combining the LFs of \\citeN{WJK83p163} and\n\\citeN{GBF96} at $M_{V} = 9$. The result is a luminosity function for the\ndisk ranging from $M_{V} = -1$ to $M_{V} \\sim 14$. We assume that the disk\nis well modeled by the standard double exponential disk density profile\ngiven by,\n\n\\begin{equation}\\label{disk}\n\\rho_{d} = \\rho_{d0}\\;{\\rm exp}\\left (\\frac{-|z^{\\prime}|}{h} \n+ \\frac{(R_{0}-s)}{s_{d}} \\right).\n\\end{equation}\n\n\n\\nocite{ZSR95}\n\\nocite{VA95}\\noindent\nFor, the disk cylindrical galactocentric coordinates ($s,z^{\\prime}$),\nwe have $s^{2} = 1 + z^{2}\\cos^{2}{\\theta} - 2z \\cos{b}\\cos{l}$,\nand $z^{\\prime} = z\\sin{b}$,\nwhere $z = D_{d}/D_{8.5}$ and $s_{d}$ is the disk scale length \n($2.5-3.5$ kpc). The mass density in the solar neighbourhood \nis given by $\\rho_{d0}$ (0.05 $\\rm M_{\\odot}pc^{-3}$), and $h$ is \nthe scale height of the disk.\n\n%\\placetable{tab1}\n\nWe use this double exponential disk model to determine the form of the\ndisk-LF that we expect in each of our fields, with proper consideration of\nthe reddening and the number of stars within the observed volume. Here we\nassumed a disk scale height $h$ of 325 pc and a scale length $s_{d}$ of 3\nkpc. We normalised the disk-model luminosity function for each field to the\nnumber of disk stars, observed in the magnitude range $14 < V < 18$. This\nallowed us to approximate how many disk stars there are in each field down\nto the source star magnitude limit ($V=23$). The percentage of disk stars at\nthis limit, $p$, was given in Table \\ref{tab3}. By subtracting the\ndisk-model LF for the Baade's Window field from the HST+MACHO combined LF,\nwe were able to determine the bulge luminosity function. We then assumed\nthat the number of bulge stars in our fields were traced by the number of\nclump giant stars. The detection completeness for these bright clump stars\nis nearly $100\\%$, so these stars serve as a good tracer of the number of\nfainter bulge main sequence stars, where our detection completeness is low\nand uncertain. We scaled our disk-subtracted bulge luminosity function by\nthe ratio of the number of clump stars in each field relative to our Baade's\nWindow field, $r$ (Table \\ref{tab3}). We finally determined the total number\nof stars in each field by combining the number of disk stars from our disk\nmodel, and bulge stars from the scaled Baade's Window bulge luminosity\nfunction.\n\nThe optical depth for each individual field (given in Table \\ref{tab3}) was\ndetermined using the total number of stars in each field. From the tabulated\ndata it seems there is a trend in optical depth with Galactic latitude and a\nweaker trend with longitude, as is expected. However, the observed optical\ndepth for field 104 appears to break from the general trend, although the\nuncertainties for this field are quite large. The density of stars in field\n104 is not significantly higher than the other fields ($N_{104} \\approx\nN_{108}$), and this evidence naturally leads us to believe that the number\nof faint lensing stars should be similar. A group or cluster of low-mass,\nfaint stars in the foreground of the field could go undetected in our\nphotometry and would act as efficient lenses. There is a globular cluster in\nfield 104 and another nearby, but the locations of these do not appear to\ncoincide with the observed microlensing events.\n\n\n\\subsection{\\it Uncertainties in the Optical Depth}\n\nThe major cause of uncertainty with the determined optical depth could be\nerroneous values from the microlensing fits. For low S/N events it is\ndifficult to determine whether the correct amplifications and $\\hat{t}$\nvalues have been found. For such events there are similarities between the\nlight curves of events with different timescales and amplifications\n\\cite{WP97}. However, \\citeN{Han99} has recently shown\nthat fits to high amplification pixel lensing events do give accurate\nresults.\n\nIn this analysis we have used a single CMD to determine the microlensing\ndetection efficiency of the analysis. This CMD is artificially reddened\nto the average value determined for each field, which assumes that the\nmorphology of the CMD does not vary much between these fields. From our\ndisk model it appears that the number of disk stars relative to bulge stars\nvaries little from field to field. The gradient in the observed optical\ndepth is in part due to the dependence of the numbers of disk stars on the\nGalactic latitude.\nIf our assumed model of the disk is in error, the number of stars we\ndetermine for the individual field will be incorrect. This will also affect\nour disk-subtracted bulge luminosity function. However, the calculated\nnumber of disk stars is only about $10\\%$ of the total number of stars in\nany field, in good agreement with \\citeN{ZCFGOR99}, so this effect\nshould be small. Our total optical depth result is less dependent on the\nassumed disk model than the individual fields, since the ratio of the total\nnumber of disk stars ($14 < V < 18$) to clump stars in all 8 fields\ncombined, is very close to the ratio of disk to clump stars in Baade's\nWindow.\n\nIn addition, the extinction in each observed field is taken into account\nusing values determined from the RR Lyrae stars in each field\n\\shortcite{ALC98a}. There are small uncertainties in the efficiencies due\nto variations in reddening within a field. This should not affect the\noverall optical depth but may be important for estimates in individual\nfields.\n\n\n\\section{\\sc The Structure of the Galaxy}\n\nWe will now review what has been reported about each of the Galactic\nfeatures that have the largest effect on the observed microlensing optical\ndepth. For each of these we will discuss whether our results are consistent\nwith models and previous determinations.\n\n\\subsection{\\it Bar Orientation}\n\nThe optical depth is highly dependent on the position angle of the bar\n\\shortcite{Peale98} and bulge mass \\cite{Gyuk99}. The\nobservational results for the bulge inclination based on a number of\ndifferent types of observations give conflicting values ranging from\n$16\\arcdeg$ to $44\\arcdeg$, see Table \\ref{tab7}. A bar inclined at the\nlarge angle reported by \\citeN{SSV99} is not an efficient source of\nlensing events. The size of our observed optical depth favours the smallest\npossible bar inclination angle. However, the bar is insufficient to produce\nan optical depth greater than $\\sim 2.5 \\times 10^{-6}$ even for models with\na small bar inclination angle, and a large bar mass (see \\citeNP{Peale98}).\n%\\placetable{orient}\n\n\n\\subsection{\\it Bar Mass}\n\nIdeas to explain the observed microlensing optical depth with values of\nbulge mass are also not clear-cut. Based on COBE map data \\citeN{ZM96}\nfound $M_{bulge} =(2.2\\pm0.2) \\times 10^{10}\\,\\rm M_{\\odot}$, yet\n\\citeN{DAH95} found a mass of only $1.3 \\times 10^{10}\\,\\rm M_{\\odot}$.\n\\citeN{BEBG97} found that, based on DIRBE results, within 2.4 kpc of\nthe Galactic center the combined bulge plus disk mass is $1.9 \\times 10^{10}\\,\n\\rm M_{\\odot}$. However, only $0.72-0.86 \\times 10^{10}\\,\\rm M_{\\odot}$ of this is\nattributed to the bulge mass. This is consistent with \\citeN{HWBG98}'s\nresults ($0.74 - 1.5 \\times 10^{10}\\, \\rm M_{\\odot}$).\n\nPredictions of the Galactic bar mass have also been discrepant. Based on the virial\ntheorem, \\citeN{HC95b} predicted that $M_{bar}= 1.6 \\times\n10^{10}\\,\\rm M_{\\odot}$, as did \\citeN{Kent92} based on a simple oblate\nrotator model. But \\shortciteN{Blum95} predicted $M_{bar}= 2.8 \\times\n10^{10}\\,\\rm M_{\\odot}$ when pattern rotation of the Galactic bar is included. \n\\citeN{ZM96} determined that a bar mass of at least \n$> 2.0 \\times 10^{10}\\,\\rm M_{\\odot}$ is required for the COBE G1 model \n(with $\\theta=11$ degrees) to account for observed amount of lensing. \nThe \\shortciteNP{ZM96} model is consistent with \\citeN{ALC97a}'s\nmicrolensing data at the 2 $\\sigma$ level, if a bar mass of $2.8 \\times 10^{10}\\,\\rm M_{\\odot}$ \nis used. \\citeN{Gyuk99} advocated $M_{bulge}= 2.5 \\times 10^{10}\\,\\rm M_{\\odot}$\nbased on a maximum likelihood estimate of the COBE G2 model, where a small\ninclination angle of $\\theta= 12$ degrees was assumed. However, if $\\theta$\nwas instead taken to be 20 degrees (consistent with most of the\nvalues in Table \\ref{tab7}), the most likely bulge mass rises to \n$\\sim 3.6 \\times 10^{10}\\,\\rm M_{\\odot}$. A heavy bar is favoured for our \nobserved optical depth, but as yet there is no evidence that the bar is \nmassive enough to produce the observed optical depth. It is clear that more \naccurate measurements are necessary to better constrain the bar mass \nused in models. In Table \\ref{tab8} we present the optical depths for a \nnumber of Galactic models for comparison with our result.\n\n%\\placetable{Modopt}\n\n\\subsection{\\it The Disk}\n\nThe estimates of the optical depth due to the disk also exhibit a range of\nvalues. Spiral arms may contribute $0.5 \\times 10^{-6}$ \\shortcite{Fux97}. \nA truncated disk would contribute $0.37-0.47 \\times 10^{-6}$\n\\shortcite{PAC91}, whereas a full disk would contribute\n$0.63-0.87 \\times 10^{-6}$ \\cite{ZSR95}. However, from these values it\nis clear that the disk is expected to be a less important contributor to\nthe optical depth than bar mass or bar orientation. The only measurement \nof the contribution of disk lensing comes from the EROS II analysis\n($0.38^{+0.58}_{-0.15} \\times 10^{-6}$, \\shortciteNP{DAF99}). \nThis is in good agreement with predictions, but is based on just three \nmicrolensing events.\n\nWe have estimated a disk lensing contribution to the optical depth of\n$f_{disk} \\sim 25\\%$. This gives \n$\\tau_{_{bulge}}= 3.23^{+0.52}_{-0.50}\\times 10^{-6}[0.75/(1-f_{disk})]$. \nDisk lenses are not expected to contribute much more than $25\\%$ since there\nis little evidence of any disk-disk lensing of the foreground main sequence\nstars in Figure \\ref{figCMD}. However, the optical depths we observed in\nthe individual Galactic bulge fields are quite high and are thus consistent \nwith a large disk optical depth.\n\n\\subsection{\\it The Timescale Distribution}\n\nThe mass function of the compact objects in the Galaxy has a direct effect\non the timescales of microlensing events. To date, few authors have\nattempted to reproduce simultaneously both the optical depth and the\ntimescale event distribution. The agreement between models and observations\nunder these two constraints is necessary, if any confidence is to be put in\nthe models. \\citeN{Peale98} and \\shortciteN{MER98a}\nhave found that the Galactic models poorly reproduce the\nobserved timescale distribution of \\citeN{ALC97a}.\n\n\\subsubsection{Short Timescale Events}\n\nThe geometry of microlensing produces short timescale events where the\nlenses are either near the observer or the source. However, variation in\nthe lens location has only a small effect, unless either the observer-lens\ndistance or lens-source distance is very small. Short timescale events can\nalso be produced by low mass lenses, but the timescale goes as $\\rm\nM^{1/2}$, so the effect is small. Distributions with a significant number of\nshort timescale events and a number of long timescale events, point to a\nlarge range of lens masses.\n\nIn an attempt to reproduce the observed short timescale events of\n\\citeN{ALC97e}, mass functions with large numbers of low mass\nobjects, such as brown dwarfs, have been produced \\shortcite{MER98a,Peale98,HC98}. This is supported by recent discoveries of\nfree floating brown dwarfs \\cite{RKLBG99,CSMDL99}. But there is still\ncontroversy surrounding whether this will \\cite{HC98}, or will not\n\\cite{Peale99} reproduce the expected timescales correctly.\nIf the short timescale events are not brown dwarf lensing events, \nthen this indicates a large population of low mass stars in the bulge.\nIt is possible that a large number of M-type stars might \nbetter explain the timescale distribution \\shortcite{Peale98}, \nbut this does not appear to be consistent with the shallow slope\nof the mass function found in recent deep observations of the \nGalactic bulge \\cite{HWBG98,ZCFGOR99}.\n\n\n\\subsubsection{Long Timescale Events}\n\nThe timescale distributions from contemporary Galactic models also do not\nreproduce the observed number of long timescale events, $\\hat{t} > 140$ days\n(for example, see \\shortciteNP{ALC97e,HC96b,MER98a,Peale98}). With the data\nof \\shortciteN{ALC97e} it was unclear whether these long timescale events\nwere a real population or a statistical anomaly. For long timescale events\nwe expect either large lens masses, low transverse velocities, or equal\nobserver-lens and lens-source distances. Disk-disk lensing events are\nexpected to give long timescale events because of the low velocity\ndispersion of the disk. Long timescale events have been observed for known\ndisk-disk lensing events \\shortcite{DAF99}. However, disk-disk lensing\nevents are considered in the Galactic models and are constrained by star\ncounts. The variation in lens-observer and observer-source distances has a\nrelatively small effect on an events timescale (factor of $\\sim 2$). If the\nlenses are normal main sequence stars, their luminosities can easily be\nrelated to their masses. Owing to this relation, for nearby lenses we can\nimpose an upper-limit to the lens mass, given an upper-limit to the lens'\nluminosity (from the microlensing fits). This constrains the proximity of\nbright lenses to the observer \\shortcite{ALC97a}. However, reddening\ntowards the Galactic bulge is patchy and weakens this argument. One\npossibility is that there exists some unknown population of dynamically cold\nor massive, dark objects in the Galactic disk.\n\n\\subsubsection{The Observed Timescale Distribution}\n\n\\begin{figure}[ht]\n\\epsscale{1.0}\n\\plottwo{./Figures/fig10a.ps}{./Figures/fig10b.ps}\n\\figcaption{Left: the timescale ($\\hat{t}$) of the 99 candidate microlensing\n events compared to predictions from four mass models, normalised to the\n observed number of events. The mass functions are: a $\\delta$ function at\n 0.1M$_{\\odot}$ (long-dashed line); a $\\delta$ function at 1M$_{\\odot}$\n (the short-dashed line); a Scalo (1986) PDMF (solid line); the Han \\&\n Gould (1996) power-law model with $\\alpha = -2.3$ and $m_{lo} = 0.1$\n (dash-dotted line). Right: the cumulative timescale distributions of the\n events from this work (solid line). The Alcock {et~al.\\ }(1997e) full\n sample of 41 events (short-dashed line) and the Alcock {et~al.\\ }(1997e)\n 13 clump giants (long-dashed line). The results are consistent with the\n 13 clump sample but not with the 41 event sample which are affected by\n blending problems.\\label{figdis}}\n\\end{figure}\n\nThe microlensing event timescale distribution is plotted in Figure\n\\ref{figdis}. Here we have over-plotted the timescale distribution expected\nfor four mass functions assuming the barred bulge, given in equations\n(\\ref{pt1}) and (\\ref{pt2}), plus a double exponential disk density model\ngiven in \\shortcite{ALC97e}. These four mass models are: a \\citeN{Scalo86}\nmain sequence present day mass function (hereafter PDMF), two $\\delta$\nfunction distributions (0.1M$_{\\odot}$ \\& 1M$_{\\odot}$), and the power-law\ndistribution of \\citeN{HC96b} ($\\alpha = -2.3$, $0.1 < m < 1.4M_{\\odot}$).\nThe agreement between the \\citeN{Scalo86} mass model and event timescale\ndistributions appears quite reasonable. This might imply that the\nmicrolensing events seen towards the Galactic bulge can be explained by the\nobserved distribution of stars, if they follow a \\citeN{Scalo86} mass model.\nHowever, this does not explain the large observed optical depth. The\ntimescale distribution also appears to be somewhat broader and much less\npeaked than expected. This difference could possibly be due to uncertainties\nin the timescales of the fainter events. There does not appear to be a\nlarge population of short timescale events in the distribution. This implies\nthere is probably not a large population of brown dwarfs along the bulge\nline-of-sight. However, there is some evidence for a population of long\ntimescale microlensing events, as there are seven events with timescales\n$\\hat{t} > 140$ days.\n\nIn the right panel of Figure \\ref{figdis} we present the cumulative\ndistribution of event timescale for these results compared with our previous\nresults for the Galactic bulge. Here we find relatively few short timescale\nevents compared to the previous results of \\shortciteN{ALC97e}, where fits\ndid not include parameters for the blended flux component. A K-S statistic\ncomparison between the cumulative distributions of the events from\n\\shortciteN{ALC97e} and this analysis gives a K-S statistic $D =0.296$,\ncorresponding to a probability $P(KS)= 0.0154$. This indicates that there is\na significant difference between the two timescale distributions. The fact\nthat blending will bias the event timescale distribution to shorter values\nhas been known for many years \\cite{Nemiroff94} and has been studied in\ndetail by a number of authors (\\citeNP{ALC96d,WP97,Alard97,HAN97a};\n\\shortciteNP{HJK98}). Therefor, this result is not surprising. We note that a\nsmall fraction of the difference between these and the previous results is\nalso due to difference between the detection efficiencies of\n\\shortciteN{ALC97e} and those of this analysis. In a comparison with the 13\nclump giant sample of \\shortciteN{ALC97e} we get a K-S statistic of $D =0.25$\ncorresponding to $P(KS)= 0.37$. Our results are thus consistent with the\nclump giant microlensing events. Clump giant microlensing events are less\naffected by blending since they generally are much brighter than the stars\nwith which stars they are blended. This suggests that, as expected, our\nmicrolensing timescale distribution is less biased by blending than the\ndistribution of 41 events given in \\shortciteN{ALC97e}.\n\n%\\placefigure{figcor}\n\\begin{figure}[ht]\n\\epsscale{1.0} \n\\plottwo{./Figures/fig11a.ps}{./Figures/fig11b.ps}\n\\figcaption{Left: a histogram of the $\\hat{t}$ distribution corrected to\n $100\\%$ efficiency (The expected true event timescale\n distribution.)\\label{figcor}. The first bin in the distribution is a\n lower estimate as the efficiency has been truncated at 2 days where\n events are too short to be detected in this analysis. Right: the\n contribution to the overall microlensing optical depth ($\\tau$) of the\n observed event timescale distribution. }\n\\end{figure}\n\n\nThe efficiency corrected timescale distributions are presented in Figure \\ref{figcor}.\nFrom the right panel of Figure \\ref{figcor} one can also see that the optical\ndepth for this sample of events is not dominated by the long timescale\nevents as it was in \\citeN{ALC97e}. The present optical depth\ndetermination is less dependent on a small number of long timescale events.\nHowever, the relative contribution of each individual long timescale event\nis still large compared to short ones because of the detection efficiency.\n\n\n\\section{\\sc Summary}\n\nWe have presented the results from the DIA survey of MACHO Galactic bulge\ndata. In this analysis we detect 99 microlensing events in eight fields.\nThis survey covers three years of data for $\\sim$ 17 million stars to a\nlimiting magnitude of $V \\sim$23. Our result is consistent with the\ndetection of $75-85\\%$ more events than an analysis performed with PSF\nphotometry on the same data.\n\nWe have measured a microlensing optical depth of $\\tau= 2.43^{+0.39}_{-0.38}\n\\times 10^{-6}$ for events with timescales between 2 and 300 days. With\nconsideration of the disk-disk component we find a Galactic bulge\nmicrolensing optical depth of $\\tau_{_{bulge}}= 3.23^{+0.52}_{-0.50}\\times\n10^{-6}[0.75/(1-f_{disk})]$.\n%$\\tau_{giants}=2.29^{+1.33}_{-0.95}\\times 10^{-6}$\nThese optical depth determinations are consistent with the previous 45 event\nanalysis and 13 event clump giant sub-sample of \\shortciteN{ALC97e}, and the\nvalue determined by the OGLE group \\shortcite{USKK94d}.\n\nFor the individual fields we find that there is a trend in optical\ndepth with longitude and latitude as expected, although there is some\nevidence of fine structure within the optical depth spatial distribution.\nHowever, it is difficult to set limits on this as the uncertainties for each\nfield are quite large.\n\nWe find that our timescale distribution is compatible with lenses having\nmasses distributed in the same way as the PDMF of Scalo (1986). We\nnote that the timescale distribution of \\shortciteN{ALC97e} was biased\ntowards shorter events by blending, making it appear that many low mass\nobjects were required to explain the observed distribution. With our new\nunbiased sample, we do not require a large population of brown dwarfs towards\nthe Galactic centre to reproduce the measured timescale distribution. It is\nstill unclear whether or not there is an anomalous population of long\ntimescale events. \n%as was found by \\shortciteN{ALC97e}.\n\nThe measured microlensing optical depth is a lower limit to the true value\nas events shorter than few days or longer than a few hundred would not be\ndetected in this analysis. Nevertheless, the values presented here\nstill appear to be larger than those predicted by most Galactic models with a bar\nmass and inclination consistent with observations. Such models might be\nbetter constrained by attempting to reproduce the optical depth, the\nobserved timescale distribution and the observed optical depth gradient\nmeasured here.\n\nWe are grateful to S.~Chan, M.~MacDonald, S.~Sabine and the technical staff\nat the Research School of Astronomy and Astrophysics\\footnote{Formerly Mount\nStromlo Observatory.} for their skilled support of the project. \nThis work was in part performed under the auspices of the U.S. Department \nof Energy by University of California Lawrence Livermore National \nLaboratory under contract No. W-7405-Eng-48.\n%This work was performed under the auspices of the U.S. DOE by \n%LLNL under contract no. W-7405-Eng-48\n%Work at Lawrence Livermore National Laboratory is supported by DOE \n%contract %W7405-ENG-48.\nWork at the Center for Particle Astrophysics at the\nUniversity of California, Berkeley is supported by NSF grants AST 88-09616\nand AST 91-20005. Work at Mount Stromlo and Siding Spring Observatories is\nsupported by the Australian Department of Industry, Technology and Regional\nDevelopment. K.~G. and T.~V. acknowledge support from DoE under grant\nDEF0390-ER 40546. W.~J.~S. is supported by a PPARC Advanced Fellowship.\nC.~W.~S. is grateful for support from the Sloan, Packard and Seaver\nFoundations. D.~M. is supported by Fondecyt 1990440. 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},
{
"name": "table1.tex",
"string": "\\begin{deluxetable}{llll}\n\\footnotesize\n%\\scriptsize\n\\tablecaption{Parameters of Level-1.0 Cuts.\\label{tab1}}\n\\tablewidth{0pt}\n\\tablehead{\\colhead{Cut type} & \\colhead{} & \\colhead{Cut level} & \\colhead{}}\n\\startdata\ngeneric var. & $66.6\\%$ pts, $\\rm F_{i} < (F_{med} + 4 \\sigma_{i})$\\tablenotemark{a} &\n$54.5\\%$ pts, $\\rm F_{i} < (F_{med} + 2.5 \\sigma_{i})$\\tablenotemark{a} &\n$31\\%$ pts, $\\rm F_{i} < (F_{med} + 1.5 \\sigma_{i})$\\tablenotemark{a} \\nl\ntransient var. & $90\\%$ pts, $\\rm F_{i} > (F_{med} - 4.5 \\sigma_{i})$\\tablenotemark{b} &\n$93\\%$ pts, $\\rm F_{i} > (F_{med} - 8 \\sigma_{i})$\\tablenotemark{b} &\n$<$11 pts, $\\rm F_{i} < (F_{med} - 11 \\sigma_{i})$\\tablenotemark{b} \\nl\npulsating var. & $<$5 pts\\tablenotemark{d}$\\;$, $\\rm F_{i} < (F_{med} - 3.5 \\sigma_{i})$\\tablenotemark{b} &\n$<$ 4 pts\\tablenotemark{d}$\\;$, $\\rm F_{i} < (F_{med} - 5.5 \\sigma_{i})$\\tablenotemark{b} &\n$<$5 pts\\tablenotemark{d}$\\;$, $\\rm F_{i} < (F_{med} - 5.5 \\sigma_{i})$\\tablenotemark{c}\\nl\n2nd peak & $SN_{p3} < 13 \\sigma_{i}$, $t_{i} > (t_{max} + 110)$\\tablenotemark{b} &\n$SN_{p4} < 16 \\sigma_{i}$, $t_{i} > (t_{max} + 110)$\\tablenotemark{c}\\nl \nphotometry std & $95\\%$ pts, $\\rm F_{i} < \\rm F_{p}$\\tablenotemark{a} &\n$>$ 100 good photometry pts\\tablenotemark{a}\\nl\n\\enddata\n\\tablecomments{Microlensing event selection cuts.\nWe denote the uncertainty in an individual flux measurement \n$\\rm F_{i}$, taken at time $t_{i}$, as $\\sigma_{i}$. \nEvent peaks occur with flux measurement $\\rm F_{p}$ at $\\approx t_{max}$.\nThe median flux value for a lightcurve is $\\rm F_{med}$.\nThe total signal to ratio of three points in a second peak is denoted $SN_{p3}$.\nFor four points this is denoted $SN_{p4}$.\n}\n\\tablenotetext{a}{ In $R_{m}$ or $B_{m}$ bands.}\n\\tablenotetext{b}{ In $R_{m}$ \\& $B_{m}$ bands.}\n\\tablenotetext{c}{ In $R_{m}$ band.}\n\\tablenotetext{d}{ Consecutive good photometry points.}\n\\end{deluxetable}\n"
},
{
"name": "table2.tex",
"string": "\\begin{deluxetable}{ccccccc}\n\\footnotesize\n%\\tablewidth{7pt}\n%\\scriptsize\n\\tablecaption{Parameters of level-1.5 \\& 2.0 Cuts.\\label{tab2}}\n\\tablewidth{0pt}\n\\tablehead{\\colhead{Level} & \\colhead{$\\rm F_{pr}$} & \\colhead{$\\rm A_{F(>)}$} &\n\\colhead{$\\chi_{c(<)}$} & \\colhead{$\\chi_{m(<)}$} & \\colhead{$\\rm SN_{r(>)}$}}\n\\startdata\n1.5 & \\nodata & 1.34 & 9.3 & 3.0 & \\nodata\\nl\n1.5 & $>$ 2.5e4 & 1.34 & 15 & \\nodata & \\nodata\\nl\n1.5 & $>$ 5e4 & 1.34 & 30 & \\nodata & \\nodata\\nl \n1.5 & $>$ 3e2 & 10 & 9.5 & 4.0 & \\nodata\\nl\n1.5 & $>$ 3e2 & 20 & 15 & 4.0 & \\nodata \\nl\n1.5 & $>$ 3e3 & 20 & 15 & \\nodata & \\nodata \\nl\n1.5 & $>$ 3e4 & 10 & 30 & \\nodata & 100\\nl\n2.0 & $<$ 8e3 & 1.34 & \\nodata & $1.65 + 1.8e$-$4 \\rm\\, F_{pr}$ & \\nodata\\nl\n2.0 & $<$ 1e4 & 1.34 & $3.75 + 6.0e$-$4 \\rm\\, F_{pr}$ & \\nodata & \\nodata\\nl\n2.0 & 1e4 $< \\rm\\, F_{pr} <$ 1e5 & 1.34 & \\nodata & \\nodata & $2 + 2e$-$4 \\rm\\, F_{pr}$ \\nl\n2.0 & $>$ 1e4 & 1.34 & \\nodata & \\nodata & \\nodata \\nl\n\\enddata\n%\\tablenotemark{a}\n%\\tablenotetext{a}{Level 1.5 cuts.}\n%\\tablenotetext{a}{Level 2.0 cuts.}\n\\tablecomments{Microlensing event selection cuts. \nAll level 1.5 events must pass the cuts in the text\n($V-R > 0.55$, $A > 1.34$, $\\Omega\\chi^{2} < 3.6$, $\\hat{t} < 365$ days, \n$t_{max_{ij}} \\neq t_{max_{kj}}$, $\\chi^{2}_{c} < 30$) in addition to one line \nof level 1.5 cuts.\nAll events must also pass all 2.0 cuts for which their red fit peak flux $\\rm F_{pr}$ \nis in the range of the cut. Unlisted values specify no cut is applied on that parameter.\nHere $\\rm A_{F>}$ specifies that the fitted microlensing event amplitudes $\\rm A_{F}$ must be \ngreater to pass this cut. Likewise for the other columns, signal to noise ratio \nin the red band ($SN_{r}$), microlensing fit reduced chi square\n$\\chi_{m}$ and constant baseline fit reduced chi square $\\chi_{c}$.\n}\n\\end{deluxetable}\n\n"
},
{
"name": "table3.tex",
"string": "% TABLE3.TEX -- Stuff.\n\n\\begin{deluxetable}{cccccccccc}\n\\tablecaption{Individual Fields.\\label{tab3}}\n\\footnotesize\n%\\small\n\\tablewidth{0pt}\n\\tablehead{\\colhead{Field} & \\colhead{SOD} & \\colhead{DIA} & \n\\colhead{New}& \\colhead{$l$ $(\\arcdeg)$} &\n\\colhead{$b$ $(\\arcdeg)$} & \\colhead{$r$} & \\colhead{$p$ ($\\%$)} & \\colhead{$\\tau$ ($10^{6}$)}}\n\\startdata\n$101$ & 6 & 11 & 6 & 3.728 & $-$3.021 & 1.12 & 11.8 & 1.72$^{_{+0.60}}_{^{-0.48}}$ \\nl\n$104$ & 10 & 16 & 10 & 3.109 & $-$3.008 & 1.17 & 10.6 & 4.18$^{_{+1.62}}_{^{-1.35}}$ \\nl\n$108$ & 12 & 16 & 8 & 2.304 & $-$2.649 & 1.43 & 11.4 & 2.39$^{_{+0.67}}_{^{-0.57}}$ \\nl\n$113$ & 9 & 17 & 9 & 1.629 & $-$2.781 & 1.57 & 8.9 & 1.96$^{_{+0.54}}_{^{-0.45}}$ \\nl\n$118$ & 8 & 13 & 8 & 0.833 & $-$3.074 & 1.41 & 10.0 & 2.64$^{_{+0.89}}_{^{-0.78}}$ \\nl\n$119$ & 6 & 12 & 7 & 1.065 & $-$3.831 & 1.00 & 9.3 & 2.43$^{_{+0.86}}_{^{-0.72}}$ \\nl\n$128$ & 4 & 10 & 7 & 2.433 & $-$4.029 & 0.91 & 8.4 & 1.62$^{_{+0.62}}_{^{-0.47}}$ \\nl\n$159$ & 3 & 4 & 2 & 6.353 & $-$4.402 & 0.48 & 9.3 & 1.06$^{_{+0.84}}_{^{-0.66}}$ \\nl\n%$101$ & 6 & 11 & 6 & 3.728 & $-$3.021 & 1.12 & 11.8 & 1.92$^{_{+0.65}}_{^{-0.57}}$ \\nl\n%$104$ & 10 & 16 & 10 & 3.109 & $-$3.008 & 1.17 & 10.6 & 5.65$^{_{+2.19}}_{^{-1.82}}$ \\nl\n%$108$ & 12 & 16 & 8 & 2.304 & $-$2.649 & 1.43 & 11.4 & 2.72$^{_{+0.75}}_{^{-0.68}}$ \\nl\n%$113$ & 9 & 17 & 9 & 1.629 & $-$2.781 & 1.57 & 8.9 & 2.15$^{_{+0.58}}_{^{-0.52}}$ \\nl\n%$118$ & 8 & 13 & 8 & 0.833 & $-$3.074 & 1.41 & 10.0 & 3.23$^{_{+1.16}}_{^{-1.01}}$ \\nl\n%$119$ & 6 & 12 & 7 & 1.065 & $-$3.831 & 1 & 9.3 & 2.89$^{_{+1.09}}_{^{-0.92}}$ \\nl\n%$128$ & 4 & 10 & 7 & 2.433 & $-$4.029 & 0.91 & 8.4 & 1.84$^{_{+0.66}}_{^{-0.58}}$ \\nl\n%$159$ & 3 & 4 & 2 & 6.353 & $-$4.402 & 0.48 & 9.3 & 1.26$^{_{+0.88}}_{^{-0.67}}$ \\nl\n\\enddata\n\\tablecomments{\n%{\\large\\it This is dependent on what is done with the cuts so I might\n%just drop the comparison, although it is quite important for my work.}\\\\\nCol. (1), field number Id.\nCol. (2), number of events which pass cuts performed on SodoPhot photometry as of \nAugust 1999.\nCol. (3), number of events from the Difference Image Analysis.\nCol. (4), number of new events from this analysis.\nCols. (5) \\& (6), field location.\nCol. (7), Number of clump stars relative to field 119 (BW).\nCol. (8), Percentage of disk stars in each field to $V=23$.\nCol. (9), microlensing optical depth.}\n\\end{deluxetable}\n"
},
{
"name": "table4.tex",
"string": "% Table4 Stuff\n\n% The following gobbledygook is done so that this particular table\n% paginates in a way that \"looks nice\" regardless of whether apjpt4\n% or aj_pt4 styles are chosen. The \\tablebreak commands that appear\n% after some of the lines force page breaks when the apjpt4 style is\n% selected, but act simply as line delimiters for aj_pt4.\n\n\\makeatletter\n\\def\\jnl@aj{AJ}\n\\ifx\\revtex@jnl\\jnl@aj\\let\\tablebreak=\\nl\\fi\n\\makeatother\n\n\n\\begin{deluxetable}{lccccc}\n\\footnotesize\n\\tablecaption{Candidate Microlensing Events.\\label{tab4}}\n\\tablewidth{0pt}\n\\tablehead{\\colhead{Event Id} & \\colhead{DIA Id} & \\colhead{MACHO Id} &\n\\colhead{RA (J2000)} & \\colhead{Dec (J2000)} & \\colhead{S ($\\arcsec$)}}\n\\startdata\n95-BLG-d6\\tablenotemark{v} & 101.14.1893 & 101.21688.5320 & 18 07 02.02 & -27 32 40.2 & 1.47\\nl\n95-BLG-30\\tablenotemark{f} & 101.15.3933 & 101.21821.128 & 18 07 04.27 & -27 22 06.3 & 0.16\\nl\n95-BLG-d7 & 101.15.3935 & 101.21950.1897 & 18 07 25.07 & -27 24 41.1 & 1.28\\nl\n95-BLG-s8 & 101.15.3936 & 101.21691.836 & 18 06 52.77 & -27 23 19.8 & 0.55\\nl\n96-BLG-d2 & 101.19.3670 & 101.21171.4799 & 18 05 38.12 & -27 23 07.8 & 0.51\\nl\n97-BLG-24 & 101.21.3714 & 101.20650.1216 & 18 04 20.26 & -27 24 45.8 & 0.51\\nl\n97-BLG-42 & 101.22.3422 & 101.20914.3873 & 18 04 56.56 & -27 10 43.2 & 3.75\\nl\n95-BLG-5 & 101.23.3319 & 101.20658.2639 & 18 04 22.40 & -26 53 15.8 & 0.56\\nl\n97-BLG-s4 & 101.24.2939 & 101.21174.3417 & 18 05 38.80 & -27 08 29.5 & 1.06\\nl\n96-BLG-d3 & 101.24.2940 & 101.21174.2131 & 18 05 47.03 & -27 08 54.8 & 0.43\\nl\n95-BLG-15 & 101.26.2507 & 101.21564.4657 & 18 06 28.79 & -27 09 35.9 & 0.85\\nl\n95-BLG-s9 & 104.14.5859 & 104.21161.1997 & 18 05 34.46 & -28 02 51.7 & 0.24\\nl\n96-BLG-d4 & 104.15.7362 & 104.21162.3642 & 18 05 47.75 & -27 56 32.9 & 0.01\\nl\n96-BLG-d5 & 104.15.7365 & 104.21423.530 & 18 06 09.00 & -27 53 39.1 & 1.42\\nl\n96-BLG-d6 & 104.15.7366 & 104.21033.4316 & 18 05 29.53 & -27 54 00.6 & 2.38\\nl\n96-BLG-14 & 104.16.4493 & 104.21032.4118 & 18 05 15.39 & -27 58 24.4 & 0.71\\nl\n97-BLG-d5 & 104.16.4494 & 104.20901.1319 & 18 05 02.68 & -28 00 47.7 & 0.56\\nl\n96-BLG-12\\tablenotemark{p} & 104.19.5184 & 104.20382.803 & 18 03 53.20 & -27 57 35.7 & 0.14\\nl\n96-BLG-1 & 104.19.5185 & 104.20645.3129 & 18 04 26.19 & -27 47 35.0 & 0.25\\nl\n97-BLG-38 & 104.19.5186 & 104.20514.1500 & 18 04 06.10 & -27 48 26.9 & 0.72\\nl\n95-BLG-d10 & 104.19.5187 & 104.20643.299 & 18 04 25.12 & -27 54 31.6 & 0.93\\nl\n97-BLG-18 & 104.20.5880 & 104.20121.1692 & 18 03 15.26 & -28 00 13.9 & 0.13\\nl\n97-BLG-58 & 104.24.4584 & 104.20515.498 & 18 04 09.68 & -27 44 35.1 & 0.27\\nl\n95-BLG-d11 & 104.24.4585 & 104.20517.707 & 18 04 06.15 & -27 39 21.4 & 0.21\\nl\n96-BLG-26 & 104.25.4571 & 104.20388.2766 & 18 03 53.97 & -27 33 30.5 & 0.08\\nl\n97-BLG-2 & 104.26.4393 & 104.20775.2644 & 18 04 50.73 & -27 45 57.3 & 0.09\\nl\n96-BLG-d7\\tablenotemark{b} & 104.27.4089 & 101.20908.1433 & 18 04 57.73 & -27 33 18.3 & 0.71\\nl\n97-BLG-d6\\tablenotemark{v} & 113.14.6365 & 113.19454.768 & 18 01 35.64 & -29 08 39.3 & 0.87\\nl\n95-BLG-d12 & 113.14.6367 & 113.19322.2128 & 18 01 15.47 & -29 18 06.6 & 1.33\\nl\n95-BLG-s13 & 113.16.6650 & 113.18934.4131 & 18 00 28.87 & -29 09 34.9 & 0.63\\nl\n96-BLG-d8 & 113.16.6651 & 113.19192.228 & 18 01 04.67 & -29 17 31.1 & 0.52\\nl\n96-BLG-d9 & 113.16.6652 & 113.18932.3227 & 18 00 26.34 & -29 17 36.1 & 3.01\\nl\n97-BLG-1\\tablenotemark{b} & 113.18.6227 & 113.18674.756 & 17 59 53.38 & -29 09 07.8 & 0.47\\nl\n95-BLG-4 & 113.18.6228 & 113.18804.1061 & 18 00 03.41 & -29 11 04.3 & 0.73\\nl\n96-BLG-21 & 113.21.5667 & 113.18156.1823 & 17 58 43.15 & -29 00 30.0 & 1.83\\nl\n97-BLG-d7\\tablenotemark{b} & 113.21.5669 & 113.18286.536 & 17 59 02.71 & -29 03 02.5 & 0.24\\nl\n96-BLG-s10 & 113.22.6004 & 113.18420.5494 & 17 59 25.01 & -28 46 32.9 & 0.61\\nl\n95-BLG-1 & 113.23.5372 & 113.18292.2374 & 17 59 00.57 & -28 36 57.3 & 0.53\\nl\n96-BLG-20 & 113.24.6037 & 113.18550.1664 & 17 59 40.59 & -28 47 24.9 & 0.53\\nl\n96-BLG-10 & 113.25.5974 & 113.18680.3511 & 18 00 02.01 & -28 45 17.6 & 0.68\\nl\n95-BLG-23 & 113.25.5975 & 113.18812.4511 & 18 00 03.61 & -28 39 14.8 & 1.15\\nl\n97-BLG-d8 & 113.26.5353 & 113.18938.3003 & 18 00 32.48 & -28 53 22.7 & 0.83\\nl\n95-BLG-d14 & 113.26.5354 & 113.18940.3399 & 18 00 34.56 & -28 47 06.5 & 2.35\\nl\n95-BLG-d15 & 113.26.5357 & 113.19070.2853 & 18 00 46.72 & -28 46 45.0 & 0.79\\nl\n96-BLG-8 & 118.15.7509 & 118.19184.3770 & 18 00 58.17 & -29 49 50.5 & 0.50\\nl\n96-BLG-d11 & 118.17.3390 & 118.18663.1884 & 18 00 01.63 & -29 52 19.7 & 0.76\\nl\n97-BLG-16 & 118.17.6294 & 118.18662.2180 & 17 59 56.37 & -29 56 37.5 & 0.70\\nl\n97-BLG-4 & 118.18.5693 & 118.18270.3615 & 17 59 04.71 & -30 07 06.5 & 0.90\\nl\n96-BLG-d12 & 118.18.6885 & 118.18531.1816 & 17 59 37.67 & -30 00 53.1 & 0.53\\nl\n97-BLG-8\\tablenotemark{p} & 118.18.6886 & 118.18529.538 & 17 59 35.35 & -30 08 48.1 & 0.36\\nl\n95-BLG-d16 & 118.19.7905 & 118.18404.992 & 17 59 18.58 & -29 47 49.4 & 0.52\\nl\n95-BLG-d17\\tablenotemark{v} & 118.20.1711 & 118.19050.2888 & 18 00 42.46 & -30 06 37.2 & 2.16\\nl\n96-BLG-d12 & 118.21.3144 & 118.18143.4794 & 17 58 34.55 & -29 53 13.4 & 2.98\\nl\n97-BLG-d9 & 118.23.7346 & 118.18019.3386 & 17 58 28.69 & -29 29 11.5 & 1.44\\nl\n95-BLG-10 & 118.23.7347 & 118.18018.2379 & 17 58 16.01 & -29 32 10.9 & 0.06\\nl\n96-BLG-d13 & 118.25.5470 & 118.18539.3614 & 17 59 34.95 & -29 30 04.2 & 0.86\\nl\n97-BLG-26\\tablenotemark{p} & 118.26.5695 & 118.18797.1397 & 18 00 06.94 & -29 38 06.0 & 0.01\\nl\n97-BLG-d10 & 119.14.4347 & 119.20479.459 & 18 04 13.52 & -30 09 25.9 & 0.16\\nl\n95-BLG-11 & 119.14.4348 & 119.20738.3418 & 18 04 37.26 & -30 12 11.6 & 0.41\\nl\n97-BLG-14 & 119.15.5936 & 119.20480.2914 & 18 04 16.36 & -30 07 23.2 & 0.18\\nl\n97-BLG-37 & 119.17.5482 & 119.20352.2589 & 18 03 58.66 & -29 58 48.8 & 0.08\\nl\n95-OGLE-16 & 119.21.1551 & 119.19571.1616 & 18 02 07.62 & -30 01 12.7 & 0.46\\nl\n96-BLG-3\\tablenotemark{b} & 119.22.1686 & 119.19444.2055 & 18 01 45.54 & -29 49 47.1 & 0.28\\nl\n95-BLG-d18 & 119.22.4857 & 119.19704.1819 & 18 02 22.70 & -29 50 35.2 & 0.74\\nl\n95-BLG-39 & 119.23.4960 & 119.19576.2024 & 18 02 04.76 & -29 43 15.6 & 0.27\\nl\n95-BLG-d19 & 119.23.541 & 119.19447.724 & 18 01 41.23 & -29 37 23.2 & 0.47\\nl\n95-BLG-d20 & 119.25.1896 & 119.20096.2073 & 18 03 07.84 & -29 40 09.7 & 1.04\\nl\n95-BLG-3 & 119.25.5509 & 119.19837.1072 & 18 02 37.52 & -29 39 35.9 & 0.12\\nl\n97-BLG-d11 & 119.26.5056 & 119.20223.2492 & 18 03 33.85 & -29 53 30.0 & 2.73\\nl\n96-BLG-d14 & 128.15.5153 & 110.22318.4078 & 18 08 24.14 & -28 54 59.9 & 2.21\\nl\n96-BLG-d15 & 128.16.4751 & 128.21923.1479 & 18 07 32.70 & -29 13 06.8 & 2.81\\nl\n96-BLG-s16 & 128.17.5429 & 128.22057.2384 & 18 07 38.96 & -28 57 11.8 & 0.05\\nl\n96-BLG-d17 & 128.20.3810 & 128.21145.1300 & 18 05 40.51 & -29 05 59.6 & 1.12\\nl\n96-BLG-d18 & 128.22.4279 & 128.21410.1924 & 18 06 07.41 & -28 47 25.5 & 0.25\\nl\n95-BLG-d21 & 128.23.4933 & 128.21153.867 & 18 05 36.73 & -28 32 41.5 & 0.08\\nl\n96-BLG-31 & 128.24.4809 & 128.21541.1133 & 18 06 42.39 & -28 41 15.9 & 0.12\\nl\n97-BLG-d12 & 128.24.4810 & 128.21671.4941 & 18 06 52.95 & -28 42 19.2 & 2.87\\nl\n95-BLG-d22 & 128.26.4587 & 128.21800.522 & 18 07 11.15 & -28 46 59.8 & 0.89\\nl\n95-BLG-18\\tablenotemark{p} & 128.27.4562 & 128.21932.1362 & 18 07 20.56 & -28 36 51.1 & 0.57\\nl\n97-BLG-d13 & 159.14.3680 & 159.26652.168 & 18 18 21.08 & -25 56 37.1 & 0.43\\nl\n95-BLG-22 & 159.16.3693 & 159.26132.3182 & 18 17 14.80 & -25 55 58.2 & 0.20\\nl\n97-BLG-s14 & 159.21.3328 & 159.25486.1627 & 18 15 42.76 & -25 41 01.9 & 1.34\\nl\n97-BLG-d15\\tablenotemark{v} & 159.25.4026 & 177.26012.459 & 18 16 51.10 & -25 19 03.8 & 1.10\\nl\n\\enddata\n\\tablecomments{Col. (1) gives the event ID. Col. (2) gives the ID for DIA light curve. Col. (3)\ngives the identification number of the nearest monitored SoDoPhot object. Cols. (4) \\& (5) are \nthe centroid location coordinates of the event. Col. (6), the seperation $S$ of the \nevent centroid from the SoDoPhot object centroid.\nWe have not included events from field 108 as these have been presented in \n\\shortciteN{ALC99c}. Events where the centroid offset $S$ is $> 1$ pixel from any SoDoPhot object are likely\nPixel lensing events.\n}\n\\tablenotetext{f}{ Finite source event}\n\\tablenotetext{b}{ Binary lens event}\n\\tablenotetext{p}{ Parallax event}\n\\tablenotetext{v}{ Possible variable}\n\\end{deluxetable}\n"
},
{
"name": "table5.tex",
"string": "% TABLE5.TEX -- Stuff.\n\n% The following gobbledygook is done so that this particular table\n% paginates in a way that \"looks nice\" regardless of whether apjpt4\n% or aj_pt4 styles are chosen. The \\tablebreak commands that appear\n% after some of the lines force page breaks when the apjpt4 style is\n% selected, but act simply as line delimiters for aj_pt4.\n\n\\makeatletter\n\\def\\jnl@aj{AJ}\n\\ifx\\revtex@jnl\\jnl@aj\\let\\tablebreak=\\nl\\fi\n\\makeatother\n\n\\begin{deluxetable}{lccccccccc}\n\\footnotesize\n\\tablecaption{Parameters of Classical Microlensing Events.\\label{tab5}}\n\\tablewidth{0pt}\n\\tablehead{\\colhead{Event Id} & \\colhead{$\\hat t_{ns}$ (days)} & \\colhead{$\\hat t_{s}$ (days)} & \\colhead{$t_{max}$} & \\colhead{$\\rm A_{ns}$} & \\colhead{$\\rm A_{s}$} & \\colhead{$\\rm V_{ns}$} & \\colhead{$\\rm V_{s}$} & \\colhead{$(\\rm V-R)_{ns}$} & \\colhead{$(\\rm V-R)_{s}$}}\n\\startdata\n95-BLG-30 & $ 60.62\\pm{0.03 }$ & $ 72.2\\pm{0.3 }$ & $1321.38\\pm{0.01 }$ & $ 14.06$ & $ 24.77$& $ 16.27$ & $ 16.59^{_{+0.01 }}_{^{-0.01 }}$ & $ 1.37$ & $ 1.39$\\nl\n95-BLG-s8 & $ 28.6\\pm{0.2 }$ & $ 41.9\\pm{1.8 }$ & $ 836.14\\pm{0.06 }$ & $ 1.76$ & $ 2.94$& $ 17.17$ & $ 18.37^{_{+0.11 }}_{^{-0.10 }}$ & $ 0.55$ & $ 0.72$ \\nl\n97-BLG-24 & $ 9.7\\pm{0.2 }$ & $ 25.4\\pm{4.5 }$ & $1594.29\\pm{0.11 }$ & $ 2.99$ & $ 42.80$& $ 17.98$ & $ 20.00^{_{+0.34 }}_{^{-0.26 }}$ & $ 0.59$ & $ 0.64$ \\nl\n95-BLG-5 & $ 17.7\\pm{0.4 }$ & $ 17.6\\pm{2.7 }$ & $ 827.90\\pm{0.06 }$ & $ 6.80$ & $ 6.73$& $ 20.71$ & $ 20.62^{_{+0.29 }}_{^{-0.23 }}$ & $ 0.91$ & $ 0.84$ \\nl\n95-BLG-s9 & $ 30.4\\pm{0.2 }$ & $ 41.0\\pm{1.4 }$ & $ 925.21\\pm{0.02 }$ & $ 5.99$ & $ 8.77$& $ 18.63$ & $ 19.10^{_{+0.05 }}_{^{-0.05 }}$ & $ 0.74$ & $ 0.74$ \\nl\n97-BLG-d5 & $ 59.1\\pm{2.3 }$ & $111.3\\pm{21.8}$ & $1527.99\\pm{0.89 }$ & $ 1.36$ & $ 3.09$& $ 18.20 $ & $ 19.96^{_{+0.54 }}_{^{-0.36 }}$ & $ 0.76 $ & $ 0.72$ \\nl\n96-BLG-12 & $236.7\\pm{0.9 }$ & $297.9\\pm{11.0}$ & $1399.31\\pm{0.32 }$ & $ 16.46$ & $ 70.65$& $ 17.77$ & $ 18.19^{_{+0.09 }}_{^{-0.08 }}$ & $ 0.89$ & $ 0.86$ \\nl\n96-BLG-1 & $147.4\\pm{1.3 }$ & $158.2\\pm{10.0}$ & $1180.88\\pm{0.42 }$ & $ 1.69$ & $ 1.83$& $ 17.70$ & $ 17.89^{_{+0.20 }}_{^{-0.17 }}$ & $ 1.04$ & $ 1.03$ \\nl\n97-BLG-18 & $140.7\\pm{0.9 }$ & $161.1\\pm{11.4}$ & $1609.24\\pm{0.20 }$ & $ 2.14$ & $ 2.56$& $ 18.57$ & $ 18.96^{_{+0.17 }}_{^{-0.15 }}$ & $ 0.82$ & $ 0.88$ \\nl\n97-BLG-58 & $ 52.2\\pm{5.5 }$ & $ 61.9\\pm{278.5}$ & $1683.67\\pm{7.40 }$ & $ 1.64$ & $ 2.19$& $ 17.63$ & $ 18.42^{_{\\; \\dots}}_{^{-3.39 }}$\\tablenotemark{a} & $ 0.93$ & $ 0.90$ \\nl\n95-BLG-d11 & $111.7\\pm{21.4}$ & $124.8\\pm{632.2}$ & $ 715.01\\pm{41.88}$ & $ 1.51$ & $ 1.54$& $ 17.34$ & $ 18.35^{_{\\; \\dots}}_{^{-3.45 }}$ & $ 0.85$ & $ 0.97$ \\nl\n96-BLG-26 & $ 55.4\\pm{0.7 }$ & $105.1\\pm{8.3 }$ & $1315.03\\pm{0.08 }$ & $ 5.03$ & $ 11.15$& $ 19.83$ & $ 20.54^{_{+0.12 }}_{^{-0.11 }}$ & $ 0.90$ & $ 0.56$ \\nl\n97-BLG-2 & $ 38.8\\pm{0.7 }$ & $ 55.4\\pm{6.9 }$ & $1523.27\\pm{0.07 }$ & $ 4.61$ & $ 7.14$& $ 20.03$ & $ 20.29^{_{+0.21 }}_{^{-0.17 }}$ & $ 0.86$ & $ 0.64$ \\nl\n%104.26.4394 & $626.8\\pm{7.6 }$ & $1445.4\\pm{171.3}$ & $1360.88\\pm{2.02 }$ & $ 1.95$ & $ 5.01$& $ 19.75$ & $ 21.16^{_{+0.22 }}_{^{-0.18 }}$ & $ 0.80$ & $ 0.73$ \\nl\n97-BLG-1 & $ 76.0\\pm{0.2 }$ & $ 44.6\\pm{0.9 }$ & $1511.65\\pm{0.03 }$ & $ 4.76$ & $ 2.18$& $ 17.39$ & $ 16.05^{_{+0.06 }}_{^{-0.06 }}$ & $ 0.90$ & $ 0.84$ \\nl\n97-BLG-d7 & $ 14.7\\pm{0.2 }$ & $ 22.1\\pm{3.5 }$ & $1618.44\\pm{0.14 }$ & $ 1.14$ & $ 1.45$& $ 17.04$ & $ 18.21^{_{+0.72 }}_{^{-0.43 }}$ & $ 0.83$ & $ 0.74$ \\nl\n95-BLG-1 & $ 40.6\\pm{0.2 }$ & $ 57.5\\pm{2.8 }$ & $ 816.80\\pm{0.03 }$ & $ 8.56$ & $ 13.13$& $ 19.16$ & $ 19.61^{_{+0.07 }}_{^{-0.07 }}$ & $ 1.00$ & $ 0.94$ \\nl\n96-BLG-20 & $ 40.6\\pm{0.6 }$ & $ 61.0\\pm{6.0 }$ & $1273.57\\pm{0.26 }$ & $ 1.83$ & $ 3.21$& $ 18.43$ & $ 19.33^{_{+0.25 }}_{^{-0.20 }}$ & $ 0.89$ & $ 0.69$ \\nl\n97-BLG-8 & $ 98.2\\pm{0.2 }$ & $159.9\\pm{2.1 }$ & $1575.50\\pm{0.01 }$ & $ 17.24$ & $ 31.48$& $ 19.53$ & $ 20.18^{_{+0.02 }}_{^{-0.02 }}$ & $ 1.02$ & $ 1.02$ \\nl\n95-BLG-d16 & $ 8.1\\pm{0.6 }$ & $ 18.1\\pm{12.6}$ & $ 896.33\\pm{0.40 }$ & $ 1.26$ & $ 4.16$& $ 18.23$ & $ 20.94^{_{\\; \\dots}}_{^{-0.89 }}$ & $ 0.85$ & $ 0.72$ \\nl\n95-BLG-10 & $ 79.3\\pm{0.7 }$ & $ 89.3\\pm{5.8 }$ & $ 869.15\\pm{0.17 }$ & $ 2.31$ & $ 2.71$& $ 19.04$ & $ 19.29^{_{+0.16 }}_{^{-0.14 }}$ & $ 0.77$ & $ 0.70$ \\nl\n97-BLG-26 & $111.4\\pm{0.2 }$ & $127.7\\pm{1.9 }$ & $1636.63\\pm{0.02 }$ & $ 7.05$ & $ 8.38$& $ 19.38$ & $ 19.47^{_{+0.02 }}_{^{-0.02 }}$ & $ 1.26$ & $ 1.16$ \\nl\n97-BLG-d10 & $ 70.0\\pm{0.6 }$ & $ 87.2\\pm{7.8 }$ & $1636.49\\pm{0.24 }$ & $ 1.22$ & $ 1.45$& $ 16.91$ & $ 17.73^{_{+0.34 }}_{^{-0.26 }}$ & $ 0.73$ & $ 0.75$ \\nl\n95-BLG-11 & $ 22.5\\pm{0.2 }$ & $ 31.2\\pm{1.9 }$ & $ 853.36\\pm{0.02 }$ & $ 22.89$ & $ 33.02$& $ 20.04$ & $ 20.58^{_{+0.09 }}_{^{-0.08 }}$ & $ 0.90$ & $ 0.98$ \\nl\n97-BLG-14 & $ 47.4\\pm{1.4 }$ & $ 60.6\\pm{12.1}$ & $1561.32\\pm{0.36 }$ & $ 1.83$ & $ 2.40$& $ 19.72$ & $ 20.37^{_{+0.66 }}_{^{-0.41 }}$ & $ 0.79$ & $ 0.91$ \\nl\n97-BLG-37 & $ 21.6\\pm{0.3 }$ & $103.9\\pm{9.7 }$ & $1610.86\\pm{0.08 }$ & $ 1.74$ & $ 13.88$& $ 17.48$ & $ 20.62^{_{+0.14 }}_{^{-0.12 }}$ & $ 0.54$ & $ 0.69$ \\nl\n95-OGLE-16 & \\nodata\\tablenotemark{b} & $ 41.8\\pm{26.3}$ & $ 884.38\\pm{0.68 }$ & \\nodata & $ 1.35$ & \\nodata & $ 19.45^{_{\\; \\dots}}_{^{-1.20 }}$ & \\nodata & $ 0.89$ \\nl\n96-BLG-3 & $ 46.4\\pm{0.2 }$ & $ 71.7\\pm{3.8 }$ & $1167.33\\pm{0.02 }$ & $ 14.38$ & $ 24.48$& $ 19.15$ & $ 19.85^{_{+0.07 }}_{^{-0.07 }}$ & $ 0.70$ & $ 0.81$ \\nl\n95-BLG-39 & $ 39.3\\pm{0.7 }$ & $ 53.5\\pm{5.2 }$ & $ 993.97\\pm{0.19 }$ & $ 2.21$ & $ 3.66$& $ 18.77$ & $ 19.51^{_{+0.22 }}_{^{-0.19 }}$ & $ 0.65$ & $ 0.53$ \\nl\n95-BLG-d19 & $ 55.7\\pm{1.6 }$ & $104.2\\pm{31.7}$ & $ 936.83\\pm{0.82 }$ & $ 1.15$ & $ 1.98$& $ 17.92$ & $ 19.84^{_{+1.51 }}_{^{-0.61 }}$ & $ 0.75$ & $ 0.53$ \\nl\n95-BLG-3 & $ 2.28\\pm{0.04 }$ & $ 2.9\\pm{0.6 }$ & $ 809.29\\pm{0.02 }$ & $ 4.80$ & $ 7.05$& $ 18.93$ & $ 19.47^{_{+0.52 }}_{^{-0.35 }}$ & $ 1.01$ & $ 1.05$ \\nl\n96-BLG-s16 & $ 74.8\\pm{2.2 }$ & $ 73.3\\pm{18.0}$ & $1339.17\\pm{0.77 }$ & $ 1.55$ & $ 1.51$& $ 19.12$ & $ 19.09^{_{+1.43 }}_{^{-0.60 }}$ & $ 0.51$ & $ 0.56$ \\nl\n96-BLG-d18 & $ 5.4\\pm{0.2 }$ & $ 6.5\\pm{2.7 }$ & $1255.38\\pm{0.07 }$ & $ 1.69$ & $ 2.11$& $ 19.11$ & $ 19.62^{_{\\; \\dots}}_{^{-0.77 }}$ & $ 0.82$ & $ 0.63$ \\nl\n95-BLG-d21 & $ 31.8\\pm{0.6 }$ & $ 35.6\\pm{9.8 }$ & $ 872.37\\pm{0.29 }$ & $ 1.32$ & $ 1.46$& $ 18.02$ & $ 18.35^{_{+1.87 }}_{^{-0.65 }}$ & $ 0.85$ & $ 0.79$ \\nl\n96-BLG-31 & $ 49.6\\pm{0.4 }$ & $ 54.4\\pm{1.8 }$ & $1373.14\\pm{0.05 }$ & $ 4.80$ & $ 6.09$& $ 18.80$ & $ 18.88^{_{+0.08 }}_{^{-0.07 }}$ & $ 0.92$ & $ 0.80$ \\nl\n95-BLG-18 & $ 83.1\\pm{0.9 }$ & $ 98.7\\pm{10.1}$ & $ 904.18\\pm{0.29 }$ & $ 1.70$ & $ 2.18$& $ 18.93$ & $ 19.55^{_{+0.31 }}_{^{-0.24 }}$ & $ 0.79$ & $ 0.93$ \\nl\n97-BLG-d13 & $ 5.82\\pm{0.09 }$ & $ 25.7\\pm{3.2 }$ & $1607.01\\pm{0.05 }$ & $ 1.13$ & $ 6.39$& $ 16.75$ & $ 20.65^{_{+0.23 }}_{^{-0.19 }}$ & $ 0.74$ & $ 0.70$ \\nl\n95-BLG-22 & $ 19.9\\pm{0.7 }$ & $ 20.1\\pm{4.3 }$ & $ 899.82\\pm{0.13 }$ & $ 4.97$ & $ 4.85$& $ 20.75$ & $ 20.80^{_{+0.54 }}_{^{-0.36 }}$ & $ 0.70$ & $ 0.79$ \\nl\n\\enddata\n\\tablecomments{The results of two types of fits. One where the source flux is assumed to be that \nof the nearest SoDoPhot source. The other where the source flux is also used as a fit parameter. The parameters from\nthe fits are denoted by the subscipts $ns$ (no source fit) ans $s$ (source fit) respectively.\nWe have not include events from field 108 as these have been presented in \\shortciteN{ALC99c}.\nCol. (1), the DIA light curve identification number. \nCols. (2) \\& (3), the fit timescale of the event. \nCol. (4), the time of maximum amplifiction (JD $- 2449000$).\nCols (5) \\& (6), the amplificiations of these two fits. \nCol. (7), the baseline magnitude of the SoDoPhot object. \nCol. (8), the fitted source magnitude. \nCol. (9), the colour of the nearest SoDoPhot object.\nCol. (10), the fit colour of the object.\nIn all cases the presented uncertainties are the formal $1 \\sigma$ errors from the fitting process.\n}\n\\tablenotetext{a}{ Events where a lower limit to the source flux was not determined.}\n\\tablenotetext{b}{ Nearest SoDoPhot source has $R_{m}$ band data only.}\n\\end{deluxetable}\n"
},
{
"name": "table6.tex",
"string": "% Table3 Stuff\n\n% The following gobbledygook is done so that this particular table\n% paginates in a way that \"looks nice\" regardless of whether apjpt4\n% or aj_pt4 styles are chosen. The \\tablebreak commands that appear\n% after some of the lines force page breaks when the apjpt4 style is\n% selected, but act simply as line delimiters for aj_pt4.\n\n\\makeatletter\n\\def\\jnl@aj{AJ}\n\\ifx\\revtex@jnl\\jnl@aj\\let\\tablebreak=\\nl\\fi\n\\makeatother\n\\begin{deluxetable}{lccccc}\n\\footnotesize\n\\tablecaption{Parameters of Pixel Microlensing Events.\\label{tab6}}\n\\tablewidth{0pt}\n\\tablehead{\\colhead{Event Id} & \\colhead{$\\hat t$ (days)} & \\colhead{$t_{max}$} &\n\\colhead{A} & \\colhead{V$_{fit}$} & \\colhead{(V-R)$_{fit}$}}\n\\startdata\n95-BLG-d6 & $ 9.5\\pm{13.8}$ & $ 873.46\\pm{0.60 }$ & $ 1.73$ & $ 20.56^{_{\\; \\dots}}_{^{-1.76 }}$ & $ 0.56$ \\nl\n95-BLG-d7 & $ 34.4\\pm{16.8}$ & $ 872.24\\pm{0.06 }$ & $ 17.36$ & $ 22.59^{_{+0.93 }}_{^{-0.49 }}$ & $ 0.91$ \\nl\n96-BLG-d2 & $ 30.0\\pm{11.9}$ & $1222.84\\pm{0.23 }$ & $ 3.99$ & $ 21.51^{_{+1.25 }}_{^{-0.57 }}$ & $ 1.02$ \\nl\n97-BLG-42 & $ 37.5\\pm{12.5}$ & $1635.06\\pm{0.34 }$ & $ 2.21$ & $ 20.41^{_{+1.67 }}_{^{-0.63 }}$ & $ 0.67$ \\nl\n97-BLG-s4 & $ 25.8\\pm{5.2 }$ & $1533.26\\pm{0.15 }$ & $ 7.68$ & $ 20.21^{_{+0.42 }}_{^{-0.30 }}$ & $ 0.73$ \\nl\n96-BLG-d3 & $107.7\\pm{14.6}$ & $1172.75\\pm{0.09 }$ & $ 19.03$ & $ 21.20^{_{+0.21 }}_{^{-0.18 }}$ & $ 0.57$ \\nl\n95-BLG-15 & $ 37.6\\pm{4.0 }$ & $ 853.84\\pm{0.04 }$ & $ 22.80$ & $ 21.16^{_{+0.16 }}_{^{-0.14 }}$ & $ 0.67$ \\nl\n96-BLG-d4 & $202.5\\pm{72.2}$ & $1319.32\\pm{0.78 }$ & $ 9.66$ & $ 22.10^{_{+0.73 }}_{^{-0.43 }}$ & $ 0.82$ \\nl\n96-BLG-d5 & $ 62.6\\pm{15.8}$ & $1258.99\\pm{0.39 }$ & $ 4.04$ & $ 21.15^{_{+0.64 }}_{^{-0.40 }}$ & $ 0.85$ \\nl\n96-BLG-d6 & $ 10.0\\pm{2.2 }$ & $1247.00\\pm{0.03 }$ & $ 49.41$ & $ 21.43^{_{+0.36 }}_{^{-0.27 }}$ & $ 0.83$ \\nl\n96-BLG-14 & $ 37.9\\pm{6.7 }$ & $1244.09\\pm{0.17 }$ & $ 3.26$ & $ 20.13^{_{+0.46 }}_{^{-0.32 }}$ & $ 0.68$ \\nl\n97-BLG-38 & $ 12.2\\pm{3.3 }$ & $1616.52\\pm{0.01 }$ & $ 33.85$ & $ 21.13^{_{+0.51 }}_{^{-0.35 }}$ & $ 0.86$ \\nl\n95-BLG-d10 & $ 84.7\\pm{7.1 }$ & $ 866.39\\pm{0.08 }$ & $ 10.93$ & $ 20.45^{_{+0.13 }}_{^{-0.12 }}$ & $ 0.73$ \\nl\n96-BLG-d7 & $ 18.9\\pm{10.2}$ & $1221.76\\pm{0.25 }$ & $ 2.12$ & $ 20.22^{_{\\; \\dots}}_{^{-0.88 }}$ & $ 0.86$ \\nl\n97-BLG-d6 & $ 12.4\\pm{13.0}$ & $1632.81\\pm{0.39 }$ & $ 1.58$ & $ 19.96^{_{\\; \\dots}}_{^{-1.51 }}$ & $ 0.58$ \\nl\n95-BLG-d12 & $ 91.4\\pm{23.5}$ & $ 804.21\\pm{0.90 }$ & $ 2.58$ & $ 20.15^{_{+0.92 }}_{^{-0.49 }}$ & $ 0.96$ \\nl\n95-BLG-s13 & $ 25.0\\pm{15.2}$ & $806.19\\pm{0.14 }$ & $ 87.45$ & $ 22.25^{_{+2.11 }}_{^{-0.67 }}$ & $ 0.96$ \\nl\n96-BLG-d8 & $ 31.4\\pm{3.6 }$ & $1225.88\\pm{0.02 }$ & $ 50.90$ & $ 22.51^{_{+0.15 }}_{^{-0.13 }}$ & $ 1.32$ \\nl\n96-BLG-d9 & $ 10.3\\pm{3.9 }$ & $ 895.32\\pm{0.02 }$ & $ 18.94$ & $ 22.07^{_{+0.62 }}_{^{-0.39 }}$ & $ 0.94$ \\nl\n95-BLG-4 & $ 13.7\\pm{7.3 }$ & $ 790.09\\pm{0.10 }$ & $ 4.44$ & $ 20.31^{_{+1.97 }}_{^{-0.66 }}$ & $ 0.91$ \\nl\n96-BLG-21 & $ 41.9\\pm{7.2 }$ & $1272.15\\pm{0.14 }$ & $ 6.56$ & $ 20.96^{_{+0.33 }}_{^{-0.25 }}$ & $ 0.78$ \\nl\n96-BLG-s10 & $ 35.5\\pm{4.3 }$ & $1169.70\\pm{0.03 }$ & $ 9.80$ & $ 20.40^{_{+0.19 }}_{^{-0.16 }}$ & $ 0.59$ \\nl\n96-BLG-10 & $ 62.0\\pm{7.7 }$ & $1236.97\\pm{0.07 }$ & $ 10.92$ & $ 21.05^{_{+0.21 }}_{^{-0.18 }}$ & $ 0.81$ \\nl\n95-BLG-23 & $ 21.1\\pm{7.8 }$ & $ 900.93\\pm{0.25 }$ & $ 4.03$ & $ 21.15^{_{+1.24 }}_{^{-0.56 }}$ & $ 0.83$ \\nl\n97-BLG-d8 & $ 18.0\\pm{8.7 }$ & $1525.98\\pm{0.21 }$ & $ 3.51$ & $ 20.87^{_{+2.61 }}_{^{-0.70 }}$ & $ 0.86$ \\nl\n95-BLG-d14 & $ 24.5\\pm{3.4 }$ & $ 816.36\\pm{0.04 }$ & $ 11.70$ & $ 20.10^{_{+0.24 }}_{^{-0.20 }}$ & $ 0.71$ \\nl\n95-BLG-d15 & $ 47.6\\pm{10.0}$ & $ 886.35\\pm{0.35 }$ & $ 1.90$ & $ 19.44^{_{+0.81 }}_{^{-0.46 }}$ & $ 0.77$ \\nl\n96-BLG-8 & $ 39.7\\pm{9.8 }$ & $1224.63\\pm{0.05 }$ & $ 20.82$ & $ 22.24^{_{+0.39 }}_{^{-0.29 }}$ & $ 1.00$ \\nl\n96-BLG-d11 & $ 49.3\\pm{28.1}$ & $1249.74\\pm{0.38 }$ & $ 5.98$ & $ 22.50^{_{+2.15 }}_{^{-0.67 }}$ & $ 0.79$ \\nl\n97-BLG-16 & $ 38.3\\pm{8.6 }$ & $1567.20\\pm{0.30 }$ & $ 3.13$ & $ 21.05^{_{+0.67 }}_{^{-0.41 }}$ & $ 1.12$ \\nl\n97-BLG-4 & $ 25.4\\pm{9.0 }$ & $1518.15\\pm{0.04 }$ & $ 51.38$ & $ 22.62^{_{+0.56 }}_{^{-0.37 }}$ & $ 0.97$ \\nl\n96-BLG-d12 & $171.0\\pm{65.4}$ & $1183.51\\pm{1.29 }$ & $ 4.42$ & $ 21.92^{_{+1.17 }}_{^{-0.55 }}$ & $ 0.64$ \\nl\n95-BLG-d17 & $ 12.0\\pm{13.3}$ & $ 853.74\\pm{0.50 }$ & $ 1.40$ & $ 18.67^{_{\\; \\dots}}_{^{-1.66 }}$ & $ 0.72$ \\nl\n96-BLG-d12 & $ 27.8\\pm{13.1}$ & $1250.07\\pm{0.19 }$ & $ 5.69$ & $ 21.78^{_{+1.38 }}_{^{-0.59 }}$ & $ 0.62$ \\nl\n97-BLG-d9 & $ 90.1\\pm{32.3}$ & $1546.12\\pm{0.30 }$ & $ 10.96$ & $ 22.84^{_{+0.75 }}_{^{-0.44 }}$ & $ 0.86$ \\nl\n96-BLG-d13 & $ 33.5\\pm{10.1}$ & $1317.54\\pm{0.15 }$ & $ 7.94$ & $ 21.71^{_{+0.62 }}_{^{-0.39 }}$ & $ 0.87$ \\nl\n\\tablebreak\n95-BLG-d18 & $ 20.2\\pm{13.1}$ & $ 954.71\\pm{0.24 }$ & $ 4.12$ & $ 21.75^{_{\\; \\dots}}_{^{-0.83 }}$ & $ 0.65$ \\nl\n95-BLG-d20 & $ 26.7\\pm{4.7 }$ & $ 869.10\\pm{0.06 }$ & $ 12.31$ & $ 20.76^{_{+0.30 }}_{^{-0.23 }}$ & $ 0.75$ \\nl\n97-BLG-d11 & $ 49.3\\pm{10.1}$ & $1607.01\\pm{0.31 }$ & $ 2.90$ & $ 20.13^{_{+0.60 }}_{^{-0.38 }}$ & $ 0.63$ \\nl\n96-BLG-d14 & $ 93.5\\pm{15.7}$ & $1140.66\\pm{12.00}$ & $ 14.31$ & $ 19.43^{_{+1.70 }}_{^{-0.63 }}$ & $ 0.82$ \\nl\n96-BLG-d15 & $ 26.3\\pm{6.1 }$ & $1332.95\\pm{0.06 }$ & $ 12.74$ & $ 21.57^{_{+0.46 }}_{^{-0.32 }}$ & $ 0.81$ \\nl\n96-BLG-d17 & $ 42.5\\pm{7.3 }$ & $1257.79\\pm{0.23 }$ & $ 2.70$ & $ 19.73^{_{+0.49 }}_{^{-0.34 }}$ & $ 0.93$ \\nl\n97-BLG-d12 & $ 11.5\\pm{6.3 }$ & $1630.10\\pm{0.11 }$ & $ 4.47$ & $ 21.72^{_{+2.75 }}_{^{-0.71 }}$ & $ 0.74$ \\nl\n95-BLG-d22 & $ 24.5\\pm{5.3 }$ & $ 938.76\\pm{0.04 }$ & $ 19.08$ & $ 21.12^{_{+0.40 }}_{^{-0.29 }}$ & $ 0.62$ \\nl\n97-BLG-s14 & $134.8\\pm{16.9}$ & $1633.64\\pm{0.19 }$ & $ 6.71$ & $ 21.48^{_{+0.22 }}_{^{-0.18 }}$ & $ 0.73$ \\nl\n97-BLG-d15 & $ 27.7\\pm{19.4}$ & $1569.43\\pm{0.65 }$ & $ 2.84$ & $ 21.41^{_{\\; \\dots}}_{^{-1.00 }}$ & $ 0.68$ \\nl\n\\enddata\n\\tablecomments{Events where offset $> 1$ pixel from any SoDoPhot object or the fitted baseline magnitude \nis below the detection threshold for SoDoPhot. Parameters come from fits to the DIA photometry\nlight curves. In all cases the presented uncertainties are the \nformal $1 \\sigma$ errors from the fitting process. Values of $t_{max}$ are (JD $- 2449000$).\nEvents for field 108 have been presented in \\shortciteN{ALC99c}.\n}\n\\tablenotetext{a}{ Events where a lower limit to the source flux was not determined.}\n\\end{deluxetable}\n"
},
{
"name": "table7.tex",
"string": "\\begin{deluxetable}{clc}\n\\footnotesize\n\\notetoeditor{Table \\ref{pars} should be rotated and occupy an entire page\nif too big.}\n\\tablewidth{0pt}\n\\tablecaption{Measurements of Bar Orientation.\\label{tab7}}\n\\tablehead{\\colhead{Ref} & \\colhead{Method} & \\colhead{Inclination Angle ($\\theta$)}\n}\n\\scriptsize\n\\startdata\n1 & Gas Dynamics ($\\rm H_{I}$) & 30$-$45 \\nl\n2 & Gas Dynamics ($\\rm H_{I}$,CO,CS) & 16 $\\pm$ 2 \\nl\n3 & Main Sequence Star CMD & 18 $\\pm$ 3 \\nl\n4 & Dirbe non-parametric deprojection & 10$-$40\\nl\n5 & 2-D Gas Simulations ($\\rm H_{I}$, CO) & $>25$\\nl\n6 & Dirbe L,M-band deprojection & 15$-$35\\nl\n7 & COBE K-band constrained N-Body Sim& 28 $\\pm$ 8\\nl\n8 & Star Counts & 19$\\pm$1\\nl\n8 & Star Counts & 24$\\pm$2\\nl\n9 & Red-clump giant numbers & 10$-$45\\nl\n10 & OGLE+MACHO Microlensing & $<$ 30\\nl %\\tablenotemark{a}\n11 & Fux N-body \\& OH/IR stars & 44 $\\pm \\sim$ 5\\nl %\\tablenotemark{b}\n12 & 2-D simulations \\& $\\rm H_{I}$ data & 35$\\pm$5 \\nl \n\\enddata\n\\tablerefs{\n(1) \\citeNP{DEV64}; (2) \\citeNP{BGSBU91};\n(3) \\citeNP{BERT95}; (4) \\citeNP{DAH95};\n(5) \\citeNP{WES96}; (6) \\citeNP{BGS97};\n(7) \\citeNP{Fux97}; (8) \\citeNP{NL99};\n(9) \\citeNP{SUSK97}; (10) \\citeNP{Gyuk99};\n(11) \\citeNP{SSV99}; (12) \\citeNP{WES99}}.\n\\tablecomments{Col. (1), reference. \nCol. (2), the observation or method used in determination. \nCol. (3), the bar inclination angle relative to Sun-GC line-of-sight with \nuncertainties if given.}\n\\end{deluxetable}\n\n"
},
{
"name": "table8.tex",
"string": "\\begin{deluxetable}{clcc}\n\\footnotesize\n\\tablecaption{Optical Depths from Models\\label{tab8}}\n\\tablewidth{0pt}\n\\tablehead{\\colhead{Ref} & \\colhead{Model} \n& \\colhead{$\\rm M_{b} (10^{10} M_{\\odot})$} & \\colhead{ $\\tau\\; (10^{-6})$}\n}\n\\startdata\n1 & double exp disk & \\nodata & 0.4$-$0.8\\nl %\\tablenotemark{a}\n2 & double exp disk + halo & \\nodata & 0.5$-$1.1\\nl\n3 & symetric bulge + massive disk & 1.9 & 1.9\\nl\n4 & bar + truncated disk & 2.0& 2.2 $\\pm$ 0.45\\nl %\\tablenotemark{b} \n5 & bar + double exp disk & 1.8 & 1.9\\nl\n6 & N-body model (m08t3200) & 3.0\\tablenotemark{a} & 1.8\\nl\n6 & N-body model (m04t3000) & 5.0\\tablenotemark{a} & 2.0\\nl\n7 & non-bisymmetric disk & 1.65 & 1.1$-$1.8\\nl\n7 & bisymmetric disk & 1.65 & 1.1$-$1.6\\nl\n8 & bar + nucleus + dble exp disk & 2.2 & 1.54\\nl %\\tablenotemark{b}\n8 & bar + nucleus + dble exp disk & 3.3 & 2.14\\nl %\\tablenotemark{b}\n9 & Fux N-body \\& OH/IR stars & 2.0 & 2.2\\nl %\\tablenotemark{c}\n10 & maximum likelihood & \\nodata & $1.93\\pm0.39$\\nl\n11 & thick \\& thin disk & 1.8 & 1.9\\nl\n12 & Schwarzchild orbits & \\nodata & 1.4\\nl\n\\enddata\n\\tablerefs{\n(1) \\citeNP{PAC91}; (2) \\citeNP{GAAB91};\n(3) \\citeNP{Evans94};\n(4) \\citeNP{ZSR95}; (5) \\citeNP{ALC97e};\n(6) \\citeNP{Fux97} (7) \\citeNP{NL99};\n(8) \\citeNP{Peale98}; (9) \\citeNP{SSV99};\n(10) \\citeNP{Gyuk99}; (11) \\citeNP{GJSDP99};\n(12) \\citeNP{HAF99}.}\n\\tablecomments{Col. (1), reference.\nCol. (2), characteristic feature of model. Col. (3), Bulge/Bar\nmass used in model (if known). Col. (4), optical depth obtain from model.\n}\n\\tablenotetext{a}{ Spheroid plus nucleus mass}\n\\end{deluxetable}\n\n"
}
] |
[
{
"name": "astro-ph0002510.extracted_bib",
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astro-ph0002511
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Aluminum Abundances, Deep Mixing and the Blue-Tail Second-Parameter Effect in the Globular Clusters M~3 and M~13
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[
{
"author": "Robert M. Cavallo\\altaffilmark{1,2}"
}
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\begin{small} We analyze high resolution, high signal-to-noise spectra of six red-giant-branch (RGB) stars in the globular cluster M~3 (NGC~5272) and three in M~13 (NGC~6205) that were obtained with the Mayall 4-meter telescope and echelle spectrometer on Kitt Peak. The spectra include lines of O, Na, Mg, Al, Si, Ca, Ti, V, Mn, Fe and Ni. We also analyze the [Al/Fe] values of 96 RGB stars in M~13 covering the brightest 3.5 magnitudes, which include 66 measurements that were derived from moderate resolution, low signal-to-noise spectra obtained with the WIYN 3.5-meter telescope and Hydra multi-object spectrograph, also on Kitt Peak. In addition, we compile from the literature and inspect the [Na/Fe] values of 119 RGB stars in M~13. We test for bimodality in the [Al/Fe] and [Na/Fe] distributions using the KMM algorithm and find that the [Al/Fe] values in M~13 are distributed bimodally at all points along the RGB that were observed, while the [Na/Fe] values are bimodal only over the brightest two magnitudes. The ratios of Al-enhanced to Al-normal and Na-enhanced to Na-normal giants increase towards the tip of the RGB in M~13, which is suggestive of deep mixing in this cluster. The limited M~3 data exhibit a bimodal distribution of [Al/Fe] values and are suggestive of no deep mixing; however, they are too few to be conclusive. We further test for a relationship between deep mixing on the RGB and a second parameter that can create the extended blue tail seen along the horizontal-branches of some clusters by using an ``instantaneous'' mixing algorithm, which we develop here. We conclude that the data for both clusters are consistent with deep mixing as a ``blue-tail second parameter'', and we suggest future observations to further constrain the results. Finally, we offer a solution to the problem of over producing sodium during deep mixing that is based on the depletion of $^{22}$Ne in asymptotic-giant-branch stars and suggest that pollution might best be traced by {\spr} elements in the Sr-Y-Zr peak. \end{small}
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[
{
"name": "Cavallo.tex",
"string": "\\documentclass[11pt,preprint]{aastex}\n\n\\newcommand{\\teff}{$T_{\\rm eff}$}\n\\newcommand{\\vi}{$V - I$}\n\\newcommand{\\LL}{${\\lambda}{\\lambda}$}\n\\newcommand{\\lgf}{log $gf$}\n\\newcommand{\\spr}{$s$-process}\n\\newcommand{\\rpr}{$r$-process}\n\\newcommand{\\cratio}{$^{12}$C/$^{13}$C ratio}\n\n\\received{28 February 2000}\n\\accepted{17 May 2000}\n\n\\slugcomment{To appear in {\\em The Astrophysical Journal}, September 2000}\n\n\\shortauthors{Cavallo \\& Nagar}\n\\shorttitle{Abundances in M3 and M13 Giants}\n\n\n\\begin{document}\n\n\\title{Aluminum Abundances, Deep Mixing and the Blue-Tail Second-Parameter\n Effect in the Globular Clusters M~3 and M~13}\n\n\\author{Robert M. Cavallo\\altaffilmark{1,2}}\n\\affil{Laboratory for Astronomy and Solar Physics, NASA/Goddard Space Flight\n Center, Greenbelt, MD 20771}\n\\email{rob@shemesh.gsfc.nasa.gov}\n\n\\and\n\n\\author{Neil M. Nagar\\altaffilmark{1}}\n\\affil{Department of Astronomy, University of Maryland, College Park, MD 20742}\n\n\\altaffiltext{1}{Visiting Astronomer, Kitt Peak National Observatory. \nKPNO is operated by AURA, Inc.\\ under contract to the National Science\nFoundation.} \n\n\\altaffiltext{2}{NRC Research Associate} \n\n\n\\begin{abstract}\n\\begin{small}\nWe analyze high resolution, high signal-to-noise spectra\n of six red-giant-branch (RGB) stars in the globular cluster M~3\n (NGC~5272) and three in M~13 (NGC~6205) that were obtained with the\n Mayall 4-meter telescope and echelle spectrometer on Kitt Peak.\nThe spectra include lines of O, Na, Mg, Al, Si, Ca, Ti, V, Mn, Fe and Ni.\nWe also analyze the [Al/Fe] values of 96 RGB stars in M~13 covering\n the brightest 3.5 magnitudes, which include 66 measurements that were\n derived from moderate resolution, low signal-to-noise spectra \n obtained with the WIYN 3.5-meter telescope and Hydra multi-object\n spectrograph, also on Kitt Peak.\nIn addition, we compile from the literature and inspect the [Na/Fe] values\n of 119 RGB stars in M~13.\nWe test for bimodality in the [Al/Fe] and [Na/Fe] distributions using the KMM\n algorithm and find that the [Al/Fe] values in M~13 are distributed\n bimodally at all points along the RGB that were observed,\n while the [Na/Fe] values are bimodal only over the brightest two magnitudes.\nThe ratios of Al-enhanced to Al-normal and Na-enhanced to Na-normal giants\n increase towards the tip of the RGB in M~13, which is suggestive of\n deep mixing in this cluster.\nThe limited M~3 data exhibit a bimodal distribution of [Al/Fe] values \n and are suggestive of no deep mixing; however, they are too few to be \n conclusive.\nWe further test for a relationship between deep mixing on the RGB and a\n second parameter that can create the extended blue tail seen along\n the horizontal-branches of some clusters by using an ``instantaneous''\n mixing algorithm, which we develop here.\nWe conclude that the data for both clusters are consistent with\n deep mixing as a ``blue-tail second parameter'', and we suggest future\n observations to further constrain the results.\nFinally, we offer a solution to the problem of over producing sodium during\n deep mixing that is based on the depletion of $^{22}$Ne in\n asymptotic-giant-branch stars and suggest that pollution might best be\n traced by {\\spr} elements in the Sr-Y-Zr peak.\n\\end{small}\n\\end{abstract}\n\n\\keywords{globular clusters: individual (M~3,M~13)\n --- stars: abundances --- stars: horizontal branch --- stars: late-type\n --- stars: Population II}\n\n\\section{INTRODUCTION}\n\n%Abundance variations\nAccording to canonical stellar evolution models, the by-products of the\n nuclear processing around the hydrogen-burning shell (H~shell)\n of low-mass red-giant-branch (RGB) stars should remain confined to the stellar\n interior; however, observations over the past 25 years have shown\n star-to-star variations in the elements C, N, O, Na, Mg and Al, among\n others, on the surfaces of globular cluster red giants (see Kraft 1994,\n Briley et al. 1994 and Cavallo 1998a for detailed reviews of the observations).\nIn particular, the data show evidence of the CNO cycle that dominates the\n energy production in such stars: C and O are anticorrelated with N, while\n the {\\cratio} is near the equilibrium value of 4 in many clusters\n \\citep{SS91,S96b,BSKL97,BSSBN97,ZWB96}.\nWhile the first dredge-up phenomenon \\citep{Iben67} does alter the carbon\n and nitrogen abundances slightly, it cannot account for the observed large\n variations of these elements and their isotopic ratios, nor can it account\n for the variations of the other elements.\nIn addition, some elements show evidence for gradual changes along\n the RGB, indicating that something is occurring during the course of\n evolution to facilitate these alterations.\nFor example, C becomes more depleted with decreasing $V$ in\n the clusters M~15, M~55, M~92 and NGC~6397 \\citep{BDG79,Carbon1982,TCLSK83,\n BBHD90}.\n\n%Approach # 1\nTwo separate approaches have been developed to address the observations.\nOne assumes that some form of non-canonical mixing occurs along the\n RGB, which gradually brings material from around the H~shell to the stellar\n surface \\citep[hereafter, SM79]{SM79}.\nModels by SM79, \\citet{DD90}, \\citet{LHS93}, \\citet{CSB96}, \\citet{DW96} and \n \\citet[hereafter, CSB98]{CSB98} have shown that most variations \n along the brighter part of the RGB can be explained by nuclear processing\n around the H~shell combined with mixing.\nThe source of mixing is generally assumed to be rotationally induced\n meridional circulation currents (SM79); although, other theories abound\n \\citep{LHZ97,FAK99}.\nThe observations by \\citet{Peterson83} that show the horizontal-branch\n (HB) stars in M~13, a cluster with large variations of oxygen and\n aluminum on the RGB, rotating nearly a factor of two faster than the HB stars\n in M~3, a cluster with a composition similar to M~13, but with\n less extreme abundance variations along it's RGB, support the SM79\n hypothesis.\n\n%Approach #2\nThe second approach assumes that some of the variations,\n particularly those of the heavier elements, are primordial in\n nature, perhaps originating in the processed envelopes\n of intermediate-mass asymptotic-giant-branch (AGB) stars that were\n shed into the nascent cluster environment \\citep{CD81}.\nWhile it has been shown that this scenario cannot account for all the\n variations \\citep{DWW97}, some aspects of it are plausible in light of\n the data.\nFor example, observations of CN-band strength and sodium variations on the\n upper main sequence of 47~Tuc, sodium enhancements on the subgiant branch \n of M~92 and enhancements in the neutron-capture elements in some clusters\n all point to primordial origins\n \\citep{BBSH89,BHB91,S96a,BSSLBH96,CCBHS98,KSB98,Ivans99}.\n%Approach 1 + 2\nThe most likely solution to the abundance anomaly problem probably\n involves a combination of both scenarios, where primordial pollution\n is present in the cluster, but mixing later plays a role in adjusting \n the abundance patterns (see, e.g., Denissenkov et al. 1998 and Briley et al.\n 1999), an approach we examine here.\n\n%M13/M3: Second-Parameter \nThis paper focuses on determining the chemical abundances\n in the red giants of the globular clusters M~3 (NGC~5272) and M~13\n (NGC~6205) from high resolution,\n high signal-to-noise echelle spectra obtained with the Mayall 4-meter\n telescope on Kitt Peak.\nWe choose these two clusters because they are often considered a classical\n ``second-parameter'' pair since they have markedly different HB's,\n despite having similar [Fe/H]\\footnote{\n We use the usual notation: [X/Y]~=log(X/Y)$_{\\star}$~$-$~log(X/Y)$_{\\sun}$.}\n values, the first parameter.\nWe discuss the hypothesis that deep mixing along the RGB, which we define \n as mixing that penetrates the H~shell, brings helium to the surface and\n affects the HB morphology as a second parameter that creates the extended\n blue tail in M~13 \\citep{AVS97a,AVS97b}.\nOne oft-quoted choice for the second parameter is a relative age difference\n between M~3 and M~13 \\citep{FPB97,SCD97,Chaboyer98}; however, this is not\n borne out by the photometry \\citep{JB98,vdB99,FG99} and leaves open the\n need for a qualified alternative.\n\nWhile M~13 is by far the most well-studied cluster for abundance variations,\n the data for M~3 are lacking.\nOne goal of this paper is to increase our knowledge of the chemical \n abundances in this latter cluster so that it can be compared with M~13 in\n greater detail.\n\n%Outline\nThe outline of this paper is as follows:\n we describe the observations, data reduction approach, abundance analysis\n technique and abundance results in sections 1, 2, 3 and 4, respectively.\nIn section 5 we discuss evidence for and the implications of deep mixing\n along the RGB, and we give our conclusions in section 6.\nIn the appendix we derive the instantaneous mixing algorithm that is used \n in section 5.\n\n\\section{OBSERVATIONS}\n\nWe chose five bright giants in M~3 (MB~4, vZ~205, vZ~297, vZ~1000 and vZ~1127)\n and three in M~13 (L~262, L~324 and L~414) from previous studies by the\n ``Lick/Texas'' group \\citep{KSLP92,KSLS93,KSLSB95} that show\n evidence for abundance variations but have no information regarding aluminum.\nThe observations were made on the nights of 29-31 May 1998 using the\n echelle spectrograph on the Mayall 4-meter telescope.\nThe echelle setup used - echelle grating 31.6-63{\\arcdeg}, \n cross grating 226-1, long-focus camera and the T2KB CCD - \n resulted in continuous spectral coverage between 5500~{\\AA}\n and 8800~{\\AA} at a dispersion of 0.08 {\\AA} per CCD pixel,\n i.e. resolution, R$\\sim$ 30,000 at 6000 {\\AA} over a 2.5 pixel resolution\n element. \nThe central wavelength on the CCD was chosen to maximize the signal-to-nose \n ratio near the {\\LL}6696, 6698 {\\AA} Al I lines. \nThe seeing was typically between 1{\\arcsec} and 1{\\farcs}2 on all three nights,\n causing us to use a slit-width of 1\\farcs5, and a slit length of 7\\farcs5.\nEach 4 ${\\times}$ 30 min. exposure of a target star was sandwiched between\n two 15 s exposures of a ThAr comparison lamp, which was observed\n at the same telescope position and slit position angle (P.A.).\nTo facilitate subtraction of telluric lines we observed one or two fast\n rotating B stars each night at various airmasses.\n\nIn addition to our data, Dr. M. Briley provided us with the reduced spectrum\n of the M~3 giant AA and a fast rotating B star, taken on 21 March 1998 \n with the McDonald Observatory 2.7-m telescope and 2D-Coude echelle\n spectrograph.\nHis set-up yielded an non-continuous wavelength coverage from 4070 {\\AA} to\n 10,500 {\\AA} with $R ~{\\sim} ~$ 50,000 over a resolution element.\n\nA log of all the observations with estimates of the signal-to-noise ratio\n around the {\\LL}6696, 6698 {\\AA} Al I region is given in Table 1.\nThe photometry is from \\citet{Ferr97} for M~3 and from \\citet{CM79} for M~13.\nThe locations of the program stars in their respective color-magnitude diagrams\n are given in Figure 1 for M~3 and Figure 2 for M~13.\nIn the following discussions, we refer to each star by its most commonly\n used identification, which in some cases is its alternate name.\n\n\\begin{figure}\n\\epsscale{0.50}\n\\plotone{Cavallo.fig01.eps}\n\\caption[Cavallo.fig01.eps]{ Color-magnitude diagram for M~3,\n with the program stars circled and labeled.\n The data are from \\citet{Ferr97}.}\n\\end{figure}\n\\begin{figure}\n\\plotone{Cavallo.fig02.eps}\n\\caption[Cavallo.fig02.eps]{As Figure 1, except for M~13 with the data from\n \\citet{CM79}.}\n\\end{figure}\n\n\\section{CCD PROCESSING AND SPECTRA EXTRACTION}\n\nThe data were reduced using standard IRAF\\footnote{IRAF is distributed by\n the National Optical Astronomy Observatories, which are operated by the\n Association of Universities for Research in Astronomy, Inc., under\n cooperative agreement with the National Science Foundation.}\\citep{IRAF}\n tasks (version 2.11.1), following the reduction procedure outlined in ``Users\n guide to reducing Echelle spectra with IRAF'', by D. Wilmarth \\& J.\n Barnes\\footnote{available by anonymous ftp to \\url{ftp://iraf.noao.edu/iraf/docs}}. \nZero frames were taken on each night, but since $>$ 99.9\\% of the pixels \n had zero values within the 5 e$^-$ r.m.s. noise of the T2KB CCD, we only\n zero-corrected ``hot'' (zero values $>$ 5 e$^-$) pixels.\nQuartz flat exposures were taken on each of the three nights and used\n to flat field the target stars and to determine ``dead'' pixels.\n\nAfter initial processing of the CCD target star data, we used the IRAF task\n APSCATTER to correct for scattered light between the orders.\nThe orders were then extracted to single dimensional spectra using the\n task APSUM with variance weighting, and then wavelength calibrated using\n the closest (in time) ThAr spectrum, taken at the same telescope position\n and slit P.A. as the target star.\nOrders that contained telluric lines were corrected using the task\n TELLURIC and a comparison spectrum of a fast-rotating B star that\n was observed at an airmass similar to the program star.\nFinally, each spectrum was shifted into the rest frame and flattened\n by fitting a spline through the continuum.\n\nThe spectrum for the star M~3 AA was given to us in its extracted form\n and required only correction for telluric lines in some orders.\n\nFigure 3 shows the spectra around the Al~I region for all the M~3 and\n M~13 giants.\nThe variation in the Al~I line strengths is quite apparent from one star to\n the next.\n\n\\begin{deluxetable}{lllllll}\n\\tabletypesize{\\footnotesize}\n\\tablewidth{0pc}\n\\tablenum{1}\n\\tablecaption{Observing Log}\n%\\rotate\n\\tablecolumns{7}\n\\tablehead{\n \\colhead{Star } & \n \\colhead{Alt.} & \n \\colhead{$V$} & \n \\colhead{{\\bv}} & \n \\colhead{Date Obs.} &\n \\colhead{Exposure} & \n \\colhead{S/N} \\\\ \n \\multicolumn{2}{c}{} & \n \\colhead{mag.} &\n \\colhead{mag.} &\n \\colhead{U.T.} &\n \\colhead{min.} &\n \\colhead{est.}\n}\n\n% Star Alt. Name V (B-V) Date Obs. Exp SNR\n\\startdata\n\\cutinhead{M~3}\n% Photometry from Ferraro et al (1997)\nvZ~238 & AA, SK 586, F 12959 & 12.72 & 1.58 & 21-23 March 1998{\\tablenotemark{a}} & 180 & 110 \\\\\n% & SK 586 & & & & & \\\\\n% & F 12959 & & & & & \\\\\nvZ~752 & MB 4, F 14194 & 12.74 & 1.44 & 1 June 1998 & 120 & 115 \\\\\n% & F 14194 & & & & & \\\\\nvZ~205 & III-28, SK 617, F 16682 & 12.75 & 1.44 & 31 May 1998 & 120 & 95 \\\\\n% & SK 617 & & & & & \\\\\n% & F 16682 & & & & & \\\\\nvZ~297 & SK 525, F 9 & 12.84 & 1.44 & 30 May 1998 & 120 & 105 \\\\\n% & F 9 & & & & & \\\\\nvZ~1127 & MB 26, F 14246 & 13.09 & 1.29 & 31 May/1 June 1998 & 180 & 135 \\\\\n% & F 14246 & & & & & \\\\\nvZ~1000 & SK 297, F 17307 & 13.10 & 1.29 & 31 May/1 June 1998 & 180 & 90 \\\\\n% & F 17307 & & & & & \\\\\n\\\\\n\\cutinhead{M~13}\nL~324 & V 11, CM~425 & 12.00 & 1.60 & 1 June 1998 & 120 & 160 \\\\\n% & CM~425 & & & & & \\\\\nL~414 & III-56, CM~176 & 12.14 & 1.47 & 30 May 1998 & 120 & 140 \\\\\n% & CM~176 & & & & & \\\\\nL~262 & CM~476 & 12.25 & 1.39 & 31 May 1998 & 120 & 140 \\\\\n\\enddata\n\\tablenotetext{a}{Observed by M. Briley}\n\\tablecomments{Star names taken from the catalogs of \\citet[vZ]{vZ08} and\n \\citet[L]{L05}. Alternate names are from \\citet[AA, III-28]{San53},\n \\citet[III-56]{Arp55}, \\citet[V11]{Hogg73}, \\citet[CM]{CM79}, \\citet[SK]{SK82},\n \\citet[F]{Ferr97} and Michael Bolte (1998, priv comm. MB).}\n\\end{deluxetable}\n\n\n\\begin{figure}\n\\plotone{Cavallo.fig03.eps}\n\\caption[Cavallo.fig03.eps]{The Al I spectral region for all M~3 and M~13\n stars observed.\n The ordinate is in relative units, with each spectrum offset from\n the others. \n The Al I lines at ${\\lambda\\lambda}6696,6698$~{\\AA} are labeled.}\n\\end{figure}\n\n\\section{ABUNDANCE ANALYSIS}\n\\subsection{Equivalent Widths, Line Lists and Line Parameters}\nThe free spectral range for the Mayall 4-meter data is less than the width\n of the CCD so that each line appears at least twice in our spectra,\n with one line usually towards the center of the chip where the noise \n is minimized.\nUnfortunately, combining adjacent orders to increase the number of\n photon counts resulted in a lowered signal-to-noise ratio since the\n edges of the chip contained much lower quality spectra than the center.\nWe measured the equivalent widths of the line closer to the center of the\n CCD and present the results in Table 2 (found after {\\bf REFERENCES}).\nThe equivalent widths were determined by summing the flux in a line if\n the line was cleanly separated from other lines or by fitting a Gaussian\n in more crowded regions.\nSome data are excluded from the table for one of several reasons:\n 1) In the case of M~3 AA, some lines fell between the orders;\n 2) the line was too weak to be measured (our minimum measurable equivalent\n width was about 5 m{\\AA}, depending on the signal-to-noise ratio);\n 3) the line was contaminated by either a bad column or a cosmic ray hit; or\n 4) in very few cases, the line gave abundance results that were anomalously\n discordant with the mean and r.m.s. deviation of the rest of the lines \n of the same species for unknown reasons and was rejected.\n\n\n\\subsubsection{Fe lines}\n\nThe iron lines were chosen from Kurucz's CD-ROM~\\# 23\\footnote{\n \\url{http://cfa-www.harvard.edu/amdata/ampdata/amdata.html}}\n and the solar atlas of \\citet{MMH66}.\nThey were used to determine both the iron abundances and the model-atmosphere\n parameters and were thus selected to have a broad range in excitation \n potentials and oscillator strengths, which were adopted without modification\n from the empirically derived tables of \\citet{Nav94} and \\citet{Bie91} for\n the Fe~I and Fe~II lines, respectively.\nWe rejected Fe I lines that were listed by \\citet{Nav94} as showing blends with\n other iron lines or uncertainties in their energy levels of more than\n 0.005 cm$^{-1}$.\nFor the sake of comparison, we measured the equivalent widths of our\n chosen iron lines in the visible and near-infrared solar\n spectra\\footnote{NSO/Kitt Peak FTS data used here were produced by NSF/NOAO.}\n of \\citet{WHL93,WHL98}\\footnote{\n available at \\url{http://www.nso.noao.edu/diglib/ftp.html}}\n and folded them through the \\citet{HM74} solar model atmosphere with a\n microturbulent velocity of 1.0~km s$^{-1}$.\nWe adjusted the {\\lgf}'s to reproduce an assumed solar iron abundance of\n log~${\\epsilon}({\\rm Fe})_\\odot ~= ~7.52$\\footnote{log~${\\epsilon}$(X) =\n log N(X/H) + 12.00, where N is the number abundance. For an informative\n debate about the preferred solar iron abundance we refer the reader to\n the papers of \\citet{BLS95}, \\citet{HKB95}, \\citet{BSL95} and \\citet{KSR96}.\n For consistency with the Lick/Texas studies we adopt the 7.52 value.}.\nThis results in an average difference between the two sets of oscillator\n strengths (in the sense of solar model $-$ laboratory) of $+0.09 ~{\\pm} \n~0.12$ for Fe I and $+0.05 ~{\\pm} ~0.08$ for Fe II (see Table 3).\nBoth differences show a slight trend toward lower oscillator strengths from\n the laboratory measurements ($< ~1 ~{\\sigma}$), which in\n turn would cause the determined iron abundances to be overestimated by 0.09 \n and 0.05 dex from the Fe I and Fe II lines, respectively.\n\n\\subsubsection{EW comparisons}\n\nIn the main panel of Figure 4 we compare the equivalent widths of our\n iron lines with those from the earlier Lick/Texas studies.\nThe solid line in the figure has a 45 degree slope and represents perfect\n agreement between the two data sets.\nThe Lick/Texas data tend toward higher equivalent widths relative\n to our data, especially above ${\\sim}$100 m{\\AA}.\nThe average differences between the two data sets (in the sense of \n Lick/Texas $-$ present work) are 6.9 ${\\pm}$ 9.7 m{\\AA} for Fe I and 5.9\n ${\\pm}$ 9.7 m{\\AA} for Fe II.\nWe attribute the differences to several factors.\nFirst, our present data have higher signal-to-noise ratios\n than the Lick/Texas data (90 to 160 compared with 40 - 100), which reduces the\n level of uncertainty in placing the continuum.\nSecond, the Lick/Texas spectra have higher resolution, $R ~{\\sim}\n ~48,000$ compared with $R ~{\\sim} ~30,000$, which helps separate lines\n from the continuum and other lines.\nThird, the two datasets were reduced using two separate software packages\n that apply scattered-light corrections differently (see, e.g., Sneden et al.\n 1991), which can affect the continuum levels and depths of each line.\nFourth, measuring equivalent widths is subjective and two different \n observers can get different results from the same data.\nFor example, the inset in Figure 4 shows the equivalent widths for two\n separate observations of the M~13 giant III-56 by the Lick/Texas group and\n demonstrates how much variability can be present in the data even\n with consistent reduction techniques (see also Kraft et al. 1993, Figure 1).\nFifth, the methods of measuring equivalent widths differ: we use both\n a Simpson's rule technique (i.e., direct integration) and Gaussian fits,\n while the Lick/Texas results come from Gaussian fits for their earlier\n papers and both techniques for \\citet{Kraft97}.\n\n\n%\\placefigure{Cavallo.fig04}\n\\begin{figure}\n\\plotone{Cavallo.fig04.eps}\n\\caption[Cavallo.fig04.eps]{Main panel: Our Fe I and Fe II equivalent widths\n compared with the measurements from the Lick/Texas group.\n Inset: Comparison of Fe I and Fe II equivalent width measurements for the\n M~13 giant III-56 from two different studies by the Lick/Texas group\n [Kraft et al. 1992 (KSLP) and Kraft et al. 1993 (KSLP)].\n The scale on the ordinate is identical to the abscissa.\n The solid line in both plots has a slope of 1. }\n\\end{figure}\n\n\\subsubsection{Other lines}\n\nWe determined the oscillator strengths for the other elements in our\n study by measuring the equivalent widths in the solar spectrum \n and adjusting the {\\lgf} values until we derived the \\citet{AG89}\n solar abundances using the \\citet{HM74} solar model.\nTable 3 lists the average differences between the values we determined\n and the literature values, which are mostly from \\citet{Thev90}, who\n did a similar differential analysis using an older solar spectrum with\n a MARCS \\citep{MARCS} solar model and a microturbulent velocity of 0.6 km\n s$^{-1}$.\nWith the exception of the Mn I lines, we used our derived {\\lgf}'s to derive\n the elemental abundances.\nThe two Mn I lines in the solar spectrum (we discard a third line at\n ${\\lambda}$6022 {\\AA} since it blends with a nearby Fe I feature) suffer \n from hyperfine splitting effects.\nTo avoid detailed calculations, we adopt the recommended {\\lgf}'s \n from \\citet{Thev90} for these two manganese lines.\n\n\\begin{deluxetable}{lllcc}\n\\tabletypesize{\\small}\n\\tablenum{3}\n\\tablewidth{0pc}\n\\tablecaption{Comparison of Oscillator Strengths}\n\\tablecolumns{5}\n\\tablehead{\n \\colhead{Species} &\n \\colhead{Avg. Diff} &\n \\colhead{${\\pm}{\\sigma}$} &\n \\colhead{\\# Lines} &\n \\colhead{Ref.} \n}\n%Species Diff Sigma Lines Ref\n\\startdata\nNa I & $+0.015$ & 0.021 & 2 & T90 \\\\\nMg I & $+0.263$ & 0.349 & 3 & T90, K \\\\\nAl I & $+0.035$ & 0.052 & 6 & T90, K \\\\\nSi I & $+0.059$ & 0.108 & 9 & T90, K \\\\\nCa I & $+0.203$ & 0.059 & 8 & T90, K \\\\\nTi I & $-0.025$ & 0.113 & 8 & T90, K \\\\\nTi II & $-0.120$ & 0.282 & 2 & T90 \\\\\nV I & $+0.048$ & 0.084 & 18 & T89, T90\\\\\nMn I & $+0.615$\\tablenotemark{a} & 0.064\\tablenotemark{a} & 2 & T90 \\\\\nNi I & $+0.203$ & 0.097 & 12 & T90 \\\\\nFe I & $+0.088$ & 0.123 & 24 & N \\\\\nFe II & $+0.048$ & 0.077 & 8 & B\\\\\n\\enddata\n\\tablenotetext{a}{Used values from \\citet{Thev90}.}\n\\tablerefs{\n B = \\citet{Bie91}; K = Kurucz CD-ROM~\\#23; N = \\citet{Nav94};\n T89 = \\citet{Thev89}; T90 = \\citet{Thev90}.\n }\n\\end{deluxetable}\n\n\n\\subsection{Model Parameters: Effective Temperature, Gravity and\n Microturbulence}\n\\subsubsection{Spectroscopic models}\n\nWe constructed models using the Fe I and Fe II lines with the MARCS model\n atmosphere code and the MOOG abundance-analysis code \\citep{MOOG}.\nThe initial models were built with the parameters determined by the\n Lick/Texas group and were constructed with the alpha elements enhanced \n by 0.4 dex in accordance with previous observations of cluster giants \n (see, e.g., Kraft et al. 1997).\nWe iteratively ran MOOG and MARCS to refine the models until the\n derived abundances were independent of excitation potential, line width\n and ionization level.\nWe checked our final choice of model parameters by independently\n using the Ni~I lines to determine {\\teff}, the Ni~I and Ti~I lines to \n determine the microturbulent velocity, $v_t$, and the Ti~I and Ti~II lines\n to determine log~$g$.\nThe results were generally in agreement with the more numerous iron lines\n and allowed us to estimate systematic errors in the model parameters\n determined from spectroscopy: ${\\Delta}T_{\\rm eff} ~{\\sim} ~{\\pm}30 ~{\\rm K},\n ~{\\Delta}{\\rm log}(g) ~{\\sim} ~{\\pm}0.2 ~{\\rm cm~s^{-2}}, ~{\\rm and} \n ~{\\Delta}v_{t} ~{\\sim} ~{\\pm}0.15 {\\rm km~s^{-1}}$.\n\nOur final spectroscopically determined model parameters are given in Table 4,\n along with the original Lick/Texas model parameters, and are the parameters\n we used to derive the elemental abundances.\nThe effective temperatures generally agree to less than 100 K, while \n our gravities are typically lower than the Lick/Texas values by 0.15 \n ${\\pm}$ 0.27 dex and our microturbulent velocities are lower\n by 0.09 ${\\pm}$ 0.11 km s$^{-1}$.\nThe differences are consistent with the error estimates derived from the\n nickel and titanium lines and are not surprising given the differences in\n the equivalent widths and the choices of lines and line parameters\n between the two studies.\n\n\\begin{deluxetable}{lccccccc}\n\\tablewidth{0pc}\n\\tablenum{4}\n\\tablecolumns{8}\n\\tablecaption{Spectroscopic Models}\n\\tablehead{\n \\colhead{} & \n \\multicolumn{3}{c}{Present Work} & \n \\colhead{} &\n \\multicolumn{3}{c}{Lick/Texas} \\\\\n \\cline{2-4} \\cline{6-8}\n \\colhead{Star} &\n \\colhead{\\teff} &\n \\colhead{log $g$} &\n \\colhead{$v_t$} &\n \\colhead{} &\n \\colhead{\\teff} &\n \\colhead{log $g$} &\n \\colhead{$v_t$}\n}\n\\startdata\n\\cutinhead{M~3}\nAA & 4050 & 0.40 & 2.27 && 4000 & 0.40 & 2.25 \\\\\nMB 4 & 4060 & 0.45 & 2.05 && 3925 & 0.30 & 2.15 \\\\\nIII-28 & 4175 & 0.55 & 1.84 && 4160 & 0.75 & 1.75 \\\\\nvZ~297 & 4050 & 0.25 & 1.98 && 4070 & 0.70 & 2.25 \\\\\nvZ~1127 & 4300 & 1.00 & 1.98 && 4225 & 0.90 & 2.00 \\\\\nvZ~1000 & 4200 & 0.65 & 1.94 && 4175 & 0.45 & 2.10 \\\\\n\\cutinhead{M~13}\nL~324 & 3990 & 0.10 & 2.34 && 4050 & 0.50 & 2.50 \\\\\nIII-56 & 4030 & 0.20 & 2.13 && 4100 & 0.65 & 2.25 \\\\\nL~262 & 4160 & 0.50 & 1.89 && 4180 & 0.80 & 2.00 \\\\\n\\enddata\n\\end{deluxetable}\n\n\\subsubsection{Photometric models}\n\nWe used recent photometry of our cluster giants, given in Tables 5a and 5b,\n to derive alternative model atmosphere parameters.\nThe {\\bv} and {\\vi} data were used to derive effective temperatures\n based on a 12 Gyr isochrone that was constructed with [Fe/H] = $-1.54$\n and [${\\alpha}$/Fe] = $+0.3$ dex.\nThe luminosities and {\\teff}'s for the models in the isochrone were computed\n by Dr. D. VandenBerg while Dr. R. Bell performed the luminosity-M$_V$ \n and the color-temperature transformations.\nThe age of the isochrone (mass) fixes the gravities, while the microturbulent\n velocities are still determined from spectroscopy.\nThe ${V - K}$ calibrations are from \\citet{CFP78}.\nWe present the results of the photometric calibrations in Tables 6a and\n 6b.\nUsing the extrema from the photometric and spectroscopic parameters we\n derived new model atmospheres from which we determined a range in the\n abundances allowed by the uncertainties in the models.\n\n\\begin{deluxetable}{lcccccccccccccccccc}\n\\tabletypesize{\\footnotesize}\n\\rotate\n\\tablewidth{0pc}\n\\tablenum{5a}\n\\tablecaption{Photometry for M 3 Giants}\n\\tablehead{\n \\colhead{} & \n \\multicolumn{3}{c}{Ferraro {\\em et al.}} & \\colhead{} & \n \\colhead{Rood} & \\colhead{} &\n \\multicolumn{2}{c}{MB} & \\colhead{} & \n \\colhead{vB} & \\colhead{} & \n \\multicolumn{2}{c}{SK} & \\colhead{} & \n \\multicolumn{2}{c}{Cudworth} & \\colhead{} &\n \\colhead{CFP} \\\\\n \\cline{2-4} \\cline{6-6} \\cline{8-9} \\cline{11-11} \\cline{13-14} \\cline{16-17} \\cline{19-19} \\\\\n \\colhead{Star} &\n \\colhead{$V$} &\n \\colhead{\\bv} &\n \\colhead{$V - I$} & \\colhead{} & \\colhead{$V - I$} & \\colhead{} & \\colhead{$V$} & \\colhead{\\bv} & \\colhead{} & \\colhead{$V - I$} & \\colhead{} & \\colhead{$V$} & \\colhead{\\bv} & \\colhead{} & \\colhead{$V$} & \\colhead{\\bv} & \\colhead{} & \\colhead{$V - K$} } % Star Ferraro Bolte SK C CFP % V (B-V) (V-I) V (B-V) V (B-V) V (B-V) (V-K)\n\\startdata\nAA & 12.72 & 1.58 & \\nodata && 1.46 && \\nodata & \\nodata && \\nodata && 12.71 & 1.56 && 12.69 & 1.57 && 3.43 \\\\\nMB 4 & 12.74 & 1.44 & 1.47 && 1.45 && 12.75 & 1.64 && \\nodata && \\nodata & \\nodata && \\nodata & \\nodata && \\nodata \\\\\nIII-28 & 12.75 & 1.44 & \\nodata && 1.45 && \\nodata & \\nodata && 1.32 && 12.80 & 1.37 && 12.81 & 1.37 && 3.21 \\\\\nvZ 297 & 12.84 & 1.44 & \\nodata && 1.42 && \\nodata & \\nodata && \\nodata && 12.85 & 1.43 && 12.89 & 1.42 && \\nodata \\\\\nvZ 1127 & 13.09 & 1.29 & 1.47 && 1.35 && \\nodata & \\nodata && \\nodata && \\nodata & \\nodata && 12.93 & 1.23 && \\nodata \\\\\nvZ 1000 & 13.10 & 1.29 & 1.39 && 1.35 && \\nodata & \\nodata && \\nodata && 13.03 & 1.29 && 13.01 & 1.40 && \\nodata \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\begin{deluxetable}{lccccc}\n\\tablewidth{0pc}\n\\tablenum{5b}\n\\tablecaption{Photometry for M 13 Giants}\n\\tablehead{\n \\colhead{} & \n \\multicolumn{2}{c}{CM79} & \\colhead{} &\n \\colhead{CFP} \\\\\n \\cline{2-3} \\cline{5-5} \\\\\n \\colhead{Star} &\n \\colhead{$V$} &\n \\colhead{\\bv} &\n \\colhead{} &\n \\colhead{$V - K$}\n}\n% Star C CFP\n% V (B-V) (V-K)\n\\startdata\nL 324 & 12.00 & 1.60 && \\nodata \\\\\nIII-56 & 12.14 & 1.47 && 3.31 \\\\\nL 262 & 12.25 & 1.39 && \\nodata \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\begin{deluxetable}{lcccccccccccccc}\n\\tabletypesize{\\footnotesize}\n\\tablewidth{0pc}\n\\tablenum{6a}\n%\\rotate\n\\tablecaption{Photometric Models for M 3 Giants}\n\\tablehead{\n \\colhead{} & \n \\multicolumn{2}{c}{$V - I_{\\rm F97}$} & \n \\colhead{} &\n \\multicolumn{2}{c}{\\bv$_{\\rm F97}$} & \n \\colhead{} &\n \\multicolumn{2}{c}{$V - I_{\\rm FR}$} & \n \\colhead{} &\n \\multicolumn{2}{c}{$V - I_{\\rm vB}$} & \n \\colhead{} &\n \\multicolumn{2}{c}{$V - K_{\\rm CFP}$} \\\\\n \\cline{2-3} \\cline{5-6} \\cline{8-9} \\cline{11-12} \\cline{14-15}\\\\\n \\colhead{Star} &\n \\colhead{$T_{\\rm eff}$} &\n \\colhead{log $g$} &\n \\colhead{} &\n \\colhead{$T_{\\rm eff}$} &\n \\colhead{log $g$} &\n \\colhead{} &\n \\colhead{$T_{\\rm eff}$} &\n \\colhead{log $g$} &\n \\colhead{} &\n \\colhead{$T_{\\rm eff}$} &\n \\colhead{log $g$} &\n \\colhead{} &\n \\colhead{$T_{\\rm eff}$} &\n \\colhead{log $g$}\n}\n\\startdata\nAA & \\nodata & \\nodata && 3950 & 0.31 && 4061 & 0.57 && \\nodata & \\nodata && 4000 & 0.70 \\\\\nMB 4 & 4060 & 0.57 && 4046 & 0.54 && 4067 & 0.58 && \\nodata & \\nodata && \\nodata & \\nodata \\\\\nIII-28 & \\nodata & \\nodata && 4048 & 0.54 && 4070 & 0.59 && 4232 & 0.90 && 4100 & 0.80 \\\\\nvZ 297 & \\nodata & \\nodata && 4043 & 0.54 && 4102 & 0.65 && \\nodata & \\nodata && \\nodata & \\nodata \\\\\nvZ 1127 & 4059 & 0.56 && 4182 & 0.80 && 4189 & 0.81 && \\nodata & \\nodata && \\nodata & \\nodata \\\\\nvZ 1000 & 4088 & 0.62 && 4182 & 0.80 && 4190 & 0.81 && \\nodata & \\nodata && \\nodata & \\nodata \\\\\n\\enddata\n\\end{deluxetable}\n\n\\begin{deluxetable}{lcc}\n\\tablewidth{0pc}\n\\tablenum{6b}\n\\tablecaption{Photometric Models for M 13 Giants}\n\\tablehead{\n \\colhead{Star} &\n \\colhead{$T_{\\rm eff}$} &\n \\colhead{log $g$} \\\\\n}\n\\startdata\nL 324 & 3905 & 0.27 \\\\\nIII-56 & 4020 & 0.49 \\\\\nL 262 & 4089 & 0.62 \\\\\n\\enddata\n\\end{deluxetable}\n\n\\subsection{Results}\n\nTables 7a, 7b and 7c present the final results of our abundance analysis\n for the iron-peak elements, alpha elements and proton-capture elements\n (those that can be altered in the CNO, NeNa and MgAl nuclear burning cycles),\n respectively.\nThe numbers in parentheses are the line-to-line scatter for each element\n while the numbers in the super- and subscripts give the estimated ranges\n based on variations in the models.\nWe include these latter estimates of uncertainties so that our data can\n be compared with other observations that use photometric-based atmospheres\n in the abundance analysis derivations.\nClearly, the uncertainty in the abundance determination is dominated more by\n the uncertainty in the models than by the line-to-line scatter, which we\n use here as the ``error'' under the assumption that our models are\n well-determined.\n\nThe Fe-peak elements are consistent with the solar ratio with the exception\n of nickel, which seems to be under-abundant in all the giants by 0.22 ${\\pm}$\n 0.03 dex on average.\nThis would be expected if the oscillator strengths for the Ni I lines were\n overestimated by a similar amount, as indicated in Table 3, which\n shows our derived oscillator strengths to be 0.20 ${\\pm}$ 0.10 dex larger\n than the literature value.\nThus, using the oscillator strengths from the literature would put [Ni/H]\n closer to zero in our sample.\nWhy it should be that the oscillator strengths for Ni would be inconsistent\n with the published values while those of the other elements are more\n agreeable remains uncertain.\nIn fact the difference in {\\lgf}'s is even larger since our assumed solar\n Ni abundance is 0.07 dex higher than what \\citet{Thev90} assumes; to\n force agreement with the lower Ni abundance value would cause us to increase\n our {\\lgf}'s by another 0.07 dex.\n\n\\begin{deluxetable}{lcccccc}\n\\tabletypesize{\\footnotesize}\n\\tablewidth{0pc}\n\\rotate\n\\tablenum{7a}\n\\tablecaption{Fe-Peak Abundances}\n\\tablehead{\n \\colhead{Star} &\n \\colhead{[Fe/H]$_{\\rm Fe I}$} &\n \\colhead{[Fe/H]$_{\\rm Fe II}$} &\n \\colhead{[Fe/H]$_{\\rm av}$} &\n \\colhead{[V/Fe]} &\n \\colhead{[Mn/Fe]} &\n \\colhead{[Ni/Fe]}\n}\n\n%Star [Fe/H] [Fe/H] [Fe/H] [V/Fe] [Mn/Fe] [Ni/Fe]\n% I II av \n%===============================================================================\n\\startdata\n\\cutinhead{M~3}\nAA & $-1.59(0.09)^{+0.11}_{-0.20}$ & $-1.53(0.08)^{+0.08}_{-0.11}$ & $-1.57(0.12)^{+0.12}_{-0.22}$ & $+0.25(0.14)^{+0.17}_{-0.36}$ & $-0.02(0.16)^{+0.17}_{-0.26}$ & $-0.22(0.15)^{+0.15}_{-0.25}$ \\\\[0.in]\nMB 4 & $-1.55(0.07)^{+0.07}_{-0.11}$ & $-1.53(0.06)^{+0.06}_{-0.14}$ & $-1.55(0.09)^{+0.08}_{-0.12}$ & $+0.19(0.11)^{+0.14}_{-0.13}$ & $-0.03(0.14)^{+0.15}_{-0.16}$ & $-0.22(0.14)^{+0.14}_{-0.20}$ \\\\[0.in]\nIII-28 & $-1.62(0.07)^{+0.13}_{-0.22}$ & $-1.59(0.07)^{+0.09}_{-0.09}$ & $-1.61(0.10)^{+0.12}_{-0.23}$ & $-0.04(0.13)^{+0.25}_{-0.39}$ & $-0.26(0.11)^{+0.18}_{-0.24}$ & $-0.28(0.12)^{+0.16}_{-0.24}$ \\\\[0.in]\nvZ~297 & $-1.57(0.08)^{+0.14}_{-0.09}$ & $-1.49(0.07)^{+0.08}_{-0.07}$ & $-1.55(0.11)^{+0.14}_{-0.11}$ & $+0.09(0.14)^{+0.26}_{-0.31}$ & $+0.04(0.16)^{+0.22}_{-0.17}$ & $-0.25(0.13)^{+0.15}_{-0.14}$ \\\\[0.in]\nvZ~1127 & $-1.44(0.09)^{+0.09}_{-0.40}$ & $-1.50(0.05)^{+0.05}_{-0.15}$ & $-1.45(0.10)^{+0.10}_{-0.39}$ & $+0.19(0.13)^{+0.13}_{-0.63}$ & $-0.06(0.13)^{+0.13}_{-0.41}$ & $-0.25(0.15)^{+0.15}_{-0.39}$ \\\\[0.in]\nvZ~1000 & $-1.49(0.10)^{+0.10}_{-0.24}$ & $-1.50(0.07)^{+0.07}_{-0.20}$ & $-1.49(0.12)^{+0.10}_{-0.24}$ & $+0.13(0.15)^{+0.15}_{-0.37}$ & $-0.14(0.14)^{+0.14}_{-0.27}$ & $-0.24(0.21)^{+0.21}_{-0.32}$ \\\\\n\\cutinhead{M~13}\nL~324 & $-1.68(0.08)^{+0.08}_{-0.12}$ & $-1.64(0.07)^{+0.28}_{-0.07}$ & $-1.67(0.11)^{+0.31}_{-0.13}$ & $+0.07(0.14)^{+0.14}_{-0.36}$ & $-0.16(0.11)^{+0.11}_{-0.22}$ & $-0.24(0.15)^{+0.17}_{-0.15}$ \\\\[0.in]\nIII-56 & $-1.72(0.07)^{+0.14}_{-0.08}$ & $-1.64(0.03)^{+0.05}_{-0.03}$ & $-1.70(0.08)^{+0.12}_{-0.10}$ & $+0.14(0.11)^{+0.25}_{-0.15}$ & $-0.09(0.08)^{+0.16}_{-0.09}$ & $-0.17(0.12)^{+0.17}_{-0.13}$ \\\\[0.in]\nL~262 & $-1.61(0.07)^{+0.07}_{-0.14}$ & $-1.62(0.08)^{+0.10}_{-0.08}$ & $-1.61(0.11)^{+0.09}_{-0.14}$ & $+0.15(0.14)^{+0.14}_{-0.28}$ & $-0.11(0.14)^{+0.14}_{-0.22}$ & $-0.18(0.14)^{+0.14}_{-0.20}$ \\\\\n\\enddata\n\\end{deluxetable}\n\nFigure 5 shows the Fe-peak abundances as a function of {\\teff} for M~3 and M~13.\nThe [Fe/H] ratios for M~13 are 0.12 dex lower on average than those for M~3;\n although, the difference is only at the 1.5~${\\sigma}$ level.\nDespite the marginal disparity in [Fe/H], the trends for [V/Fe], [Mn/Fe] and\n [Ni/Fe] are the same for each cluster.\n\n%\\placefigure{Cavallo.fig05}\n\\epsscale{1.00}\n\\begin{figure}\n\\plotone{Cavallo.fig05.eps}\n\\caption[Cavallo.fig05.eps]{Fe-peak abundances as a function of {\\teff} \n in M~3 (filled triangles) and M~13 (open circles).\n The dotted and dashed lines are the means for each element for\n M~3 and M~13, respectively, while the error bars are representative of\n the line-to-line scatter. }\n\\end{figure}\n\nFigure 6 shows the alpha elements as a function of {\\teff} for M~3\n and M~13.\nThe alpha element enhancements are consistent with other observations:\n $[{\\alpha}/{\\rm Fe}]~=~+0.28 ~{\\pm} ~0.08$ for M~3 and $+0.30 ~{\\pm} ~0.07$\n for M~13.\nThe low dispersions probably indicate that the upper and lower limits \n from the various model atmospheres overestimate the actual errors in the\n abundances.\nThe two clusters have very similar alpha enhancements, despite the differences\n in their iron abundances.\n\n\\begin{deluxetable}{lcccccc}\n\\tabletypesize{\\footnotesize}\n\\tablewidth{0pc}\n%\\rotate\n\\tablenum{7b}\n\\tablecaption{${\\alpha}$ Abundances}\n\\tablehead{\n \\colhead{Star} &\n \\colhead{[Fe/H]$_{\\rm av}$} &\n \\colhead{[Si/Fe]} &\n \\colhead{[Ca/Fe]} &\n \\colhead{[Ti/Fe]$_{\\rm Ti I}$} &\n \\colhead{[Ti/Fe]$_{\\rm Ti II}$} &\n \\colhead{[Ti/Fe]$_{\\rm av}$}\n}\n\n%Star [Fe/H] [Si/Fe] [Ca/Fe] [Ti/Fe] [Ti/Fe] [Ti/Fe]\n% av I II av\n%==================================================================================================================================== \n\\startdata\n\\cutinhead{M~3}\nAA & $-$1.57 & $+0.42(0.14)^{+0.14}_{-0.15}$ & $+0.22(0.13)^{+0.14}_{-0.23}$ & $+0.42(0.15)^{+0.18}_{-0.34}$ & $+0.40(0.12)^{+0.12}_{-0.21}$ & $+0.42(0.19)^{+0.18}_{-0.34}$ \\\\[0.in]\nMB 4 & $-$1.55 & $+0.20(0.12)^{+0.12}_{-0.16}$ & $+0.26(0.12)^{+0.14}_{-0.13}$ & $+0.45(0.11)^{+0.13}_{-0.13}$ & $+0.28(0.10)^{+0.10}_{-0.18}$ & $+0.36(0.15)^{+0.22}_{-0.26}$ \\\\[0.in]\nIII-28 & $-$1.61 & $+0.26(0.11)^{+0.12}_{-0.12}$ & $+0.23(0.14)^{+0.21}_{-0.28}$ & $+0.25(0.13)^{+0.25}_{-0.37}$ & $+0.29(0.13)^{+0.14}_{-0.18}$ & $+0.26(0.18)^{+0.38}_{-0.24}$ \\\\[0.in]\nvZ~297 & $-$1.55 & $+0.37(0.16)^{+0.16}_{-0.17}$ & $+0.22(0.14)^{+0.20}_{-0.15}$ & $+0.26(0.15)^{+0.26}_{-0.16}$ & $+0.21(0.11)^{+0.13}_{-0.11}$ & $+0.25(0.19)^{+0.27}_{-0.15}$ \\\\[0.in]\nvZ~1127 & $-$1.45 & $+0.17(0.12)^{+0.12}_{-0.17}$ & $+0.21(0.13)^{+0.13}_{-0.41}$ & $+0.40(0.11)^{+0.11}_{-0.57}$ & $+0.35(0.10)^{+0.10}_{-0.29}$ & $+0.39(0.15)^{+0.12}_{-0.56}$ \\\\[0.in]\nvZ~1000 & $-$1.49 & $+0.29(0.13)^{+0.13}_{-0.15}$ & $+0.28(0.13)^{+0.13}_{-0.26}$ & $+0.29(0.14)^{+0.14}_{-0.35}$ & $+0.20(0.17)^{+0.17}_{-0.23}$ & $+0.27(0.22)^{+0.16}_{-0.33}$ \\\\\n\\cutinhead{M~13}\nL~324 & $-$1.67 & $+0.31(0.12)^{+0.21}_{-0.12}$ & $+0.21(0.13)^{+0.13}_{-0.28}$ & $+0.34(0.13)^{+0.13}_{-0.32}$ & $+0.24(0.11)^{+0.22}_{-0.11}$ & $+0.32(0.17)^{+0.15}_{-0.30}$ \\\\[0.in]\nIII-56 & $-$1.70 & $+0.31(0.11)^{+0.11}_{-0.11}$ & $+0.27(0.12)^{+0.20}_{-0.12}$ & $+0.28(0.11)^{+0.26}_{-0.12}$ & $+0.48(0.09)^{+0.16}_{-0.09}$ & $+0.32(0.14)^{+0.32}_{-0.16}$ \\\\[0.in]\nL~262 & $-$1.61 & $+0.45(0.13)^{+0.14}_{-0.13}$ & $+0.22(0.14)^{+0.14}_{-0.21}$ & $+0.29(0.12)^{+0.12}_{-0.25}$ & $+0.40(0.11)^{+0.11}_{-0.14}$ & $+0.31(0.16)^{+0.20}_{-0.27}$ \\\\\n\\enddata\n\\end{deluxetable}\n\n%\\placefigure{Cavallo.fig06}\n\\begin{figure}\n\\plotone{Cavallo.fig06.eps}\n\\caption[Cavallo.fig06.eps]{The alpha elements as a function of {\\teff}\n in M~3 (filled triangles) and M~13 (open circles).\n The dotted and dashed lines are the means for each element for\n M~3 and M~13, respectively, while the error bars are representative of\n the line-to-line scatter. }\n\\end{figure}\n\nFigure 7 shows the trends in the proton-capture element abundances\n for all observed stars in M~3 and M~13 as a function of {\\teff}.\nThe oxygen abundances were calculated via the synthetic-spectra fitting \n package in MOOG and are presented without error bars because 1) they come\n only from the [O I] line at ${\\lambda}$6300 {\\AA} so there is no\n line-to-line scatter, and 2) the spectral resolution around the line\n is too low to accurately deblend the line from the nearby Sc II line,\n making attempted variations in the models less meaningful.\nWe estimate the error in [O/Fe] to be ${\\sim}$ 0.15 dex.\nThe magnesium abundances are derived from only three lines, all near the\n edges of the CCD where the noise is high, and are susceptible to large\n uncertainties.\nFor the M~3 star III-28, we fit a synthetic spectrum to the data around\n the {\\LL}6696,6698 and {\\LL}7835,7836 Al~I regions to determine the\n abundance and estimate the error to be ${\\pm}$0.15 dex.\n\n\\begin{deluxetable}{lccccl}\n\\tabletypesize{\\footnotesize}\n\\tablewidth{0pc}\n%\\rotate\n\\tablenum{7c}\n\\tablecaption{Proton-Capture Abundances}\n\\tablehead{\n \\colhead{Star} &\n \\colhead{[Fe/H]$_{\\rm av}$} &\n \\colhead{[O/Fe]} &\n \\colhead{[Na/Fe]} &\n \\colhead{[Mg/Fe]} &\n \\colhead{[Al/Fe]}\n}\n\n%Star [Fe/H] [O/Fe] [Na/Fe] [Mg/Fe] [Al/Fe]\n% av \n%=================================================================\n\\startdata\n\\cutinhead{M~3}\nAA & $-$1.57 & $-$0.03 & $+0.42(0.13)^{+0.14}_{-0.19}$ & $+0.25(0.12)^{+0.13}_{-0.14}$ & $+0.87(0.14)^{+0.14}_{-0.20}$ \\\\[0.in]\nMB 4 & $-$1.55 & $-$0.05 & $+0.42(0.10)^{+0.13}_{-0.10}$ & $+0.18(0.13)^{+0.13}_{-0.14}$ & $+0.83(0.13)^{+0.14}_{-0.13}$ \\\\[0.in]\nIII-28 & $-$1.61 & $+$0.36 & \\nodata & $-0.01(0.11)^{+0.13}_{-0.18}$ & $-0.19(0.15)$ \\\\[0.in]\nvZ~297 & $-$1.55 & $-$0.01 & $+0.13(0.15)^{+0.19}_{-0.16}$ & $+0.05(0.11)^{+0.14}_{-0.11}$ & $+0.71(0.13)^{+0.17}_{-0.13}$ \\\\[0.in]\nvZ~1127 & $-$1.45 & $+$0.33 & $-0.22(0.10)^{+0.10}_{-0.28}$ & $+0.01(0.13)^{+0.13}_{-0.25}$ & $-0.01(0.11)^{+0.11}_{-0.28}$ \\\\[0.in]\nvZ~1000 & $-$1.49 & $-$0.01 & $+0.17(0.12)^{+0.13}_{-0.20}$ & $+0.18(0.13)^{+0.13}_{-0.19}$ & $+0.72(0.14)^{+0.14}_{-0.21}$ \\\\\n\\cutinhead{M~13}\nL~324 & $-$1.67 & $-$0.38 & $+0.54(0.12)^{+0.12}_{-0.23}$ & $-0.02(0.13)^{+0.13}_{-0.15}$ & $+0.99(0.12)^{+0.12}_{-0.19}$ \\\\[0.in]\nIII-56 & $-$1.70 & $-$0.05 & $+0.50(0.08)^{+0.13}_{-0.09}$ & $+0.23(0.16)^{+0.19}_{-0.17}$ & $+0.74(0.11)^{+0.16}_{-0.11}$ \\\\[0.in]\nL~262 & $-$1.61 & $+$0.11 & $+0.34(0.11)^{+0.11}_{-0.16}$ & $+0.10(0.14)^{+0.14}_{-0.17}$ & $+0.61(0.13)^{+0.13}_{-0.17}$ \\\\\n\\enddata\n\\end{deluxetable}\n\n\nIn M~3 [Al/Fe] spans a range of 1.1 dex over 250 K and, while the three\n cooler stars appear to be more enhanced than the three hotter ones on\n average, it must be cautioned that the sample is biased since the stars\n were chosen from the Lick/Texas studies based on evidence for or against\n mixing and is not close to being complete.\nIn addition, the oxygen abundances exhibit a strong anticorrelation with\n aluminum, but aren't as depleted as in the so-called \n ``super-oxygen-poor'' giants in M~13 \\citep{KSLP92}.\nThe sodium abundances likewise show an increasing trend with decreasing\n {\\teff} and are correlated with [Al/Fe].\nFinally, [Mg/Fe] seems fairly independent of {\\teff} and doesn't appear\n to show any correlations with the other proton-capture elements.\n\nIn M~13 it is impossible to make any firm conclusions since the\n data are so few; however, we note several trends.\nFirst, [O/Fe] and [Al/Fe] are strongly anticorrelated.\nSecond, the aluminum abundance is high ($>$ 0.6 dex) for all three stars,\n while the oxygen varies from slightly enhanced to fairly depleted \n(but again, not super oxygen poor).\nThird, [Na/Fe] shows a slight trend of increasing with decreasing {\\teff}.\nFourth, the one giant with the strongest aluminum enhancement (L~324) has\n the strongest magnesium depletion and is also the brightest star in the sample.\n\n%\\placefigure{Cavallo.fig07}\n\\begin{figure}\n\\plotone{Cavallo.fig07.eps}\n\\caption[Cavallo.fig07.eps]{The proton-capture elements as a function of\n {\\teff} in M~3 (filled triangles) and M~13 (open circles). \n The dotted and dashed lines are the means for each element for\n M~3 and M~13, respectively, while the error bars are representative of\n the line-to-line scatter. }\n\\end{figure}\n\n\\section{DISCUSSION}\n\\subsection{Evidence for Deep Mixing on the RGB}\n\\subsubsection{Theoretical predictions}\n\nIn \\citet{CSB96} and CSB98 we explored the development of the abundance\n profiles around the H~shell of four canonical stellar evolutionary sequences.\nAlthough the models were unmixed, we can infer some predictions with regard to\n deep mixing, which, we remind the reader, is defined as mixing that\n penetrates the H shell.\n\\begin{itemize}\n\\item Carbon, nitrogen and oxygen are not good tracers of deep mixing since \n they are easily altered above the H~shell in the CN and ON nuclear reaction \n cycles.\n\\item Sodium is altered above the H~shell from $^{22}$Ne and inside the\n H~shell from $^{20}$Ne through the NeNa cycle as shown in Figure 8.\n The proton-capture rates for the NeNa cycle are still uncertain and the\n initial neon abundance in real RGB stars is impossible to measure,\n making the theoretical prediction of actual sodium enhancements difficult. \n\\item As shown in Figure 8, $^{27}$Al, the only stable aluminum isotope, \n is enhanced only inside the H~shell at the expense of mostly $^{25}$Mg plus\n $^{26}$Mg, but also some $^{24}$Mg deep inside the H~shell\n for very bright, metal-poor models. The reaction rates for the MgAl cycle\n are still subject to large uncertainties; although, the $^{24}$Mg proton\n capture rates are now well-determined \\citep{Powell99}.\n\\item Aluminum enhancements are temperature sensitive, indicating that they \n should not be expected until higher luminosities are achieved in lower \n metallicity giants ([Fe/H]~${\\lesssim}~-1.2$).\n In addition, the creation of sodium from $^{20}$Ne also requires the high\n temperatures found only inside the H~shell of the same bright, metal-poor\n giants, indicating that large sodium enhancements that originate with\n this neon isotope occur only towards the RGB tip.\n\\end{itemize}\n\n%\\placefigure{Cavallo.fig08}\n\\begin{figure}\n\\plotone{Cavallo.fig08.eps}\n\\caption[Cavallo.fig08.eps]{The abundance variations around the H shell of a\n red-giant-branch tip model with [Fe/H]~=~$-1.67$ and scaled solar\n composition.\n The abscissa is the mass difference between any point and the\n center of the H shell.\n Hydrogen and helium are given in mass fractions while the other elements\n are scaled relative to the total number of metals.}\n\\end{figure}\n\nAll these inferences are still subject to uncertainties in\n the source of mixing, the initial abundances and the nuclear \n reaction rates; nonetheless, we now venture to compare the observational \n data with the theoretical predictions.\n\n\\subsubsection{Observational results: the aluminum data}\n\nIn addition to the M~13 aluminum abundances determined above, we were\n provided with the equivalent-width data of the Al~I ${\\lambda}$6696 {\\AA}\n line for 78 more giants in this cluster that were obtained from spectra\n taken with the WIYN telescope and Hydra multi-object spectrograph by \n Dr. C. Pilachowski in an attempt to find spurious Li~I\n features at ${\\lambda}$6707 {\\AA} \\citep{PSKHW2000}, as found in the M~3\n giant IV-101 by \\citet{Kraft99}\\footnote{We find no evidence of this feature\n in any of our M~3 or M~13 spectra.}.\nThe details of the data reduction and search results for her study\n will be reported on elsewhere \\citep{PSKHW2000}.\nSince the Li~I line is so strong, the exposures were short in order\n to probe as many stars as possible.\nUnfortunately, this resulted in a lower signal-to-noise ratio than preferred\n for the nearby Al~I lines, but the data are reliable for stars with\n strong aluminum lines: of the 78 giants in the WIYN dataset, 66 had\n measurable equivalent widths.\nTo derive the aluminum abundances, the equivalent widths were folded through\n models that were built using the MARCS code based on the atmospheric parameters\n that were initially derived via photometry \\citep{PSKL96}.\nThe models were assumed to have [Fe/H]~=~$-1.50$, while, as with the models\n from this present study, the [${\\alpha}$/Fe] ratio was also assumed to be \n enhanced by $+$0.4 dex.\nAs shown in Tables 5, 6 and 7, models based on photometric indices can\n lead to a wider range of abundances; although, we believe the\n WIYN models to be well-determined since they were iteratively corrected\n with the spectra.\n\nOf the twelve other stars in the WIYN sample that did not have measurable\n lines, one, II-76, had an aluminum abundance previously determined in the\n literature; [Al/Fe]~=~$-0.19$ \\citep{S96a}.\nThe other eleven were assumed to have [Al/Fe]~=~0 for the purposes of the\n statistics discussed below, an assumption that seems verified by II-76\n having such a low aluminum abundance; although, this star is rather\n bright while the other eleven are much lower magnitude examples.\nThe three M~13 stars observed by us were also present in the WIYN sample and\n the abundances agreed to within the errors from the line-to-line scatter.\nAlthough the differences between the 4-m data and the WIYN data are small,\n systematic errors can arise since the WIYN models are rooted in photometry\n with corrections to {\\teff} and $v_t$ from lower resolution spectra\n (see Pilachowski et al. 1996 for details).\nFor example, our models are 40-60~K hotter, have gravities that are\n 0.35-0.45 dex lower and microturbulent velocities that are 0.09-0.13\n km~s$^{-1}$ higher.\nIn addition, our spectra have a factor of two higher resolution than the\n WIYN data and our signal-to-noise ratios are significantly higher.\n\nFinally, the sample was then augmented with [Al/Fe] values taken from the\n literature \\citep{WLO87,S96a,Kraft97}, bringing the total number of stars\n with determined [Al/Fe] values to 85.\nWe believe that the systematic errors that might be present in the\n data are mostly removed before combination since, with the exception\n of \\citet{WLO87} for two stars, they are derived by the same group of\n Lick/Texas observers who practice consistent reduction techniques.\nSmall differences will arise as the telescopes and instruments are varied,\n but the Lick/Texas observers do compare their various observations\n and show little scatter among their results.\nIf one adds the eleven stars with the assumed low [Al/Fe] values, the total\n sample size is 96, covering a range from the tip of the RGB at $V~=~11.9$ to\n $V~=~15.5$, with complete coverage of the \\citet{CM79} photometry for\n $V~{\\lesssim}~13.7$.\nThe data are plotted in Figure 9a as a function of $V$ magnitude, where\n the eleven stars with [Al/Fe] assumed to be 0 are shown as open circles.\nThe [Al/Fe] values of giants with multiple measurements were averaged\n together after being normalized by [Fe/H].\n\n%\\placefigure{Cavallo.fig09}\n\\begin{figure}\n\\plotone{Cavallo.fig09.eps}\n\\caption[Cavallo.fig09.eps]{\n(a) [Al/Fe] values of M~13 giants as a function of $V$ magnitude.\n Measured values are shown as filled squares, while those with assumed\n values are shown as open circles. In the cases where multiple measurements\n for the same star exist, the values are averaged together after being\n normalized by [Fe/H]. \n(b) Same as (a), except for [Na/Fe].\n(c) A histogram of the [Al/Fe] distribution in M~13.\n(d) Same a (c) except for [Na/Fe].\n(e) The Al ratio in M~13 as a function of magnitude as determined by\n the KMM algorithm.\n The thick solid line is the mean of the 15 points at $V~>~13.0$.\n(f) The Na ratio in M~13 as a function of magnitude as determined by\n the KMM algorithm. \n The open triangles indicate where the distribution is likely unimodal.\n The thick solid line is the mean of the 14 points at $V~>~13.1$.}\n\\end{figure}\n\nIn Figure 9c we show a histogram of the distribution of [Al/Fe] for the \n entire sample of 96 M~13 giants.\nFrom this figure, we see an apparent gap between [Al/Fe]~=~0.2 and 0.4,\n indicating that the distribution might be bimodal.\nWe test for bimodality by applying the KMM algorithm of \\citet{KMM}, which\n tests the null hypothesis that a single Gaussian is a good description of\n the data by comparing the fit of a single-peaked distribution to one\n with multiple modes.\nThe algorithm returns a $P$-value that describes the confidence level of \n the single-mode fit, where $P~<~0.05$ indicates that a single Gaussian\n can be rejected at better than the 95\\% confidence level, which is generally\n accepted as strongly consistent with the multimodal distribution.\nTesting for a bimodal distribution in our sample gives $P~=~0.000$ with\n means in [Al/Fe] of $0.12~{\\pm}~0.25$ for the ``Al-normal'' peak and\n $1.03~{\\pm}~0.25$ for the ``Al-enhanced'' peak.\nThe number of stars in each distribution is 65 and 31 for the Al-enhanced\n and Al-normal groups, respectively, giving a ratio of Al-enhanced\n to Al-normal stars (hereafter, the ``Al ratio'') of $2.10~{\\pm}~0.46$,\n where the error is estimated from Poissonian statistics.\n\nIt is possible that our assumption concerning the actual [Al/Fe] values\n of stars with no measurable Al~I lines can introduce a bias into our\n statistics.\nFor example, \\citet{PSKHW2000} report an upper limit of 20~m{\\AA} for \n equivalent width measurements of lower luminosity giants.\nApplying this measurement to the star K~272, which has $V~=~15.47$, and\n using the model parameters supplied by \\citet{PSKHW2000}, we obtain \n [Al/Fe]~${\\leq}$~0.88, assuming [Fe/H]~=~$-$1.49.\nWe test the effect of this bias by subjecting just the 85 giants with actual\n aluminum measurements to the KMM algorithm,\n yielding a $P$-value of 0.002, with an Al-ratio of $3.72~{\\pm}~0.98$.\nThis clearly demonstrates that removal of the uncertain data still results in \n a strongly bimodal distribution.\nThe only real solution to correct this possible bias is to make higher\n signal-to-noise observations.\n\nIf deep mixing is occurring on the RGB, then the Al ratio should be a\n function of magnitude, increasing with decreasing $V$.\nTo test this hypothesis we bin the data by magnitude and apply the KMM algorithm\n to each bin to determine whether the distribution in each bin is bimodal and,\n if it is, the Al ratio.\nThe KMM algorithm requires that the number of data points be greater than\n 50, forcing the size of magnitude bins to be rather wide (${\\Delta}V~=2$)\n in order to ensure that enough stars are included for reliable statistics.\nWe began our binning at $V~=~12.9$, one magnitude lower than the brightest\n star in the sample, \n and shift each bin by 0.1 magnitude up to $V~=~14.5$, one magnitude\n brighter than the lowest luminosity star in the sample.\nThis choice of bins avoids adding empty points along the RGB into our \n statistics; although, it reaches magnitudes where the sample is incomplete.\nAccording to the KMM algorithm, the aluminum distribution in each magnitude\n bin is bimodal, with all $P$ values less than 0.013 and most less than 0.004.\nIn Figure 9e we show the Al ratio (with Poissonian error bars)\n as a function of the central magnitude of each bin.\nThe mean of the 15 points between $V~=~13.1$ and 14.5 is 1.99,\n which is shown as the solid horizontal line in Figure 9e.\nThe Al ratios in the second brightest and brightest magnitude bins are\n 2.9 and 3.\nThe upturn at the brighter magnitudes is due to both an increase in the number\n of Al-enhanced stars and a decrease in the number of Al-normal stars, as\n can be seen in Figure 9a, and is consistent with mixing occurring along\n the RGB; although, the error bars do not allow for a definitive conclusion\n in this regard.\n\nWe now compare the M~13 data with those from M~3, where the number of\n giants with measured Al abundances is substantially smaller.\nAlthough we augment our sample of six giants in M~3 with an additional\n four that were observed by \\citet[none in common with our sample]{Kraft99},\n the numbers are still too small to apply the KMM algorithm; however, \n as shown in Figure 10a, the M~3 [Al/Fe] distribution\n% visual inspection of the data indicates that the M~3 [Al/Fe] distribution\n appears bimodal, with an Al ratio of $1.5~{\\pm}~1.0$.\nThis is consistent with the presumably unmixed dimmer giants in M~13, \n suggesting that deep mixing is not occurring in M~3.\n\n%\\placefigure{Cavallo.fig10}\n\\epsscale{0.50}\n\\begin{figure}\n\\plotone{Cavallo.fig10.eps}\n\\caption[Cavallo.fig10.eps]{A histogram of (a) the [Al/Fe] and (b) the \n [Na/Fe] distributions in M~3.}\n\\end{figure}\n\n\\subsubsection{Observational results: the sodium data}\n\nWe incorporated the [Na/Fe] data from our three M~13 giants with the literature\n values \\citep{LBC91,KSLP92,KSLS93,KSLSB95,PSKL96,S96a,S96b,Kraft97} to\n build a database containing 119 M~13 RGB stars with measured sodium abundances,\n which we show in Figure 9b as a function of $V$.\nAgain, we believe the systematic errors associated with the combination of\n various data sets to be minimized for the reasons outlines in the previous\n section.\nAs demonstrated in Figures 9a and 9b, and in Figure 11 for a subset of stars\n that have both [Al/Fe] and [Na/Fe] values determined, the range in [Na/Fe]\n is not as wide as in the [Al/Fe] data, with the [Na/Fe] values being both\n more negative for ``low'' sodium stars and not as enhanced for the ``high''\n sodium stars.\nHowever, from Figure 9b it is clear that the tip of the RGB does lack\n sodium-poor giants, if [Na/Fe]~=~0 can be considered a high value relative\n to the rest of the low-Na distribution.\n\nA histogram of the total sodium distribution is shown in Figure 9d,\n where two peaks are apparent, but with no obvious gap between them.\nHowever, application of the KMM algorithm does indicate a bimodal\n distribution ($P~=~0.001$) with two peaks at [Na/Fe]~=~$-$0.09~${\\pm}$~0.13 \n and $+$0.29~${\\pm}~$0.13.\nThe low sodium group has 33 members, while the high sodium group has 86,\n for a ratio of Na-enhanced to Na-poor giants (hereafter, the ``Na ratio\"\")\n of 2.61~${\\pm}~0.53$.\nSince the peaks are not widely separated (2.93 ${\\sigma}$), it is\n difficult to determine to which group stars between the peaks belong,\n making the above ratio less certain.\nThe KMM algorithm provides two group membership probabilities (GMPs) for each \n star, which give the percentage probability that a value belongs to ``high''\n and ``low'' mode (the sum of the GMPs for each star equals 100\\%).\nFor the sample of M~13 giants, 35 stars (30\\%) have both GMPs between 10\\%\n and 90\\%, indicating that these stars cannot be assigned to either mode with\n high confidence.\n \nWhen we bin the [Na/Fe] values by magnitude, as with the [Al/Fe] data, the\n bimodality of the distribution within each bin is not as certain as with\n the entire sample.\nWe demonstrate this in Figure 9f, where we show the Na ratio for M~13 in \n two-magnitude wide bins.\nThe open triangles in this figure represent ratios where the $P$-values are\n greater than 0.05, indicating that a unimodal Gaussian is not easily\n ruled out as the ``true'' parent distribution.\nThe filled-in triangles at $V~>~14.2$ represent apparently bimodal \n distributions; however, the stars at these lower magnitudes are\n undersampled, confounding efforts to determine the nature of the [Na/Fe]\n distribution in M~13.\n\nDespite the uncertainty at lower magnitudes, the [Na/Fe] distribution for the \n brighter M~13 giants is likely bimodal according the KMM statistics\n and is similar to that of the [Al/Fe] distribution.\nThe mean of the 14 dimmest points is 2.33, represented by the solid line \n in Figure 9f.\nRelative to the mean, the upturn at $V~=~13.1$ appears real and is due to\n the lack of low-sodium stars $V~{\\lesssim}~12.5$, as seen in Figure 9b.\n\nWe compare the distribution of sodium in M~3 with that of M~13.\nAs with the aluminum data, the sample size is small, with only fourteen M~3\n stars having determined [Na/Fe] values \\citep{KSLP92,KSLS93,KSLSB95,Kraft99},\n six of which are also determined above.\nWe believe the systematic differences among the Lick/Texas results and\n between our data and theirs that arise from the use of different telescopes,\n instruments and reduction techniques are not significant enough to affect the\n interpretation of the results.\nAs can be seen in Figure 10b the distribution is fairly flat with eight stars\n having [Na/Fe]~${\\le}~0.00$ and only three with [Na/Fe]~$>~+0.3$.\nThis is more consistent with limited mixing, but the numbers are much\n too small to draw realistic conclusions.\n\n\\subsubsection{Observational results: sodium and aluminum}\n\nReferring again to Figure 11, we look for a correlation between sodium and\n aluminum in M~13 by comparing the 62 giants with measured values of both\n [Al/Fe] and [Na/Fe].\nThe data appear correlated, with a linear correlation coefficient of 0.723,\n which according the probability coefficient given in Appendix C\n of \\citet{Taylor}, is ``highly significant'' for this sample size.\nWe do note, however, that at [Al/Fe]$~{\\sim}~+0.5$, the full range of\n [Na/Fe] values is present, making the correlation suspicious around\n this narrow range of [Al/Fe] values.\n\n%\\placefigure{Cavallo.fig11}\n\\begin{figure}\n\\plotone{Cavallo.fig11.eps}\n\\caption[Cavallo.fig11.eps]{[Na/Fe] versus [Al/Fe] for 62 giants in M~13.}\n\\end{figure}\n\nApplying the KMM algorithm to just these 62 stars reveals that both the\n aluminum and sodium distributions are bimodal, with\n each having $P~=~0.000$, and an Al ratio and a Na ratio both equal \n to 3.13~${\\pm}~0.93$.\nTo test whether the identical ratios are just coincidence or if indeed,\n a star with high [Al/Fe] is likely to have high [Na/Fe] and vice-versa,\n we examine the difference in the GMPs between the aluminum and sodium data\n for the ``high'' modes, as shown in the histogram in Figure 12.\n\nIf the abundances of aluminum and sodium are correlated then\n the difference between the GMPs will be close to zero, as seems to be the\n case for most giants since 52 stars fall between $-$0.10 and $+$0.10.\nHowever, ten deviate from zero by more than 0.25; so that while Figure 12 \n indicates that only two or three giants do not fit into the correlation\n around [Al/Fe]~=~$+0.5$, the KMM algorithm actually shows that this number \n is larger and that around 16\\% of the sample are not statistically correlated.\nWe must also reiterate that the [Al/Fe] values from the WIYN\n sample are not always well-determined and these numbers are likely\n to change with better data.\nIn general, the correlation between [Al/Fe] and [Na/Fe] seems fairly\n constrained; however, we suggest that when testing for deep mixing, \n aluminum is a better element to observe than sodium since the data show\n that the distinction between high and low [Al/Fe] is clearer and the\n models suggest that aluminum is produced much closer to the H shell\n than sodium.\nAccording to the models, the appearance of {\\em mixing-enhanced} aluminum\n on the surface of a star should imply the existence of extra sodium, \n while the converse is not necessarily true.\nThe fact that the data show that 84\\% of the time the abundance of one\n element is a predictor of the other indicates just how\n deeply mixed the M~13 giants probably are.\n\n%\\placefigure{Cavallo.fig12}\n\\begin{figure}\n\\plotone{Cavallo.fig12.eps}\n\\caption[Cavallo.fig12.eps]{A histogram of the differences between the group\n membership probabilities for aluminum and sodium for the stars in Figure 11.}\n\\end{figure}\n\n\\subsection{Hot Flashes and Primordial Influences}\n\nEnhanced aluminum abundances at magnitudes much lower than the tip of RGB\n are difficult to explain with mixing models since the peak temperature of\n the H~shell is not high enough to produce a significant amount of\n aluminum at this stage of evolution \\citep{CSB98}.\nWhat then is the source of the high aluminum abundances at these lower\n luminosities?\nSome have suggested that the H~shell might become unstable at lower \n magnitudes due to rotation and can undergo flashes that result in peak\n temperatures near 70 MK or higher \\citep{LHZ97,FAK99}, as opposed to\n the canonical temperatures below 60 MK (see, e.g., CSB98).\nThis hot temperature was chosen specifically by \\citet{LHZ97} because\n it reproduces the observed abundance anomalies in some M~13 giants,\n particularly the $^{24}$Mg depletions and aluminum enhancements observed\n by \\citet{S96b}.\nUnfortunately, such an exercise depends strongly on the\n accuracy of the nuclear reaction rates, which, in many cases, are\n not very well determined \n [see, e.g., the NACRE \\citep{NACRE} compilation\\footnote{Also available\n at \\url{http://pntpm.ulb.ac.be/nacre.htm}}].\nAlso, in addition to the fact that the H-shell instabilities have yet\n to be demonstrated in RGB models, it is not clear what the effects of such\n flashes would be on the structure and evolution of a star.\nFor example, would these flashes have observable consequences that affect the\n RGB luminosity function, which is generally well-reproduced by canonical\n evolutionary models? \nFurthermore, the instability scenario of \\citet{LHZ97} only applies to the\n lower RGB \\citep{VVH88}, which is at odds with the data reported here that\n show aluminum and sodium enhancements occur towards the tip of the RGB;\n if these elements were being produced on the lower RGB via hot flashes,\n the Al and Na ratios would vary at lower magnitudes.\nIn the case of the \\citet{FAK99} scenario, which involves continually peeling\n off layers of the core and completely disrupting the H~shell, it is not clear\n how stars can evolve up the RGB and not have serious consequences for the\n observed luminosity functions of clusters in which the RGB members experience\n deep mixing.\n\nWe suggest that a more favorable location for hot hydrogen burning is around\n the H~shell of intermediate-mass ($M~>~4 {\\rm M}_{\\odot}$) AGB stars \n (referred to as IMS), which undergo hot bottom burning (HBB), so called because\n the convective envelope is in contact with the H~shell.\nThe IMS could have shed their nuclearly processed envelopes that include\n enhanced aluminum and sodium abundances into the early cluster environment\n \\citep{CD81,DWW97,DDNW98}, creating the observed abundance distributions.\nFor example, the bimodality of [Al/Fe] values on the lower RGB\n would be created if the ejecta were distributed locally.\nLikewise, the [Na/Fe] distribution in both clusters is explained\n if the IMS envelopes were also enriched in sodium.\nUnfortunately, detailed and accurate aluminum and sodium abundance yields\n from metal-poor AGB evolution models are non-existent; but\n the high temperatures of HBB in IMS, and the observations themselves,\n lend some weight to this hypothesis.\n\n\\subsection{Results from a Deep Mixing Algorithm}\n\nWe have taken the models described in CSB98 and subjected them to a deep \n mixing algorithm that assumes that the mixing is instantaneous; \n i.e., the mixing timescale is the same as the nuclear burning timescale.\nA complete derivation of our algorithm is given in the appendix.\nWhile this simplified approach is unable to mimic a realistic\n mixing timescale, it does have several advantages:\n 1) it can give an upper limit to the amount of variation an element can \n experience due to nuclear processing, \n 2) it can show the lowest point on the RGB where an element can \n be processed and \n 3) it can be used to check the effect of the uncertainties in the \n nuclear reaction rates on the envelope abundances \\citep{Cavallo98}.\nWe discuss the first two points after a brief description of the algorithm.\n\nThe nuclear reaction network employed in the mixing algorithm is the same as\n the one used in CSB98 with the following modifications.\nWe use updated rates for the $^{26}$Mg$(p,{\\gamma})^{27}$Al reaction\n that have been provided to us by C. Rowland.\nHer rates are 1) approximately 10 - 16 times faster than the NACRE rates\n \\citep{NACRE}, 2) 1.5 to 4.5 times faster than the rates used in\n CSB98 and 3) commensurate with the \\citet{CF88} tables in the range of\n $T_9~=~0.4-0.6$, where $T_9~=~10^{9}$~K.\nThe $^{26}$Al proton-capture rate has been separated into the short-lived\n isomeric state, $^{26m}$Al, and the meta-stable ground state,\n $^{26g}$Al (this had a negligible impact on the conclusions drawn in CSB98).\nThe NACRE compilation shows that the $^{26g}$Al proton-capture rate is\n uncertain by three orders of magnitude, the effects of which are \n discussed in the conclusions. \nWe use the rates for the $^{24}$Mg$(p,{\\gamma})^{25}$Al reaction that\n have been updated by \\citet{Powell99}, who measured the resonance \n parameters of the E$_{R}$~=~223~keV resonance to show that the low-energy\n contribution to the total rate does not significantly \n increase this rate as suggested by \\citet{ZL97}.\nThe new rates show a 32\\% increase over the commonly used \\citet{CF88} rates in\n the range of $T_9~=~0.4-0.6$, which is not enough to account for the observed\n depletions of $^{24}$Mg in a handful of M~13 and NGC~6752 giants observed by\n \\citet{S96b,S97}.\n \nThe initial abundances that we put into our algorithm are those of\n \\citet{DDNW98}, who suggest using [$^{25}$Mg/Fe]$~=~+1.1$~dex as the result\n of AGB contamination, while the initial [$^{24}$Mg/Fe] and [$^{26}$Mg/Fe]\n both equal 0.\nThis suggestion is further backed by recent results of \\citet{LFC99} who\n find the overproduction of $^{25}$Mg and $^{26}$Mg relative to\n $^{24}$Mg in metal-poor AGB models.\nIn addition, we enhance the other ${\\alpha}$~elements by $+0.4$ dex.\n\nWe assume mixing begins on the part of the RGB where the H~shell\n burns through the hydrogen discontinuity left behind after the first\n dredge-up, in accordance with the assumption that \n large ${\\mu}$ barriers can prevent mixing at earlier epochs (SM79).\nThis point along the RGB corresponds to the well-known ``bump'' in the\n luminosity function \\citep{BUMP}.\nSupporting this choice are the theoretical mixing models by\n \\citet{Charbonnel94,Charbonnel95} that also assumed\n mixing begins at this point and reproduced the observed\n variations of the {\\cratio}, $^{7}$Li and the $^{12}$C/$^{14}$N ratio\n in both open and globular clusters.\nIn addition, the observations of \\citet{BDG79}, \\citet{NBS81}, \\citet{GB91},\n \\citet{FG99} and \\citet{CGSB99} also support this choice for the onset of\n mixing.\nWe do point out that not all observations support this choice as\n the onset mixing \\citep{Carbon1982,TCLSK83,LKCF86,BBHD90}, however,\n the exact start of deep mixing will little affect our final results.\n\nThe timestep for nuclear processing fixes the timestep for mixing and\n is controlled by setting a limit on how fast any element above a minimum\n abundance threshold may vary.\nSince this timestep is much shorter than the time difference between the models\n used in CSB98, new models were interpolated along the RGB until\n the He flash was encountered.\nThe free parameters in our code are ${\\Delta}X$, the mixing depth defined\n by a change in the H-mass fraction, $X$, within the H~shell, and ${\\eta}$, \n Reimers' mass-loss parameter \\citep{Reimers75}.\nThe algorithm was run with various combinations of mixing depths and\n mass-loss rates for a stellar evolution sequence having [Fe/H]$~=~-1.67$.\n\nFigure 13 shows the [Al/Fe] values derived from our algorithm \n as a function of absolute magnitude, parameterized by various mixing \n depths and mass-loss rates.\nThe absolute magnitude scale was derived from the bolometric luminosity, $L$,\n and {\\teff} provided by the models.\nWe first converted log ($L/{\\rm L}_{\\odot}$) into the bolometric magnitude,\n M$_{bol}$, using the sun as a reference with a value of 4.75 for \n M$_{bol\\odot}$.\nNext, we used the following relationship between gravity, $g$, mass, $M$,\n {\\teff} and $L$ to obtain log $g$ for each model:\n\\begin{equation}\n {\\rm log}~g~=~{\\rm log}~(M/{\\rm M}_{\\odot})~+~4{\\rm log}~T_{\\rm eff}~\n -~{\\rm log}~(L/{\\rm L}_{\\odot})~-~10.61028,\n\\end{equation}\n where a value of T$_{{\\rm eff}{\\odot}}~=~5780~{\\rm K}$ is assumed.\nMass loss is implicitly accounted for via the first term on the right hand \n side of equation 1.\nUsing a 12 Gyr isochrone with [Fe/H]~=~$-$1.67, also constructed by Drs.\n VandenBerg and Bell,\n we interpolated according to log $g$ to find a bolometric correction then\n converted M$_{bol}$ into M$_{V}$.\nWe show the results for ${\\Delta}X$ = 0.05, 0.10, 0.15 and 0.20, and for\n ${\\eta}$ = 0.0, 0.2, 0.4 and 0.6 at each ${\\Delta}X$, as described\n in the figure caption.\n\n%\\placefigure{Cavallo.fig13}\n\\begin{figure}\n\\plotone{Cavallo.fig13.eps}\n\\caption[Cavallo.fig13.eps]{The predicted variation in [Al/Fe] with absolute\n magnitude for different mixing depths and mass-loss parameters. The\n groups of lines are labeled by the mixing depth, ${\\Delta}X$.\n The long-dashed, short-dashed, dotted and solid lines are for Reimers' (1975)\n ${\\eta}$ = 0.6, 0.4, 0.2 and 0.0, respectively.\n The vertical dotted lines represents the low end of ``high'' [Al/Fe] values \n in M~13.}\n\\end{figure}\n\nAccording to Figure 13, the dominant parameter for determining [Al/Fe] at\n the RGB tip is the mixing depth.\nMass loss plays a secondary role for deeply mixed models but is more\n important for less deeply mixed models.\nThe importance of ${\\eta}$ is controlled by a competition between the timescale\n for mass loss and the timescale for converting magnesium into aluminum:\n for a given sequence, the mass-loss timescale is fixed by ${\\eta}$, so\n that with deeper mixing, the Mg-burning timescale decreases, bringing\n the two closer together and limiting the influence of mass loss.\nThus, in the limit of instantaneous mixing, the distribution of [Al/Fe] near\n the tip of the RGB is due primarily to variations in the mixing depth; \n although, for the deeply mixed sequences, the value of [Al/Fe] at the tip\n begins to saturate.\n\nOne factor that depends strongly on the mixing depth is the earliest point\n along the RGB where mixing-induced aluminum variations can occur.\nIn Figure 13 we draw a dashed vertical line at [Al/Fe]~=~$+0.4$ to indicate\n where the aluminum enhancements cross the into the ``high'' aluminum \n distribution for M~13 giants.\nOur results show that for the mixing depths shown in Figure 13, \n large aluminum enhancements should appear along the brightest\n ${\\sim}$~1.5 to 0.5 magnitudes of the RGB.\nA change in ${\\Delta}X$ from 0.05 to 0.20 results in a one magnitude \n difference in where the aluminum abundance rises on the RGB.\nWe apply these estimates to the M~13 sample by binning the data according to\n the magnitudes at which the various mixing depths predict the aluminum\n abundances will cross the [Al/Fe]~=~$+0.4$ threshold.\nThe results are shown in Table 8, where the first two columns describe the\n models and the next five discuss the data. \nThe first column gives the mixing depth and the second column describes how\n far down the RGB the models predict a star will cross into the ``high''\n aluminum group for that mixing depth, while\n the third column gives the fraction of stars in the bin (out of 96).\nThe fourth and fifth columns give the $P$-value and the Al ratio for stars in\n the bin, respectively, while final two columns give the $P$-value and the \n Al ratio for the remaining stars outside the bin.\n\\footnote{It must be emphasized again that the KMM algorithm is only accurate\n for sample sizes greater than 50; nonetheless, we include the $P$-values for\n the sake of completeness.}.\n\nIf the assumption that mixing is the cause of the aluminum enhancements in\n the bright giants of M~13 holds, then it is apparent from Table 8\n that it must turn on somewhere during the brightest magnitude of the RGB,\n as the Al ratio changes from 67:33 to 88:12 as one approaches the RGB tip.\nThis signifies that at least 21\\% of the giants are experiencing deep mixing.\nWe call this a lower limit because our technique for measuring aluminum \n enhancements cannot detect mixing in stars with initially high [Al/Fe] values.\nSince two-thirds of the giants in M~13 appear to have high aluminum\n abundances before mixing takes effect, we could be missing a substantial\n number of stars undergoing deep mixing.\nIf the same relative number of giants with initially high aluminum abundances\n undergo deep mixing as the relative number of giants with initially low\n aluminum abundances, the percentage of all stars undergoing deep mixing jumps\n to 63\\%.\n\n\\begin{deluxetable}{ccccccc}\n\\tablewidth{0pc}\n\\tablenum{8}\n\\tablecaption{Ratio of Al-enhanced to Al-normal Giants in M~13}\n\\tablehead{\n \\colhead{${\\Delta}X$} &\n \\colhead{${\\Delta}M_V$} &\n \\colhead{\\% Stars} &\n \\colhead{$P$-value} &\n \\colhead{Al Ratio} &\n \\colhead{$P$-value} &\n \\colhead{Al Ratio} \\\\\n \\colhead{} &\n \\colhead{from RGB tip} &\n \\colhead{IN} &\n \\colhead{IN} &\n \\colhead{IN} &\n \\colhead{OUT} &\n \\colhead{OUT}\n}\n% Del M_V Del X #Stars P-value Ratio\n%\n\\startdata\n0.05 & 0.50 & 18 & 0.005 & 7.5${\\pm}$5.6 & 0.000 & 1.7${\\pm}$0.2 \\\\\n0.10 & 1.00 & 35 & 0.000 & 2.8${\\pm}$1.1 & 0.007 & 1.7${\\pm}$0.5 \\\\\n0.15 & 1.30 & 44 & 0.001 & 2.5${\\pm}$0.9 & 0.005 & 1.8${\\pm}$0.5 \\\\\n0.20 & 1.50 & 46 & 0.001 & 2.4${\\pm}$0.8 & 0.006 & 1.9${\\pm}$0.6 \\\\\n\\enddata\n\\end{deluxetable}\n\n\\subsection{Deep Mixing, the Blue-Tail Parameter and the Signatures of\n AGB Pollution}\n\nDeep mixing in red giants might have an effect on their future evolution.\nFor example, \\citet{CG96} noticed a relationship between the HB\n morphology and the amount of depletion of oxygen in RGB stars.\nThis does not necessarily imply that oxygen is a second parameter,\n but rather that whatever is responsible for the oxygen depletions might\n also be causing the blueward shift in the HB.\nOne such mechanism that can do both is rotation,\n which has several effects: 1) it can extend the life of a red giant,\n causing it to lose more mass and ultimately end up on the blue HB,\n 2) it can drive meridional circulation currents, which can deplete the\n oxygen, and 3) if fast enough, it can cause the circulation\n currents to penetrate the H~shell and bring helium to the surface.\nThis extra helium causes RGB stars to evolve to the blue HB at brighter\n luminosities than their unmixed counterparts, mimicking the second-parameter\n effect and reproducing the upward slope of the HB with decreasing color\n \\citep{SC98} that is observed in some metal-rich clusters\n \\citep{PIOTTO97,RICH97}.\n\nIn this section, we examine the suggestion by \\citet{Ferr98} that a so-called\n ``blue-tail second parameter'' (BTP) exists in M~13.\nSuch a parameter differs from the more commonly sought after second parameter\n in the sense that the latter typically deals with difference in HB\n morphology on the flat part of the HB, whereas the former describes\n how clusters like M~13, M~80 and NGC~6752 develop extended blue tails.\nWe attempt to discover whether or not deep mixing can be a/the BTP by\n adding extra helium into the atmosphere of RGB stars.\nUnfortunately, helium cannot be measured spectroscopically in cool giants;\n however, as shown in Figure 8, aluminum is made from magnesium inside\n the H~shell where helium is being produced, so that the mixing of helium\n is accompanied by the mixing of aluminum; i.e., aluminum can be a good\n tracer of helium mixing.\nWe conclude that if deep mixing is a BTP and if aluminum\n traces helium mixing, then there should exist a correlation\n between the Al ratio and the HB morphology. \nTo describe the HB morphology quantitatively, we suggest using at the ratio \n of blue to red HB stars (hereafter, the ``HB ratio''), which, of course,\n require definitions of their own.\nPerhaps a solution can be found in the corollary assumption that if\n cluster giants do not mix, then the cluster should not have an\n extended blue tail on the HB.\nTherefore, by assumption, a cluster like M~3, whose giants appear not to\n experience deep mixing, defines the ``red'' HB, so that for clusters\n like M~13, any star on the HB that is hotter than the M~3 HB is defined\n as ``blue,'' provided, of course, that the clusters are similar in all\n other ways (e.g., metallicity, age, environment, etc.).\n\nTo make the comparison between M~3 and M~13, we obtained high-quality Hubble\n Space Telescope photometric data for both clusters from Dr. F. Ferraro and\n shifted the M~3 HB by ${\\delta}V~=~-0.6$ and ${\\delta}(U-V)~=~-0.03$\n to align it with that of M~13, as done by \\citet{Ferr98}.\nWe then plotted histograms of the distributions of colors along the HB's for\n each cluster, as shown in Figure 14, and compared the HB ratio in M~13 with\n its Al ratio.\nWe define stars with $U-V~<~-0.3$ as being blue, which gives a HB ratio\n of 58:42.\nWe note that while this choice of color coincides with the apparent gap \n in Figure 14, it is chosen because this is where the M~3 distribution drops \n off and not because of the appearance of bimodality in the M~13 distribution.\nTo estimate an error in the HB ratio, we fit a Gaussian to the M~3\n distribution to determine the standard deviation, ${\\sigma}$,\n and call blue all M~13 stars hotter than the mean minus $~3{\\sigma}$ in M~3,\n resulting in an HB ratio of 74:26.\nCompared with the 21\\% to 63\\% deeply mixed stars on the RGB,\n the 58\\% to 74\\% blue HB stars is consistent with deep mixing as the BTP;\n although, the uncertainties in the number of RGB stars that have undergone\n deep mixing makes the results less robust than desired.\n\n%\\placefigure{Cavallo.fig14}\n\\begin{figure}\n\\plotone{Cavallo.fig14.eps}\n\\caption[Cavallo.fig14.eps]{Histograms of the HB distributions in M~3 (dashed line)\n and M~13 (solid line). The photometry is from \\citet{Ferr98} and the\n M~3 data have been shifted by ${\\delta}V~=~-0.6$ and ${\\delta}(U-V)~=~-0.03$.}\n\\end{figure}\n\nIt would be helpful if we could discriminate between stars that have undergone\n deep mixing and those that have been polluted.\nOne way to do this might be by using the $s$-process abundances, which are\n also created in intermediate-mass AGB stars (IMS) with aluminum, albeit at \n different locations within the stars \\citep{DDNW98,Lattanzio99,BS99}. \nStars above $M~{\\gtrsim}4~{\\rm M}_{\\odot}$ experience HBB, which, as discussed\n above, result in the production of $^{27}$Al from magnesium.\nThese same stars also create a neutron exposure during the thermal pulses\n in the He-burning shell that favors the production of the Sr-Y-Zr peak\n elements and $^{25}$Mg, all through the $^{22}$Ne$({\\alpha},n)^{25}$Mg\n reaction \\citep{GBLTS99}.\nIf the $n$-rich material is mixed with the HBB by-products and ejected\n into the cluster medium, then one should trace the other in the\n polluted stars.\nSpecifically, we expect stars with high $s$-process abundances to also have \n high [Al/Fe] values, but not vice-versa.\nThe key is to choose the best {\\spr} elements that trace the Al-rich\n IMS ejecta.\nWe suggest using zirconium since many lines are available in the\n optical spectrum and it's abundance easily computed \\citep{CN2000}.\nConversely, elements near the barium peak would not be good choices\n to represent the {\\spr}/HBB enhancement from IMS since they are produced in\n low-mass AGB stars ($M~{\\sim}2~{\\rm M}_{\\odot}$).\n\n\\subsection{On the Overproduction of Sodium}\n\nRecently, \\citet{CDW99} pointed out that mixing into the H~shell to enrich\n the stellar atmosphere in aluminum and helium would result in an\n overproduction of sodium by ${\\gtrsim}~0.3$ dex (see their Figure 3) \n relative to the M~13 data, essentially precluding deep mixing.\nThis constraint keeps the change in the atmospheric helium abundance\n to less than 0.06, much less than the 0.20 found by \\citet{AVS97a,AVS97b} \n that is needed to account for the most extended HB's seen in clusters \n like M~13.\nWe submit that the solution to this discrepancy might be found in the initial \n abundances; i.e., primordial contamination from the IMS plays a role.\nOur algorithm shows that the overproduction of sodium is avoided, even with\n deep mixing, if the initial $^{22}$Ne abundance were depleted as a result\n of $^{22}$Ne$({\\alpha},n)^{25}$Mg reaction in the IMS, which also \n enhances the initial $^{25}$Mg, as needed to produce the large aluminum\n enhancements in the RGB stars.\nThe calculations of \\citet{GBLTS99} indicate that the $^{22}$Ne abundance\n is depleted by approximately 30\\% during the thermal pulses in\n IMS, but it is not clear how HBB and interpulse burning affects\n the net $^{22}$Ne abundance.\nOne would assume that HBB would deplete the $^{22}$Ne reserves in the\n convective envelope and produce $^{23}$Na as is done on the RGB.\nIn contradistinction, the AGB yields calculated by \\citet{DDNW98} actually \n enhance the net $^{22}$Ne abundance from a series of ${\\alpha}$ captures \n on $^{14}$N.\nClearly, a more complete and detailed look into the yields \n of all abundances from primordial AGB stars is necessary to determine a\n more translucent picture of how pollution plays a role on the RGB.\n\nOur hypothesis is consistent with the sodium data, which raise two\n important questions: 1) why does the Na ratio increase only at the same\n magnitudes as the Al ratio when sodium is very easily produced from\n $^{22}$Ne above the H~shell at luminosities far below the RGB tip, and\n 2) why does the sodium abundance vary without oxygen abundance variations\n for ``oxygen-normal'' giants (see, e.g., KSLP).\nThe answer to both these questions might be found on the AGB: if\n $^{22}$Ne is depleted to build up $^{25}$Mg during the He-shell flashes\n and $^{23}$Na during HBB, then sodium will not be produced at lower\n magnitudes on the RGB, but will be made at brighter magnitudes\n from $^{20}$Ne with deep mixing.\nThe extra sodium produced during HBB could be distributed locally within\n the cluster creating the [Na/Fe]-rich stars that are independent of their\n oxygen abundances.\nAlthough oxygen is likely depleted during HBB, this is unlikely to create\n an oxygen-poor RGB atmosphere since it is easier to enhance elements\n in an atmosphere than to deplete them by pollution.\nA primordial pollution scenario is consistent with the data that show\n aluminum and sodium enhancements on the lower RGB and, for some clusters, \n on the main sequence and subgiant branch, and can help prevent the \n overproduction of sodium during deep mixing.\n\n\\section{CONCLUSIONS}\n\n\nBefore we discuss our final conclusions, we first remind the reader of the \n number of assumptions that have gone into our analysis.\nFirst, there are errors associated with the abundance determinations that\n we tried to characterize by allowing for significant variations in the\n model atmosphere parameters, which contribute the most to the uncertainty\n in the analysis.\nSecond, the inclusion of the WIYN data into our analysis comes at a price:\n the data have poor signal-to-noise ratios, come from only one line and\n require the assumption that, for some stars with indeterminate line strengths,\n the [Al/Fe] value is ``low.''\nThird, despite that fact that this is the largest compilation of [Al/Fe] \n and [Na/Fe] values in one globular cluster to be analyzed in a single\n paper, the data are still subject to small number effects, particularly\n at the RGB tip.\nUnfortunately, there are only so many tip stars that can be spectroscopically\n measured from the ground, leaving this problem difficult to solve.\nWe suggest the best way to handle the small numbers is to expand this\n analysis to other clusters for a broad comparative study.\nFourth, the models have many assumptions in them: we assume that canonical\n evolution holds and add in our mixing algorithm after the fact, we assume\n that mixing is instantaneous, we assume that the abundances are distributed\n as per \\citet{DDNW98}, we assume that our reaction rates are accurate, and\n we assume that we adequately searched the parameter space allowed by\n the uncertainties in the initial abundances, nuclear reaction rates and\n mass-loss rates.\nFifth, we assume that no other second parameter affects the relationship\n between the M~3 and M~13 HB morphologies.\n%Since the M~13 RGB stars in our small sample seem to have [Fe/H] values\n% around 0.2 dex lower than those of our M~3 giants, the HB's should not\n% be identical even if all other parameters were equal.\n%Estimating from Table 3 of \\citet{LD90}, this difference in [Fe/H] is\n% roughly 90 K at the red end of the HB, which will give the M~13 HB\n% a slightly bluer color.\nSixth, we make no attempt to correct for blending of the AGB with the RGB\n when performing our analysis.\nApproximately 20\\% of the red giants above the point where giant branches\n merge are supposedly AGB interlopers based on comparative lifetimes:\n the problem is to determine which ones are really AGB stars.\nThis might not be as much of a problem for the M~13 sample, however, since\n blue HB stars tend to evolve away from the AGB.\nThe best workaround for this problem is also an extension of our analysis to\n other clusters to look for consistent trends despite this, and the other,\n uncertainties.\n\nThe importance of having accurate nuclear reaction rates cannot be overstated.\nThis is particularly true when using aluminum as a diagnostic of deep mixing.\nIf we were to vary, for example, the $^{26}$Mg proton-capture rate to its\n upper limit in range of $T_9~=~0.05-0.06$, the production of Al can move\n outside the H shell, although, just barely.\nDepending on the initial abundances of $^{25}$Mg and $^{26}$Mg, this might\n be able to account for the full enhancements of aluminum that we observed.\nIn addition, according to the NACRE compilation, the rate for\n $^{26}$Al$^{g}(p,{\\gamma})^{27}$Si is uncertain by as much as three orders of\n magnitude in the same temperature range.\nIncreasing these rates might help solve the problem presented by \\citet{CDW99}\n who show that, if mixing occurs below the top of the H shell, sodium is \n overproduced due to the extra enhancement from $^{20}$Ne in the NeNa cycle,\n a result we confirm with our instantaneous mixing algorithm.\nIf the $^{26}$Mg proton-capture rate is near its upper limit, then deep\n mixing is not required to produce the observed aluminum abundances and\n sodium is not over enhanced compared to the observations.\n\nOur general results for the M~3 and M~13 abundances obtained in this\n work show the usual trends in the proton-capture, ${\\alpha}$ and\n iron-peak elements:\n the sodium and aluminum abundances are anticorrelated with oxygen, the\n ${\\alpha}$ elements are enhanced by approximately 0.3 dex and the\n iron-peak elements remain constant.\n\nOur analysis shows that the variation in both the [Al/Fe] and [Na/Fe] ratios\n are consistent with deep mixing occurring on the RGB in M~13 and not in M~3.\nThe aluminum and sodium data are correlated for the M~13 giants;\n although, the Al ratio is probably a better indicator of deep mixing\n since it is more easily separated into ``high'' and ``low'' groups.\nWe would not expect such a similar tight correlation between aluminum and\n sodium in the M~3 giants since sodium can be enhanced without increasing\n the aluminum abundance if the mixing currents do not penetrate the H shell,\n as seems indicated in M~3 from the low Al ratio.\nHowever, some semblance of a correlation between aluminum and sodium might be\n set up by primordial effects in this cluster.\nIn addition, the Na ratio increases near the same magnitude as the Al ratio,\n which is contrary to the previous predictions that sodium should be\n enhanced further down the RGB from $^{22}$Ne (CSB98).\nOur models show that this would be expected if the $^{22}$Ne were depleted\n in primordial intermediate-mass AGB stars.\n\nWhen comparing the Al ratio with the HB ratio, it seems that the\n assumption of deep mixing as a blue-tail parameter is self-consistent;\n however, the large range allowed in the actual number of mixed RGB stars\n and the empirical definitions of ``blue'' and ``red'' HB stars do\n not constrain the results enough to be firmly conclusive.\nAgain we suggest that a similar analysis as the one presented here be extended\n to other clusters to determine the Al ratio as a function of $V$ and\n to compare this with the HB ratio.\nIf the Al ratio at the RGB tip can be shown to be a predictor of the \n HB ratio, then helium mixing would certainly be given greater credence\n as a blue-tail second parameter, supplanting the oft-assumed cluster age\n differences that have been shown to fail for this classical pair of clusters.\nIn particular, we suggest the study of metal-rich clusters to see if the\n aluminum distribution is bimodal, and if it is, if the Al ratio varies.\nAccording to our models, it should not vary since aluminum cannot be produced\n in metal-rich cluster giants on the same scale as it can in the intermediate\n metallicity and metal-poor giants.\nIn addition, we suggest further examination of the sodium abundance in\n clusters to search for similar behavior as in M~13.\nAlso, we suggest a more extensive comparison of the {\\spr} abundances with\n the aluminum data as a test of primordial contamination.\n\nFinally, we conclude that the problem of abundance anomalies in globular\n cluster red giants requires detailed study of the abundance yields\n from primordial AGB stars as well as an in-depth and complete study\n of the hydrodynamical evolution of rotating RGB stars.\nIn the meantime, aluminum, and to a lesser extent, sodium, give the\n best diagnostics of deep mixing during the evolution up the RGB and the \n {\\spr} elements near the Sr-Y-Zr peak are the best tracers of AGB pollution\n from IMS.\n\n\\acknowledgments\nThe authors expressly acknowledge and thank Caty Pilachowski for allowing\n us to use her data and for her valuable help in analyzing them.\nWe also extend our gratitude to Mike Briley, who supplied us \n with the unpublished data for the star M~3 AA and who acted as our\n referee with many thoughtful and valuable comments.\nWe thank Bob Kraft and Chris Sneden, who sent us their original Lick spectra\n for the stars in common between our two studies, and to Michael Bolte, who\n provided us with finding charts for the star MB~4.\nWe express our gratitude to Francesco Ferraro for making \n his recent HB photometry for M~13 and M~3 available to us.\nWe thank Carrie Rowland for giving us updated data for\n the $^{26}$Mg proton capture rates and we look forward to her\n experimental results regarding this reaction.\nOur thanks are given to Keith Ashman for supplying us with a FORTRAN version\n of his KMM algorithm.\nIn addition, we are indebted to Roger Bell for providing us with\n his latest isochrones describing color-{\\teff} relations.\nWe also thank Allen Sweigart for his many invaluable discussions throughout the\n development of this project.\nWe wish to thank Daryl Wilmarth for his aid in gathering and\n reducing the data and Frank Hill for providing us with the Kitt Peak\n solar spectrum.\nWe likewise acknowledge Jennifer Johnson and Peter Stetson for making \n their separate photometric data sets available to us.\nN. M. N. acknowledges travel support from the Astronomy Department at UMD and\nR. M. C acknowledges KPNO for travel support while he was a visiting graduate\n student.\nThis work was performed while R. M. C. held a National Research Council-GSFC\n Research Associateship.\nFunding for publication was provided in part by a Small Research Grant from\n the American Astronomical Society.\n\n\\appendix\n\\section{INSTANTANEOUS MIXING ALGORITHM}\n\nThis derivation of instantaneous mixing begins with the form of the nuclear\n reaction equation that involves the proton-capture reactions and \n ${\\beta}$-decays; although, it is easily extended to other rates and is\n applied in its most general form in our code:\n\\begin{equation}\n \\frac{{\\rm d}n_{i}}{\\rm dt}~=~\\sum_{j} ({\\pm}n_{j}n_{\\rm H}<{\\sigma}v>_{j}~\n {\\pm}~n_{j}\\frac{{\\ln}2}{{\\tau}_{j}}),\n\\end{equation}\n where $n_{i}$ is the number of nuclei of type $i$ cm$^{-3}$ that are\n being produced or destroyed, $n_{j}$ is the number cm$^{-3}$ of nuclei that\n produce ($+$ sign) or, when $j~=~i$, destroy ($-$ sign) nuclei of type ${i}$,\n $n_{\\rm H}$ is the number cm$^{-3}$ of protons, $<{\\sigma}v>_{j}$ is the \n velocity-averaged cross section of the proton-capture reaction and\n ${\\tau}_{j}$ is the mean lifetime of radioactive isotopes that destroy\n or produce element $i$.\n\nIf we integrate equation A1 over a mass interval from some mixing depth, \n $M_{\\rm d}$, to the surface,\n $M$, and substitute the molar fraction $Y~=~n/{\\rho}{\\rm N_A}$, where \n ${\\rho}$ is the density and N$_{\\rm A}$ is Avogadro's number, so that\n $Y_{\\rm H}~=~X$, the hydrogen-mass fraction, we get\n\\begin{equation}\n \\int_{M_{\\rm d}}^{M} \\frac{{\\rm d}Y_{i}}{\\rm dt} {\\rm d}M_{r}~=~\n \\sum_{j} \n ({\\pm} \\int_{M_{\\rm d}}^{M} \n Y_{j}X{\\rm N_A}<{\\sigma}v>_{j}{\\rho}{\\rm d}M_{r}~\n {\\pm}~\\frac{{\\ln}2}{{\\tau}_{j}} \\int_{M_{\\rm d}}^{M} Y_{j}{\\rm d}M_{r}),\n\\end{equation}\n which is equivalent to spreading the nuclearly processed material over the\n whole mixing zone.\nMass loss can be accounted for my modifying the total integrated mass by\n some mass-loss recipe such as given by \\citet{Reimers75}.\n\nNow assuming that the mixing is instantaneous so that d$Y_{i}$/dt, $Y_{j}$ \n and $X$ vary little over the whole mixing zone, we can rewrite equation\n A2 as\n\\begin{equation}\n \\frac{{\\rm d}Y_{i}}{\\rm dt} \\int_{M_{\\rm d}}^{M}{\\rm d}M_{r}\n ~=~\n \\sum_{j} \n ({\\pm} Y_{j}X\n \\int_{M_{\\rm d}}^{M} \n {\\rm N_A}<{\\sigma}v>_{j}{\\rho}{\\rm d}M_{r}\n ~{\\pm}~\n \\frac{{\\ln}2}{{\\tau}_{j}} Y_{j} \\int_{M_{\\rm d}}^{M}{\\rm d}M_{r}).\n\\end{equation}\n\nThe first integral on the right hand side is just the mass-averaged \n reaction rate that can be substituted as an effective reaction rate, \n $<{\\sigma}v>_{j}^{\\rm eff}$, while the other two integrals in equation\n A3 are just the total mixed mass, $M_{\\rm mix}$.\nThus, equation A3 can now be written \n\\begin{equation}\n \\frac{{\\rm d}Y_{i}}{\\rm dt}\n ~=~\n \\frac{1}{M_{\\rm mix}} \n \\sum_{j} \n ({\\pm} Y_{j}X<{\\sigma}v>_{j}^{\\rm eff}\n ~{\\pm}~\n \\frac{{\\ln}2}{{\\tau}_{j}} Y_{j}),\n\\end{equation}\n the final form of equation A1 under the assumptions of instantaneous mixing.\n\nThe implementation of this equation in our nuclear reaction network\n is straightforward.\nWe average the reaction rates together by weighting the reaction\n rate determined at each mesh point by the mass contained between that\n mesh point and the one below it and summing over all the mass intervals.\nThe temperature and density for calculating the reaction rate for each\n mass interval are taken at the top mesh point (towards the surface).\nSince the spacing between mesh points becomes closer as the temperature\n profile steepens, the differences between the temperature and density at\n the top and bottom of the mass intervals has negligible influence on the\n effective reaction rates.\nThe effective rates are then applied to the initial abundances and\n integrated (``mixed'') over some determined mass interval where the\n mixing depth and mass-loss rate are the free parameters chosen by the user.\nThe burning timescale is controlled by limiting how much the fastest \n changing isotope with some chosen minimum abundance can vary in a\n single timestep.\nNew models are interpolated from a sequence at timesteps\n according to this nuclear burning timescale.\nEach new model contains the output abundances derived from the previous model\n for its input abundances.\nThe mixing algorithm can begin anywhere on the RGB and proceeds until the\n helium flash is encountered at the tip.\nThe code outputs the new abundances, the mass-averaged reaction rates and\n information regarding the position of the model on the theoretical RGB.\n\n\\clearpage\n\n\\begin{thebibliography}{}\n\\begin{small}\n\\bibitem[Anders \\& Grevesse(1989)]{AG89} Anders, E. \\& Grevesse, N. \n 1989, \\gca, 53, 197\n\\bibitem[Angulo et al.(1999)]{NACRE} Angulo, C. et al. 1999, \\nphysa, A656, 3\n\\bibitem[Armosky et al.(1994)]{ASLK94} Armosky, B. 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A. 1996, \\pasp, 108, 911\n\\end{small}\n\\end{thebibliography}\n\n\\begin{deluxetable}{lllllllllllll}\n\\tabletypesize{\\footnotesize}\n\\tablenum{2}\n\\tablewidth{0pc}\n\\tablecaption{Measured Equivalent Widths (m{\\AA})}\n\\tablecolumns{13}\n\\tablehead{\n \\multicolumn{3}{c}{} & \n \\multicolumn{6}{c}{M~3 Giants} & \n \\colhead{} & \n \\multicolumn{3}{c}{M~13 Giants} \\\\\n \\cline{4-9} \\cline{11-13}\n \\colhead{Wavelength} & \n \\colhead{E. P.} & \n \\colhead{log ($gf$)} & \n \\colhead{AA} & \n \\colhead{vZ~297} & \n \\colhead{III-28} & \n \\colhead{vZ~1000} & \n \\colhead{vZ~1127} & \n \\colhead{MB 4} & \n \\colhead{} & \n \\colhead{L~262} & \n \\colhead{L~324} & \n \\colhead{III-56} \\\\\n \\colhead{(\\AA)} & \n \\colhead{(eV)} & \n \\multicolumn{11}{c}{}\n}\n\\startdata\n% M~3 Giants M~13 Giants\n% ----------------------------------------------------- ----------------------------\n% lambda EP Log gf AA VZ297 VZ205 VZ1000 VZ1127 MB4 L262 L324 L414\n% (A) (eV) (III-28) (III-56)\n\\cutinhead{Na I}\n6154.23 & 2.10 & $-1.66$ & 33.5 & 20.7 & \\nodata & \\nodata & \\nodata & 34.1 & & 20.0 & 34.5 & 29.1 \\\\*\n6160.75 & 2.10 & $-1.32$ & 51.7 & 29.8 & \\nodata & 33.2 & 14.9 & 52.1 & & 38.6 & 53.1 & 48.9 \\\\*\n\\cutinhead{Mg I}\n5711.09 & 4.35 & $-1.58$ & 110.7 & 93.2 & 73.9 & 93.8 & 83.5 & 96.4 & & 91.5 & 87.2 & 100.4 \\\\*\n8717.83 & 5.93 & $-0.96$ & \\nodata & \\nodata & \\nodata & 18.8 & 15.7 & 21.5 & & 11.9 & 10.4 & 18.4 \\\\*\n8736.02 & 5.95 & $-0.04$ & \\nodata & 59.1 & 50.7 & 74.0 & 54.1 & 68.2 & & 51.6 & 45.5 & 52.7 \\\\*\n\\cutinhead{Al I}\n6696.03 & 3.14 & $-1.54$ & 69.0 & 52.4 & \\nodata & 51.1 & 13.2 & 58.8 & & 41.1 & 71.1 & 52.8 \\\\*\n6698.67 & 3.14 & $-1.91$ & 40.1 & 24.6 & \\nodata & 25.9 & 5.1 & 34.5 & & 18.2 & 36.9 & 23.7 \\\\*\n7835.31 & 4.02 & $-0.72$ & 36.0 & 28.3 & \\nodata & 26.3 & \\nodata & 31.3 & & 18.3 & 38.7 & 23.4 \\\\*\n7836.13 & 4.02 & $-0.63$ & 50.9 & 36.3 & \\nodata & 30.2 & \\nodata & 45.8 & & 24.2 & 47.3 & 32.7 \\\\*\n8772.86 & 4.02 & $-0.35$ & 63.1 & 60.1 & \\nodata & 59.8 & \\nodata & 64.0 & & 38.0 & 80.1 & 51.2 \\\\*\n8773.90 & 4.02 & $-0.15$ & 91.3 & 83.8 & \\nodata & 81.0 & \\nodata & 98.1 & & 62.1 & 98.8 & 62.2 \\\\*\n\\cutinhead{Si I}\n6142.48 & 5.62 & $-1.53$ & 10.2 & \\nodata & \\nodata & \\nodata & \\nodata & 6.1 & & 11.4 & 9.3 & 6.2 \\\\*\n6145.02 & 5.61 & $-1.41$ & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata & & \\nodata & 10.1 & 9.9 \\\\*\n6243.82 & 5.61 & $-1.32$ & 17.1 & 14.8 & 11.7 & 16.7 & 14.0 & 15.4 & & 15.3 & 11.9 & \\nodata \\\\*\n6721.85 & 5.86 & $-1.16$ & \\nodata & \\nodata & 9.0 & 14.0 & 12.3 & \\nodata & & 14.5 & 9.3 & 13.4 \\\\*\n7405.77 & 5.61 & $-0.57$ & \\nodata & 60.2 & 50.0 & 53.2 & 45.6 & 42.0 & & \\nodata & 55.0 & 41.7 \\\\*\n7415.95 & 5.68 & $-0.77$ & \\nodata & 44.7 & 31.5 & 43.6 & 33.4 & 28.6 & & 45.2 & 36.8 & 34.6 \\\\*\n8648.47 & 6.21 & $-0.10$ & 47.9 & \\nodata & \\nodata & \\nodata & \\nodata & \\nodata & & \\nodata & \\nodata & \\nodata \\\\*\n8742.45 & 5.87 & $-0.41$ & \\nodata & 41.6 & 45.3 & 45.0 & 38.0 & 42.0 & & \\nodata & 41.1 & 36.3 \\\\*\n8752.01 & 5.87 & $-0.42$ & \\nodata & 56.2 & 36.2 & 47.4 & 47.4 & 35.8 & & 54.1 & 47.2 & 37.9 \\\\*\n\\cutinhead{Ca I}\n6161.30 & 2.52 & $-1.22$ & 87.7 & 88.0 & 70.5 & 78.2 & 78.5 & 91.8 & & 69.9 & 78.4 & 85.8 \\\\*\n6166.44 & 2.52 & $-1.04$ & 95.5 & 90.1 & 64.6 & \\nodata & 74.6 & 96.2 & & 74.5 & 83.3 & 78.1 \\\\*\n6169.04 & 2.52 & $-0.67$ & 115.9 & 112.5 & \\nodata & 108.0 & 96.7 & 113.8 & & 96.3 & 115.7 & 113.1 \\\\*\n6439.08 & 2.53 & {\\phs}0.26 & 197.3 & 182.1 & 166.7 & 176.5 & 169.3 & 182.9 & & 166.7 & 196.8 & 185.8 \\\\*\n6455.60 & 2.52 & $-1.34$ & 78.4 & 71.6 & 52.5 & 62.9 & 61.2 & 74.7 & & 57.6 & 64.8 & 68.5 \\\\*\n6471.66 & 2.53 & $-0.67$ & 123.1 & 113.5 & 99.2 & 112.9 & 102.2 & 123.8 & & 108.3 & 131.0 & 120.3 \\\\*\n6493.78 & 2.52 & $-0.22$ & 162.3 & 153.6 & 128.6 & 145.9 & 132.0 & 159.7 & & 129.4 & 163.6 & 152.7 \\\\*\n6499.65 & 2.52 & $-0.79$ & 119.7 & 103.0 & 96.4 & 105.0 & 91.8 & 109.7 & & 89.3 & 107.8 & 104.4 \\\\*\n% M~3 Giants M~13 Giants\n% ----------------------------------------------------- ----------------------------\n% lambda EP Log gf AA VZ297 VZ205 VZ1000 VZ1127 MB4 L262 L324 L414\n% (A) (eV) (III-28) (III-56)\n\\cutinhead{Ti I}\n5866.45 & 1.07 & $-0.82$ & 154.8 & 137.2 & 110.1 & 115.4 & 113.9 & 144.5 & & 112.8 & 156.0 & 131.9 \\\\*\n6064.63 & 1.05 & $-1.99$ & 72.4 & 58.0 & 36.3 & 42.9 & 38.0 & \\nodata & & 40.0 & 65.3 & 59.1 \\\\*\n6126.22 & 1.07 & $-1.43$ & 113.9 & 95.4 & 75.3 & 77.6 & 84.8 & 108.9 & & 81.2 & 104.0 & 93.2 \\\\*\n6258.10 & 1.44 & $-0.37$ & 149.1 & 140.9 & 99.9 & 115.8 & 115.3 & 156.6 & & 109.9 & 162.7 & 136.4 \\\\*\n6261.10 & 1.43 & $-0.46$ & 159.1 & 138.4 & 96.9 & 119.6 & 114.4 & 147.4 & & 108.4 & 154.9 & 136.2 \\\\*\n6336.10 & 1.44 & $-1.80$ & 36.6 & 35.4 & 17.3 & 28.1 & 20.8 & 41.9 & & 20.7 & 31.5 & 22.9 \\\\*\n6497.68 & 1.44 & $-2.10$ & 29.5 & 13.0 & 13.5 & 10.8 & 14.0 & 28.6 & & 11.1 & 19.8 & 14.3 \\\\*\n6508.12 & 1.43 & $-2.18$ & 28.9 & 14.1 & 10.2 & 14.5 & 11.7 & 23.7 & & 12.5 & 20.6 & 14.8 \\\\*\n\\cutinhead{Ti II}\n6219.94 & 2.06 & $-3.14$ & \\nodata & \\nodata & 9.7 & 9.2 & \\nodata & 12.7 & & 14.3 & 12.7 & 16.7 \\\\*\n6491.56 & 2.06 & $-2.07$ & 75.1 & 63.5 & 63.0 & 65.9 & 66.4 & 61.7 & & 66.3 & 72.0 & 71.8 \\\\*\n\\cutinhead{V I}\n5670.85 & 1.08 & $-0.47$ & 100.8 & 78.2 & 50.6 & 67.9 & 49.2 & 85.1 & & 60.8 & 81.5 & 70.8 \\\\*\n5727.05 & 1.08 & {\\phs}0.02 & 142.4 & 120.3 & 80.2 & 105.3 & 99.9 & \\nodata & & 98.3 & 128.4 & 117.3 \\\\*\n5727.65 & 1.05 & $-0.92$ & 68.8 & 50.0 & 23.8 & 35.6 & 32.7 & 53.3 & & 35.6 & 47.2 & 49.5 \\\\*\n5737.06 & 1.06 & $-0.81$ & 69.6 & 54.9 & 25.7 & 39.5 & 32.2 & 67.7 & & 39.7 & 55.4 & 53.0 \\\\*\n6039.72 & 1.06 & $-0.73$ & 70.7 & 57.0 & 33.4 & 43.7 & 49.3 & 64.0 & & 45.0 & 61.0 & 60.3 \\\\*\n6058.14 & 1.04 & $-1.37$ & 24.6 & 20.1 & \\nodata & 12.4 & 12.8 & 26.3 & & 14.5 & 16.8 & 15.7 \\\\*\n6090.21 & 1.08 & $-0.13$ & \\nodata & 96.7 & 69.7 & 90.0 & 76.3 & 107.2 & & 82.1 & 103.0 & 96.4 \\\\*\n6111.65 & 1.04 & $-0.81$ & 74.5 & 59.9 & 37.4 & 50.7 & 35.6 & 58.4 & & 37.5 & 58.0 & 55.8 \\\\*\n6150.16 & 0.30 & $-1.56$ & 102.6 & 84.5 & 40.6 & 58.8 & 58.8 & 98.7 & & 59.8 & 90.1 & 77.7 \\\\*\n6199.20 & 0.29 & $-1.49$ & \\nodata & 112.3 & 57.1 & 88.6 & 76.3 & \\nodata & & 84.4 & 117.7 & 106.6 \\\\*\n6224.53 & 0.29 & $-1.89$ & 80.5 & 64.4 & 28.9 & 43.0 & 40.6 & 70.5 & & 50.2 & 68.7 & 60.6 \\\\*\n6233.16 & 0.28 & $-1.94$ & 69.2 & 59.4 & 21.5 & 41.7 & 28.4 & 69.7 & & 33.6 & 57.2 & 52.2 \\\\*\n6251.83 & 0.29 & $-1.40$ & 127.6 & 101.6 & 57.4 & 80.4 & 76.0 & 119.9 & & 80.6 & 108.0 & 96.3 \\\\*\n6266.31 & 0.28 & $-2.27$ & 55.0 & 43.2 & 18.7 & 28.7 & 24.1 & 54.8 & & 29.8 & 46.9 & 37.8 \\\\*\n6274.65 & 0.27 & $-1.76$ & 95.6 & 79.0 & 40.9 & 53.1 & 44.5 & 83.4 & & 51.9 & 82.8 & 74.2 \\\\*\n6285.15 & 0.28 & $-1.63$ & 102.5 & 89.7 & 51.7 & 63.9 & 55.6 & 98.2 & & 70.8 & 91.6 & \\nodata \\\\*\n6292.83 & 0.29 & $-1.61$ & 103.4 & 83.3 & 44.3 & 74.1 & 60.9 & 91.3 & & 64.9 & 88.3 & 85.0 \\\\*\n6531.42 & 1.22 & $-0.97$ & 44.7 & 33.0 & 17.2 & 25.3 & 20.3 & 41.8 & & 23.8 & 32.4 & 32.0 \\\\*\n\\cutinhead{Mn I}\n% M~3 Giants M~13 Giants\n% ----------------------------------------------------- ----------------------------\n% lambda EP Log gf AA VZ297 VZ205 VZ1000 VZ1127 MB4 L262 L324 L414\n% (A) (eV) (III-28) (III-56)\n6013.51 & 3.07 & $-0.34$ & 96.9 & 85.5 & 53.7 & 74.5 & 75.1 & 92.5 & & 62.9 & 78.3 & 75.4 \\\\*\n6016.67 & 3.07 & $-0.24$ & 94.7 & 101.9 & 63.5 & 73.5 & 74.2 & 88.3 & & 77.1 & 85.7 & 84.3 \\\\*\n\\cutinhead{Fe I}\n6096.67 & 3.98 & $-1.93$ & 35.8 & 31.6 & 25.2 & 29.3 & 30.3 & 33.6 & & 23.3 & 27.8 & 22.3 \\\\*\n6127.91 & 4.14 & $-1.40$ & 42.1 & 42.7 & 40.6 & 34.3 & 45.6 & 45.0 & & 34.5 & 37.4 & 32.7 \\\\*\n6151.62 & 2.18 & $-3.30$ & 110.5 & 104.9 & 82.9 & 95.9 & 89.3 & 102.8 & & 89.8 & 109.4 & 102.4 \\\\*\n6157.73 & 4.07 & $-1.26$ & \\nodata & 70.6 & 44.0 & 64.2 & 58.5 & \\nodata & & 57.3 & 70.0 & \\nodata \\\\*\n6165.36 & 4.14 & $-1.47$ & 42.4 & 35.8 & \\nodata & 43.1 & 29.5 & 42.5 & & 31.9 & 33.7 & 36.7 \\\\*\n6187.99 & 3.94 & $-1.72$ & 50.2 & 45.6 & 34.0 & 39.6 & 32.4 & 39.1 & & 38.3 & 43.7 & 40.6 \\\\*\n6200.31 & 2.61 & $-2.44$ & \\nodata & 125.6 & 108.6 & 111.5 & 112.4 & 118.4 & & 106.1 & 131.9 & 113.8 \\\\*\n6213.43 & 2.22 & $-2.48$ & 157.3 & 147.5 & 127.2 & 139.1 & 132.8 & 152.3 & & 132.7 & 160.4 & 160.5 \\\\*\n6219.28 & 2.20 & $-2.43$ & 164.7 & 149.0 & 136.0 & 147.2 & 137.0 & 161.2 & & 136.3 & 171.5 & 158.1 \\\\*\n6226.74 & 3.88 & $-2.22$ & 17.7 & 23.3 & 13.4 & 22.1 & \\nodata & 21.7 & & 19.6 & 21.1 & 16.7 \\\\*\n6229.23 & 2.84 & $-2.97$ & 69.6 & 58.0 & 56.3 & 57.0 & 50.0 & 55.9 & & 53.7 & 60.8 & 56.4 \\\\*\n6246.32 & 3.60 & $-0.88$ & 133.5 & 118.9 & 115.6 & 111.6 & 124.5 & 123.2 & & 114.3 & 126.5 & 124.0 \\\\*\n6270.23 & 2.86 & $-2.61$ & 89.1 & 88.4 & 71.6 & 89.0 & 82.7 & 90.4 & & 75.8 & 87.5 & 87.9 \\\\*\n6297.80 & 2.22 & $-2.74$ & 152.3 & 129.2 & 116.2 & 117.6 & 120.2 & 131.5 & & 121.5 & 156.4 & 142.2 \\\\*\n6301.50 & 3.65 & $-0.75$ & 130.8 & 126.0 & 117.6 & 125.3 & 126.4 & 124.6 & & 112.2 & \\nodata & 123.9 \\\\*\n6311.51 & 2.83 & $-3.23$ & \\nodata & 51.6 & 37.5 & 50.6 & 44.1 & 52.7 & & 39.7 & 50.6 & 34.9 \\\\*\n6355.04 & 2.84 & $-2.29$ & 121.2 & 116.6 & 91.4 & 112.2 & 98.5 & 117.8 & & 98.8 & 122.9 & 105.1 \\\\*\n6380.75 & 4.19 & $-1.38$ & 51.9 & 48.3 & 32.0 & 42.8 & 35.3 & 48.1 & & 42.5 & 47.2 & 38.8 \\\\*\n6392.54 & 2.23 & $-4.03$ & 43.2 & 48.2 & 34.5 & 38.0 & 33.6 & 48.6 & & 37.2 & 50.8 & 44.5 \\\\*\n6393.60 & 2.43 & $-1.58$ & 196.4 & 191.5 & 166.8 & 182.0 & 171.3 & 206.3 & & 187.0 & 222.5 & 183.0 \\\\*\n6421.35 & 2.28 & $-2.03$ & 194.2 & 182.4 & 162.3 & 174.8 & 162.9 & 182.8 & & 160.8 & 198.3 & 173.4 \\\\*\n6430.85 & 2.18 & $-2.01$ & \\nodata & 197.3 & 163.3 & 171.2 & 167.6 & 192.9 & & 168.2 & 219.5 & 186.6 \\\\*\n6495.74 & 4.84 & $-0.94$ & 16.7 & 17.8 & 16.5 & 16.0 & 20.1 & 20.5 & & 13.3 & 15.8 & 15.3 \\\\*\n6498.94 & 0.96 & $-4.70$ & 156.8 & 144.4 & 119.4 & 122.8 & 123.9 & \\nodata & & 122.9 & 163.3 & 136.9 \\\\*\n% M~3 Giants M~13 Giants\n% ----------------------------------------------------- ----------------------------\n% lambda EP Log gf AA VZ297 VZ205 VZ1000 VZ1127 MB4 L262 L324 L414\n% (A) (eV) (III-28) (III-56)\n\\cutinhead{Fe II}\n5991.38 & 3.15 & $-3.56$ & 28.9 & 31.3 & 29.5 & 30.6 & 25.0 & 27.7 & & 21.2 & 31.2 & 23.4 \\\\*\n6149.25 & 3.89 & $-2.72$ & 22.0 & 23.3 & 21.0 & 21.3 & 21.6 & 20.4 & & 19.7 & 23.6 & 21.6 \\\\*\n6247.56 & 3.89 & $-2.33$ & 34.6 & 37.0 & 35.2 & \\nodata & \\nodata & 35.6 & & 40.0 & 40.7 & 35.3 \\\\*\n6369.46 & 2.89 & $-4.25$ & 12.1 & 16.0 & 13.2 & 17.3 & 15.8 & \\nodata & & 13.2 & 14.3 & \\nodata \\\\*\n6416.93 & 3.89 & $-2.74$ & 21.9 & 26.9 & 18.2 & 24.6 & 24.0 & 20.3 & & 18.5 & 25.1 & 18.4 \\\\*\n6432.68 & 2.89 & $-3.71$ & \\nodata & 35.8 & 35.1 & 33.4 & 31.7 & 29.0 & & 30.1 & 33.6 & 33.1 \\\\*\n6456.39 & 3.90 & $-2.08$ & 47.3 & 53.0 & 49.1 & 58.0 & 49.3 & 48.1 & & 51.9 & 57.8 & 44.6 \\\\*\n6516.08 & 2.89 & $-3.45$ & 55.4 & 56.5 & 45.7 & 48.2 & 45.3 & 48.4 & & 49.8 & 55.5 & 45.5 \\\\*\n\\\\\n\\\\\n\\cutinhead{Ni I}\n6086.28 & 4.27 & $-0.39$ & \\nodata & 18.9 & 16.8 & 36.3 & 15.7 & 17.1 & & 18.7 & 19.1 & 17.1 \\\\*\n6108.11 & 1.68 & $-2.44$ & 137.0 & 115.8 & 90.7 & 106.7 & 102.2 & 124.0 & & 107.7 & 133.0 & 120.5 \\\\*\n6111.07 & 4.09 & $-0.77$ & 13.9 & 14.1 & \\nodata & 23.0 & 18.0 & \\nodata & & 10.3 & 9.7 & 14.8 \\\\*\n6175.36 & 4.09 & $-0.45$ & 29.3 & 33.4 & 17.6 & 21.8 & 26.7 & \\nodata & & 25.6 & 25.8 & 22.1 \\\\*\n6176.81 & 4.09 & $-0.13$ & 43.8 & 42.7 & 35.0 & 33.4 & 34.4 & 50.5 & & 37.2 & 38.0 & 35.8 \\\\*\n6586.31 & 1.95 & $-2.73$ & 74.5 & 78.5 & 63.1 & 70.3 & 70.5 & 79.6 & & 74.4 & 92.9 & 83.7 \\\\*\n6643.63 & 1.68 & $-1.85$ & 169.9 & 157.5 & 138.7 & 155.9 & 145.9 & 169.1 & & 143.5 & 179.0 & 160.6 \\\\*\n6772.31 & 3.66 & $-0.84$ & 45.1 & 36.2 & 30.9 & 38.0 & 33.2 & 40.8 & & 39.2 & 42.0 & 43.0 \\\\*\n7525.11 & 3.64 & $-0.53$ & 65.7 & 67.1 & 57.6 & 63.5 & 64.6 & 72.7 & & 71.2 & 73.0 & 70.8 \\\\*\n7555.60 & 3.85 & {\\phs}0.09 & \\nodata & 85.2 & 72.6 & 77.9 & 72.0 & 81.3 & & 75.7 & 95.3 & 82.5 \\\\*\n7788.94 & 1.95 & $-1.70$ & 159.0 & 152.3 & 127.8 & 134.9 & 128.9 & 146.0 & & 141.0 & 165.4 & 150.1 \\\\*\n7797.59 & 3.90 & $-0.14$ & 74.6 & 70.4 & \\nodata & 53.6 & 59.7 & 59.1 & & 63.9 & 79.4 & 65.1 \\\\*\n\\enddata\n\\end{deluxetable}\n\n\\end{document}\n"
}
] |
[
{
"name": "astro-ph0002511.extracted_bib",
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D. 1993, \\aj, 106, 1490\n\\bibitem[Kraft et al.(1995)]{KSLSB95} Kraft, R. P., Sneden, C., Langer, G. E.,\n Shetrone, M. D. \\& Bolte, M. 1995, \\aj, 109, 2586\n\\bibitem[Kraft et al.(1999)]{Kraft99} Kraft, R. P., Peterson, R. C.,\n Guhathakurta, P., Sneden, C., Fulbright, J. P. \\& Langer, G. E.\n 1999 \\apjl, 518, L53\n\\bibitem[Kraft et al.(1997)]{Kraft97} Kraft, R. P., Sneden, C., Smith, G. H.,\n Shetrone, M. D., Langer, G. E. \\& Pilachowski, C. A. 1997, \\aj, 113,\n 279\n\\bibitem[Langer et al.(1993)]{LHS93} Langer, G. E., Hoffman, R. \\& Sneden, C.\n 1993, \\pasp, 105, 301\n\\bibitem[Langer et al.(1997)]{LHZ97} Langer, G. E., Hoffman, R. D., \\& Zaidins,\n C. S. 1997, \\pasp, 109, 244\n\\bibitem[Langer et al.(1986)]{LKCF86} Langer, G. E., Kraft, R. P., Carbon, D.\n F. \\ Friel, E. 1986, \\pasp, 98, 473\n\\bibitem[Lattanzio(1999)]{Lattanzio99} Lattanzio, J. C. 1999, in Proceedings\n of Nuclei in the Cosmos V, eds. N. Orantzos and S. Harissopulos,\n (Editions Frontieres), p~163\n\\bibitem[Lattanzio et al.(1999)]{LFC99} Lattanzio, J., Forestini, M. \\&\n Charbonnel, C. 1999, to appear in The Changes in Abundances in \n Asymptotic Giant Branch Stars (astro-ph/9912298)\n\\bibitem[Lehnert et al.(1991)]{LBC91} Lehnert, M. D., Bell, R. A., \\& Cohen,\n J. G. 1991, \\apj, 367, 514\n\\bibitem[Ludendorf(1905)]{L05} Ludendorf 1905, Publ. Astrophys. Obs. Potsdam,\n 15, No. 50\n\\bibitem[Moore et al.(1966)]{MMH66} Moore, C. E., Minnaert, M. G. J. \\&\n Houtgast, J. 1966, The Solar Spectrum 2935{\\AA} to 8770{\\AA},\n 2nd Ed., (Washington, D.C.: National Bureau of Standards)\n\\bibitem[Nave et al.(1994)]{Nav94} Nave, G., Johansson, S., Learner, R. C. M.,\n Thorne, A. P. \\& Brault, J. W. 1994, \\apjs, 94, 221\n\\bibitem[Peterson(1983)]{Peterson83} Peterson, R. C. 1983, \\apj, 275, 737\n\\bibitem[Pilachowski et al.(2000)]{PSKHW2000} Pilachowski, C. A., Sneden, C.,\n Kraft, R. P., Harmer, D. \\& Willmarth, D. 2000, \\aj, in press\n\\bibitem[Pilachowski et al.(1996)]{PSKL96} Pilachowski, C. A., Sneden, C.,\n Kraft, R. P., \\& Langer, G. E. 1996, \\aj, 112, 545\n\\bibitem[Piotto et al.(1997)]{PIOTTO97} Piotto, G. et al. 1997, in Advances\n in Stellar Evolution, ed. R. T. Rood and A. Renzini (Cambridge:\n Cambridge Univ. Press), p~84\n\\bibitem[Powell et al.(1999)]{Powell99} Powell, D. C., Iliadis, C.,\n Champagne, A. E., Grossman, C. A., Hale, S. E., Hansper, V. Y.\n \\& McLean, L. K. 1999, \\nphysa, 660, 349\n\\bibitem[Reimers(1975)]{Reimers75} Reimers, D. 1975, in M\\'{e}moires de la\n Societ\\'{e} Royale des Sciences de Li\\`{e}ge, 6e serie, tome VIII,\n Probl\\`{e}mes D'Hydrodynamique Stellaire, p.~369\n\\bibitem[Rich et al.(1997)]{RICH97} Rich, R. M. et al. 1997, \\apjl, 484, L25\n\\bibitem[Sandage(1953)]{San53} Sandage, A. R. 1953, \\aj, 58, 61\n\\bibitem[Sandage \\& Katem(1982)]{SK82} Sandage, A. \\& Katem, B. 1982, \\aj, 87,\n 537\n\\bibitem[Sarajedini et al.(1997)]{SCD97} Sarajedini, A., Chaboyer, B. \\&\n Demarque, P. 1997, \\pasp, 109, 1321\n\\bibitem[Shetrone(1996a)]{S96a} Shetrone, M. D. 1996a, \\aj, 112, 1517\n\\bibitem[Shetrone(1996b)]{S96b} Shetrone, M. D. 1996b, \\aj, 112, 2639\n\\bibitem[Shetrone(1997)]{S97} Shetrone, M. D. 1997, in Poster Proc of IAU\n Symp. 189: The Interaction between Observations and Theory, ed.\n T. R. Bedding (Univ. of Sydney: Sydney), p.~158\n\\bibitem[Sneden(1973)]{MOOG} Sneden, C. 1973, \\apj, 184, 839\n\\bibitem[Sneden et al.(1991)]{SKPL91} Sneden, C., Kraft, R. P., Prosser, C. F.\n \\& Langer, G. E. 1991, \\aj, 102, 2001\n\\bibitem[Suntzeff(1981)]{NBS81} Suntzeff, N. B. 1981, \\apjs, 41, 1\n\\bibitem[Suntzeff \\& Smith(1991)]{SS91} Suntzeff, N. B. \\& Smith, V. V. 1991, \n \\apj, 381, 160\n\\bibitem[Sweigart(1997a)]{AVS97a} Sweigart, A. V. 1997a, \\apjl, 474, L23\n\\bibitem[Sweigart(1997b)]{AVS97b} Sweigart, A. V. 1997b, in The Third Conference\n on Faint Blue Stars, eds. A. G. D. Philips, J. W. Liebert and\n R. A. Saffer (L. Davis: Schenectady), p.~3\n\\bibitem[Sweigart \\& Catelan(1998)]{SC98} Sweigart, A. V. \\& Catelan, M. 1998,\n \\apjl, 501, L63\n\\bibitem[Sweigart \\& Mengel(1979)]{SM79} Sweigart, A. V. \\& Mengel, J. G.\n 1979, \\apj, 229, 624\n\\bibitem[Taylor(1982)]{Taylor} Taylor, J. R. 1982, An Introduction to Error\n Analysis: The Study of Uncertainties in Physical Measurements, ed.\n E. D. Commins (University Science Books: USA)\n\\bibitem[Th\\'{e}venin(1989)]{Thev89} Th\\'{e}venin, F. 1989, \\aaps, 77, 137\n\\bibitem[Th\\'{e}venin(1990)]{Thev90} Th\\'{e}venin, F. 1990, \\aaps, 82, 179\n\\bibitem[Tody(1986)]{IRAF} Tody, D. 1986, \"The IRAF Data Reduction and \n Analysis System\" in Proc. SPIE Instrumentation in Astronomy VI, ed.\n D.L. Crawford, 627, 733\n\\bibitem[Trefzger et al.(1983)]{TCLSK83} Trefzger, C. F., Carbon, D. F., \n Langer, G. E., Suntzeff, N. B. \\& Kraft, R. P. 1983, \\apj, 266, 144\n\\bibitem[VandenBerg(1999)]{vdB99} VandenBerg, D. A. 1999, in The Third\n Stromlo Symposium: The Galactic Halo, eds. B. K. Gibson, T. S.\n Axelrod and M. E. Putnam (San Francisco :ASP) p.~46\n\\bibitem[Von Rudloff et al.(1988)]{VVH88} Von Rudloff, I. R., VandenBerg, D.\n A. \\& Hartwick, F. D. A. 1988, \\apj, 324, 840\n\\bibitem[von Zeipel(1908)]{vZ08} von Zeipel, M. H. 1908, Ann. Obs. Paris, 25,\n F1\n\\bibitem[Wallace et al.(1993)]{WHL93} Wallace, L., Hinkle, K. \\& Livingston,\n W. 1993, N. S. ). Technical Report \\#93-001\n\\bibitem[Wallace et al.(1998)]{WHL98} Wallace, L., Hinkle, K. \\& Livingston,\n W. 1998, N. S. ). Technical Report \\#98-001\n\\bibitem[Wallerstein et al.(1987)]{WLO87} Wallerstein, G., Leep, E. M., \\&\n Oke, J. B. 1987, \\aj, 93, 1137\n\\bibitem[Zaidins \\& Langer(1997)]{ZL97} Zaidins, C. S. \\& Langer, G. E. 1997,\n \\pasp, 109, 252\n\\bibitem[Zucker et al.(1996)]{ZWB96} Zucker, D., Wallerstein, G., \\& Brown,\n J. A. 1996, \\pasp, 108, 911\n\\end{small}\n\\end{thebibliography}"
}
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astro-ph0002512
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The Detection of Multimodal Oscillations on $\alpha$ UMa
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[
{
"author": "D. Buzasi"
}
] |
We have used the star camera on the WIRE satellite to observe the K0~III star $\alpha$ UMa, and we report the apparent detection of 10 oscillation modes. The lowest frequency mode is at $1.82~\mu Hz$, and appears to be the fundamental mode. The mean spacing between the mode frequencies is $2.94~\mu Hz$, which implies that all detected modes are radial. The mode frequencies are consistent with the physical parameters of a K0~III star, if we assume that only radial modes are excited. Mode amplitudes are $100 - 400~\mu mag$, which is consistent with the scaling relation of Kjeldsen \& Bedding \markcite{kandb95}(1995).
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"name": "lapp.tex",
"string": "\n%\\documentstyle[12pt,aasms4]{article}\n\\documentstyle[11pt,aaspp4]{article}\n%\\documentstyle[aas2pp4]{article}\n\n\\received{}\n\\accepted{}\n%\\journalid{}{}\n%\\articleid{}{}\n\n%\\comment{Submitted to Astrophysical Journal Letters}\n\n\\lefthead{Buzasi et al.}\n\\righthead{Seismology of $\\alpha$ UMa}\n\n\\begin{document}\n\n\\title{The Detection of Multimodal Oscillations on $\\alpha$ UMa}\n\n\\author{D. Buzasi}\n\\affil{Space Sciences Laboratory, University of California, \nBerkeley, CA 94720}\n\n\\author{J. Catanzarite, R. Laher, T. Conrow, D. Shupe}\n\\affil{Infrared Processing and Analysis Center, California Institute of\nTechnology, MS 100-22,\nPasadena, CA 91125}\n\n\\author{T. N. Gautier III}\n\\affil{Jet Propulsion Laboratory, California Institute of Technology, MS 100-22,\nPasadena, CA 91125}\n\n\\author{T. Kreidl}\n\\affil{Information Technology Services and \nDepartment of Physics and Astronomy, Box 5100, Northern Arizona University, \nFlagstaff, AZ 86011}\n\n\\and\n\n\\author{D. Everett}\n\\affil{NASA/Goddard Space Flight Center, Greenbelt, MD 20771}\n\n% Notice that each of these authors has alternate affiliations, which\n% are identified by the \\altaffilmark after each name. The actual alternate\n% affiliation information is typeset in footnotes at the bottom of the\n% first page, and the text itself is specified in \\altaffiltext commands.\n% There is a separate \\altaffiltext for each alternate affiliation\n% indicated above.\n\n% The abstract environment prints out the receipt and acceptance dates\n% if they are relevant for the journal style. For the aasms style, they\n% will print out as horizontal rules for the editorial staff to type\n% on, so long as the author does not include \\received and \\accepted\n% commands. This should not be done, since \\received and \\accepted dates\n% are not known to the author.\n\n\\begin{abstract}\nWe have used the star camera on the WIRE satellite to observe the K0~III\nstar $\\alpha$ UMa, and we report the apparent detection of 10 oscillation modes. The\nlowest frequency mode is at $1.82~\\mu \\rm Hz$, and appears to be the fundamental\nmode. The mean spacing between the mode frequencies is $2.94~\\mu \\rm Hz$,\nwhich implies that all detected modes are radial. The mode frequencies are\nconsistent with the physical parameters of a K0~III star, if we assume that\nonly radial modes are excited. Mode amplitudes are \n$100 - 400~\\mu \\rm mag$, which is consistent with the scaling relation of\nKjeldsen \\& Bedding \\markcite{kandb95}(1995).\n\\end{abstract}\n\n\\keywords{star --- oscillations: stars --- individual}\n\n\\section{Introduction}\n\nOver the past few decades, our understanding of the interior of the Sun -- its\nthermodynamic structure, internal rotation, and dynamics -- has been revolutionized\nby the technique of helioseismology, the study of the frequencies and amplitudes\nof seismic waves that penetrate deep into the solar interior (Leibacher et al. \n\\markcite{leibacher85}1985;\nDuvall et al. \\markcite{duvall88}1988; \nSchou et al. \\markcite{schou98}1998). The high quality of the modeling \nin the solar case is made possible by the large number ($\\approx 10^7$) of\nmodes visible in the Sun. Unfortunately, the lack of spatial resolution \ninherent in stellar observations limits the number of detectable modes in stars\nto only a few (those with low degree $l$). Nonetheless, successful detection of\neven a few modes has the potential to provide greatly improved values for\nfundamental stellar parameters such as mass, abundance, and \nage (Gough \\markcite{gough87}1987; Brown et al. \\markcite{brown94}1994)\n\nWhile oscillations have been\nsuccessfully detected on roAp stars, $\\delta$\nScuti stars, and white dwarfs, these stars all show oscillation amplitudes\nseveral orders of magnitude larger than is expected from solar-like stars.\nMore recently, however, a number of authors \n(Hatzes \\& Cochran \\markcite{hatzes94a}1994a, \\markcite{hatzes94b}1994b;\nEdmonds \\& Gilliland \\markcite{edmonds96}1996) have reported the\ndetection of periodic variability at levels of a few hundred $\\rm m~s^{-1}$\nand/or several millimagnitudes in a number of K giants. However, each\ndetection is of only a single mode,\nand multimodal oscillations have yet to be unambiguously detected in any \ncool star other than the Sun.\n\nIn March 1999, NASA launched the Wide-Field Infrared Explorer satellite, with\nthe intent of carrying out an infrared sky survey to better understand \ngalaxy evolution.\nUnfortunately, within days of launch the primary science instrument on WIRE failed\ndue to loss of coolant. However, the satellite itself continues to function\nnearly perfectly, and in May we began a program of asteroseismology using the \nWIRE's onboard 52 mm aperture star camera. Below we report WIRE's probable detection\nof multimodal oscillations on a cool star, which is the first of its kind.\n\n\\section{Instrument Description and Data Reduction Technique}\n\nThe WIRE satellite star camera, a Ball Aerospace model CT-601, \nconsists of a $512 \\times 512$ SITe CCD with \n$27 \\mu$m pixels (1 arc minute on the sky) and gain of $15 \\rm~e^-/ADU$, \nfed by a 52 mm, f/1.75 refractive optical system. The read noise of the system\nis 30 electrons. The pixel\ndata is digitized with a 16 bit ADC, and up to five $8 \\times 8$ pixel fields\ncan be digitized and transmitted to the ground. For this work, we used only\none field, which permitted us to read out the CCD at a rate of 10 Hz. The\nstellar image is somewhat defocused, but essentially all of the light falls\non the central $2 \\times 2$ pixel spot. The spectral response of the system is \ngoverned entirely by the response of the CCD plus the optical system, and\nis approximately equivalent to the V+R bandpass.\n\nThe WIRE satellite is in a sun-synchronous orbit which, when combined with\nconstraints imposed by the solar panels, limits pointing to two strips, each approximately\n$\\pm 30^\\circ$ wide,\nlocated perpendicular to the Earth-Sun line.\nIn addition, continuous observing is not possible. Early in the program, we\nwere able to obtain only 7 or 8 minutes worth of data during every 96-minute\norbit. Later, after viewing constraints were relaxed (which involved\nscheduling software and onboard data-table modifications), observing efficiency\nrose to as much as 40 minutes per target per orbit -- up to two targets\nare possible during any orbit. Thus, with integrations\nevery 0.1 s, we acquired continuous data segments of up to 24,000 observations.\n\nBias correction was performed on board the satellite, and further \ndata reduction was accomplished using software developed at IPAC.\nEach $8 \\times 8$ pixel \nfield was extracted by summing the flux in the central $4 \\times 4$ pixel\nregion. Although scattered light in the field is limited by the one-meter\nsun-shield mounted on the star camera, we performed a background\nsubtraction using the flux from a 20-pixel octagonal annulus\nsurrounding the central region\nof the image. Finally, we converted our fluxes to an instrumental magnitude.\nAfter removal of thermal effects (see below), the rms noise in the final \nreduced time series was typically comparable to\nthe 1.8 mmag noise expected from pure photon statistics, although non-Poisson\nnoise is certainly present as well.\nThe lack of a good\nflat field for the instrument was a concern, which we dealt with by\nrejecting those frames in which the mean centroid of the stellar image lay\nmore than $4\\sigma$ from the mean position, where $\\sigma$ is the mean \nstandard deviation of the image centroid. This criterion applied to approximately\n3.8\\% of the observations, and the vast majority of these observations were at the\nstart or end of an orbital segment.\nOverall, the satellite displayed excellent attitude stability during our run, \nwith $\\sigma$\nmeasured to be typically 0.7 arc sec or less (Laher et al. 2000).\n\n\\section{Observations and Data Analysis}\n\n$\\alpha$ UMa was the primary target for WIRE from 18 May through 23 June 1999.\nIt is a K0 III star (Taylor \\markcite{taylor99}1999)\nwith an angular diameter of\n$6.79 \\rm~mas$ (Hall \\markcite{hall96}1996; \nBell \\markcite{bell93}1993). At the Hipparcos distance of 38 pc,\nthis corresponds to a stellar radius of $28 \\rm~R_{\\sun}$, in substantial\nagreement with the earlier value of $25 \\rm~R_{\\sun}$ derived by Bell (1993).\nThe effective temperature of the star is variously reported as \n$T_e = 3970 \\rm~K$ to $T_e = 4660 \\rm~K$ (Cayrel de Stroebel \\markcite{cayrel92}\n1992), with the\nlatter value being the most recent (Taylor \\markcite{taylor99}1999).\n\n$\\alpha$ UMa is a member of a binary system, with a total system mass of\n$5.94 \\rm~M_{\\sun}$ (Soderhjelm \\markcite{soderhjelm99}1999). \nThe spectral type of $\\alpha$ UMa B\nis somewhat unclear, with F7~V most often cited in the literature (see, e.g.,\nthe SAO catalog). However, on the basis of IUE observations, Kondo, Morgan,\n\\& Modisette \\markcite{kondo77}(1977) estimate the \nsecondary to be ``late A'' (see also\nAyres, Marstad, \\& Linsky \\markcite{ayres81}1981). \nIn either case, however, the secondary makes\nonly a small contribution ($\\approx 6\\%$) to the total system luminosity, and\nshould show oscillation frequencies quite different from those of the primary.\n\nDuring the observation period, WIRE made a\ntotal of $4,036,448$ observations of $\\alpha$ UMa, after removal of those observations with\npoor pointing characteristics.\nThe data were first phased at the period of the spacecraft, to determine if\nany obvious instrumental periodicities existed. This phasing showed the existence\nof a strong sinusoidal variation, with amplitude $\\approx 8$ mmag. A thermistor\nis mounted on the star camera, and examination of data from this thermistor \nshowed that these variations were in fact correlated with temperature variations\nof a few tenths of a degree. Although a thermoelectric cooler (TEC) is mounted\non the CCD, the star tracker thermal design lowered the CCD temperature below the\ndefault setpoint, so the TEC never actually turned on.\nIn order to reduce the impact of this signal\non our data analysis, we prewhitened the data by fitting and subtracting from the\nentire time series a sinusoid constrained\nto have the satellite orbital period (the phase was allowed to vary). The best-fit\nsinusoid has an amplitude of 8.63 mmag, and its subtraction results in an rms residual\nof 2.05 mmag, compared to the 1.84 mmag expected from photon statistics.\nWe also explored other means of fitting and removing the thermal signature,\nincluding high-order polynomial fits to the phased data. Such approaches, \nwhile more complex than sine fitting, did not lead to any appreciable\nimprovement in the fit, and were therefore discarded. The use of\nany of these thermal fitting procedures did not affect the peaks in the amplitude\nspectrum described below. In fact, these peaks were visible even before \nthe application of any fit to the thermal variation, though removal of the \nthermal signature, by removing the largest single peak, did help to render them more easily visible.\n\nData from the first portion of the run (18 May - 7 June) were all in short\n($t < 8 \\rm~minutes$) segments, which obviously extended over a short range of\norbital phase ($\\approx 0.08$). We were concerned that including these data in our\nsine fit would bias the result, so we decided to exclude them from our analysis. In addition, \nNASA responded to our expressed concern about the TEC by lowering the set\npoint on 18 June, and the CCD behavior subsequent to this had not yet \nreached an equilibrium point before the end of our run. Thus, we excluded\ndata taken after 18 June from the analysis as well, leaving only the 8 June\n-- 18 June window of approximately $9.2 \\times 10^5 \\rm~s$. Shortening the\ntime series clearly adversely affects our frequency resolution, but we felt\nthat confidence in the data quality was the overriding issue. The data that\nwere subjected to analysis are shown in Figure 1, after removal of the \nthermal variation. \n\nData were searched for periodicities using Discrete Fourier Transform (DFT;\nFoster \\markcite{foster96}1996),\nLomb-Scargle periodogram (Scargle \\markcite{scargle82}1982; \nHorne \\& Baliunas \\markcite{horne86}1986), \nand epoch-folding techniques (Davies \\markcite{davies90}1990)\nwhich are essentially equivalent to phase dispersion modulation\n(PDM; see Stellingwerf \\markcite{stellingwerf78}1978; \nSchwarzenberg-Czerny \\markcite{schwarzenberg98}1998). \nThe Scargle periodogram analysis was conducted as described in Scargle (1982;\nsee also Hatzes \\& Cochrane 1994ab); the data were windowed using a Parzen function\nand the resulting periodogram was oversampled\nby a factor of 8 in frequency space. The DFT analysis was similarly oversampled,\nand implemented the CLEAN algorithm of Roberts, \nLehar, \\& Dreher \\markcite{roberts87}(1987) to\nremove alias peaks. When using the DFT, the data were windowed using a \nParzen function, and 100\niterations of CLEAN were performed. Due to its relative slowness, epoch-folding\nanalysis was conducted only within the frequency range of interest, as \nidentified by the Scargle and DFT analysis, and was used only to aid\nin interpretation of the\nresults from the other two algorithms.\nIn the discussion that follows, we will concentrate on the\nLomb-Scargle periodogram results, though, in general, the three techniques\ngave similar results. \n\nFigure 2 shows the window function for the time series. The upper frequency limit\nfor the figure has been arbitrarily set at $5~\\rm mHz$ to enhance the visibility of the\namplitude spectrum at frequencies near 1 mHz, while the lowest frequencies are shown\nin the inset. No significant features are present in the window function above\n5 mHz. The large evenly spaced peaks correspond to the satellite orbital\nfrequency and its aliases. \n\nThe Lomb-Scargle periodogram\nfor the time series is shown in Figure 3 on the same frequency scale as\nthe window function. \nThe low-frequency inset shows ten significant peaks, \nand the frequencies and amplitudes of these peaks are given in \nTable~1 (derived from Lorentzian fits),\nalong with conservative formal error estimates derived from the half-width \nof the \nperiodogram peaks. We note that periodograms of portions\nof the time series (halves and thirds)\ngive results similar to that of the whole, with decreased frequency resolution,\nand that sine fits to these frequencies show coherent phasing in these different\nportions.\n\n\\placetable{table-1}\n\nUnfortunately, the implementation of the on-board data collection on WIRE means that we lack \nsimultaneous\nobservations of a comparison star, and are thus essentially performing absolute photometry\nwith an instrument not designed for that purpose. However, we have \nobserved stars\nother than $\\alpha$ UMa, and the sun-synchronous orbit of WIRE implies that most\ninstrumental effects should be similar for all sources.\nThe dashed line in the Figure~3 inset shows the periodogram from a time series\nof $\\alpha \\rm~Leo$, a B7~V star not expected to show significant low-frequency\noscillations. The $\\alpha$ Leo data set, which was obtained from 23 May through\n3 June 1999, consists of segments similar in length to those of $\\alpha$ UMa, has similar\nrms noise to the $\\alpha$ UMa time series, and was reduced in exactly the\nsame manner. Our object here is not to perform analysis of the $\\alpha$\nLeo data (which might well benefit from a different approach than we have\nused for $\\alpha$ UMa), but rather to show that the particular low frequency\npeaks in the periodogram of $\\alpha$ UMa do not arise from either\ninstrumental effects or the data reduction procedure itself. The larger\npeaks in the $\\alpha$ Leo periodogram may arise from an imperfect\nremoval of both long-term and orbital variations; none show coherent\nphasing across different segments of the time series.\nThe dissimilarity of the two periodograms increases our confidence\nthat the peaks\nvisible in the $\\alpha$ UMa periodogram are due to the star itself,\nalthough it is of course possible that the instrumental \nbehavior changes significantly for different targets.\n\nAs is apparent from Figures 2 and 3, the family of peaks\nvisible at low frequencies is repeated at higher frequencies, which leads to the difficulty\nof determining which set of peaks is the correct one. We can easily eliminate peaks above\n$\\approx 200 ~\\mu \\rm Hz$ by examining the summed power spectrum of the individual orbital\nsegments, which shows no significant power at these frequencies. We are therefore left with\nthe problem of selecting between the set of peaks in the $\\approx 2 - 50 ~\\mu \\rm Hz$ range\nand the similar set around $200 ~\\mu \\rm Hz$. We believe that the low-frequency peaks\nare the physical solution and the higher-frequency set an alias for the following\ntwo reasons:\n\n\\begin{enumerate}\n\n\\item The low-frequency peaks are always of larger amplitude than the alias \npeaks. While a resonance of stochastic noise with an alias frequency can enhance\nan individual alias peak such that it is larger than the corresponding true\npeak, this is\nunlikely to occur simultaneously for multiple peaks. \nWe have performed simulations which\nconfirm this reasoning. \n\n\\item{Hatzes (1999, private communication) has searched for oscillations in\n$\\alpha$ UMa using ground-based spectroscopic methods. He reports finding\nfrequencies of 1.36 and 6.0 $\\mu \\rm Hz$ (the latter less convincingly), \nalthough his observing run was too\nshort to lend much confidence to the exact values. While not identical to the\nfrequencies we report here, these frequencies are certainly comparable to\nour lower-frequency peaks rather than to the alias peaks.} \n\n\\end{enumerate}\n\nThough neither of these factors is conclusive on its own, we believe that\ntogether they indicate that we are on solid ground interpreting the\nobserved peaks in the amplitude spectrum as stellar oscillations. Of course, it \nremains possible that the observed variations are due to instrumental effects\nor to non-oscillatory stellar phenomena such as granulation.\n\n\\section{Interpretation}\n\nBelow we discuss the astrophysical implications of our results. \nA more detailed discussion, in the context\nof a complete stellar interiors model for $\\alpha$ UMa A, can be found\nin Guenther et al. \\markcite{guenther99}(1999).\n\n\\subsection{Mode Frequencies and Spacings}\n\nThe frequency of the fundamental mode is determined primarily by structure in\nthe envelope and so its determination requires a complete stellar model. However,\nwe can easily determine the range in which it should lie. The fundamental \nperiod $P_0$ is given \nby (Christy \\markcite{christy66}1966; \\markcite{christy68}1968)\n\\begin{equation}\nQ_0 = P_0 \\sqrt{\\frac{\\rho}{\\rho_{\\sun}}}\n\\end{equation}\nwhere $Q_0$ should lie between the value of 0.038 for a polytrope with $\\gamma\n= 4/3$ and 0.116 (for $\\gamma = 5/3$). Using the values of $R = 28 R_{\\sun}$\nfrom interferometry and the mass $M \\approx 4 M_{\\sun}$ appropriate to a\nK0 giant yields a fundamental mode of between 2.8 and 8.6 days. The lowest\nfrequency that we see corresponds to a period of 6.35 days, so we identify it\nas the fundamental mode for $\\alpha$ UMa.\n\nAs noted above, the average mode spacing for the first 8 modes is \n$2.94 ~\\mu \\rm Hz$. The last two modes have much larger spacings, which we interpret\nas signifying that we are not detecting all of the possible oscillation modes\nfor the star, presumably because they are not excited to large amplitudes.\nThe large separation $\\Delta \\nu_0$ is related to the mean stellar density, \nas shown by Cox \\markcite{cox80}(1980):\n\\begin{equation}\n\\Delta \\nu_0 = 135 \\left( \\frac{\\rho}{\\rho_{\\sun}} \\right)^{1/2} ~\\mu \\rm Hz\n\\end{equation}\nUsing the values appropriate to a typical K0 giant yields a predicted spacing of \n$1.82 ~\\mu \\rm Hz$, about half the observed value. Once again, this \ndiscrepancy can be accounted for by assuming that all modes are not\nexcited. In particular, if only even or odd-valued radial $n$ modes are \nexcited, we\nwould expect the observed large separation to be twice the predicted value.\nThe simplest explanation is then that only the $l=0$ modes are excited, and\nthe frequencies we observe correspond to radial oscillations of the star.\n\n\\subsection{Mode Amplitudes}\n\nThe Kjeldsen \\& Bedding \\markcite{kandb95}(1995) scaling law\n\\begin{equation}\n\\delta L / L = \\frac{L/L_{\\sun}}{(\\lambda/550 {\\rm~nm})(T_e/5777{\\rm~K})^2\n(M/M_{\\sun})} \\times 5.1 \\mu \\rm mag\n\\end{equation}\npredicts oscillation amplitudes of $\\approx 500 ~\\mu \\rm mag$ for $\\alpha$\nUMa, which is essentially in agreement with our results. It should be noted\nthat the WIRE data are obtained in white light and, consequently, phase differences\nin the oscillation amplitudes as a function of wavelength would tend to combine to\nreduce the observed amplitude. Consequently, it is not surprising that the\nobserved amplitudes are somewhat smaller than those predicted by theory.\nIn addition, of course, extending a relationship derived for lower main sequence\nstars to giants is a risky enterprise!\nNonetheless, the near-agreement between theory and observation may imply that the\nexcitation mechanism for oscillations in $\\alpha$ UMa is fundamentally \nsimilar to the solar mechanism (presumably convection; see, e.g. Bogdan\net al \\markcite{bogdan93}1993), unlike oscillations observed in other K giants,\nwhich show amplitudes an order of magnitude greater than those we have \ndetected.\n\n\\acknowledgments\n\nWe gratefully acknowledge the support of Dr. Harley Thronson and Dr. Phillipe Crane\nat NASA Headquarters for making this unusual use of WIRE possible.\nJ.C., T.C., R.L., T.G., and D.S. would like to thank Drs. Carol Lonsdale and Perry Hacking\nfor the opportunity to work with them on the WIRE project,\nand their support of the WIRE asteroseismology effort. The hard work of many\npeople, including the WIRE operations and spacecraft teams at GSFC, and\nthe timeline generation team\nteam at IPAC, was essential to making this project a reality.\nWe would also like to acknowledge the contributions of the anonymous referee,\nwhose criticisms helped to greatly improve the presentation of our results.\n\n\\clearpage\n\n\\begin{references}\n\\reference{ayres81}{Ayres, T.R., Marstad, N.C., \\& Linsky, J.L. 1981,\n\\apj 247, 545.}\n\\reference{bell93}{Bell, R.A. 1993, \\mnras, 264, 345.}\n\\reference{bogdan93}{Bogdan, T.J., Cattaneo, F., \\& Malagoli, A. 1993,\n\\apj, 307, 316.}\n\\reference{brown94}{Brown, T.M., Christensen-Dalsgaard, J., Weibel-Mihalas,\nB., \\& Gilliland, R.L. 1994, \\apj, 427, 1013.}\n\\reference{brown97}{Brown, T., Kennelly, E., Korzennik, S., Nisenson, P.,\nNoyes, R., \\& Horner, S. 1997, \\apj, 475,322.}\n\\reference{cayrel92}{Cayrel de Stroebel, G., Hauck, B., Francois, P., \nThevenin, F., Friel, E., Mermilliod, M., \\& Borde, S. 1992, A\\&AS, 95, 273.}\n\\reference{christy66}{Christy, R.F. 1966, \\araa, 4, 353.}\n\\reference{christy68}{Christy, R.F. 1968, \\qjras, 9, 13.}\n\\reference{cox80}{Cox, J.P. 1980, Theory of Stellar Pulsations (Princeton:\nPrinceton University Press).}\n\\reference{davies90}{Davies, S.R. 1990, \\mnras, 244, 93.}\n\\reference{duvall88}{Duvall, T.L., Harvey, J.W., Libbrecht, K.G., Popp, B.D.,\n\\& Pomerantz, M.A. 1988, \\apj, 3234, 1158}\n\\reference{edmonds96}{Edmonds, P.D. \\& Gilliland, R.L. 1996, \\apjl, 464, 157.}\n\\reference{foster96}{Foster, G. 1996, \\apj, 111, 541.}\n\\reference{gough87}{Gough, D.O. 1987, \\nat, 326, 257.}\n\\reference{guenter99}{Guenther, D.B, Demarque, P., Buzasi, D., Catanzarite, T., \nLaher, R., Conrow, T., \\& Kreidl, T. 1999, \\apjl, submitted.}\n\\reference{hall96}{Hall, J.C. 1996, \\pasp, 108, 313.}\n\\reference{hatzes94a}{Hatzes, A.P. \\& Cochran, W.D. 1994, \\apj, 422, 366.}\n\\reference{hatzes94b}{Hatzes, A.P. \\& Cochran, W.D. 1994, \\apj, 432, 763.}\n\\reference{horne86}{Horne, J.H. \\& Baliunas, S.L. 1986, \\apj, 302, 757.}\n\\reference{kandb95}{Kjeldsen, H, \\& Bedding, T.R. 1995, \\aap, 293, 87.}\n\\reference{kjeldsen95}{Kjeldsen, H., Bedding, T.R., Vishum, M, \\& Frandsen, S.\n1995, \\apj, 109, 1313.}\n\\reference{kjeldsen99}{Kjeldsen, H., Bedding, T.R., Frandsen, S., \\& Dall, T.H.\n1999, \\mnras, 303, 579.}\n\\reference{kondo77}{Kondo, Y., Morgan, T.H., \\& Modisette, J.L. 1977, \\pasp, 89, 163.}\n\\reference{laher00}{Laher, R., et al. 2000, in Proceedings, 10th AAS/AIAA Space Flight\nMechanics Meeting, in press.}\n\\reference{leibacher85}{Leibacher, J.W., Noyes, R.W., Toomre, J., \\& Ulrich,\nR.K. 1985, Scientific American, 253, 34.}\n\\reference{roberts87}{Roberts, D.H., Lehar, J., \\& Derher, J.W. 1987,\n\\apj, 93, 968.}\n\\reference{scargle82}{Scargle, J.D. 1982, \\apj, 263, 835.}\n\\reference{schou98}{Schou, J., et al. 1998, \\apj, 505, 390.}\n\\reference{schwarzenberg98}{Schwarzenberg-Czerny, A., 1998, \\mnras, 301, 831.}\n\\reference{soderhjelm99}{Soderhjelm, S. 1999, \\aap, 341, 121.}\n\\reference{stellingwerf78}{Stellingwerf, R.F. 1978, \\apj, 224, 953.}\n\\reference{taylor99}{Taylor, B.J. 1999, A\\&AS, 134, 523.}\n\\end{references}\n%\\end{document}\n\n\\clearpage\n\n\\figcaption[figure1.eps]{The thermally corrected and mean subtracted time series \nof observations of $\\alpha$ UMa.\nEach vertical stripe visible in the figure corresponds to one spacecraft orbit\n(about 96 minutes). The error bar shown represents the $\\pm 3 \\sigma$ \nPoisson noise\nbased on pure photon statistics. \\label{fig1}}\n\n\\figcaption[figure2.eps]{The window function for the $\\alpha$ UMa time series\nshown in Figure~1. \\label{fig2}}\n\n\\figcaption[figure3.eps]{The Scargle periodogram of the time series in \nFigure~1. The dashed line in the inset represents the periodogram of a\ncomparable time series of $\\alpha$ Leo. \\label{fig3}}\n\n\\clearpage\n\n\\plotone{figure1.eps}\n\n\\clearpage\n\n\\plotone{figure2.eps}\n\n\\clearpage\n\n\\plotone{figure3.eps}\n\n\\clearpage \n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{cccc}\nFrequency ($\\mu\\rm Hz$)& Error ($\\mu\\rm Hz$) & $\\Delta \\nu$ \n($\\mu\\rm Hz$)\\tablenotemark{1} & Amplitude ($\\mu\\rm mag$)\\\\\n\\tableline\n1.82 & 0.74\\tablenotemark{2}& & 390 \\\\\n4.84 & 0.98 & 3.02 & 400 \\\\\n7.86 & 0.77 & 3.02 & 390 \\\\\n11.41 & 0.59 & 3.55 & 230 \\\\\n15.00 & 0.71 & 3.59 & 330 \\\\\n18.25 & 0.71 & 3.25 & 220 \\\\\n20.90 & 0.67 & 2.65 & 200 \\\\\n22.37 & 0.73 & 1.47 & 210 \\\\\n34.93 & 0.63 & 12.56 & 210 \\\\\n43.56 & 0.92 & 8.63 & 180 \\\\\n\\end{tabular}\n\\end{center}\n\n\\tablenotetext{1}{The difference in frequency between each peak and the previous\none. The mean separation for the first 8 modes is 2.94 $\\mu$Hz.}\n\\tablenotetext{2}{Since this frequency is poorly sampled by the time series, the \nerror estimate for the lowest\nfrequency is probably an underestimate\nof the true error, which may be as much as twice this value.}\n\\tablenum{1}\n\\end{table*}\n\n\\end{document}\n"
}
] |
[] |
astro-ph0002513
|
METALLICITY EVOLUTION IN THE EARLY UNIVERSE
|
[
{
"author": "JASON X. PROCHASKA\\altaffilmark{1}"
},
{
"author": "Pasadena"
},
{
"author": "CA 91101"
},
{
"author": "and"
},
{
"author": "ARTHUR M. WOLFE\\altaffilmark{1}"
},
{
"author": "Department of Physics"
},
{
"author": "and Center for Astrophysics and Space Sciences"
},
{
"author": "San Diego"
},
{
"author": "C--0424; La Jolla; CA 92093"
}
] |
Observations of the damped \lya systems provide direct measurements on the chemical enrichment history of neutral gas in the early universe. In this Letter, we present new measurements for four damped \lya systems at high redshift. Combining these data with $\lbrack$Fe/H$\rbrack$ values culled from the literature, we investigate the metallicity evolution of the universe from $z \approx 1.5-4.5$. Contrary to our expectations and the predictions of essentially every chemical evolution model, the $\N{HI}$-weighted mean $\lbrack$Fe/H$\rbrack$ metallicity exhibits minimal evolution over this epoch. For the individual systems, we report tentative evidence for an evolution in the unweighted $\lbrack$Fe/H$\rbrack$ mean and the scatter in $\lbrack$Fe/H$\rbrack$ with the higher redshift systems showing lower scatter and lower typical $\lbrack$Fe/H$\rbrack$ values. We also note that no damped \lya system has $\lbrack$Fe/H$\rbrack$ $< -2.7$~dex. Finally, we discuss the potential impact of small number statistics and dust on our conclusions and consider the implications of these results on chemical evolution in the early universe.
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[
{
"name": "pprt.tex",
"string": "\\documentclass{article}\n\n\\usepackage{emulateapj}\n\\usepackage{onecolfloat}\n\\usepackage{graphicx} \n\\usepackage{fancyheadings} \n%\\usepackage{amstex}\n\\usepackage{ulem}\n\\usepackage{rotating}\n\\usepackage{lscape}\n\n\n\\newcommand{\\tskip}{\\tablevspace{3pt}}\n\\newcommand{\\citep}{\\cite}\n\\newcommand{\\feh}{[{\\rm Fe/H}]}\n\\newcommand{\\znh}{[{\\rm Zn/H}]}\n\\newcommand{\\ndla}{19~}\n\\newcommand{\\msol}{M_\\odot}\n\\newcommand{\\pkts}{P_{KS}}\n\\newcommand{\\etal}{et al.\\ }\n\\newcommand{\\delv}{\\Delta v}\n\\newcommand{\\Lya}{Ly$\\alpha$ }\n\\newcommand{\\lya}{Ly$\\alpha$ }\n\\newcommand{\\ndmp}{31 }\n\\newcommand{\\nnew}{14 }\n\\newcommand{\\nrun}{8,500}\n\\newcommand{\\kms}{km~s$^{-1}$ }\n\\newcommand{\\cm}[1]{\\, {\\rm cm^{#1}}}\n\\newcommand{\\N}[1]{{N({\\rm #1})}}\n\\newcommand{\\f}[1]{{f_{\\rm #1}}}\n\\newcommand{\\rAA}{{\\AA \\enskip}}\n\\newcommand{\\sci}[1]{{\\rm \\; \\times \\; 10^{#1}}}\n\\newcommand{\\ltk}{\\left [ \\,}\n\\newcommand{\\ltp}{\\left ( \\,}\n\\newcommand{\\ltb}{\\left \\{ \\,}\n\\newcommand{\\rtk}{\\, \\right ] }\n\\newcommand{\\rtp}{\\, \\right ) }\n\\newcommand{\\rtb}{\\, \\right \\} }\n\\newcommand{\\ohf}{{1 \\over 2}}\n\\newcommand{\\nohf}{{-1 \\over 2}}\n\\newcommand{\\rhf}{{3 \\over 2}}\n\\newcommand{\\smm}{\\sum\\limits}\n\\newcommand{\\perd}{\\;\\;\\; .}\n\\newcommand{\\cmma}{\\;\\;\\; ,}\n\\newcommand{\\sgint}{\\sigma_{int}}\n\\newcommand{\\Nperp}{N_{\\perp} (0)}\n\\newcommand{\\intl}{\\int\\limits}\n\\newcommand{\\und}[1]{{\\rm \\underline{#1}}}\n\\newcommand{\\sphr}{\\sqrt{R^2 + Z^2}}\n\\newcommand{\\vrot}{v_{rot}}\n\\newcommand{\\btau}{\\bar\\tau (v_{pk}) / \\sigma (\\bar \\tau)}\n\\newcommand{\\Ipk}{I(v_{pk})/I_{c}}\n\\newcommand{\\mkms}{{\\rm \\; km\\;s^{-1}}}\n\n\\special{papersize=8.5in,11in}\n\n\\begin{document}\n\n\\twocolumn[%\n%\\submitted{AJ: December 15, 1998}\n\\accepted{ApJ: February 28, 2000}\n\n\\title{METALLICITY EVOLUTION IN THE EARLY UNIVERSE}\n\n\\author{ JASON X. PROCHASKA\\altaffilmark{1} \\\\\nThe Observatories of the Carnegie Institute of Washington\n813 Santa Barbara St. \\\\\nPasadena, CA 91101 \\\\\nand \\\\\nARTHUR M. WOLFE\\altaffilmark{1} \\\\\nDepartment of Physics, and Center for Astrophysics and Space Sciences \\\\\nUniversity of California, San Diego \\\\\nC--0424; La Jolla; CA 92093}\n\n\n\n\\begin{abstract} \n\nObservations of the damped \\lya systems provide\ndirect measurements on the chemical enrichment history of neutral gas in the \nearly universe. In this Letter, we present new measurements for four damped\n\\lya systems at high redshift. Combining these data\nwith $\\lbrack$Fe/H$\\rbrack$ values culled from the literature, we investigate\nthe metallicity evolution of the universe from $z \\approx 1.5-4.5$.\nContrary to our expectations and the predictions of essentially every \nchemical evolution model, the $\\N{HI}$-weighted mean $\\lbrack$Fe/H$\\rbrack$ \nmetallicity exhibits minimal evolution over this epoch.\nFor the individual systems,\nwe report tentative evidence for an evolution in \nthe unweighted $\\lbrack$Fe/H$\\rbrack$\nmean and the scatter in $\\lbrack$Fe/H$\\rbrack$ with the higher \nredshift systems showing\nlower scatter and lower typical $\\lbrack$Fe/H$\\rbrack$ values. \nWe also note that no damped \\lya system has $\\lbrack$Fe/H$\\rbrack$ \n$< -2.7$~dex. Finally, we discuss the potential\nimpact of small number statistics and dust on our conclusions and \nconsider the implications of these results\non chemical evolution in the early universe.\n\n\\end{abstract}\n\n\\keywords{galaxies: abundances --- \ngalaxies: chemical evolution --- quasars : absorption lines}]\n\n\n\\altaffiltext{1}{Visiting Astronomer, W.M. Keck Telescope.\nThe Keck Observatory is a joint facility of the University\nof California and the California Institute of Technology.}\n\n\\pagestyle{fancyplain}\n\\lhead[\\fancyplain{}{\\thepage}]{\\fancyplain{}{PROCHASKA \\& WOLFE}}\n\\rhead[\\fancyplain{}{METALLICITY EVOLUTION IN THE EARLY UNIVERSE}]\n{\\fancyplain{}{\\thepage}}\n\\setlength{\\headrulewidth=0pt}\n\\cfoot{}\n\n\\section{INTRODUCTION}\n\nThe damped \\lya systems, neutral hydrogen gas layers identified in the\nabsorption line spectra of background quasars, \ndominate the neutral hydrogen \ncontent of the Universe at all epochs. At high redshift,\nthese systems are widely\naccepted as the progenitors of present-day galaxies for the following\nreasons:\n(i) their very large HI\ncolumn densities, $\\N{HI} > N_{thresh} = 2 \\times 10^{20} \\cm{-2}$, imply\noverdensities $\\delta \\rho / \\rho \\gg 100$, i.e., these are virialized\nsystems at high redshift; \n(ii) they contain the majority of neutral gas \nin the early Universe and are therefore the reservoirs for galaxy\nformation; (iii) their gas density\n$\\Omega_{gas}$ at redshift $z \\approx 2 -3$ is consistent with the\nmass density of stars today (\\citep{wol95}). \nWhile the physical nature\nof the damped \\lya systems is still controversial \n(\\citep{pro97,hae98,maller99,lebr97}), \nby studying the chemical abundances of the damped \\lya system one directly\ntraces the chemical enrichment history of the Universe at high redshift.\nObserving damped \\lya systems is equivalent to\npoking sightlines through the ISM of protogalaxies. \nBecause these observations are biased to HI cross-section\nand the HI gas mass of a system is proportional to\n$\\int \\sigma \\cdot N$,\none can measure global properties of the universe \nsimply by weighting the measurement from \neach damped system by $\\N{HI}$. \nAt the same time, the observations afford an efficient means for\nexamining the characteristics of individual protogalaxies\nin the early Universe.\nIn this Letter, we examine the metallicity of\nthe damped \\lya systems from $z \\approx 2 - 4.5$ which places\ntight constraints on chemical evolution models (e.g.\\ Pei et al.\\ 1999),\nas well as a valuable consistency check on SFR observations\n(\\citep{ptt99a}).\n\nOver the past decade, several groups have surveyed the metallicity of the\ndamped \\lya systems from $z \\approx 1 - 4$ \n(\\citep{ptt94,ptt97,ptt99a,lu96,pro99}). \nTo date, the chemical abundances of over 40 systems have been measured, \nthe majority with $z = 1.5 - 3$\nwhere the identification and follow-up observations\nof damped \\lya systems is most efficient.\nThese studies argue that at $z \\approx 2$, the mean metallicity\nof the damped systems is approximately 1/10 $-$ 1/30 solar metallicity\n([Zn/H]~$\\approx -1.1$,\n[Fe/H] $\\approx -1.5$) with a large scatter from nearly solar to less than\n$1/100$ solar metallicity. At very high redshift ($z>3$), the picture\nis far less certain. Focusing on a sample of seven $z>3$ damped \\lya\nsystems, Lu et al.\\ (1996,1997) noted\nthat the metallicity of these systems is significantly\nlower than the $z<3$ observations. \nIn turn, the authors argued that $z \\approx 3$ marked the epoch \nwhere significant star formation begins,\na claim with important implications for the processes of galaxy formation.\n\nIn this Letter we present new measurements on the metallicity of\nfour damped \\lya systems (including three at $z>3.5$) and together \nwith the data from Prochaska \\& Wolfe (1999) double the sample of $z>3$\nsystems. The new full sample -- including the systems from \nLu et al.\\ (1996,1997) --\nreveals evidence for little change in the $\\N{HI}$-weighted\nmean metallicity of the neutral Universe from \n$z \\approx 1.5-4.5$, contrary to the predictions of essentially every chemical\nevolution model. On the other hand, we find tentative evidence for an\nevolution in the unweighted mean and scatter of [Fe/H] for individual damped \n\\lya systems. Finally, we comment on the\nrobustness of these results (particularly in the light of small number\nstatistics and dust), speculate on the implications\nfor chemical enrichment, and discuss the prospects for future advances.\n\n\n\\section{OBSERVATIONS AND ANALYSIS}\n\\label{sec-obs}\n\nTo determine the metallicity of a damped \\lya system, one must accurately\nmeasure the neutral hydrogen column density $\\N{HI}$ and a metallicity\nindicator, typically either Zn or Fe.\nIn stellar population studies of the Galaxy one traditionally uses Fe\nas the metallicity indicator, primarily as a matter of convenience. \nAs we are\nstudying gas-phase abundances, however, we must account for the\npossible depletion of Fe onto dust grains or instead choose an\nelement like Zn which is minimally affected by depletion. Unfortunately,\nthere are both\ntheoretical and observational disadvantages to using Zn as the\nmetallicity indicator. Theoretically,\nZn has a very uncertain chemical origin.\nIt is referred to as an Fe peak element because it traces\nFe in Galactic stars (\\citep{sne91}), yet the leading theory on the production\nof Zn proposes it forms in the neutrino-driven \nwinds of Type~II SN (\\citep{hff96}). \nFurthermore, recent measurements of [Zn/Fe] in\nmetal-poor stars (\\citep{jhnsn99}) and thick disk stars\n(\\citep{pro00}) suggest Zn/Fe is enhanced relative to the Sun by\n+0.1 to +0.3~dex, perhaps consistent with a Type~II origin.\nObservationally there are complications with measuring Zn in the \ndamped \\lya systems, where one must rely on two\nweak ZnII transitions with $\\lambda_{rest} \\approx 2000$\\AA.\nThe transitions are so weak that \neven at high resolution and high S/N, Zn can only be detected in damped\nsystems when $\\log [\\N{HI}] + [{\\rm Zn/H}] > 19.0$ (e.g.\\ [Zn/H]~$> -1.3$ for\nsystems with $\\N{HI} \\approx N_{thresh}$). Most important to this study,\nhowever, the large rest wavelength of the ZnII transitions prevents one\nfrom readily measuring Zn in $z>3$ damped \\lya systems as it is \ndifficult to make sensitive observations at $\\lambda \\approx 8000$\\rAA\nwith current high resolution spectrographs. \nIn fact, at the time of publication {\\it we are not aware of a single accurate\nZn measurement for any $z>3$ damped \\lya system}.\nTherefore, we will focus\non Fe in this Letter, which has two singly ionized transitions at \n$\\lambda_{rest} \\approx 1600$\\rAA with a complement of $f$-values\nideal for measuring the abundance of Fe in \ndamped systems. We restrict the analysis\nto Fe measurements made with HIRES on the Keck~I telescope (\\citep{vgt92}), \nspecifically the systems\nobserved by Prochaska \\& Wolfe (1999) and Lu et al.\\ (1996,1997) \nand the additional systems\nintroduced here. In addition to providing a homogeneous data set which has\nbeen reduced and analyzed with the same techniques,\nthese observations account for nearly every damped \\lya system\nwith an accurate Fe abundance at $z>1.5$ and every system with $z>3$.\n\n\\begin{table}[ht] \\footnotesize\n\\begin{center}\n\\caption{ \\label{newobs}}\n{\\sc NEW METALLICITY MEASUREMENTS} \n\\begin{tabular}{llccccc}\n\\tskip\n\\tableline\n\\tableline \\tskip\nQSO & $z_{abs}$& $\\N{HI}$ & $\\N{Fe^+}$ & [Fe/H] \\\\\n\\tableline \\tskip\nBRI0952$-$0115 & 4.024 & $20.50 \\pm 0.1$ & $14.054 \\pm 0.07$\\tablenotemark{a} \n& $-1.95$ \\\\\nBRI1108$-$0747 & 3.608 & $20.50 \\pm 0.1$ & $13.860 \\pm 0.03$\\tablenotemark{b} \n& $-2.14$ \\\\\nQ1223$+$1753 & 2.466 & $21.50 \\pm 0.1$ & $15.279 \\pm 0.03$\\tablenotemark{c} \n& $-1.72$ \\\\\nPSS1443+2724 & 4.224 & $20.80 \\pm 0.1$ & $15.325 \\pm 0.10$\\tablenotemark{c} \n& $-0.98$ \\\\\n\\tskip \\tableline\n\\end{tabular}\n\\end{center}\n\\centerline{$^a$Average of FeII 1144 ($\\log gf = 0.105$) and FeII 1608}\n\\centerline{$^b$FeII 1608}\n\\centerline{$^c$FeII 1611}\n\\end{table}\n\nTable~\\ref{newobs} summarizes the new $\\N{Fe^+}$ measurements derived from\nobservations acquired by the authors in February 1998 and March 1999 \nwith HIRES on the Keck~I 10m telescope. The data was reduced with the\n{\\it makee} software package developed by T. Barlow\nand the column densities were \nderived primarily from the FeII~1608 and/or FeII~1611 \ntransitions with the apparent optical\ndepth method (\\citep{sav91}). We adopt the oscillator strengths from \nCardelli \\& Savage (1995) \nnoting that our conclusions on the evolution of [Fe/H] in\nthe damped \\lya systems are not sensitive to their values.\nThe $\\N{HI}$ values for these systems are taken from the literature\n(\\citep{wol95,storr99}) and are the dominant source of error in the \n[Fe/H] values.\nFinally, we evaluate [Fe/H] assuming the meteoritic Fe abundance\n($\\epsilon$(Fe)=7.50; Grevesse \\& Sauval 1999)\nwithout adopting any ionization corrections\nwhich is an excellent assumption for all but possibly the lowest \n$\\N{HI}$ damped \\lya systems (\\citep{pro96}).\nTogether with the published measurements of Prochaska \\& Wolfe (1999) and \nLu et al.\\ (1996,1997) the total [Fe/H] sample is 37 systems, 15 with \n$z>3$. The systems were chosen independent of any prior metallicity\nmeasurements; the only possible biases are due to the magnitude limited\nselection of the quasars (e.g.\\ Fall \\& Pei 1993)\nwhich will be discussed in the following section. \n\n\\section{RESULTS AND DISCUSSION}\n\\label{sec-disc}\n\nFigure~\\ref{Fevsz} plots the 37 [Fe/H] values versus $z_{abs}$ for \nthe Wolfe \\& Prochaska\nsample (dark squares) and (light stars)\nthe sample of damped \\lya systems observed by\nLu et al.\\ (1996,1997). To explore evolution in the metallicity of the damped\n\\lya systems,\nwe consider three moments of the metallicity data \nin two redshift intervals, $z_{low} = [1.5,3]$ and\n$z_{high} = (3,4.5]$. These are: \n(1) the $\\N{HI}$-weighted mean metallicity of neutral gas, $<Z>$;\n(2) the unweighted mean metallicity, $<\\feh>$,\nof the set of damped \\lya systems at $z_{low}$ and $z_{high}$; and \n(3) the standard deviation of [Fe/H] in these protogalaxies, \n$\\sigma(\\feh)$. \n\n\\begin{figure*}[ht]\n\\begin{center}\n\\includegraphics[height=5.3in, width=3.7in,angle=-90]{fig1.ps}\n\\caption{Thirty-nine [Fe/H], $z_{abs}$ pairs for the damped \\lya systems\nobserved by Prochaska \\& Wolfe (1999, this paper; squares) \nand Lu et al.\\ (1996,1997; \nstars) with HIRES on the Keck~I telescope. The open circles correspond to\nthe $\\N{HI}$-weighted mean metallicity for the systems at \n$z_{abs} = [1.5,3]$ and $z_{abs} = (3,4.5]$. Note that the difference in these\nmeans is small, $\\approx 0.1$~dex. Also observe that the scatter in the [Fe/H]\nvalues appears to increase at lower redshift.}\n\\label{Fevsz}\n\\end{center}\n\\end{figure*}\n\nThe first moment represents the global metallicity\nof all neutral gas at a given epoch, $\\Omega_{metals}/\\Omega_{HI}$. \nIt is evaluated by weighting each\n[Fe/H] measurement by the corresponding HI column density, \n$<Z> \\; \\equiv \\, \\log [\\smm \\N{Fe^+} / \\smm \\N{HI}] - \n\\log ( {\\rm Fe/H} )_\\odot$.\nComputing\nthe mean for the damped \\lya systems at the two\nepochs, we find $<Z>_{low} = -1.532 \\pm 0.036$ and \n$<Z>_{high} = -1.634 \\pm 0.049$. \nThe errors on the $<Z>$ values reflect only\nthe statistical uncertainty in measuring\n$\\N{Fe^+}$ and $\\N{HI}$ and\nwere derived with standard error propagation techniques. \nBelow we estimate the uncertainty due to small number statistics.\nComparing the $<Z>$ values, we note that they\nfavor no significant evolution\nin the mean metallicity of neutral gas from $z = 1.5 - 4.5$.\nIf we include the tentative result from Pettini et al.\\ (2000)\nthat the Zn mean metallicity does not change from $z \\approx 1 - 3$, then\none concludes there is no evidence for\nsignificant metallicity evolution from $z=1-4.5$,\nan interval spanning more than 3~Gyr. The other two\nmoments, the unweighted mean\n$<\\feh> = \\frac{1}{n} \\smm^n \\feh$ and the scatter $\\sigma(\\feh)$, \nare more sensitive to the chemical enrichment history within \nindividual protogalaxies as each damped system is given equal weight.\nFor the two intervals we find that the mean logarithmic\nabundance, $<\\feh>_{low} = -1.61$ and $<\\feh>_{high} = -1.83$.\nMeanwhile, the scatter in [Fe/H] is $\\sigma(\\feh) = 0.50$ and\n$\\sigma(\\feh) = 0.35$ for the $z_{low}$ and $z_{high}$ samples\nrespectively. Performing the Student's t-test and the F-test \non the two moments, we find that the\n$<\\feh>$ and $\\sigma(\\feh)$ statistics for the two epochs\nare inconsistent at the 90$\\%$\nand 80$\\%$ c.l. Therefore, there is tentative evidence for chemical \nevolution in the individual damped \\lya systems with the $z<3$ sample\nshowing a higher typical metallicity and a larger scatter in [Fe/H]\nfrom system to system.\n\n\\begin{figure*}[ht]\n\\begin{center}\n\\includegraphics[height=5.3in, width=3.7in,angle=-90]{fig2.ps}\n\\caption{Thirty-nine [Fe/H], $\\N{HI}$ pairs for the damped \\lya systems\nin the full sample. The circles correspond to $z_{abs} \\leq 3$ systems and\nthe triangles are for $z_{abs} > 3$. While the systems with \n$\\N{HI} < 10^{21} \\cm{-2}$ show a large scatter in [Fe/H], the large\n$\\N{HI}$ systems all have $\\feh \\approx -1.8$.}\n\\label{FeHvsNHI}\n\\end{center}\n\\end{figure*}\n\n\nTo address the robustness of these results, one must consider several issues.\nFirst, because $<Z>$ is dominated by the systems with the largest\n$\\N{HI}$ values, this mean is robust only in so far as the total $\\N{HI}$,\nHI$_T$~$\\equiv \\smm_n \\N{HI}$, \nwell exceeds that of a single damped \\lya system. \nFigure~\\ref{FeHvsNHI} plots the $\\feh$, $\\N{HI}$ pairs for all 37 \nsystems where the dark circles are members of the $z_{low}$ sample and the\nlight triangles those of the $z_{high}$ sample.\nNote that there are four systems with $\\N{HI}>10^{21} \\cm{-2}$ at\n$z<3$ but only a single system in the $z_{high}$ sample. For the\n$z_{low}$ sample, HI$_T = 10^{22.30} \\cm{-2}$, which is a factor of\n4 larger than the largest $\\N{HI}$ measured for any damped \\lya system\n(Q0458$-$0203; $\\N{HI} = 10^{21.7}$) and 10 times greater than \nmost of the known damped \\lya systems. As such, we consider the mean\nderived from the $z_{low}$ sample to be reasonably robust. The primary\npotential\npitfall is if the optical surveys have systematically missed damped systems\nwith $\\N{HI} > 10^{22} \\cm{-2}$, a possibility if dust obscuration is\nsignificant (discussed further below).\nThe situation is far more uncertain for the $z>3$ sample where \nHI$_T = 10^{21.84} \\cm{-2}$, comparable to the $\\N{HI}$ of\nthe Q0458$-$0203 system from the $z_{low}$ sample\nand only three times greater than the largest $\\N{HI}$ system in the\n$z_{high}$ sample. While the most recent surveys \nsuggest there are very few $z>3$ damped systems with \n$\\N{HI} > 10^{21} \\cm{-2}$ Storrie-Lombardi \\& Wolfe (2000), \nwe caution that the mean\nwe have derived for the $z_{high}$ sample is a tentative result. \nFor example, the system toward Q0000$-$2619 has significant bearing on \n$<Z>_{high}$ and its Fe abundance has been difficult to determine\n(\\citep{pro99,lu96}). Ironically, removing it from the $z_{high}$\nsample would actually increase $<Z>_{high}$ into exact\nagreement with $<Z>_{low}$ because we have adopted \n[Fe/H]~$= -1.77$ based on the FeII~1611 profile from this system. \nMeanwhile, lowering [Fe/H] \nby 0.6~dex to establish consistency with the Ni and Cr abundances, \nwould decrease $<Z>_{high}$ by 0.1~dex. \nIn short, while we have confidence in the $<Z>_{low}$ value,\nwe caution the reader that \nsmall number statistics are still important\nin evaluating the $<Z>_{high}$.\nOne can estimate the uncertainty associated with the small number statistics\nof the two samples by performing a bootstrap statistical analysis.\nFor each sample, we independently\ncalculated $<Z>$ 500 times by randomly drawing \n$n$ objects ($n$ is the number of damped systems in a given redshift \ninterval) from each interval. In turn, we can estimate the effects of cosmic\nvariance on our results by calculating the standard deviation of the\ntwo bootstrap $<Z>$ distributions: $\\sigma^{<Z>}_{low} = 0.088$~dex and\n$\\sigma^{<Z>}_{high} = 0.155$~dex.\nAs one would expect, the $<Z>_{high}$ value, which is based on \nonly 15 systems, is \nconsiderably less certain than the $<Z>_{low}$ measurement.\nThe difference in the $\\sigma^{<Z>}$ values\nstresses the outstanding need for\nfuture observational programs to focus on $z>3$ damped systems.\n\nAny study on the chemical abundances of the damped \\lya systems must\nassess the potential effects of dust. \nWith respect to this analysis, where\nwe have taken Fe as the metallicity indicator, there are two important\naspects to consider: (1) if we need to correct the\nobserved [Fe/H] values by some factor to obtain the true metallicity of\neach system, does the mean correction evolve in\ntime and/or differ from system to system at the same epoch?; \nand (2) dust obscuration could remove \ndamped \\lya systems from the magnitude limited \nsamples which would significantly alter the conclusions (e.g.\\ metal-rich,\nhigh $\\N{HI}$ systems). With respect to the first concern, we can estimate\nthe maximum dust correction to [Fe/H] via the measured Zn/Fe ratio.\nAgain, Zn is essentially undepleted in the gas-phase so that\n[Zn/H] = [Fe/H] + [Zn/Fe] may be more representative of the true metallicity\nin the damped \\lya systems.\nThis practice is limited, however, by the fact that Zn may be produced\nin Type~II SN (\\citep{hff96}) such that super-solar Zn/Fe ratios would be \nrepresentative of nucleosynthesis, not dust depletion.\nFurthermore, recent results on the [Zn/Fe] ratio measured in Galactic stars \nshows that [Zn/Fe] $> +0.2$~dex in very metal-poor stars ($\\feh < -2.5$;\nJohnson 1999) and even exhibits super-solar values \n(average [Zn/Fe]~$\\approx +0.13$ in 10 stars; Prochaska et al.\\ 2000) \nin thick disk stars with $\\feh > -1$. \nTherefore, while the typical [Zn/Fe] value in the damped \\lya systems is\n+0.4~dex with relatively small scatter (\\citep{ptt97,pro99}), \nit is unclear what fraction \nis due to dust depletion. Nonetheless, if we take [Zn/H] as the\ntrue metallicity indicator, $<Z>$ and the\nunweighted mean are enhanced \nby $\\approx 0.4$~dex, but there is\nvery little change in the observed scatter.\nThe potential effects of dust depletion on \nthe statistical moments for the $z_{high}$ sample are more\nspeculative as there is {\\it no accurate Zn determination} for any\n$z>3$ damped \\lya system. To estimate the depletion level, we can compare\nthe relative abundance patterns \n(in particular the Si/Fe ratio) of these systems with the\n$z_{low}$ sample. \nIn the few cases where Si/Fe has been measured in the $z>3$ systems\none finds [Si/Fe]~$\\approx +0.3$~dex, nearly identical to\nthe typical value of the $z<3$ sample.\nWhile the similarity of a metal ratio like Si/Fe does not require \nsimilar dust depletion levels, the $z>3$ [Si/Fe] values do\nimply depletion levels of at least 0.3~dex. \nTherefore, unless one takes the unlikely stance that\nthe $z>3$ systems are significantly more depleted than the $z_{low}$ sample,\nwe expect very minimal evolution in the depletion levels of the\ndamped systems and no significant impact \non any of our conclusions. The effects of biasing due to dust obscuration\nare more difficult to address. Note in Figure~\\ref{FeHvsNHI} the \nabsence of any $\\N{HI} > 10^{21} \\cm{-2}$ systems with $\\feh \\sim -1$.\nWhile this may be due to small number statistics or \nthat very few regions exist in the early universe with large\n$\\N{HI}$ and $\\feh \\gtrsim -1$, the trend could also be explained by dust \nobscuration.\nFall \\& Pei (1993) have presented an excellent framework for addressing the\neffects of dust depletion on damped \\lya statistics. \nTheir calculations indicate that if \nthe logarithmic scatter in the dust-to-gas ratio $k$ is small\n(less than 1 dex), then only a small correction to the mean optical\ndepth and in turn to $<Z>$ is required (Fall 1999).\nFor a constant dust-to-metals ratio -- implied by the nearly constant\n[Zn/Fe] values -- the logarithmic scatter in $k \\approx \\sigma(\\feh)$\nand we have shown $\\sigma(\\feh) \\leq 0.5$ for the two samples.\nTherefore, we expect dust obscuration to have a minimal effect\n($<0.2$~dex) on our results.\n\nThe results in this Letter on the evolution of the metallicity\nof neutral gas in the Universe \nand individual protogalaxies present an unexpected picture.\nA number of groups have estimated the chemical evolution of neutral gas\nat high redshift (\\citep{mny96,edmns97,pei99}) and essentially every\ntreatment predicts a substantial ($> 0.5$~dex) increase in the mean\nmetallicity from $z = 4$ to $z = 2$. \nWhile a 0.5~dex evolution is consistent with our results at the 3$\\sigma$\nlevel, the current observations favor a very mild evolution in $<Z>$.\nIf future observations lend further support for this conclusion,\nthe theoretical models will require significant revision.\nOf course, these theoretical treatments\ndepend sensitively on a number of factors which are uncertain:\n(i) the star formation rate, (ii) the IMF, (iii) the mass distribution of\nprotogalaxies, (iv) the loss of metals to the IGM, (v) the yield of\nvarious elements, etc.\nTherefore, there is considerable theoretical freedom to bring the models\ninto agreement with the observed lack of evolution. Nonetheless,\nthe Lyman break galaxies offer incontrovertible evidence \nthat significant star formation is taking place from $z = 3 - 4$\n(\\citep{std98})\nsuch that the total metal content of the Universe must be increasing.\nUnless these metals are enriching only ionized regions\n(an unlikely scenario), then to explain the minimal evolution in $<Z>$\nthe total HI content of the Universe must be increasing at nearly the same\nrate as the metal content. It is intriguing to note \nthat this is qualitatively consistent with \nthe evolution of $\\Omega_{gas}$ observed by\nStorrie-Lombardi \\& Wolfe (2000) for the damped \\lya systems. \n\nThe other statistical moments are sensitive to the chemical\nenrichment history within individual galaxies.\nComparing the unweighted mean with the weighted mean\nwe find that $<\\feh>$ is less than $<Z>$ at both epochs. \nWhile the difference is\nnot large ($\\approx 0.1 - 0.2$~dex), it does highlight the fact that\nmany of the $\\N{HI} < 10^{21} \\cm{-2}$ systems exhibit low\nmetallicity. In particular, in the $z>3$ sample \nonly 2 of 15 damped systems show [Fe/H] $> -1.5$~dex. \nOne possible explanation for the difference\nis systems which have just formed have preferentially \nlow $\\N{HI}$ and [Fe/H].\nThe trend is also suggestive of the correlation Cen \\& Ostriker (1999)\nfind between overdensity and metallicity in their numerical simulations.\nThe problem remains, however, in explaining why the highest metallicity\nsystems of the $z_{low}$ sample also have low $\\N{HI}$.\nLastly, recall that there is tentative support for an evolution in both\nthe scatter and $<\\feh>$ with the $z_{low}$ sample yielding larger values. \nIf more recently formed systems \ntend to have lower metallicity, then the evolution\nmay easily be explained by a larger mean and scatter in the age\nof the damped \\lya systems at $z \\approx 2$. Furthermore, the\nsystems at $z \\approx 2$ may have larger masses and a greater variety of\nmorphologies. \n\nFinally, we stress that only two systems from the full sample\nhave [Fe/H]~$< -2.5$ and the large majority show [Fe/H]~$> -2$. \nAs first noted by Lu et al.\\ (1997), there appears to be a threshold\nto the minimum metallicity of the damped \\lya systems at $\\approx$ 1/100\nsolar metallicity.\nTherefore, even out to $z \\approx 4.5$\nthere is no evidence for damped \\lya systems with primordial abundance.\nThis places a further constraint on chemical evolution models.\nAs we probe higher and higher redshift without detecting primordial gas,\none may be forced toward one of the following conclusions:\n(i) star formation proceeds rapidly\n($< 10^7$ years) to bring the metallicity to 1/100 solar\nafter the formation of a damped system; \n(ii) either the damped system or its progenitors have been undergoing\nstar formation for a lengthy time; and/or\n(iii) a generation of Population III stars has pre-enriched all of the gas.\n\nThe future prospects for improving the $z>3$ observational sample are \nexcellent. While further progress with HIRES and similar instruments\nis limited by the faintness of $z>4$ quasars, \nthe new ESI instrument on Keck~II will be ideal for surveying the $\\N{HI}$\nand metal content of a large ($N>20$) sample of very high redshift \ndamped \\lya systems.\nWe intend to pursue such a program over the next few years, taking full\nadvantage of the ever increasing sample of known $z>4$ quasars\n(\\citep{fan99a}).\n\n\n\\acknowledgments\n\nWe would like to thank A. McWilliam, E. Gawiser, M. Pettini,\nand M. Fall for insightful discussion and comments.\nWe thank T. Barlow for providing the HIRES reduction package.\nWe acknowledge the very helpful Keck support staff for their efforts\nin performing these observations. J.X.P. acknowledges support from a\nCarnegie postdoctoral fellowship. \n\n\\begin{thebibliography}{}\n\n\\bibitem[Cardelli \\& Savage 1995]{car95}\t% Fe oscillator strengths\nCardelli, J.A. and Savage, B.D. 1995, \\apj, 452, 275\n\n\\bibitem[Cen \\& Ostriker 1999]{cen99}\nCen, R. \\& Ostriker, J.P. 1999, \\apj, 519, 109L\n\n\\bibitem[Edmunds \\& Phillips 1997]{edmns97}\nEdmunds, M.G. \\& Phillips, S. 1997, \\mnras, 292, 733\n\n\\bibitem[Fall \\& Pei 1993]{fal93}\nFall, S.M. \\& Pei, Y.C. 1993, \\apj, 402, 479\n\n\\bibitem[Fall 1999]{fall99}\nFall, S.M. 1999, private communication\n\n\\bibitem[Fan et al.\\ 1999a]{fan99a}\nFan, X., SDSS collaboration, 1999a, AJ, submitted (astro-ph/9903237)\n\n\\bibitem[Grevesse \\& Sauval 1999]{grvss99}\nGrevesse, N. \\& Sauval, A.J. 1999, \\aa, 347, 348\n\n\\bibitem[Haehnelt et al.\\ 1998]{hae98}\nHaehnelt, M.G., Steinmetz, M. \\& Rauch, M. 1998, \n\\apj, 495, 647\n\n\\bibitem[Hoffman et al.\\ 1996]{hff96}\nHoffman, R.D. et al.\\ 1996, \\apj, 460, 478\n\n\\bibitem[Johnson 1999]{jhnsn99}\nJohnson, J. 1999, PhD thesis, UC Santa Cruz\n\n\\bibitem[Le Brun et al.\\ 1996]{lebr97}\nLe Brun, V., Bergeron, J., Boiss$\\rm \\acute e$, P., \\& Deharveng, J.M. 1997,\n\\aap, 321, 733\n\n\\bibitem[Lu et al.\\ 1996]{lu96}\nLu, L., Sargent, W.L.W., Barlow, T.A.,\nChurchill, C.W., \\& Vogt, S. 1996, \\apjsupp, 107, 475\n\n\\bibitem[Lu et al.\\ 1997]{lu97}\nLu, L., Sargent, W.L.W., \\& Barlow, T.A. 1997, astro-ph/9711298 % Radio\n\n\\bibitem[Malaney and Chaboyer 1996]{mny96}\nMalaney, R.A. and Chaboyer, B. 1996, \\apj, 462, 57\n\n\\bibitem[Maller et al.\\ 1999]{maller99}\nMaller, A.H., Somerville, R.S., Prochaska, J.X., \\& Primack, J.R.\n1999, {\\it After the Dark Ages: The 9th Annual October Astrophysics\nConference}, in press (astro-ph/9812369)\n\n\\bibitem[Pei et al.\\ 1999]{pei99}\nPei, Y.C., Fall, S.M., \\& Hauser, M.G. 1999, \\apj, submitted, \n(astro-ph/9812182)\n\n\\bibitem[Pettini et al.\\ 1994]{ptt94}\nPettini, M., Smith, L. J., Hunstead, R. W., and King,\nD. L. 1994, \\apj, 426, 79\n\n\\bibitem[Pettini et al.\\ 1997]{ptt97}\nPettini, M., Smith, L.J., King, D.L., \\& Hunstead, R.W. 1997,\n\\apj, 486, 665\n\n\\bibitem[Pettini 1999]{ptt99a}\t\t% High z Metallicity\nPettini, M. 1999, in Lecture Notes in Physics, Proceedings of ESO Workshop,\ned.\\ J. Walsh and M. Rosa (Berlin: Springer), in press,\nastro-ph/9902173\n\n\\bibitem[Pettini et al.\\ 2000]{ptt99b}\t% Low z DLA II\nPettini, M., Ellison, S.L., Steidel, C.C., Shapley, A.E., \\& Bowen, D.V. \n2000, \\apj, in press\n\n\\bibitem[Prochaska \\& Wolfe 1996]{pro96}\nProchaska, J. X. \\& Wolfe, A. M. 1996, \\apj, 470, 403\n\n\\bibitem[Prochaska \\& Wolfe 1997]{pro97}\nProchaska, J. X. \\& Wolfe, A. M. 1997, \\apj, 486, 73\n\n\\bibitem[Prochaska \\& Wolfe 1999]{pro99}\nProchaska, J. X. \\& Wolfe, A. M. 1999, \\apjs, 121, 369\n\n\\bibitem[Prochaska et al.\\ 2000]{pro00}\nProchaska, J. X. et al.\\, 2000, in preparation\n\n\\bibitem[Savage and Sembach 1991]{sav91}\nSavage, B. D. and Sembach, K. R. 1991, \\apj, 379, 245\n\n\\bibitem[Sneden et al.\\ 1991]{sne91}\nSneden, C., Gratton, R.G., \\& Crocker, D.A. 1991,\nA \\& A, 246, 354\n\n\\bibitem[Steidel et al.\\ 1998]{std98}\nSteidel, C.C., Adelberger, K., Dickinson, M., Giavalisco, M.,\nPettini, M., \\& Kellogg, M. 1998, \\apj, 492, 428\n\n\\bibitem[Storrie-Lombardi \\& Wolfe 2000]{storr99}\nStorrie-Lombardi, L.J. \\& Wolfe, A.M. 2000, \\apj, submitted\n\n\\bibitem[Vogt 1992]{vgt92}\nVogt, S. S. 1992, in {\\em ESO Conf. and Workshop Proc. 40,\nHigh Resolution Spectroscopy with the VLT}, ed. M.-H. Ulrich (Garching:\nESO), p. 223\n\n\\bibitem[Wolfe et al.\\ 1995]{wol95}\nWolfe, A. M., Lanzetta, K. M., Foltz, C. B., and\nChaffee, F. H. 1995, \\apj, 454, 698\n\n\\bibitem[Wolfe \\& Prochaska 1998]{wol98}\nWolfe, A.M. \\& Prochaska, J.X. 1998, \\apj, 494, 15L\n\n\n\\end{thebibliography}\n\n\\end{document}\n"
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{
"name": "astro-ph0002513.extracted_bib",
"string": "\\begin{thebibliography}{}\n\n\\bibitem[Cardelli \\& Savage 1995]{car95}\t% Fe oscillator strengths\nCardelli, J.A. and Savage, B.D. 1995, \\apj, 452, 275\n\n\\bibitem[Cen \\& Ostriker 1999]{cen99}\nCen, R. \\& Ostriker, J.P. 1999, \\apj, 519, 109L\n\n\\bibitem[Edmunds \\& Phillips 1997]{edmns97}\nEdmunds, M.G. \\& Phillips, S. 1997, \\mnras, 292, 733\n\n\\bibitem[Fall \\& Pei 1993]{fal93}\nFall, S.M. \\& Pei, Y.C. 1993, \\apj, 402, 479\n\n\\bibitem[Fall 1999]{fall99}\nFall, S.M. 1999, private communication\n\n\\bibitem[Fan et al.\\ 1999a]{fan99a}\nFan, X., SDSS collaboration, 1999a, AJ, submitted (astro-ph/9903237)\n\n\\bibitem[Grevesse \\& Sauval 1999]{grvss99}\nGrevesse, N. \\& Sauval, A.J. 1999, \\aa, 347, 348\n\n\\bibitem[Haehnelt et al.\\ 1998]{hae98}\nHaehnelt, M.G., Steinmetz, M. \\& Rauch, M. 1998, \n\\apj, 495, 647\n\n\\bibitem[Hoffman et al.\\ 1996]{hff96}\nHoffman, R.D. et al.\\ 1996, \\apj, 460, 478\n\n\\bibitem[Johnson 1999]{jhnsn99}\nJohnson, J. 1999, PhD thesis, UC Santa Cruz\n\n\\bibitem[Le Brun et al.\\ 1996]{lebr97}\nLe Brun, V., Bergeron, J., Boiss$\\rm \\acute e$, P., \\& Deharveng, J.M. 1997,\n\\aap, 321, 733\n\n\\bibitem[Lu et al.\\ 1996]{lu96}\nLu, L., Sargent, W.L.W., Barlow, T.A.,\nChurchill, C.W., \\& Vogt, S. 1996, \\apjsupp, 107, 475\n\n\\bibitem[Lu et al.\\ 1997]{lu97}\nLu, L., Sargent, W.L.W., \\& Barlow, T.A. 1997, astro-ph/9711298 % Radio\n\n\\bibitem[Malaney and Chaboyer 1996]{mny96}\nMalaney, R.A. and Chaboyer, B. 1996, \\apj, 462, 57\n\n\\bibitem[Maller et al.\\ 1999]{maller99}\nMaller, A.H., Somerville, R.S., Prochaska, J.X., \\& Primack, J.R.\n1999, {\\it After the Dark Ages: The 9th Annual October Astrophysics\nConference}, in press (astro-ph/9812369)\n\n\\bibitem[Pei et al.\\ 1999]{pei99}\nPei, Y.C., Fall, S.M., \\& Hauser, M.G. 1999, \\apj, submitted, \n(astro-ph/9812182)\n\n\\bibitem[Pettini et al.\\ 1994]{ptt94}\nPettini, M., Smith, L. J., Hunstead, R. W., and King,\nD. L. 1994, \\apj, 426, 79\n\n\\bibitem[Pettini et al.\\ 1997]{ptt97}\nPettini, M., Smith, L.J., King, D.L., \\& Hunstead, R.W. 1997,\n\\apj, 486, 665\n\n\\bibitem[Pettini 1999]{ptt99a}\t\t% High z Metallicity\nPettini, M. 1999, in Lecture Notes in Physics, Proceedings of ESO Workshop,\ned.\\ J. Walsh and M. Rosa (Berlin: Springer), in press,\nastro-ph/9902173\n\n\\bibitem[Pettini et al.\\ 2000]{ptt99b}\t% Low z DLA II\nPettini, M., Ellison, S.L., Steidel, C.C., Shapley, A.E., \\& Bowen, D.V. \n2000, \\apj, in press\n\n\\bibitem[Prochaska \\& Wolfe 1996]{pro96}\nProchaska, J. X. \\& Wolfe, A. M. 1996, \\apj, 470, 403\n\n\\bibitem[Prochaska \\& Wolfe 1997]{pro97}\nProchaska, J. X. \\& Wolfe, A. M. 1997, \\apj, 486, 73\n\n\\bibitem[Prochaska \\& Wolfe 1999]{pro99}\nProchaska, J. X. \\& Wolfe, A. M. 1999, \\apjs, 121, 369\n\n\\bibitem[Prochaska et al.\\ 2000]{pro00}\nProchaska, J. X. et al.\\, 2000, in preparation\n\n\\bibitem[Savage and Sembach 1991]{sav91}\nSavage, B. D. and Sembach, K. R. 1991, \\apj, 379, 245\n\n\\bibitem[Sneden et al.\\ 1991]{sne91}\nSneden, C., Gratton, R.G., \\& Crocker, D.A. 1991,\nA \\& A, 246, 354\n\n\\bibitem[Steidel et al.\\ 1998]{std98}\nSteidel, C.C., Adelberger, K., Dickinson, M., Giavalisco, M.,\nPettini, M., \\& Kellogg, M. 1998, \\apj, 492, 428\n\n\\bibitem[Storrie-Lombardi \\& Wolfe 2000]{storr99}\nStorrie-Lombardi, L.J. \\& Wolfe, A.M. 2000, \\apj, submitted\n\n\\bibitem[Vogt 1992]{vgt92}\nVogt, S. S. 1992, in {\\em ESO Conf. and Workshop Proc. 40,\nHigh Resolution Spectroscopy with the VLT}, ed. M.-H. Ulrich (Garching:\nESO), p. 223\n\n\\bibitem[Wolfe et al.\\ 1995]{wol95}\nWolfe, A. M., Lanzetta, K. M., Foltz, C. B., and\nChaffee, F. H. 1995, \\apj, 454, 698\n\n\\bibitem[Wolfe \\& Prochaska 1998]{wol98}\nWolfe, A.M. \\& Prochaska, J.X. 1998, \\apj, 494, 15L\n\n\n\\end{thebibliography}"
}
] |
astro-ph0002514
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The Future of Radio Astronomy: Options for Dealing with Human Generated Interference
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[
{
"author": "R. D. Ekers and J. F. Bell"
}
] |
--
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[
{
"name": "ekersr.tex",
"string": "% PSAMPLE2.TEX -- PASP Conference Proceedings macro package tutorial paper.\n\n% Lines starting with \"%\" are comments; they will be ignored by LaTeX.\n\n% This is a comprehensive example, meaning thyat we have made use of each\n% of the capabilities of the LaTeX + the PASP macro package that we think\n% you may need to use. If you want to see a \"base-bones\" sample paper,\n% take a look at psample1.tex.\n\n% The first item in a LaTeX file must be a \\documentstyle command to\n% declare the overall style of the paper.\n\n\\documentstyle[11pt,paspconf,epsf]{article}\n\n\\markboth{Ekers & Bell}{The Future of Radio Astronomy: Options for Dealing\nwith Human Generated Interference}\n\\setcounter{page}{33}\n\n% There is no more markup in the \"preamble\" for paspconf papers. You should\n% not define any \"personal\" LaTeX commands, in the preamble or anyplace else,\n% for that matter. Use only standard LaTeX commands or the additional ones\n% provided as part of the paspconf package.\n%\n% Now start with the real material for the paper, which is indicated with\n% \\begin{document}. Following the \\begin{document} command is the \"front\n% matter\" for the paper, viz., the title, author and address data, the\n% abstract, etc.\n\n\\def\\plotone#1{\\centering \\leavevmode\n\\epsfxsize=\\columnwidth \\epsfbox{#1}}\n\n\\begin{document}\n\n\\title{The Future of Radio Astronomy: Options for Dealing\nwith Human Generated Interference}\n\n\\author{R. D. Ekers and J. F. Bell}\n\\affil{ATNF CSIRO, PO Box 76 Epping NSW 1710, Sydney Australia; rekers@atnf.csiro.au~~~jbell@atnf.csiro.au}\n\n% Notice that some of these authors have alternate affiliations, which\n% are identified by the \\altaffilmark after each name. The actual alternate\n% affiliation information is typeset in footnotes at the bottom of the\n% first page, and the text itself is specified in \\altaffiltext commands.\n% There is a separate \\altaffiltext for each alternate affiliation\n% indicated above.\n\n% The abstract is entered in a LaTeX \"environment\", designated with paired\n% \\begin{abstract} -- \\end{abstract} commands. Other environments are\n% identified by the name in the curly braces.\n\n% Poster authors ONLY may omit the abstract in order to gain a little\n% more page space for the text of the poster.\n\n\\begin{abstract}\nRadio astronomy provides a unique window on the universe, allowing us\nto study: non-thermal processes (galactic nuclei, quasars, pulsars) at\nthe highest angular resolution using VLBI, with low opacity. It is the\nmost interesting wave band for SETI searches. To date it has yielded 3\nNobel prizes (microwave background, pulsars, gravitational radiation).\nThere are both exciting possibilities and substantial challenges for\nradio astronomy to remain at the cutting edge over the next 3\ndecades. New instruments like ALMA and the SKA will open up new\nscience if the challenge of dealing human generated interference can\nbe met. We summarise some of the issues and technological developments\nthat will be essential to the future success of radio astronomy.\n\\end{abstract}\n\n% Keywords should be included, but they are not printed in the hardcopy.\n\n\\keywords{SKA, radio astronomy, interference, mitigation, OECD, }\n\n% That's it for the front matter. On to the main body of the paper.\n% We'll only put in tutorial remarks at the beginning of each section\n% so you can see entire sections together.\n\n\n\n\\section{Telescope Sensitivity}\n\nMoore's law for the growth of computing power with time (ie a doubling every\n18 months) is often quoted as being vitally important for the success of the\nnext generation of radio telescopes (working at cm wavelengths) such as the\nSquare Kilometre Array (SKA). It is worth noting that radio astronomy has\nenjoyed a Moore's law of it own, having a exponentially improvement in\nsensitivity with time as shown in Figure \\ref{fig-1}. In fact the doubling\ntime is approximately 3 years, and has been in progress since 1940, giving\nan overall improvement in sensitivity of $10^5$. However, we are approaching\nthe fundamental limits of large mechanical dishes and noise limits of broad\nband receivers systems. We need to look to other means of extending this\ngrowth into the future.\n\n\\begin{figure}[htbp]\n%\\vspace{1.75in}\n\\plotone{sensitivity_curve3.ps}\n\\caption{Exponential growth of Radio Telescope sensitivity. Boxes indicate\nthe sensitivity attained when the systems were first commissioned. Since\nthen substantial improvements have been made to the system temperature and\nbandwidth of many telescopes. This diagram does not convey anything about\ntheir present capabilities. Projected capabilities are shown for ALMA,\nLOFAR, SKA (with 1,10,100 beams) and Arecibo (with 100 beams). The curent 13\nbeams system on Parkes is also shown.} \\label{fig-1}\n\\end{figure}\n\nWhy do we want to be on this exponential growth curve ? Fields of research\ncontinue to produce scientific advances while they maintain and exponential\ngrowth in some fundamentally limiting parameter. For radio astronomy\nsensitivity is definitely fundamentally limiting. An interesting question is\nwhether there are other parameters for which as exponential growth could be\nmaintained for a period of time. The first and most obvious point about\nexponential growth is that it cannot be sustained indefinitely. Can we\nmaintain it for sometime into the future ? There are 2 basic ways to stay on\nthe exponential curve: 1) Spend more money, 2) Take advantage of\ntechnological advances in other areas.\n\n\\subsection{International Mega Science Projects}\n\nInternational cooperation is now needed for dramatic improvements in\nsensitivity, because no one country can afford to do it alone. ALMA, the\nAtacama Large MM Array being developed by the USA, Europe and Japan is an\nexample. It will cost around \\$US700M spread over 1999-2007 and will provide\nan unprecedented opportunity to study redshifted molecular lines. The Square\nKilometre Array (SKA) which will work at centimetre wavelengths is likely to\nbe a collaboration of 10 or more nations, spending \\$US500M over 2008 to\n2015. \n\nThere are 2 basic approaches to funding such large projects: 1) User pays,\nwhere member countries pay for slice of the time, or 2) The member countries\nbuild the facility and make it openly accessible to all. Optical astronomy\nhas moved very much down path 1) as the Keck, VLT, Gemini type facilities\ndemonstrate. Radio astronomy has traditionally followed path 2) as have\nprojects like CERN. For future facilities there may be pressure to move more\ninto the user pays regime, possibly bringing about substantial change in the\ndynamics of the radio astronomy community.\n\nIn the past considerable extensibility was attained by each country\nlearning from the last one to build a telescope. That evolution is\nvery clear from Figure \\ref{fig-1}. Since we are likely to be moving\nto internationally funded projects, that path to extensibility is much\nmore restricted and designers must think very carefully about\ndesigning extensibility into the next generation telescopes.\n\n\\subsection{Extensibility Through Improved Technologies}\n\nFor a given telescope, past and present extensibility has been\nachieved by improvements in 3 main areas:\n\n\\begin{description}\n\n\\item{\\bf System Temperature:} Reber started out with a 5000 K system\ntemperature. Modern systems now run at around 20 K, meaning that if\neverything else was kept constant, Reber's telescope would now be 250\ntimes more sensitive than when first built. There are possibilities of\nsome improvements in future, but nothing like what was possible in the\npast.\n\n\\item{\\bf Band Width:} Telescopes like the GBT (Green bank Telescope)\nhaving bandwidths some 500 times greater than Reber's, will give\nfactors of 20--25 improvement in sensitivity. Some future improvements\nwill be possible, but again they will not be as large as in the past.\n\n\\item{\\bf Multiple Beams:} Whether in the focal or aperture plane, multiple\nbeam systems provide an excellent extensibility path, allowing vastly deeper\nsurveys than were possible in the past. Although multiple beam systems have\nbeen used for a number of experiments in the past, the full potential of\nthis approach is yet to be exploited. A notable example that has made a\nstride forward in this direction is the Parkes L band system (Stavely-Smith\net al. 1996, PASA, 13, 243). The fully sampled focal plane phased array\nsystem being developed at NRAO by Fisher and Bradley highlights the likely\npath for the future.\n\n\\end{description}\n\nUsing these three methods, a small telescope like the Parkes 64m has\nremained on the exponential curve and the forefront of scientific discovery\nfor 35 years (shown in Figure \\ref{fig-1} in 1962 and in 1997). Other\ntelescopes have of course undergone similar evolution and we only high light\nParkes as an example. Scope for continuing this evolution looks good for the\nnext decade, but beyond that more collecting area will be needed. A 64 beam\nsystem installed in 2010, would allow Parkes to stay on the curve for some\ntime. The technology to do this is probably only 3-4 years away from having\na realisable system, making it possible to jump well ahead of the\ncurve. Putting a 100 beam system on Arecibo by 2005 is possible and would\nallow Arecibo to jump way out in front of the curve as it did when first\nbuilt in 1964.\n\nThe relevance of Moore's law in this context, is that if it continues to\nhold true for the next 1-2 decades, it will provide the necessary back end\ncomputational power to realise the gains possible with multiple beams.\n\n\\section{Key Technologies, driving future developments:}\n\n\\begin{description}\n\\item{\\bf HEMT receivers} which are wide band, cheap, small, reliable, and\nlow noise systems with many elements.\n\n\\item{\\bf Focal plane arrays} giving large fully sampled fields of view\nwill allow rapid sky coverage for survey applications and great\nflexibility for targetted observations, including novel possibilities\nfor calibration and interference excision.\n\n\\item{\\bf Interference rejection} allowing passive use of spectrum,\noutside the bands allocated to passive uses. High dynamic range linear\nsystems, coupled with high temperature superconducting or photonic\nfilters will allow use of the spectrum between communication\nsignals. Adaptive techniques may allow some cochannel experiments, by\nremoving the undesired signals, so that astronomy signals can be seen\nbehind them.\n\n\\item{\\bf More computing capacity} may result in much more of the system\nbeing defined in software rather than hardware. This may lead to a\nvery different expenditure structure, where software is a capital\nexpense and computing harware is a considered as a consumable or\nrunning cost.\n\n\\item{\\bf Fibre/photonic} based beamforming and transmission of recorded\nsignals will revolutionise bandwidths and signal quality, especially\nfor high resolution science.\n\n\\item{\\bf Software radio and smart antenna} techniques which will\nallow great flexibilty in signal processing and signal selection.\n\n\\end{description}\n\n\\section{Summary of New Facilities}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lllll}\\hline\n\t& D(m)\t\t& Area(m2)\t & Freq(GHz)\t& Date \\\\ \\hline\nALMA\t& 64 x 12m\t& 7.2 x $10^3$ \t & 30.0 - 900 \t& 2007 \\\\\nGBT\t& 100m\t \t& 7.8 x $10^3$ \t & 0.30 - 86\t& 2000 \\\\ \n1hT\t& 512 x 5m\t& 1.0 x $10^4$ \t & 1.00 - 12\t& 2003 \\\\\nVLA \t& 27 x 25m\t& 1.3 x $10^4$ \t & 0.20 - 50\t& 2002 \\\\\nGMRT\t& 30 x 45m\t& 4.8 x $10^4$\t & 0.03 - 1.5\t& 1999 \\\\ \nSKA\t& undecided\t& 1.0 x $10^6$\t & 0.20 - 20\t& 2015 \\\\\nLOFAR\t& $10^6$ x 1m\t& 1.0 x $10^6$\t & 0.03 - 0.2\t& 2003 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Interference Sources and Spectrum Management}\n\nIt is important to be clear of what we mean when we talk about\ninterference. Radio astronomers make passive use of many parts of the\nspectrum legally allocated to communication and other services. As a\nresult, many of the unwanted signals are entirely legal and\nlegitimate. We will adopt the working definition that interference is\nany unwanted signal, getting into the receiving system.\n\nIf future telescopes like the SKA are developed with sensitivities up\nto 100 times greater than present sensitivities, it is quite likely\nthat current regulations will not provide the necessary\nprotection. There is also a range of experiments (eg redshifted\nhydrogen or molecular lines) which require use of the whole spectrum,\nbut only from a few locations, and at particular times, suggesting\nthat a very flexible approach may be beneficial. Other experiments\nrequire very large bandwidths, in order to have enough\nsensitivity. presently only 1-2\\% of the spectrum in the metre and\ncentimetre bands is reserved for passive uses, such as radio\nastronomy. In the millimetre band, much larger pieces of the spectrum\nare available for passive use, but the existing allocations are not\nnecessarily at the most useful frequencies.\n\n\\subsection{Terrestrial Sources of Interference}\n\nInterference can arise from a wide variety of terrestrial sources,\nincluding communications signals and services, electric fences, car\nignitions, computing equipment, domestic appliances and many others.\nAll of these are regulated by national authorities and the ITU\n(International Telecommunications Union). In the case of Australia,\nthere is a single communcations authority for whole country and\ntherefore for the whole continent. As a result there is a single\ndatabase containing information on the frequency, strength, location,\netc of every licensed transmitter. This makes negotiations\nover terrestrial spectrum use simpler in principle.\n\n\\subsection{Space \\& Air Borne Sources of Interference}\n\nRadio astronomy could deal with most terrestrial interfering signals,\nby moving to a remote location, where the density and strength of\nunwanted signals is greatly reduced. However with the increasing\nnumber of space borne telecom and other communications systems in low\n(and mid) Earth orbits, a new class of interference mitigation\nchallenges are arising - radio astronomy can run, but it cant hide !\nThe are several new aspects introduced to the interference mitigation\nproblem by this and they include: rapid motion of the transmitter,\nmore strong transmitters in dish sidelobes and possibly in primary\nbeam, and different spectrum management challenges.\n\nThere is an upside to the space borne communication systems in that\nthey help to develop the technology that make space VLBI possible,\nwhich leads to the greatest possible resolution.\n\nA classic example of the problems that can arise is provided by\nIrridium mobile communications system, which has a constellation of\nsatellites transmitting signals to every point on the surface of the\nEarth. Unfortunately in this case, there is some leakage into the\npassive band around 1612 MHz, with signals levels up to $10^{11}$ times\nas strong as signals from early universe.\n\n\\subsection{Radio Quiet Reserves}\n\nRadio quiet reserves have been employed in a number of places, with\nGreen Bank being a notable success. For future facilties such as the\nSKA and ALMA, the opportunity exists to set radio quiet reserve\nplanning in process a deceade before the instruments are actully\nbuilt. Radio quiet reserves of the future may take advantage not only\nof spatial and frequency orthogonality to human generated signals, but\nalso time, coding and other means of multiplexing. These later\nparameters may be particularly important for obtaining protection from\nspace borne undesired signals, a number of which illuminate most of\nthe Earths surface.\n\n\\section{Radio Wavelength Fundamentals}\n\nUndesired interfering signals and astronomy signals can differ (be\northogonal) in a range of parameters:\n\n\\noindent\n$\\bullet$ Frequency\\\\\n$\\bullet$ Time\\\\\n$\\bullet$ Position\\\\ \n$\\bullet$ Polarisation\\\\\n$\\bullet$ Distance\\\\\n$\\bullet$ Coding\\\\\n\n\nIt is extremely rare that interfering and astronomy signals do not possess\nsome level of orthogonality in this 6 dimensional parameter space. We\ntherefore need to develop sufficiently flexible back end systems to take\nadvantage of the orthogonality and separate the signals. This is of course\nvery similar to the kinds of problems faced by mobile communication\nservices, which are being addressed with smart antennas and software radio\ntechnologies.\n\n\\section{Interference Excision Approaches}\n\nThere is no silver bullet for detecting weak astronomical signals in the\npresence of undesired human generated signals. Spectral bands allocated for\npassive use, provide a vital window, which cannot be achieved in any other\nway. There are a range of techniques that can make some passive use of other\nbands possible and in general these need to be used in combined or\ncomplimentary way.\n\n\\begin{description}\n\n\\item{\\bf Screening} to prevent signals entering the primary elements\nof receivers.\n\n\\item{\\bf Front end filtering} (possibly using high temperature super\nconductors) to remove strong signals as soon as they enter the signal\npath.\n\n\\item{\\bf High dynamic range linear receivers} to allow appropriate\ndetection of both astronomy (signals below the noise) and interfering\nsignals.\n\n\\item{\\bf Notch filters} (digital or ananlog) to excise particularly\nbad spectral regions.\n\n\\item{\\bf Decoding} to remove multiplexed signals. Blanking of period\nor time dependent signals is a very succesful but simple case of this\nmore general approach.\n\n\\item{\\bf Calibration} to provide the best possible characterisation\nof interfering and astronomy signals.\n\n\\item{\\bf Cancellation} of undesired signals, before correlation using\nadaptive filters and after taking advantage of phase closure techniques\n(Sault et al. 1997)\n\n\\item{\\bf Adaptive beam forming} to steer nulls onto interfering\nsources. Conceptually, this is equivalent to cancellation, but it provides a\nway of taking advantage of the spatial orthogonality of astronomy and\ninterfering signals.\n\n\\end{description}\n\n\\subsection{Adaptive Systems}\n\nOf all the approaches listed above, the nulling or cancellation systems (may\nbe adaptive) are the most likely to permit the observation of weak astronomy\nsignals that are coincident in frequency with undesired signals. These\ntechniques have been used extensively in military, communications, sonar,\nradar, medicine and others (Widrow \\& Stearns 1985, Haykin 1995). Radio\nastronomers have not kept pace with these developments and in this case need\nto infuse rather diffuse technology in this area. A prototype cancellation\nsystem developed at NRAO (shown in Figure \\ref{fig-2}) has demonstrated 70dB\nof rejection on the lab bench and 30dB of rejection on real signals when\nattached to the 140 foot at Green Bank (Barnbaum \\& Bradley 1998). Adaptive\nNulling systems are being prototyped by NFRA in the Netherlands. However\ntheir application in the presence of real radio astronomy signals is yet to\nbe demonstrated and their toxicity to the weak astronomy signals needs to be\nquantified. The best prospect for doing this in the near future is recording\nbaseband data from exisiting telescope, containing both interferring and\nastronomy signals (Bell et al. 1999). A number of algorithms can then be\nimplemented is software and assessed relative to each other.\n\n\\begin{figure}[htbp]\n%\\vspace{1.75in}\n\\plotone{fisher2.ps}\n\\caption{Example of an adaptive cancellation system. From Barnbaum \\&\nBradley (1998).} \\label{fig-2}\n\\end{figure}\n\n\\section{The Telecommunications Revolution}\n\nWe cannot (and dont want to) impede this revolution, but we can try to\nminimise its impact on passive users of the radio spectrum and maximise the\nbenefits of technological advances. The deregulation of this industry has\nhad some impact on the politics. Major companies now play a prominent\n(dominate ?) role in the ITU (International Telecommunications\nunion). Protection of the bands for passive use must therefore addressed and\npromoted by government.\n\n\\section{Spectrum Pricing}\n\nThere may be some novel ways in which spectrum pricing could evolve in\norder to provide incentives for careful use of a precious\nresource. Radio astronomy and other passive users cannot in general\nafford commercial rates and therefore need government support. One\npossibility would be to have a green tax which could be used to fund\ninterference management and research. \n\nSuch strategies do not come without cost. While the long term economic\ncost may be relatively small, upfront R\\&D costs to an individual\ncompany may compromise their competitiveness. This issue must\ntherefore be addressed at national or international policy level.\n\nUnlike many other environmental resource use problems, spectrum over use is\nboth reversible and possible to curtail. This leads to certain political\nadvantages because politicians like to have problems which they can solve\nand this is a more soluble problem than many other environemntal problems.\n\n\\section{Remedies}\n\n\\begin{description}\n\\item {\\bf Siting Radio Telescopes} Choose remote sites with natural\nshielding helps but doesn't protect against satellite\ninterference. Establish radio quiet zones, using National government\nregulations. This is easier for fixed than mobile transmitters. Far side of\nmoon or L2 Lagrangian point are naturally occurring radio quite zones but\nare very expensive to use.\n\n\\item {\\bf OECD Mega Science Forum: Task Force on Radio Astronomy} The goals\nof the OECD mega science forum are complimentary to IAU efforts, providing a\npath for top down influence of governments which would otherwise not be\npossible. This task force aims to promote constructive dialog between:\nregulatory bodies, international radio astronomy community,\ntelecommunications companies, and government science agencies. It will\ninvestigate three approaches favoured by Megascience Forum: Technological\nsolutions, regulation, and radio quiet reserves.\n\n\\item {\\bf Environmental Impact} In the meter and cm band $<$1\\% is\nallocated to passive use! 99\\% already used, resource use has been\nextravagant. Almost all the spectrum at wavelenghts $>$ 1cm are now\npolluted, and situation is rapidly deteriorating at shorter wavelengths.\nIt is reversable and sheareable in more creative ways! in contrast to\nmost other pollution problems.\n \n\n\\end{description}\n\n\\section{Funding of radio astronomy}\n\nUniversity based radio astronomy research in the USA has suffered relative\nto other wavelengths for 2 reasons: 1) The centralised development at NRAO\nhas made it difficult for many universities to remain involved in technical\ndevelopments. 2) Space based programs in infrared, optical, UV, X-ray, and\nGamma ray bands have a rather different funding structure, where access to\nresearch funds is based on successful observing proposals. Radio astronomy\nhas no access to such funds and therefore is a relatively uneconomical\npursuit for astronomers. This method of funding is being taken up in other\ncountries and radio astronomy needs to find a way to join the scheme. More\nglobally, radio astronomy has suffered relative to other wavelengths,\nbecause data acquisition, reduction and analysis is unnecessarily\ncomplex. Researchers have to spend a lot more effort in data processing than\nother areas of astronomy. For example, in many other bands, fully calibrated\ndata lands on the researchers desk a few days after the observations were\ntaken. Radio astronomy needs to finds ways to move into this regime (as\nWesterbork have done to some degree), but at the same time preserve the vast\nflexibility that can be derived from measuring the electric field at the\naperture.\n\n\\section{Conclusions}\n\nThe possibilities for the future of radio astronomy are good, but\nthere are some challenges issues for the community to consider and\naddress:\n\n\\noindent\n$\\bullet$ Whole of radio spectrum needed for redshifted lines\\\\\n$\\bullet$ About 2\\% of spectrum is reserved for passive use by regulation\nso must develop other approaches\\\\\n$\\bullet$ We cannot (and don't want to) impede the telecommunications revolution\\\\\n$\\bullet$ Radio astronomy has low credibility until we use advanced techniques\\\\\n$\\bullet$ Essential to influence government policy\\\\\n$\\bullet$ Astronomers should have a uniform position\\\\\n$\\bullet$ Threatening language doesn't help\\\\\n$\\bullet$ Is interference harmful?\\\\\n\n% In this section, we see the use of the \\subsection command to set off\n% an independent subsection. We only have one here; usually there would\n% be several.\n%\n% We show the use of several of the displayed math environments described\n% in the User Guide, and you get a healthy dose of mathematical typesetting\n% examples. Also, observe the use of the LaTeX \\label command after the\n% \\subsection to give a symbolic tag to the subsection for cross-referencing\n% in a \\ref command. LaTeX automatically numbers the sections, equations,\n% tables, etc. as it goes, so in general you don't know what number something\n% is going to have. We'll refer to the \"hairymath\" section a little later.\n\n% Now comes the reference list. Since we typed out the citations ourselves,\n% the reference list is enclosed in a \"references\" environment. Each\n% new reference begins with a \\reference command which sets up the proper\n% indentation. Typography that may be required in the reference list by\n% the editorial staff must be included by the author.\n%\n% Observe the \"standard\" order for bibliographic material: author name(s),\n% publication year, journal name, volume, and page number for articles.\n% Some journal names are available as macros; see the WGAS markup\n% instructions for a listing of which ones have been \"macro-ized\".\n% Note the use of curly braces to delimit the font changes: it is essential\n% that this be done to limit the scope of the font declaration.\n%\n% There is no need to engage in any other typographic manipulation.\n\n\\begin{references}\n\n\\reference Barnbaum, c. \\& Bradley, R., 1998, AJ, 116, 2598.\n\n\\reference Bell J.~F., et al. 1999 \"Software radio telescope: interference\n mitigation atlas and mitigation strategies\", in Perspectives in Radio\n Astronomy: Scientific Imperatives at cm and m Wavelengths (Dwingeloo:\n NFRA), Edited by: M.P. van Haarlem \\& J.M. van der Hulst.\n\n\\reference Haykin, S., 1995, ``Adaptive Filter Theory'' Prentice Hall.\n\n\\reference Sault, B., Ekers, R., Kewley, L. 1997 ``Cross-correlation\napproaches to interference mitigation'' Sydney SKA workshop\nhttp://www.atnf.csiro.au/SKA/WS/\n\n\\reference Smolders, A.~B., 1999, ``Phased-array system for the next\ngeneration of radio telescopes'', in Perspectives in Radio Astronomy:\nScientific Imperatives at cm and m Wavelengths (Dwingeloo: NFRA), Edited by:\nM.P. van Haarlem \\& J.M. van der Hulst.\n\n\\reference Staveley-Smith L. et al. 1996, PASA, 13, 243 \"The Parkes 21cm\nMultibeam Receiver\". An overview of the science and overall system design of\nthe multibeam receiver.\n\n\\reference Widrow, B. \\& Stearns, S., 1985, ``Adaptive Signal Processing''\nPrentice Hall\n\n\\reference Interference Mitigation Web pages\nhttp://www.atnf.csiro.au/SKA/intmit/\n\n\\reference AN SKA web site http://www.atnf.csiro.au/SKA/\n\\end{references}\n\n% That's all, folks.\n%\n% The technique of segregating major semantic components of the document\n% within \"environments\" is a very good one, but you as an author have to\n% come up with a way of making sure each \\begin{whatzit} has a corresponding\n% \\end{whatzit}. If you miss one, LaTeX will probably complain a great\n% deal during the composition of the document. Occasionally, you get away\n% with it right up to the \\end{document}, in which case, you will see\n% \"\\begin{whatzit} ended by \\end{document}\".\n\n\n\\end{document}\n"
}
] |
[] |
astro-ph0002515
|
Radio Frequency Interference
|
[
{
"author": "R. D. Ekers and J. F. Bell"
}
] |
We describe the nature of the interference challenges facing radio astronomy in the next decade. These challenges will not be solved by regulation only, negotiation and mitigation will become vital. There is no silver bullet for mitigating against interference. A successful mitigation approach is most likely to be a hierarchical or progressive approach throughout the telescope and signal conditioning and processing systems. We summarise some of the approaches, including adaptive systems.
|
[
{
"name": "pune99.tex",
"string": "\\documentstyle[11pt,paspconf,epsf]{article}\n\n\\markboth{Ekers & Bell}{Radio Frequency Interference}\n\n\\def\\plotone#1{\\centering \\leavevmode\n\\epsfxsize=\\columnwidth \\epsfbox{#1}}\n\n\\def\\plot08#1{\\centering \\leavevmode\n\\epsfxsize=0.8\\columnwidth \\epsfbox{#1}}\n\n\\begin{document}\n\n\\title{Radio Frequency Interference}\n\n\\author{R. D. Ekers and J. F. Bell}\n\\affil{ATNF CSIRO, PO Box 76 Epping NSW 1710, Sydney Australia; rekers@atnf.csiro.au~~~jbell@atnf.csiro.au}\n\n\\begin{abstract}\nWe describe the nature of the interference challenges facing radio astronomy\nin the next decade. These challenges will not be solved by regulation only,\nnegotiation and mitigation will become vital. There is no silver bullet for\nmitigating against interference. A successful mitigation approach is most\nlikely to be a hierarchical or progressive approach throughout the telescope\nand signal conditioning and processing systems. We summarise some of the\napproaches, including adaptive systems.\n\n\\end{abstract}\n\n\\keywords{radio astronomy, interference, mitigation, OECD, adaptive systems,\ncommunications}\n\n\\section{The RFI Challenge and Spectrum Management}\n\nIf future telescopes like the SKA are developed with sensitivities up to 100\ntimes greater than present sensitivities, it is quite likely that current\nregulations will not provide the necessary protection against\ninterference. There is a range of experiments (eg redshifted hydrogen or\nmolecular lines) which require use of arbitrary parts of the spectrum, but\nonly at a few locations, and at particular times, suggesting that a very\nflexible approach may be beneficial. Other experiments require very large\nbandwidths, in order to achieve enough sensitivity. As shown in Figure\n\\ref{alloc}, presently only 1-2\\% of the spectrum in the metre and\ncentimetre bands is reserved for passive uses, such as radio astronomy\n(Morimoto 1993). In the millimetre band, much larger pieces of the spectrum\nare available for passive use, but the existing allocations are not\nnecessarily at the most useful frequencies. Current regulations alone will\nbe inadequate, we need technology as well as regulation. We cannot (and do\nnot want to) impede the telecommunications revolution, but we can try to\nminimise its impact on passive users of the radio spectrum and maximise the\nbenefits of technological advances. Further information on many of the\ntopics discussed below is available on http://www.atnf.csiro.au/SKA/intmit/.\n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\plotone{spectrum_allocation2.ps}\n\\caption{Spectral lines (at zero redshift) are indicated in absorption or\nemission over the 0--30 GHz band. The boxes indicate the bands allocated for\npassive radio astronomy uses. Figure from Morimoto 1993.}\n\\label{alloc}\n\\end{center}\n\\end{figure}\n\n\\section{Classes of Interference}\n\nIt is important to be clear of what we mean when we talk about\ninterference. Radio astronomers make passive use of many parts of the\nspectrum legally allocated to communication and other services. As a\nresult, many of the unwanted signals are entirely legal and legitimate. We\nwill adopt the working definition that interference is any unwanted signal\nentering the receiving system.\n\nInterfering signals vary a great deal in their source and nature. This\nnaturally leads to different mitigation approaches. Local sources of\ninterference include things internal to telescope instruments, networking\nfor IT systems, and general and special purpose digital processors in the\nobservatory. Interference compliance testing, shielding, separate power\ncircuits, minimising nearby equipment are key steps that need to be taken to\nminimise this kind of interference.\n\nExternal interference may arise from fixed or moving sources. Not all\nmethods of mitigation apply to both: in fact methods that work well for\nfixed sources, may not work at all for moving sources, due to problems like\nside lobe rumble. Interference may be naturally occurring or human\ngenerated. Examples of naturally occurring interference include: the\nground, sun, other bright radio sources, and lightening. Human\ngenerated interference may come from broadcast services (eg TV, radio),\nvoice and data communications (eg mobile telephones, two-way radio, wireless\nIT networks), navigation systems (eg GPS, GLONASS), radar, remote sensing,\nmilitary systems, electric fences, car ignitions, and domestic appliances\n(eg microwave ovens) (Goris 1998).\n\nThe vast majority of these operate legally with in their allocated bands,\nregulated by national authorities and the ITU (International\nTelecommunications Union). However there are sources of interference, such\nas the Iridium mobile communications systems, whose signals leak into bands\nprotected for passive use. In this case, these interfering signals are\n$10^{11}$ times stronger than the signal from the early universe. In the\ncase of Australia, there is a single communications authority for whole\ncountry and therefore for the whole continent. As a result there is a single\ndatabase containing information on the frequency, strength, location, etc of\nevery licensed transmitter (Sarkissian 2000). A key point therefore, is that\nthe modulation schemes and other characteristics of the vast majority of\nthese signal are known. Their effect on radio telescopes is not only\npredictable, but can be modelled and used to excise the unwanted signals.\n\nRadio astronomy could deal with most terrestrial interfering signals, by\nmoving to a remote location, where the density and strength of unwanted\nsignals is greatly reduced. As shown in Figure \\ref{forte}, this is getting\nmore and more difficult, but there are still some possibilities. However\nwith the increasing number of space borne telecom and other communications\nsystems in low (and mid) Earth orbits, a new class of interference\nmitigation challenges are arising - radio astronomy can run, but it cannot\nhide ! The are several new aspects introduced into the interference\nmitigation problem by this and they include: rapid motion of the transmitter\non satellites, more strong transmitters in dish side lobes and possibly in\nprimary beam, and different spectrum management challenges, because no place\non Earth is free from interference from the sky.\n\n\n \n\\begin{figure}[htbp]\n\\begin{center}\n\\plotone{forte.ps}\n\\caption{Forte satellite: 131MHz, RF background data for calibration of\nlightening monitors provides a map of signals emanating from the Earths\nsurface, many of which are human generated. Acquired by the Los Alamos\nNational Laboratory FORTE spacecraft, Principal Investigator\nA. R. Jacobson. Data courtesy of R. J. Strangeway, UCLA.}\n\\label{forte}\n\\end{center}\n\\end{figure}\n\n\n\\section{RFI fundamentals}\nUndesired interfering signals and astronomy signals can differ (be\northogonal) in a range of parameters, including: frequency, time, position,\npolarisation, distance, coding, positivity, and multi path. It is extremely\nrare that interfering and astronomy signals do not possess some level of\northogonality in this $\\geq 8$ dimensional parameter space. We therefore\nneed to develop sufficiently flexible signal processing systems to take\nadvantage of the orthogonality and separate the signals. This is of course\nvery similar to the kinds of problems faced by mobile communication\nservices, which are being addressed with smart antennas and software radio\ntechnologies. Examples in radio astronomy to date include the use of the\ntime or frequency phase space or even better both, as in pulsar studies\nwherer the time/frequency dispersion relation can be used, and the\nrequirement that signals are positive in very low frequency studies. Antenna\narrays could take advantage of the position and distance (curvature of\nwavefront) phase space. Human generated interference is normally polarised,\nso unpolarised astronomical signals can be observed by measuring the\nunpolarised component $(I - (U^2 + Q^2 + V^2)^{1/2})$ (R. Fisher, private\ncommunication.\n\n\\subsection{Mitigation Strategies and Issues}\n\n\nThere is no silver bullet for detecting weak astronomical signals in the\npresence of strong undesired naturally occurring or human generated\nsignals. Spectral bands allocated for passive use provide a vital window,\nwhich cannot be achieved in any other way. It is important to characterise\nthe RFI so that the number, strength, band width, duty cycle, spatial and\nfrequency distributions, and modulation and coding schemes can all be used\nto advantage in modelling and mitigating RFI. Doing this at low frequencies\ngives greater sensitivity due to the effects of harmonic content and ease of\npropagation. In order to do this, the telescope and instruments must be\ncalibrated to provide the best possible characterisation of interfering and\nastronomy signals. A/D converters must be fast enough to give sufficient\nbandwidth, with a sufficient number of bits so that both strong and weak\nsignals are well sampled. There are a range of techniques that can make\npassive use of other bands possible and in general these need to be used in\na progressive or hierarchical way.\n\n\\noindent\n$\\bullet~~~${\\bf Remove at source} is obviously best, but that is often not possible,\\\\\n$\\bullet~~~${\\bf Regulation} providing radio quiet frequencies or regions,\\\\\n$\\bullet~~~${\\bf Negotiation} with owners users can lead to win-win solutions,\nfor example replacing nearby radio links with underground fibres, removes\ninterference and improves voice and data connectivity for users,\\\\\n$\\bullet~~~${\\bf Avoid interference} by choosing appropriate locations with\nterrain screening or radio quiet zones,\\\\\n$\\bullet~~~${\\bf Move to another frequency},\\\\\n$\\bullet~~~${\\bf Screening} to prevent signals entering the primary elements\nof receivers,\\\\\n$\\bullet~~~${\\bf Far Side lobes} of primary and secondary elements must be both\nminimised and well characterised,\\\\\n$\\bullet~~~${\\bf Minimise coherent signals} through out the array and thereby\nallow the natural rejection of the array to deal with the incoherent\nsignals,\\\\\n$\\bullet~~~${\\bf Front end filtering}, using for example high temperature\nsuper conducting filters with high Q to reject strong signals in narrow\nbands, before they cause saturation effects.\\\\\n$\\bullet~~~${\\bf High dynamic range linear receivers} to allow appropriate\ndetection of both astronomy (signals below the noise) and very strong\ninterfering signals,\\\\ \n$\\bullet~~~${\\bf Notch filters} (analog, digital or photonic) to excise\nbad spectral regions,\\\\\n$\\bullet~~~${\\bf Clip} samples from data streams to mitigate burst type\ninterference,\\\\\n$\\bullet~~~${\\bf Decoding} to remove signals with complex modulation and\nmultiplexing schemes. Blanking of period\nor time dependent signals is a very successful but simple case of this\nmore general approach,\\\\\n$\\bullet~~~${\\bf Cancellation} of undesired signals, before correlation using\nfixed and adaptive signal processing (harris, 2000),\\\\\n$\\bullet~~~${\\bf Post correlation cancellation} of undesired signals,\ntaking advantage of phase closure techniques (Sault 2000)\\\\\n$\\bullet~~~${\\bf Parametric techniques} allow the possibility of taking\nadvantage of known interference characteristics to excise it (Ellingson,\n2000),\\\\ \n$\\bullet~~~${\\bf Adaptive beam forming} to steer one or more nulls onto\ninterfering \nsources. This is equivalent to cancellation, but it provides a\nway of taking advantage of the spatial orthogonality of astronomy and\ninterfering signals,\\\\\n$\\bullet~~~${\\bf Use of Robust statistics} in data processing to minimise the\neffects of outliers.\n\n\\subsection{Which signal processing regime: traditional analog, digital, photonic ?}\n\nIn most applications of signal processing, there is a strong trend towards\nthe use of digital techniques, as well as photonic techniques. The\nfundamental reason for this is that digital and photonic devices have cost\ncurves which are evolving much more rapidly than traditional analog\nsystems. In addition to that, they open up new techniques and offer\nsubstantial reduction in computational effort in many cases. The inherent\nimmunity of photonic approaches to radio interference also creates\nfunctional advantages (Minasian 2000). Astronomers are joining these trends\nfor exactly the same reasons. The jury is still out on what the appropriate\nbalance or mixture of these techniques will be.\n\n\\section{Adaptive EMI rejection}\n\nAdaptive rejection algorithms can be either constrained or\nunconstrained. Constrained algorithms generally incorporate either a model\nor a copy of the desired or interfering signal, which is used control the\nadaption. For example it may be constrained so that only those signals with\na certain coding or chip sequence are removed (Ellingson, 2000). Most\nastronomy signals are expected to be pure noise, so one could envisage a\nconstraint that rejects all non noise like signals. In the case of\nunconstrained adaption, some algorithms (predictive adaptive algorithms)\nsimply assume that the interference is much stronger than the signal and\njust use previous data samples to predict the following data samples for\ncancellation (harris, 2000). If that assumption cannot be made, another\napproach is to block the desired signal and let the unconstrained algorithm\nwork on the remaining signals. This is often done using a blocking matrix,\nwhich can be thought of as an operator that applies a set of complex weights\nwhich block certain signals, while passing everything else. Advantages of\nthis approach are that it can deal with multiple interfering signals which\nare changing in time and space, without affecting the signal to noise of the\ndesired signal.\n\nA key ingredient of constrained adaptive algorithms is a reference channel\nthat maximises interference to noise ratio for the ensemble of\ninterferers. One way of achieving this is using additional omni-directional\nantennas or arrays which at least matches the gain of the side lobe response\nof the main array.\n\n\\subsection{Adaptive Interference Cancelling}\n\nOf all the approaches listed above, the nulling or cancellation systems (may\nbe adaptive or predictive) are the most likely to permit the observation of\nweak astronomy signals that are coincident in frequency or space with\nundesired signals. There is an important space-time duality with cancelling\nalgorithms. Any algorithm that works in the time domain can also be applied\nin the space domain.\n\nThese techniques have been used extensively in communications, sonar, radar,\nmedicine and others (Widrow \\& Stearns 1985, Haykin 1995). Radio astronomers\nhave not kept pace with these developments and in this case need to infuse\nrather than diffuse technology in this area. A prototype time based\ncancellation system developed at NRAO (shown in Figure \\ref{fig-2}) has\ndemonstrated 70dB of rejection on the lab bench and 30dB of rejection on\nreal signals when attached to the 140 foot at Green Bank (Barnbaum \\&\nBradley 1998). Adaptive nulling systems are being prototyped by NFRA in the\nNetherlands (van Ardenne, these proceedings). Combined space-time approaches\nhave been used to cancel interference in GPS receivers (Trinkle, 2000).\nHowever, in all cases, the application in the presence of real radio\nastronomy signals is yet to be demonstrated and their effects on the weak\nastronomy signals needs to be quantified. A good prospect for doing this in\nthe near future is recording baseband data from existing telescopes,\ncontaining both interfering and astronomy signals and simulating the\nreceiver system in software (Bell et al. 1999). A number of algorithms can\nthen be implemented is software and assessed relative to each other.\n\n\\begin{figure}[htbp]\n%\\vspace{1.75in}\n\\plot08{fisher2.ps}\n\\caption{A Conceptual View of Adaptive Interference Cancelling. From\nBarnbaum \\& Bradley (1998).} \\label{fig-2}\n\\end{figure}\n\nBeam forming and adaptive nulling as are not necessarily being done\nsequentially, but rather in parallel. While there are some sequential\nschemes (genetic algorithms for example), most approaches simultaneously\nsolve for the coefficients that give the desired beams and nulls. One can\nthink of this as an optimisation problem. For example, in the minimum\nvariance beam former, the ``goal'' is to minimise output power, and the\n``constraint'' is maintain constant gain in a certain direction. The goal\nforces the nulls onto all interfering signals not coming from the direction\nof the astronomical source, and the constraint protects the beam gain\n(Ellingson, 2000). Of course, it is only protected for one direction, so\nthere is still shape distortion. In general for an N element array, you can\nform up to N-1 nulls. However if more control over the main beam shape is\nrequired, one may use other beam formers, which form a smaller number of\nnulls, and use more degrees of freedom to control the main beam shape.\n\nA physical interpretation of why you can form nulls without wrecking the\nbeam might go as follows: Phased arrays with many elements (not equally\nspaced), have lots of nulls, and they are all over the sky once you get a\nreasonable distance from the main beam. Imagine changing the coefficients a\nlittle to get the closest one on to an interferer. Very little variation in\nthe coefficients is required. Since the difference is so small, the main\nbeam hardly notices. The other nulls will shift around, of course, because\nthey are sensitive to small changes in the coefficients. Close to the main\nbeam, the nulls are further apart, so you need a bigger variation in the\ncoefficients to nudge the closest null into place - hence the increased\ndistortion in the main beam in this case. It may be necessary to record the\nweights applied to generate the nulls, so that the beam shape changes can be\ncalibrated out later (Cram, 2000).\n\n\\section{Real time v post correlation}\n\nReal time systems permit full recovery of temporal information in the signal\nrequired. Real time systems have been well studied and numerous examples\ncan be found in other fields such as: radar, sonar, communications, defence\nanti jam, speech processing, and medicine. For example, there are existing\nsystems which are capable of nulling up to 7 simultaneous moving jammers. In\nradio astronomy we operate in a totally different regime in which the\nastronomical signals are weak and noise like. We only wish to measure the\ntime averaged statistical properties of the signals. For example in\naperture synthesis the time averaged coherence between two antennas. Since\nwe don't have to recover the signal modulation, radio astronomy does not have\nto use the real time algorithms developed for communications and radar. In\nsuch post correlation systems (Sault 2000) the information is only contained\nin the statistical properties of the signal which may vary slowly in time,\nfrequency, space or direction. Both the signal of interest and the\ninterference obey phase and amplitude closure relations. This results in an\nover determined set of equations which form a closed set which can be used\nto self calibrate the array for both the source and the interferer.\n\nWe conjecture that: {\\bf {\\it ``post correlation processing of time averaged\nsignals can achieve the same RFI rejection as in real-time algorithms and\nthat self calibration (phase closure) techniques provide powerful additional\nconstraints''}}.\n\n\\begin{references}\n\n{\\small\n\n\\reference Barnbaum, c. \\& Bradley, R., 1998, AJ, 116, 2598.\n\n\\reference Bell J., et al. 1999 ``Software radio telescope: interference\n mitigation atlas and mitigation strategies'', in Perspectives in Radio\n Astronomy: Scientific Imperatives at cm and m Wavelengths (Dwingeloo:\n NFRA), Edited by: M.P. van Haarlem \\& J.M. van der Hulst.\n\n\\reference Cram, L., 2000, ``System Considerations'', Radio Frequency\nInterference Mitigation Strategies.\nhttp://www.atnf.csiro.au/SKA/intmit/atnf/conf/\n\n\\reference Ellingson, S., 2000, ``Interference Mitigation techniques'',\nRadio Frequency Interference\nMitigation Strategies. http://www.atnf.csiro.au/SKA/intmit/atnf/conf/\n\n\\reference harris, f., 2000, ``Radio Frequency Interference Mitigation'',\nRadio Frequency Interference\nMitigation Strategies. http://www.atnf.csiro.au/SKA/intmit/atnf/conf/\n\n\\reference Goris, M., 1998, ``Categories of Radio Interference'', NFRA\nTechnical Report - 415/MG/V2.3,\nhttp://www.nfra.nl/skai/archive/technical/index.shtml\n\n\\reference Haykin, S., 1995, ``Adaptive Filter Theory'' Prentice Hall.\n\n\\reference Minasian, R., 2000, ``Photonics in radio astronomy'', Radio\nFrequency Interference Mitigation Strategies.\nhttp://www.atnf.csiro.au/SKA/intmit/atnf/conf/\n\n\\reference Morimoto, M., 1993, ``Charm \\& Crisis in Radio Astronomy'',\nModern Radio Science.\n\n\\reference Sarkissian, J., 2000, ``Transmitter database\nvisualisation'',Radio Frequency\nInterference Mitigation Strategies.\nhttp://www.atnf.csiro.au/SKA/intmit/atnf/conf/\n\n\\reference Sault, R., 2000, ``Synthesis Arrays'', Radio Frequency\nInterference Mitigation Strategies.\nhttp://www.atnf.csiro.au/SKA/intmit/atnf/conf/\n\n\\reference Trinkle, M., 2000, ``A DSP testbed for interference mitigation in\nGPS'',Radio Frequency\nInterference Mitigation Strategies.\nhttp://www.atnf.csiro.au/SKA/intmit/atnf/conf/\n\n\\reference Widrow, B. \\& Stearns, S., 1985, ``Adaptive Signal Processing''\nPrentice Hall.\n\n}\n\n\\end{references}\n\n\n\\end{document}\n"
}
] |
[] |
astro-ph0002516
|
Cancellation of GLONASS signals from Radio Astronomy Data
|
[
{
"author": "Steven W. Ellingson\\supit{a}"
},
{
"author": "John D. Bunton\\supit{b}"
},
{
"author": "Jon F. Bell\\supit{c} \\skiplinehalf \\supit{a}OSU ElectroScience Laboratory 1320 Kinnear Road"
},
{
"author": "Columbus OH 43212 USA"
},
{
"author": "\\supit{b}CSIRO Telecommunications and Industrial Physics"
},
{
"author": "PO Box 76 Epping NSW 1710"
},
{
"author": "AUSTRALIA"
},
{
"author": "\\supit{c}CSIRO Australia Telescope National Facility"
},
{
"author": "PO Box 76 Epping NS W 1710"
}
] |
Astronomers use the 1612 MHz OH spectral line emission as a unique window on properties of evolved stars, galactic dynamics, and putative proto-planetary disk systems around young stars. In recent years, experiments using this OH line have become more difficult because radio telescopes are very sensitive to transmissions from the GLONASS satellite system. The weak astronomical signals are often undetectable in the presence of these unwanted human generated signals. In this paper we demonstrate that GLONASS narrow band signals may be removed using digital signal processing in a manner that is robust and non-toxic to the weak astronomy signals, without using a reference antenna. We present results using real astronomy data and outline the steps required to implement useful systems on radio telescopes.
|
[
{
"name": "spie2000b.tex",
"string": "\\documentstyle[spie]{article} \n\\input{psfig} \n\\title{Cancellation of GLONASS signals from Radio Astronomy Data} \n\n\\author{Steven W. Ellingson\\supit{a}, John D. Bunton\\supit{b}, Jon\nF. Bell\\supit{c} \n\\skiplinehalf \n\\supit{a}OSU ElectroScience Laboratory 1320 Kinnear Road, Columbus OH 43212\nUSA \\\\\n\\supit{b}CSIRO Telecommunications and Industrial Physics, PO Box 76 Epping\nNSW 1710, AUSTRALIA\\\\\n\\supit{c}CSIRO Australia Telescope National Facility, PO Box 76 Epping NS\nW 1710, AUSTRALIA \\\\\n}\n\n\\authorinfo{Further author information: (Send correspondence to S.W.E. or\nJ.F.B.)\\\\S.W.E. : E-mail: ellingson.1@osu.edu~~~~~J.F.B. : E-mail:\njbell@atnf.csiro.au~~~~~J.D.B. : E-mail: john.bunton@tip.csiro.au}\n\n\\begin{document} \n\\maketitle \n\n\\begin{abstract}\nAstronomers use the 1612 MHz OH spectral line emission as a unique window on\nproperties of evolved stars, galactic dynamics, and putative proto-planetary\ndisk systems around young stars. In recent years, experiments using this OH\nline have become more difficult because radio telescopes are very sensitive\nto transmissions from the GLONASS satellite system. The weak astronomical\nsignals are often undetectable in the presence of these unwanted human\ngenerated signals. In this paper we demonstrate that GLONASS narrow band\nsignals may be removed using digital signal processing in a manner that is\nrobust and non-toxic to the weak astronomy signals, without using a reference\nantenna. We present results using real astronomy data and outline the steps\nrequired to implement useful systems on radio telescopes.\n\\end{abstract}\n\n%>>>> Please include a list of keywords after the abstract \n\n\n\\keywords{GLONASS, interference, cancellation, radio astronomy}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{INTRODUCTION}\n\\label{sec:intro} % \\label{} allows reference to this section\n\nMany papers in the astronomical literature cite problems with interference\nfrom the Russian {\\em Global'naya Navigatsionnaya Sputnikovaya Sistema}\n(GLONASS) system of navigational satellites when trying to observe 1612 MHz\nOH spectral line emission. Galt (1991) \\cite{gal91} and Combrinck et\nal. (1994) \\cite{cwg94} both present data demonstrating the damaging effect\nof GLONASS signals on astronomy data. Some reports have stated that up to\n50\\% of all observations have had to be discarded \\cite{gal91}. The\nscientific merits of OH spectral line observations are discussed in detail\nelsewhere \\cite{coh89,hh85}; however, there is no question that this is\nextremely valuable spectrum whose continued use is essential to radio\nastronomy.\n\nOne possible solution to the problem is regulation; this is being addressed\nwithin international organisations such as the ITU and URSI. However,\nregulation cannot be expected to recover the spectrum into which the GLONASS\nsystem already transmits. The solution most often employed by radio\nastronomers in dealing with unwanted signals is to put their telescopes in\nremote locations. However, when dealing with signals that emanate from\nEarth-orbiting satellites, that method obviously fails. The next most\nobvious solution is not to observe when interfering signals are present, or\nsimply throw away affected data \\cite{gal91}. Some ``GLONASS aware'' tools\nhave been developed that allow dynamic scheduling observations in order to\nminimise interference \\cite{cwg94}. However, the strategy of avoidance\nresults in the loss of valuable telescope time, which often amounts\nthousands of dollars per day, a better solution is desired.\n\nHere we present a direct, technical solution to the problem. We have\ndeveloped and demonstrated a parametric signal processing algorithm which\nidentifies GLONASS signals present in the pre-detection, complex baseband\ntelescope output, and removes them. This algorithm results in a high degree\nof suppression with negligible distortion of radio astronomical signals. We\nbelieve this approach can be applied to interference from the U.S. Global\nPositioning System (GPS) and possibly other sources as well. This technique\nis presented in Section~3 of this paper. First (Section~2), we describe the\nproperties of GLONASS that are relevant to the operation of the canceller.\nIn Sections~4--5, we present experimental results demonstrating the\neffectiveness of this approach. Section~4 describes the procedure used to\ncollect GLONASS-corrupted data; whereas Section~5 shows the results before\nand after application of the canceller. In section~6 we consider how this\napproach may be implemented on existing telescope systems.\n\n\\section{Properties of GLONASS signals}\n\nGLONASS satellites transmit at frequencies between 1602--1616~MHz and have\nshared primary user status with radio astronomy for the 1610.6--1613.8~MHz\nband\\cite{cwg94}. There are 24 carriers spread over the 14~MHz band at\nintervals of 0.5625 MHz. The carrier is modulated by a pair of noise like,\nequal power, pseudo noise (PN) codes of 0.511 and 5.11 MHz. Figure~1. of\nCombrinck et al. (1994)\\cite{cwg94} shows time averaged spectra of these\nsignals. The unfiltered sinc$^{2} $ side lobes of these signals have\nrelative power levels as high as $-25$~dB extending out to 20 MHz either\nside of the main carrier in some cases\\cite{gal91}. GLONASS satellites\nlaunched more recently do have some band-limiting filters.\n\n\nGLONASS, despite its wide band spectrum, actually has a very simple\nstructure\\cite{GLONASS_ICD}. Consider the narrow band (0.511~MHz) GLONASS\nmodulation. This signal is simply a sinusoidal carrier which experiences a\nphase shift of $0^{\\circ}$ or $180^{\\circ}$ every\n$(0.511~\\mbox{MHz})^{-1}$. Each phase shift represents a modulation symbol,\nor {\\em chip}. Each group of 511 chips represents a PN code, which is\npublic knowledge, never changes, and is the same for every GLONASS\nsatellite. GLONASS data bits are represented by changing the sign of a block\nof 10 PN codes, with 10~ms period. Parameters of the signal which are\nunknown when received are (1) the Doppler shift due to satellite motion, (2)\nthe {\\em code phase}, that is, the relative position within the 1~ms PN code\nperiod, and (3) the carrier phase, which rotates because the satellite is\nmoving and the transmitter's LO is not perfectly stable. However the\ncarrier phase, the current value of the data bit, and the complex gain due\nto the antenna pattern can all be combined into a single unknown complex\nmagnitude parameter. Thus three parameters are sufficient to describe the\nGLONASS signal with high accuracy.\n\n\nFinally, we note that the modulation used by the course/acquisition (C/A)\nmode of the U.S. Global Positioning System (GPS) is very similar to\nmodulation used in the GLONASS 0.511~MHz transmission. The main differences\nare longer code (1023~chips) and higher chip rate (1.023~MHz); also, all GPS\nsatellites transmit on the same centre frequency, but with different (but\nknown) PN codes. Thus, techniques which are effective against 0.511~MHz\nGLONASS modulation may be effective against GPS C/A transmissions as well.\n\n\n\n\\section{CANCELLATION ALGORITHM}\n\n\\subsection{Theory}\n\nOur technique for suppressing GLONASS signals in radio astronomy data is\nbased on {\\em parametric signal modelling}. Recall that the GLONASS signal\ncan be described using a model consisting of just three parameters: Doppler,\ncode phase, and complex magnitude. Given a block of data containing a\nGLONASS signal, one can then estimate the parameters. Given the parameters,\nit is possible to synthesise a noise-free copy of the GLONASS signal. This\ncopy is then subtracted from the telescope output to achieve the\nsuppression. This procedure is illustrated in Figure~\\ref{fPSM}.\n%\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{c}\n\\psfig{figure=ellingson7.eps,height=9.5cm} \n\\end{tabular}\n\\end{center}\n\\caption{Parametric cancelling technique. The ESTIMATE DOPPLER, CODE PHASE\nblock uses the down converted telescope signal to generates an estimate of\nthe time aligned PN code and the offset in frequency between the GLONASS\nCENTER FREQ oscillator an the down converted GLONASS signal. These two\nestimates and the down converted signals are used by the ESTIMATE COMPLEX\nMAGNITUDE block to calculate a magnitude and phase correction. These\nestimates of time aligned PN code, frequency, magnitude and phase are then\nused to synthesise a zero-IF GLONASS signal.}\n\\label{fPSM} \n\n\\end{figure} \n\n\nThe parametric solution proceeds on two time scales. Doppler frequency and\ncode phase are difficult to estimate, but change slowly. Complex magnitude,\non the other hand, is simple to estimate, and changes quickly. Our approach\nis to first {\\em acquire} the GLONASS signal. This involves a joint search\nover the possible Doppler frequencies and the code phases. For each\nDoppler/code phase pair, a complex baseband (zero-IF) version of the signal\nis cross-correlated with the PN sequence. The correct Doppler/code phase\npair is the one which maximises the magnitude of the cross-correlation.\nAlthough tedious, this is a simple procedure, and is essentially the same\nacquisition procedure used by hand held GPS receivers. Once acquired, the\nDoppler and code phase can be tracked simply by sensing the drift in the\ncorrelation peak and adjusting the Doppler and code phase parameters\naccordingly. It appears that the Doppler and code phase estimates can be\nfrozen for at least 0.1s between updates without any significant effect on\nthe results.\n\nOnce the signal is acquired, we estimate the complex magnitude by\ncross-correlating the time- and frequency-aligned PN code with the complex\nzero-IF representation of the GLONASS signal. The magnitude and phase of\nthe cross-correlation then represents the desired complex gain. The complex\ngain is expected to change quickly, so this procedure must be updated often.\nIn the example presented below, the complex gain update rate is 128~$\\mu s$,\nusing 1024 samples at 8~Msamples per second (conversion to a complex\nbaseband signal has halved the sample rate).\n\nGiven the Doppler frequency, code phase, and complex magnitude, one can then\nsynthesise a noise-free estimate of the GLONASS signal. However, it has been\nfound by experience that better cancellation is achieved by low-pass\nfiltering the zero-IF version of the synthesised GLONASS signal before\nsubtraction from the telescope output. This models the real-world low-pass\neffect which smoothes discontinuities in band limited signals. This also has\nthe desirable effect of suppressing the high-order side lobes of the\nsynthesised signal, which may not be accurately represented by the proposed\nsignal model. A suitable filter was found to be a 32-tap finite impulse\nresponse (FIR) filter based on the Hamming window, with cutoff frequency\nequal to $0.05F_{S}$ at $F_{S}=8$~MSPS. Such a filter can be obtained\nusing the MATLAB command {\\tt fir1(32,0.1)}.\n\n\n\\subsection{Implementation}\n\nThe results presented below were obtained using non-real-time\npost-processing software, written in MATLAB. On a 400MHz pentium the\nprocessing presently runs at 1000 times real time. The MATLAB source code is\nfreely available from the authors. Any practical system would, of course,\nrequire real time implementation. The maturity of GPS technology means that\nthe techniques and hardware for the acquisition of GLONASS and GPS signal\nparameters are well developed. The design of the signal modulators in the\nsatellites is also known. With the knowledge of these two areas a practical\nreal time implementation is within reach and is discussed in Section~6.\n\n%However, the processing described\n%above is very similar to the processing now commonly performed within\n%handheld GPS units. Suitable hardware techniques and\n%computationally-efficient algorithms are available in the open\n%literature\\cite{Kaplan}. Thus, practical real-time implementation seems to\n%be within reach even now.\n\n\\section{DATA COLLECTION} \n\nThe astronomy data used in testing these algorithms is a single linear\npolarisation, 4-bit data stream from each antenna of the 6x22m antenna,\nCSIRO Australia Telescope Compact Array at Narrabri in Australia. The data\nwas 4-bit sampled at 16MHz and recorded on an S2 recorder\\cite{s2ref}. The\nresulting 8MHz bandpass centred on 1610 MHz was wide enough to include\nsignals from GLONASS-69 at $\\sim$1609~MHz, an OH maser source (IRAS 1731-33)\nat $\\sim$1612~MHz and some flat spectrum. The data were then extracted using\nthe S2TCI system\\cite{s2ref} and demultiplexed. More details on this dataset\nand others that are freely available for conducting these kind of\nexperiments are in reports by Smegal et al.\\cite{sme99} and Bell et\nal.\\cite{bel99}. The algorithm works on a single polarisation data stream\nfrom one antenna only. However, data from a second antenna were also used in\ncross correlations as a test of how well the GLONASS signals were removed.\n\n\\section{RESULTS TO DATE}\n\nThe results so far are encouraging, with GLONASS narrow band signals being\neffectively removed in a way that is non toxic to astronomy signals. Figure\n\\ref{fig:skeptics} (left plot, top curve) shows a spectrum of the raw data,\nwith test tones added in software. The bottom curve shows the same 0.1s\n($1.6 \\times 10^6$ samples) of data with cancellation technique applied,\nwith no apparent GLONASS signals left. There is an OH maser source at\n$\\sim$1612~MHz. The top two curves in the right plot of Figure\n\\ref{fig:skeptics} show a blow up of this region. The bottom curve in the\nsame plot shows the difference multiplied by a factor of 1000, indicating\nthat no damage has been done in the spectral region of the OH source.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{c}\n\\psfig{figure=skeptics.ps,height=13cm,angle=-90} \n\\end{tabular}\n\\end{center}\n\\caption{Left plot: top curve: raw data, OH maser source at 1612 MHz, test\ntones inserted (in software) at 1609.3 and 1611.3 MHz. Left plot: lower\ncurve: data with GLONASS removed. Right plot: blow up of region around OH\nsource. The bottom curve shows the difference of the pre and post\ncancellation spectra multiplied by 1000. This indicates that this region of\nthe spectrum is not changed by more that a few parts in 1000. (curves offset\nfor clarity)}\n\\label{fig:skeptics} \n\\end{figure} \n\nIn order to examine the toxicity of this algorithm in the same frequency\nrange as the GLONASS signal we added a test tone to the data before the\ncancellation was applied and examined how it was affected by the\nprocessing. The left plot of Figure \\ref{fig:tt_xcorr} shows the result of\nsubtracting the test tone again, after the cancellation. There is no\nevidence that the test tone has been affected by the cancellation. However,\nthere is a small spike left right under the middle of the GLONASS signal,\nthat is unrelated to the test tones. This seems to be a result of some break\nthrough, or leakage of the GLONASS carrier signal from the GLONASS imbalance\nin the GLONASS phase modulator. It should be possible to model and remove\nthis as well, but we have not addressed that yet.\n\nA more sensitive test of the supression is to cross correlate with signals\nfrom another antenna. The Right plot of Figure \\ref{fig:tt_xcorr} show some\ncross correlations. The top curve is the cross correlation of raw data from\ntwo antennas. The bottom curve is the cross correlation of raw data from one\nantenna and data with GLONASS cancelled from data from another\nantenna. There are some extra ripples here that are not apparent in the\nother spectra, suggesting that there are some inaccuracies in the algorithm\nthat need further investigation. The majority of the signal seen in this\ncross correlation is probably due to the GLONASS wide band signal, which we\nhave not tried to suppress yet.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular}{c}\n\\psfig{figure=tt_xcorr.ps,height=13cm,angle=-90} \n\\end{tabular}\n\\end{center}\n\\caption{Left plot: Top: Offset test tone added before cancellation. Middle:\nCarrier signal break through apparent after GLONASS signal has been\ncancelled. Bottom: Offset test tone subtracted after cancellation shows the\ntest tone is unaffected. Right plot: top curve: Cross correlation of raw\ndata from 2 antennas. Bottom curve: Cross correlation of data with GLONASS\nremoved and raw data from another antenna. (Curves offset for clarity).}\n\\label{fig:tt_xcorr} \n\\end{figure} \n\n\nThe addition and subtraction of test tones give us some indication of how\ntoxic the algorithm is to astronomy signals. However, astronomy signals are\nnot coherent sine waves, but are more like band limited noise in the case of\nspectral lines and wide band noise in the case of continuum sources. We\nreplaced the test tones with some synthetic band limited noise, added it\nbefore the cancellation and subtracted it again after. As shown in Figure\n\\ref{fig:blnoise} the band limited noise is not affected by the cancellation\nalgorithm, down to a part in 1000, in other words, we have achieved 30~dBs\nof dynamic range in the supression. Processing with and without the presence\nof an astronomy sources make very little difference to the effectiveness of\nthe cancellation.\n\n\\section{LIMITATIONS AND FUTURE IMPROVEMENTS}\n\nThe carrier break through described earlier needs to be modelled and\ncancelled. While we have not investigated this carefully yet, we believe\nthis should be possible. We do not know the cause of the ripples in the\ncross correlation spectra and some more investigation there may lead to an\nimprovement in the algorithm. At present we have modelled the band limiting\nfilters of the GLONASS transmission system by a low-pass filter when the\nsignal is centered on zero-IF. This is limiting the accuracy with which we\nestimate the GLONASS spectrum and therefore possibly limiting the\ncancellation. We aim to either obtain details of the band limiting filters\nused on the GLONASS transmission system, or use the existing data to\nestimate them.\n\nNote that both GPS and GLONASS have in-band secondary channels that are\nspread 10 times wider, and therefore have power spectral density about 10\ntimes weaker. This secondary channel for GPS and GLONASS is much harder to\ndeal with, because the PN codes are more complex and may not be completely\nknown. We do need to find a way to mitigate against these signals, because\nthey do still cause substantial problems for radio astronomy.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\begin{tabular}{c}\n\\psfig{figure=blnoise.ps,height=13cm,angle=-90} \n\\end{tabular}\n\\end{center}\n\\caption{Left plot: Left plot: Top: Band limited noise added before\ncancellation. Middle: GLONASS signal has been cancelled. Bottom: Band\nlimited noise subtracted after cancellation shows the band limited noise is\nunaffected by the cancellation (the carrier break through is still\npresent). Right plot: Top: Result of cancellation with no test tone or\nnoise added. Middle: Result of cancellation with bland limited noise added\nbefore and subtracted after cancellation. Bottom: Difference of top 2 curves\nmultiplied by 1000 shows that the band limited noise was not affected by more\nthan 1 part in 1000. (Curves offset vertically for clarity in several\ncases).}\n\\label{fig:blnoise} \n\\end{figure} \n\nIn addition to GLONASS and GPS, many other satellite systems have\nwell-specified modulation and coding schemes. This opens up the\npossibility of removing these other classes of signals using digital signal\nprocessing. The techniques described here will be useful for some others,\nbut not all. At a recent meeting\\cite{efwhite} a wide range of interference\nmitigation techniques applicable to many different undesired signals were\npresented and discussed.\n\nThe usefulness of the method presented in this paper must ultimately be\nshown by conducting astronomy in the presence of GLONASS or GPS. Ideally\nthe astronomy results with and without interference present would be\nindistinguishable. The use of recorded data and post processing is useful\nfor demonstrating the method. But with processing currently running at one\n1/1000 real time the amount of astronomy that can be demonstrated is\nvery limited. It is therefore desirable to explore the use of dedicated\nhardware to achieve real time processing of the signal.\n\nThe most comprehensive approach is to build a complete receiver that\nincorporates the interference cancellation method described in this paper.\nThis is the approach that must be taken in designing new interference\nresistant receivers but currently the best option is to build an 'add on' to\nexisting receivers. The problem with this method is that the quantiser in\ncurrent receivers is designed for adequate performance when processing noise\nlike signals. Typically a 1 or 2 bit quantiser is used. Techniques like\nadaptive noise cancellation will cause the number of bits needed to\nrepresent the signal to grow when the interference is suppressed. For\nexample, consider an interferer whose peak amplitude is equivalent to one\neighth of the least significant bit in the quantiser and assume that 2 bits\ncan accurately represent the interference. To generate an interference free\nestimate of the signal it is necessary to subtract the estimate of the\ninterferer from the signal. In doing this, the number of bits needed to\nrepresent the signal grows by 4 lower order bits. The signal now has too\nmany bits to be processed by receivers normally used for radioastronomy.\nThe solution to this problem is to form auto and cross correlations of the\nmeasured signal and the estimate of the interference. If the estimate e(t)\nand the interference i(t) are related to each other by a linear transfer\nfunction then in frequency domain this can be written as I(f) = H(f)E(f)\nwhere H(f) is the frequency domain transfer function.\n\nIn this context the astronomy signal can be considered to noise. The system\ncan be redrawn as\n\n\\begin{center}\n\\begin{tabular}{c}\n\\psfig{figure=glo_hard2.ps,height=2.5cm,angle=-90} \n\\end{tabular}\n\\end{center}\n\nwhere o(t) is the wanted astronomy signal plus interference. The cross-spectrum method can now be used to estimate the transfer function\nH(f) . If the astronomy signal a(t) is uncorrelated with i(t) then\nH(f)\\cite{bp71,pap89} equal to the ratio of the cross-spectrum $S_{oe}(f)$\nbetween o(t) and e(t) and the power spectrum $S_{ee}(f)$ of e(t).\n\n$H(f) = S_{oe}(f)/ S_{ee}(f)$ \n\nThe power spectrum $S_{aa}(f)$ of a(t) is now equal to: \n\n$S_{aa}(f) = S_{oo}(f)-|H(f)|^{2}.S_{ee}(f)$\n\nThus measurement of the power spectrum of a(t) and e(t) plus the cross\nspectrum of the two gives enough information to derive the power spectrum of\nthe uncorrupted astronomical signal. These spectra can be obtained from the\nauto and cross correlations of the two signal. Correlators used for\nastronomy normally process at most 2-bit data. Straight 2-bit sampling may\nnot be sufficient to accurately represent the estimated GLONASS/GPS\nsignal. The addition of dither and the use of noise shaped oversampling\nshould solve this problem. The hardware itself will internally generate a\nmulti-bit accurate representation of the interferer.\n\nThe hardware used to synthesise the baseband GLONASS or GPS is comparatively\nsimple and easy to emulate in an FPGA. The main difficulty is the\nestimation of and tracking of signal phase and amplitude. This task is best\nleft to software. Data needed to perform this task are correlations between\nthe input and the current estimate of the interferer. If a reasonable\ninitial estimate of the interferer has been found then very few correlations\nare needed to maintain tracking of carrier phase, carrier Doppler and code\nphase. The hardware could also be used to generate these correlations. The\noperations performed by the hardware are:\n\n\\begin{enumerate}\n\\item Generate the carrier digitally with adjustable carrier phase and Doppler.\n\\item Generate the GLONASS/GPS modulation with adjustable code phase\n\\item Modulate the carrier with the GLONASS code. This gives an unscaled\n'noise free' estimate of the interference.\n\\item Generate the zero delay and $\\pm1/2$ chip correlations between the\ninput signal and the 'noise free' estimate. \n\\item Scale the magnitude of the 'noise free' estimate to match the\ninterference.\n\\item Optionally delay data and estimate. This allows the magnitude and phase corrections to be applied to the estimate used in generating the corrections. This extra item is needed to fully emulate the current software. In practice it may be unnecessary.\n\\end{enumerate}\n\nThis hardware removes most of the intensive tasks from software and\nleaves the software to monitor the correlations. From this monitoring the\nsoftware then needs to generate updates for the carrier phase, carrier\nDoppler, code phase and amplitude. With these updates the output of the\nhardware is a 'noise free' estimate of the interference. \n\n\\subsection*{Acknowledgements}\nThe authors gratefully acknowledge discussions with and help in\nobtaining data from Ron Ekers, Rick Smegal, Peter Hall, Bob Sault,\nMatthew Bailes, Willem van Stratten, Frank Briggs, Mike Kesteven, Warwick\nWilson, Dick Ferris\n\n\\begin{thebibliography}{99} \n\n%% the last item specifies width of reference number column\n\n\\bibitem{bel99} J.~F.~Bell et al. {\\em Base band data for testing\ninterference mitigation algorithms}, ATNF technical\ndocument. http://www.atnf.csiro.au/SKA/\n\n\\bibitem{bp71} J.~S.~Bendat, and A.~G.~Piersol, {\\em Random Data: Analysis\nand Measurement Procedures}, Wiley-Interscience, New York, 1971. ISBN\n0-471-06470-X \n\n\\bibitem{coh89} R.~J.~Cohen, ``The threat to radio astronomy from radio\npollution'', {\\em Space Policy}, {\\bf 5}, pp. 91-93, 1989.\n\n\\bibitem{cwg94} W.~L.~Combrinck, M.~E.~West, and M.~J.~Gaylard, ``Coexisting\nwith GLONASS: Observing the 1612 MHz Hydroxyl Line'', {\\em PASP}, {\\bf 10\n6}, pp. 807-812, 1994.\n\n\\bibitem{eb00} R.~D.~Ekers and J.~F.~Bell, ``Radio Frequency Interference\n'' to appear in {\\em The Universe at Low Radio Frequencies}, IAU Symposium 1\n99, Pune, Dec. 1999.\n\n\\bibitem{efwhite} ``The Elizabeth and Frederick White Conference on Radio\n\nFrequency Interference Mitigation Strategies'',\nhttp://www.atnf.csiro.au/SKA/intmit/atnf/conf/, 2000.\n\n\\bibitem{gal91} J.~Galt, ``Interference with Astronomical Observations of\n OH\nMasers from the Soviet Union's GLONASS Satellites'', in {\\em Light\nPollution, Radio Interference, and Space Debris}, ASP Conference Series,\nVol. 17, IAU Colloquium 112, 1991, D.L. Crawford, Ed., p13, 1991.\n\n\\bibitem{hh85} J.~Herman and H.~J.~Habing, ``Time variations and Shell Si\nzes of OH Masers in Late-Type Atars'', {\\em AASS}, {\\bf 59}, p. 523, 1985.\n\n\\bibitem{GLONASS_ICD} {\\em GLONASS Interface Control Document}, Coordinat\nion Scientific Information Center, Moscow, Russia. Available in PDF form \nfrom www.nz.dlr.de/gps/glonass7.html. \n\n\\bibitem{RFpc} R.~Fisher, NRAO, personal communication, Feb. 21, 2000.\n\n\\bibitem{Kaplan} E.~Kaplan (ed.), {\\em Understanding GPS: Principles and \nApplications}, Artech House, 1996.\n\n\\bibitem{pap89} A.~Papoulis, {\\em Probability, Random Variables, and\nStochastic Processes}, international Edition, McGraw-Hill Singapore\n1989. ISBN 0-07-Y66465-X \n\n\n\\bibitem{sme99} R.~Smegal et al. {\\em Array of\nIndependent Element Observations}, http://www.atnf.csiro.au/SKA/intmit/\n\n\\bibitem{s2ref} Weitfeldt et al. `` The S2 baseband processing system for\nphase-coherent pulsar applications'', {\\em AASS}, {\\bf 131}, pp. 594-554, 1998.\n\n\\end{thebibliography} \n\n\\end{document}\n"
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[
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"name": "astro-ph0002516.extracted_bib",
"string": "\\begin{thebibliography}{99} \n\n%% the last item specifies width of reference number column\n\n\\bibitem{bel99} J.~F.~Bell et al. {\\em Base band data for testing\ninterference mitigation algorithms}, ATNF technical\ndocument. http://www.atnf.csiro.au/SKA/\n\n\\bibitem{bp71} J.~S.~Bendat, and A.~G.~Piersol, {\\em Random Data: Analysis\nand Measurement Procedures}, Wiley-Interscience, New York, 1971. ISBN\n0-471-06470-X \n\n\\bibitem{coh89} R.~J.~Cohen, ``The threat to radio astronomy from radio\npollution'', {\\em Space Policy}, {\\bf 5}, pp. 91-93, 1989.\n\n\\bibitem{cwg94} W.~L.~Combrinck, M.~E.~West, and M.~J.~Gaylard, ``Coexisting\nwith GLONASS: Observing the 1612 MHz Hydroxyl Line'', {\\em PASP}, {\\bf 10\n6}, pp. 807-812, 1994.\n\n\\bibitem{eb00} R.~D.~Ekers and J.~F.~Bell, ``Radio Frequency Interference\n'' to appear in {\\em The Universe at Low Radio Frequencies}, IAU Symposium 1\n99, Pune, Dec. 1999.\n\n\\bibitem{efwhite} ``The Elizabeth and Frederick White Conference on Radio\n\nFrequency Interference Mitigation Strategies'',\nhttp://www.atnf.csiro.au/SKA/intmit/atnf/conf/, 2000.\n\n\\bibitem{gal91} J.~Galt, ``Interference with Astronomical Observations of\n OH\nMasers from the Soviet Union's GLONASS Satellites'', in {\\em Light\nPollution, Radio Interference, and Space Debris}, ASP Conference Series,\nVol. 17, IAU Colloquium 112, 1991, D.L. Crawford, Ed., p13, 1991.\n\n\\bibitem{hh85} J.~Herman and H.~J.~Habing, ``Time variations and Shell Si\nzes of OH Masers in Late-Type Atars'', {\\em AASS}, {\\bf 59}, p. 523, 1985.\n\n\\bibitem{GLONASS_ICD} {\\em GLONASS Interface Control Document}, Coordinat\nion Scientific Information Center, Moscow, Russia. Available in PDF form \nfrom www.nz.dlr.de/gps/glonass7.html. \n\n\\bibitem{RFpc} R.~Fisher, NRAO, personal communication, Feb. 21, 2000.\n\n\\bibitem{Kaplan} E.~Kaplan (ed.), {\\em Understanding GPS: Principles and \nApplications}, Artech House, 1996.\n\n\\bibitem{pap89} A.~Papoulis, {\\em Probability, Random Variables, and\nStochastic Processes}, international Edition, McGraw-Hill Singapore\n1989. ISBN 0-07-Y66465-X \n\n\n\\bibitem{sme99} R.~Smegal et al. {\\em Array of\nIndependent Element Observations}, http://www.atnf.csiro.au/SKA/intmit/\n\n\\bibitem{s2ref} Weitfeldt et al. `` The S2 baseband processing system for\nphase-coherent pulsar applications'', {\\em AASS}, {\\bf 131}, pp. 594-554, 1998.\n\n\\end{thebibliography}"
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astro-ph0002517
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A Broad 22 $\mu$m Emission Feature in the \\ Carina Nebula \ion{H}{2} Region\altaffilmark{1}
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"author": "Kin-Wing Chan and Takashi Onaka"
}
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We report the detection of a broad 22 $\mu$m emission feature in the Carina nebula \ion{H}{2} region by the Infrared Space Observatory (ISO) Short Wavelength Spectrometer. The feature shape is similar to that of the 22 $\mu$m emission feature of newly synthesized dust observed in the Cassiopeia A supernova remnant. This finding suggests that both of the features are arising from the same carrier, and that supernovae are probably the dominant production source of this new interstellar grain. A similar broad emission dust feature is also found in the spectra of two starburst galaxies from the ISO archival data. This new dust grain could be an abundant component of interstellar grains and can be used to trace the supernova rate or star formation rate in external galaxies. The existence of the broad 22 $\mu$m emission feature complicates the dust model for starburst galaxies and must be taken into account correctly in the derivation of dust color temperature. Mg protosilicate has been suggested as the carrier of the 22 $\mu$m emission dust feature observed in Cassiopeia A. The present results provide useful information in studies on chemical composition and emission mechanism of the carrier.
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[
{
"name": "22um_eprint.tex",
"string": "%%\n%% Beginning of file 'sample.tex'\n%%\n%% Modified 03 Nov 99\n%%\n%% This is a sample manuscript marked up using the\n%% AASTeX v5.0 LaTeX 2e macros.\n\n%% The first piece of markup in an AASTeX v5.0 document\n%% is the \\documentclass command. LaTeX will ignore\n%% any data that comes before this command.\n\n%% The command below calls the default manuscript style,\n%% which will produce a double-spaced document on one column.\n%% Examples of commands for other substyles follow. Use\n%% whichever is most appropriate for your purposes.\n\n%%\\documentclass{aastex}\n\n%% preprint produces a one-column, single-spaced document:\n\n \\documentclass[preprint]{aastex}\n\n%% preprint2 produces a double-column, single-spaced document:\n\n% \\documentclass[preprint2]{aastex}\n\n%% If you want to create your own macros, you can do so\n%% using \\newcommand. Your macros should appear before\n%% the \\begin{document} command.\n%%\n%% If you are submitting to a journal that translates manuscripts\n%% into SGML, you need to follow certain guidelines when preparing\n%% your macros. See the AASTeX v5.0 Author Guide\n%% for information.\n\n\\newcommand{\\vdag}{(v)^\\dagger}\n\\newcommand{\\myemail}{skywalker@galaxy.far.far.away}\n\n%% You can insert a short comment on the title page using the command below.\n\n\\slugcomment{}\n\n%% If you wish, you may supply running head information, although\n%% this information may be modified by the editorial offices.\n%% The left head contains a list of authors,\n%% usually a maximum of three (otherwise use et al.). The right\n%% head is a modified title of up to roughly 44 characters. Running heads\n%% will not print in the manuscript style.\n\n\n%% This is the end of the preamble. Indicate the beginning of the\n%% paper itself with \\begin{document}.\n\n\\begin{document}\n\n%% LaTeX will automatically break titles if they run longer than\n%% one line. However, you may use \\\\ to force a line break if\n%% you desire.\n\n\\title{A Broad 22 $\\mu$m Emission Feature in the \\\\\n Carina Nebula \\ion{H}{2} Region\\altaffilmark{1}}\n\n\n%% Use \\author, \\affil, and the \\and command to format\n%% author and affiliation information.\n%% Note that \\email has replaced the old \\authoremail command\n%% from AASTeX v4.0. You can use \\email to mark an email address\n%% anywhere in the paper, not just in the front matter.\n%% As in the title, you can use \\\\ to force line breaks.\n\n\\author{Kin-Wing Chan and Takashi Onaka}\n\\affil{Department of Astronomy, School of Science, University of Tokyo,\\\\\n Bunkyo-ku, Tokyo 113-0033, Japan}\n\\email{kwc@astron.s.u-tokyo.ac.jp, onaka@astron.s.u-tokyo.ac.jp}\n\n\n%% Notice that each of these authors has alternate affiliations, which\n%% are identified by the \\altaffilmark after each name. Specify alternate\n%% affiliation information with \\altaffiltext, with one command per each\n%% affiliation.\n\n\\altaffiltext{1}{ Based on observations with ISO, an ESA project with\ninstruments funded by ESA members states (especially the PI countries France, \nGermany, the Netherlands, and the United Kingdom) and with the participation \nof ISAS and NASA.}\n\n\n%% Mark off your abstract in the ``abstract'' environment. In the manuscript\n%% style, abstract will output a Received/Accepted line after the\n%% title and affiliation information. No date will appear since the author\n%% does not have this information. The dates will be filled in by the\n%% editorial office after submission.\n\n\n\\begin{abstract}\nWe report the detection of a broad 22 $\\mu$m emission feature in the Carina\nnebula \\ion{H}{2} region by the Infrared Space Observatory (ISO) Short \nWavelength Spectrometer. The feature shape is similar to that of the 22 $\\mu$m\nemission feature of newly synthesized dust observed in the Cassiopeia A \nsupernova remnant. This finding suggests that both of the features are arising \nfrom the same carrier, and that supernovae are probably the dominant production \nsource of this new interstellar grain. A similar broad emission dust feature is \nalso found in the spectra of two starburst galaxies from the ISO archival data. \nThis new dust grain could be an abundant component of interstellar grains and\ncan be used to trace the supernova rate or star formation rate in external\ngalaxies. The existence of the broad 22 $\\mu$m emission feature complicates the\ndust model for starburst galaxies and must be taken into account correctly in \nthe derivation of dust color temperature. Mg protosilicate has been suggested \nas the carrier of the 22 $\\mu$m emission dust feature observed in \nCassiopeia A. The present results provide useful information in studies on\nchemical composition and emission mechanism of the carrier.\n\\end{abstract}\n\n%% Keywords should appear after the \\end{abstract} command. The uncommented\n%% example has been keyed in ApJ style. See the instructions to authors\n%% for the journal to which you are submitting your paper to determine\n%% what keyword punctuation is appropriate.\n\n\\keywords{dust extinction---infrared: ISM: lines and bands---ISM: \\ion{H}{2} \nregions}\n\n\n%% From the front matter, we move on to the body of the paper.\n%% In the first two sections, notice the use of the natbib \\citep\n%% and \\citet commands to identify citations. The citations are\n%% tied to the reference list via symbolic KEYs. The KEY corresponds\n%% to the KEY in the \\bibitem in the reference list below. We have\n%% chosen the first three characters of the first author's name plus\n%% the last two numeral of the year of publication as our KEY for\n%% each reference.\n\n\\section{Introduction}\nSupernovae have been suggested besides evolved stars as one of the major sources\n of interstellar dust (see Gehrz 1989, Jones and Tielens 1994, Dwek 1998 for \nreview). Supporting evidence includes observations of dust condensation in the \nejecta of SN 1987A (Moseley et al. 1989, Whitelock et al. 1989, Dwek et al. \n1992, Wooden et al. 1993), and those of the newly synthesized dust in the \nCassiopeia A (Cas A) supernova remnant (Arendt, Dwek, \\& Moseley 1999). The dust\n formation mechanism and the amount of dust that is formed in supernovae are\nstill poorly known. Observations of SN 1987A and Cas A showed that the mass of\nthe newly formed dust is much less than expected, and the discrepancy may be \ndue to the fact that most of the dust is cold and cannot be detected in the \nfar-infrared (Dwek 1998, Arendt et al. 1999). Finding an abundant dust \ncomponent in the interstellar medium (ISM) which is formed only in supernovae \nwill support the hypothesis that supernovae are a major source of interstellar \ndust. Furthermore, since the amount of this specific grain is proportional to \nthe number of supernova, its total mass in the ISM can be used as a tracer of\nthe supernova rate or star formation rate in external galaxies. In this Letter \nwe report the detection of a broad 22 $\\mu$m emission dust feature in the\nCarina nebula \\ion{H}{2} region by the ISO guaranteed time observations. We \nfound that the shape of the present 22 $\\mu$m emission dust feature is similar\nto the 22 $\\mu$m emission feature observed in Cas A. We also found a similar \nemission feature in two starburst galaxies from the ISO archival data.\n\n\\section{Observations}\nThe observations were made as part of the ISO guaranteed time program \n(TONAKA.WDISM1) using the Short Wavelength Spectrometer (SWS; de Graauw et al.\n 1996). All the observations were made with the SWS AOT01 mode with scan speed\n of 1, which provided full grating spectra of 2.38 to 45.2 $\\mu$m with a\nresolution of $\\lambda$/$\\Delta$$\\lambda$ = 300. The data have been processed\nthrough the Off-Line Processing (OLP) 8.4, and reduced with the SWS Interactive\nAnalysis (IA) package developed by the SWS Instrument Dedicated Team. \nWe observed the Car I \\ion{H}{2} region in the Carina nebula and regions away\nfrom it to the nearby molecular clouds (see de Grauuw et al. 1981 for\ndiscussions of molecular clouds in the Carina nebula). The Car I \\ion{H}{2}\n region is excited by the Trumpler 14, an open cluster containing numerous\n O-type stars. Totally four positions were observed. Pos 1 is at the Car I\n\\ion{H}{2} region with ${l}$ = 287$^{\\circ}$.399 and ${b}$ = --0$^{\\circ}$.633.\n Pos 2 (${l}$ = 287$^{\\circ}$.349 and ${b}$ = --0$^{\\circ}$.633), Pos 3 (${l}$\n = 287$^{\\circ}$.299 and ${b}$ = --0$^{\\circ}$.633), and Pos 4 (${l}$ =\n 287$^{\\circ}$.249 and ${b}$ = --0$^{\\circ}$.633) are at a distance of 2.4,\n 4.7, and 7.1 pc away from Pos 1, respectively. Throughout our Letter, we\nadopt a Sun-to-Carina nebula distance of 2.7 kpc (Grabelsky et al. 1988). \nSince the SWS aperture size varies across the wavelength ranges, we adjusted \nthe difference in fluxes at the SWS band boundaries by scaling the spectra \nto the shortest band. The above adjustment does not affect the results\npresented here.\n\n\n%% In this section, we use the \\subsection command to set off\n%% a subsection. \\footnote is used to insert a footnote to the text.\n\n%% Observe the use of the LaTeX \\label\n%% command after the \\subsection to give a symbolic KEY to the\n%% subsection for cross-referencing in a \\ref command.\n%% You can use LaTeX's \\ref and \\label commands to keep track of\n%% cross-references to sections, equations, tables, and figures.\n%% That way, if you change the order of any elements, LaTeX will\n%% automatically renumber them.\n\n%% This section also includes several of the displayed math environments\n%% mentioned in the Author Guide.\n\n\\section{Results}\nFigure 1a shows the observed SWS spectrum of the Carina nebula at Pos 1. A \nbroad feature from $\\sim$ 18 to 28 $\\mu$m is clearly seen in the spectrum. \nThe adjustment of the observed fluxes due to the different aperture sizes of\n SWS has no effect on this feature, since the SWS has the same aperture size\n from 12 -- 27.5 $\\mu$m. It is difficult, however, to derive the spectral \nshape of this feature correctly since the underlying continuum emission is\nvery strong. We derived the feature shape by assuming the feature starts at \n18 $\\mu$m and ends at 28 $\\mu$m. Then the assumed underlying continuum \nemission, as shown in Figure 1a by the dashed line, is subtracted from the\n observed spectrum. The continuum emission comprises grains of graphite with\n temperature of 157 K and silicate with temperature of 40 K. Dust optical\n constants are adopted from Draine (1985).\nThe resultant feature shape is shown in Figure 1b, in which a peak around \n22 $\\mu$m is clearly seen. This new 22 $\\mu$m feature is distinctly \ndifferent than the 21 $\\mu$m feature that was discovered \nby Kwok, Volk, \\& Hrivnak (1989). The 21 $\\mu$m feature, \nwhich was only observed in carbon-rich post asymptotic\ngiant branch stars, has a much narrow feature width\nof $\\sim$ 4 $\\mu$m (Volk, Kwok, \\& Hrivnak 1999) compared to\nthat of the present 22 $\\mu$m feature (with width of $\\sim$ 10 $\\mu$m).\nThis suggests that the 21 and 22 $\\mu$m emission features \nare arising from different kinds of dust grain. \nFigure 2a shows another Carina nebula spectrum at Pos 2. The broad feature\n from 18 to 28 $\\mu$m is also seen in this spectrum. The continuum emission\n comprising graphite with temperature of 135 K and silicate with temperature \nof 42 K is assumed (the dashed line in Fig. 2a) and subtracted from the\n observed spectrum. The excess emission is shown in Figure 2b. The unidentified\n infrared (UIR) emission features become stronger at Pos 2, a position farther\n away from the Car I \\ion{H}{2} region compared to Pos 1. The slight difference\n in feature shape between figures 1b and 2b is probably due to the dust\n temperature effect. The 22 $\\mu$m emission feature is also seen in Pos 3 and\n Pos 4 (not shown) but in weaker intensity. \n\nThe same broad feature has been reported in the SWS spectra of M17--SW\n \\ion{H}{2} region by Jones et al. (1999). They found in their spectra that\nthe intensity of this emission feature decreases with distance from the\n exciting stars, the same phenomenon we see in the present four observed\n spectra. The decrease of feature intensity may be due to: (1) dilution by the\n cool dust emission from the nearby molecular clouds; (2) emission of the\n feature requiring very high UV radiation intensity to be excited; and/or (3)\n decrease in the abundance of this specific grain with distance from the\n exciting stars. Identification of the carrier of the feature will help us to\n understand the observed decrease in the feature intensity.\n\nVery recently, a broad emission dust feature with peak at 22 $\\mu$m was \nreported in Cas A (Arendt et al. 1999). We compare this 22 $\\mu$m feature \nand the present feature to see whether there is a similarity in feature shape.\n The comparison is shown in Figure 3. The Cas A spectrum was observed in the\n optical knot called N3 (see Arendt et al. 1999 for details), and is obtained\n from the ISO archival data. In order to obtain a better fit at wavelengths\n longer than 28 $\\mu$m, we choose a new continuum emission, as shown in Figure\n 1a by the dotted line, to give the 22 $\\mu$m feature more long wavelength\n emission. The new continuum emission comprises graphite with temperature of\n 160 K and silicate with temperature of 45 K. In Figure 3 we can see that the\n feature present in the Carina nebula shows a good agreement with that\n observed in Cas A. The origin of the excess emission around 13 $\\mu$m is\n unknown. It should be noted, however, that the emission in Cas A at \nwavelengths between 20 and 50 $\\mu$m may arise mostly from a warm ($\\sim$ \n90 K) silicate component that originates from the diffuse shell (see Tuffs \net al. 1999 for discussions of the spectral energy distribution of Cas A). \nIf this warm silicate component is subtracted from the Cas A N3 spectrum, \nthe resultant 22 $\\mu$m feature (without emission at wavelengths longer than\n 30 $\\mu$m) will give a good fit to our observed 22 $\\mu$m feature shown \nin Figure 1b.\n\n%% This section contains more display math examples, including unnumbered\n%% equations (displaymath environment). The last paragraph includes some\n%% examples of in-line math featuring a couple of the AASTeX symbol macros.\n\n\\section{Discussion}\nEvolved stars and supernovae have been suggested as the major production \nsources of interstellar dust. Past observations of evolved stars have found\n a number of dust features in the near to far-infrared ranges (see Waters et \nal. 1999 for a recent review). However, the broad 22 $\\mu$m emission feature\n that we found in Carina nebula \\ion{H}{2} region has never been reported in\n evolved stars. On the other hand, the present broad 22 $\\mu$m emission \nfeature is quite similar to the emission feature of newly synthesized dust\n observed in Cas A, suggesting that both of these features arise from the \nsame dust grain, and that supernovae are probably the major production source\n of this new interstellar grain. The non-detection of the 22 $\\mu$m feature\nin SN 1987A (Moseley et al. 1989) does not make the latter suggestion less\n convincing, since the infrared emission in SN 1987A probably arises from\n optically thick clumps. Lucy et al. (1991) and\nWooden et al. (1993) suggest that the infrared emission in SN 1987A is\ndominated by the dust in the optically thick clumps, and the low\ndensity small grains in the interclump medium contribute\nto the visual extinction. With this model, the infrared\nemission in SN 1987A is a graybody emission, but the visual \nextinction is not. \n\nWe would expect to find the 22 $\\mu$m dust feature in astronomical sources\n with high supernova rate if supernovae are the major production source of\n this new interstellar grain. Starburst galaxies are an ideal place to \nsearch for. From the ISO archival data we found that two starburst galaxies,\nM82 and NGC7582, show a similar 22 $\\mu$m emission feature. Figure 4 shows \nthe SWS spectrum of the nuclear region of NGC7582, a narrow-line X-ray \ngalaxy with strong starburst in the central kpc (Radovich et al. 1999, \nand references therein). The 20 to 30 $\\mu$m emission is mostly or \ncompletely arising from the broad 22 $\\mu$m emission feature. The spectrum\nof NGC7582 was taken by the SWS AOT01 with the speed of 2. We processed the\ndata through the OLP 8.4 and reduced with the SWS IA package in a way similar\nto the Carina nebula spectra. The feature intensity in M82 (not shown) is\n much weaker, about 10$\\%$ of the 18 -- 28 $\\mu$m emission if the continuum\nis assumed to pass through the 18 and 28 $\\mu$m data points. Two other \nstarburst galaxies, NGC253 and Circinus may also have a 22 $\\mu$m feature,\nbut they are further weak in intensity and more observations are needed to\n confirm it.\n\nThe findings of the 22 $\\mu$m dust feature in \\ion{H}{2} regions and \nstarburst galaxies suggest that this new grain could be an abundant component\n of interstellar dust. If the amount of this interstellar grain in the ISM\n is supposed to be proportional to the number of supernovae, its total mass \nin the ISM can be used as a tracer of the supernova rate or star formation \nrate in external galaxies. Studies of a large sample of starburst galaxies \nare required to confirm the above relationship. Only a limited number \nof galaxies have been observed by the SWS full grating scan mode, and a\n statistically useful sample of starburst galaxies for this study is not\n available at present. Future space missions like the Space InfraRed\nTelescope Facility (SIRFT) and Infrared Imaging Surveyor (IRIS), and the\n Stratospheric Observatory for Infrared Astronomy (SOFIA) are expected\n to provide the necessary data base.\n\nThe existence of this broad 22 $\\mu$m emission feature complicates the\ndust model used in the study of the spectral energy distribution of \nstarburst galaxies. Dust grains like graphite, amorphous carbon, silicates,\n and polycyclic aromatic hydrocarbons may not be representative of all the \ndust properties in starburst galaxies. Particularly, this broad 22 $\\mu$m\n emission feature could have significant effects in the derivation of the \ndust color temperature based on the 20 -- 30 $\\mu$m photometric flux (e.g., \nthe Infrared Astronomical Satellite 25 $\\mu$m data) as well as the number \ncounts of deep surveys in the infrared spectral range to be carried out by\n SIRTF and IRIS observations, and must be taken into account appropriately. \n\nArendt et al. (1999) suggested that the carrier of the 22 $\\mu$m feature\n observed in Cas A is Mg protosilicate based on the good agreement between\nthe observed feature shape and the laboratory spectrum of the Mg \nprotosilicate taken by Dorschner et al. (1980). They found that FeO can\nalso give a good fit to their observed 22 $\\mu$m feature, but the required \ndust temperature higher than expected and the deficient of emission at\n wavelegths longer than 30 $\\mu$m led them to rule it out as a promising\n candidate. If the identification of Mg protosilicate is true, it is the\n second silicate grain besides the astronomical silicates found in the ISM. \nMore observations are needed to confirm (or test) the suggested identification.\n Observing the 22 $\\mu$m feature in a variety of astronomical environments will\n provide useful information in studies on chemical composition and emission\n mechanism of the carrier.\n\nThe major results of this Letter are: (1) a broad 22 $\\mu$m emission dust\n feature is detected in \\ion{H}{2} regions and starburst galaxies; (2) the \n22 $\\mu$m emission feature is similar in shape with the emission feature \nof newly synthesized dust observed in the ejecta of Cas A, and both of \nthese features arise from the same carrier; and (3) supernovae are probably \nthe major production source of this new interstellar dust.\n\n\n\\acknowledgments\nWe would like to thank the SWS IDT for providing the SWS IA software, and \nISO project members for their efforts and help. We thank Robert Gehrz for \nuseful comments. We also thank Issei Yamamura for useful discussions on the\n data reduction, and K. Kawara, Y. Satoh, H. Okuda, and the Japanese ISO team\n for their continuous help and encouragement. K. W. C. is supported by the\n JSPS Postdoctoral Fellowship for Foreign Researchers. This work was \nsupported in part by Grant-in-Aids for Scientific Research from JSPS.\n\n\n%% The reference list follows the main body and any appendices.\n%% Use LaTeX's thebibliography environment to mark up your reference list.\n%% Note \\begin{thebibliography} is followed by an empty set of\n%% curly braces. If you forget this, LaTeX will generate the error\n%% \"Perhaps a missing \\item?\".\n%%\n%% thebibliography produces citations in the text using \\bibitem-\\cite\n%% cross-referencing. Each reference is preceded by a\n%% \\bibitem command that defines in curly braces the KEY that corresponds\n%% to the KEY in the \\cite commands (see the first section above).\n%% Make sure that you provide a unique KEY for every \\bibitem or else the\n%% paper will not LaTeX. The square brackets should contain\n%% the citation text that LaTeX will insert in\n%% place of the \\cite commands.\n\n%% We have used macros to produce journal name abbreviations.\n%% AASTeX provides a number of these for the more frequently-cited journals.\n%% See the Author Guide for a list of them.\n\n%% Note that the style of the \\bibitem labels (in []) is slightly\n%% different from previous examples. The natbib system solves a host\n%% of citation expression problems, but it is necessary to clearly\n%% delimit the year from the author name used in the citation.\n%% See the natbib documentation for more details and options.\n\n\\begin{thebibliography}{}\n\\bibitem[Arendt et al.(1999)]{are99} Arendt, R. G., Dwek, E., and Moseley,\n S. H. 1999, \\apj, 521, 234\n\\bibitem[de Graauw et al.(1996)]{deG96} de Graauw, T. et al. 1996,\n \\aap, 315, L49\n\\bibitem[de Graauw et al.(1981)]{deG81} de Graauw, T., Lidholm, S., Fitton, B.,\n Beckman, J., Israel, F. P., Nieuwenhuijzen, H., and Vermue, J. 1981,\n \\aap, 102, 257\n\\bibitem[Dorschner et al.(1980)]{dor80} Dorschner, J., Friedemann, C., \n G\\\"{u}rtler, J., and Duley, W. W. 1980, \\apss, 68, 159\n\\bibitem[Draine(1985)]{dra85} Draine, B. T. 1985, \\apjs, 57, 587\n\\bibitem[Dwek(1998)]{dwe98} Dwek, E. 1998, \\apj, 501, 643\n\\bibitem[Dwek et al.(1992)]{dwe92} Dwek, E., Moseley, S. H., Glaccum, W., \n Graham, J. R., Loewenstein, R. F., Silverberg, R. F., and Smith, R. K.\n 1992, \\apj, 389, L21\n\\bibitem[Gehrz(1989)]{geh89} Gehrz, R. D. 1989, in IAU Symp. 135, \n Interstellar Dust, ed. L. J. Allamandola and A. G. G. M. Tielens \n (Dordrecht: Kluwer), 445\n\\bibitem[Grabelsky et al.(1988)]{gra88} Grabelsky, D. A., Cohen, R. S.,\n Bronfman, L., and Thaddeus, P. 1988, \\apj, 331, 181\n\\bibitem[Jones et al.(1999)]{jon99} Jones, A. P., Frey, V., Verstraete,\n L., Cox, P., and Demyk, K. 1999, in The Universe as seen by ISO,\n ed. P. Cox and M. F. Kessler (ESA SP-427), 679\n\\bibitem[Jones and Tielens(1994)]{jon94} Jones, A. P. and Tielens, A. G. G. M.\n 1994, in The Cold Universe, ed. Th. Montmerle, Ch. J. Lada, I. F. Mirabel,\n J. Tr\\^{a}n Thanh V\\^{a}n (Gif-sur-Yvette: Editions Frontieres), 35\n\\bibitem[Kwok et al.(1989)]{kwo89} Kwok, S., Volk, K. M., and \n Hrivnak, B. J. 1989, \\apjl, 345, L51\n\\bibitem[Lucy et al.(1991)]{luc91} Lucy, L. B., Danziger, I. J., Gouiffes,\n C., and Bouchet, P. 1991, in Supernovae, ed. S. E. Woosley \n (Berlin: Springer), 82\n\\bibitem[Moseley et al.(1989)]{mos89} Moseley, S. H., Dwek, E., Glaccum, W.,\n Graham, J. R., Loewenstein, R. F., and Silverberg, R. F. 1989,\n \\nat, 340, 697\n\\bibitem[Radovich et al.(1999)]{rad99} Radovich, M., Klaas, U., Acosta-Pulido,\n J., and Lemke, D. 1999, \\aap, 348, 705\n\\bibitem[Tuffs et al.(1999)]{tuf99} Tuffs, R. J., Fischera, J., O'C Drury,\n L., Gabriel, C., Heinrichsen, I., Rasmussen, I., and V\\\"{o}lk, H. J.\n 1999, in The Universe as seen by ISO, ed. P. Cox and M. F. Kessler\n (ESA SP-427), 241\n\\bibitem[Volk et al.(1999)]{vol99} Volk, K., Kwok, S., and \n Hrivnak, B. J. 1999, \\apjl, 516, L99\n\\bibitem[Waters et al.(1999)]{wat99} Waters, L. B. F. M. et al. 1999, in \n The Universe as seen by ISO, ed. P. Cox and M. F. Kessler (ESA SP-427), 219\n\\bibitem[Whitelock et al.(1989)]{whi89} Whitelock, P. A. et al.\n 1989, \\mnras, 240, 7\n\\bibitem[Wooden et al.(1993)]{woo93} Wooden, D. H., Rank, D. M.,\n Bregman, J. D., Witteborn, F. C., Tielens, A. G. G. M., Cohen, M., \n Pinto, P. A., and Axelrod, T. S. 1993, \\apjs, 88, 477\n\\end{thebibliography}\n\n\n%% Generally speaking, only the figure captions, and not the figures\n%% themselves, are included in electronic manuscript submissions.\n%% Use \\figcaption to format your figure captions. They should begin on a\n%% new page.\n\n\\clearpage\n\n%% No more than seven \\figcaption commands are allowed per page,\n%% so if you have more than seven captions, insert a \\clearpage\n%% after every seventh one.\n\n%% There must be a \\figcaption command for each legend. Key the text of the\n%% legend and the optional \\label in curly braces. If you wish, you may\n%% include the name of the corresponding figure file in square brackets.\n%% The label is for identification purposes only. It will not insert the\n%% figures themselves into the document.\n%% If you want to include your art in the paper, use \\plotone.\n%% Refer to the on-line documentation for details.\n\n\n\\figcaption{(a) The observed SWS spectrum of the Carina nebula at Pos 1. The\n dashed (dotted) line represents the continuum emission with graphite and\n silicate at 157 (160) and 40 (45) K, respectively. The spectrum was scaled\n to the shortest band in order to adjust the difference in the aperture size.\n The flux unit is Jansky (Jy) per beam with beam size of \\(14''\\) $\\times$\n \\(20''\\) in the shortest band. (b) The resultant 22 $\\mu$m feature after\n subtraction of the dashed line continuum emission. \\label{fig1}}\n\n\\figcaption{(a) The observed SWS spectrum at Pos 2. The dashed line\n represents the continuum emission with graphite and silicate at 135 and\n 42 K, respectively. (b) The resultant 22 $\\mu$m feature after subtraction\n of the continuum. \\label{fig2}}\n\n\\figcaption{The Cas A N3 spectrum (the solid line) multiplied by a factor \n 5.2, together with the 22 $\\mu$m feature (the + symbol) after subtraction of\n the dotted line continuum emission shown in Figure 1(a). For easy \n comparison, the Cas A spectrum has been smoothed to a resolution of\n $\\lambda$/$\\Delta$$\\lambda$ = 100. \\label{fig3}} \n\n\\figcaption{The SWS spectrum of NGC7582, which has been smoothed to a\n resolution of $\\lambda$/$\\Delta$$\\lambda$ = 300. \\label{fig4}}\n\n\t\t\n\n%% The following command ends your manuscript. LaTeX will ignore any text\n%% that appears after it.\n\n\\end{document}\n\n%%\n%% End of file `sample.tex'.\n"
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"name": "astro-ph0002517.extracted_bib",
"string": "\\begin{thebibliography} is followed by an empty set of\n%% curly braces. If you forget this, LaTeX will generate the error\n%% \"Perhaps a missing \\item?\".\n%%\n%% thebibliography produces citations in the text using \\bibitem-\\cite\n%% cross-referencing. Each reference is preceded by a\n%% \\bibitem command that defines in curly braces the KEY that corresponds\n%% to the KEY in the \\cite commands (see the first section above).\n%% Make sure that you provide a unique KEY for every \\bibitem or else the\n%% paper will not LaTeX. The square brackets should contain\n%% the citation text that LaTeX will insert in\n%% place of the \\cite commands.\n\n%% We have used macros to produce journal name abbreviations.\n%% AASTeX provides a number of these for the more frequently-cited journals.\n%% See the Author Guide for a list of them.\n\n%% Note that the style of the \\bibitem labels (in []) is slightly\n%% different from previous examples. The natbib system solves a host\n%% of citation expression problems, but it is necessary to clearly\n%% delimit the year from the author name used in the citation.\n%% See the natbib documentation for more details and options.\n\n\\begin{thebibliography}{}\n\\bibitem[Arendt et al.(1999)]{are99} Arendt, R. G., Dwek, E., and Moseley,\n S. H. 1999, \\apj, 521, 234\n\\bibitem[de Graauw et al.(1996)]{deG96} de Graauw, T. et al. 1996,\n \\aap, 315, L49\n\\bibitem[de Graauw et al.(1981)]{deG81} de Graauw, T., Lidholm, S., Fitton, B.,\n Beckman, J., Israel, F. P., Nieuwenhuijzen, H., and Vermue, J. 1981,\n \\aap, 102, 257\n\\bibitem[Dorschner et al.(1980)]{dor80} Dorschner, J., Friedemann, C., \n G\\\"{u}rtler, J., and Duley, W. W. 1980, \\apss, 68, 159\n\\bibitem[Draine(1985)]{dra85} Draine, B. T. 1985, \\apjs, 57, 587\n\\bibitem[Dwek(1998)]{dwe98} Dwek, E. 1998, \\apj, 501, 643\n\\bibitem[Dwek et al.(1992)]{dwe92} Dwek, E., Moseley, S. H., Glaccum, W., \n Graham, J. R., Loewenstein, R. F., Silverberg, R. F., and Smith, R. K.\n 1992, \\apj, 389, L21\n\\bibitem[Gehrz(1989)]{geh89} Gehrz, R. D. 1989, in IAU Symp. 135, \n Interstellar Dust, ed. L. J. Allamandola and A. G. G. M. Tielens \n (Dordrecht: Kluwer), 445\n\\bibitem[Grabelsky et al.(1988)]{gra88} Grabelsky, D. A., Cohen, R. S.,\n Bronfman, L., and Thaddeus, P. 1988, \\apj, 331, 181\n\\bibitem[Jones et al.(1999)]{jon99} Jones, A. P., Frey, V., Verstraete,\n L., Cox, P., and Demyk, K. 1999, in The Universe as seen by ISO,\n ed. P. Cox and M. F. Kessler (ESA SP-427), 679\n\\bibitem[Jones and Tielens(1994)]{jon94} Jones, A. P. and Tielens, A. G. G. M.\n 1994, in The Cold Universe, ed. Th. Montmerle, Ch. J. Lada, I. F. Mirabel,\n J. Tr\\^{a}n Thanh V\\^{a}n (Gif-sur-Yvette: Editions Frontieres), 35\n\\bibitem[Kwok et al.(1989)]{kwo89} Kwok, S., Volk, K. M., and \n Hrivnak, B. J. 1989, \\apjl, 345, L51\n\\bibitem[Lucy et al.(1991)]{luc91} Lucy, L. B., Danziger, I. J., Gouiffes,\n C., and Bouchet, P. 1991, in Supernovae, ed. S. E. Woosley \n (Berlin: Springer), 82\n\\bibitem[Moseley et al.(1989)]{mos89} Moseley, S. H., Dwek, E., Glaccum, W.,\n Graham, J. R., Loewenstein, R. F., and Silverberg, R. F. 1989,\n \\nat, 340, 697\n\\bibitem[Radovich et al.(1999)]{rad99} Radovich, M., Klaas, U., Acosta-Pulido,\n J., and Lemke, D. 1999, \\aap, 348, 705\n\\bibitem[Tuffs et al.(1999)]{tuf99} Tuffs, R. J., Fischera, J., O'C Drury,\n L., Gabriel, C., Heinrichsen, I., Rasmussen, I., and V\\\"{o}lk, H. J.\n 1999, in The Universe as seen by ISO, ed. P. Cox and M. F. Kessler\n (ESA SP-427), 241\n\\bibitem[Volk et al.(1999)]{vol99} Volk, K., Kwok, S., and \n Hrivnak, B. J. 1999, \\apjl, 516, L99\n\\bibitem[Waters et al.(1999)]{wat99} Waters, L. B. F. M. et al. 1999, in \n The Universe as seen by ISO, ed. P. Cox and M. F. Kessler (ESA SP-427), 219\n\\bibitem[Whitelock et al.(1989)]{whi89} Whitelock, P. A. et al.\n 1989, \\mnras, 240, 7\n\\bibitem[Wooden et al.(1993)]{woo93} Wooden, D. H., Rank, D. M.,\n Bregman, J. D., Witteborn, F. C., Tielens, A. G. G. M., Cohen, M., \n Pinto, P. A., and Axelrod, T. S. 1993, \\apjs, 88, 477\n\\end{thebibliography}"
}
] |
astro-ph0002518
|
X-ray observation of the super-luminal quasar 4C\,+73.18 (1928+738) by ASCA
|
[
{
"author": "W. Yuan$^{1,2,3,4}$"
},
{
"author": "W. Brinkmann$^{1}$"
},
{
"author": "M. Gliozzi$^{1}$"
},
{
"author": "Y. Zhao$^{3}$"
},
{
"author": "M. Matsuoka$^{4}$"
}
] |
The results of an X-ray observation of the nearby core-dominated, super-luminal quasar 4C\,+73.18 (1928+738) with ASCA are reported. Variations of the 1--10\,keV flux on time-scales of the order of 10 hours were found while the 0.5--1\,keV flux remained almost unchanged. The measured flux at 1\,keV had increased by $\sim 50\%$ compared with previous observations 18 years ago. The quasar shows a curved spectrum in the 0.8--10\,keV band, which can be modeled as a power law plus a soft component. Both, the spectrum and the uncorrelated flux variations indicate the presence of a soft X-ray excess over the extrapolation of the hard band power law, which might be the high-energy extension of the intense UV bump emission found in this quasar. The hard X-ray emission is dominated by the flat power law (photon index 1.1--1.5). The results are consistent with models in which the hard X-rays are emitted from the relativistic jet. An iron line emission as reported in a previous Ginga observation (Lawson \& Turner 1997) was not detected. We suggest that the line detected by Ginga might be not associated with the quasar but, most likely, with a distant cluster of galaxies. There are indications for a weak emission line-like feature around 1.25\,keV (1.63\,keV in the quasar frame) in both, the SIS and GIS spectra. The energy is surprisingly close to that of an unexplained line found in the flat-spectrum quasar PKS\,0637-752 by Yaqoob \eta (1998). No evidence for the proposed binary black holes (Roos \eta 1993) is found in the current X-ray data. % \keywords{Galaxies: quasars: individual: 4C\,+73.18 (1928+738)-- X-rays: galaxies } %
|
[
{
"name": "paper.tex",
"string": "%&latex209\n\\documentstyle[psfig]{l-aa}\n%\n\\def\\eta{et al. }\n\\def\\etae{et al.}\n\\def\\ergs{${\\rm erg\\,cm^{-2}\\,s^{-1}}$ }\n\\def\\ergse{${\\rm erg\\,cm^{-2}\\,s^{-1}}$}\n\\def\\mlum{${\\rm erg\\,s^{-1}\\,Hz^{-1}}$ }\n\\def\\mlume{${\\rm erg\\,s^{-1}\\,Hz^{-1}}$}\n\\def\\nh{$N_{\\rm H}$ }\n\\def\\nhe{$N_{\\rm H}$}\n\\def\\src{4C\\,+73.18 }\n\\def\\srce{4C\\,+73.18}\n%\n\\begin{document}\n\\thesaurus{11.17.4:4C\\,+73.18 (1928+738) -- 13.25.2}\n\\title{X-ray observation of the super-luminal quasar 4C\\,+73.18 \n(1928+738) by ASCA} \n\\author{W. Yuan$^{1,2,3,4}$ \\and W. Brinkmann$^{1}$ \n\\and M. Gliozzi$^{1}$ \\and Y. Zhao$^{3}$ \\and M. Matsuoka$^{4}$ }\n%\\offprints{W. Yuan}\n\\institute{\n$^1$ Max--Planck--Institut f\\\"ur extraterrestrische Physik,\nGiessenbachstrasse, D-85740 Garching, Germany \\\\\n$^2$ Beijing Astrophysics Center (BAC),\nNo.1 Yifu Building 5501, Peking University, Beijing 100871 China\\\\\n$^3$ Beijing Astronomical Observatory, National Astronomical Observatories,\n Chinese Academy of Sciences, Beijing 100012, China\\\\\n$^4$ National Space Development Agency of Japan (NASDA), \nTsukuba Space Center, 2-1-1 Sengen, Tsukuba, Ibaraki 305 Japan}\n\\date{Received Sep. 1, 1999; accepted Feb. 24, 2000}\n\\maketitle\n\\markboth{W. Yuan et al.: ASCA observation of 4C\\,+73.18 (1928+738)}{}\n\\begin{abstract}\nThe results of an X-ray observation of the nearby \ncore-dominated, super-luminal quasar\n4C\\,+73.18 (1928+738) with ASCA are reported.\nVariations of the 1--10\\,keV flux on time-scales of the order of 10 hours\nwere found while the 0.5--1\\,keV flux remained almost unchanged.\nThe measured flux at 1\\,keV had increased by $\\sim 50\\%$ \ncompared with previous observations 18 years ago. \nThe quasar shows a curved spectrum in the 0.8--10\\,keV band,\nwhich can be modeled as a power law plus a soft component. \nBoth, the spectrum and the uncorrelated flux variations\nindicate the presence of a soft X-ray excess\nover the extrapolation of the hard band power law,\nwhich might be the high-energy extension of \nthe intense UV bump emission found in this quasar.\nThe hard X-ray emission is dominated by the flat power law \n(photon index 1.1--1.5).\nThe results are consistent with models in which \nthe hard X-rays are emitted from the relativistic jet. \nAn iron line emission as reported \nin a previous Ginga observation (Lawson \\& Turner 1997) was not detected. \nWe suggest that the line detected by Ginga might be \nnot associated with the quasar but, most likely, with a distant cluster\nof galaxies.\nThere are indications for a weak emission line-like feature around 1.25\\,keV \n(1.63\\,keV in the quasar frame) \nin both, the SIS and GIS spectra. \nThe energy is surprisingly close to \nthat of an unexplained line found\nin the flat-spectrum quasar PKS\\,0637-752 by Yaqoob \\eta (1998).\nNo evidence for the proposed binary black holes (Roos \\eta 1993) is found in\nthe current X-ray data.\n% \n\\keywords{Galaxies: quasars: individual: 4C\\,+73.18 (1928+738)-- X-rays: galaxies }\n%\n\\end{abstract}\n\n\\section{Introduction}\n\n4C\\,+73.18 (1928+738) is an extremely core-dominated, \nsuper-luminal quasar at a redshift of 0.302.\nIt is a flat-spectrum radio source in the S\\,5 survey (K\\\"uhr \\eta 1981),\nwhich has been well studied with VLBI and \nshows unusual jet properties. \nOn arc-second (kpc) scales, the source exhibits two-sided curved jets, lobes, \nand a dominant core (Johnston \\eta 1987, Hummel \\eta 1992, Murphy \\eta 1993). \nOn pc scales, VLBI observations of the core reveal an one-sided jet,\nwhich exhibits apparent super-luminal motion with $v_{\\rm app}/c \\sim 4-7$\n(Eckart \\eta 1985, Witzel et al.\\ 1988).\nThe jet is estimated to be aligned within \n$7^{\\circ}-12^{\\circ}$ to the line-of-sight\n(e.g.\\ Ghisellini \\eta 1993, Jiang \\eta 1998).\nNo $\\gamma$-ray emission was detected by EGRET (Fichtel \\eta 1994).\nTo explain the mis-alignment between the VLBI jet\nand the kpc-scale jet, Hummel \\eta (1992) suggested that the \nobserved VLBI jet might\nbe only one of the Doppler-boosted filaments.\nInterestingly, the most recent results from \nthe space VLBI (VSOP) monitoring of 4C\\,+73.18\nreveal substantial jet bending at a distance of\n6.3\\,pc from the core, and dramatic temporal \nvariations of the bend angle \nover a few months (Murphy et al.\\ 1999).\n\nIt is intriguing that the sub-milliarcsec jet \nexhibits ballistic motion along a sinusoidal curve \nwith a period of about 3 years (Hummel \\eta\\ 1992).\nThe wiggles over such short period were modeled \nby Roos \\eta (1993) as the orbital motion of a binary black hole system,\nwhich has a mass of the order of $10^8\\,M_{\\odot}$, a mass ratio $>0.1$, and \na separation of $\\sim 10^{16}$\\,cm. \nNo other observational evidence for the binary black hole model \nhas been reported so far. \nX-ray observations, having the advantage of looking into the \nregion closed to the black hole, provide a potential tool to find signatures\nof binary black holes.\n\nIn the X-ray band,\n4C\\,+73.18 has been observed previously by {\\em Einstein} \n(Biermann et al.\\ 1981), \n{\\em EXOSAT} (Biermann \\eta 1992, Ghosh \\& Soundararajaperumal 1992), \nGinga (Lawson \\& Turner 1997), \nand by ROSAT in the Survey (Voges et al.\\ 1999) \nand pointed observations with PSPC\n(Brunner \\eta 1994, and Sambruna 1997) and HRI, respectively.\nThe measured spectra were generally described by a power law.\nOf particular interest is the detection of \na significant iron K$_\\alpha$ emission line in the Ginga observation, \nmaking this object one of the few \nflat-spectrum radio quasars showing iron K$_\\alpha$ line. \nFurthermore, the ROSAT data \nseem to suggest an absorption column in excess of the Galactic value. \n\nIn this paper, we report on results of an X-ray observation\nof 4C\\,+73.18 in the 0.8--10\\,keV band \nperformed by the ASCA satellite.\nThe spectral and temporal analysis of the data \nare presented in \\S\\,2 and 3, respectively. \nA re-analysis of the archival ROSAT\nPSPC spectrum is also given (\\S\\,\\ref{spec:rosat}).\nWe discuss the implications of the results and summarize the main\nconclusions in \\S\\,4 and \\S\\,5, respectively. \nErrors are quoted at 68\\% level throughout the paper,\n unless mentioned otherwise.\n\n\\section{X-ray spectrum}\n\\subsection{ASCA observation and data reduction}\n4C\\,+73.18 was observed with ASCA (Tanaka et al.\\ 1994) on 12 August, 1997.\nThe Solid-state Imaging Spectrometer (SIS) was operated in 1-CCD faint mode.\nThe data reduction and spectral analysis was performed using FTOOLS (v.4.2) \nand XSPEC (v.10), respectively.\nWe used the `bright~2' mode data for SIS with the corrections for \ndark frame error and echo effect applied. \nAfter removing hot and flickering pixels,\nthe data screening was performed in the standard way using the following \ncriteria: an elevation angle of $10^{\\circ}$, a magnetic cutoff rigidity\nof 6\\,GeV/c, and bright Earth angles of $25^{\\circ}$ for the SIS0 and \n$20^{\\circ}$ for the SIS1.\nThe effective exposures for good data interval were \n16.8/17.6\\,ksec for the SIS0/1 and 17.8\\,ksec \nfor the Gas Imaging Spectrometer (GIS).\nThe source counts were extracted from circular regions of $\\sim 3.5$ and 6 arcmin\nradius for SIS and GIS, respectively. \nThe background counts were determined in two ways,\nfrom blank sky observations at the same region of the detector \nand from a `local' off-source region of the same observation \n(at the same off-axis position with \nthe same radius as the source region in the case of GIS).\nThe count rates of the two kinds of background data agree \nwith each other within about 10\\% relative errors.\nIn the analysis we always used both backgrounds\nand compared the results but only those obtained by using blank sky\ndata are presented here, for they have a higher signal-to-noise ratio. \nThe averaged net source count rates \nof the SIS1/2 and GIS3/4 are 0.14\\,cts/s and 0.13\\,cts/s, respectively.\nThe spectra of both the two GIS and SIS detectors were \ncombined\\footnote{We found that, for relatively weak sources, \nthe spectrum combining applications\nin the standard FTOOLS package resulted in unreasonably large \nstatistical errors and thus unrealistically small $\\chi^2$\nin the spectral fits. We therefore used pure Poissonian errors\nfor the counts in each bin for the combined spectrum.\nThis might slightly overestimate the resulting $\\chi^2$ of the fits\nsince systematic uncertainties were not taken into account;\nhowever, for relatively weak sources, the uncertainties were dominanted\nby photon statistics.}\nand re-binned to have at least 30 counts in each energy bin. \n \n\\subsection{The ASCA spectrum}\n\\label{spec:asca}\n\\subsubsection{A single power law fit}\nDue to the increasing degradation of the response at low \nenergies the calibration below $\\sim 1$\\,keV is \nuncertain for the SIS\\footnote{ASCA Calibration Uncertainties, 1999, \nASCA web page at GSFC, NASA.}.\nAs a conservative criterion, we ignored the energy band below\n0.9\\,keV\\footnote{Fitting an absorbed power law to the 0.6--10\\,keV SIS spectra\ngave \\nh significantly larger than the Galactic value ($7.4\\times 10^{20}$),\n$16.0\\times 10^{20}$ for SIS1 and $9.7\\times 10^{20}$ for SIS0, which\nare inconsistent with the GIS spectra.}\n(Ikebe, private communication).\nA simultaneous single power law fit to the combined SIS and GIS spectra \nis not acceptable (see Table\\,1) either for free or for fixed Galactic absorption\n($N_{\\rm H}^{\\rm Gal}=7.40 \\times 10^{20}$\\,cm$^{-2}$, Dickey \\& Lockman 1990). \nNoticeable deviations from a power law can be seen from the residuals \nin Fig.\\,\\ref{fig:one-pl} (a) and (b), which show\na spectral hardening above 6\\,keV and a line-like feature around 1.3\\,keV.\nEven ignoring the 1.1--1.5\\,keV data which covers the line-like feature \nyielded no acceptable fits. \nRepeating the same fits (fixing \\nh at the Galactic value) \nto the 2--10, 3--10, and 4--10\\,keV band data\ngave progressively flattening photon indices \n$\\Gamma =1.61\\pm 0.06$, $1.48^{+0.08}_{-0.10}$, and $1.37^{+0.14}_{-0.18}$, \nrespectively), indicative of a curved spectral shape in the 0.8--10\\,keV band\n(1--13\\,keV in the quasar frame).\nThis suggests composite spectral models, i.e.\\, \na power law plus a soft X-ray component. \n\nThe integrated 2--10\\,keV flux is \n$6.2 \\times 10^{-12}$ \\ergse, \nusing the mean value of the SIS and GIS measurements, which \ncorresponds to a K-corrected luminosity of \n$2.9\\times 10^{45}$\\,erg\\,s$^{-1}$ ($q_{0}=0.5, H_{0}=50$).\n\n\\begin{figure}\n\\psfig{file=figure1.ps,width=8.5cm,height=9cm,angle=270}\n\\caption{\\label{fig:one-pl} \n({\\bf a}) Simultaneous fit to the SIS (open circles) and GIS (dots) spectra with\nan absorbed power law (Galactic absorption). \n({\\bf b}) The residuals of (a) as the ratio between \nthe data and the fitted model. \n({\\bf c}) The residuals of the best fit\nmodel composed of a power law with Galactic absorption \nplus cold reflection, thermal plasma,\nand an additional Gaussian line (see text).}\n\\end{figure}\n\nNo emission line features were seen at the iron K$_{\\alpha}$ line energy.\nThe 90\\% upper limits on the equivalent width of a 6.4\\,keV line\nranges from 62 to 112\\,eV (source rest frame)\nfor line-widths ranging from 0.01 to 0.5\\,keV.\n\n\\subsubsection{Power law plus soft X-ray component}\nWe have tried the following models for the soft X-ray component: \npower law (PL), blackbody (BB), disk-blackbody (DBB),\nand thermal plasma (RS, Raymond-Smith in XSPEC).\nThe results are summarized in Table\\,1 except for DBB models. \nThe listed temperatures are in the quasar rest frame. \nIn general, the addition of a soft component improves the fits\nsignificantly\n($\\Delta \\chi^2=$16, 16, 17, 19 for PL, BB, DBB, RS, respectively); \nyet the fits remain relatively poor.\nIgnoring the 1.1--1.5\\,keV line-like feature \nimproved the fits slightly and made them marginally acceptable, \nwith a significance level \n$P\\sim$ 0.11 for PL, 0.13 for BB, 0.16 for DBB, 0.25 for RS, respectively.\nThus, the current data are insufficient \nto impose strong constraints on the spectral form of\nthe soft X-ray component, given the degrading SIS low energy response.\nOn the other hand, as will be discussed below, it is also possible\nthat none of the above models is a good representation of \nthe real spectral shape.\nThe thermal plasma model yields the minimal $\\chi^2$ and \\nh in\ngood agreement with the Galactic value, but, \nimplausibly low metal abundances and a much flatter\nphoton index than the `canonical value' $\\Gamma \\simeq 1.6$\nfor flat-spectrum radio quasars. \n\n\\begin{table}\n\\caption{Spectral fits for 4C\\,+73.18}\n\\label{tab:spec_fit}\n\\begin{tabular}{llll} \\hline\\hline\nparameters & SIS+GIS & SIS+GIS$^{(a)}$ & PSPC \\\\ \\hline\n%\n\\multicolumn{4}{l}{power law }\\\\\n$N_{\\rm H}^{(b)}$& $\\sim 0$ & $\\sim 0$ & 10.9$^{+1.2}_{-1.0}$ \\\\\n$\\Gamma$ & 1.69$^{+0.04}_{-0.02}$ & 1.67$^{+0.03}_{-0.02}$ & 2.25$\\pm$0.12 \\\\\n$N^{(c)}$ & 1.67$\\pm$0.15 & 1.63$\\pm$0.16 & 1.88$\\pm$0.13 \\\\\n$\\chi^2$/dof&1.26/140 & 1.20/121 & 0.95/23 \\\\ \\hline\n%\n\\multicolumn{4}{l}{power law (fixed \\nh = 7.4~$10^{20}$)}\\\\\n$\\Gamma$ & 1.79$\\pm$0.02 & 1.77$\\pm$0.02 & 1.82$^{+0.05}_{-0.06}$ \\\\\n$N $ & 1.90$\\pm$0.07 & 1.84$\\pm$0.09 & 1.63$\\pm$0.06 \\\\\n$\\chi^2$/dof& 1.31/141 & 1.27/122 & 1.73/24 \\\\ \\hline\n%\n\\multicolumn{4}{l}{power law + power law }\\\\\n\\nh & 26.4$\\pm$15.5 & 22.8$\\pm$21.5 & 13.6$^{+3.4}_{-2.6}$ \\\\\n$\\Gamma$ &1.14$\\pm$0.45 & 1.12$\\pm$0.60 & 1.5 (fixed) \\\\\n$\\Gamma_{\\rm s}$&3.04$\\pm$1.04&2.93$\\pm$1.44 & 3.26$^{+0.74}_{-0.85}$\\\\ \n$\\chi^2$/dof& 1.17/137 & 1.14/118 & 0.94/22 \\\\ \\hline\n%\n\\multicolumn{4}{l}{power law + zbbody} \\\\\n\\nh & $\\sim 0$ & $\\sim 0$ & 8.9$^{+1.3}_{-0.9}$ \\\\\n$\\Gamma$ & 1.42$^{+0.05}_{-0.09}$ & 1.42$\\pm$0.08 & 1.5 (fixed) \\\\\n$kT$ (keV) & 0.44$^{+0.02}_{-0.04}$ & 0.44$^{+0.04}_{-0.03}$ & 0.18$^{+0.03}_{-0.02}$ \\\\\n$\\chi^2$/dof& 1.17/137 & 1.15/118 & 0.93/22 \\\\ \\hline\n%\n\\multicolumn{4}{l}{power law + zbbody (\\nh = $7.4~10^{20}$)} \\\\\n$\\Gamma$ & 1.48$^{+0.03}_{-0.04}$ & 1.49$^{+0.04}_{-0.06}$ & 1.17$^{+0.18}_{-0.30}$ \\\\\n$kT$ & 0.38$^{+0.02}_{-0.03}$ & 0.37$^{+0.03}_{-0.04}$ & 0.21$\\pm$0.02 \\\\\n$\\chi^2$/dof& 1.18/138 & 1.16/119 & 0.97/22 \\\\ \\hline\n%\n\\multicolumn{4}{l}{power law + thermal plasma } \\\\\n\\nh & 8.1$^{+4.9}_{-2.7}$ & 12.2$^{+7.0}_{-4.2}$ & 7.3$^{+0.6}_{-0.5}$ \\\\\n$\\Gamma$ & 1.02$^{+0.22}_{-1.02}$ & 1.27$^{+0.16}_{-0.15}$ & 1.5 (fixed) \\\\\n$kT$ & 1.83$\\pm$0.43 & 1.44$^{+0.21}_{-0.26}$ & 0.84$^{+0.18}_{-0.13}$\\\\\n$A^{(d)}$ & 0.1$\\pm$0.1 &0.5$^{+1.1}_{-0.3}$ & 1.0 (fixed) \\\\\n$\\chi^2$/dof& 1.16/136 & 1.08/117 & 1.07/22 \\\\ \\hline\n%\n\\multicolumn{4}{l}{power law + thermal plasma (\\nh = $7.4~10^{20}$)} \\\\\n$\\Gamma$ & 1.10$^{+0.13}_{-0.53}$ & 1.27$^{+0.16}_{-0.26}$ & 1.50$^{+0.17}_{-0.70}$\\\\\n$kT$ & 1.76$^{+0.73}_{-0.21}$ & 1.59$^{+0.29}_{-0.21}$ & 0.86$^{+0.19}_{-0.15}$\\\\\n$A$ & 0.1$\\pm$0.1 & 0.4$^{+1.3}_{-0.2}$ & 0.2$\\pm$0.2 \\\\\n$\\chi^2$/dof& 1.15/137 & 1.08/118 & 1.08/21 \\\\ \\hline\n%\n\\multicolumn{4}{l}{perxav + thermal plasma + Gaussian (\\nh =$7.4~10^{20}$)} \\\\\n$\\Gamma$ & 1.50$^{+0.16}_{-0.50}$ & 1.41$^{+0.22}_{-0.35}$ & -- \\\\\n$kT$ & 1.83$^{+0.40}_{-0.29}$ & 1.62$^{+0.30}_{-0.23}$ & -- \\\\\n$A$ & 0.2$\\pm$0.2 & 0.4$^{+0.8}_{-0.3}$ & -- \\\\ \n$\\Omega/2\\pi^{e)}$ &0.58$^{+0.47}_{-0.58}$ & 0.46$^{+0.81}_{-0.46}$ & -- \\\\\n$\\cos \\theta ^{(f)}$ &0.97$^{+0.03}_{-0.65}$ & 0.96$^{+0.04}_{-0.90}$ & -- \\\\ \n$E_{\\rm line}^{(g)}$ SIS &1.25$^{+0.03}_{-0.04}$ & -- & -- \\\\ \n$E_{\\rm line}$ GIS &1.26$\\pm$0.05 & -- & -- \\\\\n$I_{\\rm line}^{(h)}$ SIS & $2.1^{+1.1}_{-1.0}$ & -- & -- \\\\\n$I_{\\rm line}$ GIS & $3.7^{+2.5}_{-2.2}$ & -- & -- \\\\\n$\\chi^2$/dof & 1.15/131 & 1.09/116 & -- \\\\ \\hline\n\\end{tabular}\n\n (a) Excluding 1.1-1.5\\,keV line-like structure \\\\\n (b) Column density in units of 10$^{20}$\\,cm$^{-2}$ \\\\\n (c) Normalization at 1\\,keV in units of $10^{-3}$ photon\\,cm$^{-2}$\\,s$^{-1}$\\,keV$^{-1}$\\\\\n (d) Abundances in units of solar value \\\\\n (e) Solid angle subtended by the reflecting matter\\\\\n (f) Inclination angle of the normal of the reflecting slab\\\\\n (g) line energy in units of keV\\\\ \n (h) line intensity in units of $10^{-5}$\\,photon\\,cm$^{-2}$\\,s$^{-1}$\n\\end{table}\n\n\\subsubsection{Reflection component}\n\\label{fit-refl}\nAn additional soft X-ray component seems not to be able to \nfully account for the spectral hardening.\nIn the case of thermal plasma model, \nthe power law index is too flat ($\\Gamma=1.10^{+0.13}_{-0.53}$),\nwhile for the other models the residuals show \nsome remaining excess flux in the highest energy bins.\nFitting a broken power law plus thermal plasma model \nyielded a low energy band index \n$\\Gamma=$1.4 and a flat index 0.1 in the hard energy range \nbeyond $\\sim 7$\\,keV.\nWe then tried to include a Compton-reflection component \n({\\it pexrav} in XSPEC),\nin which a Compton thick, neutral (except H and He)\nslab is irradiated by an X-ray source with \na subtended solid angle $\\Omega$.\nThe high-energy cutoff was fixed at 500\\,keV.\nThe fitted photon index became $1.50^{+0.16}_{-0.50}$.\nHowever, the improvement of the fit is insignificant and \nthe addition of the reflection component cannot be justified.\nIt should be noted that the model of\na power law plus reflection but without a soft component\nyielded no acceptable fits. \n\n\\subsubsection{Emission line like feature around 1.3\\,keV}\nWe now consider the emission line like feature around 1.3\\,keV.\nWe repeated the same fitting as in \\S\\,\\ref{fit-refl} \n(RS plus {\\it pexrav} model) by adding \na Gaussian line at $\\sim 1.3$\\,keV with fixed line width $\\sigma=0.01$.\nThe line energies and normalizations were set to be independent for\nthe SIS and the GIS.\nThe best-fit parameters are given in Table\\,\\ref{tab:spec_fit}, \nand the residuals are shown in Fig.\\,\\ref{fig:one-pl} (c).\nThe fit improved with $\\Delta\\chi^2=6.0$.\nGiven four additional parameters involved \n(the line energies and intensities for both SIS and GIS),\nthe significance for adding the line component is only marginal\n(at a confidence level slightly less than 90\\%). \nThe derived equivalent width is $22^{+12}_{-11}$\\,eV for the SIS\nand $31^{+21}_{-20}$\\,eV for the GIS data.\n\nIt seems unlikely that the excess in Fig.\\,1 is an instrumental effect\nas the feature appears in both, the SIS and GIS data at\nalmost identical energies,\n$1.25^{+0.03}_{-0.04}$ and $1.26\\pm 0.05$\\,keV, respectively.\nThe line seems to be narrow, \nwith an apparent width of the order of the detector energy resolution,\nand to be non-Gaussian, as indicated by the structure of the residuals\nin Fig.\\,\\ref{fig:one-pl} (c). \nPlotted in Fig.\\,\\ref{fig:line_cont} are the \nconfidence contours in the parameter space of the\nline energy vs.\\ strength, which were obtained by using the GIS data only. \nWe also tried absorption models instead of an emission line to see\nwhether the line-like feature can be accounted for.\nHowever, neither an absorption edge nor a notch line fits the data.\n\n\\begin{figure}\n\\psfig{file=figure2.ps,width=7.0cm,height=6.0cm,angle=270}\n\\caption{\\label{fig:line_cont}\nConfidence contours in the line energy versus strength space \nfor the line fit to the GIS data.\nThe contours are at 68\\%, 90\\%, and 99\\% levels \nfrom the center outwards, respectively.}\n\\end{figure}\n\n\\subsection{The ROSAT spectrum}\n\\label{spec:rosat}\nTo check the consistency of the spectral models derived from ASCA \nwith previous observations, we re-analyzed the archival spectrum\nof the PSPC pointed observation of 4C\\,+73.18. \nA power law with neutral absorption model yields \na steep photon index ($\\Gamma= 2.25\\pm 0.12$)\nand absorption \\nh\nsignificantly higher than the Galactic value (Table\\,1).\nFixing \\nh at the Galactic value gives an unacceptable fit\n($P=0.01$) and a flatter index. \nThese results are in agreement \nwith those given in Brunner et al.\\ (1994) and Sambruna (1997).\nIn light of the ASCA spectral modeling,\nwe added a soft X-ray component and \nperformed the same fits as in \\S\\,\\ref{spec:asca}. \nThe results are given in Table\\,\\ref{tab:spec_fit}. \nConsidering the narrow PSPC bandpass,\nwe first fix the photon index of the hard power law at $\\Gamma=1.5$.\nAdding a blackbody or thermal plasma model both yields the best fits, \nwith $kT=0.18^{+0.03}_{-0.02}$ and $0.84^{+0.18}_{-0.13}$\\,keV (source frame), \nrespectively, and an \\nh consistent with the Galactic value.\nFixing \\nh at the Galactic value and setting $\\Gamma$ as a\n free parameter gives \n$\\Gamma=1.50^{+0.17}_{-0.70}$ for the thermal plasma and \n$\\Gamma=1.17^{+0.18}_{-0.30}$ for the blackbody model, respectively. \nThe fits are insensitive to the metal abundances,\nand the best fit value is $0.17\\pm0.16$ solar.\nThe excess absorption suggested previously is not required,\nwhich apparently resulted from fitting a single power law \nto the steepening part of the curved spectrum toward low energies.\n\nNo emission line like feature is seen around 1.25\\,keV in the ROSAT spectrum.\nHowever, the poor energy resolution (40\\% at 1\\,keV) \nmakes the PSPC insensitive to such a weak line. \nThe upper limit on the equivalent width \nof a narrow emission line at 1.25\\,keV is \nfound to be 74.5\\,eV at 90\\% confidence,\nwhich is consistent with the ASCA data.\n\n\\subsection{Summary of spectral fits}\n\nThe ASCA spectra exhibit a concave curvature which \nflattens toward high energies.\nModels composed of a power law and a soft X-ray excess \nimprove the fits significantly,\nthough the spectral shape of the soft component is inconclusive. \nSuch models also fit the previous ROSAT spectrum well, \nhence the soft X-ray excess in 4C\\,+73.18 seems to be \npersistent, rather than a transient emission feature.\nNo significant excess X-ray absorption is indicated.\nNo iron K$_{\\alpha}$ line is detected. \nA weak emission line like feature at 1.25\\,keV\nis present in both, the SIS and GIS spectra,\nthough the significance is not high.\nIt should be noted that the ASCA spectra are `noisy' above 2\\,keV, \nand the spectral fits are relatively poor in general. \nThis may result, at least partly, from \nsource variability as shown below. \n\n\\section{X-ray variability}\n\\subsection{Variability on short time-scales} \nX-ray light curves were extracted from the source region and binned\nwith a bin size of 96\\,min (one ASCA orbit) for both the SIS and GIS detectors.\nThe background count rates, estimated from the source free region and \nnormalized to the source region, were subtracted.\nPlotted in Fig\\,\\ref{fig:vari_flux} are the light curves \nfrom the SIS detectors only. \nSmall amplitude, but statistically significant flux variations in \nthe 0.5--10\\,keV band are \nfound ($\\chi^2$ test, see Table\\,\\ref{tab:vari}) \nover the $\\sim40$\\,ksec observation.\nAs shown above, the X-rays in the ASCA band \nare likely composed of a hard power law, a soft component dominating\nbelow 1\\,keV, and a tentative reflection component \nabove 7\\,keV (9\\,keV in the source frame).\nWe divided the ASCA energy band into three corresponding bands, i.e.\\ \na low (0.5--1\\,keV), a medium (1--6\\,keV), and a high (6--10\\,keV) band.\nThe light curves in Fig.\\,\\ref{fig:vari_flux} show different temporal\nvariations in the medium band (the power law component) \nfrom those in the low and high bands. \nThe medium band flux reveals significantly more variability than \nthe whole band ($\\Delta \\chi^2=4$), \nwhereas the low and high energy band fluxes are consistent with \nno variations (Table\\,\\ref{tab:vari}). \nWe checked the significance of the \nrelative variations between different bands\nby testing the constancy of their counts ratios,\n$C_{\\rm 1-6keV}/C_{\\rm 0.5-1keV}$ and $C_{\\rm 1-6keV}/C_{\\rm 6-10keV}$.\nThe probabilities for the $\\chi^2$ tests \nare listed in Table\\,\\ref{tab:vari}.\nNon-synchronized variations between the medium and \nlow, and the medium and high bands, however, cannot be confirmed. \nThis is likely due to the poor photon statistics in the low and high bands,\nas well as to the contribution from the underlying power law flux\nin these two bands.\nThe GIS light curves show very similar variations;\nhowever, the large counting errors due to the higher background level\nmakes the GIS insensitive to variability with such small amplitudes.\n\\begin{figure}\n\\psfig{file=figure3.ps,width=8.5cm,height=9.0cm,angle=0}\n\\caption{\\label{fig:vari_flux} Background subtracted X-ray light curves of \\src \nfrom the SIS detectors (SIS0+SIS1) in the \nwhole band (upper panel) as well as\nin the low, medium, and high passbands\nThe bin size is 5760\\,s and the errors are of 1\\,$\\sigma$.}\n\\end{figure}\n%\n\\begin{table}\n\\caption{$\\chi^2$ test of variability for the SIS fluxes}\n\\label{tab:vari}\n\\begin{tabular}{llll} \\hline\\hline\nband (keV) & $\\chi^2$ (dof=6) & $P (\\chi^2)$ & significance \\\\ \\hline\n0.5--10 & 13.5 & 0.035 & yes \\\\\n0.5--1 & 5.9 & 0.43 & no \\\\\n1--6 & 17.4 & 0.008 & yes \\\\\n6-10 & 4.3 & 0.64 & no \\\\ \\hline\n\\multicolumn{4}{l}{Variability of counts ratio} \\\\ \\hline\n$C_{\\rm 1-6keV}/C_{\\rm 0.5-1keV}$ & 8.1 & 0.23 & no \\\\\n$C_{\\rm 1-6keV}/C_{\\rm 6-10keV}$ & 5.3 & 0.50 & no \\\\ \\hline\n\\end{tabular}\nNote: we use the critical significance level of $P=0.05$.\n\\end{table}\n\nWe conclude that the observed short time scale flux variability in \\src\nis mainly attributable to variations of the hard band power law.\nThe variability time scale is not obtainable from \nthe relatively short\nobservation interval; however, it could be of the order of the \nduration of the observation, i.e.\\ 10 hours in the source frame. \nNo statistically significant variations are found for the flux in \nthe 0.1--1\\,keV band.\nThis result is supported by the lack of variations of\nthe flux in the 0.1--2.4\\,keV PSPC band\non similar time scales during the ROSAT observation\n($\\chi^2=1.06$ for 37 d.o.f., Brunner et al.\\ 1994).\nThe uncorrelated, energy dependent flux variations are consistent with the\nabove models of multi-component X-ray emission.\n\n\\subsection{Long-term variability} \n\nWe present in Fig.\\,\\ref{fig:hist_lc} \nthe long-term X-ray light curve of 4C\\,+73.18, compiled from \nthe flux densities at 1\\,keV from previous missions.\nThe estimated Ginga flux is an extrapolation of\nthe observed 2--10\\,keV spectrum \nand, as discussed below, should be taken as \nan upper limit due to possible \nflux contamination from nearby sources. \nThe light curve tends to suggest an overall increase of \nthe 1\\,keV flux density \nby a factor of 50\\% over 18 years of observations.\n\nIn addition, prominent flux variations in the hard X-ray band\nwere reported from the {\\em EXOSAT} observations:\nthe 2--10\\,keV flux varied by a factor of 2--3 \n(from 1.5--5.1$\\times 10^{-12}$\\,\\ergs) \non a time scale of about one month\n(Ghosh \\& Soundararajaperumal 1992).\n\n\\begin{figure}\n\\psfig{file=figure4.ps,width=8.5cm,height=6.5cm,angle=0}\n\\caption{\\label{fig:hist_lc} Historical light curve of the X-ray \nflux densities of 4C\\,+73.18 measured at 1\\,keV.\nThe fluxes of {\\em Einstein} and {\\em EXOSAT} \nobservations are taken from Brunner et al.\\\n(1994). The two ROSAT data points represent the Survey and pointed\nobservations, respectively.\nThe estimated Ginga flux (taken as an upper limit here; see text) \nis an extrapolation of the measured \n2--10\\,keV spectrum from Lawson \\& Turner (1997).}\n\\end{figure}\n\n\\section{Discussion}\n\n\\subsection{Emission from host cluster of galaxies?}\nAs mentioned by Brunner \\eta (1994, quoting a private communication) \nthere are indications for a \ncluster of galaxies around 4C\\,+73.18, which would affect \nthe X-ray spectrum determined from instruments with limited spatial resolution.\nTo investigate the possible role played by an \nextended X-ray emission component, we fitted a point source plus\na $ \\beta$-model (e.g. Gorenstein et al. 1978) of the form\n$$\nS(r)=S_0\\left(1+{r^2\\over r_c^2}\\right)^{-3\\beta+1/2}\n$$\nto the surface brightness profile of the ROSAT HRI source \n(with an exposure of 38660\\,sec).\nThe result is inconclusive; however, a noticeable contribution\nfrom a cluster at a flux level greater than a few percent\nin the HRI energy band can be ruled out. \n\nIn the hard X-ray band (0.8--10\\,keV), \nno evidence for extended emission \nis found in the ASCA GIS image.\nMoreover, no iron K$_{\\alpha}$ emission line is detected in the ASCA spectra.\nIt should be noted that the GIS source extraction radius of 6\\,arcmin \ncorresponds to $\\sim 1.3$\\,Mpc at the quasar distance, \nwhich is about 2--3 times the typical X-ray core radius\nand hence may well encompass the X-ray emitting region of the host\ncluster.\nTaking the typical value of 1\\,keV for \nthe equivalent width of the iron K$_{\\alpha}$ line of cluster emission, \nthe derived $EW<50$\\,eV places an upper limit on the flux\ndensity of a host cluster of being 7\\% of the quasar flux at about 5\\,keV\nin the observer frame.\nThus, we do not find evidence for \nnoticeable X-ray emission from a host cluster of galaxies.\n\n\\subsection{The iron line puzzle}\n\\label{dis:line}\nOne of the interesting results of this work is the non-detection of\nthe iron K$_{\\alpha}$ line, which was previously reported \nfrom the Ginga observation (Lawson \\& Turner 1997).\nThe upper limits of the equivalent width is estimated to be \n62\\,eV, in contrast to the measured\n$185^{+79}_{-74}$\\,eV by Ginga.\nMoreover, the spectrum ($\\Gamma=2.08^{+0.05}_{-0.04}$)\nis steeper and the flux \n($8.7\\times10^{-12}$\\,\\ergse) higher\nfor Ginga than for the ASCA observations \n($\\Gamma=1.61\\pm0.06$ and flux $6.2\\times10^{-12}$\\,\\ergse)\nin the 2--10\\,keV band.\nThe difference of the two continuum spectra can be seen clearly in \nFig.\\,\\ref{fig:sed}.\nThere are two possible explanations to this observed discrepancy, namely,\neither genuine spectral variations of both the iron line and continuum, \nor contamination of the Ginga spectrum by other sources.\nWe consider the latter to be more plausible, \nbased on the following arguments.\n\n\\subsubsection{Iron line variability}\nIt is rare for blazar-type radio quasars to show iron line emission \n(e.g.\\ Siebert et al.\\ 1996, Lawson \\& Turner 1997, \nReeves et al.\\ 1997, Cappi et al.\\ 1997).\nX-ray monitoring of 3C\\,273 also revealed that the iron line\nwas detected in only some but not all of the observations.\nThis effect seems to be dependent on variations of the\ncontinuum rather than of the line flux---at a high continuum flux,\nthe line equivalent width decreases or the line is even swamped\n(e.g.\\ Turner et al.\\ 1990, Cappi et al.\\ 1998).\nHowever, this was not the case for 4C\\,+73.18 during the Ginga observation\n(see below for a discussion on the Ginga flux).\nIf the line did come from the quasar,\nthere must be a variation of the line itself.\nThis may suggest that during the Ginga observation Seyfert like \nemission from the central AGN was substantially enhanced \nwhile the jet emission dimmed since \nno significant changes were found for the overall flux level.\n\n\\subsubsection{Iron line from another object?}\nGiven the large field of view (FOV) of $1.1^{\\circ}\\times2.0^{\\circ}$ (FWHM)\nand poor energy resolution of Ginga, \nthe association of the detected line with the quasar, and even \nthe presence of the line, may be questionable.\nWe examined X-ray sources in the vicinity of 4C\\,+73.18 which \nmay have fallen into the FOV of the Ginga observation.\nWithin the GIS FOV there is another source detected at a distance of\n12\\,arcmin from the quasar;\nhowever, the flux in the Ginga energy band is too low\n($2.7\\times10^{-13}$\\,\\ergse) to\nmake considerable contribution to the observed Ginga flux.\n\nWe next analyzed the data of ROSAT PSPC pointed observation of 4C\\,+73.18,\nwhose $2^{\\circ}$ FOV covers a large portion of the Ginga FOV.\nAmong 27 X-ray sources detected (with a detection likelihood larger than 10) \nin addition to 4C\\,+73.18, we found one source of particular interest.\nThe source is located \n43\\,arcmin away from the quasar and \ninside the Ginga FWHM field of view (Yamagisi, private communication).\nIt is bright (0.1 counts\\,s$^{-1}$ in the 0.1--2.4keV band---40\\% of \nthe count rate of the quasar) \nand extended (4 times of the FWHM of the point spread function).\nThis source has been selected from the ROSAT All-Sky Survey as\na candidate of, and later confirmed in optical to be, a cluster of galaxies\n(Boehringer et al.\\ 2000).\nThe ROSAT spectrum, which is of poor photon statistics, \ncan be fitted with thermal bremsstrahlung emission,\nand the temperature seems to be higher than 3\\,keV. \nThough the redshift is unknown, we suggest \nthis X-ray cluster as a likely source of the emission line detected by Ginga.\n\nAs a self-consistency check, we compare the \nline intensity predicted for the cluster with the observed value.\nThe observed line flux by Ginga is \n$2.55^{+1.08}_{-1.02}\\times10^{-5}$\\,photon\\,s$^{-1}$\\,cm$^{-2}$ \nand the line energy\n$5.13^{+0.30}_{-0.20}$\\,keV (observer frame).\nAssuming a cluster temperature of 6\\,keV and \nextrapolating the fitted ROSAT spectrum to higher energies, \nwe expect that Ginga would detect a line flux of\n$F_{\\rm line}\\simeq 4.4 f \\, EW/(1+z) \\times 10^{-5}$\\,photon\\,s$^{-1}$\\,cm$^{-2}$,\nwhere $EW$ is the line equivalent width \n(in units of keV, in the source frame)\nand $f$ the Ginga off-center detection efficiency at the source position \n($f\\simeq 0.6$ in this case).\nFor a redshift close to that of the quasar, $z\\sim 0.3$, we have \n$F_{\\rm line}\\sim 2.0 EW$ ($10^{-5}$\\,photon\\,s$^{-1}$\\,cm$^{-2}$),\nwhich is compatible with the observed value for a range of line \nequivalent width from several hundred eV to 1\\,keV. \nIn this case, the measured Ginga line energy \nimplies that the cluster is at a redshift range 0.23 $ \\leq z \\leq $ 0.36\nfor a 6.7\\,keV K$_{\\alpha}$ line.\nHowever, no firm conclusion can be drawn on the origin of the line \nuntil the redshift of the cluster is available.\n\n\\subsubsection{The Ginga continuum flux}\nThe estimated continuum flux from the cluster, using the above parameters,\nis $\\sim 2\\times 10^{-12}$\\,\\ergs in 2--10\\,keV, \nwhich may account for the measured flux excess for Ginga over the\nASCA measurement. Further,\nthough the rest of the sources in the PSPC FOV are too faint and soft\nto be considerable contributors to the observed Ginga flux, \nthey may account for the steepening of the Ginga spectrum in the soft\nX-rays.\nThe total PSPC count rate of sources in the vicinity of \n4C\\,+73.18 turned out to be $\\sim 0.36$\\,counts~ s$^{-1}$,\n1.3 times higher than that of quasar itself.\nThus we suggest that the excess fluxes of the Ginga observation\nover those of ROSAT and ASCA in the soft and medium \nX-ray band (see Fig.\\,\\ref{fig:sed})\ncould be, at least partly, accounted for \nby emission from the cluster of galaxies and other \nnearby soft X-ray sources. \nThe genuine X-ray flux of the quasar might be comparable with that\nof ASCA and ROSAT, though quantitative estimates can only be obtained \nby detailed modeling. \n\n\\subsection{The soft X-ray excess}\nThe presence of an excess flux at energies below $\\sim 1$\\,keV \nover the extrapolation of the hard band spectrum is evident\nfrom the curved spectrum in the ASCA band\nand the uncorrelated variations in the soft (0.5--1\\,keV) and \nhard (1--6\\,keV) band. \nThe latter was also reported from the {\\em EXOSAT} observations\n(Ghosh \\& Soundararajaperumal 1992).\nHowever, the spectral shape of the soft excess is uncertain. \nWe show in Fig.\\,\\ref{fig:sed} \nthe optical to X-ray energy distribution of 4C\\,+73.18, which \nis very similar to that of 3C\\,273. \nThe optical-UV spectral shape, though with sparse data points, \nis suggestive of \na prominent UV component which peaks at shortward 1300\\,$\\AA$\nor above $10^{15.3}$\\,Hz (the `blue bump'),\nand probably extends into the EUV band.\nIt is natural to regard the soft X-ray excess \nto be the high-energy tail of the UV bump,\nwhich results from Compton up-scattering disk photons \nto higher energies by hot gas.\n\\begin{figure}\n\\psfig{file=figure5.ps,width=9.0cm,height=7.0cm,angle=0}\n\\caption{\\label{fig:sed} The optical to X-ray spectral energy\ndistribution of 4C\\,+73.18 in the source rest frame ($q_{0}=0, H_{0}=50$).\nThe UV ({\\it HST}) spectrum is taken from Laor et al.\\ (1995). \nThe open circles are the ROSAT data and the filled dots the ASCA GIS data. \nFor a comparison, the power law spectrum \nobtained in the Ginga observation is over-plotted. \n}\n\\end{figure}\n\nWhen fitted with a thermal emission model,\nthe temperature $kT_{\\rm ex}$ (e.g.\\ 0.4\\,keV for blackbody)\nis somewhat higher than those obtained in a few \ncore-dominated radio quasars\nfor which $kT_{\\rm ex}$ was relatively well determined,\ne.g.\\ $\\sim 0.1$\\,keV in 3C\\,273 (e.g.\\ Turner \\eta 1990,\nB\\\"uhler \\eta 1994, Cappi \\eta 1998).\nThe higher $kT_{\\rm ex}$ could imply that the up-shift of \nthe energy of disk photons is more pronounced \nin 4C\\,+73.18 than in 3C\\,273.\nThe hot gas may be the skin-type coronae surrounding the accretion disks,\nas discussed in detail in, e.g., Czerny \\& Elvis (1987).\nIn the Czerny \\& Elvis model, \nthe high $kT_{\\rm ex}$ implies that \nthe quasar is likely radiating at a luminosity \naround the Eddington luminosity\n(see Fig.\\,7 of Czerny \\& Elvis 1987). \nIf this is true, \na UV luminosity of the order $\\sim 10^{46}$\\,ergs\\,s$^{-1}$ \n(see Fig.\\,\\ref{fig:sed}) implies a central mass of \nthe order of $8\\times 10^7$\\,M$_{\\odot}$ in 4C\\,+73.18.\nThis value is close to the mass of the binary black hole system \n($10^8$\\,M$_{\\odot}$)\nestimated by Roos et al.\\ (1993) by modeling the wiggles of the VLBI jet.\nOther sources of the hot gas might be\nthe base of the relativistic jet where it emerges from the \naccretion disk, as proposed by Mannheim \\eta (1995).\nThese scenarios all require a small inclination angle of the disk\n(or the jet, respectively) \nto the observer, in agreement with the super-luminal motion \nobserved in this source.\nFinally, the result is consistent with the findings by Brinkmann et al.\\ (1997) \nthat the soft X-ray excess tends to be more significant \nin core-dominated than in lobe-dominated quasars. \n\nIn the soft X-ray band, the spectral analyses in \\S\\,2 indicate a\nhigher temperature for the\nsoft thermal component in the ASCA than in the ROSAT observations.\nThis may well be a genuine spectral change,\nas the observations are about 6 years apart.\nThe conclusion, however, must be drawn with caution due to\nthe inter-instrument calibration uncertainties.\nThere have been reports that the ROSAT PSPC tends to give steeper spectra\nthan ASCA, even within the overlapping bandpass (e.g.\\ Iwasawa et al.\\ 1999).\nSuch an effect might play a role\nin comparing the temperatures derived from the two instruments.\n\n\\subsection{The hard X-ray continuum: emission from jet}\n\nThe observed flat power law continuum in the hard X-ray band \nis typical for flat-spectrum radio-loud quasars. \nThe inferred time scale of variations of the power law continuum\nmight be as short as of the order of hours.\nThese factors, as well as the lack of an iron $K_{\\alpha}$ line, \nare typical signatures of\nthe radio-jet-linked X-ray emission in radio-loud AGN, \nwhich is strongly beamed by relativistic bulk motion. \nThus the ASCA observation provides strong evidence for \nthe idea that the hard X-rays in 4C\\,+73.18 are emitted predominantly \nfrom the jet (via inverse Compton scattering).\nSuch mechanism has been tested in detail in a few similar super-luminal\nquasars (e.g.\\ 3C\\,345, Unwin \\eta 1994).\n\nTo investigate the contribution of possible Seyfert-like emission,\nwe tried to include a steep power law (fixing $\\Gamma=1.9$)\nas well as reflection in the above fittings.\nNo acceptable fits were obtained when a significant\nSeyfert-like emission component was included.\nThough, as shown in \\S\\,\\ref{spec:asca}, \nincluding a reflection model can account for the tentative\nexcess fluxes in the highest energy bins of the ASCA spectra,\nthe addition of this component is not justified statistically by the data.\nGiven the lack of an iron line and that the continuum is\ndominated by beamed jet emission, a significant reflection component\ncan not be physically justified either.\n\n\\subsection{The 1.3\\,keV emission line feature}\n\nA marginal, narrow line feature at 1.25\\,keV \nis suggested to be present in 4C\\,+73.18. \nIt is not clear from the current data \nwhether this is a real line or an instrumental artifact,\nthough the latter seems unlikely given that this feature appears \nin both the co-added SIS and GIS data at almost identical energies.\nIf interpreted as associated with the quasar, the line energy corresponds to\n1.63$\\pm0.05$ and 1.64$\\pm0.07$\\,keV \nin the quasar frame for the SIS and GIS, respectively. \nNo known emission lines have been found or are \nexpected to be present around these energies.\nThe closest line is the He-like MgXI transition at 1.58\\,keV.\n\nInterestingly enough, using ASCA data Yaqoob \\eta (1998) \nreported the discovery of a \npeculiar narrow emission line ($EW=59^{+38}_{-34}$\\,eV),\nwhich has never been seen in any other AGN, \nin another flat-spectrum quasar PKS\\,0637-752 (redshift 0.654).\nThe line energy in quasar frame is at 1.60$\\pm0.07$, which \nis strikingly close to what we found in this work.\nTherefore, the line-like feature in 4C\\,+73.18 provided\nthe first independent evidence for the presence of such an emission line,\nthough the statistical significance of the presence of the line is low.\nIt seems unlikely that the match of the line energies \nin two quasars with different redshifts is a coincidence. \nThe identification of this line is, however, unknown \n(see Yaqoob \\eta 1998 for a discussion of the possibilities).\nIf the emission line feature found in this work is real and identical to that\nof Yaqoob \\eta (1998), \nthe obtained $E_{\\rm line}\\sim 1.6$\\,keV \nshould be the line energy in the rest frame,\nrather than a Doppler-shifted value due to the in- or out-flow of the \nemitting gas.\nOtherwise it is hard to explain why the bulk motions\nare exactly the same in the two quasars.\n\n\\section{Conclusions}\nThe flat-spectrum, super-luminal quasar 4C\\,+73.18 \nresembles the well studied, similar object 3C\\,273 \nin some basic properties in the soft-to-medium X-ray band. \nRapid, small amplitude ($\\sim 25\\%$) flux variations \nwere detected in the 1--10\\,keV band on a tentative time scale\nof the order of 10 hours,\nwhereas no significant variations were found for \nthe soft X-rays below 1\\,keV. \nThe 0.8--10\\,keV spectrum exhibits a curvature,\nwhich can be modeled by a power law plus a soft component.\nBoth, the X-ray spectrum and the uncorrelated flux variations\nprovide strong evidence for the presence of a soft X-ray excess \nover the extrapolation of the hard band power law,\nwhich might be the high-energy extension of the UV bump emission. \nThe underlying continuum is flat, \nwith a power law photon index of 1.1--1.5,\ndepending on the spectral model adopted for the soft component.\nThe dominance of the flat power law \nover possible Seyfert-like emission, \ntogether with the short time scale variability,\nare naturally explained by beaming models for the jet,\nwhich is estimated to be observed at an inclination angle about 10 degree. \nNo iron K$_{\\alpha}$ emission line is detected, in contrast to \nthe significant line reported in a previous Ginga observation,\nwhich is most likely contamination from another source.\nA weak emission line-like feature around 1.25\\,keV\n(1.6\\,keV in the quasar frame)\nseems to be present in both the SIS and GIS spectra, \nthough the significance is not high.\nThe line energy is identical to that of a mysterious emission line\nin the flat-spectrum quasar PKS\\,0637-752 reported by Yaqoob \\eta (1998).\n\n\\begin{acknowledgements}\nWY thank Y.\\ Ikebe for the help regarding the ASCA data analysis and\ndiscussion on the X-ray cluster in the Ginga FOV, \nand also thank S.\\ Xue for useful discussions.\nThe authors thank I.\\ Yamagisi at ISAS for getting the attitude information \nof the Ginga observation.\nWe thank the Northern ROSAT All-Sky (NORAS) team for providing the\nidentification results on the cluster of galaxies before publication.\nWY is grateful to Prof.\\ J.\\ Tr\\\"umper and Dr.\\ W.\\ Voges for the \nfinancial support during the visit at MPE, and also to \nBeijing Astronomical Observatory for the financial support and\nto Prof.\\ J.\\ Chen at Beijing Astrophysics Center for hospitality,\nwhere part of the research was done.\nWY acknowledges support at NASDA by a STA fellowship.\nMG acknowledges partial support from\nthe European Commission under contract number ERBFMRX-CT98-0195\n(TMR network ``Accretion onto black holes, compact stars and\nprotostars\").\n\n\\end{acknowledgements}\n\n\\begin{thebibliography}{}\n\\bibitem{} Biermann P.L., Duerbeck H., Eckart A., et al., 1981, ApJL 247, 53 \n\\bibitem{} Biermann P.L., Schaaf R., Pietsch W., et al., 1992, A\\&AS 96, 339\n\\bibitem{} Boehringer H., et al.\\ 2000, ApJ submitted\n\\bibitem{} Brinkmann W., Yuan W., Siebert J., 1997, A\\&A 319, 413\n\\bibitem{} Brunner H., Lamer G., Staubert R., Worrall D.M., 1994, A\\&A 287, 436\n\\bibitem{} B\\\"uhler P., Courvoisier T.J.-L., Staubert R., 1994, A\\&A 287, 433\n\\bibitem{} Cappi M., Matsuoka M., Comastri A., et al., 1997, ApJ 478, 492\n\\bibitem{} Cappi M., Matsuoka M., Otani C., Leighly K.M., 1998, PASJ 50, 213\n\\bibitem{} Czerny B., Elvis M., 1987, ApJ 321, 305\n\\bibitem{} Dickey J.M., \\& Lockman F.J. 1990, ARA\\&A 28, 215\n\\bibitem{} Eckart A., Witzel A., Biermann P., et al., 1985, ApJL 296, L23\n\\bibitem{} Fichtel C.E., Bertsch T.L., Chiang J., 1994, ApJS 94, 551\n\\bibitem{} Ghisellini G., Padovani P., Celotti A., Maraschi L., 1993, ApJ 407, 65\n\\bibitem{} Ghosh K.K., Soundararajaperumal S., 1992, MNRAS 254, 563\n\\bibitem{} Gorenstein P.D., Fabrikant D., Topka K., Harnden F.R., Tucker W.H., 1978, ApJ 224, 718\n\\bibitem{} Hummel C.A., Schalinski C.J., Krichbaum T.P., et al., 1992, A\\&A 257, 489\n\\bibitem{} Iwasawa K., Fabian A.C., Nandra K., 1999, MNRAS 307, 611\n\\bibitem{} Jiang D.R., Cao X.W., Hong X., 1998, ApJ 494, 139\n\\bibitem{} Johnston K.J., Simon R.S., Eckart A., et al., 1987, ApJL 313, 85\n\\bibitem{} K\\\"uhr H., Pauliny-Toth I.I.K., Witzel A., Schmidt J., 1981, AJ 86, 854\n\\bibitem{} Laor A., Bahcall J.N., Jannuzi B.T., Schneider D.P., Green R.F., 1995, ApJS 99, 1 \n\\bibitem{} Lawson A.J., Turner M.J.L., 1997, MNRAS 288, 920\n\\bibitem{} Mannheim K, Schulte M., Rachen J., 1995, A\\&A 303, L41\n\\bibitem{} Murphy D.W., Browne I.W.A., Perley R.A., 1993, MNRAS 264, 298\n\\bibitem{} Murphy D.W., Tingay S.J., Preston R.A., et al., 1999, Elsevier preprint\n\\bibitem{} Reeves J.N., Turner M.J.L., Ohashi T., Kii T., 1997, MNRAS 292, 468\n\\bibitem{} Roos N., Kaastra J.S., Hummel C.A., 1993, ApJ 409, 130\n\\bibitem{} Sambruna R., 1997, ApJ 487, 536\n\\bibitem{} Siebert J., Matsuoka M., Brinkmann W., Cappi M., et al., 1996, A\\&A 307, 8\n\\bibitem{} Tanaka Y., Inoue H., Holt S.S., 1994, PASJ 46, L37\n\\bibitem{} Turner M.J.L., Williams O.R., Courvoisier T.J.-L, 1990, MNRAS 244, 310\n\\bibitem{} Unwin S.C., Wehrle A.E., Urry C.M., et al., 1994, ApJ 432, 103\n\\bibitem{} Voges W., Aschenbach B., Boller Th., et al., 1999, A\\&A 349, 389\n\\bibitem{} Witzel A., Schalinski C.J., Johnston K.J., et al., 1988, A\\&A 206, 245\n\\bibitem{} Yaqoob T., George I.M., Turner T.J., et al., 1998, ApJL 505, 87\n\\end{thebibliography}\n\\end{document}\n"
}
] |
[
{
"name": "astro-ph0002518.extracted_bib",
"string": "\\begin{thebibliography}{}\n\\bibitem{} Biermann P.L., Duerbeck H., Eckart A., et al., 1981, ApJL 247, 53 \n\\bibitem{} Biermann P.L., Schaaf R., Pietsch W., et al., 1992, A\\&AS 96, 339\n\\bibitem{} Boehringer H., et al.\\ 2000, ApJ submitted\n\\bibitem{} Brinkmann W., Yuan W., Siebert J., 1997, A\\&A 319, 413\n\\bibitem{} Brunner H., Lamer G., Staubert R., Worrall D.M., 1994, A\\&A 287, 436\n\\bibitem{} B\\\"uhler P., Courvoisier T.J.-L., Staubert R., 1994, A\\&A 287, 433\n\\bibitem{} Cappi M., Matsuoka M., Comastri A., et al., 1997, ApJ 478, 492\n\\bibitem{} Cappi M., Matsuoka M., Otani C., Leighly K.M., 1998, PASJ 50, 213\n\\bibitem{} Czerny B., Elvis M., 1987, ApJ 321, 305\n\\bibitem{} Dickey J.M., \\& Lockman F.J. 1990, ARA\\&A 28, 215\n\\bibitem{} Eckart A., Witzel A., Biermann P., et al., 1985, ApJL 296, L23\n\\bibitem{} Fichtel C.E., Bertsch T.L., Chiang J., 1994, ApJS 94, 551\n\\bibitem{} Ghisellini G., Padovani P., Celotti A., Maraschi L., 1993, ApJ 407, 65\n\\bibitem{} Ghosh K.K., Soundararajaperumal S., 1992, MNRAS 254, 563\n\\bibitem{} Gorenstein P.D., Fabrikant D., Topka K., Harnden F.R., Tucker W.H., 1978, ApJ 224, 718\n\\bibitem{} Hummel C.A., Schalinski C.J., Krichbaum T.P., et al., 1992, A\\&A 257, 489\n\\bibitem{} Iwasawa K., Fabian A.C., Nandra K., 1999, MNRAS 307, 611\n\\bibitem{} Jiang D.R., Cao X.W., Hong X., 1998, ApJ 494, 139\n\\bibitem{} Johnston K.J., Simon R.S., Eckart A., et al., 1987, ApJL 313, 85\n\\bibitem{} K\\\"uhr H., Pauliny-Toth I.I.K., Witzel A., Schmidt J., 1981, AJ 86, 854\n\\bibitem{} Laor A., Bahcall J.N., Jannuzi B.T., Schneider D.P., Green R.F., 1995, ApJS 99, 1 \n\\bibitem{} Lawson A.J., Turner M.J.L., 1997, MNRAS 288, 920\n\\bibitem{} Mannheim K, Schulte M., Rachen J., 1995, A\\&A 303, L41\n\\bibitem{} Murphy D.W., Browne I.W.A., Perley R.A., 1993, MNRAS 264, 298\n\\bibitem{} Murphy D.W., Tingay S.J., Preston R.A., et al., 1999, Elsevier preprint\n\\bibitem{} Reeves J.N., Turner M.J.L., Ohashi T., Kii T., 1997, MNRAS 292, 468\n\\bibitem{} Roos N., Kaastra J.S., Hummel C.A., 1993, ApJ 409, 130\n\\bibitem{} Sambruna R., 1997, ApJ 487, 536\n\\bibitem{} Siebert J., Matsuoka M., Brinkmann W., Cappi M., et al., 1996, A\\&A 307, 8\n\\bibitem{} Tanaka Y., Inoue H., Holt S.S., 1994, PASJ 46, L37\n\\bibitem{} Turner M.J.L., Williams O.R., Courvoisier T.J.-L, 1990, MNRAS 244, 310\n\\bibitem{} Unwin S.C., Wehrle A.E., Urry C.M., et al., 1994, ApJ 432, 103\n\\bibitem{} Voges W., Aschenbach B., Boller Th., et al., 1999, A\\&A 349, 389\n\\bibitem{} Witzel A., Schalinski C.J., Johnston K.J., et al., 1988, A\\&A 206, 245\n\\bibitem{} Yaqoob T., George I.M., Turner T.J., et al., 1998, ApJL 505, 87\n\\end{thebibliography}"
}
] |
astro-ph0002519
|
Optical and X-ray analysis of the cluster of galaxies Abell~496. \thanks{Based on observations collected at the European Southern Observatory, La Silla, Chile}
|
[
{
"author": "F.~Durret \\inst{1,2}"
},
{
"author": "C.~Adami \\inst{3,4}"
},
{
"author": "D.~Gerbal \\inst{1,2}"
},
{
"author": "V.~Pislar \\inst{1,5}"
}
] |
We present a detailed analysis of the cluster of galaxies \a4. The optical data include a redshift catalogue of 466 galaxies, out of which 274 belong to the main cluster and a CCD photometric catalogue in a much smaller region, with 239 and 610 galaxies in the V and R bands respectively. The X-ray analysis is based on an image obtained with the ROSAT PSPC. \\ Besides Abell 496 itself, the velocity distribution along the line of sight shows the existence of at least four structures at different redshifts, one of them seeming to be a poor cluster at a velocity of 30083 km/s. The other of these structures have a too large spatial extent to be clusters but may be filaments along the line of sight or other young structures.\\ Various independent methods show that Abell 496 appears to be a quite relaxed cluster, except perhaps for the distribution of emission line galaxies. These appear to be distributed in two samples falling on to the main cluster, one from the back (the ELGs concentrated towards the west) and one from the front (the high velocity ELGs). \\ The bright part of the galaxy luminosity function, built from the redshift catalogue, shows a flattening at R$\sim 16$ (M$_{R}\sim -20.5$), and can be accounted for by a gaussian distribution of bright galaxies and a power law or Schechter function for faint galaxies. The deeper galaxy counts derived from CCD imaging show a dip at R$\sim 19.5$ (M$_{R} \sim -17$) which can be modelled assuming a cut-off in the luminosity function such as that observed in Coma.\\ We propose a model for the X-ray gas and derive the galaxy, X-ray gas and total dynamical masses, as well as the baryon fraction in the cluster. \a4\ appears as a relaxed cluster which can be used as a prototype for further studies.\\ \keywords{Galaxies: clusters: individual: Abell~496; galaxies: clusters of}
|
[
{
"name": "a496.tex",
"string": "\\documentclass{aa} % LaTeX A&A New Fonts\n%\\documentclass[referee]{aa} \n\\usepackage{epsfig}\n\\def\\parn{\\\\ \\noindent}\n\\def\\kms{km~s$^{-1}$ }\n\\def\\a4{Abell 496}\n\\def\\bj{b$_{\\rm J}${}}\n\\thesaurus{11.03.4 Abell 496; 11.03.1 }\n\n\\title{Optical and X-ray analysis of the cluster of galaxies Abell~496.\n\\thanks{Based on observations collected at the European Southern\nObservatory, La Silla, Chile}\n}\n\n \\author {\n F.~Durret \\inst{1,2}\n\\and\n C.~Adami \\inst{3,4}\n\\and\n D.~Gerbal \\inst{1,2}\n\\and\n V.~Pislar \\inst{1,5}\n}\n\\offprints{F.~Durret, durret@iap.fr }\n\\institute{\n Institut d'Astrophysique de Paris, CNRS, \n 98bis Bd Arago, F-75014 Paris, France \n\\and \n DAEC, Observatoire de Paris, Universit\\'e Paris VII, CNRS (UA 173),\n F-92195 Meudon Cedex, France \n\\and\nIGRAP, Laboratoire d'Astronomie Spatiale, \nTraverse du Siphon, F-13012 Marseille, France \n\\and \nDepartment of Physics and Astronomy, Northwestern University, Dearborn\nObservatory, 2131 Sheridan, 60208-2900 Evanston, USA\n\\and\nUniversit\\'e du Havre, 25 rue Philippe Lebon, 76600 Le Havre, France\n}\n\\date{Received 17 December 1999/ Accepted}\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\n\nWe present a detailed analysis of the cluster of galaxies \\a4. The\noptical data include a redshift catalogue of 466 galaxies, out of\nwhich 274 belong to the main cluster and a CCD photometric catalogue\nin a much smaller region, with 239 and 610 galaxies in the V and R\nbands respectively. The X-ray analysis is based on an image obtained\nwith the ROSAT PSPC. \\\\ Besides Abell 496 itself, the velocity\ndistribution along the line of sight shows the existence of at least\nfour structures at different redshifts, one of them seeming to be a\npoor cluster at a velocity of 30083 km/s. The other of these\nstructures have a too large spatial extent to be clusters but may be\nfilaments along the line of sight or other young structures.\\\\ Various\nindependent methods show that Abell 496 appears to be a quite relaxed\ncluster, except perhaps for the distribution of emission line\ngalaxies. These appear to be distributed in two samples falling on to\nthe main cluster, one from the back (the ELGs concentrated towards the\nwest) and one from the front (the high velocity ELGs). \\\\ The bright\npart of the galaxy luminosity function, built from the redshift\ncatalogue, shows a flattening at R$\\sim 16$ (M$_{\\rm R}\\sim -20.5$),\nand can be accounted for by a gaussian distribution of bright galaxies\nand a power law or Schechter function for faint galaxies. The deeper\ngalaxy counts derived from CCD imaging show a dip at R$\\sim 19.5$\n(M$_{\\rm R} \\sim -17$) which can be modelled assuming a cut-off in the\nluminosity function such as that observed in Coma.\\\\ We propose a\nmodel for the X-ray gas and derive the galaxy, X-ray gas and total\ndynamical masses, as well as the baryon fraction in the cluster. \\a4\\\nappears as a relaxed cluster which can be used as a prototype for\nfurther studies.\\\\\n\n\\keywords{Galaxies: clusters: individual: Abell~496; galaxies: clusters\nof}\n\\end{abstract}\n\n\\section{Introduction}\n\nIn the framework of hierarchical clustering, the Universe is believed\nto be made of galaxies distributed in sheets encircling voids or\nfilaments, at the intersection of which clusters of galaxies are\nlocated. Such models can be tested through the analysis of clusters,\nwhich are likely to keep a ``memory'' of their formation. This is\nsuggested for example by the alignment effects observed in some\nclusters, such as for example Abell 3558 (Dantas et al. 1997) or Abell\n85 (Durret et al. 1998), where the cD, the brightest galaxies, the\nX-ray emitting gas and possibly even larger scale structures (in the\ncase of Abell 85) all appear aligned along the same direction.\nMulti-wavelength studies of clusters of galaxies also allow us to draw\na global and coherent portrait of these objects, which we can then use\nto address other questions of interest, such as the influence of\nmergers and environmental effects at various scales on the properties\nof both galaxies and X-ray gas. Large scale (i.e. cluster size)\nmergers are quite often observed from substructures detected in the\nX-ray gas; smaller scale mergers (i.e. group size) such as the infall\nof dwarf galaxies onto groups surrounding bright galaxies can be\nderived from various methods such as those of Serna \\& Gerbal (1996)\nor Gurzadyan \\& Mazure (1998), which require optical velocity and\nmagnitude catalogues; the existence of subclustering also has an\ninfluence on the shape of the galaxy luminosity function, which in\nsome cases appears to show a deficit of faint galaxies often\ninterpreted as due to accretion of dwarf galaxies onto larger galaxies\nor groups (e.g. in Coma, Lobo et al. 1997, Adami et al. 2000). It\ntherefore appears important to analyze cluster properties in detail\nbefore using them in other studies. Note in particular that the\nexistence of substructures can lead to overestimate cluster velocity\ndispersions, and hence M/L ratios and the value of $\\Omega _0$ in\nclusters.\n\nWith the improvement of both observational means (better X-ray\ndetectors, optical multi-object spectroscopy) and modern methods of\nanalysis (some of which are described below), an ever increasing\nnumber of clusters showing evidence for merging and environmental\neffects has been found. A rather general picture has therefore emerged\nfor clusters, with a main relaxed body on to which groups of various\nsizes can be falling.\n\nOur approach these last years has been to study a small sample of\nnearby clusters in detail. These have the advantage of being bright,\nand can therefore be observed in detail within a reasonable amount of\ntelescope time. Besides, they are free of evolution effects. We\npresent here a detailed multi-wavelength study of \\a4, based on\noptical (extensive redshift and photometric catalogues) and X-ray\n(ROSAT PSPC) data.\n\n\\a4 is a richness class 1 (Abell 1958) cD type (Struble \\& Rood 1987)\ncluster at a redshift of 0.0331. For a Hubble constant H$_0$=50 km\ns$^{-1}$ Mpc$^{-1}$, the corresponding scale is 55.0 kpc/arcmin and\nthe distance modulus is 36.52. At optical wavelengths, an adaptive\nkernel map of the central region (in a 60$\\times$60 arcmin$^2$ square)\nhas revealed a somewhat complicated structure, with a strong\nconcentration of galaxies in the north-south direction (Kriessler \\&\nBeers 1997). Note however that this map does not include redshift\ninformation. Thorough investigations of the X-ray properties of \\a4\\\ncan be found in Mohr et al. (1999) and Markevitch et al. (1999); their\nresults will be compared to ours in section 4.5.\n\nThe paper is organized as follows: we present the data in section 2;\nthe structures along the line of sight derived from the velocity\ncatalogue are described in section 3; the optical properties of the\n\\a4\\ cluster itself are described in section 4; the X-ray cluster\nproperties are described in section 5; a summary and conclusions are\ngiven in section 6.\n\n\\section{The data} \n\n\\subsection{Optical data}\n\nOur redshift catalogue includes 466 galaxies in the direction of the\ncluster \\a4, in a region covering about 160$\\times$160 arcmin$^2$\n(9.2$\\times$9.2~Mpc for an average redshift for \\a4 of 0.0331). It is\nfully described in Durret et al. (1999b).\n\nOur photometric catalogues are described in Slezak et al. (1999). The\nphotographic plate catalogue was obtained by scanning an SRC plate in\nthe b$_{\\rm J}$ band with the MAMA measuring machine at the\nObservatoire de Paris; it includes 3879 galaxies located in a region\nof roughly $\\pm$~1.3$^\\circ$ from the cluster centre. Positions are\nvery accurate in this catalogue and were used for spectroscopy; on the\nother hand, b$_{\\rm J}$ magnitudes are not accurate, so a CCD\nphotometric catalogue was obtained in the V and R bands in order to\nrecalibrate photographic plate magnitudes. The R magnitudes thus\nestimated for the photographic plate catalogue were used to estimate\nthe completeness of our redshift catalogue. The CCD imaging catalogue\nincludes 239 and 610 galaxies in the V and R bands respectively, and\nis limited to a much smaller region of $\\sim 246$ arcmin$^2$ in the\ncentre of the cluster.\n\nThe cluster center will be taken to be the position of the cD galaxy,\nwhich coincides with the X-ray maximum: $\\alpha = 4^h 33^{mn} 38.45^s,\n\\delta = -13^\\circ 15' 49.5''$ (2000.0). \n\n\\subsection{X-ray data}\n\nThe ROSAT PSPC image (\\#800024) was taken from the archive and\nanalyzed by Pislar (1998). The cluster was observed during 8972~s.\nThe effective exposure time for this image, after data reduction is\n5354~s. We have used the routines developed by Snowden (1995) to\nobtain a non cosmic background subtracted image between 0.5 and\n2~keV. We have defined the image limiting radius\n($\\sim\\,1620\\,h_{50}^{-1}$ kpc) as the radius where the surface\nbrightness reaches the surface brightness of the cosmic background\n($3\\,10^{-4}\\,{\\rm{s}}^{-1}\\,{\\rm{arcmin}}^{-2}$). The different PSPC\nbackground components are detailed in Snowden et al. (1994).\n\nThe global X-ray gas temperatures derived from Einstein and EXOSAT\nsatellite data are 3.9$\\pm$0.2 keV (David et al. 1993) and\n$4.7^{+1}_{-0.8}$~keV respectively (Edge \\& Stewart 1991).\n\n\\section{Velocity distribution along the line of sight}\n\nWe first discuss the overall properties derived from the velocity\ndistribution along the line of sight.\n\n\\subsection{Overall characteristics of the structures detected along\nthe line of sight}\n\nA wavelet reconstruction of the velocity distribution along the line\nof sight is displayed in Fig. \\ref{pdfall} (466 galaxies). We remind\nthe reader that this type of reconstruction takes into account\nstructures at a significance level of at least 3$\\sigma$, and detects\nstructures at various scales. The sample was analyzed with 256\npoints, and the smallest scale was excluded because it is very noisy.\nA more detailed description of this technique can be found in Fadda et\nal. (1998).\n\n\\begin{figure}[h]\n\\centerline{\\psfig{figure=fig1.ps,height=7cm}}\n\\caption[ ]{Wavelet reconstruction of the velocity distribution in the\ndirection of \\a4. The numbers above the peaks correspond to those of the\nstructures described in the text and in Table \\ref{tab9groupes}. The\ndensity units correspond to a total integrated galaxy density of 1.}\n\\protect\\label{pdfall}\n\\end{figure}\n\nNine ``groups'' or velocity substructures are found with this method,\nand their velocity characteristics are given in Table\n\\ref{tab9groupes}. The group number is given in col. 1, the number of\ngalaxies in col. 2, the BWT mean velocity (Beers et al. 1990) and\ncorresponding BWT velocity dispersion in cols. 3 and 4, and the\nvelocity interval in col. 5. Foreground groups 1 and 2 and background\ngroups 7 and 8 are most probably not real groups, since they are\nwidely spread both on the sky and in velocity distribution; because of\nthe small number of galaxies involved in the first three of these\ngroups, we did not calculate mean velocities or velocity dispersions\nfor these structures. For group 8 these values are only indicative,\nbut characteristic of a low mass structure. Group 3 appears to be the\ncluster \\a4\\ itself. Except for 7 objects, all the galaxies in group 4\nappear to be located north of \\a4. Group 5 has the same kind of shape\nand size as group 4 and is roughly coaligned with \\a4\\ along the line\nof sight. Group 6 appears strongly concentrated both spatially and in\nvelocity space, all but two galaxies being located west of\n\\a4. Moreover, its velocity dispersion is also low.\n\n\\begin{table}[h]\n\\caption{Structures detected along the line of sight}\n\\begin{tabular}{rrrrc}\n\\hline\nName & N$_{gal}$ & v$_{mean}$ & $\\sigma _v$~ & velocity interval \\\\\n & & (km/s)~ & (km/s) & (km/s)~~~ \\\\\n\\hline\n1 & 6 & & & [1803,3396] \\\\\n2 & 6 & & & [5124,6233] \\\\\n3 & 274 & 9885 & 715 & [7813,11860] \\\\\n4 & 59 & 15758 & 1214 & [13458,17928] \\\\\n5 & 25 & 25361 & 1627 & [23477,28209] \\\\\n6 & 29 & 30083 & 380 & [29144,30814] \\\\\n7 & 5 & & & [40606,41650] \\\\\n8 & 9 & 46349 & 263 & [45836,46660] \\\\\n9 & 22 & 52358 & 1467 & [50052,54727] \\\\\n\\hline\n\\end{tabular}\n\\protect\\label{tab9groupes}\n\\end{table}\n\nThese results are confirmed when we apply the same method as for the\nENACS clusters (Katgert et al. 1996, Mazure et al. 1996) to detect\nvelocity structures along the line of sight. This method consists in\nsorting the galaxies in order of increasing velocity, and plotting\ntheir rank as a function of velocity (hereafter the rank-velocity\nclassification). If the distribution of galaxies in redshift space is\nstrictly gaussian, we expect to see a regular S-shape in the\nsequence/gap space. When there are more than 5 galaxies between two\nsuccessive gaps, we consider that the galaxies belong to a structure.\n\n\\subsection{A finer analysis of structures 4, 5, 6 and 9}\n\n\\begin{figure}[h!]\n\\centerline{\\psfig{figure=fig2a.ps,width=8.1cm,height=4.8cm,clip=true}}\n\\centerline{\\psfig{figure=fig2b.ps,width=8.1cm,height=4.8cm,clip=true}}\n\\centerline{\\psfig{figure=fig2c.ps,width=8.1cm,height=4.8cm,clip=true}}\n\\centerline{\\psfig{figure=fig2d.ps,width=8.1cm,height=4.8cm,clip=true}}\n\\caption[ ]{Isodensity contours for galaxies in structures 4, 5, 6 and\n9 (from top to bottom); galaxy positions are superimposed with the\nfollowing symbols: structure 4: empty rectangles=4a, filled\nrectangles=4b, empty triangles=4c, filled triangles=4d, crosses=other\ngalaxies in structure 4; structure 5: empty rectangles=5a, filled\nrectangles=5b, empty triangles=5c, crosses=other galaxies in structure\n5; structure 6: all galaxies; structure 9: filled rectangles=9a,\nfilled triangles=9b, empty triangles=9c, crosses=other galaxies in\nstructure~9. Positions are relative to the cluster center defined in the text.}\n\\protect\\label{xyg4569}\n\\end{figure}\n\nIn order to understand whether groups 4, 5, 6 and 9 can be physically\ncoherent structures, we performed a Serna \\& Gerbal (1996) analysis\nfor each of these groups separately. Since this type of analysis takes\ninto account galaxy magnitudes, we had to eliminate one galaxy in\ngroup 4 and one in group 5, for which we have no magnitude in our\nredshift catalogue. We also tried keeping these galaxies and assigning\nthem an ``average'' magnitude R=17. The results in both cases were\nsimilar. Note that the redshift catalogue completeness is about\n50\\% in regions 4 and 5, and 55\\% in region 9, within the magnitude\nlimit R=18.8. It could therefore be argued that the Serna \\& Gerbal\nmethod is meaningless for these samples. However, in this type of\nanalysis it is the brightest galaxies which mainly contribute to the\ndynamics of the system (since the mass to luminosity ratio is taken to\nbe constant for all galaxies). If the samples are limited to \nmagnitudes R$\\leq$17.0, the redshift catalogue completeness then\nbecomes 72\\%, 72\\% and 79\\% for regions 4, 5 and 9 respectively, and\nthe Serna \\& Gerbal method is therefore fully applicable.\n\n\\begin{table}[h]\n\\caption{Substructures detected along the line of sight}\n\\begin{tabular}{rrrr}\n\\hline\nName & N$_{gal}$ & v$_{mean}$ & $\\sigma _v$~ \\\\\n & & (km/s)~ & (km/s) \\\\\n\\hline\n4a & 9 & 14043 & 381 \\\\\n4b & 10 & 17568 & 243 \\\\\n4c & 9 & 15302 & 201 \\\\\n4d & 9 & 15604 & 131 \\\\\n5a & 4 & 23785 & 195 \\\\ \n5b & 4 & 25159 & 174 \\\\\n5c & 5 & 26719 & 457 \\\\\n9a & 6 & 54164 & 119 \\\\\n9b & 5 & 51165 & 28 \\\\\n9c & 8 & 52013 & 348 \\\\\n\\hline\n\\end{tabular}\n\\protect\\label{tabssgroupes}\n\\end{table}\n\nThe characteristics of the substructures found with the Serna \\&\nGerbal method are given in Table \\ref{tabssgroupes}. Structure 4\nhas subgroups 4c and 4d well defined in space; they extend over 5\nand 3.6 Mpc respectively and could therefore be members of two\ndifferent clusters. Due to their large spatial extent, subgroups 4a\nand 4b respectively seem to be just forward and background galaxies,\nwith the exception of the five galaxies of group 4a at the north west\nextremity (see Fig. \\ref{xyg4569}).\n\nSubgroups 5a and 5b form structures with a small velocity dispersion,\nbut extending over about 7 Mpc, a size which appears rather large for\nthese groups to be members of two background clusters; the extent and\nthe velocity dispersion of 5c are even larger (Fig. \\ref{xyg4569}).\n\nWhile the Serna \\& Gerbal (1996) method finds dynamical sub-structures\nfor the other groups along the line of sight, the same method reveals\nno substructures in group 6, except for two pairs of galaxies; group 6\ntherefore appears well defined both in velocity distribution and in\nspace. Eighteen galaxies are included in an ellipse with a major and\nminor axes of about 8 and 3 Mpc, suggesting that this is a poor,\ndiffuse and low mass cluster (Fig. \\ref{xyg4569}).\n\nFinally, three subgroups are apparently found in structure 9, 9a and\n9b having very small velocity dispersions but spanning a rather large\nspatial region. The overall velocity field in group 9 shows an\ninteresting pattern looking like a velocity gradient\n(Fig. \\ref{vg9}). This could be a filament more or less aligned along\nthe line of sight.\n\n\\begin{figure}[h!]\n\\centerline{\\psfig{figure=fig3.ps,height=5cm,width=7.9cm}}\n\\caption[ ]{Isocontours of the velocity field in group 9, from 51000 \n(right) to 54000 km/s (left) by steps of 500 km/s. Positions are relative \nto the cluster center defined in the text.}\n\\protect\\label{vg9}\n\\end{figure}\n\nA rank-velocity classification applied to each group confirms that\ngroup 4 and possibly group 9 appear to have three substructures (two\nclear breaks in the curves), group 5 has three or four substructures\nand group 6 has no clear substructure except perhaps for the two or\nthree first galaxies which are probably infalling objects.\n\nTo summarize, groups 4 and 9 clearly appear as filaments or at least\nelongated structures along the line of sight, but not really massive\nclusters. This analysis is confirmed by the iso-velocity contours of\ngroup 9. The continuous velocity gradient could be interpreted as the\nresult of a merger, with the infalling groups not perfectly aligned\nalong the line of sight. Group 6 has a low velocity dispersion and is\nprobably a poor cluster. Group 5 exhibits two low velocity dispersion\nsub-structures and a moderately high velocity dispersion group\n(5c), but the number of galaxies in 5c is too low to provide a robust\nestimation and we assume that this structure is not a cluster.\n\nWe will now discuss the dynamical state of the main structure on the\nline of sight: the cluster \\a4\\ itself (group 3).\n\n\\section{Morphological and physical properties of Abell 496}\n\n\\subsection{Morphology of the cluster at various wavelengths}\n\nWe display in Fig. \\ref{image} the superposition of the optical image\nof the cluster with ROSAT PSPC X-ray and radio isocontours. The X-ray\ncontours are quite smooth, with no obvious substructures. However,\nthere appears to be an excess of X-ray emission towards the north\nwest, in the direction where there is also an excess of emission line\ngalaxies (see below). A radio source is visible south east of the\ncluster, probably associated with a galaxy.\n\n\\begin{figure}[h!]\n\\centerline{\\psfig{figure=fig4.ps,height=7cm}}\n\\caption[ ]{Optical POSS image of the central region of Abell 496 on to \nwhich X-ray (full lines) and radio 1400 MHz (dot-dashed lines) contours are \nsuperimposed. The radio data were taken from Condon et al. (1998). }\n\\protect\\label{image}\n\\end{figure}\n\n\\subsection{The galaxy velocity distribution in Abell 496}\n\n\\begin{figure}[b]\n\\centerline{\\psfig{figure=fig5.ps,height=7cm}}\n\\caption[ ]{Wavelet reconstruction of the velocity distribution in the\ndirection of \\a4, obtained by excluding the two smallest scales (full line)\nand the three smallest scales (dashed line). The arrow indicates the \nvelocity of the cD galaxy. The\ndensity units correspond to a total integrated galaxy density of 1.}\n\\protect\\label{pdfin}\n\\end{figure}\n\nThe cluster \\a4\\ corresponds to structure 3 in Table\n\\ref{tab9groupes}; it has a BWT mean velocity of 9885 km/s and a\nglobal velocity dispersion of 715 km/s. The corresponding velocity\ninterval is [7813,11860 km/s] and includes 274 galaxies. Note that\nthe cD galaxy has a velocity of 9831 km/s, close to the mean cluster\nvelocity, and is located very close the X-ray emission center,\nsuggesting that the cD is at the bottom of the cluster gravitational\npotential well. This is an indication of a quiescent history of the\ncluster (see e.g. Zabludoff et al. 1993, Oegerle \\& Hill 1994).\n\nThe wavelet reconstruction of the velocity distribution of \\a4\\ shown\nin Fig. \\ref{pdfin} (274 galaxies) suggests the presence of a certain\namount of substructure. The sample was analyzed with 256 points, and\nthe two smallest scales were excluded. The corresponding velocity\ndistribution is non-gaussian; it shows: a tiny feature at $\\sim$8300\nkm/s; a main asymmetric peak in the [8500, 10700 km/s] range\ncontaining 232 galaxies, with a BWT mean velocity of 9769 km/s, and a\nBWT velocity dispersion of 518 km/s; note that this velocity structure\nis not quite centered on the velocity of the cD galaxy; a smaller peak\nat 10940 km/s with 36 galaxies in the [10700, 11860 km/s] range. If\nwe only keep the largest scales, we are left with a rather symmetric\nvelocity distribution showing an excess at high velocities. This\nexcess corresponds to the peak at 10940 km/s, which contains a small\nnumber of galaxies (see section 4.4).\n\n\\begin{figure*}[t]\n\\centerline{\\psfig{figure=fig6.ps,height=16cm,width=8cm,angle=90,clip=true}}\n\\caption[ ]{Dendogram obtained by applying a Serna \\& Gerbal analysis to \nthe subsample of 96 galaxies located within a radius of 1800 arcsec \naround the cD and with magnitudes R$\\leq$17.0. The small numbers at the bottom\ncorrespond to the galaxy numbers in our velocity catalogue. }\n\\protect\\label{sernaamas}\n\\end{figure*}\n\nThese structures are also found by applying a rank-velocity\nclassification, which gives two breaks globally consistent with those\nfound by analyzing the cluster velocity distribution. Such breaks\nprobably indicate substructures with velocities coherent with the\nfiner analysis based on the wavelet technique. However, the number of\ngalaxies involved in these structures is small, and the velocity\ndistribution in the main cluster therefore appears to be quite smooth,\nsuggesting that Abell 496 is quite well relaxed.\n\nIn order to confirm the state of relaxation of Abell 496, we have\napplied a Serna \\& Gerbal (1996) analysis to the subsample of 96\ngalaxies located within a radius of 1800 arcsec around the cD and with\nmagnitudes R$\\leq$17.0; within this limited sample, the completeness\nof the redshift catalogue is 82\\% and this type of analysis is\nexpected to give robust results. Note that galaxies in this region\nwith measured velocities but without published magnitudes were\ndiscarded. Results are displayed in Fig. \\ref{sernaamas}. At the\nextreme lower right of the figure, we can see the very tight pair made\nby the cD (\\#280, R=12.2) and a satellite galaxy (\\#292, R=15.6, in\nthe Durret et al. 1999b catalogue): this confirms that the cD is at\nthe bottom of the cluster potential well. We also observe a structure\nof 11 bright galaxies (10 galaxies with R$\\leq$15.8, plus one with\nR=16.8) highly concentrated in space around the cD (mean distance to\nthe cD: 216 arcsec, with a dispersion of 48 arcsec) but not in\nvelocity (BWT mean velocity and velocity dispersion: 9745 and 375\nkm/s). This result is comparable to what is found in other clusters,\nwhere the central core is more or less well discriminated. The main\nbody of the cluster center appears quite relaxed, with no strong\nsubclustering within a radius of 1800 arcsec (1.65 Mpc). This picture\nagrees with the general shape of \\a4\\ seen in X-rays (see\nFig. \\ref{image}).\n\n\\subsection{Luminosity segregation in \\a4}\n\nAfter violent relaxation, two-body gravitational interactions lead to\na certain level of energy equipartition between galaxies of various\nmasses, and consequently to a certain segregation in velocity\ndispersion with luminosity (mass). This process concerns essentially\nmassive galaxies, and added to dynamical friction, it creates\nsegregation with distance to the cluster center. The stage of\npost-violent relaxation therefore leads to a segregation in the\n[L,$\\sigma _v$] space larger in the central regions than in the\noverall cluster.\n\nIn order to search for such effects in \\a4, we have derived the\nvelocity dispersion and the average distance of galaxies to the\ncluster center (defined by the position of the cD) in several\nmagnitude bins. We restrict our sample to galaxies belonging to the\ncluster, i.e. in the velocity range [7813,11860 km/s], and within 1000\nand 1800 arcsec from the cluster center, in order to have reasonably\ncomplete samples:100\\% and 79\\% complete for R$\\leq 18.5$\nrespectively. The completeness is estimated by comparing the number of\ngalaxies with measured redshifts to the number of galaxies in our\nphotographic plate catalogue, for the same R magnitude limit. Since\nthere are 2 galaxies with R$< 14$ and 6 with 14$<$R$<$15, we chose to\nfit the data with two different ``brightest'' bins: one including the\n8 galaxies with 12$<$R$<$15, and the other including only the 6\ngalaxies with 14$<$R$<$15.\n\n\\begin{figure}[t!]\n\\centerline{\\psfig{figure=fig7.ps,height=7cm}}\n\\caption[ ]{Velocity dispersion in several magnitude bins. The top\nfigure corresponds to a region of 1000 arcsec radius, the bottom figure\nto a 1800 arcsec radius. The grey line is the fit including all the points,\nwhile the dark line is the fit when the brightest bin is excluded.}\n\\protect\\label{sigmavR}\n\\end{figure}\n\nThe velocity dispersions estimated in several magnitude bins are\ndifferent for the two samples, as shown in Fig. \\ref{sigmavR}. In a\n1000 arcsec radius, the velocity dispersion increases more steeply\nwith magnitude: the corresponding slopes are $65 \\pm 43$ and $70 \\pm\n50$ km~s$^{-1}$~mag$^{-1}$ when the brightest bin is included or not\nrespectively. In a 1800 arcsec radius, the velocity dispersion\nincreases less with magnitude, the corresponding slopes being $52 \\pm\n30$ and $32 \\pm 41$ km~s$^{-1}$~mag$^{-1}$.\n\n\\begin{figure*}\n\\centerline{\\psfig{figure=fig8a.ps,height=6cm}\\qquad{\\psfig{figure=fig8b.ps,height=6cm}}}\n\\caption[ ]{Average distance to the cluster center for several\nmagnitude bins. The sample is restricted to galaxies in the velocity\nrange [7813,11860 km/s], within 1000 arcsec (left) and 1800 arcsec\n(right) from the cluster center. Filled squares include all galaxies, \nempty squares indicate non-emission line galaxies.}\n\\protect\\label{meandR}\n\\end{figure*}\n\nAs seen in Fig. \\ref{meandR}, the average distance to the cluster\ncenter is somewhat smaller for the galaxies located in the brightest\nbin (R$\\leq$15), then remains roughly constant with a possible\ndecrease with increasing magnitude, specially in the broadest sample\n(1800 arcsec radius). \n\nThe combination of Figs. \\ref{sigmavR} and \\ref{meandR} seems to\ncorrespond well to a post-violent relaxation stage.\n\nInterestingly, we can notice that both the distance to the cluster\ncenter and the overall velocity dispersion range are reduced when\nemission line galaxies (hereafter ELGs) are excluded. This agrees\nwith the general scheme that ELGs are often found in the outskirts of\nclusters of clusters and are not as strongly tied to the cluster\ngravitationally (e.g. Biviano et al. 1997). We now discuss in more\ndetail the properties of ELGs in \\a4.\n\n\\subsection{The emission line galaxy distribution}\n\n\\begin{figure*}\n\\centerline{\\psfig{figure=fig9a.ps,height=6cm}\\qquad{\\psfig{figure=fig9b.ps,height=6cm}}}\n\\caption[ ]{Left: positions of the 211 non-emission line galaxies (crosses)\nand of the 21 emission line galaxies (filled squares) in the \n[8500, 10700 km/s] velocity range. Right: positions of the 23 non-emission \nline galaxies (crosses) and of the 13 emission line galaxies (filled squares) \nin the [10700, 11860 km/s] velocity range.}\n\\protect\\label{xymain}\n\\end{figure*}\n\nWe now compare the distribution of emission line (ELGs) versus\nnon-emission line (NoELGs) galaxies. There are 85 ELGs and 381 NoELGs\nin our velocity catalogue, among which 34 ELGs and 241 NoELGs in the\nvelocity range of \\a4. The global percentage of ELGs in the cluster is\ntherefore $\\sim 12 \\pm 3$\\%. Note that this percentage is perfectly\ncoherent with the proportions observed by Biviano et al. (1997) in the\nENACS survey.\n\nThe spatial distribution of the 211 NoELGs and 21 ELGs belonging to\nthe main [8500, 10700 km/s] velocity peak is displayed in\nFig. \\ref{xymain}. The fraction of ELGs in this velocity range is\n9$\\pm$3\\%, consistent with the global cluster value within the error\nbars. NoELGs appear rather homogeneously distributed, except for a\nsort of linear north-south concentration towards the center. On the\nother hand, a large majority of the ELGs in this velocity range is\nlocated west of a north to south line crossing the center, and at\nleast half of these ELGs even seem to be close to the west cluster\nedge. This agrees with the fact that ELGs tend to concentrate in the\nouter regions of clusters (see e.g. results based on ENACS data by\nBiviano et al. 1997). The presence of an excess of ELGs can at least\nin some cases be interpreted as due to merging events producing shocks\nwhich trigger star formation. This was shown to be the case in the\nzone of Abell 85 where the X-ray filament merges into the main body of\nthe cluster (Durret et al. 1998): an excess of ELGs was observed in\nthat region, together with a temperature increase of the X-ray gas.\nThe ASCA X-ray gas temperature map presently available for Abell 496\n(Markevitch et al. 1999, Donnelly, private communication), does not\nshow any temperature increase for the X-ray gas in that area, so we\ncannot correlate the excess of ELGs towards the west cluster edge with\na higher gas temperature zone. However, we can note that this excess\nis located roughly in the same region as the X-ray excess emission in\nthe north west region of the cluster (Fig. \\ref{image}). Such an X-ray\nenhancement could be due to a merging event originating from the north\nwest, but our data cannot show this with certainty.\n\nOn the other hand, the spatial distributions of the 23 NoELGs and 13\nELGs in the [10700, 11860 km/s] velocity interval are comparable\n(Fig. \\ref{xymain}), while the ELG fraction seems much higher:\n36$\\pm$18\\%. Though the small number of objects may introduce errors,\nthere definitely seems to be an excess of ELGs with somewhat higher\nvelocities than the bulk of the cluster; these ELGs account at least\npartly for the peak at 10940 km s$^{-1}$ in the wavelet reconstruction\nof the velocity distribution.\n\nA general picture for the ELG distribution in \\a4\\ is that of two\nsamples of galaxies falling on to the main cluster, one from the back\n(the ELGs concentrated towards the west) and one from the front (the\nhigh velocity ELGs). \n\n\\subsection{The galaxy luminosity function}\n\nWe have seen in the previous section that \\a4\\ appears to have\nproperties common to many clusters, with a relaxed main body and ELGs\nprobably falling on to the cluster. We therefore expect its galaxy\nluminosity function not to be strongly modified by environmental\neffects, as observed in some clusters showing more prominent\nsubstructures. We discuss below its main features.\n\n\\subsubsection{The bright end of the galaxy luminosity function}\n\n\\begin{figure*}[t!]\n\\centerline{\\psfig{figure=fig10a.ps,height=6cm}\\qquad{\\psfig{figure=fig10b.ps,height=8cm,angle=90,clip=true}}}\n\\caption[ ]{Left: wavelet reconstruction of the luminosity function in\nthe R band for the 196 galaxies in the redshift catalogue with\nvelocities in the cluster range, within a radius of 1800 arcsec and a\nlimiting magnitude R=18.5 (full line). The contribution of the bright\nand faint populations accounted for by a gaussian and a power law\nrespectively are also shown (dots and short dashes), as well as their\nsum (long dashes). The top scale indicates R absolute\nmagnitudes. Right: contributions of the bright and faint populations\naccounted for by a gaussian and a power law (full line), and by a\ngaussian and a Schechter function (dashes). The sums of both functions\nare also displayed. }\n\\protect\\label{Rredshifts}\n\\end{figure*}\n\nWe have first derived the galaxy luminosity function (GLF) of \\a4\\ in\nthe R band from the redshift catalogue, within a radius of 1800 arcsec\naround the center, and for a limiting magnitude R=18.5. There are 196\ngalaxies in this sample. The completeness of the redshift catalogue\nwithin these limits is 79\\%, and it is 100\\% in that region for\nR$\\leq$16.0. The obvious interest of such a GLF is that no background\ncontribution needs to be subtracted, therefore making the results very\nrobust. We have limited the magnitude interval to the [13,18.5] range,\nbecause for R$\\leq$13 there is only one galaxy (the cD), which\nintroduces edge effects in the wavelet reconstruction of the GLF, and\nfor R$\\geq$18.5 the completeness sharply decreases. This corresponds\nto the [$-23.5,-18.0$] interval in R absolute magnitude.\n\nThe GLF obtained after a wavelet reconstruction is shown in Fig.\n\\ref{Rredshifts}. The sample was analyzed with 128 points, excluding\nthe two smallest scales. The significance level of the detected\nfeatures is at least 3$\\sigma$. A flattening is observed at R$\\sim$16,\ncorresponding to an absolute magnitude M$_{\\rm R}\\sim -20.5$. This\nshape is comparable to that found in Virgo (Binggeli et al. 1988) and\nin Abell 963 (Driver et al. 1994), where a comparable flattening was\nobserved at a common absolute magnitude of $-19.8$. The GLFs of Coma\nand Abell 85 are more complex, with a ``bump'' corresponding to the\nbrightest galaxies, followed by a ``dip'' at M$_{\\rm R}\\sim -20.5$\n(see Fig. 9 in Durret et al. 1999a). Note that the flattening of the\nGLF in \\a4\\ occurs exactly at the same absolute magnitude as the dip\nin Abell 85 and Coma.\n\nThe GLF in \\a4\\ suggests at least a bimodal galaxy distribution, with\nbright (mostly elliptical) galaxies in the bright part and dwarf\ngalaxies in the fainter part. We therefore performed a fit of the\nwavelet reconstructed GLF of \\a4\\ by summing two functions: a gaussian, to\naccount for bright galaxies, and a power law (case 1) or a Schechter\nfunction (case 2) to represent faint and/or dwarf galaxies. In case 1,\nwe fit the data as a function of apparent R magnitude; the gaussian is\nthen found to be centered on R=15.19$\\pm$0.01, with $\\sigma = 0.8 \\pm\n0.1$, and the power law varies as R$^{11.67 \\pm 0.16}$. In case 2, we\nfit the data as a function of absolute R magnitude, to allow a direct\ncomparison with other authors; the gaussian is then found to be a\nlittle broader, centered on R=15.48$\\pm$0.04, with $\\sigma = 1.0 \\pm\n0.1$; the Schechter function, defined as in Rauzy et al. (1988,\nsection 3.2.3), has $\\alpha = -1.19\\pm 0.04$ and M$_* = -19.43 \\pm 0.13$\n(in the [$-23.5,-18.0$] absolute magnitude range).\n\nThe GLF resulting from these various fits is shown in\nFig. \\ref{Rredshifts}; it obviously reproduces the data very\nwell. Except at the faint end where the sample incompleteness most\nprobably modifies the GLF shape, both fits 1 and 2 are good but we\ncannot distinguish between them. In view of the obvious quality of the\nfit, we did not attempt to estimate error bars with Monte-Carlo\nsimulations, as done previously e.g. for Abell 85 (see Durret et al.\n1999a, Fig. 12).\n\nThe various values obtained from these fits of the GLF can be compared\nto those found in other clusters. The gaussian used to fit the bright\npart of the GLF in Abell 85 has $\\sigma =1.1$, comparable to the value\nwe find in case 1. The Schechter function for \\a4\\ has a slope\ncomparable to that found by Lumsden et al. (1997), but notably flatter\nthan the values found in other surveys (e.g. Valotto et al. 1997,\nRauzy et al. 1998 and references therein). Rauzy et al. (1998) argued\nthat the flatter slope found by Lumsden was due to incompleteness at\nfaint magnitudes; this may also be true in our case, since our sample\nis 100\\% complete only to R=16.0 (M$_{\\rm R}=-20.5$), and we also find\na brighter value of M$_*$ than the above surveys, suggesting that we\nare missing faint galaxies.\n\nWe can note that the GLFs of Coma and Abell 85 were interpreted in a\nsimilar way, with the bright part mainly due to ellipticals (with a\nsmall contribution of spirals) and the faint part due to dwarfs\n(Durret et al. 1999a). Comparable shapes were found in several other\nclusters. The fact that the GLF of \\a4\\ shows a flattening at the\nsame value M$_{\\rm R}=-20.5$ indicates that the galaxy population in\n\\a4\\ is comparable to those of the above mentioned clusters.\n\nNote that Molinari et al. (1998) have analyzed the GLF of \\a4\\ from a\nphotometric catalogue in three colors, reaching magnitudes much\nfainter than those of our spectroscopic catalogue. We will therefore\ncompare our results to theirs in the next section.\n\n\\subsubsection{The faint end of the galaxy luminosity function}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig11.ps,height=7cm}}\n\\caption[ ]{Wavelet reconstruction of the number of galaxies in the R band \nfor the 411 galaxies in the CCD catalogue within the R magnitude range\n[17,21] (full line). The background contribution obtained from the LCRS and \nESS is shown as a dotted line (see text). The difference between the \nobserved number of galaxies and the background is shown as a long dashed \nline. }\n\\protect\\label{Rccd}\n\\end{figure}\n\nOur intent was also to derive the luminosity function from the CCD\ncatalogue, which corresponds to a small region of $\\sim$246 arcmin$^2$\nin the cluster center. For this we first performed a wavelet\nreconstruction of the R magnitude distribution in the R magnitude\nrange [17,23]. Since we have no background exposure, we estimated the\nbackground contribution by connecting the counts from the Las Campanas\nRedshift Survey (LCRS, Lin et al. 1996) and from the ESO-Sculptor\nSurvey (ESS, Arnouts et al. 1997), as described in our study of Abell\n85 (Durret et al. 1999a, Fig. 10 and text), and we subtracted this\nbackground to the observed number of galaxies. The result is shown in\nFig. \\ref{Rccd}. We have checked that the consistency of the\nbackground number counts estimated by Tyson (1988) with those of the\nLCRS and ESS combined as described above is good.\n\nThe difference between the observed number of galaxies and the\nbackground (Fig. \\ref{Rccd}) becomes negative for magnitudes R$\\geq\n18.4$, while the CCD catalogue is complete at least up to\nR=21. Therefore, this background cannot be considered as\nrepresentative of the local background in our CCD field of view. Note\nthat this was already the case for the CCD photometric data of Abell\n85.\n\nOne notable feature is the dip in the galaxy magnitude distribution at\nR$\\sim 19.5$ (M$_{\\rm R} \\sim -17$), which is detected at a high\nconfidence level. This dip corresponds to that observed by Molinari et\nal. (1998), who found a dip at R$\\sim$19 (M$_{\\rm R} \\sim\n-17.5$). Note that they also find a similar dip in the g band, and\npossibly in the i band. Molinari et al. (1998) made a second\ndetermination of the GLF by selecting cluster members in a\ncolour-magnitude diagram. In this case, they find a small dip, or at\nleast a flattening, for R$\\sim 18$ (M$_{\\rm R} \\sim -18.5$). This\nvalue does not agree either with the bright nor with the faint GLF\nthat we derived. It is difficult to understand why, since their\ncolour-magnitude relation appears quite well defined.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig12.ps,height=7cm,angle=-90}}\n\\caption[ ]{Model of the total counts reproducing the dip observed in Fig.\n\\ref{Rccd} (see description in text).}\n\\protect\\label{fdlada}\n\\end{figure}\n\nIn order to investigate the origin of the dip seen in our data, we\npropose a toy model, which is not a fit but only illustrates how the\ndip could be accounted for. Let us first note that the contribution of\nthe other structures detected along the line of sight is negligible.\nAssuming a Gaussian + a Schechter function to model the GLF (see\nsection 4.5.1), we rescaled the number of galaxies produced by this\ncomposite function to fit the dimension of the CCD field. We then\napplied a magnitude cut-off to this GLF, as suggested by Adami et\nal. (2000), for galaxies fainter than M$_{\\rm R}=-19.75$ in the inner\ncore of the Coma cluster. This effect becomes very strong for galaxies\nfainter than M$_{\\rm R} =-17$. The exact shape of such a cut-off is\nunknown, so we applied a convenient apodization function (the choice\nof this function influences the shape and smoothness of the dip). The\nbackground counts were modeled as the background contribution from the\nLCRS and ESS described above. We then summed the cluster and\nbackground contributions, and the result is shown in\nFig. \\ref{fdlada}.\n\nSuch a toy model can reproduce the global GLF shape, with counts\nsimilar to the observed data and a dip comparable to the observed one.\nA fine-tuning of the various parameters involved could make\nFigs. \\ref{Rccd} and \\ref{fdlada} more similar, but this would push\nthe model too far. However, we can state that a cut-off in the GLF of\n\\a4\\ similar to that observed in Coma is a solution to account for the\nobserved dip.\n\n\\section{The X-ray gas}\n\n\\begin{figure}[h]\n\\centerline{\\psfig{figure=fig13.ps,height=7cm,width=7.5cm,clip=true}}\n\\caption[ ]{Observed ROSAT PSPC image with the isocontours of the best \n$\\beta$-model fit superimposed (see text). The pixel size is 30 arcsec.}\n\\protect\\label{fitx}\n\\end{figure}\n\nA pixel by pixel fit was performed on the X-ray image, as described by\nPislar et al. (1997). The pixel size is 30 arcsec. A $\\beta$-model and\na 3D S\\'ersic model (Lima Neto et al. 1999) were considered for the\nvariations of the density with radius. The global temperature\nestimated from these ROSAT data, using a Raymond-Smith spectrum and a\nGalactic absorption column density was found to be 4$\\pm$1 keV and\nassumed to be constant (Pislar 1998). This is consistent with the\ntemperatures of 3.9 and 4.7 keV previously measured with the Einstein\nand EXOSAT satellites respectively (David et al. 1993; Edge \\& Stewart\n1991). The parameters corresponding to the best fits for both models\nare given in Table \\ref{tabfitx}, and the result of the $\\beta$-model\n1 fit superimposed on the observed image is displayed in\nFig. \\ref{fitx}. We observe that in model~2 the central density is\nlower than in model~1 and that the $\\beta$ and r$_c$ parameters are\nhigher. This is because in model 2 we do not include the central\nregion, where the cooling flow lies. The effect is the same for\nmodels 3 and 4.\n\n\\begin{table}\n\\caption{Fits of the X-ray gas with a $\\beta$-model and a S\\'ersic model}\n\\begin{tabular}{lrrrrrr}\n\\hline\nModel & n$_0$ & $\\beta$ & r$_c$ & $\\epsilon$ & R$_{cool}$ & $\\dot{M}$\\\\\n &(10$^{-3}$ cm$^{-3}$) & & (kpc) & & (kpc) & (M$_\\odot$/yr)\\\\\n\\hline\n1 & 21.7 & 0.53 & 50 & 0.87 & 192 & 235 \\\\\n & 1.9 & 0.01 & 4 & 0.03 & 25 & 81 \\\\\n2 & 2.55 & 0.82 & 356 & 0.86 & & \\\\\n & 0.35 & 0.07 & 58 & 0.03 & & \\\\\n\\hline\nModel & I$_0$ & $\\nu$ & a & $\\epsilon$ & R$_{cool}$ & $\\dot{M}$\\\\\n &(10$^{-3}$ cm$^{-3}$) & & (kpc) & & (kpc) & (M$_\\odot$/yr) \\\\\n\\hline\n3 & 5.5 & 0.61 & 230 &0.86 &198 & 314 \\\\\n & 0.3 & 0.02 & 12 & 0.03 & & \\\\\n4 & 2.7 & 0.75 & 368 & 0.85 & & \\\\\n & 1.3 & 0.17 & 129 & 0.03 & & \\\\\n\\hline\n\\end{tabular}\n\nNotes to Table \\ref{tabfitx}: n$_0$, $\\beta$ and r$_c$ are the parameters of\nthe best $\\beta$-model fit, and I$_0$, $\\nu$ and a those of the best\nS\\'ersic fit; $\\epsilon$ is the cluster ellipticity, R$_{cool}$ the\ncooling radius and $\\dot{M}$ the cooling flow mass deposit rate.\\\\\nModels 1 and 2 correspond to a $\\beta$-model, \nwith the central region respectively included or not; models 3 and 4 \ncorrespond to a S\\'ersic model with the central region respectively \nincluded or not. For each model, the first line gives the parameters and the\nsecond line the corresponding 3$\\sigma$ errors.\n\\protect\\label{tabfitx}\n\\end{table}\n\nOur values of $\\beta$ and r$_c$ (in model~2) are higher than those of\nMarkevitch et al. (1999), who found $\\beta=0.7$ and\nr$_c=249$~kpc. This is due to the fact that they exclude a central\nregion smaller than ours (3 arcmin instead of 3.3 arcmin). Pislar\n(1998) has shown that in a cooling flow cluster the bigger the\nexcluded central region, the higher $\\beta$ and r$_c$, and the lower\nthe central density. Moreover, at the image limiting radius, the\nvalues of the dynamical and gas masses do not depend on the size of\nthe central region excluded.\n\nThe cooling radius R$_{cool}$ and the mass $\\dot{M}$ deposited in the\ncentre were estimated as in Pislar et al. (1997) and are given in\nTable \\ref{tabfitx} for a temperature of 4 keV. We take $2\\,10^{10}$\nyr for the cooling time.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig14.ps,height=7cm}}\n\\caption[ ]{Masses of the X-ray gas (lower curves) and total dynamical \nmasses (upper curves) derived from the X-ray data. Full lines correspond to\n$\\beta$-model fits and dot-dashed lines to S\\'ersic. The vertical\nlines indicate the cooling radius (dashes) and the limiting radius of the \nimage (dots). The filled triangle indicates the integrated galaxy mass at\nthe X-ray limiting radius.}\n\\protect\\label{masses}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=fig15.ps,height=7cm,clip=true}}\n\\caption[ ]{X-ray gas to dynamical mass ratio. The full line corresponds \nto the $\\beta$-model fit and the dot-dashed line to the S\\'ersic fit.\nThe vertical lines indicate the cooling radius (dashes) and the limiting \nradius of the image (dots).}\n\\protect\\label{baryons}\n\\end{figure}\n\nThe X-ray gas and total dynamical masses derived from the X-ray data\nassuming equilibrium are shown in Fig. \\ref{masses} as a function of\nradius. Note that these curves are only valid between the cooling\nradius and the limiting radius of the X-ray image; within this\nvalidity range, both models are in good agreement with each other.\n\nWe find, with models 2 and 4 for a prolate geometry, at the limiting\nradius of the image, a gas mass of $(6.1\\pm2.2)\\ 10^{13} M_{\\odot}$\nand a dynamical mass of $(4.2\\pm1.1)\\,10^{14} M_{\\odot}$. We\noverestimate the errors because we have supposed that the parameters\nare not correlated. The masses at $1\\,h_{50}^{-1}$ Mpc are\nrespectively $(3.45\\pm1.1)\\,10^{13} M_{\\odot}$ and\n$(2.4\\pm0.6)\\,10^{14} M_{\\odot}$. The gas mass found by Mohr et\nal. (1999) and the dynamical mass found by Markevitch et al. (1999)\nare very similar if we remember that their geometry is spherical. The\nmass calculated by the virial theorem applied to all the galaxies in\nthe redshift catalogue with velocities in the cluster is M$_{vir}=(7.2\n\\pm 0.8) 10^{14}$ M$_\\odot$. This value agrees with the dynamical mass\nderived above, providing the mass profiles can be extrapolated to\nradii larger than the X-ray image. The integrated mass of galaxies\nwithin the X-ray image limiting radius, assuming a mass to luminosity\nratio of 8 M$_\\odot$/L$_\\odot$, is $1.0\\ 10^{13}$ M$_\\odot$. Note\nthat, due to incompleteness of our redshift catalogue, in particular\nat faint magnitudes, this is only a lower limit.\n\nThe ratio of the X-ray gas mass to the dynamical mass is shown as a\nfunction of radius in Fig. \\ref{baryons}. The fraction of X-ray gas is\nabout 0.15 at the limiting radius of the image and at $1\\,h_{50}^{-1}$\nMpc with a $\\beta$-model. With a S\\'ersic model, the ratio is 0.12 at\nthe limiting radius of the image. The baryon fraction that we find at\n$1\\,h_{50}^{-1}$ Mpc with the $\\beta$-model is very similar to that\nobtained by Markevitch et al. (1999) ($0.158\\pm0.017$), and comparable\nto that found in other clusters. At the X-ray limiting radius, the\nstellar to X-ray gas mass ratio is 16\\% and the stellar to total\ndynamical mass ratio is 2.4\\%.\n\n\\section{Discussion and conclusions}\n\nThe optical analysis of the \\a4\\ field has shown the existence of\nseveral structures along the line of sight. Among these, one\n(structure 6) is likely to be a poor, diffuse and low mass cluster,\nwhile two others (structures 4 and 9) are probably filaments more or\nless aligned along the line of sight, the latter presenting a smooth\nvelocity gradient. Notice that the distances between these various\nstructures are very large.\n\nThe cluster \\a4\\ itself has quite a regular morphology. It includes\n274 galaxies in the [7813,11860 km/s] velocity range and has a\nvelocity dispersion of 715 km/s. Its velocity distribution implies a\nsmall amount of substructure. The analysis of the correlations between\nposition, luminosity and velocity dispersion indicates a post-violent\nrelaxation state. We can notice that both the distance to the cluster\ncenter and the overall velocity dispersion ranges are reduced when\nemission line galaxies (hereafter ELGs) are excluded. This agrees\nwith the general scheme that ELGs are often found in the outskirts of\nclusters and are not as strongly tied to the cluster\ngravitationally (e.g. Biviano et al. 1997). There may be two samples\nof ELGs falling on to the main cluster, one from the back (the ELGs\nconcentrated towards the west) and one from the front (the high\nvelocity ELGs).\n\nThe bright luminosity function derived from our redshift catalogue\nshows a flattening at R$\\sim 16$ (M$_{\\rm R}\\sim -20.5$), comparable\nto similar shapes found in other clusters. This suggests at least a\nbimodal distribution, one for ellipticals and one for fainter\ngalaxies. The fact that the flattening occurs at the same absolute\nmagnitude as for other clusters suggests that the galaxy populations\nin all these clusters are comparable. \n\nAt fainter magnitudes, galaxy counts derived from CCD imaging show a\ndip at R$\\sim 19.5$ (M$_{\\rm R} \\sim -17$) which can be reproduced if\nwe assume a magnitude cut-off similar to that observed in Coma (Adami\net al. 2000). Notice that such a cut-off is observed in the very\ncentral regions of both clusters. Although this result is only based\non imaging and remains to be confirmed spectroscopically, we may be\nevidencing a second example of a cut-off of the faint end of the\nluminosity function in a cluster.\n\nWe have modelled the X-ray gas and derived the X-ray gas mass and the\ndynamical mass, which we compare to the stellar mass. At the limiting\nradius (1.62 h$_{50}^{-1}$ Mpc) of the image, we find a fraction of\nX-ray gas to total mass of 0.12$-$0.15 and a stellar to X-ray gas mass\nratio of 0.16. We can note that \\a4\\ follows exactly the two by two\ncorrelations between the X-ray luminosity (L$_{\\rm X}=6.8\\ 10^{44}$\nerg/s, Wu et al. 1999), the X-ray temperature (T$_{\\rm X}=4$ keV) and\nthe galaxy velocity dispersion ($\\sigma _v = 715$ km/s) described in\nthe literature (see e.g. Wu et al. 1999 and references therein). These\nvalues are typical of a richness class 1 cluster.\n\n\\a4\\ therefore appears to be a relatively quiet and simple cluster,\nwith no strong environmental effects, although we may see an\nenhancement of the X-ray emission and of the number of emission line\ngalaxies towards the north west.\n\nWhile Coma has long been the archetype of a relaxed cluster and is\nnot believed to be relaxed any more (see Biviano 1998 and the\nproceedings of the Coma meeting), the results presented above suggest\nthat \\a4\\ may be such a prototype, and can be used as a ``template''\nin the future study of more complex (i.e. substructured) clusters.\n\n\\acknowledgements{The authors thank Andrea Biviano for help. C.~Adami\nis grateful to the staff of the Dearborn Observatory for their\nhospitality during his postdoctoral fellowship. We acknowledge\nfinancial support from the French Programme National de Cosmologie,\nCNRS.}\n\n\\begin{thebibliography}{}\n\n\\bibitem{} Abell G.O. 1958, ApJS 3, 211\n\\bibitem{} Adami C., Ulmer M., Durret F. et al. 2000, A\\&A in press,\nastro-ph/9910217\n\\bibitem {} Arnouts S., de Lapparent V., Mathez G. et al. 1997, A\\&AS 124, \n163 \n\\bibitem{} Beers T.C., Flynn K., Gebhardt K. 1990, AJ 100, 32\n\\bibitem{} Binggeli B., Sandage A., Tammann G.A. 1988, ARA\\&A 26, 509\n\\bibitem {} Biviano A., Katgert P., Mazure A. et al. 1997, A\\&A 321, 84\n\\bibitem {} Biviano A. 1998, Proc. ``A new vision of an old cluster: \nuntangling Coma Berenices'', Marseille June 17-20 1997, Eds. Mazure et al.,\nWorld Scientific, page 1\n\\bibitem{} Condon J.J., Cotton W.D., Greisen E.W. et al. 1998, AJ 115, 1693\n\\bibitem{} Dantas C.C., de Carvalho R.R., Capelato H.V., Mazure A.\n1997, ApJ 485, 447\n\\bibitem{} David L.P., Slyz A., Jones C., et al. 1993, ApJ 412, 479\n\\bibitem{} Driver S.P., Phillipps S., Davies J.I., Morgan I.,\nDisney M.J. 1994, MNRAS 268, 393\n\\bibitem{} Durret F., Forman W., Gerbal D., Jones C., Vikhlinin A. 1998,\nA\\&A 335, 41\n\\bibitem{} Durret F., Gerbal D., Lobo C., Pichon C. 1999a, A\\&A 343, 760\n\\bibitem{} Durret F., Felenbok P., Lobo C., Slezak E. 1999b, A\\&AS 139, 525\n\\bibitem{} Edge A.C., Stewart G.C. 1991, MNRAS, 252, 428\n\\bibitem{} Fadda D., Slezak E., Bijaoui A. 1998, A\\&AS 127, 335 \n\\bibitem{} Gurzadyan V.G., Mazure A. 1998, MNRAS 295, 177\n\\bibitem{} Katgert P., Mazure A., Perea J. et al. 1996, A\\&A 310, 8\n\\bibitem{} Kriessler J.R., Beers T.C. 1997, AJ 113, 80\n\\bibitem{} Lima Neto G.B., Gerbal D., M\\'arquez I. 1999, MNRAS 309, 481\n\\bibitem {} Lin H., Kirshner R.P., Schectman S.A. et al.\n 1996, ApJ 464, 60\n\\bibitem{} Lobo C., Biviano A., Durret F., et al. 1997, A\\&A 317, 385\n\\bibitem{} Lumsden S.L., Collins C.A., Nichol R.C., Eke V.R., Guzzo L. 1997,\nMNRAS 290, 119\n\\bibitem{} Markevitch M., Vikhlinin A., Forman W.R., Sarazin C.L. 1999, \nApJ 527, 545\n\\bibitem{} Mazure A., Katgert P., den Hartog R. et al. 1996, A\\&A 310, 31\n\\bibitem{} Mohr J.J., Mathiesen B., Evrard A.E. 1999, ApJ 517, 627\n\\bibitem{} Molinari E., Chincarini G., Moretti A., De Grandi S. 1998,\nA\\&A 338, 874\n\\bibitem{} Oegerle W.R., Hill J.M. 1994, AJ 107, 857\n\\bibitem{} Pislar V., Durret F., Gerbal D., Lima Neto G.B., Slezak E. 1997, \nA\\&A 322, 53\n\\bibitem{} Pislar V. 1998, PhD Thesis, Universit\\'e Paris 6\n\\bibitem{} Rauzy S., Adami C., Mazure A. 1998, A\\&A 337, 31\n\\bibitem{} Serna A., Gerbal D. 1996, A\\&A 309, 65\n\\bibitem{} Slezak E., Durret F., Guibert J., Lobo C. 1999, A\\&AS 139, 559\n\\bibitem{} Snowden S.L. 1995, {\\sl Cookbook for analysis procedures\nfor Rosat XRT/PSPC observations of extended objects and diffuse background}\n(Greenbelt: NASA USRSDC)\n\\bibitem{} Snowden S.L., McCammon D., Burrows D.N., Mendenhall J.A. 1994, ApJ\n424, 714\n\\bibitem{} Struble M.F., Rood H.J. 1987, ApJS 63, 543\n\\bibitem{} Tyson J.A., 1988, AJ 96, 1\n\\bibitem{} Valotto C.A., Nicotra M.A., Muriel H., Lambas D.G. 1997, ApJ 479,\n90\n\\bibitem{} Wu X.-P., Xue Y.-J., Fang L.-Z. 1999, ApJ 524, 22\n\\bibitem{} Zabludoff A.I., Geller M.J., Huchra J.P., Vogeley M.S. 1993, \nAJ 106, 1273\n\n\\end{thebibliography}\n\n\\end{document}\n"
}
] |
[
{
"name": "astro-ph0002519.extracted_bib",
"string": "\\begin{thebibliography}{}\n\n\\bibitem{} Abell G.O. 1958, ApJS 3, 211\n\\bibitem{} Adami C., Ulmer M., Durret F. et al. 2000, A\\&A in press,\nastro-ph/9910217\n\\bibitem {} Arnouts S., de Lapparent V., Mathez G. et al. 1997, A\\&AS 124, \n163 \n\\bibitem{} Beers T.C., Flynn K., Gebhardt K. 1990, AJ 100, 32\n\\bibitem{} Binggeli B., Sandage A., Tammann G.A. 1988, ARA\\&A 26, 509\n\\bibitem {} Biviano A., Katgert P., Mazure A. et al. 1997, A\\&A 321, 84\n\\bibitem {} Biviano A. 1998, Proc. ``A new vision of an old cluster: \nuntangling Coma Berenices'', Marseille June 17-20 1997, Eds. Mazure et al.,\nWorld Scientific, page 1\n\\bibitem{} Condon J.J., Cotton W.D., Greisen E.W. et al. 1998, AJ 115, 1693\n\\bibitem{} Dantas C.C., de Carvalho R.R., Capelato H.V., Mazure A.\n1997, ApJ 485, 447\n\\bibitem{} David L.P., Slyz A., Jones C., et al. 1993, ApJ 412, 479\n\\bibitem{} Driver S.P., Phillipps S., Davies J.I., Morgan I.,\nDisney M.J. 1994, MNRAS 268, 393\n\\bibitem{} Durret F., Forman W., Gerbal D., Jones C., Vikhlinin A. 1998,\nA\\&A 335, 41\n\\bibitem{} Durret F., Gerbal D., Lobo C., Pichon C. 1999a, A\\&A 343, 760\n\\bibitem{} Durret F., Felenbok P., Lobo C., Slezak E. 1999b, A\\&AS 139, 525\n\\bibitem{} Edge A.C., Stewart G.C. 1991, MNRAS, 252, 428\n\\bibitem{} Fadda D., Slezak E., Bijaoui A. 1998, A\\&AS 127, 335 \n\\bibitem{} Gurzadyan V.G., Mazure A. 1998, MNRAS 295, 177\n\\bibitem{} Katgert P., Mazure A., Perea J. et al. 1996, A\\&A 310, 8\n\\bibitem{} Kriessler J.R., Beers T.C. 1997, AJ 113, 80\n\\bibitem{} Lima Neto G.B., Gerbal D., M\\'arquez I. 1999, MNRAS 309, 481\n\\bibitem {} Lin H., Kirshner R.P., Schectman S.A. et al.\n 1996, ApJ 464, 60\n\\bibitem{} Lobo C., Biviano A., Durret F., et al. 1997, A\\&A 317, 385\n\\bibitem{} Lumsden S.L., Collins C.A., Nichol R.C., Eke V.R., Guzzo L. 1997,\nMNRAS 290, 119\n\\bibitem{} Markevitch M., Vikhlinin A., Forman W.R., Sarazin C.L. 1999, \nApJ 527, 545\n\\bibitem{} Mazure A., Katgert P., den Hartog R. et al. 1996, A\\&A 310, 31\n\\bibitem{} Mohr J.J., Mathiesen B., Evrard A.E. 1999, ApJ 517, 627\n\\bibitem{} Molinari E., Chincarini G., Moretti A., De Grandi S. 1998,\nA\\&A 338, 874\n\\bibitem{} Oegerle W.R., Hill J.M. 1994, AJ 107, 857\n\\bibitem{} Pislar V., Durret F., Gerbal D., Lima Neto G.B., Slezak E. 1997, \nA\\&A 322, 53\n\\bibitem{} Pislar V. 1998, PhD Thesis, Universit\\'e Paris 6\n\\bibitem{} Rauzy S., Adami C., Mazure A. 1998, A\\&A 337, 31\n\\bibitem{} Serna A., Gerbal D. 1996, A\\&A 309, 65\n\\bibitem{} Slezak E., Durret F., Guibert J., Lobo C. 1999, A\\&AS 139, 559\n\\bibitem{} Snowden S.L. 1995, {\\sl Cookbook for analysis procedures\nfor Rosat XRT/PSPC observations of extended objects and diffuse background}\n(Greenbelt: NASA USRSDC)\n\\bibitem{} Snowden S.L., McCammon D., Burrows D.N., Mendenhall J.A. 1994, ApJ\n424, 714\n\\bibitem{} Struble M.F., Rood H.J. 1987, ApJS 63, 543\n\\bibitem{} Tyson J.A., 1988, AJ 96, 1\n\\bibitem{} Valotto C.A., Nicotra M.A., Muriel H., Lambas D.G. 1997, ApJ 479,\n90\n\\bibitem{} Wu X.-P., Xue Y.-J., Fang L.-Z. 1999, ApJ 524, 22\n\\bibitem{} Zabludoff A.I., Geller M.J., Huchra J.P., Vogeley M.S. 1993, \nAJ 106, 1273\n\n\\end{thebibliography}"
}
] |
astro-ph0002520
|
% CMB Anisotropies: A Decadal Survey
|
[
{
"author": "Wayne Hu"
}
] |
[
{
"name": "decadent.tex",
"string": "%\n%_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/\n%\n% Universal Academy Perss, Inc.\n% BR-Hongo-5 Bldg., 6-16-2 Hongo Bunkyo-ku, Tokyo 113-0033, Japan\n% Tel: +81-3-3813-7232\n% Fax: +81-3-3813-5932\n% E-mail address for this symposium: resceu@uap.co.jp\n% WWW Home Page URL: http://www.uap.co.jp\n%\n%_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/\n%\n\n\n\n\\documentstyle{res}\n %\\textwidth 34.5pc\n % \\textheight 52pc\n % \\raggedbottom\n % \\oddsidemargin 1pc\n % \\evensidemargin 1pc\n % \\topmargin 1pc\n %\\pagestyle{headings}\n \\textwidth 7.25in\n \\textheight 9.75in\n \\raggedbottom\n \\oddsidemargin -0.375in\n \\evensidemargin -0.375in\n \\topmargin -0.70in\n\t\t\t\\parindent 0.5cm\n \\pagestyle{headings}\n\n\\input epsf\n\n\\begin{document}\n\\def\\smalltext#1{\n\\noindent{\\small #1}\n\\baselineskip=14pt \n}\n\\def\\spose#1{\\hbox to 0pt{#1\\hss}}\n\\def\\simlt{\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$}}\n \\raise 2.0pt\\hbox{$\\mathchar\"13C$}}}\n\\def\\simgt{\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$}}\n \\raise 2.0pt\\hbox{$\\mathchar\"13E$}}}\n%\\simpropto produces \\propto with twiddle underneath\n\\def\\simpropto{\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$}}\n \\raise 2.0pt\\hbox{$\\propto$}}}\n\n% redefine the labels \n\n\\def\\thesection {\\arabic{section}}\n\\def\\thefigure{\\arabic{figure}}\n\n\n% redefine thebibliography\n \n\\def\\thebibliography#1{\\section*{References}\\baselineskip 13pt \\parskip 0pt\n\\list\n {\\arabic{enumi}.}{\\settowidth\\labelwidth{[#1]}\\leftmargin\\labelwidth\n \\advance\\leftmargin\\labelsep\n \\usecounter{enumi}}\n \\def\\newblock{\\hskip .11em plus .33em minus -.07em}\n \\sloppy\n \\sfcode`\\.=1000\\relax \n}\n\\let\\endthebibliography=\\endlist\n\n\n\\title{%\nCMB Anisotropies: A Decadal Survey}\n\n\\author{Wayne Hu \\\\\n{\\it Institute for Advanced Study, Princeton, NJ, 08540, USA,\nwhu@ias.edu}\n}\n\n\\maketitle\n\\begin{quotation}\n \\footnotesize\n \\noindent \n Said the disciple, ``After I heard your words, one year\n and I ran wild, two years and I was tame, three years\n and positions interchanged, four years and things settled\n down, five years and things came to me, nine years\n and I had the great secret.''\n \n \\begin{center}\n --Chuang-tzu\n \\end{center}\n \n\\end{quotation}\n \n\\section*{Abstract}\n\\begin{quotation}\n\\footnotesize\\noindent\nWe review the theoretical implications of the past decade of\nCMB anisotropy measurements, which culminated in the recent\ndetection of the first feature in the power spectrum,\nand discuss the tests available to the next decade of experiments. \nThe current data already suggest that\ndensity perturbations originated in an inflationary epoch, \nthe universe is spatially flat,\nand baryonic dark matter is required. We discuss the underlying\nassumptions of these claims and outline the tests required \nto ensure they are robust. The most critical test - \nthe presence of a second feature at the predicted location \n- should soon be available. Further\nin the future, \nsecondary anisotropies and polarization should open new windows to \nthe early and low(er) redshift universe.\n\\end{quotation}\n\n\\section{Introduction}\n\nThe 1990's will be remembered as a decade of discovery for\ncosmic microwave background (CMB) anisotropies. The launch\nof the COBE satellite ushered in the decade in 1990 and lead \nto the first detection of CMB anisotropies\nat $>10^{\\circ}$ scales \\cite{COBE92}. Through the\ndecade, a combination of higher resolution experiments made the\ncase for a rise in the anisotropy level on \ndegree scales and a subsequent fall at arcminute scales \\cite{ScoSilWhi95}. \nThe final year saw experiments, notably Toco and Boomerang,\nwith sufficient angular resolution\nand sky coverage to localize a sharp peak in the anisotropy\nspectrum at approximately $0^{\\circ}\\!.5$ \\cite{TegZal00}. In this review,\nwe discuss the theoretical implications of these results \nand provide a roadmap for critical tests and uses of\nCMB anisotropies in the coming decade.\n\n\\section{Once and Future Power Spectrum}\n\nThe tiny $10^{-5}$ variations in the temperature of the CMB\nacross the sky are observed to be consistent with Gaussian\nrandom fluctuations, at least on the COBE scales ($>10^{\\circ}$),\nas expected in the simplest theories of their inflationary\norigin. Assuming Gaussianity, the fluctuations can be fully\ncharacterized by their angular power spectrum\\footnote{$\\!\\!$Conventions for \nrelating multipole number to angular\nscale include: $\\theta_{\\ell} \\approx 2\\pi/\\ell$, $\\pi/\\ell$ or\n$100^{\\circ}/\\ell$. \nTo the extent that these conventions\ndiffer, none of them are correct; we hereafter refer to power\nspectrum features by multipole number, which has a precise meaning.}\n\\begin{equation}\n T(\\hat{\\bf n}) = \\sum_{\\ell m} a_{\\ell m} Y_{\\ell m}(\\hat{\\bf n})\\,,\n \\qquad \\left< a_{\\ell m}^{*} a_{\\ell 'm'} \\right> =\n \\delta_{\\ell \\ell'}\\delta_{m m'} C_{\\ell}\\,.\n\\end{equation}\nWe will often use the shorthand $(\\Delta T)^{2} = \n\\ell (\\ell +1)C_{\\ell}/2\\pi$ which represents the power\nper logarithmic interval in $\\ell$. \n\nFig.~\\ref{fig:power} (left, $1\\sigma$ errors $\\times$ window FWHM) \nshows \nthe measurements the power spectrum to date\n(see \\cite{TegZal00} for a complete list of references). \nThe data indicate a rather sharp peak in the spectrum\nat $\\ell \\sim 200$ with a significant decline at $\\ell \\simgt 1000$.\nThis peak has profound implications for\ncosmology. The primary claims in decreasing order of\nconfidence and increasing need of verification from precision\nmeasurements (e.g. from the MAP and Planck satellite \nFig.~\\ref{fig:power} center, right) are \n\\begin{figure}[t]\n\\begin{center}\n\\epsfxsize=\\textwidth\\epsffile{fig1.eps}\n\\end{center}\n\\vspace{-0.8truecm}\n\\caption{$\\!\\!$: Power Spectrum}\n%\\smalltext{Power Spectrum. Error boxes represent $1\\sigma$ in power\n%and the FWHM of the experimental windows for the current \n%data and band power range for the MAP and Planck satellites.}\n\\label{fig:power}\n\\end{figure}\n\n\\begin{itemize}\n \n \\footnotesize\n\\item{} Early universe. The simplest inflationary cold dark matter (CDM) cosmologies\nhave correctly predicted the location and morphology of the \nfirst peak in the CMB; conversely,\nall competing {\\it ab initio} theories have failed, essentially due\nto causality. {\\it \nConfirm its acoustic nature with the second peak. \nUse polarization as a sharp test of causality.}\n\n\\item{} Geometry. The universe is flat. {\\it Lower} limits on the\ntotal density ($\\Omega_{tot}\\equiv \n\\sum \\Omega_i \\simgt 0.6$ \\cite{TegZal00}) are already robust, \nunless recombination is substantially\ndelayed or $h \\gg 1$. {\\it Calibrate\nthe ``standard rulers'' (acoustic scale and damping scale)\nin this distance measure through the higher peaks}. \n\n\\item{} Baryons. At least as much baryonic\ndark matter as indicated by big \nbang\nnucleosynthesis (BBN) is required ($\\Omega_{b} h^{2}\n\\simgt 0.01$ \\cite{TegZal00}). {\\it Confirm with relative heights of\nthe peaks, especially the third peak.}\n\n\\item{} Reionization. The Thomson optical depth is low -- how low\ndepends on the range of models considered. {\\it The optical depth,\nassuming it is low, \nwill only be accurately measured by CMB polarization at large angles.}\n\n\\item{} Dark energy. The matter density is low and combined with \nflatness, this indicates\na missing energy component, possibly the cosmological constant. \nCurrently the $95\\%$ CL includes $\\Omega_{m}=1$ but the maximum \nlikelihood model including BBN and $h$ \nconstraints has $\\Omega_{m}\\approx 0.3$ \\cite{TegZal00}.\n{\\it Measure $\\Omega_{m}h^{2}$ from the\nfirst three peaks.} \n\n\\end{itemize}\nThe early universe and geometry tests basically rely on the\nposition of the first peak and hence are more robust than the \nlater ones which rely mainly on interpreting its amplitude.\n\nMoreover, all claims are based on interpreting the\npeak at $\\ell \\sim 200$ as the first in a series of {\\it acoustic\npeaks}. Based on the sharpness of the feature, this interpretation\nis now reasonably, but not completely secure. \nThe detection of a second peak in the spectrum is critical \nsince it will\nprovide essentially incontrovertible evidence that this \ninterpretation is correct (or wrong!). \nOnce this is achieved\nand the peaks pass the morphological tests described below, the CMB\nwill become the premier laboratory for precision\ncosmology, as many studies have shown\n\\cite{Junetal96}.\nThese expectations also rely on the fact that $C_\\ell$\ncan ultimately be measured to \n\\begin{equation}\n{\\Delta C_{\\ell} \\over C_{\\ell}} = \\sqrt{2 \\over (2\\ell +1)f_{\\rm sky}}\\,,\n\\label{eqn:sample}\n\\end{equation}\nbased on\nGaussian sample variance on the \n$(2\\ell+1)f_{\\rm sky}$ independent modes of a given $\\ell$,\nfrom a fraction of sky $f_{\\rm sky}$.\nThe rest of this review will make the theoretical\ncase for the above statements.\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfxsize=\\textwidth\\epsffile{fig2.eps}\n\\end{center}\n\\vspace{-0.8truecm}\n\\caption{$\\!\\!$: Harmonic Acoustic Peaks.}\n{\\footnotesize }\n\\label{fig:harmonics}\n\\end{figure}\n\n\\section{Sound Physics}\n\nThe theory underlying the predictions of CMB anisotropies has\nessentially been in place since the 1970's \\cite{Early} and is\nbased on extraordinarily simple fluid mechanics and gravity \n\\cite{Fluid,Sel94}.\nSimplicity is ensured by the smallness of the fluctuations \nthemselves: the observed amplitude of $\\Delta T/T \\sim 10^{-5}$\nguarantees that the equations of motion can be linearized. \n\nThe fluid nature of the problem follows from simple thermal\narguments. The cooling of CMB photons due to the cosmological expansion\nimplies that before $z_*\\approx 1000$,\nwhen the CMB temperature is $T > 3000$K, the photons are hot enough to\nionize hydrogen. During this epoch, the electrons glue the baryons\nto the photons by Compton scattering and electromagnetic interactions.\nThe dynamics that result involve a single photon-baryon fluid.\n\nGravity attracts and compresses the fluid into the potential wells\nthat later seed large-scale structure.\nPhoton pressure resists this compression and sets up\nsound waves or acoustic oscillations in the fluid.\nThese sound waves are frozen into the CMB at recombination.\nRegions that have reached their maximal compression by recombination\nbecome hot spots on the sky; those that reach maximum rarefaction\nbecome cold spots. \n\n\n\\section{Math}\n\\label{sec:oscillator}\n\nMathematically, the cast of characters are: for the photons,\nthe local temperature $\\Theta=\\Delta T/T$, bulk velocity or dipole\n$v_{\\gamma}$, and anisotropic stress or quadrupole $\\pi_{\\gamma}$;\nfor the baryons, the density perturbation $\\delta_{b}$ and bulk \nvelocity $v_{b}$; for gravity, the Newtonian potential $\\Psi$ \n(time-time metric fluctuation) and the curvature fluctuation\n$\\Phi$ (space-space metric fluctuation $\\approx -\\Psi$). \nCovariant conservation of energy \nand momentum requires that the photons and baryons satisfy seperate continuity\nequations \\cite{Fluid}\n\\begin{eqnarray}\n\\dot \\Theta = -{k \\over 3} v_{\\gamma} - \\dot\\Phi \\, , \\qquad\n\\dot \\delta_b = -k v_b - 3\\dot\\Phi \\, ,\n\\label{eqn:continuity}\n\\end{eqnarray}\nand Euler equations\n\\begin{eqnarray}\n\\dot v_{\\gamma} &=& k(\\Theta + \\Psi) - {k \\over 6} [1+3(1-\\Omega_{\\rm \ntot}) {H_0^2 \\over k^2}]\n\\pi_\\gamma\n - \\dot\\tau (v_\\gamma - v_b) \\, , \\nonumber\\\\\n\\dot v_b &= &- {\\dot a \\over a} v_b + k\\Psi + \\dot\\tau(v_{\\gamma} - v_b)/R\n \\, ,\n\\label{eqn:Euler}\n\\end{eqnarray}\nin wavenumber space. $\\dot\\tau = n_e\\sigma_T a$ is the differential Thomson\noptical depth, \n$R= (p_b + \\rho_b)/(p_\\gamma + \\rho_\\gamma) \\approx\n3\\rho_b / 4 \\rho_\\gamma$ is the photon-baryon momentum density ratio, and \noverdots represent derivatives with respect to conformal\ntime $\\eta=\\int dt/a$.\n\nThe continuity equations represent particle number conservation.\nFor the baryons, $\\rho_b \\propto n_b$.\nFor the photons, $T \\propto n_\\gamma^{1/3}$, which explains the\n$1/3$ in the velocity divergence term. \nThe $\\dot\\Phi$ term represents\nthe ``metric stretching'' effect and appears because $\\Phi$ represents\na spatially varying perturbation to the scale factor $a$ and\n$n_{\\gamma,b} \\propto a^{-3}$ (see Fig.~\\ref{fig:secondaries}, left).\n\nThe Euler equation has a similar interpretation. \nThe expansion makes particle momenta\ndecay as $a^{-1}$. The cosmological redshift of $T$ accounts for\nthis effect in the photons.\nFor the baryons, it becomes the expansion drag on \n$v_b$ ($\\dot a/a$ term). Potential gradients $k\\Psi$ generate \npotential flow. For the photons, stress gradients in the fluid, both\nisotropic ($k\\delta p_\\gamma /(p_\\gamma + \\rho_\\gamma) = k\n\\Theta$) and anisotropic ($k\\pi_\\gamma$) counter infall. \nCompton scattering\nexchanges momentum between the two fluids\n($\\dot \\tau$ terms). \n\nIf scattering $(\\dot\\tau^{-1})$\nis rapid compared with the light travel time\nacross the perturbation $(k^{-1})$, the photon-baryon system\nbehaves as a perfect fluid. To lowest order in $k/\\dot\\tau$,\neqns.~(\\ref{eqn:continuity}) and (\\ref{eqn:Euler})\nbecome\n\\begin{equation}\n(m_{\\rm eff} \\dot\\Theta)\\dot{\\vphantom{A}} + {k^2 \\over 3}\\Theta\n= -{k^2 \\over 3} m_{\\rm eff}\\Psi - (m_{\\rm eff}\\dot\\Phi)\n\\dot{\\vphantom{A}} \\,,\n\\label{eqn:oscillator}\n\\end{equation}\n%\\begin{eqnarray}\n%\\dot \\Theta &=& - {k \\over 3} \\Theta_1 - \\dot\\Phi \\, ,\\nonumber\\\\\n%\\dot v_\\gamma &=& -{R \\over 1 + R} {\\dot a \\over a} v_\\gamma +\n% {1 \\over 1 + R} k\\Theta_0 + k\\Psi.\n%\\end{eqnarray}\nwhere the effective mass is $m_{\\rm eff} = 1+R$ or alternatively\n$c_s^2 = \\dot p/\\dot\\rho = 1/3m_{\\rm eff}$.\nScattering isotropizes the distribution in the\nelectron rest frame $v_\\gamma=v_b$ \nand eliminates anisotropic stress ($\\pi_\\gamma = {\\cal O}(k/\\dot\\tau) v_\\gamma$). \n\nEquation (\\ref{eqn:oscillator}) is the fundamental relation for \nacoustic oscillations; it \nreads: the change in the momentum of the photon-baryon\nfluid is determined by a competition between the\npressure restoring and gravitational driving forces.\nGiven the initial conditions and gravitational potentials,\nit predicts the phenomenology of the acoustic peaks.\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfxsize=\\textwidth\\epsffile{fig3.eps}\n\\end{center}\n\\vspace{-0.8truecm}\n\\caption{$\\!\\!$: Geometry.}\n{\\footnotesize}\n\\label{fig:geometry}\n\\end{figure}\n\n\\section{Early Universe}\n\\label{sec:early}\n\nThe simplest inflationary models are essentially unique in their\nphenomenological predictions. They possess a spectrum\nof curvature (potential) fluctuations that extends\n{\\it outside} the apparent horizon in the post-inflationary epoch. \nThese perturbations remain constant while the fluctuation is outside\nthe horizon except for a small change at matter-radiation equality. \nNeglecting this and baryon inertia $(m_{\\rm eff}=1)$ for the moment,\nthe oscillator equation (\\ref{eqn:oscillator}) \nhas the simple solution\n\\begin{equation}\n[\\Theta + \\Psi](\\eta_*) = [\\Theta + \\Psi](0) \\cos(k s)\\,,\n\\qquad \nv_\\gamma = \\sqrt{3} [\\Theta + \\Psi](0) \\sin(k s)\\,,\n\\end{equation}\nwhere $s=\\int_0^{\\eta_*} c_s d\\eta$ is the {\\it sound horizon}\nat $\\eta_*$. An initial temperature perturbation $\\Theta(0)$ \nexists since the gravitational\npotential $\\Psi$ is a time-time perturbation to the metric. Because\nof the redshift with the scale factor $a \\propto t^{2/3(1+p/\\rho)}$,\na temporal shift produces a temperature perturbation of\n$\\Theta = -2 \\Psi/3(1+p/\\rho)$ or $-\\Psi/2$ in the radiation dominated\nera. We call $\\Theta+\\Psi$ the effective temperature since it\nalso accounts for the redshift a photon experiences when climbing out\nof a potential well \\cite{SacWol67}. The matter radiation transition simply \nmakes $\\Theta +\\Psi = \\Psi /3$.\n\nThere are two important aspects of this result. First, inflation sets\nthe {\\it temporal} phase of all wavemodes by starting them all at \nthe initial epoch. Wavenumbers which hit their extrema at recombination\nare given by $k_m = m\\pi/s$ and these mark the peaks of coherent oscillation\nin the power spectrum. \nSecond, the first peak at $k=\\pi/s$ represents a\ncompression of the fluid in the gravitational potential well ($\\Psi<0$,\nsee Fig.~\\ref{fig:harmonics}).\n\n\nWithout inflation to push perturbations\nsuperluminally outside the horizon, they \nmust be generated by the causal motion of matter. One might think\nany anisotopies above the horizon scale \nat recombination projected on the sky (e.g. COBE) implies \ninflation. \nHowever these could instead be generated after\nrecombination through gravitational redshifts (\\S \\ref{sec:beyond}). To test inflation,\none needs to isolate\na particular epoch in time. The acoustic peaks provide one\nsuch opportunity; we shall see later that polarization provides \nanother.\n\n\nIf the fluctuations were generated\nby non-linear dynamics well inside the horizon,\ne.g. by a cosmic string network, the temporal coherence,\nand hence the peak structures,\nwould be lost due to random forcing of\nthe oscillators \\cite{AlbCouFerMag96}. \nCausal generation itself does not guarantee incoherence. \nCoherence requires that there is one special epoch for all modes that\nsynchs up their oscillations. One common event can \ncausally achieve this: horizon crossing when $k\\eta=1$. \nFor example, textures unwind at horizon\ncrossing and maintain some coherence in their acoustic oscillations. \nHowever it is very difficult to\nplace the first compressional peak at as large a scale as \n$k_1 = \\pi/s$ since the photons tend to first cool down due to\nmetric stretching from $\\Phi$ \nas gravitational potentials grow, thus inhibiting the\ncompressional heating \\cite{HuWhi96}. \nThe only known mechanism for doing so \nis to reverse the sign of gravity: to make gravitational\npotential wells in underdense regions so that $\\Phi \\sim \\Psi$ \n\\cite{Tur96}.\nIn principle, this can be arranged by a special choice of anisotropic\nstresses but there is no known form of matter that obeys the required\nrelations. On the other hand, {\\it inflationary} curvature \n(adiabatic) \nand isocurvature (stress) fluctuations existing outside the horizon\ncan be interconverted with physically realizable stress histories \\cite{Hu99}.\n\nIn summary, verification of an inflationary series of acoustic peaks\nwith locations in an approximate ratio of $\\ell_1:\\ell_2:\\ell_3\\ldots=\n1:2:3\\ldots$ would represent\na strong test of the inflationary origin of the perturbations and \na somewhat weaker test of their initially adiabatic nature.\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfxsize=\\textwidth\\epsffile{fig4.eps}\n\\end{center}\n\\vspace{-0.8truecm}\n\\caption{$\\!\\!$: Acoustic not Doppler peaks.}\n\\label{fig:project}\n\\footnotesize\n\\end{figure}\n\n\\section{Geometry}\n\nThe physical scale of the features is related to the distance $s$ that\nsound can travel by recombination. Specifically, one expects features\nin the spatial power spectrum of the photon temperature and dipole\nat $k> k_A = \\pi/s$. Each mode is then projected\non the sky in spherical coordinates \n$\\exp(i{\\bf k}\\cdot{\\bf x}) \\propto j_\\ell(kd) Y_{\\ell 0}$, where\n$d = \\eta_0-\\eta_*$, and summed in quadrature to form the final anisotropy,\n\\begin{equation}\nC_\\ell \\approx {2 \\over \\pi} \\int {dk \\over k} k^3\n\\left[ (\\Theta + \\Psi) j_\\ell(kd) + v_\\gamma j_\\ell'(kd)\\right]^2\\,.\n\\label{eqn:clproject}\n\\end{equation}\nThis approximation ignores the finite duration of recombination but suffices\nfor a qualitative understanding of the spectrum. We \nhave also temporarily assumed that the universe \nis flat $\\Omega_{\\rm tot}=1$.\n\nThe $v_\\gamma$ term represents the Doppler effect from the \nmotion of the\nfluid along the line of sight. It has an intrinsic\ndipole angular dependence at last scattering $Y_{10}$ in addition to\nthe ``orbital'' angular dependence $Y_{\\ell 0}$. Addition of\nangular momentum implies a coupling of $j_{\\ell \\pm 1}$ that can be\nrewritten as $j_\\ell'$. \n\nAs a consequence of eqn.~(\\ref{eqn:clproject}), features in the\nspatial power spectrum of the effective temperature at recombination\nbecome features in the angular power spectrum whereas those of\nthe bulk velocity do not (see Fig.~\\ref{fig:project} $kd=100$)\n\\cite{Fluid}. \nA plane wave temperature perturbation contributes a range of anisotropies\ncorresponds to viewing angles perpendicular ($\\ell \\approx kd$) all the\nway to parallel ($\\ell \\rightarrow 0$) to the wavevector ${\\bf k}$\n(see Fig.~\\ref{fig:project}, lobes). \nThe result is a sharp maximum around $\\ell = kd$ as expected from \nnaively converting physical to angular scale. However for\nthe Doppler effect from potential flows, velocities are directed parallel to \n${\\bf k}$, so that the peak at $\\ell = kd$ \nis eliminated. Although the Doppler effect contributes\nsignificantly to the overall anisotropy, the peak structure traces\nthe temperature fluctuations. \n\nIn a spatially curved universe, one replaces the spherical Bessel\nfunctions in eqn.~(\\ref{eqn:clproject}) with the ultraspherical \nBessel functions and these peak at $\\ell \\approx kD$ where $D$\nis the {\\it comoving angular diameter distance}\nto recombination. \nConsider first \na closed universe with radius of curvature \n${\\cal R} = H_{0}|\\Omega_{\\rm tot}-1|^{1/2}$.\nSuppressing one spatial coordinate yields\na 2-sphere geometry with the observer situated at the\npole (see Fig.~\\ref{fig:geometry}). Light travels on \nlines of longitude.\nA physical scale $\\lambda$ at fixed latitude given by\nthe polar angle $\\theta$ subtends an angle\n$\\alpha = \\lambda/{\\cal R}\\sin\\theta$.\nFor $\\alpha \\ll 1$,\na Euclidean analysis would infer a \ndistance $D={\\cal R}\\sin\\theta$, even though\nthe {\\it coordinate distance} along the arc is\n$d = \\theta {\\cal R}$; thus\n\\begin{equation}\nD = {\\cal R} \\sin( d / {\\cal R})\\,, \\qquad (\\Omega_{\\rm tot}>1)\\,.\n\\end{equation}\nFor open universes, simply replace $\\sin$ with $\\sinh$.\nA given physical scale subtends a larger (smaller) angle in\na closed (open) universe than a flat universe.\n\nWe thus expect CMB features at the \ncharacteristic scale \\cite{HuSug95}\n\\begin{eqnarray}\n \\ell_{A} &=& \\pi D/s \n\t\\approx 172 \\Omega_{\\rm tot}^{-1/2} \n\t[1+\\ln(1-\\Omega_\\Lambda)^{0.085}]\n\tf(\\Omega_m h^2,\\Omega_b h^2)\\,,\\\\\n\\label{eqn:ellA}\nf &=& \\left({z_* \\over 10^3}\\right)^{1/2} \\left( {1 \\over \\sqrt{R_*}} \\ln \n\t{\\sqrt{1+R_*} + \\sqrt{R_* +\\epsilon R_*} \n\t\\over 1 + \\sqrt{\\epsilon R_*}} \\right)^{-1} \\,,\n\\label{eqn:correction}\n\\end{eqnarray}\nwhere $\\epsilon \\equiv a_{eq}/a_* = 0.042 (\\Omega_m h^2)^{-1} (z_*/10^3)$\nand $R_* = 30\\Omega_b h^2 (z_*/10^3)$; see \\cite{HuWhi97} for\n$z_*(\\Omega_m h^2,\\Omega_b h^2)$. \n\nThe main scaling of $\\ell_{A}$\nis with $\\Omega_{\\rm tot}^{-1/2}$ \\cite{KamSpeSug94},\nbut finite $\\Omega_{\\Lambda}$\ncauses it to decrease.\nThis covariance is referred to in the literature as the {\\it angular\ndiameter distance $(D)$ degeneracy}. \nThe quantity in \nparentheses in eqn.~(\\ref{eqn:correction}) goes to unity \nas $\\epsilon,R_{*}\\rightarrow 0$. The leading order \ncorrection ($1+\\epsilon^{1/2}$) makes the $\\Omega_m h^2$ dependence\nimportant in any reasonable cosmology. \nThe other correction ($1+R_*/6$) is small for reasonable \nbaryon densities. \n\n\nFor simple inflationary models, the peaks reside at $\\ell_m \\approx m \\ell_A$.\nMore generally, $\\ell_1 \\ge \\ell_A$ (see \\S \\ref{sec:early}). \nThe detection of the first peak then\nputs a reasonably robust lower limit on $\\Omega_{\\rm tot}$.\nThe key assumptions are that we can attribute the feature\nto acoustic oscillations, bound the redshift of recombination from below and bound the sound\nhorizon from above. The last assumption amounts to \nhaving an upper limit on $\\Omega_{m} h^2$ (or $h$). \nThe $D$ degeneracy is tamed since $\\Omega_\\Lambda$ is automatically bounded \nfrom above for the $\\Omega_{\\rm tot}$ of interest by requiring\n$\\Omega_m >0$.\nConverting lower limits on $\\Omega_{\\rm tot}$ into precise measurements\nrequires independent measurements of\n$\\Omega_m h^2$ and $\\Omega_b h^2$, which calibrate the standard rulers at recombination \\cite{Fluid}, and\n$\\Omega_\\Lambda$, $\\Omega_m$ or $h$ to break the $D$ degeneracy.\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfxsize=\\textwidth\\epsffile{fig5.eps}\n\\end{center}\n\\vspace{-0.8truecm}\n\\caption{$\\!\\!$: Baryons.}\n\\label{fig:baryons}\n\\footnotesize\n\\end{figure}\n\n\\section{Baryons}\n\nBaryons add inertia to the fluid.\nConsider first the case of\n$m_{\\rm eff} = 1+R=$ const. [see\neqn.~(\\ref{eqn:oscillator})]\n\\begin{equation}\n[\\Theta + \\Psi](\\eta_*) = [\\Theta(0) + (1+R)\\Psi(0)]\n\t\\cos(ks) - R\\Psi\\,,\n\\end{equation} \nwhere $s=\\eta_*/\\sqrt{3(1+R)}$. There\nare three effects of raising the baryon content:\nan amplitude increase, a zero-point shift, and a frequency\ndecrease \\cite{Fluid}.\nBaryons drag the fluid deeper into the potential wells\n(see Fig.~\\ref{fig:baryons}). For the fixed initial conditions,\nthe resulting shift in the zero point also implies a\nlarger amplitude. Since it is the power spectrum that is observed,\nthe result of squaring implies that all compressional peaks are \nenhanced by the baryons and the rarefaction peaks suppressed. This \nis the clearest signature of the baryons and also provides a \nmeans for testing the compressional nature of the first peak predicted by inflation. \nThe fact that $R \\propto a$ due to the redshifting of the photons\nsimply means that the oscillator actually has time dependent mass. \nThe adiabatic invariant ($E/\\omega$) implies an \namplitude reduction as $(1+R)^{-1/4}$.\n\n\nBaryons also affect the fluid through dissipational processes\n\\cite{Sil68}.\nThe random walk of the photons through the baryons\ndamps the acoustic oscillation exponentially below \nthe diffusion scale $k_D$, roughly the geometric \nmean of the mean free path and the horizon scale.\nMicrophysically, the dissipation comes from viscosity\n$\\pi_\\gamma$ in eqn.~(\\ref{eqn:Euler}) and heat conduction $v_\\gamma-v_b$.\nBefore recombination it can be included by keeping terms of order\n$k/\\dot\\tau$ in the equations. At recombination, the \nmean free path increases and brings the diffusion scale to \\cite{HuWhi97}\n\\begin{eqnarray} \nk_D &\\approx& a_1(\\Omega_{m}h^{2})\\,\n(\\Omega_b h^2)^{0.291} [1+a_2(\\Omega_{m}h^{2})\\,(\\Omega_b h^2)^{1.8}]^{-1/5} \n\t{\\rm Mpc}^{-1}\\,, \n\\end{eqnarray}\t\n$a_1(x) = 0.0396x^{-0.248}(1+ 13.6x^{0.638})$,\n$a_2(x) = 1480x^{-0.0606} (1+ 10.2x^{0.553})^{-1}$.\nThe main effects can be easily understood: increasing $\\Omega_m h^2$\ndecreases the horizon at last scattering \nand hence the diffusion length.\nAt low $\\Omega_b h^2$, increasing the baryon content decreases\nthe mean free path while at high $\\Omega_b h^2$, it delays recombination\nand increases the diffusion length. \n\nDamping introduces another length scale for the curvature test, $l_D = \nk_D D$; alternately \n$l_D/l_A = k_D/k_A = f(\\Omega_m h^2,\\Omega_b h^2)$ is\nindependent of $D$ and can measure this \ncombination of parameters.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfxsize=\\textwidth\\epsffile{fig6.eps}\n\\end{center}\n\\vspace{-0.8truecm}\n\\caption{$\\!\\!$: Matter-radiation ratio.}\n\\label{fig:drive}\n\\footnotesize\n\\end{figure}\n\n\\section{Matter/Radiation}\n\\label{sec:matterradiation}\n\nWe have hitherto been considering the gravitational force \non the \noscillators as constant in time. This can only be true for \n{\\it growing} density fluctuations.\nThe Poisson equation says that $\\Phi \\propto a^2 \\rho \\delta$,\nand the density redshifts with the expansion as $\\rho \\propto \na^{-3(1+p/\\rho)}$. In the radiation era, density perturbations must\ngrow as $a^2$ for constant potentials, as they do in the comoving\ngauge when pressure gradients can be neglected. Once\nthe pressure gradients have turned infall into acoustic oscillations,\nthe potential must decay. This decay actually drives the\noscillations since the fluid is left maximally compressed with\nno gravitational potential to fight as it turns around \n(see Fig.~\\ref{fig:drive}) \\cite{Fluid}. \nThe net effect is doubled by the metric stretching effect from $\\Phi$,\nleading to fluctuations with amplitude \n$2\\Psi(0) - [\\Theta +\\Psi](0) = {3 \\over 2}\\Psi(0)$.\n\nWhen the universe becomes matter-dominated the gravitational potential\nno longer reflects photon density perturbations. As discussed in \\S \n\\ref{sec:early}, $\\Theta +\\Psi =\n{\\Psi/3} = 3\\Psi(0)/10$ here, so that across the horizon scale at\nmatter radiation equality the acoustic amplitude increases by a factor of 5. \n\nThis effect mainly measures the matter-to-radiation ratio. Density perturbations\nin any form of radiation will stop growing around horizon crossing\nand lead to this effect. For the neutrinos, the only difference is \nthat\nanisotropic stress from their quadrupole anisotropies \nalso slightly affects the cessation\nof growth. \nOne can only turn this into a measure of $\\Omega_m h^2$\nby assuming that the radiation density is known through the CMB\ntemperature and the number of neutrino species otherwise we are\nfaced with a {\\it matter-radiation}\ndegeneracy. For example, determining\nboth $\\Omega_m h^2$ and the number of neutrino species from the CMB\nalone will be difficult.\n\nPrecise measurements of $\\Omega_m h^2$ when combined with the\nangular diameter distance would constrain the universe to live on \na line in the\nclassical cosmological parameter space ($\\Omega_m$,$\\Omega_\\Lambda$,$h$).\n{\\it Any} external (non-degenerate) measurement in this space \n$(\\Omega_m, h,$ acceleration,$\\ldots$) and allows the three parameters \nto be determined independently. This fortunate situation\nhas been dubbed ``cosmic complementarity'' and currently shows\n``cosmic concordance'' around a $\\Lambda$CDM model. \nMore importantly, the combination of several checks\ncreates sharp consistency checks that may even show our universe\nto live outside this space, for example if the missing energy \nis not $\\Lambda$ but some dynamical ``quintessence'' field.\n\n\\section{And Beyond}\n\\label{sec:beyond}\n\nThe primary anisotropies from the recombination epoch contain\nonly a small fraction of the cosmological information \nlatent in the CMB.\nLet us conclude this\nsurvey with topics of future study: \nsecondary anisotropies and polarization. \nBoth are expected to be at the $\\simlt 10^{-6}$ ($\\mu$K)\nlevel and will require high sensitivity experiments with\nwide-frequency coverage to reject galactic and extragalactic \nforegrounds of comparable amplitude.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfxsize=\\textwidth\\epsffile{fig7.eps}\n\\end{center}\n\\vspace{-0.8truecm}\n\\caption{$\\!\\!$: Secondary anisotropies.}\n\\label{fig:secondaries}\n\\footnotesize\n\\end{figure}\n\n\\vskip 0.2truecm\n\\noindent\n{\\bf Secondary Anisotropies:}\nThese are generated as photons travel\nthrough the large-scale structure between us and recombination.\nThey arise from two sources: gravity and \nscattering during reionization. It is currently believed that\nthe universe reionized at $5 \\le z \\simlt 15$ leading to $\\tau_{\\rm rei} \\sim 0.01-0.1$.\n\nGravitational redshifts can change the temperature along the\nline of sight. Density perturbations cease to grow once\neither the cosmological constant or curvature dominates the\nexpansion. As discussed in \\S \\ref{sec:matterradiation}, \nthe gravitational potentials must then decay. Decay of potential\nwell both removes the gravitational redshift and heats the photons\nby ``metric stretching\" leading to an effect that is $2\\Delta\\Phi$\n(see Fig.~\\ref{fig:secondaries}). The opposite effect occurs\nin voids so that on small scales the anisotropies\nare cancelled across crests and troughs of\nmodes parallel to the line-of-sight.\nThe effect from the decay is called the ISW effect \\cite{SacWol67} and\nfrom the non-linear growth of perturbations, the Rees-Sciama \n\\cite{ResSci68} effect.\n\nThe gravitational potentials also\nlens the CMB photons \\cite{Lensing}. Since lensing conserves surface \nbrightness, it only affects\nanisotropies and hence is second order.\nThe photons are deflected according to\nthe angular gradient of the potential integrated along the\nline of sight. Again the cancellation of parallel modes\nimplies that large-scale potentials are mainly \nresponsible for lensing and cause a long-wavelength modulation of\nthe sub-degree scale anisotropies. The modulation is a power \npreserving smoothing of the power spectrum which reduces\nthe acoustic peaks to fill in the troughs. Not until\nthe primary anisotropies disappear beneath the damping scale do\nthe cancelled potentials actually generate power in\nthe CMB.\n\nThe same principles apply for scattering effects -- with one twist.\nThe Doppler effect from large-scale potential flows, which\nrun parallel to the wavevector, contribute\nnothing to the cancellation-surviving perpendicular modes\n(see Fig.~\\ref{fig:secondaries}).\nThus even though $v_{b}\\tau \\sim 10^{-4}-10^{-5}$, Doppler\ncontributions are at $10^{-6}$. \nThe main effect of reionization is to suppress power in the\nanisotropies as $e^{-2\\tau}$ below the angle subtended by the\nhorizon at the scattering. Unfortunately, given the sample\nvariance of the low-$\\ell$ multipoles [see eqn.~(\\ref{eqn:sample})], this effect is nearly\ndegenerate with the normalization and the current limits from\nthe first peak that $\\tau_{\\rm rei} \\simlt 1$\nwill not be\nimproved by more than a factor of a few from the higher peaks.\n\nSurviving the Doppler cancellation\nare higher order effects due to optical depth modulation,\nperpendicular to the line of sight, of\nthe Doppler shifts at small angular scales \nfrom linear density perturbations\n(Vishniac effect \\cite{Vis87}), \nnon-linear structures (non-linear Vishniac\neffect or kinetic SZ effect \\cite{Hu00}) and patchy or inhomogeneous\nreionization \\cite{Patch}. Another opacity-modulated signal is\nthe distortion from Compton upscattering by hot gas, the (thermal)\nSunyaev-Zel'dovich (SZ) effect \\cite{SunZel80}, especially in clusters where it\nis now routinely detected.\n\nAll of these secondary effects produce signals in the $\\mu$K\nregime. Developing methods to isolate them is currently an\nactive field of research and lies beyond the scope of\nthis review. The main lines of inquiry are to explore\nsub-arcminute scales where the primary anisotropies has fallen off,\nthe non-Gaussianity of the higher order effects \\cite{NonGaus},\ntheir frequency dependence to separate them from foregrounds and\nthe thermal SZ effect \\cite{Fore}, their\ncross correlation with other tracers of large-scale structure\n\\cite{Cross},\nand finally their polarization. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfxsize=\\textwidth\\epsffile{fig8.eps}\n\\end{center}\n\\vspace{-0.8truecm}\n\\caption{$\\!\\!$: Polarization.}\n\\label{fig:pol}\n\\footnotesize\n\\end{figure}\n\n\\vskip 0.2truecm\n\\noindent\n{\\bf Polarization}:\nThomson scattering of quadrupole anisotropies generates\nlinear polarization in the CMB by passing only one component\nof polarization of the incident radiation (see Fig.~\\ref{fig:pol}).\nThe polarization amplitude, pattern, and correlation with the temperature\nanisotropies themselves \nis thus encapsulated in the quadrupole \nanisotropies at the scattering. This information and the fact\nthat it is only generated by scattering are the useful properties \nof polarization.\n\nDensity perturbations generate quadrupole anisotropies as radiation\nfrom\ncrests of a temperature perturbation flows into troughs.\nSuch anisotropies are azimuthally symmetric around the wavevector\n($Y_{20}$ quadrupole). They generate a distinct pattern where\nthe polarization is aligned or perpendicular to the wavevector\n(``$E$'' pattern \\cite{Pol}).\n\nHowever polarization generation suffers\nfrom a catch-22: the scattering which generates polarization also\nsuppresses its quadrupole source (see~\\S \\ref{sec:oscillator}).\nThey can only be generated once the perturbation becomes\noptically thin. Primary anisotropies are only substantially\npolarized in the damping region where the finite duration of\nlast scattering allows viscous imperfections in the fluid, and\nthen only at the $\\sim 10\\%$ level ($\\mu$K level, \nFig.~\\ref{fig:pol}). Nonetheless its steep rise toward\nthis maximum is itself interesting \\cite{TAMM,SpeZal97}. Since polarization \nisolates the epoch of scattering, we can directly look \nabove the horizon scale and test the causal nature of\nthe perturbations\n(see \\S \\ref{sec:early}). \nLikewise, polarization at even larger scales\ncan be used to measure the epoch and optical depth during\nreionization \\cite{HogKaiRes82}\nbut will require the sub $\\mu$K sensitivities\nof Planck and future missions.\n\nFinally the ``$E$'' pattern of polarization discussed above\nis a special property of density perturbations in the linear\nregime. Its complement (``$B$'' pattern) has the polarization\naligned at 45$^{\\circ}$ to the wavevector. Vector\n(vorticity) and tensor (gravity wave) perturbations generate\n$B$-polarization as can be seen through the quadrupole moments\nthey generate ($Y_{2\\pm1}$ and $Y_{2\\pm 2}$ respectively \\cite{TAMM,PolPrimer}).\nMeasuring the properties of the gravity waves from inflation\nthrough the polarization is our best hope of testing the\nparticle physics aspects of inflation (see e.g. \\cite{KamKos99}).\n\n$B$-polarization is also generated by non-linear effects where\nmode coupling alters the relation between the polarization\ndirection and amplitude. In the context of the simplest inflationary\nmodels, the largest of these is the gravitational lensing\nof the primary polarization \\cite{Lensing} \nbut opacity-modulated secondary\nDoppler effects also generate $B$-polarization \\cite{Hu00}.\n\n\\section{Discussion}\n\nWe are already well on on our way to extracting the cosmological\ninformation contained in the primary temperature anisotropies,\nspecifically the angular diameter distance to recombination,\nthe baryon density, the matter-radiation ratio at\nrecombination, and the ``acausal'' (inflationary) \nnature and spectrum of the initial perturbations. \nEven if our simplest inflationary cold dark\nmatter model is not correct in detail, \nthese quantities will be measured in the\nnext few years by long-duration ballooning, interferometry and\nthe MAP satellite, {\\it if \nthe acoustic nature of the peak at $\\ell\\sim 200$ is \nconfirmed by the detection of a second peak}.\nIn the long term, the high sensitivity and wide frequency\ncoverage of the Planck satellite and other future experiments\nshould allow CMB polarization and secondary anisotropies to \nopen new windows on the early universe and large-scale structure.\n\n\\vskip 0.2truecm\n\\begin{quotation}\n \\footnotesize\\noindent\n{\\it Acknowledgements:} I would like to thank my collaborators through\nthe years, especially N. Sugiyama and M. White, the organizers\nof RESCEU 1999, and the Yukawa Institute for its hospitality. \nThis work was partially supported by NSF-9513835, the Keck\nFoundation, and a Sloan Fellowship.\n\\end{quotation}\n\n\\begin{thebibliography}{99}\n\n\\footnotesize\n\\parskip 0pt\n\\bibitem{COBE92}\nSmoot G. et al. 1992, ApJL 396, L1\n\n\\bibitem{ScoSilWhi95}\nScott D., Silk J., White M. 1995, Science 829\n\n\\bibitem{TegZal00}\nTegmark M., Zaldarriaga M. astro-ph/0002091; Knox L., Page L \nastro-ph/0002162; Melchiorri A., et al., astro-ph/9911445; \nMiller A.D.\\ et al.\\ 1999, ApJ 524, 1\n\n\\bibitem{Junetal96}\nJungman G., Kamionkowski M., Kosowsky A., Spergel D.N. 1996, \nPRD, 54, 1332;\nBond J.R., Efstathiou G., Tegmark M. 1997, MNRAS 291, L33;\nZaldarriaga M., Spergel D.N., Seljak U. 1997, ApJ 488, 1;\nEisenstein D.J., Hu W., Tegmark M. 1999, ApJ 518, 2\n\n\n\\bibitem{Early}\nPeebles P.J.E., Yu J.T. 1970, ApJ 162, 815;\nDoroshkevich A.G., Zel'dovich Ya.B., Sunyaev R.A. 1978,\nSov. Astron. 22, 523; \nBond J.R., Efstathiou G. 1984, ApJ 285, L45\n% Seljak, U. \\& Zaldarriaga, M. ApJ 285, L39\n\n\\bibitem{Fluid}\nHu W., Sugiyama N. 1995, ApJ 444, 489; \nHu W., White M. 1996, ApJ 471, 30;\nHu W., Sugiyama N., Silk J. 1997, Nature 386, 37\n\n\\bibitem{Sel94}\nSeljak U. 1994, ApJ 435, 87\n\n\\bibitem{SacWol67}\nSachs R.K., Wolfe A.M. 1967, ApJ 147, 73 \n\n\\bibitem{AlbCouFerMag96}\nAlbrecht A., Coulson D., Ferreira P., Magueijo J.,\nPRL 76, 1413.\n\n\\bibitem{HuWhi96}\nHu W., White M. 1996, PRL 77, 1687;\n\n\\bibitem{Tur96}\nTurok N.G. 1996, PRL 77, 4138; Hu W., Spergel D.N., White M. 1997,\nPRD 55, 3288\n\n\\bibitem{Hu99}\nHu W 1999, PRD 59 021301; Hu W., Peebles, P.J.E. 2000, ApJ 528, 61\n\n\\bibitem{HuSug95}\nHu W., Sugiyama N.1995, PRD 51, 2599\n\n\\bibitem{HuWhi97}\nHu W., White M. 1997, ApJ 479, 568\n\n\\bibitem{KamSpeSug94}\nKamionkowski M., Spergel D.N., Sugiyama N. 1994, ApJ 434, L1\n\n\\bibitem{Sil68}\nSilk J. 1968, ApJ 151, 459\n\n\n\\bibitem{ResSci68}\nRees M.J., Sciama D.N. 1968, Nature, 217, 511;\nSeljak U. 1996, ApJ, 460, 549\n\n\\bibitem{Lensing}\nSeljak U. 1996, ApJ 463, 1; Zaldarriaga M., Seljak U., 1998, PRD 58, 023003 (1998); Hu W. 2000, astro-ph/0001303\n\n\\bibitem{Vis87}\nVishniac E.T. 1987, ApJ, 322, 597; Hu W., White M. 1996, A\\&A, 315, 33 \n\n\\bibitem{Hu00}\nHu W. 2000, ApJ, 529, 12\n\n\\bibitem{Patch}\nAghanim N., Desert F.X., Puget J.L., Gispert R. 1996, A\\&A, 311, 1;\nGruzinov A., Hu W. 1998, ApJ, 508, 435; \nKnox L., Scoccimarro R., Dodelson S. 1998, PRL,\n81, 2004; Bruscoli M., Ferrara A., Fabbri R., Ciardi B, astro-ph/9911467\n\n\\bibitem{SunZel80}\nSunyaev R.A., Zel'dovich Ya. B. 1980, MNRAS, 190, 413\n\n\\bibitem{NonGaus}\nGoldberg D.M., Spergel D. N. 1999, PRD, 59, 103002;\nZaldarriaga M., Seljak U. 1999, PRD 59, 123507;\nCooray A.R., Hu W. 1999, ApJ in press, astro-ph/9910397\n\n\\bibitem{Fore}\nBouchet F., Gispert R., 1999, NewA, 4 443;\nTegmark M., Eisenstein D.J., Hu W., de Oliveira-Costa A.\nApJ, 1999, in press, astro-ph/9905257;\nCooray A., Hu W., Tegmark M. astro-ph/0002238\n\n\\bibitem{Cross}\nSuginohara M., Suginohara T., Spergel D.N., 1998, ApJ 495 511;\nWaerbeke L.V., Bernardeau F., Benabed K. 1999, astro-ph/9910366;\nPeiris H.V., Spergel D.N., astro-ph/0001393\n\n\\bibitem{Pol}\nKamionkowski M., Kosowsky A., Stebbins A. 1997, PRD 55, 7368;\nZaldarriaga M., Seljak U. 1997, PRD 55, 1830\n\n\n\\bibitem{SpeZal97}\nSpergel D.N., Zaldarriaga M. 1997, PRL 79, 2180\n\n\n\\bibitem{TAMM}\nHu W., White M. 1997, PRD 56, 596\n%;\n%Hu, W., Seljak, U., White, M., Zaldarriaga, M. 1998, PRD 57, 3290\n\n\\bibitem{HogKaiRes82}\nHogan C.J., Kaiser N., Rees M.J. 1982, Phil. Trans. R. Soc. Lond. A307, 97\n\n\\bibitem{PolPrimer}\nHu W., White M. 1997, NewA 2 323\n\n\\bibitem{KamKos99}\nKamionkowski M., Kosowsky A. astro-ph/9904108\n\n\n\\end{thebibliography}\n\n\\end{document}\n\n"
}
] |
[
{
"name": "astro-ph0002520.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\footnotesize\n\\parskip 0pt\n\\bibitem{COBE92}\nSmoot G. et al. 1992, ApJL 396, L1\n\n\\bibitem{ScoSilWhi95}\nScott D., Silk J., White M. 1995, Science 829\n\n\\bibitem{TegZal00}\nTegmark M., Zaldarriaga M. astro-ph/0002091; Knox L., Page L \nastro-ph/0002162; Melchiorri A., et al., astro-ph/9911445; \nMiller A.D.\\ et al.\\ 1999, ApJ 524, 1\n\n\\bibitem{Junetal96}\nJungman G., Kamionkowski M., Kosowsky A., Spergel D.N. 1996, \nPRD, 54, 1332;\nBond J.R., Efstathiou G., Tegmark M. 1997, MNRAS 291, L33;\nZaldarriaga M., Spergel D.N., Seljak U. 1997, ApJ 488, 1;\nEisenstein D.J., Hu W., Tegmark M. 1999, ApJ 518, 2\n\n\n\\bibitem{Early}\nPeebles P.J.E., Yu J.T. 1970, ApJ 162, 815;\nDoroshkevich A.G., Zel'dovich Ya.B., Sunyaev R.A. 1978,\nSov. Astron. 22, 523; \nBond J.R., Efstathiou G. 1984, ApJ 285, L45\n% Seljak, U. \\& Zaldarriaga, M. ApJ 285, L39\n\n\\bibitem{Fluid}\nHu W., Sugiyama N. 1995, ApJ 444, 489; \nHu W., White M. 1996, ApJ 471, 30;\nHu W., Sugiyama N., Silk J. 1997, Nature 386, 37\n\n\\bibitem{Sel94}\nSeljak U. 1994, ApJ 435, 87\n\n\\bibitem{SacWol67}\nSachs R.K., Wolfe A.M. 1967, ApJ 147, 73 \n\n\\bibitem{AlbCouFerMag96}\nAlbrecht A., Coulson D., Ferreira P., Magueijo J.,\nPRL 76, 1413.\n\n\\bibitem{HuWhi96}\nHu W., White M. 1996, PRL 77, 1687;\n\n\\bibitem{Tur96}\nTurok N.G. 1996, PRL 77, 4138; Hu W., Spergel D.N., White M. 1997,\nPRD 55, 3288\n\n\\bibitem{Hu99}\nHu W 1999, PRD 59 021301; Hu W., Peebles, P.J.E. 2000, ApJ 528, 61\n\n\\bibitem{HuSug95}\nHu W., Sugiyama N.1995, PRD 51, 2599\n\n\\bibitem{HuWhi97}\nHu W., White M. 1997, ApJ 479, 568\n\n\\bibitem{KamSpeSug94}\nKamionkowski M., Spergel D.N., Sugiyama N. 1994, ApJ 434, L1\n\n\\bibitem{Sil68}\nSilk J. 1968, ApJ 151, 459\n\n\n\\bibitem{ResSci68}\nRees M.J., Sciama D.N. 1968, Nature, 217, 511;\nSeljak U. 1996, ApJ, 460, 549\n\n\\bibitem{Lensing}\nSeljak U. 1996, ApJ 463, 1; Zaldarriaga M., Seljak U., 1998, PRD 58, 023003 (1998); Hu W. 2000, astro-ph/0001303\n\n\\bibitem{Vis87}\nVishniac E.T. 1987, ApJ, 322, 597; Hu W., White M. 1996, A\\&A, 315, 33 \n\n\\bibitem{Hu00}\nHu W. 2000, ApJ, 529, 12\n\n\\bibitem{Patch}\nAghanim N., Desert F.X., Puget J.L., Gispert R. 1996, A\\&A, 311, 1;\nGruzinov A., Hu W. 1998, ApJ, 508, 435; \nKnox L., Scoccimarro R., Dodelson S. 1998, PRL,\n81, 2004; Bruscoli M., Ferrara A., Fabbri R., Ciardi B, astro-ph/9911467\n\n\\bibitem{SunZel80}\nSunyaev R.A., Zel'dovich Ya. B. 1980, MNRAS, 190, 413\n\n\\bibitem{NonGaus}\nGoldberg D.M., Spergel D. N. 1999, PRD, 59, 103002;\nZaldarriaga M., Seljak U. 1999, PRD 59, 123507;\nCooray A.R., Hu W. 1999, ApJ in press, astro-ph/9910397\n\n\\bibitem{Fore}\nBouchet F., Gispert R., 1999, NewA, 4 443;\nTegmark M., Eisenstein D.J., Hu W., de Oliveira-Costa A.\nApJ, 1999, in press, astro-ph/9905257;\nCooray A., Hu W., Tegmark M. astro-ph/0002238\n\n\\bibitem{Cross}\nSuginohara M., Suginohara T., Spergel D.N., 1998, ApJ 495 511;\nWaerbeke L.V., Bernardeau F., Benabed K. 1999, astro-ph/9910366;\nPeiris H.V., Spergel D.N., astro-ph/0001393\n\n\\bibitem{Pol}\nKamionkowski M., Kosowsky A., Stebbins A. 1997, PRD 55, 7368;\nZaldarriaga M., Seljak U. 1997, PRD 55, 1830\n\n\n\\bibitem{SpeZal97}\nSpergel D.N., Zaldarriaga M. 1997, PRL 79, 2180\n\n\n\\bibitem{TAMM}\nHu W., White M. 1997, PRD 56, 596\n%;\n%Hu, W., Seljak, U., White, M., Zaldarriaga, M. 1998, PRD 57, 3290\n\n\\bibitem{HogKaiRes82}\nHogan C.J., Kaiser N., Rees M.J. 1982, Phil. Trans. R. Soc. Lond. A307, 97\n\n\\bibitem{PolPrimer}\nHu W., White M. 1997, NewA 2 323\n\n\\bibitem{KamKos99}\nKamionkowski M., Kosowsky A. astro-ph/9904108\n\n\n\\end{thebibliography}"
}
] |
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astro-ph0002521
|
Distances to Cepheid Open Clusters Via Optical and {K}-Band Imaging
|
[
{
"author": "F. Hoyle$^{1,3}$"
},
{
"author": "T. Shanks$^1$"
},
{
"author": "N. R. Tanvir$^{2}$"
},
{
"author": "1 Department of Physics"
},
{
"author": "Science Laboratories"
},
{
"author": "South Road"
},
{
"author": "Durham DH1 3LE"
},
{
"author": "2 Department of Astrophysical Sciences"
},
{
"author": "College Lane"
},
{
"author": "Hatfield"
},
{
"author": "AL10 9AB"
}
] |
We investigate the reddening and Main Sequence fitted distances to eleven young, Galactic open clusters that contain Cepheids. Each cluster contains or is associated with at least one Cepheid variable star. Reddening to the clusters is estimated using the {U-B:B-V} colours of the OB stars and the distance modulus to the cluster is estimated via {B-V:V} and {V-K:V} colour-magnitude diagrams. By main-sequence fitting we proceed to calibrate the Cepheid P-L relation and find M$_V$=-2.81$\times$logP-1.33$\pm$0.32 and M$_K$=-3.44$\times$logP-2.20$\pm$0.29 and a distance modulus to the LMC of 18.55$\pm$0.32 in the {V}-band and 18.47$\pm$0.29 in the {K}-band giving an overall distance modulus to the LMC of 18.51$\pm$0.3. In the case of two important clusters we find that the {U-B:B-V} diagram in these clusters is not well fitted by the standard Main Sequence line. In one case, NGC7790, we find that the F stars show a {UV} excess which if caused by metallicity would imply Fe/H$\sim$-1.5; this is anomalously low compared to what is expected for young open clusters. In a second case, NGC6664, the {U-B:B-V} diagram shows too red {U-B} colours for the F stars which in this case would imply a higher than solar metallicity. If these effects {are} due to metallicity then it would imply that the Cepheid PL({V}) and PL({K}) zeropoints depend on metallicity according to $\frac{\delta M}{\delta Fe/H} \sim$0.66 in the sense that lower metallicity Cepheids are intrinsically fainter. Medium-high resolution spectroscopy for the main-sequence F stars in these two clusters is needed to determine if metallicity really is the cause or whether some other explanation applies. Please see http://star-www.dur.ac.uk:80/~fhoyle/papers.html for a version with all the figures correctly inserted.
|
[
{
"name": "ceph_short.tex",
"string": "\\documentstyle[harvard,epsfig]{mn}\n\n\n\\input{epsf}\n\n\\def\\mpc {h^{-1} {\\rm{Mpc}}}\n\\def\\kpc {h^{-1} {\\rm{kpc}}}\n\\def\\msun {{\\rm M}_{\\odot}}\n\\def\\ergs {{\\rm erg} \\, {\\rm s}^{-1}}\n\\def\\cm {{\\rm cm}}\n\\def\\kms {{\\rm km} \\, {\\rm s}^{-1}}\n\\def\\Hz {{\\rm Hz}}\n\\def\\yr {{\\rm yr}}\n\\def\\gyr {{\\rm Gyr}}\n\\def\\arcmin {{\\rm arcmin}}\n\\def\\U {{ U_{\\rm n} }}\n\\def\\G {{ G }}\n\\def\\R {{\\cal R }}\n\n\\def\\K {{ K }}\n\\def\\Omegab {{ \\Omega_{\\rm b} }}\n\n\\def\\lsim{\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}}}\n\\def\\gsim{\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}}}\n\n\\def\\and {\\rm {et al.} \\rm} % Roman Font is the MNRAS/ApJ style these days\n\\def\\etal {\\rm {et al.} \\rm}\n\\def\\rmd {\\rm d}\n\n\n\\begin{document}\n\n%TITLES AND AUTHORS\n\\title[Distances to Cepheid Open Clusters Via Optical and {\\it K}-Band Imaging]\n{Distances to Cepheid Open Clusters Via Optical and {\\it K}-Band Imaging}\n\n\n\n\\author[F. Hoyle et al ]\n{\nF. Hoyle$^{1,3}$, T. Shanks$^1$, N. R. Tanvir$^{2}$\n\\\\\n1 Department of Physics, Science Laboratories, South Road, Durham DH1 3LE\n\\\\\n2 Department of Astrophysical Sciences, University of Hertfordshire, College Lane, Hatfield, AL10 9AB\\\\\n3 email fiona.hoyle@durham.ac.uk\\\\\n}\n\n\\maketitle \n\n\n\n\\begin{abstract} We investigate the reddening and Main Sequence\nfitted distances to eleven young, Galactic open clusters that contain Cepheids.\nEach cluster contains or is associated with at least one Cepheid variable\nstar. Reddening to the clusters is estimated using the {\\it U-B:B-V}\ncolours of the OB stars and the distance modulus to the cluster\nis estimated via {\\it B-V:V} and {\\it V-K:V} colour-magnitude diagrams. By\nmain-sequence fitting we proceed to calibrate the Cepheid P-L relation and\nfind M$_V$=-2.81$\\times$logP-1.33$\\pm$0.32 and M$_K$=-3.44$\\times$logP-2.20$\\pm$0.29 and a distance modulus to the LMC of 18.55$\\pm$0.32 in the {\\it V}-band and 18.47$\\pm$0.29 in the {\\it K}-band giving an overall distance modulus to the\nLMC of 18.51$\\pm$0.3.\n\nIn the case of two important clusters we find that the {\\it U-B:B-V} diagram in these clusters is not well fitted by the standard Main Sequence line. In one\ncase, NGC7790, we find that the F stars show a {\\it UV} excess which if\ncaused by metallicity would imply Fe/H$\\sim$-1.5; this is anomalously low\ncompared to what is expected for young open clusters. In a second case,\nNGC6664, the {\\it U-B:B-V} diagram shows too red {\\it U-B} colours for the F stars which in this case would imply a higher than solar metallicity. If these effects\n{\\it are} due to metallicity then it would imply that the Cepheid PL({\\it V}) and PL({\\it K})\nzeropoints depend on metallicity according to $\\frac{\\delta M}{\\delta Fe/H} \\sim$0.66 in the sense that lower metallicity Cepheids are intrinsically\nfainter. Medium-high resolution spectroscopy for the main-sequence F stars \nin these two clusters is needed to determine if metallicity really is the\n cause or whether some other explanation applies.\n\nPlease see http://star-www.dur.ac.uk:80/~fhoyle/papers.html for a version with all the figures correctly inserted.\n\\end{abstract}\n\n\n\\begin{keywords} \nCepheids - open clusters - distance scale - magellanic clouds\n\\end{keywords}\n\n\\section{Introduction} Determining the value of Hubble's constant,\nH$_{\\circ}$, has been a challenge to astronomers since the discovery of the\nuniversal expansion in 1927. It is sometimes argued that we are now at the\nfine tuning stage and many measurements give values for H$_{\\circ}$ which\nlie between the hotly argued values of 50kms$^{-1}$Mpc$^{-1}$ (Sandage) and\n100kms$^{-1}$Mpc$^{-1}$ (de Vaucouleurs), e.g. \\citeasnoun{tanvir2} calculated\nH$_{\\circ}$=67$\\pm7$kms$^{-1}$Mpc$^{-1}$. However, many of these\nmeasurements are based on secondary indicator methods which in turn are\ndependent on the accuracy of primary indicators of distance such as the\nCepheid Period-Luminosity (P-L) relation. The well-studied LMC P-L relation\nis usually calibrated via the distance modulus to the LMC and the previously\naccepted value was around 18.50. However, this has been recently challenged\nin a paper by \\citeasnoun{FC} who determined the distance modulus to the LMC\nas 18.70$\\pm0.1$. This small difference in the distance modulus causes a\n10$\\%$ decrease in estimates of the Hubble's Constant. This discrepancy has\nfurther motivated us to check the Galactic zeropoint of the P-L relation. We\ndo this by checking the values of the distance modulus and reddening of the\n11 Galactic clusters that contain Cepheids via zero age main sequence\nfitting (ZAMS).\n\n\\begin{table*} \\begin{tabular}{lcllcccc} \\hline \nTelescope & Date & Cluster & Cepheid & Wavebands & Airmass & Exposure Time(s) & Photometric \\\\ \n\\hline\nJKT 1.0m & 17/9/97 & NGC6823 & SV Vul & {\\it UBV} & 1.22 & 2x150, 2x150, 2x150 & No \\\\ \nJKT 1.0m& 17/9/97 & & WZ Sgr & {\\it UB} & 1.71 & 3x150, 2x150 & No \\\\ \nJKT 1.0m& 18/9/97 & M25 & U Sgr & {\\it UBV} & 1.61 & 2x150, 1x150, 1x150 & No \\\\ \nJKT 1.0m& 18/9/97 & NGC129 & DL Cas & {\\it UBV} & 1.36 & 4x180, 2x180, 1x180 & No \\\\ \nJKT 1.0m& 20/9/97 & NGC6649 & V367 Sct & {\\it UBV} & 1.39 & 4x300, 2x180, 2x120 & No \\\\\nJKT 1.0m& 21/9/97 & NGC6664 & EV Sct & {\\it UBV} & 1.66 & 6x300, 8x300, 10x120 & Yes \\\\ \nJKT 1.0m& 21/9/97 & Trumpler 35 & RU Sct & {\\it UBV} & 1.22 & 1x180, 1x90,1x90 & Yes \\\\ \nJKT 1.0m& 21/9/97 & NGC7790 & CEa, CEb, CF Cas & {\\it UBV} & 1.29 & 6x300, 8x180, 10x120 &Yes \\\\ \n\\hline \nCTIO 0.9m & 24/9/98 & & WZ Sgr & {\\it UBV} &1.12 & 5x300, 4x150,4x90 & No \\\\ \nCTIO 0.9m& 27/9/98 & NGC6067 & V340 Nor, QZ Nor& {\\it UBV} & 1.36 & 3x300, 1x90, 1x90 & No \\\\ \nCTIO 0.9m& 27/9/98 & NGC6649 & V367 SCT & {\\it UBV} & 1.26 & 5x600, 1x60, 1x60 & No \\\\ \nCTIO 0.9m& 28/9/98 & Lynga 6 & TW Nor & {\\it UBV} & 1.51 & 4x300, 1x90, 1x90 & Yes \\\\ \nCTIO 0.9m& 28/9/98 & NGC6067 & V340 Nor, QZ Nor & {\\it UBV} & 1.36 & 3x300, 1x90, 1x90 & Yes \\\\ \nCTIO 0.9m&28/9/98 & vdBergh 1 & CV Mon & {\\it UBV} & 1.26 & 1x300, 1x90, 1x90 & Yes \\\\\nCTIO 0.9m& 28/9/98 & NGC6649 & V367 SCT & {\\it UBV} & 1.26 & 1x300, 1x60, 1x30 & Yes \\\\ \nCTIO 0.9m& 28/9/98 & M25 & U Sgr & {\\it UBV} & 1.27 & 1x60, 1x5, 1x5&Yes \\\\\n\\hline \nUKIRT 3.8m& 16/6/97 & NGC6649 & V367 Sct & {\\it K} & 1.40 & 60x2 & Yes \\\\\nUKIRT 3.8m& 17/6/97 & M25 & U Sgr & {\\it K} & 1.74 & 60x2 & Yes \\\\ \nUKIRT 3.8m& 17/6/97 & Trumpler 35 & RU Sct & {\\it K} & 1.64 & 60x2 & Yes \\\\ \nUKIRT 3.8m& 19/6/97 & NGC6664 & EV Sct & {\\it K} & 1.14 & 60x2 &Yes \\\\ \nUKIRT 3.8m& 19/6/97 & NGC6823 & SV Vul & {\\it K} & 1.03 & 60x2 &Yes \\\\ \\hline \nCalar Alto 3.5m& 18/8/97 & NGC129 & DL Cas & {\\it K$_{\\rm short}$} & 1.25 & 10x1.5 &Yes \\\\ \\hline \nWHT 4.2m& 1/9/96 & NGC7790 & CF, CEa, CEb Cas & {\\it K$_{\\rm short}$} & 1.205 & 50x1 & Yes \\\\ \\hline \n\\end{tabular} \n\\caption{Details of the observations. The airmass is the average airmass of the exposure and the exposure time is given in seconds for each of the wavebands in column 5.} \n\\label{tab:data} \n\\end{table*}\n\n\n\nPrevious work on measuring the reddening and distance to young open clusters\nwhich contain Cepheids via ZAMS fitting has been done using photoelectric\nand photographic measurements in optical wavebands. It is time consuming to\nobserve a large number of stars using photoelectric observations as each\nstar has to be observed individually. Photographic data can give relatively\ninaccurate magnitudes and colours. However, CCD's now make it possible to\nobserve a large number of stars in many different wavebands quickly and\naccurately. Although CCD's have already been used for open cluster studies\ne.g. \\citeasnoun{walkly6}, \\citeasnoun{walk6067}, \\citeasnoun{romeo}, these\nhave mainly been carried out in {\\it BVRI}. Recently CCD's with improved {\\it U}-band\nsensitivity have become available and {\\it U}-band CCD data is included in this\nstudy. Infra-red imaging detectors are also now available and although some\nof the detectors used here do not cover as wide an area as optical CCD's, observing the full extent of an open cluster with a mosaic of pointings is a practical proposition.\n\n\nUntil fairly recently, good quality infra-red measurements of the Cepheids\nthemselves were not available and the Cepheid P-L relation has been\nprimarily calibrated in the {\\it V}-band. Laney and Stobie (1993,1994) present\ninfra-red along with {\\it V}-band magnitudes for a large number of Southern\nHemisphere Galactic Cepheids. Using data in the literature to obtain values\nfor the distance modulus and reddening to the clusters they calibrated the\nCepheid P-L relation in the {\\it V} and {\\it K}-band. Any errors in the determination\nof the distance modulus and the reddening in the previous work would cause\nan error in the PL relation as determined by Laney and Stobie.\n\nThe layout of this paper is as follows. In section \\ref{sec:data} we present\nthe observational data and we test the accuracy of the photometry and\ncalibration of the data. In section \\ref{sec:method} we describe how the\nreddenings and distances to the open clusters are obtained and in section\n\\ref{sec:clus} we discuss each cluster individually. In section \\ref{sec:PL}\nwe use these values with the magnitudes of the Cepheids to calibrate the\nCepheid Period-Luminosity relation, In section \\ref{sec:interp} we discuss the implications of the results, particularly for the clusters whose {\\it U-B:B-V} diagrams do not appear to follow the canonical locus. In section \\ref{sec:conc} we draw conclusions.\n\n\n\n\\section{Data} \\label{sec:data}\n\n\\subsection{Observations} \\label{sec:obs}\n\nThe observations of the Galactic Open clusters were taken during five\nobserving runs on the JKT, UKIRT, at CTIO, at Calar Alto and on the WHT over\na two year period. The spread in declination of the clusters and the\nmulti-wavelength nature of the study meant that many different telescopes\nwere required.\n\n\\subsubsection{JKT}\n\nOptical imaging of eight open clusters was obtained during an observing run\nfrom the 16/9/1997 to the 22/9/1997. The observations were carried out using\nthe 1024$\\times$1024 Tektronix CCD with pixel scale of 0.33 arcsec pixel$^{-1}$. Typical\nseeing was around 1.3$^{\\prime\\prime}$. Short exposures of 5s in {\\it V}, 10s in {\\it B}\nand 20s in {\\it U} were observed for calibration purposes but the main imaging\nobservations were typically 6x120s in the {\\it V}-band, 6x180s in the {\\it B}-band\nand 6x300s in the {\\it U}-band. Due to the Southerly declination of some of the objects, they\nhad to be observed at high air mass. However these observations were\nnormally used to obtain relative photometry and calibration frames were\nobserved at as low an airmass as possible or during a later observing\nrun at CTIO. The pointings are given in Table \\ref{tab:point}. For calibration purposes, standard stars from \\citeasnoun{landolt} were used. On the one\nfully photometric night (21/9/97) six Landolt fields were observed at\nregular intervals throughout the night, most of these fields containing\nseveral standard stars.\n\n\n\\begin{table}\n\\begin{tabular}{ll} \nCluster & Pointing \\\\ \\hline\nNGC6649 & Star 19 in \\protect\\citeasnoun{MVDB} \\\\\nM25 & Star 95 in \\protect\\citeasnoun{sandm25} \\\\\nNGC6664 & Star 5 in \\protect\\citeasnoun{arpEV} \\\\\nWZ Sgr & WZ Sgr \\\\\nLynga 6 & TW Nor \\\\\nNGC6067 & Star 136 in \\protect\\citeasnoun{thack} \\\\\nvdBergh 1 & CV Mon \\\\\nTR35 & 5$^{\\prime\\prime}$ south of TR35 \\\\\nNGC6823 & Star j in \\protect\\citeasnoun{guetter} \\\\\nNGC129 & Star 113 in \\protect\\citeasnoun{arp129} \\\\ \nNGC7790 & Star E in \\protect\\citeasnoun{romeo} \\\\ \\hline\n\\end{tabular}\n\\caption{Approximate pointings for the clusters in the study in all wavebands.}\n\\label{tab:point}\n\\end{table}\n\n\\subsubsection{CTIO} The observations were made using the CTIO 0.9-m during\nan observing run from the 24/9/98 to 29/9/98. These observations were\ncarried out using the 2048x2048 Tek\\#3 CCD with pixel scale 0.384 arcsec\npixel$^{-1}$. The Tek\\#3 CCD has low readout noise (4 electrons) and good quantum efficiency in the {\\it U}-band. The average seeing during the observations was\naround 1.4$^{\\prime\\prime}$. Approximate pointings are again given in Table \\ref{tab:point}. Standard E-region fields from\n\\citeasnoun{graham} and standard stars from \\citeasnoun{landolt} were\nobserved throughout the night. The only photometric night was the 28/9/1998\nand on this night six E-region standard fields and one Landolt standard\nfield containing four standard stars were observed throughout the night.\nThree new clusters were observed at CTIO and further observations of\nclusters observed at the JKT were made in cases where the clusters had been observed in non-photometric conditions only.\n\n\n\n\\subsubsection{UKIRT}\n\nThe infra-red data was mostly observed at UKIRT during the four nights\n16-19/6/97 using the IRCAM3 near-IR imaging camera with a 256x256 detector.\nThe pixel scale used was 0.286 arcsec pixel$^{-1}$ giving a field of view of\n73$^{\\prime\\prime}$. To cover a sufficient area of each open cluster we therefore had to\nmosaic images. Generally a mosaic of 9x7 images was observed. Each image was\noverlapped by half in both the x and y direction so the final image had an\napproximate area of 6$^{\\prime}\\times$ 5$^{\\prime}$. Observations were generally taken using\nthe ND-STARE mode, where the array is reset and read immediately and then\nread again after the exposure which reduces the readout noise to 35e$^{-}$.\nThe exposures were 60$\\times$2 seconds and the centre of the mosaic is approximately in the same position as the corresponding optical frame. Standards from the UKIRT faint\nstandards list were observed throughout the nights. Around 10 standards\nwere observed on the three photometric nights, some of which were observed\nearly on in the night, half way through and at the end of the night. The\nseeing throughout the run was typically 0.6$^{\\prime\\prime}$. Observations\nwere also taken in the {\\it J} and {\\it H}-band.\n\n\\subsubsection{Calar Alto}\n\nNGC129 lies further north than the declination limit of UKIRT so infra red observations were instead\ntaken at Calar Alto during another observing run. Observations were done\nusing the Rockwell 1k$\\times$1k Hawaii detector with pixel scale 0.396 arcsec pixel$^{-1}$.\nThis gives a 6.$^{\\prime}$6 field of view so there was no need for the mosaicing\ntechnique used at UKIRT. The exposure time was 10$\\times$1.5s, the seeing\nwas better than 1$^{\\prime\\prime}$ and the exposure was centred on star 113 in \\citeasnoun{arp129}. Observations were made in the {\\it K$_{\\rm short}$}-band. UKIRT faint standards and standards of Hunt et al (1998) were observed for calibration.\n\n\\subsubsection{WHT} \n\nNGC7790 also lies further north than the UKIRT declination limit and so infra red observations were made on the 4.2m WHT during another observing run. The observations were made on the 1/9/1996. The WHIRCAM 256$\\times$256 detector which was\nsituated at the Nasmyth focus and behind the MARTINI instrument was used for the observations, without MARTINI tip-tilt in operation.\nThe WHIRCAM detector was the IRCAM detector previously used at\nUKIRT. The pixel size was 0.25 arcsec pixel$^{-1}$ and the field-of-view was therefore 64$^{\\prime\\prime}$, centred on star E in \\citeasnoun{romeo}. The {\\it K$_{\\rm short}$} filter was used and UKIRT faint standards were\nobserved for calibration.\n\n\n\\subsubsection*{} All the observations are summarized in Table\n\\ref{tab:data}. The date of each observation, the wavebands observed for\neach cluster and the airmass are given in columns 2, 5 and 6 respectively.\nColumn 7 gives the exposure time of the frames used for imaging. For some of\nthe clusters a calibration frame was observed at CTIO and the exposure time\nof these clusters are also given in column 7. Column 8 indicates where a\ncluster was observed on a photometric night and hence where an independent\nzero point was obtained.\n\n\n\\begin{table*}\n\\begin{tabular}{lccccl} \nCluster & {\\it U}$_{\\rm{obs}}$ - {\\it U}$_{\\rm{previous}}$ & {\\it B}$_{\\rm{obs}}$ - {\\it B}$_{\\rm{previous}}$ & {\\it V}$_{\\rm{obs}}$ - {\\it V}$_{\\rm{previous}}$ & N(stars) &Previous Work \\\\ \\hline\nNGC6649 & 0.02$\\pm$0.025 & 0.01$\\pm$0.01 & 0.00$\\pm$0.012 & 20 & \\protect\\citeasnoun{MVDB} \\\\\nM25 & -0.01$\\pm$0.03 & -0.03$\\pm$0.02 & -0.02$\\pm$0.02 & 21 & \\protect\\citeasnoun{sandm25} \\\\\nNGC6664 & -0.01$\\pm$0.012 & -0.02$\\pm$0.01 & -0.01$\\pm$0.015 & 15 & \\protect\\citeasnoun{arpEV} \\\\\nLynga 6 & 0.02$\\pm$0.02 & 0.02$\\pm$0.02 & -0.01$\\pm$0.015 & 17 & \\protect\\citeasnoun{moff} \\\\\nNGC6067 & 0.07$\\pm$0.08 & 0.04$\\pm$0.03 & -0.06$\\pm$0.04 & 6* & \\protect\\citeasnoun{thack} \\\\\nvdBergh 1 & 0.02$\\pm$0.015 & 0.00$\\pm$0.02 & -0.005$\\pm$0.008 & 24 & \\protect\\citeasnoun{arpCV} \\\\\nTrumpler 35 & -0.04$\\pm$0.02 & -0.01$\\pm$0.015 & 0.01$\\pm$0.015 & 15 & \\protect\\citeasnoun{hoag} \\\\\nNGC7790 & -0.009$\\pm$0.01 & -0.015$\\pm$0.008 & 0.01$\\pm$0.007 & 22 & \\protect\\citeasnoun{ped} \\\\ \\hline\nWZ Sgr & 0.00$\\pm$0.01 & 0.00$\\pm$0.011 & 0.00$\\pm$0.007 & 12 & \\protect\\citeasnoun{turnWZ}\\\\\nNGC6823 & 0.00$\\pm$0.01 & 0.00$\\pm$0.005 & 0.00$\\pm$0.004 & 13 & \\protect\\citeasnoun{guetter} \\\\\nNGC129 & 0.00$\\pm$0.03 & -0.005$\\pm$0.01 & -0.008$\\pm$0.01 & 13 & \\protect\\citeasnoun{turn129} \\\\ \\hline\n\\end{tabular}\n\\caption{Comparison of the photometry of this study with previous photoelectric data. Note there are 6 stars from \\protect\\citeasnoun{thack} in common with the {\\it B} and {\\it V}-band data but only 2 in common with the {\\it U}-band data. The last three clusters were observed on non-photometric nights only so the previous work was relied upon for calibration. There is a small offset between the zero point of NGC129 and \\protect\\citeasnoun{turn129} as only the brightest stars were used for calibration.}\n\\label{tab:resids}\n\\end{table*}\n\n\n\n\\subsection{Data Reduction} \n\\subsubsection{JKT} \n\\label{sec:jktred}\n\nRemoval of the bias introduced into the data and trimming of the frames to\nremove the overscan region was done on all the frames using the IRAF task\nCCDPROC. At least eight {\\it U}-band sky flats and six {\\it B} and {\\it V}-band sky\nflats were observed on each of the nights so a separate flat field was\ncreated for every night using a combination of dust and dawn sky flats. This\nwas created within FLATCOMBINE using a median combining algorithm and a\n3$\\sigma$ clipping to remove any cosmic rays. The residual gradient in the\nflat fields is around 1\\%. The task CCDPROC then applies the flat fields to\nall the images. The same flat fields were used in the reduction of the\nstandard star frames.\n\nMany images of the same cluster were observed. These were all combined\ntogether by aligning the images with linear shifts using the task IMSHIFT.\nGenerally these shifts were small (a few pixels either way) as the\nobservation were done one after each other and in some cases no shifts were\nrequired. The images were combined using IMCOMBINE and were averaged\ntogether using a 5$\\sigma$ clipping.\n\n\\subsubsection{CTIO} \n\\label{sec:ctiored}\n\nThe data was obtained at CTIO using the four amplifier readout mode. A\npackage called QUADPROC within IRAF corrects for the different bias levels\nin the four quadrants of the CCD. The frames were also trimmed using\nQUADPROC. In a previous observing run, \\citeasnoun{croomphot} had found that\nthere was a residual gradient of 5\\% in the dome flat fields so sky flats\nwere used. At least three sky flats were observed on each night in the {\\it B} and\n{\\it V}-band and at least 5 sky flats were observed on each night in the {\\it U}-band.\nThe resulting flat fields were flat to better then 1\\%. The equivalent\nversion of FLATCOMBINE in the QUAD package was used to median combine the\nflat fields using a 3$\\sigma$ clipping. The E-region standards and\nobservations of Landolt standards were again reduced in the same manner.\n\nMultiple images of the same cluster were again combined using IMCOMBINE with\nthe same settings as for the JKT data and where any offset shifts appeared\nbetween the data frames they were again corrected for using IMSHIFT.\n\n\n\\subsubsection{UKIRT} \n\\label{sec:ukirtred}\n\nThe UKIRT data was reduced using a program called STRED within the package\nIRCAMDR. This is a fairly automated routine which reads in the data frames,\nsubtracts of the dark count and creates a flat field frame by median\nfiltering the image frames. Then the program flat fields the dark subtracted\nobject images, corrects for any bad pixels and finally creates a mosaic.\nAll the data frames were median combined to create a flat field for each night. There was no evidence of a large scale gradient greater than about 1\\% in the flat fields.\n\nTo create the final image, all the individual frames have to be mosaiced\ntogether. STRED reads in the offset from the data header, however, these\noffsets were not accurate enough. By creating a separate offsets\nfile the mosaicing could be done more accurately. To create the offsets file, one of the corner frames was fixed and the offset required for the neighbour frame were found by eye. This was built up over the whole frame, however once one offset had been determined, all the other offsets in the x and y direction from frame to frame were the same. The offsets for the standard stars were more accurate and\ncould be used to create the mosaic.\n\n\\subsubsection{Calar Alto} \\label{sec:calarred}\n\nBasic IRAF routines such as IMCOMBINE and IMARITH were used to reduce the\nCalar Alto data. A flat field was created by median combining all the data\nframes. A more detailed description of the data reduction can be found in\n\\citeasnoun{henry}. We found that the best results were obtained by first\nsubtracting a sky frame from each image, as follows: A sky frame for each\nindividual image was created using IMCOMBINE with a $5\\sigma$ clipping to\nmedian filter four data frames that were local in time to the image frame to\ncreate a sky frame for each data frame. The sky frame was then subtracted\noff the image before IMCOMBINE was used again to combine the data frames,\nforming one final image frame.\n\n\\subsubsection{WHT} \\label{sec:whtred}\n\nThe data from the WHT was reduced for us as part of another project.\nDome flats were used to flat field the data and this was divided into the science frame using IMARITH. Sky subtraction was also required. A sky frame was created by combining dedicated sky frames observed locally in time to the science frame and this was then subtracted from the science frame also using IMARITH.\n\n\n\\begin{figure*} \\begin{tabular}{ccc} \n{\\epsfxsize=5.6truecm\\epsfysize=6truecm\\epsfbox[60 170 550 620]{air_21_paper.ps}} &\n{\\epsfxsize=5.6truecm\\epsfysize=6truecm\\epsfbox[60 170 550 620]{col_21_paper.ps}} & \n{\\epsfxsize=5.6truecm\\epsfysize=6truecm\\epsfbox[60 170 550 620]{zero_21_paper.ps}} \\\\ \n\\end{tabular}\n\\caption{The airmass coefficient, colour equation and zero point, with the\ncolour equation and airmass correction applied, for the {\\it U}, {\\it B} and {\\it V} wavebands for the data from JKT. The open circles in the airmass plot show the standard stars observed at one airmass only.} \n\\label{fig:zerojkt} \n\\end{figure*}\n\n\n\n\\subsection{Image Alignment}\n\nIn order to be able to produce colour-magnitude diagrams, the magnitude of\neach star in all the different wavebands is required. To do this, all the\nframes need to be aligned. Aligning the optical frames was easy as there\nwere only linear shifts between each waveband. Rather than altering the\ndata, the alignment was just done by applying small corrections to the x and\ny positions of the stars. Aligning the optical data with the {\\it K}-band data\nwas more complicated though as there were shifts, shears and rotations\nbetween the frames. We used the IRAF routine GEOMAP to calculate the best\nspatial transformation function between any two images thus allowing the\noptical and {\\it K}-band data to be aligned. The mapping was only used to\ntransform coordinates and we did not perform photometry on the resampled\nimages.\n\n\n\\subsection{Photometric Calibration} \\label{sec:cal}\n\n\n\\begin{figure*} \n\\begin{tabular}{ccc} \n{\\epsfxsize5.6truecm \\epsfysize=6truecm\n\\epsfbox[60 170 550 620]{air_ctio_paper.ps}} &\n{\\epsfxsize=5.6truecm \\epsfysize=6truecm \\epsfbox[60 170 550\n620]{col_ctio_paper.ps}} & \n{\\epsfxsize=5.6truecm\\epsfysize=6truecm \\epsfbox[60 170 550\n620]{zero_ctio_paper.ps}} \\\\ \n\\end{tabular}\n\\caption{The airmass coefficient, colour equation and zero point, with the\ncolour equation and airmass correction applied, for the {\\it U}, {\\it B} and {\\it V}\nwavebands for the data from CTIO. The open circles in the airmass plot show the standard stars observed at one airmass only.} \n\\label{fig:zeroctio} \n\\end{figure*}\n\n\n\n\n\\subsubsection{JKT} \n\\label{sec:jktcal}\n\nObservations of the standard stars of \\citeasnoun{landolt} were taken at\nregular intervals on each of the six nights at the JKT. The CCD frames of\nthe standard stars were reduced in the same manner as the data frames with\nthe same flat fields etc as discussed in section \\ref{sec:jktred}. The\naperture size used to measure the magnitude of the standard stars was\n15$^{\\prime\\prime}$, large enough to determine accurately the total\nmagnitude of the star but not so large that sky subtraction errors dominate\nthe magnitude measurement.\n\nOnly the sixth night (21/9/97) was fully photometric and the zero point,\nairmass coefficient and colour equation are shown in Figure\n\\ref{fig:zerojkt} and given below with the rms scatter. \n\n\n\\begin{displaymath}\nU_{\\rm{jkt}} = U_{\\rm{ldt}} + 3.76 + 0.49sec(z) - 0.063(U_{\\rm{ldt}}-B_{\\rm{ldt}}) \\pm0.038\n\\end{displaymath}\n\\begin{displaymath}\nB_{\\rm{jkt}} = B_{\\rm{ldt}} + 2.00 + 0.27sec(z) - 0.013(B_{\\rm{ldt}}-V_{\\rm{ldt}}) \\pm0.033\n\\end{displaymath} \n\\begin{displaymath}\nV_{\\rm{jkt}} = V_{\\rm{ldt}} + 1.99 + 0.19sec(z) \\pm0.026\n\\end{displaymath}\nwhere the subscript ldt stands for the Landolt\nstandard star magnitude and the subscript jkt stands for the instrumental\nmagnitude, z represents the zenith distance. The errors on the airmass are $\\pm$0.0025, 0.0012 and 0.0008 in the {\\it U, B} and {\\it V} and the errors on the colour equation are $\\pm$0.0007 and 0.0005 in the {\\it U} and {\\it B}-bands respectively. There are no Landolt magnitudes available for the data frames so the colour term has to be translated in instrumental magnitudes. The colour term is negligible in the {\\it V}-band calibration so the instrumental {\\it V}-band magnitudes come directly from the above equations. The {\\it B}$_{\\rm ldt}$-{\\it V}$_{\\rm ldt}$ and {\\it U}$_{\\rm ldt}$-{\\it B}$_{\\rm ldt}$ colours are given by\n\\begin{displaymath}\n(B_{\\rm{ldt}}-V_{\\rm{ldt}}) = 1.013((B_{\\rm{jkt}}-V_{\\rm{jkt}}) - 0.01 - 0.08sec(z))\n\\end{displaymath}\n\\begin{displaymath}\n(U_{\\rm{ldt}}-B_{\\rm{ldt}}) = 1.067((U_{\\rm{jkt}}-B_{\\rm{jkt}}) - 1.76 - 0.22sec(z))\n\\end{displaymath}\nwe assume that the contribution from the colour term in the calibration of the {\\it B}-band is negligible when determining the {\\it U}$_{\\rm ldt}$-{\\it B}$_{\\rm ldt}$ colour. These exact colour terms are used to correct the instrumental magnitudes in order to make the colour-colour and colour-magnitude diagrams in Figures \\ref{fig:ubbv}, \\ref{fig:bv} and \\ref{fig:vk}.\n\n\nThe clusters NGC7790, NGC6664 and Trumpler 35 were observed on the one\nphotometric night. Short exposure observations of NGC6649 and M25 were made\nat CTIO in order to obtain an independent zero point for these frames.\nAround 20 bright (brighter than {\\it V}$\\sim$15), fairly uncrowded, unsaturated stars were\ntaken as standard stars to identify the relation between the zero point from\nthe JKT data and from CTIO data to an accuracy of 0.01 mags. The agreement between the zero point found via this method and previous, photoelectric calibrations is good (see Table \\ref{tab:resids}) with only small offsets in each case. The remaining three clusters were observed in non-photometric conditions only so previous work had to be relied upon\nfor the calibration. For NGC6823 the photoelectric observations from Table 1\nof \\citeasnoun{guetter} were used. Some of these stars were saturated on the\nCCD frame and the area of overlap between the two images was not identical but\nthirteen stars were suitable for calibration purposes. \\citeasnoun{guetter} compares his photoelectric data with that of previous work and finds good agreement. Two sources of photoelectric data are available for the cluster NGC129, \\cite{arp129} and \\cite{turn129}. There are 9 stars in common with the Arp photometry and 13 stars in common with the Turner photometry. For these samples of stars, we find that the {\\it U}-band zero point obtained from Arp is 0.04$\\pm$0.03 mags brighter than that of Turner. In the {\\it B}-band the difference is 0.02$\\pm0.01$ mags in the sense that Arp is brighter than Turner and there is 0.01$\\pm0.01$ mag difference in the same sense in the {\\it V}-band. These differences are mainly caused by the stars in the sample with {\\it V} fainter than 14 mag. We therefore use the average of the brightest two stars from Arp and Turner to calibrate NGC129. The zero point from this method agrees very well with the zero point obtained using Turner's photometry, which is shown in Figure \\ref{fig:other}. Finally the photoelectric work of \\citeasnoun{turnWZ} was used to calibrate the cluster containing the Cepheid WZ Sgr. \n\nThe magnitudes of the stars on the data frames on all nights were measured using a 5$^{\\prime\\prime}$ aperture. The standard star magnitudes were measured using a 15$^{\\prime\\prime}$ aperture so for the photometric night (21/9/97) an aperture correction had to be applied to the data. The correction in the {\\it U}, {\\it B} and\n{\\it V}-bands are -0.165$\\pm$0.02, -0.11$\\pm$0.025 and -0.11$\\pm$0.024 magnitudes respectively as determined by comparing the magnitudes of the standard stars with a 5$^{\\prime\\prime}$ and 15$^{\\prime\\prime}$ aperture. No aperture correction was required for those cases where the calibration was tied to a photometric sequence in the cluster itself. For the clusters NGC6649 and M25 the CTIO aperture correction was required.\n\n\n\n\\subsubsection{CTIO} \n\\label{sec:ctiocal}\n\nE-region standards from \\citeasnoun{graham} and Landolt standards were\nobserved for the photometric calibration of the optical CTIO data.\n\\citeasnoun{menz} compared the zero points and colour differences found from\nusing the two different standard star studies and found the offsets between\nthe two to be small, 0.004$\\pm$0.0095 offset in the sense E regions -\nLandolt. and similar sized offsets in the {\\it U-B} and {\\it B-V} colours. Any\noffsets are within the quoted error. As before, these standard frames were\nreduced in the same manner as the data frames and a 15$^{\\prime\\prime}$\naperture was used to determine the magnitude. Again, only one night was\nphotometric and this was the last night (28/9/98). The zero points, airmass\ncoefficients and colour equations for each waveband, {\\it U}, {\\it B} and {\\it V}, are\nshown in Figure \\ref{fig:zeroctio} and given below again with the rms scatter.\n\n\\begin{displaymath} \nU_{\\rm{ctio}} = U_{\\rm{std}} + 4.61 + 0.47sec(z) -\n0.036(U_{{\\rm std}}-B_{{\\rm std}}) \\pm0.035 \n\\end{displaymath}\n\\begin{displaymath} \nB_{\\rm{ctio}} = B_{\\rm{std}} + 3.15 + 0.21sec(z) +\n0.099(B_{\\rm{std}}-V_{\\rm{std}}) \\pm0.015 \n\\end{displaymath}\n\\begin{displaymath} V_{\\rm{ctio}} = V_{\\rm{std}} + 2.93 + 0.12sec(z) -\n0.018(B_{\\rm{std}}-V_{\\rm{std}}) \\pm0.007 \n\\end{displaymath} \nwhere the subscript std stands for the E-field or Landolt standard\nmagnitude and ctio stands for the instrumental magnitude. The errors on the airmass are $\\pm$0.003, 0.0011 and 0.0009 in the {\\it U, B} and {\\it V} and the errors on the colour equation are $\\pm$0.0006, 0.0007 and 0.001 in the {\\it U}, {\\it B} and {\\it V}-bands respectively. There is\ngenerally good agreement between the values for the airmass coefficients\nand colour equations found in this work and in \\citeasnoun{croomphot}. Again, a 5$^{\\prime\\prime}$ aperture was used for the data frames so an aperture correction for the photometric night was required. The aperture corrections, in the {\\it U}, {\\it B} and {\\it V}-bands, are -0.17$\\pm$0.03, -0.17$\\pm$0.02 and -0.13$\\pm$0.02 magnitudes respectively. Again, for the data frames the colour terms have to be found in terms of CCD magnitudes rather than standard magnitudes. The colour term in the {\\it B}-band is in this case non-negligible so {\\it V}-band magnitudes have to be used in the {\\it U}-band calibration. The standard colours are given by\n\\begin{displaymath}\n(B_{\\rm{std}}-V_{\\rm{std}}) = 1.088((B_{\\rm{ctio}}-V_{\\rm{ctio}}) - 0.22 - 0.09sec(z))\n\\end{displaymath} \n\\begin{displaymath}\n\\begin{array}{r}\\!\\!\\!(U_{\\rm{std}}-B_{\\rm{std}}) = 1.037((U_{\\rm{ctio}}-B_{\\rm{ctio}}) - 1.46 - 0.26sec(z) \\\\ + 0.099(1.088((B_{\\rm{ctio}}-V_{\\rm{ctio}}) - 0.22 - 0.09sec(z)))) \\end{array}\n\\end{displaymath} \nagain these colours are used in all to calculate the instrumental magnitudes which are used in the colour-colour and colour-magnitude diagrams in Figures \\ref{fig:ubbv}, \\ref{fig:bv} and \\ref{fig:vk}.\n\nThis night provided independent zero points for the clusters NGC6067, Lynga\n6 and vdBergh1, which were not observed at JKT, and also provided a zero\npoint for NGC6649 and M25, which were only observed in non-photometric\nconditions at the JKT.\n\n\n\n\\subsubsection{UKIRT} \\label{sec:ukirtcal}\n\nThe standards stars observed at UKIRT were taken from the faint standards\nlist available from the UKIRT Web page. The standard stars were observed as\na mosaic of five frames, a central frame with an overlapping frame in each\ndirection. A 15$^{\\prime\\prime}$ aperture was used to determine the\nmagnitude of the standard. Three of the nights were photometric, night1,2\nand 4 (16,17,19/6/97) so all the clusters were observed in photometric\nconditions.\n\nThe zero points for each of the nights are shown in Figure \\ref{fig:kzero}\nand the airmass coefficient and colour equation are shown in Figure\n\\ref{fig:kaircol} and given below for each night.\n\n\\noindent16/06/1997; \n\n\\begin{displaymath} K_{\\rm{U}} = K_{\\rm{std}} + 6.94 -\n0.082sec(z) + 0.005(J_{\\rm{std}} - K_{\\rm{std}}) \\pm0.024 \\end{displaymath}\n17/06/1997; \n\\begin{displaymath} K_{\\rm{U}} = K_{\\rm{std}} + 6.95 - 0.082sec(z)\n+ 0.005(J_{\\rm{std}} - K_{\\rm{std}}) \\pm0.033 \\end{displaymath} \n18/06/1997;\n\\begin{displaymath} K_{\\rm{U}} = K_{\\rm{std}} + 6.96 - 0.082sec(z) +\n0.005(J_{\\rm{std}} - K_{\\rm{std}}) \\pm0.037 \\end{displaymath} \nwhere again the subscript std refers to the standard stars. The\ndifference between each of the calibrations is just a small change in the\nzero point. The error on the airmass coefficient is $\\pm$0.002 and the error on the colour term is $\\pm$0.0003. The airmass coefficient agrees well with the values given in \\citeasnoun{kris} and those found on the UKIRT web page. An aperture correction of -0.19$\\pm$0.018 magnitudes is required for the magnitudes of the data frames measured using a 5$^{\\prime\\prime}$ aperture on each night. As the colour term is very small, it was assumed negligible in the {\\it V-K:V} CMD's.\n\n\\subsubsection{Calar Alto and WHT} UKIRT faint standards and \\citeasnoun{hunt}\nstandards were observed in order to calibrate the Calar Alto and WHT data.\nThe calibration was provided for us by Nigel Metcalfe as part of another project (see McCracken, Metcalfe and Shanks, in preparation, for more details). \n\n\\begin{figure} \n{\\epsfxsize=6truecm \\epsfysize=6truecm \\epsfbox[60 170 550\n620]{kzero_paper.ps}} \n\\caption{The {\\it K}-band zero\npoints for the three photometric nights at UKIRT. The top panel is for night\n1, (16/8/97, filled circles), centre for night 2 (17/8/97, open circles) and the lower panel for night 4 (19/8/97, filled triangles).} \n\\label{fig:kzero} \n\\end{figure}\n\n\\begin{figure} \n{\\epsfxsize=6truecm \\epsfysize=6truecm \\epsfbox[60 170 550\n620]{kaircol_paper.ps}} \n\\caption{The airmass\ncorrection and colour equation for the {\\it K}-band data taken at UKIRT. The symbols are the same as in Figure \\ref{fig:kzero}.}\n\\label{fig:kaircol} \n\\end{figure}\n\n\n\n%The {\\it K}-band data was used as a consistency check rather than to actually determine the reddening or distance modulus so the calibration is not shown in detail here.\n\n\n\n\\begin{figure}\n{\\epsfxsize=8.5truecm \\epsfysize=8.5truecm \n\\epsfbox[60 170 550 620]{mean_var_cal.ps}}\n\\caption\n{An indication of the accuracy of the {\\it U}-band photometry based on results for simulated stars (see text for details). The large errorbar indicates the total error (Poisson errors, crowding errors and also read noise), the smaller, wider errorbar indicates the Poisson error found as described in section \\ref{sec:photometry}, paragraph 2. The total error at 17th magnitude is only $\\pm$0.036 magnitudes and down to 18th magnitude the error is less than $\\pm$0.1 magnitudes.}\n\\label{fig:phot}\n\\end{figure}\n \n\n\n\\subsection{Photometry}\n\\label{sec:photometry}\n\nAutomated aperture photometry was done using PHOT within IRAF's DAOPHOT\npackage. A small, 5$^{\\prime\\prime}$, aperture was used to minimise any\ncrowding problems. PHOT was also used to obtain the magnitudes of the\nstandard stars, the magnitude of each standard star was measured individually using a 15$^{\\prime\\prime}$ aperture.\n\nFirst we establish the depth of the photometry. We define the limiting depth\nof the observations to be where the Poisson error in the electron counts is\n5\\%, i.e. when $\\sqrt{N_{obj} + N_{sky}}/N_{obj}$ = 0.05. The depth of the JKT data is 19.9 mags for a 1200s {\\it U}-band exposure, 19.7 mags for a 360s {\\it B}-band exposure and 20.4 mags for a 240s {\\it V}-band exposure. The depths of the CTIO data are similar. The depth of a typical {\\it K}-band exposure of 120s observed was around 19.2 magnitudes.\n\nThe accuracy of the zero points were tested by comparing our photometry to photoelectric observations in the literature. Table \\ref{tab:resids} shows the residual when our magnitudes for typically 10\nstars from the calibrated data frames (see section \\ref{sec:cal}) are\ncompared to the magnitudes found from previous photoelectric studies. In all\ncases except for NGC6067, the residual is small (less than 0.03 mags).\nThe errors in the comparison of our magnitudes to other photoelectric observations are slightly smaller than the errors in the zero point found from the standard stars in this work. This is probably because the stars considered when comparing the previous work were slightly brighter. \nUnfortunately for the cluster NGC6067 there is little photoelectric data\navailable and only 6 stars can be compared in the {\\it B} and {\\it V}-bands due to saturation and different parts of the cluster being observed. In the\n{\\it U}-band there are only 2 stars in common. Given that the offsets between\nthe work here and previous work are small for all the other clusters, we\nassume that the zero point obtained for NGC6067 is accurate. \n\n\nFinally, we test how the crowding of the field effects the photometry.\nTwenty ``simulated stars'' of each magnitude shown in Figure \\ref{fig:phot} were placed in a 300s {\\it U}-band image of the cluster M25 then\nPHOT was used to determine how well the magnitudes could be recovered. Shown in\nFigure \\ref{fig:phot} are the results. The x axis shows the true magnitude of the stars and the y\naxis shows the deviation of the mean magnitude from the true magnitude. This plot shows the total error (Poisson errors, read noise and also errors due to crowding) at each magnitude (full errorbar) and the Poisson error, found as described above, as the smaller, wider, errorbar. The total error for this exposure of 300s is less than 0.03 magnitudes down to {\\it U}=17 mags. For a {\\it U}-band exposure of 1800s, assuming that the crowding errors remain the same and then\ncorrecting the error shown in Figure \\ref{fig:phot} for the reduced Poisson error, the total error is estimated to be $\\pm$0.17 at {\\it U}=20, reducing to $\\pm$0.08 at 18th mag and $\\pm$0.03 at 17th mag. Figure \\ref{fig:phot} also shows that although the errors increase for fainter magnitudes there is no systematic trend. \n\n\n\n\n\n\\section{Reddening and Distance} \n\\label{sec:rd}\n\n\\subsection{Method} \n\\label{sec:method}\n\nIn order to work out the Cepheid Period-Luminosity (P-L) relation, the\ndistance to the cluster needs to be known. However, there is generally\nsignificant dust absorption along the line of sight to the cluster which\nmust be corrected for. The method for determining both of these parameters\nis done via ZAMS fitting to colour-magnitude and colour-colour diagrams. The\nZAMS used in this study is a combination of \\citeasnoun{allen} for the\noptical data, the intrinsic colours of near IR-band stars from the UKIRT Web\npage \\cite{tok} for the {\\it K}-band data and nearby local stars taken from the\nStrasbourg Catalogue to give an indication of the acceptable spread in the\nZAMS. The ZAMS of \\citeasnoun{turnZAMS} and \\citeasnoun{mermZAMS} have also\nbeen tested and would give the same results as the ZAMS of\n\\citeasnoun{allen}.\n\nThe reddening is obtained using the {\\it U-B:B-V} diagram which is \nindependent of the distance to the cluster. The {\\it U-B:B-V} diagram has always\nbeen the favoured method for estimating the reddening; however, previously,\nthe accuracy of the reddening determination was limited by the depth of the\n{\\it U}-band photoelectric photometry. The improved U sensitivity of the current\ngeneration of CCD's should allow a potential improvement in the accuracy of\nthe reddening estimated from {\\it U-B:B-V} diagrams and this is the route we\nhave adopted here.\n\nThe reddening law assumed in this work is from \\citeasnoun{sharp}.\n\n\\begin{equation} \\frac{{\\rm E}(U-B)}{{\\rm E}(B-V)} = 0.72 + 0.05{\\rm E}(B-V)\n\\end{equation}\n\nWe choose to fit the ZAMS to the ridge-line of the O and B stars \nrather than the least reddened envelope, because it helps take\naccount of differential reddening in some of the clusters. There is evidence\nfor differential reddening in NGC6823 and TR35 (see Figure \\ref{fig:ubbv})\nas the main sequence in the colour-colour diagram is substantially\nbroadened. By fitting to the centre of the data we measure the average\nreddening for the cluster which we can then apply to colour-magnitude\ndiagrams which are uncorrected for differential absorption to obtain\ndistances. To determine the error we measure the standard deviation of\nthe O and B type stars from the ZAMS via least squares fitting. \nThe errors quoted on the values for the reddening (see Table \\ref{tab:rd}) are typically 0.1 mags and include any error in the calibration. In quite a\nfew cases, the ZAMS does not fit the colour-colour data well over the whole\nrange of B-V colours. This is particularly problematic in some cases and\nthese cases are discussed below. There is also the problem that the Cepheid\ncould have a different reddening to the cluster, caused by differential\nreddening across the cluster or by the location of the Cepheid away from the\ncluster. This is discussed further in section \\ref{sec:PL}.\n\nWe use both the {\\it B-V:V} and {\\it V-K:V} colour-magnitude diagrams to determine\nthe cluster distance. {\\it V-K:V} diagrams have the advantage that the slope of\nthe ZAMS is flatter than at {\\it B-V:V}, possibly allowing more accurate distance estimates but {\\it V-K:V} diagrams are available for only 7 of the clusters and the scatter in these diagrams is greater than in the {\\it B-V:V} CMD's, particularly in the case of NGC6664 as there were difficulties aligning the {\\it V} and {\\it K} frames due to a shift in the telescope position half way through creating the {\\it K}-band mosaic for this cluster. There are also few points on the {\\it V-K:V} diagram for NGC7790 due to the small size of the IRCAM detector. This diagram was also made for us before the {\\it V}-band observations for this cluster were available as a test for the feasibility of this project. The {\\it V}-band data therefore comes from \\citeasnoun{romeo}, however there is a good match between our {\\it V}-band data and that of Romeo et al. (see Figure \\ref{fig:7790comp}). The distance modulus which best fits the {\\it B-V:V} CMD around the position of the A0V\nstars, using the method of least squares, is taken to be the distance\nmodulus, $\\mu_0$, of the cluster. The errors on the distance modulus were\nfound by measuring the standard deviation away from the ZAMS of A0V type\nstars over the range -0.1$ \\lsim ${\\it B-V} $\\lsim$0.1 in the dereddened ZAMS which covers a range of approximately 4 mags in {\\it V}. The distance modulus found from the {\\it B-V:V} CMD is then checked against the {\\it V-K:V} CMD for the 7 clusters with such a CMD for consistency. In all cases the distance modulus found from the {\\it B-V:V} diagram was consistent with the {\\it V-K:V} CMD within the errors.\n\nNo attempt has been made to remove foreground and background stars. Only stars\nwhich lie clearly off the main sequence (off in {\\it B-V} by more than 1 mag for\nexample) were removed. There is no clear recipe for how to remove the\ncontaminating stars from the colour-magnitude and colour-colour diagrams so\nthe data shown in Figures \\ref{fig:ubbv}, \\ref{fig:bv} and \\ref{fig:vk}\ncontain non-cluster members. The observations in this study only cover a field-of-view of $\\sim 6^{\\prime}$ and are pointed at the cluster centre so contamination may not be as much of an issue as for the wider field photographic plates. As an example, we discuss this further for the cluster NGC7790 in section \\ref{sec:clus}.\n\n\nAll of the {\\it U-B:B-V} diagrams, the {\\it B-V:V} and the {\\it V-K:V} diagrams are given\nin Figures \\ref{fig:ubbv}, \\ref{fig:bv} and \\ref{fig:vk} respectively and\nour results for the reddening and distances obtained are given in Table\n\\ref{tab:rd} together with previous results as summarised by\n\\citeasnoun{LS2}. All of the clusters are individually discussed below.\n\n\n\\begin{table} \\begin{tabular}{lcccc} Cluster & E({\\it B-V})$_{\\rm{clus}}$ &\nE({\\it B-V})$_{\\rm{clus}}^{\\rm{LS}}$ & $\\mu_{\\circ}$ & $\\mu_{\\circ}^{\\rm{LS}}$\\\\\n\\hline \nNGC6649 & 1.37$\\pm0.07$ & 1.35 & 11.22$\\pm0.32$ & 11.28 \\\\ \nM25 &0.49$\\pm0.08$ & 0.48 & 9.05$\\pm0.43$ & 9.03 \\\\ \nNGC6664 & 0.66$\\pm0.08$ &0.64 & 11.01$\\pm0.37$ & 10.40 \\\\ \nWZ Sgr & 0.56$\\pm0.20$ & 0.57&11.15$\\pm0.49$ & 11.22 \\\\ \nLynga 6 & 1.36$\\pm0.17$ & 1.34 & 11.1$\\pm0.45$&11.43 \\\\ \nNGC6067 & 0.42$\\pm0.02$ & 0.35 & 11.17$\\pm0.35$ & 11.13 \\\\\nvdBergh1& 0.90$\\pm0.18$ & 0.77 & 11.4$\\pm0.65$ & 11.36 \\\\ \nTrumpler 35&1.19$\\pm0.10$ & 0.92 & 11.3$\\pm0.53$ & 11.56 \\\\ \nNGC6823 & 0.85$\\pm0.09$&0.75 & 11.20$\\pm0.55$ & 11.79 \\\\ \nNGC129 & 0.57$\\pm0.06$ & 0.53*&10.90$\\pm0.37$ & 11.24* \\\\ \nNGC7790 & 0.59$\\pm0.05$ & 0.64* &12.70$\\pm0.43$& 12.39* \\\\ \\hline \n\\end{tabular}\n\n\\caption{The values in the above Table come from the work here and from\n\\protect\\citeasnoun{LS1}. The values marked * come from\n\\protect\\citeasnoun{FW} as the clusters NGC129 and NGC7790 are not included\nin the studies of Laney and Stobie.} \\label{tab:rd} \\end{table}\n\n\\subsection{Discussion of Individual Clusters} \n\\label{sec:clus}\n\n\n\\textbf{NGC6649} has been studied previously by, for example, \\citeasnoun{MVDB} and \\citeasnoun{walk6649}. The agreement between the photometry of this study and the photoelectric data of \\citeasnoun{MVDB} is good (see Table \\ref{tab:resids}), with only small offsets in the {\\it U} and {\\it B}-band. \\citeasnoun{MVDB} find E({\\it B-V}) = 1.37 (no quoted error) for the reddening towards the cluster. The distance modulus $\\mu_{\\circ}$ from \\citeasnoun{MVDB} is 11.15$\\pm$0.7. \n\n\\citeasnoun{turn6649} uses the photometry of \\citeasnoun{MVDB} and that of \\citeasnoun{talb} to study NGC6649 and find the cluster suffers from differential reddening. However for stars close to the cluster centre a value of E({\\it B-V})=1.38 is appropriate. The distance modulus is found to be 11.06$\\pm$0.03 when individual stars are dereddened.\n\n\\citeasnoun{walk6649} used {\\it U, B} and {\\it V}-band CCD data to study NGC6649. Agreement between the photoelectric data of \\citeasnoun{MVDB} and \\citeasnoun{walk6649} was found to be better than 0.03 mags in the {\\it V}-band. \\citeasnoun{walk6649} do not measure the reddening of the cluster due to the claims of differential reddening by Turner. \\citeasnoun{walk6649} deredden each star individually to find a distance modulus of 11.00$\\pm$0.15.\n\nRather than correct for differential reddening, we fit the ZAMS line to the centre of the {\\it U-B:B-V} and {\\it B-V:V} diagrams to try and measure the average values of the reddening. We find E({\\it B-V})=1.37$\\pm$0.07 and $\\mu_{\\circ}$=11.22$\\pm$0.32, both values are consistent with previous work.\nThe distance modulus used in \\citeasnoun{LS2} is slightly higher, 11.278 and the reddening,E({\\it B-V}), slightly lower, 1.35 but again these values are well within the errors. \n\n\\citeasnoun{barrell} found the radial velocity of the Cepheid V367 Sct to be -20$\\pm$6 kms$^{-1}$ and the radial velocity of the cluster NGC6649 to be -14$\\pm$5 kms$^{-1}$. This is taken to be evidence for the cluster membership of the Cepheid.\n\n\\textbf{M25} The photometry used in this study and that of \\citeasnoun{sandm25} is compared in Figure \\ref{fig:zero} and \\ref{fig:col}. The agreement in all the wavebands is good, less that 0.03 mags different from that of Sandage.\n\nM25 has a {\\it U-B:B-V} diagram where the ZAMS fit is good over a wide range of {\\it B-V} colours. \nThe value for E({\\it B-V}) of 0.49$\\pm$0.08 is in excellent agreement with the work of \\citeasnoun{sandm25} who obtained 0.49$\\pm$0.05. \\citeasnoun{johnm25} stated that E({\\it B-V}) lay in the range 0.4 to 0.56 and \\citeasnoun{SVDB} obtained E({\\it B-V})=0.51$\\pm$0.01. \n\nThe distance modulus obtained for this cluster is 9.05$\\pm0.43$ which is slightly higher than that of Sandage, who found $\\mu_{\\circ}$=8.78$\\pm$0.15. The difference in the distance modulus is probably due to where to fit was made, we fit to the A type stars to obtain 9.05$\\pm0.43$, Sandage's work contains brighter stars which may lie slightly off the main sequence. Wampler et al (1960) obtained 9.08$\\pm0.2$ and \\citeasnoun{SVDB} found $\\mu_{\\circ}$= 9.0$\\pm$0.3 which agree well. There is also good agreement between our values and those used by \\citeasnoun{LS2}, as given in Table \\ref{tab:rd}.\n\n%explained by the ZAMS fit in \\citeasnoun{sandm25} being slightly higher than the ZAMS in this work in the {\\it B-V:V} CMD.\n\n\\citeasnoun{feast} studied M25 by measuring radial velocities and spectra for stars within M25. He obtained radial velocities of around 4 kms$^{-1}$ for the\nCepheid U Sgr and for 35 stars in the cluster M25 indicating that U Sgr is a member of M25.\n\n%although the error on the measurement of the 35 stars was also $\\pm4$kms$^{-1}$, \n\n\n\n\\textbf{NGC6664} has been studied previously by \\citeasnoun{arpEV}. The agreement between his photometry and ours is good, with only small offsets between the two data sets as given in Table \\ref{tab:resids}.\n\nFigure \\ref{fig:ubbv} shows that the cluster NGC6664 has an anomalous {\\it U-B:B-V} diagram. This is a clear case where we fit the O and B type stars rather than trying to fit the F and G type stars as the F and G type stars maybe more affected by metallicity. The value for E({\\it B-V}) is then 0.66$\\pm$0.08 which is slightly larger than 0.6 obtained by \\citeasnoun{arpEV}. No error is quoted by Arp. In the previous work by Arp, the observations were not deep enough to see if the anomalous shape of the {\\it U-B:B-V} diagram would have been detected or not. Unfortunately the only previous source of photometry for more than a handful of stars is that of Arp so no other zero point comparisons can be made. \n\n\nThe distance modulus obtained is 11.01$\\pm$0.37 which is larger than 10.8 from \\citeasnoun{arpEV}, mostly due to the increased value for the reddening estimated here.\n\nThere is an larger discrepancy between the value of the distance modulus used by \\citeasnoun{LS2} who quote a reddening of 0.64 but a distance modulus of 10.405. It is not clear why their distance modulus is so low as \\citeasnoun{FW} use 10.88.\n\nThe radial velocity work of \\citeasnoun{Kraft} shows EV Sct is a member of NGC6664. \n\n\\textbf{WZ Sgr} The first problem with WZ Sgr is that its membership of an open cluster is questionable. \\citeasnoun{turnWZ} discusses the membership of WZ Sgr to the cluster C1814-190 in some detail and concludes that the strongest evidence for membership of WZ Sgr to an open cluster comes from the fact that often the Cepheid in a cluster is around 4 magnitudes more\nluminous than the B-type stars on the main sequence. This essentially means that the age of the Cepheid is consistent with the age of the cluster. \\citeasnoun{FW} assume WZ Sgr to be a cluster member.\n\nThe cluster C1814-190 is also only comparatively sparsely populated with only around 35 members brighter than B$\\sim$16 \\citeasnoun{turnWZ}. There is also patchiness in the dust obscuration which would cause differential reddening \\cite{turnWZ}. These two factors could possibly explain the odd shape of the {\\it U-B:B-V} colour-colour diagram particularly in the range 0.5 $\\lsim$ {\\it B-V} $\\lsim$ 1. Contamination from foreground and background stars could also be the source of the unusual {\\it U-B:B-V} diagram. By measuring the reddening of the O and B type stars though we get a value for E({\\it B-V})=0.56$\\pm$0.20 which is in agreement with \\citeasnoun{turnWZ} and consistent with E({\\it B-V})=0.57 used by \\citeasnoun{LS2}. \n\nThe {\\it B-V:V} diagram is surprisingly tight, giving a distance modulus of 11.15$\\pm$0.49 which is again in agreement with \\citeasnoun{turnWZ} who obtained 11.16$\\pm$0.1. The distance modulus used by \\citeasnoun{LS2} is slightly larger, 11.219 but again well within the errors of our result.\n\n\n\\textbf{Lynga 6} The photometry of Lynga 6 has been checked against that of \\citeasnoun{moff} (shown in Figure \\ref{fig:zero} and \\ref{fig:col}) and also against that of \\citeasnoun{VDBH}. Offsets between the different sets of photometry are less than 0.03 mags in each waveband.\n\n\nLike NGC6649, Lynga 6 is also very heavily reddened which makes it very difficult to observe stars over a large range of {\\it B-V} and {\\it U-B} colours.\nE({\\it B-V}) is estimated to be 1.36$\\pm$0.17 which is consistent with previous measurements by \\citeasnoun{VDBH} who obtained 1.34$\\pm$0.01, by \\citeasnoun{madly6} who obtained 1.37$\\pm$0.03 and the value of 1.34 used by \\citeasnoun{LS2}.\n\nThe distance modulus obtained here is around 11.10$\\pm$0.45 which is consistent with 11.15$\\pm$0.3 obtained by \\citeasnoun{walkly6} but is discrepant with the value used by \\citeasnoun{LS2}. They have a distance modulus of 11.429 which agrees reasonably well with the value of \\citeasnoun{madly6} who obtained ($V-M_V$)=16.2$\\pm$0.5 for the apparent distance modulus which, if a value of 3.2 is assumed for the extinction coefficient, roughly implies 11.8$\\pm$0.5 for the distance modulus (see section \\ref{sec:PL} for more details of the extinction coefficient used here). The difference between the value of Madore and the value for the distance found here is that Madore tended to fit the edge of the ZAMS.\n\nThe Cepheid TW Nor lies close to the centre of the cluster Lynga 6 and \nhas a very similar value of the reddening. This is taken as evidence of the membership of the Cepheid to the cluster \\cite{walkly6}.\n\n\\textbf{NGC6067} There is very little photoelectric data for this cluster. The {\\it B} and {\\it V}-bands have been compared to the data of \\citeasnoun{thack} but there are only 7 stars in common and in the {\\it U}-band there are only two stars in common. The comparison shows that the zero point used here is at least consistent with previous the previous work. As the photometry obtained at CTIO for other clusters such as Lynga 6 and vdBergh1 agrees well with previous results, we have to assume that the photometry for NGC6067 is also good. \\citeasnoun{walk6067} finds good agreement between their {\\it B} and {\\it V}-band CCD data and the photoelectric data of \\citeasnoun{thack}.\n\nNGC6067 has the lowest value for the reddening, of the clusters in this study, with E({\\it B-V}) estimated as 0.42$\\pm$0.02. The {\\it U-B:B-V} diagram presented here has a main sequence which agrees fairly well with the ZAMS, although there is a spread around {\\it B-V}=0.8. The value of 0.42 is slightly higher than previous values, \\citeasnoun{coulson} obtained 0.35$\\pm$0.1 and \\citeasnoun{thack} obtained 0.33 with no quoted error.\n\nAs the reddening has been estimated to be slightly higher than previously, the distance modulus is also increased to 11.17$\\pm$0.35 as opposed to 11.05$\\pm$0.1 from Walker. However \\citeasnoun{thack} found the distance modulus of the cluster to be around 11.3. This estimate is higher than our estimate, despite a smaller measured reddening, as the ZAMS fit was made to the edge of the CMD.\n\n\\citeasnoun{LS2} use 0.35 for the reddening and 11.13 for the distance modulus to the cluster.\n\nThe membership of the Cepheids to this cluster is discussed in detail by \\citeasnoun{eggen}. Eggen noted that \nthe Cepheid V340 Nor is centrally located in the cluster and has the same reddening as the cluster so is assumed to be a member. The Cepheid QZ Nor lies at a distance of two cluster radii out from the cluster centre \\cite{walk6067} but is still assumed to be a cluster member by \\citeasnoun{FW}\n\n\n\\textbf{vdBergh1} Our photometry is tested against the photoelectric data of of \\citeasnoun{arpCV} in Table \\ref{tab:resids}. There is no significant offset between the two data sets and no evidence of any scale dependent error. \\citeasnoun{turnCV} has compared their photometry to that of \\citeasnoun{arpCV} and find good agreement.\n\nThe cluster vdBergh1 was only given a short exposure of 300s in {\\it U} and 90s in {\\it B} and {\\it V}. The {\\it U-B:B-V} diagram is therefore not very well populated at faint {\\it U} magnitudes. There is a fair amount of scatter in the {\\it U-B:B-V} diagram for this cluster, the average value is E({\\it B-V})=0.9$\\pm$0.18. This is larger than the value of previous estimates. \\citeasnoun{turnCV} obtained a minimum value of E({\\it B-V})=0.66, this clearly fits the the edge of the B-type stars. \\citeasnoun{arpCV} obtained 0.76 (no error) but the spread in the {\\it U-B:B-V} diagram is such that the larger value would also have been acceptable. 0.77 was used for the cluster reddening by \\citeasnoun{LS2}\n\nThe distance modulus, $\\mu_{\\circ}$ obtained here is 11.4$\\pm$0.65. This is larger than previous results, 10.94 obtained by \\citeasnoun{arpCV} and 11.08 obtained by \\citeasnoun{turnCV}. This is mostly due to the measurement of increased reddening. The value used by \\citeasnoun{LS2} was 11.356.\n\nThe membership of CV Mon to the cluster has been determined by \\citeasnoun{turnCV} using radial velocity measurements, evolutionary arguments and by it's location in the cluster.\n\n\\textbf{Trumpler 35} There is very little photoelectric data available for this cluster. We compare our photometry against the photoelectric observations of \\citeasnoun{hoag} and find reasonable agreement in the {\\it B} and {\\it V}-bands. The agreement in the {\\it U}-band is less good but as there is only one source of comparison and the JKT photometry appears to agree well for other clusters we suggest that the {\\it U}-band data photometry is accurate. \n\nThe data in the {\\it U-B:B-V} diagram for the cluster Trumpler 35 (TR35) shows quite a large spread. This is probably caused by differential reddening. Rather than trying to correct for this, a mean value of E({\\it B-V})=1.19$\\pm$0.10 is taken for the reddening. Shown on the TR35 panel in Figure \\ref{fig:ubbv} are two lines. The dashed shows the value of E({\\it B-V})=1.03 taken from \\citeasnoun{turn35} and the solid line shows the value of 1.19 adopted here. Using this, the distance modulus to the cluster is then estimated as 11.3$\\pm$0.53 by fitting up the centre of the data. This is different to the value of \\citeasnoun{turn35} who obtained 11.6$\\pm0.16$ as Turner fitted the edge of the {\\it B-V:V} diagram rather than the centre. The values for the reddening used by \\citeasnoun{LS2} (E({\\it B-V})=0.92, $\\mu_{\\circ}$=11.56) are in better agreement with those of \\citeasnoun{turn35} rather than those obtained here.\n\nThe membership of RU Sct to Trumpler 35 using the reddening and evolutionary status of the Cepheid is discussed and supported by \\citeasnoun{turn35}. However RU Sct does lie 15$^{\\prime}$ away from TR35 \\cite{turn35}.\n\n\n\\textbf{NGC6823} was only observed in non-photometric conditions. We use the photoelectric data of \\citeasnoun{guetter} for calibration purposes. \\citeasnoun{guetter} has compared his photometry with that of Hiltner (1956) and Hoag (1961) and finds that there are only small offsets of less than 0.04mags in {\\it U-B} between the different data sets. \\citeasnoun{turn6823} finds good agreement with the photometry of Hiltner (1956) which agrees well with the photometry of \\citeasnoun{guetter} used for calibration here.\n\nNGC6823 suffers from differential reddening \\cite{turn6823}, perhaps to an even greater extent than Trumpler 35. Again shown in the panel for the cluster NGC6823 in Figure \\ref{fig:ubbv} are two lines. One is for E({\\it B-V})=0.53 taken from \\citeasnoun{FW} and the other is E({\\it B-V})=0.85 which fits the centre of the B-type stars. The value of E({\\it B-V})=0.85$\\pm$0.09 is assumed here. \n\nThe distance modulus with this reddening is then 11.20$\\pm$0.55, lower than the value of 11.81 found by \\citeasnoun{turn6823} who again fitted the edge rather than the centre of the {\\it B-V:V} diagram. \\citeasnoun{LS2} assume a reddening of E({\\it B-V})=0.44 and $\\mu_{\\circ}$=11.787. These values again agree fairly well with those of \\citeasnoun{turn6823} rather than the values found here.\n\nThe Cepheid SV Vul is only thought to be associated with the cluster NGC6823 \\cite{FW} as it lies a few arcmins away from the cluster centre.\n\n\n\\textbf{NGC129} was only observed in non-photometric conditions so previous work had to be relied upon for the calibration, as discussed in detail in section \\ref{sec:jktcal}. The reddening for this cluster is E({\\it B-V})=0.57$\\pm$0.06. This value is slightly larger than previous values, \\citeasnoun{turn129} obtained 0.47 but fits to the least reddened edge of the O and B stars rather than to the centre of these stars. \\citeasnoun{arp129} found 0.53 (with no error) for the reddening which is in agreement with the value found here.\n\nThe distance modulus obtained assuming a value of 0.57 for the reddening is $\\mu_{\\circ}$=10.90$\\pm$0.37. This value is lower than the value of 11.11 obtained by \\citeasnoun{turn129} who again fits the edge of the main sequence rather than the centre. \\citeasnoun{arp129} find 11.0$\\pm$0.15 for the distance modulus of the cluster, in agreement with the value found here. DL Cas is not included in the study by \\citeasnoun{LS2} due to the Northerly latitude of the cluster NGC129. The values for the reddening and the distance modulus used by \\citeasnoun{FW} agree better with the values of \\citeasnoun{turn129} rather than those obtained here.\n\n\\citeasnoun{Kraft} found a measurement of -14$\\pm$3 kms$^{-1}$ for the radial velocity of the cluster NGC129. He found that the Cepheid itself had a radial velocity of -11km$s^{-1}$. Given that the error on any individual measurement is estimated to be around 1.5kms$^{-1}$, DL Cas is assumed to be a member of NGC129.\n\n\\textbf{NGC7790} The {\\it U-B:B-V} diagram for this cluster appears quite clean\nand well defined. However, Fig. \\ref{fig:ubbv}\nshows that NGC7790 has a {\\it U-B:B-V} diagram where the data poorly fits\nthe ZAMS line. We show one ZAMS shifted to fit the OB stars\nwhich implies E({\\it B-V})= 0.59$\\pm$0.04 and another shifted to \nfit the F stars\nwhich would imply E({\\it B-V})=0.43$\\pm$0.04. Most previous estimates \nare closer to that for the OB stars. This poor fit of the ZAMS to the \n{\\it U-B:B-V} data in the case of this cluster is particularly significant \nsince it contains three Cepheids (see Table 1).\n\nThe {\\it U-B:B-V} diagram for NGC7790 has a history of controversy. The \noriginal {\\it UBV} photoelectric photometry of Sandage (1958) of 33 \n11$<${\\it V}$<$15 stars was\ncriticised by Pedreros et al (1984). A check of 16 stars with the KPNO CCD seemed to confirm that Sandage's {\\it U-B} and {\\it B-V} colours were too blue by +0.075 and +0.025 mag respectively, although few details were given of errors etc. However we find excellent agreement between the work here and the photometry of Sandage, see Table \\ref{tab:resids}, Fig. \\ref{fig:zero} and Fig \\ref{fig:col}. A direct comparison of the photoelectric observations of Sandage to the photographic observations in Pedreros (see Table 1 in Pedreros et al. 1984) implies that the differences in the colours of Sandage are too blue by $\\sim$ 0.04 mags for both the {\\it U-B} and {\\it B-V} colours and when we compare our CCD photometry to 22 of the brightest photographically observed stars from Pedreros, we find similar colour differences of around 0.04 mags in both the {\\it U-B} and {\\it B-V} colours. \n\nThe {\\it U-B:B-V} diagram in \\citeasnoun{ped} seemed to give the same sort of ill fitting ZAMS, throughout the range 0.3$<${\\it B-V}$<$1.2, as found in this work. The suggestion was that the problem might lie in Sandage's photometry which Pedreros et al had used for calibration. However, we have tested various zero points for the {\\it U-B} and {\\it B-V} colours. With our own zero points for the colours we find the ill fitting ZAMS and even if we apply the corrections suggested by Pedreros to the offsets found between our photometry and the photographic data of Pedreros we still find an ill fitting ZAMS. \n\\begin{figure} \n\\begin{tabular}{c} {\\epsfxsize=7.truecm \\epsfysize=7.truecm\n\\epsfbox[60 170 550 620]{membership.ps}} \\\\\n{\\epsfxsize=7.truecm \\epsfysize=7.truecm \\epsfbox[60 170 550\n620]{membership_ubbv.ps}} \\\\ \n\\end{tabular}\n\\caption{The top panel shows the {\\it B-V:V} colour-magnitude diagram for the\ncluster NGC7790 but with only the stars that fit the zero age main sequence\nshown. The lower panel shows the same stars as they appear on the {\\it U-B:B-V}\ndiagram. The U-V deficit is still clearly apparent among the stars that are\nmost likely to be members of the cluster. } \n\\label{fig:memb} \n\\end{figure}\n\n\\begin{figure}\n{\\epsfxsize=9truecm \\epsfysize=8truecm\n\\epsfbox[60 170 660 620]{convince_7790.ps}} \n\\caption{The open circles show the photometry of this work. The solid triangles show the photometry of Sandage which does not go deep enough to test the ill fitting ZAMS. The results of \\protect\\citeasnoun{ped}(filled squares), though not as deep as those presented here, show the same effect as the photometry in this work and confirms the poor fit of the ZAMS {\\it U-B:B-V} relation in NGC7790}\n\\label{fig:convince}\n\\end{figure}\n\n\n\n \nRomeo et al (1984) used CCD data to obtain {\\it BVRI} photometry for this cluster to {\\it V}=20. In the absence of {\\it U}, their only route to E({\\it B-V}) was via fitting the shape of the {\\it B-V:V} and {\\it V-I:V} CMD and they obtained E({\\it B-V})=0.54$\\pm$0.04. They used the Sandage (1958) photometry for calibration in {\\it B} and {\\it V}, subject to the small re calibration in these bands by Pedreros et al. and checked against the {\\it B,V} photoelectric photometry of 10 stars by Christian et al (1985). They tested the faint photographic photometry of Pedreros et al and found scale errors at {\\it B}$>$17 and {\\it V}$>$15 in the sense that Pedreros et al were too bright. In {\\it B-V}, however they claimed better agreement with {\\it B-V}$_{Pedreros}$ being $\\sim$0.1mag too red. A comparison of our photometry and that of Romeo suggests that there is a scale error for {\\it B}$>$17 and {\\it V}$>$16 but the extent of this is less than in the comparison of the Pedreros et al. data with the Romeo data, at {\\it V}=17 our data is brighter than Romeo's by 0.1 mag whereas the Pedreros et al. data is brighter by 0.2 mags and similar differences are found in the {\\it B} data. We conclude that the photometry in this study agrees very well with the photometry of \\citeasnoun{sand7790} and is better agreement with the CCD data of \\citeasnoun{romeo} than that of Pedreros, (see Fig. \\ref{fig:7790comp} for more details). \n\nWe have also attempted to check to see if contamination is causing the ill fitting ZAMS. Figure \\ref{fig:memb} shows the {\\it B-V:V} and the {\\it U-B:B-V} diagrams for the cluster NGC7790. The {\\it B-V:V}\ndiagram has been trimmed so that only the stars that lie very close to the\nZAMS remain. As these stars have the correct combination of distance and\nreddening to lie almost on the main sequence then it is likely that they are\nmain sequence stars. The same stars are then used to produce the {\\it U-B:B-V}\ncolour-colour diagram. The same {\\it UV} deficit around the F-type stars that\nappears in Figure \\ref{fig:ubbv}, when all the stars in the field are used,\nappears in Figure \\ref{fig:memb}. Figure \\ref{fig:convince} shows the colour-colour diagram found from this work with the points from Sandage and Pedreros included. The Sandage points do not go deep enough to test the shape of the data but the Pedreros et al. points do and the poor match of the data to the ZAMS is seen. Therefore, we believe the ill fitting ZAMS to the {\\it U-B:B-V} diagram is not caused by contamination from foreground or background stars or by errors in the photometry but is a real feature in the data. \n\nThus our estimate of the NGC7790 reddening based on the {\\it U-B:B-V} colours of OB stars, E({\\it B-V})=0.59$\\pm$0.05, is between the E({\\it B-V})=0.52 $\\pm$0.04 of Sandage (1958) and the E({\\it B-V})=0.63$\\pm$0.05 of Pedreros et al(1984) who used similar techniques. \n\nAssuming the estimate of E({\\it B-V})=0.59$\\pm$0.05 from the OB stars, the\ndistance modulus which fits {\\it V:B-V} is then $\\mu_{\\circ}$=12.72$\\pm$0.11 which\nis close to the original value of 12.8$\\pm$0.15 found by Sandage(1958),\nalthough this is in the opposite direction that would be expected from the\ndifference in reddening and slightly more than 12.65 obtained by\n\\citeasnoun{romeo} which is roughly in line with their obtaining\nE({\\it B-V})=0.54 for the reddening. \\citeasnoun{ped} obtained only 12.3 for\nthe distance modulus, no fits are shown in the paper and the reason they\nobtain this low value is unclear. The values quoted by \\citeasnoun{LS2}\nare similar to those of \\citeasnoun{ped}.\n\n\n\\citeasnoun{sand7790} states that the membership of CF Cas, CEa Cas and CEb\nCas to NGC7790 is almost certain due to the position of the Cepheids on the\nCMD.\n\n\n\\begin{table*} \\begin{tabular}{lccccccccccc} Cepheid & Log(P)\n&E({\\it B-V})$_{\\rm{ceph}}$ & $<$V$>$ & $<$K$>$ & M$_{V}$ & M$_{K}$ &\nM$_{V}^{\\rm{LS}}$ & M$_{K}^{\\rm{LS}}$ & $\\Delta(\\rm{M}_{V})$ & $\\Delta(\\rm{M}_{K)}$ \\\\ \\hline \nV367 & 0.799 & 1.27 & 11.604 & 6.662 &-3.789 & -4.937 & -3.755 & -4.992 & -0.034 & 0.055 \\\\ \nU Sgr & 0.829 & 0.45 & 6.685 & 3.952 & -3.841 & -5.232 & -3.781 & -5.214 & -0.060 & -0.018 \\\\ \nEV Sct & 0.490 & 0.62 & 10.131 & 7.028 & -2.877 & -4.164 & -2.191 & -3.553 & -0.686 & -0.611 \\\\ \nWZ Sgr & 1.339 & 0.50 & 8.023 & 4.565 &-4.810 &-6.738 & -4.657 & -6.789 & -0.153 & 0.051 \\\\ \nTW Nor & 1.033 & 1.24 & 11.670& 6.319 & -3.560 & -5.156 & -3.797 & -5.411 & 0.237 & 0.255 \\\\ \nQZ Nor & 0.730 & 0.27$\\sharp$ & 8.866 & 6.662 & -3.164 & -4.634 & -3.128 & -0.036 & -0.046 & -0.035 \\\\ \nV340 Nor & 1.053 & 0.38 & 8.375 & 5.586 & -4.056 &-5.699 & -3.797 & -5.640 & -0.259 & -0.059 \\\\ \nCV Mon & 0.731 & 0.84 & 10.306 & 6.576 & -3.821 & -5.072 & -3.305 & -4.896 & -0.516 & -0.176 \\\\ \nRU Sct & 1.294 &0.93$\\sharp$ & 9.465 & 5.071 & -4.901 & -6.508 & -5.186 &-6.775 & 0.284 & 0.267 \\\\ \nSV Vul & 1.654 & 0.44$\\sharp$ & 7.243 & 3.920 & -5.440 & -7.415 & -6.028 & -8.004 & 0.588 & 0.589 \\\\ \nDL Cas & 0.903* & 0.60 & 8.97* & 5.93$\\dagger$ & -3.648 & -5.136 & -3.860* & -5.36$\\dagger$ & -0.171 & 0.181 \\\\ \nCF Cas & 0.688* & 0.55 & 11.14* & 8.01$\\dagger$ & -3.341& -4.872 & -3.170* & -4.85$\\dagger$ & -0.171 & -0.022 \\\\ \nCEa Cas & 0.711* &0.55 & 10.92* & NA & -3.561 & NA & -3.390* & NA & -0.171 & NA \\\\ \nCEb Cas & 0.651*& 0.55 & 10.99* & NA & -3.493 & NA & -3.330* & NA & -0.163 & NA \\\\ \n\\hline\n\\end{tabular}\n \n\\caption{Comparison between the work here and that of\n\\protect\\citeasnoun{LS2}. * indicates where values are taken from\n\\protect\\citeasnoun{FW} and $\\dagger$ where the values are inferred from\n\\protect\\citeasnoun{welch}. $\\sharp$ indicates where the Cepheid reddenings\nhave been corrected (see text)} \\label{tab:diffs} \\end{table*}\n\n\n\\section{P-L Relation} \\label{sec:PL}\n\nUsing the values for the distance modulus and the reddening towards the\ncluster, we proceed to determine the P-L relation. As well as the reddening\nand the distance modulus of the cluster, the apparent magnitude of each of\nthe Cepheids is required. Where possible these come from LS (1993, 1994) who\nhave high quality {\\it V} and {\\it K}-band measurements for most of the Cepheids in\nthis study. The clusters NGC7790 and NGC129 lie at northerly latitudes so\nare unobservable from SAAO and so there are no magnitudes from Laney and\nStobie for these Cepheids. The {\\it K}-band data for DL Cas and CF Cas comes\ntherefore from \\citeasnoun{welch} and the {\\it V}-band data and the periods are\ntaken from \\citeasnoun{FW}.\n\nThe reddening obtained from the ZAMS fitting is that of the cluster OB\nstars. \\citeasnoun{SK} found that when the effect of the colour difference\nbetween the OB stars and the Cepheid is taken into account,\n\n\\begin{figure} \n{\\epsfxsize=8.5truecm \\epsfysize=8.5truecm \\epsfbox[60 170\n550 620]{MvMk.ps}} \n\\caption {The Galactic Cepheids are\nshown by the solid circles, the LMC Cepheids by the triangles and the SMC\nCepheids by the squares, taken from \\protect\\citeasnoun{LS2}. The three solid \ncircles in a box are QZ Nor, RU Sct\nand SV Vul. The reddenings of these Cepheids are corrected to the values\ngiven in \\protect\\citeasnoun{LS1}, indicated here by the boxed open\ncircles.} \n\\label{fig:MvMk} \n\\end{figure}\n\n\n\\begin{eqnarray} \n{\\rm E}(B-V)_{\\rm{ceph}}& = & {\\rm E}(B-V)_{{\\rm\nclus}}[0.98- \\nonumber \\\\ \n& & 0.09(<B_o>-<V_o>)_{{\\rm ceph}}]\n\\label{eq:cephred} \n\\end{eqnarray} gives a good approximation to the\nreddening of the Cepheid. The values for $<B_o>-<V_o>)_{\\rm{ceph}}$ \ncome from \\citeasnoun{FW}.\n\nFigure \\ref{fig:MvMk} shows the M$_{V}$-M$_{K}$ - Log(P) relation\nwith the reddenings found for the Galactic Cepheids (filled circles) found \nfrom the cluster reddenings obtained in this work via equation \\ref{eq:cephred}.\nThe triangles show the same relation for the LMC Cepheids and the squares\nare for the SMC Cepheids. These are taken from Tables 2 and 3 in \\citeasnoun{LS2}.\nThree of the Cepheids, (from left to right in Figure \\ref{fig:MvMk})\nQZ Nor, RU Sct and SV Vul seem to have the wrong M$_{\\rm v}$-M$_{\\rm k}$\ncolours for their periods. The clusters containing RU Sct and SV Vul \nsuffer both from the presence of differential reddening and from the fact \nthat the Cepheids lie at some distance away from the cluster. Figure \\ref{fig:MvMk}\nindicates that the reddening local to RU Sct and SV Vul may be somewhat \ndifferent from the average value of the cluster reddening. We correct for \nthis using the space reddenings given in \\citeasnoun{LS1} which are more \nlocal to the Cepheids (see for example Turner 1980). Note that because \nthere is difficulty obtaining the Cepheids true reddening from the cluster \nreddening for these two Cepheids we do not include them in the best sample \n(later in this section).\n\\citeasnoun{walk6067} notes that the Cepheid QZ Nor also lies\naway from the centre of the cluster NGC6067, at a distance of two cluster\nradii so again the reddening of the cluster may not be appropriate for\nthe reddening of the Cepheid. \\citeasnoun{LS1} take the value for the\nCepheid reddening from \\citeasnoun{coulson} of E({\\it B-V}) = 0.265, derived\nfrom {\\it BVI}$_c$ reddenings. This is the reddening used to calculate the\nposition of the open circle in Figure \\ref{fig:MvMk} and which we assume for\nQZ Nor henceforth. However, the effect of changing the reddening of the cluster\nto E({\\it B-V}) = 0.265 changes the distance modulus to the LMC by less than\n0.02, less than the error quoted here.\n\n\nTo determine the absolute magnitude of the Cepheid, the apparent magnitude\nhas to be corrected for reddening and distance. First of all the extinction\ncoefficient is required. We follow \\citeasnoun{LS2} and use\n\n\\begin{equation} \\Re({\\rm ceph})= 3.07 + 0.28(B-V)_{\\circ} + 0.04{\\rm\nE}(B-V)_{{\\rm ceph}} \\end{equation} to take into account the effect of\nCepheid colour on the ratio of total to selective extinction. The\nCepheid reddening comes from Eq. \\ref{eq:cephred} except for the three\ncorrected values. Then the reddening free magnitudes of the Cepheids are\n\n\\begin{eqnarray} \nV_{\\circ} & = & V - \\Re({\\rm ceph}){\\rm E}(B-V)_{{\\rm\nceph}} \\nonumber \n\\\\ \n%K_{\\circ} & = & K - 0.1 \\Re({\\rm ceph}){\\rm E}(B-V)_{{\\rm ceph}} \nK_{\\circ} & = & V_{\\circ} - V + K + \\frac{\\Re({\\rm ceph}){\\rm E}(B-V)_{{\\rm ceph}}}{1.1} \n\\end{eqnarray} The expression for\n$K_{\\circ}$ has the form given above as the extinction coefficient in\nthe {\\it K}-band is one tenth of that in the {\\it V}-band. To obtain\nfinally the absolute magnitude, the distance modulus, $\\mu_{\\circ}$, given\nin Table \\ref{tab:rd} has to be subtracted off.\n\n\n\nThe P-L relation can now be determined. We consider two samples, one where\nwe consider all the Cepheids available to us and another where the Cepheids\nRU Sct and SV Vul are removed due to the problem of differential reddening\nand the question of cluster membership. The zero points are obtained by\nfixing the slope and obtaining the least squares solution using the Galactic Cepheids in this study. These are summarised in Table \\ref{tab:zp}. The slopes that are considered are the slopes from\n\\citeasnoun{LS2} which are the best fitting slopes to all the Cepheid data\n(Galactic open cluster Cepheids, LMC and SMC Cepheids) in their study. The\nslopes are -2.874 in the {\\it V}-band and -3.443 in the {\\it K}-band. Also\nconsidered is -2.81 in the {\\it V}-band as this is the slope of the LMC Cepheids\nand the slope used by \\citeasnoun{FC}.\n\nOnce the slope and zero point of the PL relation is fixed, the distance\nmodulus to the LMC can be calculated. \\citeasnoun{LS2} give the period and\nthe dereddened {\\it V} and {\\it K}-band magnitude, {\\it V}$_{\\circ}$ and {\\it K}$_{\\circ}$ of 45\nLMC Cepheids. The distance to the LMC is then given by\n\n\\begin{equation} \n<\\mu_{\\circ}> = \\frac{1}{\\rm{N}} \\sum_i^{\\rm{N}} (m_{\\circ} - \\delta*\\rm{log}(P)) - \\rho \n\\end{equation} where $m_{\\circ}$ represents the\ndereddened apparent magnitude in each waveband and $\\delta$ and $\\rho$ are the\nvalues for the slope and zero point as given in Table \\ref{tab:zp}. Our PL({\\it V})\nand PL({\\it K}) relations are shown in Figure \\ref{fig:PL} along with the equivalent\nrelation for the LMC Cepheids from \\citeasnoun{LS2} using the distance moduli in\nTable \\ref{tab:zp} to determine the absolute magnitude.\n\n\\begin{table} \\begin{tabular}{clccc} Band & Sample & Slope($\\delta$) & Zeropoint($\\rho$)\n& $\\mu_{\\circ}$(LMC) \\\\ \\hline \n{\\it V} & All(14) & -2.874 &-1.229$\\pm$0.34 & 18.53$\\pm$0.036 \\\\ \n{\\it V} & All(14) & -2.810 &-1.289$\\pm$0.33 & 18.50$\\pm$0.036 \\\\ \n{\\it K} & All(12) & -3.443 &-2.148$\\pm$0.30 & 18.42$\\pm$0.019 \\\\ \\hline \n{\\it V} & Best(12) & -2.874 &-1.279$\\pm$0.33 & 18.58$\\pm$0.036 \\\\ \n{\\it V} & Best(12) & -2.810 &-1.332$\\pm$0.32 & 18.55$\\pm$0.036 \\\\ \n{\\it K} & Best(10) & -3.443 &-2.200$\\pm$0.29 & 18.47$\\pm$0.019 \\\\ \\hline \n{\\it V} & logP$<$1.0(9)& -2.874 &-1.418$\\pm$0.19 & 18.71$\\pm$0.036 \\\\ \n{\\it V} & logP$<$1.0(9)& -2.810 &-1.465$\\pm$0.19 & 18.68$\\pm$0.036 \\\\ \n{\\it K} & logP$<$1.0(7)& -3.443 &-2.314$\\pm$0.21 & 18.58$\\pm$0.019 \\\\ \\hline\n\\end{tabular} \\caption{The zeropoints for the P-L relation and distance\nmodulus to the LMC} \\label{tab:zp} \\end{table}\n\n\\begin{figure} \n\\begin{tabular}{c} \n{\\epsfxsize=7.7truecm \\epsfysize=6.6truecm\n\\epsfbox[50 170 590 620]{PLv.ps}} \\\\\n{\\epsfxsize=7.7truecm \\epsfysize=6.6truecm\n\\epsfbox[50 170 590 620]{PLv8.ps}} \\\\ \n{\\epsfxsize=7.7truecm \\epsfysize=6.6truecm \n\\epsfbox[50 170 590 620]{PLk.ps}} \\\\\n\\end{tabular}\n\\caption{The Cepheid P-L relation. (a) shows the {\\it V}-band P-L relation with a slope of -2.874 and the (b) shows the {\\it V}-band P-L relation with a slope of -2.810. (c) shows the {\\it K}-band P-L relation with a slope of -3.443. The solid symbols are the Galactic Cepheids, marked specifically are U Sgr (diamond), EV Sct (filled triangle), CF Cas, CEa Cas and CEb Cas (filled squares). There is no {\\it K}-band data for the Cepheids CEa Cas and CEb Cas. The zeropoints and slopes are shown in each panel. The open triangles show the P-L relation for the LMC Cepheids. The periods and magnitudes are taken from Table 2 and 3 in \\protect\\citeasnoun{LS2} and the distance moduli to the LMC are the 'best' values from Table \\ref{tab:zp} in this work.}\n\\label{fig:PL} \n\\end{figure}\n\n\nTaking the value for the PL({\\it V}) zeropoint for the best sample with the\n-2.874 slope used by Laney \\& Stobie (1994) gives $\\rho$=-1.279$\\pm$0.33\nwhich is in good agreement with the value of $\\rho$=-1.197$\\pm$0.09 found by\nthese authors and implies a distance modulus of 18.57 for the LMC as\ncompared to 18.50 found by Laney \\& Stobie. Figs. 8(a,b) show that the best\nfitting zeropoint gives a somewhat poor fit to the majority of the Galactic\nCepheids which lie at log(P)$<$1.0. This is because the 12 Galactic Cepheids\nin the best sample give a slope of $\\delta$=-1.85$\\pm$0.33 which is much\nflatter than the LMC data which gives $\\delta$=-2.79$\\pm$0.1, close to -2.81.\nIndeed, if only the 9 Galactic Cepheids with log(P)$<$1.0 are used, the\nzeropoint rises to $\\rho$=-1.418$\\pm$0.19 and the LMC distance modulus would\nrise to 18.70, indicating why the errors on the PL({\\it V}) zeropoint are as large\nas they appear in Table 5.\n\nOur PL({\\it V}) zeropoint is also consistent with the zeropoint and LMC distance\nobtained from an analysis of Hipparcos trigonometrical parallaxes of nearby\nGalactic Cepheids by \\citeasnoun{FC}. They obtained $\\rho$=-1.43$\\pm$0.1 for\nthe Galactic PL({\\it V}) zeropoint for an assumed slope of $\\delta$=-2.81. This\ncan be compared to the $\\rho$=-1.332$\\pm$0.32 obtained for our best sample\nwith the same slope. They used the same 45 Laney \\& Stobie (1994) Cepheids\nas used here to obtain a metallicity corrected LMC distance\nmodulus $\\mu_{\\circ}$=18.70$\\pm$0.10. We note that their semi-theoretical\nmetallicity correction to the LMC Cepheid {\\it V} magnitudes increases the\ndistance to the LMC, which is in the opposite sense to most empirically\ndetermined estimates of the effects of metallicity on Cepheids (eg Kennicutt et al 1998). Subtracting their metallicity correction leads to an LMC distance modulus $\\mu_{\\circ}$=18.66$\\pm$0.10 which can be\ndirectly compared with our best value of $\\mu_{\\circ}$=18.55$\\pm$0.036 from Table \\ref{tab:zp}. We\nconclude that our PL({\\it V}) estimates the LMC distance modulus are between those\nof Laney \\& Stobie and Feast \\& Catchpole but have too little statistical\npower to discriminate between these previous estimates.\n\n\nThe {\\it K}-band P-L relation is tighter for the LMC Cepheids and for the Galactic\nCepheids, see Figure \\ref{fig:PL}(c), and the slopes are closer with the LMC Cepheids giving\n$\\delta$=-3.27$\\pm$0.04 and the best sample of Galactic Cepheids giving\n$\\delta$=-2.81$\\pm$0.21. We note in passing that the slope of the Galactic\nCepheid PL({\\it K}) relation is now much flatter than the $\\delta$=-3.79$\\pm$0.1\nslope found in the Galactic Cepheid sample of Laney \\& Stobie. Assuming the\n-3.443 slope used by Laney \\& Stobie our best sample in Table 5 gives a\nPL({\\it K}) zeropoint of $\\rho$=-2.200$\\pm$0.29 which implies an LMC distance of\n$\\mu_{\\circ}$=18.47$\\pm$0.29 which remains in good agreement with the value\n$\\mu_{\\circ}$=18.56$\\pm$0.07 found by Laney \\& Stobie. (1994). The smaller error\nof Laney \\& Stobie is due to their larger numbers of calibrators although it\nmust be said that many of their extra calibrators (8/12) are in associations\nrather than clusters and frequently given half-weight in P-L fits. Indeed,\nHipparcos proper motion data has shown that one of their further cluster\nCepheids, S Nor, is also unlikely to be a member of its cluster (Haguenau conference, September 1998).\nMoreover, they have not included the 4 cluster Cepheids in NGC129\n and NGC7790. Therefore we believe that our result supercedes the Laney \\& Stobie\nresult with our bigger error estimate perhaps being a more realistic\nindication of the actual errors. Certainly in our best sample the biggest\nchanges in M$_K$ between Laney \\& Stobie and ourselves, which contribute most\nto our 50\\% increased scatter, are for EV Sct and TW Nor (see Table 4)\nwhere the reasons for the distances used by Laney \\& Stobie are unclear.\n\nFinally, as our overall estimate of the LMC distance, we take the average\nof the PL({\\it V}) and PL({\\it K}) estimates in the best sample of Table 5 which gives\n$\\mu_{\\circ}$=18.51$\\pm$0.3. We conclude that although in the case of individual\nclusters we have markedly improved the distance and reddening estimates, our\nnew estimates of the zeropoint of the PL relation and thus the distance to\nthe LMC are close to previous values.\n\n\n\\section{Discussion} \n\\label{sec:interp}\n\n\n\\begin{figure} \n{\\epsfxsize=8.5truecm \\epsfysize=8.5truecm \\epsfbox[125 210\n560 600]{positions.ps}} \n\\caption{The approximate positions of all the clusters considered here. The clusters NGC6664 (triangle), M25 (diamond), NGC129 and NGC7790 (square) are highlighted. These symbols are the same as in Figure \\ref{fig:PL}} \n\\label{fig:pos} \n\\end{figure}\n\nWe now discuss the most intriguing new result in this study, which is that\nthe Solar metallicity ZAMS may not always fit the {\\it U-B:B-V} data in individual clusters.\nThis is not the first time an effect like this has been seen. \\citeasnoun{turnros} saw\npoorly fitting {\\it U-B:B-V} ZAMS for the open cluster Roslund 3. Turner interpreted\nthis as evidence for the young, B-type stars having a cocoon of\ncircumstellar dust around them. This cocoon of dust then increases the\nreddening of the B-type stars as compared to the F and G type stars, causing\nthe ill-fitting {\\it U-B:B-V} ZAMS. The O and B type stars in the {\\it U-B:B-V}\ndiagram of Roslund 3 show a large spread around the ZAMS as the amount of\nexcess dust would probably vary from star to star. Excess reddening could perhaps also be so strong that it was causing some O and B stars to be so reddened that they appeared as F type stars. However, the O and B type\nstars in the {\\it U-B:B-V} diagram for NGC7790 are very tight so the shape of the {\\it U-B:B-V} diagram is unlikely to be caused by\nexcess dust. \n%Also this could not explain the cluster NGC6664. To fit the G-type stars for this cluster, a low reddening of E({\\it B-V})$\\sim$0.4 would be required but this would mean that stars as late as early F-type stars would have to suffer from differential reddening to fit the ZAMS.\n\n%Another solution could be that differential reddening is having a far more\n%wider effect than previously thought. If the stars in the {\\it U-B:B-V} diagram\n%of NGC7790 that show a UV excess are really O or B type stars,\n%then by tracing these back along the reddening vector the discrepant ZAMS\n%could be solved. However, there seems little evidence for differential\n%reddening of the OB stars in NGC7790, since they present a\n%tight {\\it U-B:B-V} sequence.\n\nThe next possibility we consider is that the effect might be due to \nstellar evolution, However, the CMD for the clusters look unevolved \neven at AOV as might be expected for clusters which have Cepheid variables\nand are expected to be less than 10$^8$ years old.\n\nWe also considered whether the discrepancy between the main sequence fitted\ndistance and the Hipparcos parallax to the Pleiades could explain our result.\nIf it were assumed that all open clusters had roughly the same composition\nthen the different forms for {\\it U-B:B-V} that we find might be taken as evidence\nthat the colours of main sequence stars may not be unique. This possibility has also been discussed as an explanation of the problem with the MS\nfitted distance to the Pleiades \\cite{floor} and if it proves relevant in that case it will certainly also be worthy of further consideration here.\n\n\n%This possibility\n%has also been discussed as a possible explanation of the problem with the MS\n%fitted distance to the Pleiades \\cite{floor} but since it seems to contradict the Vogt-Russell theorem which guarantees the uniqueness of the ZAMS at fixed\n%composition we do not consider it further here.\n\nIf the reddening vector in {\\it U-B:B-V} varied as a function of Galactic\nposition then this would also affect our results. However, at least in the \ncase of NGC7790 it seems that for whatever relative shift in U-B and B-V,\nthe ZAMS still has the wrong shape to fit the observed colour-colour relation.\n\n\nThe final possibility is that metallicity is affecting the F stars' U-B\ncolours in some of these clusters. Qualitatively there is some evidence\nsupporting this suggestion. First, `line blanketing' is well known to\nredden the U-B colours of metal rich stars at F and G and low metallicity\nsub-dwarfs are known to show UV- excess as the reverse of this case (e.g.\n\\citeasnoun{camFG}). Second, there is some suggestion that the cluster,\nNGC7790, that shows a UV excess lies outside the solar\nradius while NGC6664 which is redder in U-B at F tends to lie inside (see\nFigure \\ref{fig:pos}). M25 which lies closest to the Sun also fits the solar\nmetallicity {\\it U-B:B-V} diagram as well as any of the clusters. Given that\nmetallicity in the Galaxy is known to decrease with Galactocentric radius,\nthis is suggestive of a metallicity explanation. Many of the other clusters'\n{\\it U-B:B-V} diagrams are either too noisy due to differential reddening (Tr35,\nNGC6823) or too obscured to reach the F stars (NGC6649, Lynga 6, vdBergh1) to\nfurther test this hypothesis. However, NGC6067 forms a counter-example to\nany simple gradient explanation, since it seems to have a normal UBV plot\nand lies inside the solar position This would have to be accommodated by\nallowing a substantial variation on top of any average metallicity gradient.\n\nHowever, quantitatively the case for metallicity is less clear. The size of\nthe UV excess seen is much larger in the case of NGC7790 than\nexpected on the basis of previous metallicity estimates of these clusters,\nor of any measurement of the amplitude of the Galactic metallicity gradient.\nUsing the Fe/H vs $\\Delta$ U-B relations of \\citeasnoun{carney} or \\citeasnoun{camgrad} it\nwould be concluded that NGC7790 showed $\\Delta$ U-B $\\sim$ 0.2mag\nwhich corresponds to Fe/H$\\sim$-1.5. Thus clusters which on the basis of\ntheir Main Sequences and the presence of Cepheids, must be less than 10$^8$\nyr old, would be implied to have near halo metallicity. Previously, \\citeasnoun{pantos} find Fe/H$\\sim$-0.3 for these 2 clusters. Also \\citeasnoun{frycar} find Fe/H=-0.2$\\pm$0.02 for NGC7790 while finding Fe/H=-0.37$\\pm$0.03 for NGC6664 based on spectroscopy\nof the Cepheids in these clusters themselves. Also according to the\nGalactocentric metallicity gradient which is usually taken to lie in the\nrange -0.02-0.1dex kpc$^{-1}$ \\cite{rana}, there should only be on the\naverage $\\Delta$ Fe/H $\\sim$ 0.3 in the range of metallicity covering these\nclusters.\n\n\nOn the other hand, it should be noted that Panagia \\& Tosi's Fe/H estimates\nare based on more poorly measured estimates of UV excess than those\npresented here and also that there is little agreement between the\nmetallicity estimates of Fry and Carney and those of Panagia \\& Tosi.\nMeasuring the metallicity of the Cepheids themselves as attempted by Fry\nand Carney is difficult since the effective temperature is a function of the\nlight curve phase and a small difference in estimated temperature can make\na large difference in metallicity. Also the Galactocentric metallicity\ngradient at least as measured for open clusters depends on relatively poor\nU-B photometry at the limit of previous data from \\citeasnoun{janes},\n\\citeasnoun{camgrad} and \\citeasnoun{pantos}. In any case, it is well accepted\nthat the dispersion in metallicity around the mean gradient is indeed high,\nwith the range -0.6$<$Fe/H$<$+0.3 at the Solar position. Further \\citeasnoun{geisler} using Washington photometry to estimate metallicity also found an\nexample of a cluster, NGC 2112, only $\\sim$ 0.8kpc outside the solar radius\nwith Fe/H=-1.2, although this cluster is older than those discussed here.\n\nHowever, it would also seem that the tightness of the P-L relations in Fig. \\ref{fig:PL}\ncould form a final argument against the idea that NGC7790 has\nFe/H$\\sim$-1.5. If the\nmetallicity of the cluster NGC7790 was really Fe/H$\\sim$-1.5 then at given B-V, MS\nstars would be sub-dwarfs with $\\sim$1mag fainter absolute V magnitudes than\nnormal solar metallicity main sequence stars (see Cameron, 1984, Figs. 5,6). Thus since we \nhave used a normal Main Sequence to derive the distance to NGC7790,\nis it not surprising that the Cepheids in these clusters lie so tight on the\nP-L relation when they should be a magnitude too bright if the low metallicity\nhypothesis is correct. The only way that the low metallicity hypothesis for\nNGC7790 could survive this argument is if it were postulated that\n{\\it the effect of metallicity on Main sequence star magnitude and Cepheid\nmagnitude were the same - then the effect of our derived distance modulus\nbeing $\\sim$1 magnitude too high would be cancelled out by the fact that the Cepheid is\nactually sub-luminous by 1 magnitude because of metallicity which would leave\nthe Cepheid tight on the P-L relation as observed.} This might not be too\ncontrived if a low metallicity Cepheid prefers to oscillate about its\nsubdwarf, rather than solar metallicity, zero-age luminosity (at fixed effective temperature) on the Main Sequence. \nThis would lead to a strong implied metallicity effect on the Cepheid PL({\\it V})\nand PL({\\it K}) zeropoints; the implication would be that $\\frac{\\delta M}{\\delta Fe/H} \\sim$0.66 in the sense that lower metallicity Cepheids are fainter. This\ncoefficient is within the range that has been discussed for the empirical\neffects of metallicity on Cepheids by \\citeasnoun{kenn} and \\citeasnoun{gould} although the\nmost recent work by \\citeasnoun{kenn} appears to give a lower value of\n$\\frac{\\delta M}{\\delta Fe/H} \\sim$0.24$\\pm$0.16 again in the same sense.\n\nThe immediate effect on the distance to the LMC with Fe/H=-0.3 is that our\nestimate of its distance modulus would decrease from 18.5 to 18.3. However,\nsince all that is determined at the LMC is the slope of the P-L relation,\nthen the zeropoints we have derived in Table 5 from the Galactic Cepheids\nwould still refer the P-L relation to the Galactic zeropoint. The ultimate\neffect on H$_{\\circ}$ would then be decided by the metallicity of the Cepheids, for\nexample, in the galaxies observed by the HST for the Distance Scale Key\nproject \\citeasnoun{ferr}, by \\citeasnoun{tanvir} in the case of the Leo I Group and by\n\\citeasnoun{saha} in the case of SNIa. \\citeasnoun{zarit} have measured metallicities\nin these galaxies already but if the dispersion in Cepheid metallicity is as\nlarge as it is implied to be in the Galaxy then there may be some signature in\na wider dispersion in the Cepheid P-L relations in at least the high\nmetallicity cases. The possibility of detecting this signature is currently\nbeing investigated (Shanks et al 2000 in prep.)\n\n% The fact that HST Cepheid distances to SNIa galaxies which frequently have\n%quite low metallicities also produce a quite tight Hubble Diagram is also\n%either a problem for the metallicity hypothesis or an indication that\n%there is also a cancelling metallicity effect on SNIa Bmax! But\n%think it is too early to discuss this here...\n\nObviously the most direct route to checking the metallicity explanation for\nthe anomalous behaviour seen in the {\\it U-B:B-V} diagrams is to obtain\nmedium-high dispersion spectroscopy for a sample of F stars in \nNGC7790 and NGC6664 to determine the metallicity directly for these main\nsequence stars. Currently proposals are in to use WHT ISIS spectrograph for this purpose.\n\n%Currently, these observations are being attempted using the\n%AUTOFIB+WYFFOS spectrograph at the WHT (Hoyle et al 1999).\n\n\\section{Conclusions} \\label{sec:conc}\n\n\nWe have presented colour-colour diagrams and colour-magnitude diagrams of a\nsample of galactic clusters which contain or are associated with Cepheids.\nAll the clusters have been observed using similar methods and the data\nreduction and extraction has also been done with similar techniques. The use\nof the improved {\\it U}-band data has allowed powerful new checks of previous E({\\it B-V}) estimates over a wide range of magnitudes.\n%the ZAMS to be populated over a wide range of E({\\it B-V}) values. \nIn order to estimate the reddenings and\ndistance moduli, we have fitted all the clusters in the same manner and have not\nattempted to correct for differential reddening but instead taken a simpler\napproach and fitted the average reddening value of the cluster. In most cases the differences that we have found between values for the reddening and distance\nmodulus are small and where there are significant differences these can\nmostly be explained by comparing whether the ZAMS fit was made to the centre or to the edge of the colour-colour and colour-magnitude diagrams.\n\n\nThe Cepheid P-L relations found from fitting the best sample are\nM$_V$=-2.81$\\times$log(P)-1.332 and M$_K$=-3.44$\\times$log(P)-2.20 and a distance\nmodulus to the LMC of 18.54$\\pm$0.32 in the {\\it V}-band and 18.46$\\pm$0.29 in the\n{\\it K}-band giving an overall distance modulus to the LMC of 18.50$\\pm$0.3,\nignoring any possible effect of metallicity. These results for both the PL\nrelations and the LMC distance are consistent with the previous results of\nLaney \\& Stobie (1994) although the improved distances and reddenings have\nincreased the errors over what was previously claimed. These increased errors\nmean that our result for the PL({\\it V}) relation are also consistent with the\nresult of Feast \\& Catchpole (1997) from Hipparcos measurements of Cepheid\nparallaxes, although this gives rise to an LMC distance modulus of $\\mu_{\\circ}$=\n18.66$\\pm$0.1 as opposed to our $\\mu_{\\circ}$=18.51$\\pm$0.3.\n\n\n\nWith the improved {\\it U}-band data, we find that for at least two of the clusters, the\ndata in the {\\it U-B:B-V} two-colour diagram is not well fitted by the solar\nmetallicity ZAMS. One possibility is that significant metallicity variations\nfrom cluster to cluster may be affecting the {\\it U-B} colours of F- and G-type\nstars. The problem is that the metallicity variations this would require are\nmuch larger than expected for young, open clusters. More work is therefore\nrequired to determine the metallicity of the individual main sequence stars\nin each of the clusters NGC7790 and NGC6664. If metallicity is\nproven to be the cause of the anomalous {\\it U-B:B-V} relations, then it would imply that the Cepheid P-L relation \nin both the visible and the near-infrared is strongly affected by metallicity.\n\n\n\\vspace{-0.6cm}\n\n\\section*{Acknowledgments} FH acknowledges the receipt of a PPARC\nstudentship. We thank Floor van Leeuwen for useful discussions and we thank\nHenry McCracken and Nigel Metcalfe for help with the Calar Alto data\nreduction and Patrick Morris for the WHT reduction. We thank the PATT\ntelescope committee for the time on the UKIRT, JKT and WHT telescopes.\nWe thank CTIO for supporting this project and \nthe Calar Alto time allocation committee.\n\n\n\\begin{thebibliography}{}\n\\harvarditem{Allen}{1973}{allen}Allen, C. 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J., Metcalfe, N. \\& Shanks, T. in preparation\n\\harvarditem{Madore}{1975}{madly6}Madore, B. F., 1975, A\\&A, 38, 471\n\\harvarditem{Madore \\& van den Bergh}{1975}{MVDB}Madore, B. F., Van den\nBergh, S., 1975, ApJ, 197, 55 \n\\harvarditem {Menzies et al}{1991}{menz} Menzies, J. W., Marang, F., Laing, J. D., Coulson, I. M. \\& Engelbrecht, C.\nA., 1991, MNRAS, 248, 642 \n\\harvarditem{Mermilliod}{1981}{mermZAMS}Mermilliod, J. C., 1981, A\\&A, 97, 235\n\\harvarditem{Moffat \\& Vogt}{1975}{moff}Moffat, A. F. J. \\& Vogt, N., 1975, A\\&AS, 20, 155\n\\harvarditem{Panagia and Tossi}{1991}{pantos} Panagia, N. \\& Tosi, M., 1980,\nA\\&As, 81, 375 \n\\harvarditem{Pedreros et al}{1984}{ped} Pedreros H., Madore\nB.F. \\& Freedman W.L., 1984, ApJ, 286, 563 \n\\harvarditem{Rana}{1991}{rana}Rana, N. C., 1991, ARA\\&A, 29, 129 \n\\harvarditem{Romeo et al}{1989}{romeo}Romeo, G., Bonifazi, A., Fusi Pecci, F. \\& Tosi, M., 1989,\nMNRAS, 240,459 \n\\harvarditem{Saha}{1999}{saha} Saha, A., Sandage, S., Tamman, G.A., Labhardt, L., Macchetto, F.D. \\& Panagia, N., 1999, ApJ, 522, 802\n\\harvarditem{Sandage}{1958}{sand7790}Sandage, A., 1958, ApJ,128, 150 \n\\harvarditem{Sandage}{1960}{sandm25}Sandage, A., 1960, ApJ, 131, 610 \n\\harvarditem{Schmidt-Kaler}{1982}{SK} Schmidt-Kaler, T., 1982, in\nShaifers K., Voigt H. H., eds, Landolt-Bornstein VI 2b, Springer-Verlag,\nBerlin, p. 12 \n\\harvarditem{Shanks et al.}{2000}{Shanks20} Shanks, T, et al. in preperation\n\\harvarditem{Sharpless}{1963}{sharp}Sharpless, S., 1963, in\nBasic Astronomical Data, ed. K. Aa. Strand (Chicago: University of Chicago\nPress), p. 225 \n\\harvarditem{Talbert}{1975}{talb} Talbert, F. D., 1975, Publ. Astron. Soc. Pac. 87, 341\n\\harvarditem{Tanvir, Ferguson and Shanks}{1999}{tanvir2}Tanvir, N. 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The arrow\nindicates the direction of the reddening vector.} \\label{fig:ubbv}\n\\end{figure*}\n\n\n\\begin{figure*} \n\\begin{tabular}{ccc} \n{\\epsfxsize=5truecm \\epsfysize=5truecm\n\\epsfbox[60 170 550 620]{fig12.ps}} \n\\end{tabular}\n\\caption{The {\\it V-B:V} diagrams for the clusters used in the study}\n\\label{fig:bv} \\end{figure*}\n\n\n\n\\begin{figure*} \\begin{tabular}{c} \n{\\epsfxsize=5truecm \\epsfysize=5truecm\n\\epsfbox[60 170 550 620]{fig13.ps}}\n\\end{tabular}\n\\caption{The {\\it V-K:V} diagrams for the clusters used in the study}\n\\label{fig:vk} \\end{figure*}\n\n\n\\begin{figure*} \n\\begin{tabular}{c} \n{\\epsfxsize=5truecm \\epsfysize=5truecm\n\\epsfbox[60 170 660 620]{fig14.ps}}\n\\end{tabular}\n\\caption{A comparison between the zero point compared in this work with the\nzero point obtained in previous photoelectric studies. The full reference\nfor the comparison is given in Table \\ref{tab:resids}. Only the clusters\nwhere an independent zero points was obtained are shown.} \\label{fig:zero}\n\\end{figure*}\n\n\n\\begin{figure*} \n\\begin{tabular}{c} \n{\\epsfxsize=5truecm \\epsfysize=5truecm\n\\epsfbox[60 170 660 620]{fig15.ps}} \n\\end{tabular}\n\\caption{A check for any colour dependent relationship between the work in\nthis study and previous photoelectric work. The full reference for the\ncomparison is given in Table \\ref{tab:resids}. Only the clusters where an\nindependent zero points was obtained are shown.} \\label{fig:col}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{tabular}{c}\n{\\epsfxsize=5truecm \\epsfysize=5truecm\n\\epsfbox[60 170 660 620]{fig16.ps}}\n\\end{tabular}\n\\caption{The zero points and check against colour for the clusters where previous work had to be relied upon for calibration purposes.}\n\\label{fig:other}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{tabular}{c}\n{\\epsfxsize=5truecm \\epsfysize=5truecm\n\\epsfbox[60 170 660 620]{fig17.ps}}\n\\end{tabular}\n\\caption{Comparison of all the different sources of photometry for NGC7790. Note there is no {\\it U}-band data in the study by \\protect\\citeasnoun{romeo}}\n\\label{fig:7790comp}\n\\end{figure*}\n\n\n\\end{document}\n\n\n\n"
}
] |
[
{
"name": "astro-ph0002521.extracted_bib",
"string": "\\begin{thebibliography}{}\n\\harvarditem{Allen}{1973}{allen}Allen, C. W., Astrophysical Quantities, 3rd\nEdition, 1973, The Athlone Press \n\\harvarditem{Arp}{1958}{arpEV}Arp, H., 1958, ApJ, 128, 166 \n\\harvarditem{Arp}{1960}{arpCV}Arp, H., 1960, ApJ, 131,321 \n\\harvarditem{Arp et al}{1959} {arp129} Arp, H., Sandage, A., \\& Stephens, C., 1959, ApJ, 130, 80 \n\\harvarditem{Barrell}{1980}{barrell} Barrell, S., 1980, ApJ, 240, 145\n\\harvarditem{Cameron}{1984}{camFG}Cameron, L. M., 1984, A\\&A 146, 59\n\\harvarditem{Cameron}{1985}{camgrad}Cameron, L. M., 1985, A\\&A 147, 39\n\\harvarditem{Carney}{1979}{carney}Carney, B. W., 1979, ApJ, 233, 211\n\\harvarditem{Coulson and Caldwell}{1984}{coulson} Coulson, I. M. \\&\nCaldwell, J. A. R., 1984, SAAO Preprint No. 337 \n\\harvarditem{Croom et al}{1998}{croomphot}Croom, S. M., Ratcliffe, A., Parker, Q. A., Shanks, T., Boyle, B. J. \\& Smith, R. J., 1999, MNRAS, 306, 592 \n\\harvarditem{Eggen}{1983}{eggen} Eggen, O.J., 1983, AJ, 88, 379\n\\harvarditem{Feast}{1957}{feast} Feast, M. W., 1957, 117, 193\n\\harvarditem{Feast \\& Catchpole}{1997}{FC}Feast, M. W. \\& Catchpole, R. M.,\n1997, MNRAS, 286, L1 \n\\harvarditem{Feast \\& Walker}{1987}{FW}Feast, M. W. \\& Walker, A. R., 1987, ARA\\&A, 25, 345\n\\harvarditem{Ferrarese et al.}{1999}{ferr} Ferrarese, L. et al., 1999, astro-ph/9911193\n\\harvarditem{Fry \\& Carney}{1997}{frycar} Fry, A. M. \\& Carney, B. W., 1997, AJ, 113, 1073 \n\\harvarditem{Geisler}{1987}{geisler} Geisler, D., 1987, AJ, 94, 84\n\\harvarditem{Gould}{1994}{gould} Gould, A., 1994, ApJ, 426, 542\n\\harvarditem{Graham}{1982}{graham} Graham, J. A., 1982, PASP, 94, 244 \n\\harvarditem {Guetter}{1991}{guetter}Guetter, H. H., 1991, AJ, 103, 197 \n\\harvarditem{Hunt et al}{1998}{hunt} Hunt, L. K., Mannucci, F., Testi,\nL., Migliorini, S., Stanga, R. M., Baffa, C., Lisi, F. \\& Vanzi, L.,1998,\nAJ, 115, 2594 \n\\harvarditem{Hoag et al}{1961}{hoag}Hoag A.A., Johnson H.L.,\nIriarte B., Mitchell R.I., Hallam K.L. \\& Sharpless S., 1961, Publ. US. Nav.\nObs, 17, 347 \n\\harvarditem{Janes}{1979}{janes} Janes, K. A., 1979, ApJS, 39, 135\n\\harvarditem {Johnson}{1960}{johnm25}Johnson, H. L., 1960, ApJ, 131, 620 \n\\harvarditem{Kennicutt et al}{1998}{kenn} Kennicutt, R., et al., ApJ, 498, 181\n\\harvarditem{Krisciunas et al}{1987}{kris} Krisciunas, K., Sinton, W., Tholen, K., Tokunaga, A., Golisch, W., Griep, D., Kaminski, C., Impey, C. \\& Christian, C., 1987, PASP, 99, 887\n\\harvarditem {Kraft}{1958}{Kraft} Kraft, R. P., 1958, ApJ, 128, 1610 \n\\harvarditem{Laney \\& Stobie}{1993}{LS1}Laney, C. D. \\& Stobie, R. S.,\n1993, MNRAS, 263, 921 \n\\harvarditem{Laney \\& Stobie}{1994}{LS2}Laney, C. D. \\& Stobie, R. S., 1994, MNRAS, 266, 441\n\\harvarditem{Landolt}{1992}{landolt}Landolt, A. U., 1992, AJ, 104, 372\n\\harvarditem{McCracken}{1999}{henry}McCracken, H. J., 1999, PhD Thesis, University of Durham\n\\harvarditem{McCracken et al.}{2000}{HNT}McCracken, H. J., Metcalfe, N. \\& Shanks, T. in preparation\n\\harvarditem{Madore}{1975}{madly6}Madore, B. F., 1975, A\\&A, 38, 471\n\\harvarditem{Madore \\& van den Bergh}{1975}{MVDB}Madore, B. F., Van den\nBergh, S., 1975, ApJ, 197, 55 \n\\harvarditem {Menzies et al}{1991}{menz} Menzies, J. W., Marang, F., Laing, J. D., Coulson, I. M. \\& Engelbrecht, C.\nA., 1991, MNRAS, 248, 642 \n\\harvarditem{Mermilliod}{1981}{mermZAMS}Mermilliod, J. C., 1981, A\\&A, 97, 235\n\\harvarditem{Moffat \\& Vogt}{1975}{moff}Moffat, A. F. J. \\& Vogt, N., 1975, A\\&AS, 20, 155\n\\harvarditem{Panagia and Tossi}{1991}{pantos} Panagia, N. \\& Tosi, M., 1980,\nA\\&As, 81, 375 \n\\harvarditem{Pedreros et al}{1984}{ped} Pedreros H., Madore\nB.F. \\& Freedman W.L., 1984, ApJ, 286, 563 \n\\harvarditem{Rana}{1991}{rana}Rana, N. C., 1991, ARA\\&A, 29, 129 \n\\harvarditem{Romeo et al}{1989}{romeo}Romeo, G., Bonifazi, A., Fusi Pecci, F. \\& Tosi, M., 1989,\nMNRAS, 240,459 \n\\harvarditem{Saha}{1999}{saha} Saha, A., Sandage, S., Tamman, G.A., Labhardt, L., Macchetto, F.D. \\& Panagia, N., 1999, ApJ, 522, 802\n\\harvarditem{Sandage}{1958}{sand7790}Sandage, A., 1958, ApJ,128, 150 \n\\harvarditem{Sandage}{1960}{sandm25}Sandage, A., 1960, ApJ, 131, 610 \n\\harvarditem{Schmidt-Kaler}{1982}{SK} Schmidt-Kaler, T., 1982, in\nShaifers K., Voigt H. H., eds, Landolt-Bornstein VI 2b, Springer-Verlag,\nBerlin, p. 12 \n\\harvarditem{Shanks et al.}{2000}{Shanks20} Shanks, T, et al. in preperation\n\\harvarditem{Sharpless}{1963}{sharp}Sharpless, S., 1963, in\nBasic Astronomical Data, ed. K. Aa. Strand (Chicago: University of Chicago\nPress), p. 225 \n\\harvarditem{Talbert}{1975}{talb} Talbert, F. D., 1975, Publ. Astron. Soc. Pac. 87, 341\n\\harvarditem{Tanvir, Ferguson and Shanks}{1999}{tanvir2}Tanvir, N. R., Ferguson, H.C. \\& Shanks, T., 1999, MNRAS, 310, 175\n\\harvarditem{Tanvir et al}{1995}{tanvir}Tanvir, N. R., Shanks, T., Ferguson, H.C. \\& Robinson, D. R. T., 1995, Nature, 377, 27\n\\harvarditem{Thackeray et al}{1962}{thack}Thackeray, A. D., Wesselink, A. J., \\& Harding, G. A., 1962,\nMNRAS, 124, 445 \n\\harvarditem{Tokunaga}{1998}{tok}Tokunaga, A. T., 1998, in\nAstrophysical Quantities (Revised), 4th edition, submitted - see UKIRT web\npage. \n\\harvarditem{Turner}{1979a}{turn6823}Turner, D. G., 1979, JRASC, 73,\n2, 74 (a) \n\\harvarditem{Turner}{1979b}{turnZAMS}Turner, D. G., 1979, Publ. Astron. Soc. Pacific, 91, 642 (b) \n\\harvarditem{Turner}{1980}{turn35}Turner, D. G., 1980, ApJ, 240, 137 \n\\harvarditem{Turner}{1981}{turn6649} Turner, D. G., 1981, AJ, 86, 231\n\\harvarditem{Turner}{1984}{turnWZ}Turner, D. G.,\n1984, Publication of the Astronomical Society of the Pacific, 96, 422\n\\harvarditem{Turner et al}{1992}{turn129}Turner, D. 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astro-ph0002522
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Determination of cosmological parameters from large scale structure observations
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% The possibility of determining cosmological parameters on the basis of a wide set of observational data including the Abell-ACO cluster power spectrum and mass function, peculiar velocities of galaxies, the distribution of Ly-$\alpha$ clouds and CMB temperature fluctuations is analyzed. Using a $\chi^2$ minimization method, assuming $\Omega_{\Lambda}+\Omega_m =1 $ and no contribution from gravity waves, we found that a tilted $\Lambda$MDM model with one sort of massive neutrinos and the parameters $n=1.12\pm 0.10$, $\Omega_m=0.41\pm 0.11$ ($\Omega_{\Lambda}=0.59\pm0.11$), $\Omega_{cdm}=0.31\pm 0.15$, $\Omega_{\nu}=0.059\pm 0.028$, $\Omega_b=0.039\pm 0.014$ and $h=0.70\pm 0.12$ (standard errors) matches observational data best. $\Omega_{\nu}$ is higher for more species of massive neutrinos, $\sim 0.1$ for two and $\sim 0.13$ for three species. $\Omega_m$ raises by $\sim 0.08$ and $\sim 0.15$ respectively. The 1$\sigma$ (68.3\%) confidence limits on each cosmological parameter, which are obtained by marginalizing over the other parameters, are $0.82\le n\le1.39$, $0.19\le\Omega_m\le 1$ ($0\le\Omega_{\Lambda}\le 0.81$), $0\le\Omega_{\nu}\le 0.17$, $0.021\le \Omega_b\le 0.13$ and $0.38\le h\le 0.85$. Varying only a subset of parameters and fixing the others shows also that the observational data set used here rules out pure CDM models with $h\ge 0.5$, scale invariant primordial power spectrum, zero cosmological constant and spatial curvature at a very high confidence level, $>99.99\%$. The corresponding class of MDM models are ruled out at $\sim 95\%$ C.L. It is notable also that this data set determines the amplitude of scalar fluctuations approximately at the same level as COBE four-year data. It indicates that a possible tensor component in the COBE data cannot be very substantial.
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[
{
"name": "novosc.tex",
"string": "\\documentstyle[twoside,cosmion,epsf]{article}\n\n\n% Some personal definitions that make life easier:\n\n\\newcommand {\\Mpc} {\\mbox{$h^{-1}$ Mpc \\,}}\n\\newcommand {\\kpc} {\\mbox{$h^{-1}$ kpc \\,}}\n\\newcommand{\\mincir}{\\raise -2.truept\\hbox{\\rlap{\\hbox{$\\sim$}}\\raise5.truept\n\\hbox{$?$}\\ }}\n\\newcommand{\\gr}{\\kern 2pt\\hbox{}^\\circ{\\kern -2pt K}} % ====? GRADI KELVIN\n\\newcommand{\\magcir}{\\raise -2.truept\\hbox{\\rlap{\\hbox{$\\sim$}}\\raise5.truept\n\\hbox{$?$}\\ }}\n\\newcommand{\\oml}{\\Omega_{\\Lambda}}\n\\newcommand{\\Om}{\\Omega}\n\\newcommand{\\apj}{ApJ}\n\\newcommand{\\apjl}{ApJL}\n\\newcommand{\\mnras}{MNRAS}\n\\newcommand{\\aap}{AA}\n\\newcommand{\\apjs}{ApJS}\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\bea}{\\begin{eqnarray}}\n\\newcommand{\\eea}{\\end{eqnarray}}\n\\newcommand{\\De}{\\Delta}\n\\newcommand{\\si}{\\sigma}\n\\newcommand{\\La}{\\Lambda}\n\\newcommand{\\etal}{{et al.}}\n\n\\heads{Determination of cosmological parameters} \n{Large scale structure of the Universe \\ \\ {\\rm B. Novosyadlyj et al.}}\n\n\\begin{document}\n\n\\Arthead{1}{1}% Do not change this line\n\n\n\\Title{Determination of cosmological parameters from\n \tlarge scale structure observations}\n{B.~Novosyadlyj$^1$, R.~Durrer$^2$,\nS.~Gottl\\\"ober$^3$, V.N.~Lukash$^4$, S.~Apunevych$^1$}\n{$^1$Astronomical Observatory of L'viv State University, Kyryla and\nMephodia str.8, 290005, L'viv, Ukraine \\\\\n$^2$Department de Physique Th\\'eorique, Universit\\'e de Gen\\`eve,\nQuai Ernest Ansermet 24, CH-1211 Gen\\`eve 4, Switzerland \\\\\n$^3$ Astrophysikalisches Institut Potsdam, An der Sternwarte 16,\nD-14482 Potsdam, Germany \\\\\n$^4$Astro Space Center of Lebedev Physical Institute of RAS,\nProfsoyuznaya 84/32, 117810 Moscow, Russia}\n\n\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\Abstract{%\n\n\nThe possibility of determining cosmological parameters on the basis of a wide set\nof observational data including the Abell-ACO cluster power spectrum and mass\nfunction, peculiar velocities of galaxies, the distribution of Ly-$\\alpha$ clouds\nand CMB temperature fluctuations is analyzed. Using a $\\chi^2$ minimization method,\nassuming $\\Omega_{\\Lambda}+\\Omega_m =1 $ and no contribution from gravity\nwaves, we found that a tilted $\\Lambda$MDM model with one sort\nof massive neutrinos and the parameters $n=1.12\\pm 0.10$, $\\Omega_m=0.41\\pm 0.11$\n($\\Omega_{\\Lambda}=0.59\\pm0.11$), $\\Omega_{cdm}=0.31\\pm 0.15$,\n$\\Omega_{\\nu}=0.059\\pm 0.028$, $\\Omega_b=0.039\\pm 0.014$ and $h=0.70\\pm 0.12$\n(standard errors) matches observational data best. \n$\\Omega_{\\nu}$ is higher for more species of massive neutrinos, $\\sim 0.1$\nfor two and $\\sim 0.13$ for three species. $\\Omega_m$ raises by $\\sim 0.08$\nand $\\sim 0.15$ respectively.\nThe 1$\\sigma$ (68.3\\%) confidence limits on each cosmological parameter, which are\nobtained by marginalizing over the other parameters, are $0.82\\le n\\le1.39$,\n$0.19\\le\\Omega_m\\le 1$ ($0\\le\\Omega_{\\Lambda}\\le 0.81$),\n$0\\le\\Omega_{\\nu}\\le 0.17$, $0.021\\le \\Omega_b\\le 0.13$ and\n$0.38\\le h\\le 0.85$.\nVarying only a subset of parameters and fixing the others\nshows also that the observational data set used here rules out\npure CDM models with $h\\ge 0.5$, scale invariant primordial power\nspectrum, zero cosmological constant and spatial curvature at a very\nhigh confidence level, $>99.99\\%$.\nThe corresponding class of MDM models are ruled out at $\\sim 95\\%$ C.L.\nIt is notable also that this data set \ndetermines the amplitude of scalar fluctuations approximately at the\nsame level as COBE four-year data. It indicates that a possible\ntensor component in the COBE data cannot be very substantial. }\n\n\\section{Introduction}\n\nThe last years of the past century are marked by huge efforts of the\ncommunity of astronomers, physicists and astrophysicists devoted to determine\nthe most fundamental parameters of our Universe, the cosmological parameters.\nThe most important among them are the\nmass densities of baryons $\\Omega_b$ (in units of the critical density) and \nof cold dark matter $\\Omega_{cdm}$, the neutrino rest masses $m_{\\nu}$\nand their total density $\\Omega_{\\nu}$, the value of cosmological\nterm $\\Lambda$ (or $\\Omega_{\\Lambda}$), the Hubble constant $H_0$,\nthe spatial curvature parameter $\\Omega_k$ and the\nslopes $n$ and amplitudes $A$ of the primordial power spectra of \nscalar and tensor fluctuations.\n\nThe primordial ratio of the number of deuterium to hydrogen nuclei (D/H) \ncreated in Big Bang nucleosynthesis is the most sensitive measure of the \ncosmological density of baryons $\\Omega_b$. Quasar absorption systems give\ndefinite measurements of the primordial deuterium and the most accurate value \nof baryon density obtained recently in this way is $\\Omega_bh^2=0.019\\pm0.0024$\n\\cite{bur99}.\n\nThe measurements of the neutrino rest mass is not so certain, unfortunately.\nUp-to-day we have only some indications for the range where it may be found.\nThe oscillations of solar and atmospheric neutrinos registered by the \nSuperKamiokande experiment show that the difference of rest masses between \n$\\tau -$ and $\\mu$-neutrinos is $0.02<\\Delta m_{\\tau \\mu} < 0.08eV$\n\\cite{fu98,pr98}. This also provides a lower limit for the neutrino\nmass, $m_{\\nu}\\ge\n|\\Delta m|$ and does not exclude models with cosmologically\nsignificant values $\\sim 1-20eV$. Therefore, at least two species of\nneutrinos can have approximately equal masses in this range. Some\nversions of elementary particle theories predict $m_{\\nu _e}\\approx\nm_{\\nu _\\tau}\\approx 2.5eV$ and $m_{\\nu _{\\mu}}\\approx m_{\\nu _s}\\sim\n10^{-5}eV$, where ${\\nu _e}$, ${\\nu _\\tau}$, ${\\nu _{\\mu}}$ and ${\\nu\n_s}$ denote the electron, $\\tau -$, $\\mu -$ and sterile neutrinos\naccordingly (e.g. \\cite{dol95}). The strongest \nupper limit for the neutrino mass comes from the observed large scale \nstructure of our Universe: $\\sum_{i} m_{\\nu_i}/94{\\rm eV}\\le 0.3h^2$.\nSince observations give for the Hubble parameter an upper limit of\n$h=0.8$ one gets $\\sum_{i} m_{\\nu_i}\\le 18eV$. It is interesting to note that\nthis upper limit coincides roughly with the upper limit for the electron\nneutrino mass obtained from the supernova explosion \nSN1987A and tritium $\\beta$-decay experiments.\n \nImportant conclusions about measurements of matter density\n$\\Omega_m$ ($\\equiv \\Omega_b+\\Omega_{cdm}+\\Omega_{\\nu}$)\ncome from the Supernova Cosmology Project and the High-z Supernova Search.\nIn particular, the relation of observed brightness vs. redshift for SNeIa\nshows that distant supernovae are fainter than expected for a\ndecelerating Universe, and, thus, more distant. This can be\ninterpreted as an accelerated expansion\nrate, or $\\Omega_{\\Lambda}>0$. The best-fit value is\n$\\Omega_{\\Lambda}={4\\over 3}\\Omega_m + {1\\over 3}\\pm 0.1$\n(1$\\sigma$ error) and $\\Omega_m=1$ models are ruled out\nat the 8$\\sigma$ level \\cite{per98}. For a flat Universe\n$\\Omega_m + \\Omega_{\\Lambda}=1$ ($\\Omega_k=0$) the best-fit\nvalues are $\\Omega_m=0.25\\pm 0.1$ and $\\Omega_{\\Lambda}=0.75\\pm 0.1$\n\\cite{per98,rie98,bah99}.\n\nAn upper limit of $\\Omega_{\\Lambda}<0.7$ (95\\% C.L.) follows from\ngravitational lensing statistics\\cite{bar98,fal98}, just\n consistent with distant supernovas results.\n\nStrong evidence against an open Universe can be derived from recent\nmeasurements of the position of the first acoustic peak in\nthe cosmic microwave background (CMB) power spectrum by\nthe Boomerang experiment\\cite{mau99}. The $1\\sigma$ range for the curvature\nparameter derived from this experiment is \n$-0.25\\le \\Omega_k\\le 0.15$ \\cite{mel99}\nand the mean value is close to the flat Universe, $\\Omega_k\\approx 0$.\n\nCurrently there are a few completely independent and broad\nroutes to the determination of the Hubble constant $H_0$. The\ndirect experiments can be divided into three\ngroups: the gravitational lens time delay methods,\nthe Sunyaev-Zel'dovich method for clusters and extra-galactic\ndistance measurements. Almost all observations yield values of $H_0$ \nin the range 50-80 km/sec/Mpc.\n\nOther independent methods for the determination of cosmological\nparameters are based on large scale structure (LSS)\nobservations. Their advantage is that all parameters mentioned\nabove can be determined together because the form and amplitude\nof the power spectrum of density fluctuations are rather sensitive\nto all of them. Their disadvantage is that they are model dependent.\nThis approach has been carried out in several\npapers (e.g. \\cite{atr97,lin97,Teg99,bri99,nov99,phill} and references therein)\nand it is also the goal of this paper. The papers on this subject differ by the\nnumber of parameters and the set of observational data included into the\nanalysis. In this paper a total of 23 measurements from sub-galaxy\nscales (Ly-$\\alpha$ clouds) over cluster scales up to horizon scale\n(CMB quadrupole) is used to determine eight cosmological parameters,\nnamely the tilt of the primordial spectrum $n$, the densities of\ncold dark matter $\\Omega_{cdm}$, hot dark matter $\\Omega_{\\nu}$,\nbaryons $\\Omega_b$ and cosmological constant $\\Omega_{\\Lambda}$,\nthe number of massive neutrino species $N_{\\nu}$, the Hubble parameter\n$H_0$ and, in addition, the bias parameter $b_{cl}$ for rich clusters \nof galaxies.\n We restrict ourselves to the analysis of spatially flat cosmological\nmodels with $\\Omega_{\\Lambda}+\\Omega_m=1$ ($\\Omega_k=0$) and to an inflationary\nscenario without tensor mode. We also neglect the effect of a possible\nearly reionization which could reduce the amplitude of the first\nacoustic peak in the CMB anisotropy spectrum.\n\nIn comparison to the companion paper\\cite{nov00} the influence\nof the uncertainties in the normalization of the scalar mode amplitude\ncaused by experimental errors on determination of cosmological parameters\nis also taken into account here.\n\n\\section{The experimental data set and our methods}\n\nWe use the power spectrum of Abell-ACO clusters \n\\cite{ein97,ret97}, measured in the range\n$0.03\\le k\\le 0.2h/$Mpc, as observational input. \nIts amplitude and slope at lower and larger scales\nare quite sensitive to baryon content $\\Omega_{b}$,\nHubble constant $h$, neutrino mass $m_{\\nu}$ and number of species of\nmassive neutrinos $N_{\\nu}$ \\cite{nov99}.\nThe total number of Abell-ACO data points with their errors used for minimization\nis 13, but not all of these points can be considered as independent \nmeasurements. Since we can accurately fit the power spectrum by an analytic \nexpression depending on three parameters only (the amplitude at large \nscales, the slope at small scales and the scale of the bend); we\nassign to the power spectrum 3 effective degrees of freedom.\n\nThe second observational data set which we use are \nthe position and amplitude of the first acoustic peak derived \nfrom the data on the angular power spectrum of CMB temperature fluctuations.\nTo determine the position and amplitude of the\nfirst acoustic peak we use a 6-th order polynomial fit to\nthe data set on CMB\ntemperature anisotropy, accumulated in Table 2 of our accompanied paper \n\\cite{nov00}, 51 data points in total. \nThe amplitude $A_p$ and position $\\ell_p$ of first acoustic peak\ndetermined from this fit are $79.6\\pm 16.5\\mu \\rm K$ and $253\\pm 70$\ncorrespondingly.\nThe statistical errors are estimated by edges of the $\\chi^2$-hyper-surface \nin the space of polynomial coefficients which corresponds to \n68.3\\% ($1\\sigma$) probability level under the assumption of\nGaussian statistics. \nAlso the mean weighted bandwidth of each experiment around \n$\\ell_p$ is added to obtain total $\\Delta \\ell_{p}$.\n\nA constraint on the amplitude of the matter density fluctuation power spectrum at\ncluster scale can be derived from the cluster mass and X-ray temperature\nfunctions. It is usually formulated in terms of the density\nfluctuation in a top-hat sphere of 8\\Mpc radius, $\\sigma_{8}$, which\ncan be easily calculated for the given initial power spectrum.\nAccording to the recent optical determination of the mass function of nearby \ngalaxy clusters \\cite{gir98} and taking into account the results from other authors\n(for references see \\cite{borg99}) we use the value \n$\\tilde \\sigma_{8}\\tilde\\Omega_m^{0.46-0.09\\Omega_m}=0.60\\pm 0.08$.\n>From the existence of three most massive clusters of galaxies observed\nat $z>0.5$ a further constraint has been established by Bahcall \\& Fan\n\\cite{bah98}: $\\tilde \\sigma_8\\tilde\\Omega_m^{\\alpha}=0.8\\pm 0.1\\;,$\nwhere $\\alpha =0.24$ if $\\Omega_{\\Lambda}=0$ and $\\alpha =0.29$ if\n$\\Omega_{\\Lambda}>0$ with $\\Omega_{\\Lambda}+\\Omega_m=1.$ \n\nA constraint on the amplitude of the linear power spectrum of\ndensity fluctuations in our vicinity comes from the study of galaxy\nbulk flow, the mean peculiar velocity of galaxies in sphere of radius \n$50h^{-1}$Mpc around our\nposition. We use the data given by Kollat \\& Dekel \\cite{kol97}, $\\tilde V_{50}=(375\\pm 85)$ km/s.\n\nA further essential constraint on the linear power spectrum of matter\nclustering at galactic and sub-galactic scales \n$k\\sim (2-40)h/$Mpc can be obtained\nfrom the Ly-$\\alpha$ forest of absorption lines seen in quasar spectra\n\\cite{gn98,cr98}. Assuming that the\nLy-$\\alpha$ forest is formed by discrete clouds of a physical extent\nnear Jeans scale in the reionized inter-galactic medium at $z\\sim 2-4$,\nGnedin \\cite{gn98} has obtained a constraint on the value of the\nr.m.s. linear density fluctuations $1.6<\\tilde \\sigma_{F}(z=3)<2.6$\n(95\\% C.L.) at Jeans scale for $z=3$ equal\n to $k_{F}\\approx 38\\Omega_m^{1/2}h/$Mpc \\cite{gn99}.\n\nThe procedure to\nrecover the linear power spectrum from the Ly-$\\alpha$ forest has been\nelaborated by Croft et al.\\cite{cr98}. Analyzing the absorption lines in a sample\nof 19 QSO spectra, they have obtained the following 95\\% C.L. constraint on the\namplitude and slope of the linear power spectrum at $z=2.5$ and\n$k_{p}=1.5\\Omega_m^{1/2}h/$Mpc\n\\begin{equation}\n\\tilde \\Delta_{\\rho}^2(k_p)\\equiv k_p^3P(k_p)/2\\pi^2=0.57\\pm 0.26,\n\\label{Deltarho}\n\\end{equation}\n\\begin{equation}\n\\tilde n_p\\equiv {\\Delta \\log\\;P(k)\\over \\Delta \\log\\;k}\\mid_{k_p}=-2.25\\pm 0.1.\n\\label{n_p}\n\\end{equation}\n\nIn addition to the power spectrum measurements we use\nthe constraints on the value of Hubble constant $\\tilde h=0.65\\pm\n0.15$ which is a compromise between measurements made by two groups:\n\\cite{Tamm99} and \\cite{mad98}. We also employ the \nnucleosynthesis\nconstraints on the baryon density of $\\tilde{\\Omega_bh^2} =\n0.019\\pm 0.0024$ (95\\% C.L.) \\cite{bur99}.\n\n\nIn order to find the best fit model we must evaluate the \nabove mentioned quantities for a given cosmological model. \n\nTo this end we use the accurate analytic\napproximations of the MDM transfer function $T(k;z)$ depending on\nthe parameters $\\Omega_m$, $\\Omega_b$, $\\Omega_{\\nu}$, $N_{\\nu}$, \n$h$ by Eisenstein \\& Hu \\cite{eh3}.\n\nThe linear power spectrum of matter density fluctuations is\n\\begin{equation}\nP(k;z)=Ak^nT^2(k;z)D_1^2(z)/D_1^2(0),\\label{pkz}\n\\end{equation}\nwhere $A$ is the normalization constant and \n$D_1(z)$ is the growth factor,\nuseful analytical approximation for which has been given by Carrol et al.\\cite{car92}.\n\nWe normalize the spectra using the 4-year COBE data which can be\nexpressed by the value of the density perturbation at the horizon crossing\nscale, $\\delta_h$ \\cite{lid96,bun97}. \nThe normalization constant is related to $\\delta_h$ by \n\\be\nA=2\\pi^{2}\\delta_{h}^{2}(3000/h)^{3+n}\n\\;{\\rm Mpc}^{3+n}.\\label{anorm}\n\\ee\n\nThe Abell-ACO power spectrum is given by the matter power\nspectrum at $z=0$ multiplied by a linear and scale independent\ncluster biasing parameter $b_{cl}$, which we include as a free parameter\n\\begin{equation}\nP_{A+ACO}(k)=b_{cl}^2 P(k;0).\\label{bias}\n\\end{equation}\n\nFor a given set of parameters $n$, $\\Omega_m$, $\\Omega_b$, $h$,\n$\\Omega_{\\nu}$, $N_{\\nu}$ and $b_{cl}$ the theoretical value of\n$P_{A+ACO}(k_j)$ can now be calculated for each observed scale $k_j$.\nLet's denote these values by $y_j$ ($j=1,...,13$).\n\nThe dependence of position and amplitude of the first acoustic\npeak in the CMB power spectrum on cosmological\nparameters has been investigated using the public code CMBfast by Seljak \\&\nZaldarriaga\\cite{sz96}. As expected, these characteristics are\nindependent on the hot dark matter content.\nWe determine the values $\\ell_p$ and $A_p$ for given parameters \n($n$, $h$, $\\Omega_b$ and $\\Omega_{\\Lambda}$) on a\n4-dimensional grid for parameter values in between the grid points we\ndetermine $\\ell_p$ and $A_p$ by linear interpolation.\nWe denote $\\ell_p$ and $A_p$ by $y_{14}$ and $y_{15}$ respectively.\n\nThe theoretical values of the other experimental constraints are\nobtained as follows: The density fluctuation $\\sigma_8$ is\ncalculated according to\n\\begin{equation}\n\\sigma_{8}^{2}={1\\over2\\pi^{2}}\\int_{0}^{\\infty}k^{2}P(k;0)W^{2}(8{\\rm Mpc}\\;k/h)dk,\n\\end{equation}\nwith $P(k;z)$ from Eq.~(\\ref{pkz}). We\nset $y_{16}=\\sigma_{8}\\Omega_m^{0.46-0.09\\Omega_m}$ and $y_{17} =\n\\sigma_{8}\\Omega^{\\alpha}$, where $\\alpha =0.24$ for\n$\\Omega_{\\Lambda}=0$ and $\\alpha =0.29$ for $\\Omega_{\\Lambda}>0$,\nrespectively.\n\nThe r.m.s. peculiar velocity of\ngalaxies in a sphere of radius $R=50h^{-1}$Mpc is\n\\begin{equation}\nV^{2}_{50}={1\\over 2\\pi^{2}}\n\\int_{0}^{\\infty}\nk^2P^{(v)}(k)e^{-k^{2}R_{f}^{2}}W^{2}(50{\\rm Mpc}~k/h)dk, \\label{V50th}\n\\end{equation}\nwhere $P^{(v)}(k)$ is the density-weighted power spectrum for the \nvelocity field \\cite{eh3}, \n$W(50{\\rm Mpc}~k/h)$ is a top-hat window function, and \n$R_{f}=12h^{-1}$Mpc is the radius of a Gaussian filter used for smoothing\nof the raw data.\nFor the scales considered $P^{(v)}(k)\\approx (\\Omega^{0.6}H_0)^2P(k;0)/k^2$.\nWe denote the r.m.s. peculiar velocity by $y_{18}$.\n\nThe value of the r.m.s. linear density perturbation from the formation of \nLy-$\\alpha$ clouds at redshift $z$ and scale $k_{F}$ is given by\n\\begin{equation}\n\\sigma_{F}^{2}(z)={1\\over\n2\\pi^{2}}\\int_{0}^{\\infty}k^{2}P(k;z)e^{(-k/k_F)^2}dk \\label{siF}.\n\\end{equation}\nWe set $\\sigma_{F}^{2}(z=3) y_{19}$. \n\nThe value of\n$\\Delta_{\\rho}^2(k_p,z)$ and the slope $n(z)$ are obtained from the\nlinear power spectrum $P(k;z)$ by \nEq.~(\\ref{Deltarho}) and Eq.~(\\ref{n_p}) at \n$z=2.5$ and $k_{p}=0.008H(z)/(1+z)({\\rm km/s})^{-1}$, and are denoted by\n $y_{20}$ and $y_{21}$ accordingly.\n\nFor all tests except Gnedin's Ly-$\\alpha$ test\nwe use the density weighted transfer function $T_{cb\\nu}(k,z)$\nfrom \\cite{eh3}. For $\\sigma_F$ the function $T_{cb}(k,z)$ is used\naccording to the prescription given by Gnedin \\cite{gn98}. Note, however, that\neven in the model with maximal $\\Omega_{\\nu}$ ($\\sim0.2$) the difference\nbetween $T_{cb}(k,z)$ and $T_{cb\\nu}(k,z)$ is less than \n$ 12\\%$ for $k\\le k_p$.\n\nFinally, the values of $\\Omega_bh^2$\nand $h$ are denoted by $y_{22}$ and $y_{23}$ respectively.\n\nUnder the assumption that the errors on the data points are Gaussian,\nthe deviations of the\ntheoretical values from their observational counterparts can be\ncharacterized by $\\chi^2$:\n\\begin{equation}\n\\chi^{2}=\\sum_{j=1}^{23}\\left({\\tilde y_j-y_j \\over \\Delta \\tilde y_j}\n\\right)^2, \\label{chi2}\n\\end{equation}\nwhere $\\tilde y_j$ and $\\Delta \\tilde y_j$ are the experimental data\nand their dispersions, respectively. The set of parameters $n$,\n$\\Omega_m$, $\\Omega_b$, $h$, $\\Omega_{\\nu}$, $N_{\\nu}$\nand $b_{cl}$ are then determined by minimizing $\\chi^2$ using the\nLevenberg-Marquardt method \\cite{nr92}. The derivatives of the predicted\nvalues w.r.t the search parameters required by this method are\ncalculated numerically using a relative step size of $10^{-5}$.\n\nThis method has been tested and has proven to be reliable,\nindependent on the initial values of parameters and it has \ngood convergence.\\\\[2mm]\n\n\\section{Results}\n\nThe determination of the parameters $n$, $\\Omega_m$, $\\Omega_b$, $h$,\n$\\Omega_{\\nu}$, $N_{\\nu}$ and $b_{cl}$ by the Levenberg-Marquardt $\\chi^2$\nminimization method is realized in the following way: we vary the\nset of parameters $n$, $\\Omega_m$, $\\Omega_b$, $h$,\n$\\Omega_{\\nu}$ and $b_{cl}$ and find the minimum of\n$\\chi^2$, using all observational data described in previous section.\n Since the $N_{\\nu}$ is discrete, we repeat the\nprocedure three times for $N_{\\nu}$=1, 2, and 3. The lowest of the\nthree minima is the minimum of $\\chi^2$ for the\ncomplete set of free parameters. \n\n\\begin{table*}[th]\n\\caption{Cosmological parameters determined for the tilted\n$\\Lambda$MDM model with one, two and three species of massive\nneutrinos.}\n\\begin{center}\n\\def\\onerule{\\noalign{\\medskip\\hrule\\medskip}}\n\\medskip\n\\begin{tabular}{cccccccc}\n\\hline\n&&&&&&&\\\\\n$N_{\\nu}$ & $\\chi^2_{min}$ &$n$ & $\\Omega_m$&$\\Omega_{\\nu}$& $\\Omega_b$ & $h$ & $b_{cl}$ \\\\ [4pt]\n\\hline\n&&&&&&&\\\\\n1 & 4.64&1.12$\\pm$0.09&0.41$\\pm$0.11&0.059$\\pm$0.028&0.039$\\pm$0.014&0.70$\\pm$0.12&2.23$\\pm$0.33\\\\\n2 & 4.82&1.13$\\pm$0.10&0.49$\\pm$0.13&0.103$\\pm$0.042&0.039$\\pm$0.014&0.70$\\pm$0.13&2.33$\\pm$0.36\\\\\n3 & 5.09&1.13$\\pm$0.10&0.56$\\pm$0.14&0.132$\\pm$0.053&0.040$\\pm$0.015&0.69$\\pm$0.13&2.45$\\pm$0.37\\\\ [4pt]\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThe results are presented in the Table 1. The errors in the determined\nparameters are the square roots of diagonal elements of the covariance\nmatrix of standard errors. More information about the\naccuracy of the determination of parameters and their sensitivity to\nthe data used can be obtained from the contours of confidence\nlevels presented in Fig.~\\ref{Lcm1} for the tilted $\\Lambda$MDM model with\nparameters from Table 1 (case $N_{\\nu}=1$). These\ncontours show the confidence regions which contain 68.3\\% (solid line),\n95.4\\% (dashed line) and 99.73\\% (dotted line) of the total probability\ndistribution in the two dimensional sections of the six-dimensional\nparameter space, if the probability distribution is Gaussian.\nSince the number of degrees of freedom is\n7 they correspond to $\\Delta\\chi^2=$8.2, 14.3 and 21.8\nrespectively. The parameters not shown in a given diagram are set to their best-fit value.\n\n\\begin{figure*}[tp]\n\\epsfxsize=16truecm\n\\epsfbox{fig6.eps}\n\\caption{Likelihood contours (solid line - 68.3\\%, dashed - 95.4\\%, dotted\n- 99.73\\%) of the tilted $\\Lambda$MDM model with $N_{\\nu}=1$ and\nparameters from Table 1 ($N_{\\nu}=1$) in the different planes of\n$n-\\Omega_m-\\Omega_{\\nu}-\\Omega_b-h$ space. The parameters\nnot shown in a given diagram are set to their best fit value.}\n\\label{Lcm1}\n\\end{figure*}\n\nAs one can see in Fig.1a the iso-$\\chi^2$ surface is rather prolate from\nthe low-$\\Omega_m$ - high-$n$ corner to high-$\\Omega_m$ - low-$n$.\nThis indicates a degeneracy in $n-\\Omega_m$ parameter plane. Within\nthe $1\\si$ the 'maximum likelihood ridge' in this plane can be\napproximated by the equation $n\\sqrt{\\Omega_m}=0.73$. A similar \ndegeneracy is observed in the $\\Omega_{\\nu}- \\Omega_m$ plane\nin the range $0\\le\\Omega_{\\nu}\\le 0.17$, $0.25\\le\\Omega_m \\le 0.6$\n(Fig.1c). The equation for the 'maximum likelihood ridge'\nor 'degeneracy equation' has here the form:\n$\\Omega_{\\nu}=0.023-0.44\\Omega_m+1.3\\Omega_m^2~$.\n\nThe important question is: which is the confidence limit of each parameter\nmarginalized over the other ones. The straightforward answer is the\nintegral of the likelihood function over the allowed range of all\nthe other parameters.\nBut for a 6-dimensional parameter space this is computationally\ntime consuming.\nTherefore, we have estimated the 1$\\sigma$ confidence limits for\nall parameters in\nthe following way. By variation of all parameter we determine\n the 6-dimensional $\\chi^2$ surface which\ncontains 68.3\\% of the total probability distribution.\nWe then project the surface onto each axis\nof parameter space. Its shadow on the parameter axes gives us the 1$\\sigma$\nconfidence limits on cosmological parameters. For the best\n$\\Lambda$MDM model with one sort of massive neutrinos the 1$\\sigma$\nconfidence limits on parameters obtained in this way are presented\nin Table~\\ref{tabmax}.\n\\begin{table}\n\\caption{\\label{tabmax} The best fit values of all the parameters with\nerrors obtain by maximizing the (Gaussian) 68\\% confidence contours\nover all other parameters.}\n\\begin{center}\n\\begin{tabular}{||c|c||}\n\\hline\n&\\\\\nparameter & central value and errors\\\\ [4pt]\n\\hline\n&\\\\\n $\\Omega_m$ & $0.41^{+0.59}_{-0.22}$ \\\\[4pt]\n$\\Omega_{\\nu}$ & $0.06^{+0.11}_{-0.06}$ \\\\[4pt]\n $\\Omega_b$ & $0.039^{+0.09}_{-0.018}$ \\\\[4pt]\n$h^{*)}$ & $0.70^{+0.15(+0.31)}_{-0.32}$ \\\\[4pt]\n$n$ & $1.12^{+0.27}_{-0.30}$ \\\\[4pt]\n $b_{cl}$ & $2.22^{+1.3}_{-0.7}$\\\\[6pt]\n\\hline\n\\end{tabular}\n\\end{center}\n $ ^{*)}$ - the upper limit is obtained by including the\nlower limit on the age of the Universe due to the age of oldest stars,\n$t_0\\ge13.2\\pm 3.0$ \\cite{car99}. The value obtained without this\nconstraint is given in parenthesis.\n\\end{table}\n\nIt must be noted that, using the observational data described above,\nthe upper $1\\sigma$ edge for $h$ is equal 1.08 when we marginalized\nover all other parameters. But this contradicts\nthe age of the oldest globular clusters $t_0=13.2\\pm 3.0$\\cite{car99}. Thus\nwe have included this value into the marginalization procedure for\nthe upper limit of $h$. We then have 8 degrees of freedom (24 data points)\nand the 6-dimensional $\\chi^2$ surface which contains\n68.3\\% of the probability is confined by the value 13.95.\nWe did not use the age of oldest\nglobular cluster for searching of best fit parameters in general case because it is\nonly a lower limit for age of the Universe.\n\nThe errors given in Table~\\ref{tabmax} represent 68\\% likelihood, of course,\nonly when the probability distribution is Gaussian. As one can see\nfrom Fig.1 (all panels without degeneracy) the ellipticity of the likelihood\ncontours in most of planes is close to what is expected from a Gaussian\ndistribution. This indicates that around their maxima the likelihood\nfunctions are close to Gaussian. However, the asymmetry of the error\nbars obtained, shows that away from the maxima this is no longer the\ncase. Therefore, our estimates of the confidence limits have to be\ntaken with a grain of salt.\nThe errors define the range of each parameter within which \nthe best-fit values obtained for the remaining parameters lead to\n$\\chi^2_{min}\\le 12.84$. Of course, the best-fit values of the \nremaining parameters lay within the range \ngiven in Table~\\ref{tabmax}. However clearly not every \nset of parameters from these ranges satisfies the condition,\n$\\chi^2_{min}\\le 12.84$. For example,\nstandard CDM model ($\\Omega_m=1$, $h=0.5$, $\\Omega_b=0.05$, $n=1$ and best-fit value of\ncluster biasing parameter $b_{cl}=2.17$ ($\\sigma_8=1.2$)) has $\\chi^2_{min}=142$ (!),\nwhich excludes it at very high confidence level, $>99.999\\%$. When we use the\nbaryon density inferred from nucleosynthesis \n($h^2\\Omega_b=0.019$ ($b_{cl}=2.25$, $\\sigma_8=1.14$))\nthe situation does not improve much, $\\chi^2_{min}=112$. Furthermore,\neven if we leave $h$ as free parameter we still find $\\chi^2_{min}=16$ \n($>1\\sigma$) with the best-fit values $h=0.37$ and $b_{cl}=3.28$ \n($\\sigma_8=0.74$); this variant of CDM is ruled\nout by direct measurements of the Hubble constant.\n\nThe standard MDM model\n($\\Omega_m=1$, $h=0.5$, $\\Omega_b=0.5$, $n=1$, $\\Omega_{\\nu}=0.2$, $N_{\\nu}=1$\nwith a best value of the cluster biasing parameter \n$b_{cl}=2.74$ ($\\sigma_8=0.83$)) does significantly better: it has \n$\\chi^2_{min}=23.1$ ($99\\%$ C.L.) which is out of the\n$2\\sigma$ confidence contour but inside $3\\sigma$. With the\nnucleosynthesis constraint the situation does not change:\n$\\chi^2_{min}=22$; also if we leave $h$ as free parameter: \n$\\chi^2_{min}=21$, $h=0.48$. But if, in addition, we let vary\n$\\Omega_{\\nu}$, we obtain $\\chi^2_{min}=13$\nwith best-fit values of $\\Omega_{\\nu}=0.09$, $h=0.43$, \n$b_{cl}=3.2$ ($\\sigma_8=0.73$). This means that the model is ruled out by the \ndata set considered in this work at $\\sim 70\\%$ confidence level only. But also here the \nbest-fit value for $h$ is very low. If we fix it at lower observational\nlimit $h=0.5$ then $\\chi^2_{min}=18.9$ (the best fit values are: \n$\\Omega_{\\nu}=0.15$, $b_{cl}=2.8$ ($\\sigma_8=0.83$)), which\ncorresponds to a confidence level of 95\\% .\n\nTherefore, we conclude\nthat the observational data set used here rules out CDM models with \n$h\\ge 0.5$, a scale invariant primordial\npower spectrum ($n=1$) and $\\Omega_k=\\Omega_{\\Lambda}=0$\nat very high confidence level, $>99.99\\%$. MDM models\nwith $h\\ge 0.5$, $n=1$ and $\\Omega_k=\\Omega_{\\Lambda}=0$\nare ruled out at $\\sim 95\\%$ C.L.\n\nOne can see the \nmodel with one sort of massive neutrinos provides the best fit to \nthe data, $\\chi^2_{min}\\approx 4.6$. Note, however, that there are \nonly marginal differences in $\\chi^2_{min}$ for $N_\\nu = \n1,2,3$. Therefore, with the given accuracy of the data we cannot conclude \nwhether -- if massive neutrinos are present \nat all -- their number of species is one, two, or three. \n\nThe number of degrees of \nfreedom is $N_F= N_{\\rm exp}-N_{\\rm par}= 7$. The\n $\\chi^2_{min}$ for all cases is within the expected range,\n$N_F-\\sqrt{2N_F}\\le \\chi^2_{\\min}\\le N_F+\\sqrt{2N_F}$ for the given\nnumber of degrees of freedom. This means that the cosmological paradigm which\nhas been assumed is consistent with the data.\n\nOne important question is how each point of the data influences our\nresult. To estimate this we have\nexcluded some data points from the searching procedure. \nExcluding any part of observable data results only in a change\nof the best-fit values of $n$, $\\Omega_m$ and $h$ within the range of\ntheir corresponding\nstandard errors. This indicates that the data are mutually in\nagreement, implying consistent cosmological parameters (within the still\nconsiderable error bars).\nThe small scale constraints, the Ly-$\\alpha$ tests reduce the hot dark\nmatter content from\n$\\Omega_{\\nu}\\sim 0.22$ to $\\sim 0.075$. The $\\sigma_8$-tests further reduce\n$\\Omega_{\\nu}$ to $\\sim 0.06$. Including of the Abell-ACO power\nspectrum in the search\nprocedure, tends to enhance $\\Omega_{\\nu}$ slightly.\nThe most crucial test for the baryon content is of course the nucleosynthesis\nconstraint. Its $\\sim 6\\%-1\\sigma$-accuracy safely keeps\n$h^2\\Omega_b$ near its median value 0.019. The parameter $\\Om_b$ in\nturn is only known to $\\sim 36\\%$ accuracy due to the large\nerrors of other experimental data used here, especially of the Hubble constant.\nThe accuracy of $h$ ($\\sim17\\%$) is better than the one\nassumed from direct measurements, $\\sim 23\\%$.\n Summarizing, we conclude that all data points used here\nare important for searching the best-fit cosmological parameters and do not contradict\neach other.\n\nUp to this point we ignored the uncertainties in the COBE normalization.\nIndeed, the statistical uncertainty of the fit to the\nfour-year COBE data, $\\delta_h$, is 7\\% (1$\\sigma$) \\cite{bun97} and\nwe want to study how this uncertainty\ninfluences the accuracy of cosmological parameters which we determine? \n\nVarying $\\delta_h$ in the 1$\\sigma$ range we found that the best-fit values of all \nparameters except $\\Omega_{\\nu}$ do not vary by more than 2\\% from the values presented\nin Table 1. Only $\\Omega_\\nu$ varies in a range of 12\\% . These \nuncertainties are significantly smaller than the standard errors\ngiven in Table 1 and neglecting them is thus justified.\nThe normalization constant\n$A$, which for best model with one species of massive neutrinos, \n$A=4.68\\cdot10^7(\\Mpc)^{3+n}$, \nvaries in a range of 8\\%, and not 14\\% as one might expect\n from (\\ref{anorm}) at the first sight. \nThe reason is that variation of $\\delta_h$ is somewhat compensated \nby correlated variation of $n$ and $h$. Moreover, if we disregard the COBE \nnormalization and treat the normalization constant $A$ as a free\nparameter to be determined like the others, its best-fit value\nbecomes $4.82 \\cdot10^7(\\Mpc)^{3+n}$ (for $N_{\\nu}=1$), consistent\nwith COBE normalization. The best-fit values of the other parameters\ncorrespondingly do not vary substantially: $n=1.09$, $\\Omega_m=0.40$,\n$\\Omega_{\\nu}=0.052$, $\\Omega_b=0.041$, $h=0.68$ and $b_{cl}=2.19$\n(this is less than 5\\% except for $\\Omega_{\\nu}$, which is reduced by\n$\\sim 12\\%$). \nThis implies that determinations of the amplitude of scalar\nfluctuations by the COBE measurement of the large scale CMB\nanisotropies and by large scale structure data at much smaller scales\nare in good agreement. It also indicates that a possible tensor\ncomponent in the COBE data cannot be very substantial.\n\n\\section{Conclusions}\n\nWe summarize, that the observational data of the LSS of the Universe\nconsidered here can be\nexplained by a tilted $\\Lambda$MDM inflationary model without tensor\nmode. The best fit parameters are: \n$n=1.12\\pm 0.09$, $\\Omega_m=0.41\\pm 0.11$, $\\Omega_{\\nu}=0.06\\pm 0.028$,\n$\\Omega_b=0.039\\pm 0.014$ and $h=0.70\\pm 0.12$. \nAll predictions of measurements are close to the experimental \nvalues given above and within the error bars of the data. \nThe CDM density parameter is $\\Omega_{cdm} = 0.31\\pm0.12$ and\n$\\Omega_{\\Lambda}$ is moderate, $\\Omega_{\\Lambda}=0.59\\pm0.11$.\nThe neutrino matter density \ncorresponds to a neutrino mass $m_{\\nu}=94\\Omega_{\\nu}h^2\\approx2.7\\pm1.2$ eV.\nThe value of the Hubble constant is close to the measurements by \nMadore et al.\\cite{mad98}.\nThe age of the Universe for this model equals 12.3 Gyrs which is in\ngood agreement with the age of the oldest objects in our galaxy \\cite{car99}.\nThe spectral index coincides with the \nCOBE prediction. The relation between the matter density \n$\\Omega_m$ and the cosmological\nconstant $\\Omega_{\\Lambda}$ agrees well with the independent \nmeasurements of cosmic \ndeceleration and global curvature based on the SNIa observation.\n\nThe 1$\\sigma$ (68.3\\%) confidence limits on each cosmological\nparameter, obtained by marginalizing over the other parameters, are\n $0.82\\le n\\le 1.39$, $0.19\\le\\Omega_m\\le 1$, $0\\le\\Omega_{\\Lambda}\\le0.81$,\n$0\\le\\Omega_{\\nu}\\le 0.17$, $0.021\\le\\Omega_b\\le 0.13$ and\n$0.38\\le h\\le 0.85$.\n\nThe observational data set used here rules out the CDM models with $h\\ge 0.5$, \nscale invariant primordial power spectrum $n=1$ and $\\Omega_{\\Lambda}=\\Omega_k=0$\nat very high confidence level, $>99.99\\%$. Also pure MDM models are ruled out \nat $\\sim 95\\%$ C.L. \n\nIt is remarkable also that this data set\ndetermines the value of normalization constant for scalar\nfluctuations which approximately equals the value deduced from\nCOBE four-year data. It indicates that a possible tensor component\nin the COBE data cannot be very substantial. \n\nThe coincidence of the values of cosmological parameters\nobtained by different methods indicates that a wide set of \ncosmological measurements are correct and that their theoretical \ninterpretation is consistent. However, we must also \nnote that the accuracy of present\nobservational data on the large scale structure of the Universe is\nstill insufficient to determine a set of cosmological parameters with\nhigh accuracy. \n\n\\begin{thebibliography}{1} \\itemsep=-5pt\n\n\\bibitem {atr97} Atrio-Barandela, F. \\etal, 1997, Pis'ma Zh. Eksp. Teor. Fiz. 66, 373\n\\bibitem {bah98} Bahcall, N.A., Fan, X., 1998, ApJ 504, 1\n\\bibitem {bah99} Bahcall, N.A., Ostriker, J.P., Perlmutter, S., Steinhardt, P.J.,\n 1999, Science 284, 1481\n\\bibitem {bar98} Bartelmann, M. \\etal, 1998, A\\&A, 330, 1\n\\bibitem {ben96} Bennett, C.L. \\etal, 1996, ApJ 464, L1\n\\bibitem {dol95} Berezhiani, Z.G., Dolgov, A.D. and Mohapatra, R.N. 1996, \n Phys. Lett., B375, 26\n\\bibitem {borg99} Borgani, S. \\etal, 1999, astro-ph/9907323\n\\bibitem {bri99} Bridle, S.L., \\etal, 1999, astro-ph/9903472\n\\bibitem {bun97} Bunn E.F., White M., 1997, ApJ 480, 6\n\\bibitem {bur99} Burles, S., Nollett, K.M., Truran, J.N., Turner, M.S., 1999,\n Phys.Rev.Lett. 82, 4176\n\\bibitem {car99} Carretta, E., Gratton, R.C., Clementini, G., Fusi Pecci, F., 1999, ApJ (in\nprint), astro-ph/9902086\n\\bibitem {car92} Carroll, S.M., Press, W.H., Turner, E.L., 1992, ARA\\&A 30, 499\n\\bibitem {cr98} Croft, R.A.C. \\etal, 1998, ApJ 495, 44\n\\bibitem {ein97} Einasto, J. \\etal, 1997, Nature 385, 139\n\\bibitem {eh3} Eisenstein, D.J., Hu, W., 1999, ApJ 511, 5\n\\bibitem {fal98} Falco, E.E., Kochanek, C.S. and Munoz, J.A., 1998, ApJ, 494, 47\n\\bibitem {fu98} Fukuda, Y. et al. 1998, Phys. Rev. Lett. 81, 1562\n\\bibitem {gir98} Girardi, M. \\etal, 1998, ApJ 506, 45\n\\bibitem {gn98} Gnedin N.Y., 1998, MNRAS 299, 392\n\\bibitem {gn99} Gnedin N.Y., 1999, private communication\n\\bibitem {kol97} Kolatt, T., Dekel, A., 1997, ApJ 479, 592\n\\bibitem {lid96} Liddle, A.R., Lyth, D.H., Viana, P.T.P., White, M., 1996, MNRAS 282, 281\n\\bibitem {lin97} Lineweaver, C.A., Barbosa, D., 1998, ApJ 496, 624\n\\bibitem {mad98} Madore, B.F. \\etal, 1998, astro-ph/9812157\n\\bibitem {mau99} Mauskopf, P. \\etal, 1999, astro-ph/9911444\n\\bibitem {mel99} Melchiorri A., \\etal, 1999, astro-ph/9911445\n\\bibitem {nov99} Novosyadlyj B., 1999, Journal of Physical Studies V.3, No1, 122\n\\bibitem {nov00} Novosyadlyj, B., Durrer, R., Gottl\\\"ober, S., Lukash, V.N., Apunevych, S.,\n 1999, astro-ph/9912511\n\\bibitem {per98} Perlmutter, S. \\etal, 1998, Nature 391, 51\n\\bibitem {phill} Phillips, J. \\etal, 2000, astro-ph/0001089\n\\bibitem {nr92} Press, W.H., Flannery, B.P., Teukolsky, S.A., Vettrling, W.T., 1992, Numerical recipes in FORTRAN. New York: Cambridge Univ.Press\n\\bibitem {pr98} Primak, J.R., Gross, M.A., 1998, astro-ph/9810204\n\\bibitem {ret97} Retzlaff, J. \\etal, 1998, New Astronomy v.3, 631\n\\bibitem {rie98} Riess, A. \\etal, astro-ph/9805201\n\\bibitem {sz96} Seljak U., Zaldarriaga M., 1996, ApJ 469, 437\n\\bibitem {Tamm99} Tammann, G.A., Sandage, A., Reindl, B., 1999, astro-ph/9904360\n\\bibitem {Teg99} Tegmark, M., 1999, ApJ 514, L69\n\\end{thebibliography}\n\n\n\\end{document}\n\n\n"
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[
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"name": "astro-ph0002522.extracted_bib",
"string": "\\begin{thebibliography}{1} \\itemsep=-5pt\n\n\\bibitem {atr97} Atrio-Barandela, F. \\etal, 1997, Pis'ma Zh. Eksp. Teor. Fiz. 66, 373\n\\bibitem {bah98} Bahcall, N.A., Fan, X., 1998, ApJ 504, 1\n\\bibitem {bah99} Bahcall, N.A., Ostriker, J.P., Perlmutter, S., Steinhardt, P.J.,\n 1999, Science 284, 1481\n\\bibitem {bar98} Bartelmann, M. \\etal, 1998, A\\&A, 330, 1\n\\bibitem {ben96} Bennett, C.L. \\etal, 1996, ApJ 464, L1\n\\bibitem {dol95} Berezhiani, Z.G., Dolgov, A.D. and Mohapatra, R.N. 1996, \n Phys. Lett., B375, 26\n\\bibitem {borg99} Borgani, S. \\etal, 1999, astro-ph/9907323\n\\bibitem {bri99} Bridle, S.L., \\etal, 1999, astro-ph/9903472\n\\bibitem {bun97} Bunn E.F., White M., 1997, ApJ 480, 6\n\\bibitem {bur99} Burles, S., Nollett, K.M., Truran, J.N., Turner, M.S., 1999,\n Phys.Rev.Lett. 82, 4176\n\\bibitem {car99} Carretta, E., Gratton, R.C., Clementini, G., Fusi Pecci, F., 1999, ApJ (in\nprint), astro-ph/9902086\n\\bibitem {car92} Carroll, S.M., Press, W.H., Turner, E.L., 1992, ARA\\&A 30, 499\n\\bibitem {cr98} Croft, R.A.C. \\etal, 1998, ApJ 495, 44\n\\bibitem {ein97} Einasto, J. \\etal, 1997, Nature 385, 139\n\\bibitem {eh3} Eisenstein, D.J., Hu, W., 1999, ApJ 511, 5\n\\bibitem {fal98} Falco, E.E., Kochanek, C.S. and Munoz, J.A., 1998, ApJ, 494, 47\n\\bibitem {fu98} Fukuda, Y. et al. 1998, Phys. Rev. Lett. 81, 1562\n\\bibitem {gir98} Girardi, M. \\etal, 1998, ApJ 506, 45\n\\bibitem {gn98} Gnedin N.Y., 1998, MNRAS 299, 392\n\\bibitem {gn99} Gnedin N.Y., 1999, private communication\n\\bibitem {kol97} Kolatt, T., Dekel, A., 1997, ApJ 479, 592\n\\bibitem {lid96} Liddle, A.R., Lyth, D.H., Viana, P.T.P., White, M., 1996, MNRAS 282, 281\n\\bibitem {lin97} Lineweaver, C.A., Barbosa, D., 1998, ApJ 496, 624\n\\bibitem {mad98} Madore, B.F. \\etal, 1998, astro-ph/9812157\n\\bibitem {mau99} Mauskopf, P. \\etal, 1999, astro-ph/9911444\n\\bibitem {mel99} Melchiorri A., \\etal, 1999, astro-ph/9911445\n\\bibitem {nov99} Novosyadlyj B., 1999, Journal of Physical Studies V.3, No1, 122\n\\bibitem {nov00} Novosyadlyj, B., Durrer, R., Gottl\\\"ober, S., Lukash, V.N., Apunevych, S.,\n 1999, astro-ph/9912511\n\\bibitem {per98} Perlmutter, S. \\etal, 1998, Nature 391, 51\n\\bibitem {phill} Phillips, J. \\etal, 2000, astro-ph/0001089\n\\bibitem {nr92} Press, W.H., Flannery, B.P., Teukolsky, S.A., Vettrling, W.T., 1992, Numerical recipes in FORTRAN. New York: Cambridge Univ.Press\n\\bibitem {pr98} Primak, J.R., Gross, M.A., 1998, astro-ph/9810204\n\\bibitem {ret97} Retzlaff, J. \\etal, 1998, New Astronomy v.3, 631\n\\bibitem {rie98} Riess, A. \\etal, astro-ph/9805201\n\\bibitem {sz96} Seljak U., Zaldarriaga M., 1996, ApJ 469, 437\n\\bibitem {Tamm99} Tammann, G.A., Sandage, A., Reindl, B., 1999, astro-ph/9904360\n\\bibitem {Teg99} Tegmark, M., 1999, ApJ 514, L69\n\\end{thebibliography}"
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astro-ph0002523
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Radially truncated galactic discs\thanks{Based on observations obtained at the European Southern Observatory, La Silla, Chile}
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"author": "PO Box 3818"
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"author": "Charlottesville"
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"author": "VA 22903"
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"author": "USA"
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"author": "Madingley Road"
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"author": "Cambridge CB3 0HA"
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"author": "PO Box 800"
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"author": "9700 AV Groningen"
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"author": "the Netherlands"
}
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We present the first results of a systematic analysis of radially truncated exponential discs for four galaxies of a complete sample of disc-dominated edge-on spiral galaxies. \\ The discs of our sample galaxies are truncated at similar radii on either side of their centres. With possible the exception of the disc of ESO 416-G25, it appears that the truncations in our sample galaxies are closely symmetric, in terms of both their sharpness and the truncation length. However, the truncations occur over a larger region and not as abruptly as found in previous studies. \\ We show that the truncated luminosity distributions of our sample galaxies, if also present in the mass distributions, comfortably meet the requirements for longevity. The formation and maintenance of disc truncations are likely closely related to stability requirements for galactic discs.
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"name": "revised3.tex",
"string": "\\documentstyle[psfig]{mn}\n%\\documentstyle[psfig,referee]{mn}\n\n\\title[Radially truncated galactic discs]{Radially truncated galactic\ndiscs\\thanks{Based on observations obtained at the European Southern\nObservatory, La Silla, Chile}}\n\n\\author[Richard de Grijs, Michiel Kregel and Karen H. Wesson]{Richard\nde Grijs$^{1,2,}$\\thanks{E-mail: grijs@ast.cam.ac.uk}, Michiel\nKregel$^3$ and Karen H. Wesson$^{1,}$\\thanks{Present address: Center\nfor Hydrologic Science, Duke University, 106 Old Chemistry, Box 90230,\nDurham, NC 27708, USA}\n\\\\ \n$^1$ Astronomy Department, University of Virginia, PO Box 3818,\nCharlottesville, VA 22903, USA \\\\ \n$^2$ Institute of Astronomy, University of Cambridge, Madingley Road,\nCambridge CB3 0HA \\\\\n$^3$ Kapteyn Astronomical Institute, University of Groningen, PO Box\n800, 9700 AV Groningen, the Netherlands\n}\n\n\\date{Received date; accepted date}\n\\pubyear{2000}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nWe present the first results of a systematic analysis of radially\ntruncated exponential discs for four galaxies of a complete sample of\ndisc-dominated edge-on spiral galaxies. \\\\\nThe discs of our sample galaxies are truncated at similar radii on\neither side of their centres. With possible the exception of the disc\nof ESO 416-G25, it appears that the truncations in our sample galaxies\nare closely symmetric, in terms of both their sharpness and the\ntruncation length. However, the truncations occur over a larger region\nand not as abruptly as found in previous studies. \\\\\nWe show that the truncated luminosity distributions of our sample\ngalaxies, if also present in the mass distributions, comfortably meet\nthe requirements for longevity. The formation and maintenance of disc\ntruncations are likely closely related to stability requirements for\ngalactic discs. \n\\end{abstract}\n\n\\begin{keywords}\ngalaxies: formation -- galaxies: fundamental parameters -- galaxies:\nphotometry -- galaxies: spiral -- galaxies: structure \n\\end{keywords}\n\n\\section{Introduction}\n\\label{exptrunc.sec}\n\nIt is well-known from Freeman's (1970) work that the radial light\ndistribution of the stellar component of high surface brightness\ngalactic discs can be approximated by an exponential law of the form\n\\begin{equation}\n\\label{freeman.eq}\nL(R) = L_0 \\exp (- R / h_R)\n\\end{equation}\nwhere $L_0$ is the luminosity density in the galactic centre, {\\it R}\nthe galactocentric distance and $h_R$ the disc scalelength. \n\nHowever, for a few prominent high surface-brightness edge-on galaxies,\nit has initially been shown by van der Kruit \\& Searle (1981a,b,\n1982a,b, hereinafter KS1--4), that at some radius $R_{\\rm max}$ (the\ntruncation or cut-off radius of the galactic disc) the stellar\nluminosity distribution disappears asymptotically into the background\nnoise (see also Jensen \\& Thuan 1982, Sasaki 1987, Morrison, Boroson \\&\nHarding 1994, Bottema 1995, Lequeux, Dantel-Fort \\& Fort 1995, N\\\"aslund\n\\& J\\\"ors\\\"ater 1997, Fry et al. 1999, Pohlen et al. 2000a,b). In\nfact, the truncation of galactic discs does not occur instantly but over\na small region, where the luminosity decrease becomes much steeper,\nhaving exponential scalelengths of order or less than a kiloparsec,\nopposed to several kpc in the exponential disc part (e.g., KS1--4,\nJensen \\& Thuan 1982, Sasaki 1987, Abe et al. 1999, Fry et al. 1999). \n\nAn independent approach to obtain the statistics of truncated galactic\ndiscs, using a sample of galaxies selected in a uniform way, is needed\nin order to better understand the overall properties and physical\nimplications of this feature. In this paper we present the first\nresults of a systematic analysis of disc truncations for a pilot sample\nof four ``normal'' spiral galaxies, drawn from the statistically\ncomplete sample of edge-on disc-dominated galaxies of de Grijs (1998). \nIn Kregel, van der Kruit \\& de Grijs (2001) we will present a\nre-analysis of the global disc structures of the entire de Grijs (1998)\nsample, including a systematic analysis of the occurrence of truncated\ngalactic discs. \n\nEdge-on galaxies are particularly useful for the study of truncated\ngalactic discs: since these disc cut-offs usually occur at very low\nsurface brightness levels, they are more readily detected in highly\ninclined galaxies, where we can follow the light distributions out to\nlarger radii. In Sect. \\ref{technical.sect} we outline the sample\nselection and our approach and methodology. \n\nA detailed error discussion for our pilot sample is given in Sect. \n\\ref{results.sect}. The most important science driver for the study of\ntruncated galactic discs, discussed in Sect. \\ref{dynamics.sect}, is\nthat if the truncations seen in the stellar light are also present in\nthe mass distribution, they would have important dynamical consequences\nat the disc's outer edges. \n\n\\section{Global Approach}\n\\label{technical.sect}\n\n\\subsection{Pilot sample}\n\nWe selected four galaxies from the statistically complete sample of\ndisc-dominated edge-on galaxies of de Grijs (1998) for this pilot study. \n\nThe galaxies in the parent sample were selected to:\n\\begin{itemize}\n\\item have inclinations $i \\ge 87^\\circ$;\n\\item have blue angular diameters $D_{25}^B \\ge 2.'2$; and\n\\item be non-interacting and undisturbed S0 -- Sd galaxies.\n\\end{itemize}\n\nWe required these pilot galaxies to have relatively high signal-to-noise\n(S/N) ratios out to large galactocentric distances, thus allowing us to\nbetter determine the occurrence of a possible truncated disc at large\nradii. Of the parent sample of 48 edge-on disc galaxies, $\\sim 25$ met\nour overall high-S/N selection criteria in {\\em all} of the {\\it B, V}\nand {\\it I} observations. The four galaxies selected for our pilot\nsample were chosen randomly from among the larger disc galaxies because\nof their small bulge-to-disc ratio and well-defined, regular disc\ncomponent, which was only negligibly or minimally affected by a central\ndust lane (cf. Fig. 2 in Chapter 9 of de Grijs 1997). \n\nThe basic physical properties of these pilot sample galaxies are\nsummarised in Table \\ref{pilot.tab}. The observational properties were\ntaken from de Grijs (1997, 1998). The derived properties are obtained\nin Sect. \\ref{indiv.sect}. They are based on the detailed modelling of\nthe galactic luminosity density distributions projected on the plane of\nthe sky using the method described in Kregel et al. (2001). The\ndetailed observational logs of and data reduction techniques applied to\nour {\\it B, V} and {\\it I}-band observations are summarised in de Grijs\n(1998). Fig. \\ref{contours.fig} displays the {\\it I}-band contours of\nthese galaxies. \n\n{\n\\begin{table*}\n\n\\caption[]{\\label{pilot.tab}{\\bf Basic properties of the pilot sample\ngalaxies}\\\\\n\nColumns: (1) Galaxy name (Lauberts \\& Valentijn 1989; ESO-LV); (2) and\n(3) Coordinates; (4) Revised Hubble Type; (5) Blue major axis diameter,\n$D_{25}^B$; (6) Passband observed in; (7)\napparent magnitude, corrected for foreground\nextinction; (8) extrapolated edge-on disc central surface\nbrightness; (9) Exponential scaleheight; (10) Exponential scalelength.}\n\n\\begin{center}\n\\tabcolsep=1mm\n\n\\begin{tabular}{cccccccccc}\n\\hline\nGalaxy & \\multicolumn{2}{c}{RA (J2000) Dec} & Type & $D_{25}^B$ &\nPassband & $m_0$ & $\\mu_0$ & $h_z$ & $h_R$ \\\\\n\\noalign{\\vspace{2pt}}\n\\cline{2-3}\\cline{9-10}\n\\noalign{\\vspace{1pt}}\n & $(^{\\rm h \\; m \\; s})$ & $(^{\\circ}$ $'$ $'')$ & (T) & $(')$ & &\n(mag) & (mag arcsec$^{-2}$) & \\multicolumn{2}{c}{$('')$} \\\\\n(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) \\\\\nESO 201-G22 & 04 08 59.3 & $-$48 43 42 & 5.0 & 2.52 & $B$ & $14.06 \\pm 0.07$ & $20.54 \\pm 0.07$ & $2.4 \\pm 0.1$ & $26.9 \\pm 1.5$ \\\\\n & & & & & $V$ & $ $ & $20.39 \\pm 0.10$ & $2.4 \\pm 0.1$ & $25.2 \\pm 1.1$ \\\\\n & & & & & $I$ & $13.10 \\pm 0.03$ & $19.23 \\pm 0.08$ & $2.6 \\pm 0.2$ & $23.3 \\pm 1.3$ \\\\\nESO 416-G25 & 02 48 40.8 & $-$31 32 07 & 3.0 & 2.35 & $B$ & $14.42 \\pm 0.11$ & $21.41 \\pm 0.03$ & $3.1 \\pm 0.1$ & $29.2 \\pm 1.4$ \\\\\n & & & & & $V$ & $ $ & $20.90 \\pm 0.08$ & $3.5 \\pm 0.2$ & $25.8 \\pm 1.3$ \\\\\n & & & & & $I$ & $12.52 \\pm 0.03$ & $19.87 \\pm 0.05$ & $3.6 \\pm 0.2$ & $22.2 \\pm 2.0$ \\\\\nESO 446-G18 & 14 08 37.9 & $-$29 34 20 & 3.0 & 2.52 & $B$ & $15.12 \\pm 0.05$ & $20.72 \\pm 0.10$ & $2.2 \\pm 0.1$ & $22.5 \\pm 1.3$ \\\\\n & & & & & $V$ & $ $ & $20.03 \\pm 0.08$ & $2.2 \\pm 0.1$ & $19.5 \\pm 0.9$ \\\\\n & & & & & $I$ & $12.71 \\pm 0.02$ & $18.49 \\pm 0.10$ & $2.0 \\pm 0.2$ & $18.1 \\pm 1.4$ \\\\\nESO 446-G44 & 14 17 49.3 & $-$31 20 55 & 6.0 & 2.67 & $B$ & $14.83 \\pm 0.03$ & $20.39 \\pm 0.10$ & $2.1 \\pm 0.1$ & $25.1 \\pm 0.7$ \\\\\n(= IC 4393) & & & & & $V$ & $ $ & $19.76 \\pm 0.07$ & $2.0 \\pm 0.1$ & $24.5 \\pm 0.6$ \\\\\n & & & & & $I$ & $12.51 \\pm 0.04$ & $18.90 \\pm 0.15$ & $2.7 \\pm 0.2$ & $29.5 \\pm 1.6$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n}\n\n\\begin{figure*}\n\\vspace*{1.9cm}\n\\psfig{figure=4panels.ps}\n\\vspace*{0.8cm}\n\\caption[]{\\label{contours.fig}Optical {\\it I}-band isophotes of the\nfour edge-on disc galaxies in our pilot sample. North is up, East to\nthe left; each panel is approx. $3'$ on a side. The contours are\nspaced by 1.0 mag, starting from the lowest contours at 23.5 {\\it I}-mag\narcsec$^{-2}$. For reasons of clarity, the brightest foreground stars\n(that may be potentially contaminating the galactic discs) have been\nmasked out. Note that an indication of the presence of disc truncations\nis already given by the rounded contours at the discs' outer edges.}\n\\end{figure*}\n\n\\subsection{The adopted three-dimensional model}\n\\label{3Dmodel.sect}\n\nWe will approximate the three-dimensional (3D) luminosity density of the\ndiscs of ``ideal'' spiral galaxies as a combination of independent\nexponential light distributions in both the radial and the vertical\ndirections (see, e.g., de Grijs, Peletier \\& van der Kruit [1997] for a\nstatistical approach to determine the latter behaviour), for all radii\nexcluding the region of truncation. In view of the finite extent of the\ncut-off region, $\\delta$, and to avoid discontinuities in the luminosity\nand density distributions, we will adopt a slightly modified version of\nCasertano's (1983) mathematically convenient description for a ``soft\ncut-off'' of the radial density distribution in the truncation region. \nHis model assumes that in the region beyond ($R_{\\rm max} - \\delta$),\nthe radial luminosity density decreases linearly to zero:\n\n\\[\nL(R) = L_0 \\left\\{ \\begin{array}{ll}\n\n\\exp(-R / h_R), \\\\\n\n\\qquad\\qquad\\qquad \\mbox{if } R \\le (R_{\\rm max} - \\delta) \\\\\n\n\\exp\\Bigl( -\\Bigl( {{R_{\\rm max} - \\delta} \\over h_R} \\Bigr) \\Bigl[ {{1 - \\Bigl( R -\n(R_{\\rm max} - \\delta \\bigr) \\Bigr)} \\over \\delta} \\Bigr], \\\\\n\n\\qquad\\qquad\\qquad \\mbox{if } (R_{\\rm max} - \\delta) \\le R \\le R_{\\rm max} \\\\\n\n0, \\qquad\\qquad\\quad \\mbox{if } R > R_{\\rm max}. \\qquad\\qquad\\qquad\\quad (2)\n\\end{array}\n\\right. \\]\n\nA model radial profile with a ``soft cut-off'' will therefore show an\nexponentially declining disc component, displaying an abrupt steepening\nat the onset of the truncation region at $(R_{\\rm max} - \\delta)$,\ndecreasing linearly until the background noise level is reached. We\nwill use a slightly modified version of Eq. (2), in which the radial\nluminosity density decreases exponentially instead of linearly until it\ndisappears into the background noise. Examples of our modified fitting\nfunction are shown in Fig. \\ref{Iband.fig} for the current sample. \n\nThe exact functional form of the radial luminosity density in the\ncut-off region is unknown because of the limited spatial resolution and\nlow S/N ratio at these large galactocentric distances. Thus, within the\nobservational uncertainties, Casertano's (1983) description of the\n``soft cut-off'' does not deviate significantly from this simple\nexponential form, also adopted by Jensen \\& Thuan (1982) and N\\\"aslund\n\\& J\\\"ors\\\"ater (1997) to fit the truncated light profiles of NGC 4565. \nFurthermore, our approach, using the exponentially decreasing radial\nfunctionality in the truncation region, will provide us with an\nadditional constraint on the shape of the truncation region compared to\nCasertano's (1983), namely the scale length in the truncation region,\n$h_{R,\\delta}$. \n\nIn the case of an edge-on orientation, the projection onto the plane of\nthe sky of our model radial exponential luminosity density distribution\nis, for $R \\le (R_{\\rm max} - \\delta)$, closely approximated by (KS1):\n\\addtocounter{equation}{+1}\n\\begin{equation}\n\\label{bessel.eq}\nL(R) = L_0 {R \\over h_R} K_1\\Bigl({R \\over h_R}\\Bigr),\n\\end{equation}\nwhere $K_1$ is the modified Bessel function of the first order.\n\nThe total projected 3D luminosity density is now given by\n\\begin{equation}\n\\label{2dmodel.eq}\nL(R,z) = L(R) \\exp(-z/h_z) \\quad ,\n\\end{equation}\nwhere {\\it z} is the (vertical) distance from the galactic plane and\n$h_z$ the exponential scaleheight.\n\n\\subsubsection{Differences with respect to earlier work}\n\\label{diffs.sect}\n\nExcept for a few detailed studies of individual large edge-on galaxies\n(e.g., KS1, Jensen \\& Thuan 1981, Sasaki 1987, N\\\"aslund \\& J\\\"ors\\\"ater\n1997, Abe et al. 1999, Fry et al. 1999), most previous analyses aimed\nat determining disc truncations for statistically more meaningful\nsamples (e.g., KS3, Barteldrees \\& Dettmar 1994, Pohlen et al. 2000a,b)\nhave adopted a number of {\\it a priori} assumptions that may not be\nfully justified. In particular, these studies assumed that:\n\n\\begin{enumerate}\n\\item galactic discs are truncated at equal radii on either side of the\ngalactic centre, and\n\\item the radial surface brightness distributions disappear\nasymptotically (``vertically'') into the background noise at $R_{\\rm\nmax}$. \n\\end{enumerate}\n\nWhile the first assumption may approximate the observational situation\nrelatively closely (cf. Sect. \\ref{asymm.sect} for the current pilot\nsample), the surface brightness generally does not disappear\nasymptotically into the noise for most of the well-resolved galaxies\nstudied to date. \n\nForcing a fitting routine to satisfy both of these assumptions in\ngalaxies with slightly different truncation radii on either side of the\ncentre will often overpredict the surface brightness in one of the\ntruncation regions significantly (depending on the extent of this\nregion), even if the full projected 3D surface brightness distribution\nis used. This is clearly illustrated in Pohlen et al. (2000b), in\nparticular in their {\\it i}-band fits to the brightest profiles parallel\nto the major axis of the galaxies IC2207, IC4393, ESO446-G18, ESO466-G01\nand ESO578-G25. Alternatively, if the truncation radii on either side\nof the centre are similar but the truncation scalelength is relatively\nlong, forcing a 3D fitting routine to adopt sharp cut-offs as in the\nlatter assumption, will {\\it under\\ }predict the actual radius where the\ngalactic luminosity density disappears into the noise. \n\nIn view of these considerations, we will avoid such {\\it a priori}\nassumptions in our approach; we will determine the actual truncation\nradii independently on either side of the galactic centre, and\napproximate the luminosity density distribution in the truncation region\nby an exponentially decreasing function of radius. \n\nExamples of the fitting method for our pilot sample are presented in\nSect. \\ref{properties.sect}. \n\n\\subsection{Surface brightness modelling}\n\\label{2D.sect}\n\nAlthough the determination of the actual truncation radius $R_{\\rm max}$\nis relatively model-independent, for the detailed modelling of the\ntruncation region $(R_{\\rm max} - \\delta)$, it is crucially important to\ndetermine accurate scale parameters (in particular scale lengths) for\nthe main, exponential disc component, cf. Eqs. (2) and (3). \n\nThe surface brightness distributions of spiral galaxies often show\nsignificant local deviations from the assumed smooth, large-scale model\ndistribution (3) (e.g., Seiden, Schulman \\& Elmegreen 1984, Shaw \\&\nGilmore 1990, de Jong 1995, and references therein). This makes the\nglobal applicability of radial disc scalelengths obtained from radial\nprofiles parallel to the major axes of edge-on galaxies very uncertain\n(e.g., Knapen \\& van der Kruit 1991, Giovanelli et al. 1994). \n\nTherefore, we will model the global disc structures using the full\nprojected luminosity density distributions of the galactic discs in our\nsample, using Eqs. (3) and (4), in the linear regime. An elaborate \ndescription of the method, as well as extensive tests on artificial\nimages, will be presented in Kregel et al. (2001). This paper will\nonly address the occurrence of disc truncations in the de Grijs (1998)\nsample, without embarking on a detailed analysis of their shapes,\nhowever. \n\nA number of potential problems are foreseen regarding the applicability\nof our simple model, Eq. (4), to the observed luminosity distributions\nof edge-on galaxies. \n\nFirst, it does not include a description of the truncation region. A\nphysically motivated, or even an empirical description of this region\nis, at this point, too premature and therefore this region will be\nmasked out before least-squares minimization. \n\nSecondly, the regions near the galactic planes are affected by\nextinction. Whilst these effects can be included by using a 3D\nradiative transfer code (e.g., Kylafis \\& Bahcall 1987, Xilouris et al. \n1997), we choose to mask the data in the regions near the planes\ninstead.\n\nThirdly, by adopting our simple model, Eq. (4), we do not include the\neffects of a truncated galactic disc along the line of sight. However,\nthe effects of neglecting a line-of-sight truncation are small (see\nKregel et al. 2001): while it may cause us to underestimate the disc\nscalelength of ESO446-G44 by $\\sim 15$\\%, the effect is $\\le 8$\\% for\nthe other galaxies in our sample, and is in practice counteracted by\nresidual extinction by dust, if any, at the {\\it z}-heights used in our\nstudy. In addition, this effect is negligible for the accurate\ndetermination of the disc {\\it truncations} as such, the main aim of our\nstudy. \n\nBefore applying our fitting routine to the full observed luminosity\ndensity distributions, the sky-subtracted images needed to be prepared. \nFirst, foreground stars and background galaxies were masked out. \nSecondly, profiles parallel to the minor axis were taken at various\ndistances along the major axis. These were subsequently inspected for\nextinction effects, leading to the masking out of the regions described\nbelow for the individual galaxies. \n\n\\subsubsection{Fits to the individual galaxies}\n\\label{indiv.sect}\n\nWe will now discuss the fits to the luminosity density of each galaxy\nindividually, thereby addressing a number of problems encountered in\neach case. In all fits the galactic centres were fixed at the values\ndetermined by fitting ellipses to the {\\it I}-band isophotes, using a\ncustom-written IRAF\\footnote{The Image Reduction and Analysis Facility\n(IRAF) is distributed by the National Optical Astronomy Observatories,\nwhich is operated by the Association of Universities for Research in\nAstronomy, Inc., under cooperative agreement with the National Science\nFoundation.} package for galactic surface photometry (``{\\sc\ngalphot}''). \n\nIn each case, conservative estimates of the onset of the truncation\nregion and of the region near the galactic plane most affected by\nextinction were made based on detailed visual examinations of the radial\nand vertical luminosity distributions, respectively. The inner\nboundaries used for the radial fitting were chosen to minimize possible\nbulge effects and will be discussed individually below. Table\n\\ref{boundaries.tab} summarizes the radial ranges adopted for the disc\nfits, as well as the vertical ranges excluded to avoid extinction\neffects. \n\nFig. \\ref{2D.fig} shows the {\\it I}-band images and residual emission\n(disc model subtracted from the observations) for ESO 201-G22 and ESO\n416-G25. Table \\ref{pilot.tab} contains the resulting global scale\nparameters. The associated uncertainties are the observational errors,\nestimated by comparing results from several similar fits in which we\nadjusted the boundaries of the radial fitting range by 10--20\\%; the\nformal errors were in general less than 1\\%. \n\n{\n\\begin{table}\n\n\\caption[]{\\label{boundaries.tab}{\\bf Fitting regions}\\\\ \nColumns: (1) Galaxy name; (2) and (3) Radial fitting range (arcsec); (4)\nand (5) Vertical region around the galactic plane excluded from the fits\n$(h_z)$.}\n\n\\begin{center}\n\\tabcolsep=1mm\n\n\\begin{tabular}{cccccc}\n\\hline\nGalaxy & \\multicolumn{2}{c}{Radial fitting} & &\n\\multicolumn{2}{c}{Galactic plane} \\\\\n\\cline{2-3}\\cline{5-6}\n\\noalign{\\vspace{2pt}}\n& $r_{\\rm min}$ & $r_{\\rm max}$ & & $z_{\\rm min}$ & $z_{\\rm max}$ \\\\\n(1) & (2) & (3) & & (4) & (5) \\\\\nESO 201-G22 & 15 & 50 & & --1.0 (S) & 1.0 (N) \\\\\nESO 416-G25 & 25 & 50 & & --1.0 (E) & 1.0 (W) \\\\\nESO 446-G18 & 18 & 55 & & --2.0 (E) & 0.5 (W) \\\\\nESO 446-G44 & ~~0 & 36 && --1.5 (S) & 1.5 (N) \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n}\n\n\\noindent\n{\\it ESO 201-G22} -- Figs. \\ref{2D.fig}a and b clearly show the bulge\ncomponent and, just outside the masked region, the effects of either\nresidual extinction near the galactic plane or, more likely, an\nadditional disc component. The residuals in the fitted region do not\nshow large systematic effects; the bulge contribution to the\ndisc-dominated fitting range is negligible. The negative residuals\nextending to the edges of these figures clearly show the presence of a\ntruncation in the galactic light distribution. \n\n\\noindent \n{\\it ESO 416-G25} -- This is the earliest-type galaxy in our pilot\nsample. The residuals after subtracting the disc-only fit (Fig. \n\\ref{2D.fig}d) do not appear to be systematic in the region where the\nfit was done, and their amplitude is small (r.m.s. residual $\\approx\n1.7\\ \\sigma_{\\rm background}$). Although we included a bulge component,\nthe 6-parameter fit proved to be very unstable. \n\n\\noindent\n{\\it ESO 446-G18} -- Figs. \\ref{contours.fig}c and \\ref{bgcheck.fig}\nshow that this galaxy is not exactly edge-on. Extinction predominantly\naffects the eastern side. A dust mask placed symmetrically with respect\nto the major axis is therefore not appropriate. The residuals are\nrelatively large (r.m.s. residual $\\approx 3.3\\ \\sigma_{\\rm\nbackground}$) but do not appear to be systematic in the region where the\nfit was done. A fit including an exponential bulge did not converge. \n\n\\noindent\n{\\it ESO 446-G44} -- Surface brightness profiles of ESO 446-G44 do not\nreveal any bulge component. They do show, however, that ESO 446-G44 may\nnot be exactly edge-on. Again, the residuals in the region where we\napplied our fitting routine are large (r.m.s. residual $\\approx 5.3\\\n\\sigma_{\\rm background}$) but do not appear to be systematic. \n\nHaving just obtained reliable global scale parameters for our sample\ngalaxies, we are now ready to quantify the disc truncations occurring in\nthese galaxies. \n\n\\begin{figure}\n\\psfig{figure=method2D.ps,width=9.0cm}\n\\caption[]{\\label{2D.fig}{\\it (a)} -- Negative {\\it I}-band image of ESO\n201-G22, after removal of foreground stars and background galaxies. The\nradial fitting boundaries are indicated by the dashed lines; the dust\nmask covering the region close to the plane affected by extinction is\nbracketed by the solid lines; {\\it (b)} -- Residuals for ESO 201-G22\nafter subtracting the model, greyscale levels range from $-6\n\\sigma_{\\rm background}$ (black) to $+6 \\sigma_{\\rm background}$\n(white) {\\it (c)} and {\\it (d)} -- ESO 416-G25, as (a) and (b)}\n\\end{figure}\n\n\\section{Truncated discs}\n\\label{results.sect}\n\n\\subsection{Approach}\n\\label{radlum.sect}\n\nIn the analysis of edge-on galaxies, the inner disc region closest to\nthe plane often needs to be avoided because of the presence of either a\nprominent dust lane, or a patchy dust distribution with its highest\ndensity towards the galactic plane. In many cases, the dust component\nextends all the way to the edge of the disc, thus making the luminosity\ndistribution near the galactic planes useless for our study. \n\nThe exponential scale{\\it height} of galactic discs is -- to first order\n-- constant as a function of galactocentric distance, at least for\nlater-type disc galaxies (see, e.g., KS1--4, Kylafis \\& Bahcall 1987,\nShaw \\& Gilmore 1990, Barnaby \\& Thronson 1992, de Grijs \\& Peletier\n1997). Therefore, the radial luminosity distributions parallel to the\ngalactic planes show a similar functional behaviour as the luminosity\nprofiles {\\it in} the plane in the absence of the dust component. \nConsequently, by extracting light profiles parallel to the galactic\nplanes, we will also be able to study the occurrence and properties of\nradially truncated discs, provided that the S/N ratio allows us to\ndetect such a truncation. \n\nTo find the optimum balance between avoiding contamination by the\nin-plane dust component on the one hand, and retaining a sufficiently\nhigh S/N ratio at large galactocentric distances on the other, numerous\nexperiments were performed. In Fig. \\ref{zrange.fig} we show ESO\n416-G25 as an example, where we compare several radial profiles obtained\non either side of the galactic plane. The solid lines in both panels\nrepresent the total profiles, vertically averaged over the entire half\nof the galactic disc (effectively for $|z| \\le 8 h_z$). They are\nobviously significantly affected by patchy dust and/or low S/N regions\nthroughout the disc and are therefore discarded from further use. From\nthis figure (and similar figures for the other galaxies in our sample)\nit follows that either the range $(1.0 \\le |z| \\le 2.0 h_z)$ or $(1.5\n\\le |z| \\le 3.5 h_z)$ is to be preferred for our detailed analysis. \nSince the former region has in general a higher S/N ratio we conclude\nthat the most representative, unobscured light profiles suitable for a\nfurther study of disc truncations are obtained by vertically collapsing\nthe surface brightness distribution between 1 and 2 $h_z$ {\\it on the\nleast obscured side of the galactic plane}. (Even though this galaxy\ndoes not have a prominent dust lane nor a very large amount of patchy\nextinction throughout its disc, Fig. \\ref{zrange.fig} clearly shows the\nrationale behind choosing the least obscured side of the galactic disc\n[top panel].)\n\nIn the radial direction, we apply a semi-logarithmic binning algorithm,\nin order to retain an approximately constant overall S/N ratio (cf. de\nGrijs et al. 1997, de Grijs 1998), where the binning at the outermost\ndisc radii never exceeds 2 resolution elements. \n\nIn the following sections, we will use profiles thus obtained for the\ndetailed analysis of the truncated discs in our pilot sample.\n\n\\begin{figure}\n\\psfig{figure=zrange.ps,width=8.8cm}\n\\caption[]{\\label{zrange.fig}Vertically averaged radial surface\nbrightness profiles of ESO 416-G25, taken at various {\\it z}-heights\nfrom the galactic plane. To retain an approximately constant S/N ratio,\na semi-logarithmic intensity-weighted radial binning algorithm has been\napplied to the individual profiles. An indication of the observational\nuncertainites is given by the background noise dominating the individual\nprofiles at large radii.}\n\\end{figure}\n\nA potential problem of this method is that the S/N ratios in the outer\ndisc regions are often significantly lower at some (vertical) distance\nfrom the galactic planes compared to those in the planes. From Fig. \n\\ref{zrange.fig} (and similar figures for the other sample galaxies) we\nconclude, however, that although the radial extent of the cut-off region\nappears to be a (weak) function of the height from the planes, the\nactual radii at which the luminosity profiles disappear asymptotically\ninto the background noise converge to the same value of $R_{\\rm max}$,\nwithin the observational uncertainties. \n\nSecondly, some evidence exists that galactic discs thicken with\nincreasing galactocentric distance (e.g., KS1, Kent, Dame \\& Fazio 1991,\nBarnaby \\& Thronson 1992, de Grijs \\& van der Kruit 1996, de Grijs \\&\nPeletier 1997). The signature of such a thickening of the discs on\nlight profiles extracted parallel to the galaxies' major axes is either\na flattening of the radial surface brightness profiles (if the\nthickening occurs gradually; e.g., Kent et al. 1991, Barnaby \\&\nThronson 1992, de Grijs \\& Peletier 1997) or a locally enhanced surface\nbrightness level at these large galactocentric distances (if only the\noutermost profiles are affected; e.g., KS1, de Grijs \\& van der Kruit\n1996). However, de Grijs \\& Peletier (1997) have shown that the effects\nof disc thickening are largest for the earliest-type spiral galaxies and\nalmost zero for the later types, {\\it including those examined in detail\nin this paper.} Moreover, even though the discs of our sample galaxies\nmay have larger scaleheights with increasing radii, the low S/N ratios\nand deviations from exponentially decreasing light distributions due to,\ne.g., spiral arms will likely hide such observational signatures. \nFinally, the possible thickening of galactic discs will {\\it not} affect\nthe determination of the actual truncation {\\it radius} $R_{\\rm max}$,\nsince the {\\it radial} surface brightness distribution remains\nunaffected. \n\nAlthough line-of-sight projection affects the observed functional form\nof the radial luminosity distribution, we will use a version of Eq. (2)\nwith an exponentially decreasing functionality in the truncation region,\nbut not including line-of-sight projection effects to fit {\\it the\ntruncation regions} of our sample galaxies. We chose to do so, because\n(i) Eq. (2) is a mathematically convenient function, and (ii) the\nactual profile shapes in the truncation region are erratic due to low\nS/N ratios and therefore the assumption of any more complicated\nfunctionality than an exponentially decreasing luminosity density cannot\nbe taken seriously. We will perform the actual fits to our\nsky-subtracted images {\\it in the linear regime} (i.e., in luminosity\ninstead of surface brightness space), to avoid undefined surface\nbrightness values due to ``negative'' noise peaks. \n\n\\subsection{Artificial truncations?}\n\nIn interpreting the steep luminosity decline as truly representative of\na decrease in either the light or density distributions of a galactic\ndisc, one has to make sure that the observed cut-off is not an artifact\ndue to inaccurate sky subtraction. De Vaucouleurs (1948), de\nVaucouleurs \\& Capaccioli (1979), and van Dokkum et al. (1994) have\nshown that inaccurate sky subtraction (i.e., oversubtraction) causes a\nfalse cut-off in the luminosity distribution of a galactic disc. This\ncan easily be checked, because the artificial cut-off would not only be\npresent in a major axis cut, but also in cuts taken in other directions. \n\nIn all cases, the background emission in the field of view of our sample\ngalaxies could be well represented by a plane, of which the slope was\ndetermined by the flux in regions sufficiently far away from the\ngalaxies in order not to be affected by residual galactic light. For\nmost of our observations, these planes could be closely approximated by\nconstant flux levels across the CCD field. The remaining uncertainties\nin the background are largely due to poisson noise. \n\nFig. \\ref{bgcheck.fig} illustrates the quality of the background\nsubtraction in the {\\it I} band, where the background contribution is\ngreater than in our other optical passbands. The left-hand panels show\nthe minor-axis (vertical) surface brightness profiles of all sample\ngalaxies (solid lines), radially averaged over $\\sim 20''$ in order to\nbe able to reach similar or fainter light levels as for the profiles\nalong the major axes, shown in the right-hand panels. We determined the\nsky noise, $\\sigma$, in the regions used for the background subtraction\nand created new images by subtracting (background $- 2 \\sigma$),\n(background $- 1 \\sigma$), (background $+ 1 \\sigma$), and (background $+\n2 \\sigma$), where ``background'' represents our best estimate of the sky\nbackground in the individual images. The under and oversubtracted\nprofiles are shown offset from the solid profiles for reasons of\nclarity. \n\nThe effects of oversubtraction can clearly be seen in the minor-axis\nsurface brightness profiles: they show artificial cut-offs and the\nnegative background values result in undefined surface brightnesses at\nthese {\\it z} heights, above or below $\\approx 15''$ (i.e. the profiles\ncould not be plotted beyond $\\approx 15''$ due to undefined surface\nbrightness values resulting from oversubtraction of the sky background). \nAlthough the effects of oversubtraction on the major-axis profiles\n(right-hand panels; vertically averaged between $-1.0$ and $1.0 h_z$)\nshow similar false cut-offs as for the minor-axis profiles, it appears\nthat most of the features seen in the light profiles represented by the\nsolid lines are real, since they are also observed in the {\\it\nunder$\\,$}subtracted light profiles. Moreover, a qualitative comparison\nof both the major and the minor-axis profiles (solid lines) shows that\nthe apparent truncations in or steepening of the major-axis light\nprofiles do not correspond to similar cut-offs at the same surface\nbrightness levels in the minor-axis profiles in any of our sample\ngalaxies. We thus conclude that these features are not due to\ninaccurate sky subtractions, but represent real deviations from the\nradial exponential light profiles. \n\n\\begin{figure}\n\\psfig{figure=bgcheck.ps,width=9cm}\n\\caption[]{\\label{bgcheck.fig}Minor-axis (left-hand panels) and\nmajor-axis (right-hand panels) {\\it I}-band luminosity profiles of our\nsample galaxies. From top to bottom, the lines in each panel represent\nthe minor axis profiles $- 2 \\sigma$ (dashed), $- 1 \\sigma$ (dotted),\nthe profiles after subtraction of our best estimates for the sky\nbackground, $+ 1 \\sigma$ (dotted lines), and $+ 2 \\sigma$ (dashed\nlines), where $\\sigma$ corresponds to the sky noise in the regions that\nwere used to determine the sky background levels. For reasons of\nclarity, the dashed and dotted profiles are displaced by, respectively,\n$\\pm 1.0$ and $\\pm 0.5$ mag from the solid lines.}\n\\end{figure}\n\nAlternatively, the detection of radially truncated discs can be\nartificially enhanced if the discs are strongly warped. In fact, it\nappears that for a number of our sample galaxies the locus of maximum\nintensity at large radii may slightly deviate from the main galactic\nplane direction (de Grijs 1997). However, this effect is negligible for\nthe determination of their radial truncations, because the deviations\nare {\\it almost insignificant} and we average the radial luminosity\nprofiles over a sufficiently large vertical range to avoid such\nproblems. \n\nIn addition, effects due to the galaxies' positions near the CCD edge or\nto scattered light from foreground stars are potentially serious. For\nour four sample galaxies, the former are non-existent. As one can see\nin Fig. \\ref{contours.fig}, both ESO 201-G22 and ESO 446-G18 suffer\nfrom the superposition of foreground stars, but only in the case of ESO\n446-G18 this precludes us from determining its maximum disc radius on\nthe northern end; the superposed foreground star near the eastern edge\nof ESO 201-G22 is located sufficiently far away from the truncation\nregion, {\\it and} is found on the most obscured side of the galactic\nplane. \n\n\\subsection{Properties of truncated discs}\n\\label{properties.sect}\n\n\\subsubsection{Where do the truncations occur?}\n\nIn Fig. \\ref{cutoffs.fig} we show the radial surface brightness\nprofiles parallel to the major axis of our sample galaxies in all\npassbands and out to either the edge of the CCD frames or to those radii\nwhere the background noise dominates. The dashed lines indicate\ninfinite exponential model discs, using the scale parameters obtained in\nSect. \\ref{2D.sect}. It is immediately clear that the observed surface\nbrightness profiles are significantly fainter in the outer regions than\nthe model discs for all galaxies in our sample and for all passbands. \n(Note that since these profiles are only meant to guide the eye, we have\nnot computed the full line-of-sight integrated model profiles, although\nsuch models were, in fact, used to obtain the actual scale parameters.)\n\nThe values for both $R_{\\max}$ and $\\delta$, resulting from the fitting\nof our modified version of Eq. (2) to the observed profiles using a\nstandard linear least-squares fitting technique, are listed in Table\n\\ref{Rmax.tab}. Although the formal measurement errors in $R_{\\rm max}$\nare $\\le 2''$, the relatively large uncertainties associated with\n$R_{\\rm max}$ are due to the fact that S/N $< 1$ at the truncation\nradius (by definition), and to the difficulty in the unambiguous\ndetermination of the start of the truncation region. In fact, the\nuncertainties in the truncation lengths are {\\it predominantly}\ndetermined by the nature of $R_{\\rm max}$ as lower limit. \n\nHowever, this method will at least produce objective estimates of\n$R_{\\rm max}$, as opposed to visually extrapolating the observed surface\nbrightness profiles to those radii where they would (supposedly)\ndisappear asymptotically into the noise, as has been done previously by\nother workers in this field. \n\nFor a comparison with previously published results for other galaxies,\nwe have also included the estimated truncation radii in units of the\ngalaxies' {\\it I}-band scalelengths. We chose to use the {\\it I}-band\nscalelengths to determine the $R_{\\rm max} / h_R$ ratios, because these\nrepresent our longest-wavelength observations, which most likely best\napproximate the dominant stellar luminosity (and presumably mass)\ndistributions (de Grijs et al. 1997, de Grijs 1998). The corresponding\nerrors reflect the uncertainties in the determinations of both the\nscalelengths and the truncation radii. \n\nVan der Kruit \\& Searle (KS3) determined, for their small sample of\nlarge edge-ons, that the mean truncation radius $R_{\\rm max} \\simeq (4.2\n\\pm 0.6) h_R$ (cf. Bottema 1995 for NGC 4013: $R_{\\rm max} \\simeq 4.1\nh_R$). Barteldrees \\& Dettmar (1994) found, for a sample of 27 edge-on\ngalaxies, a mean truncation radius of $\\simeq (3.7 \\pm 1.0) h_R$. \nHowever, this result is based on a different definition of $R_{\\rm\nmax}$: they interpreted the truncation radius as the galactocentric\ndistance at which the observed projected radial profiles start to\ndeviate significantly from the model exponential profiles. If we keep\nin mind that the truncation occurs over a finite region, then the\ndiscrepancy between these determinations can be understood. A direct\ncomparison is therefore impossible. \n\nRecently, Pohlen et al. (2000a) largely reanalysed Barteldrees \\&\nDettmar's (1994) sample, assuming infinitely sharply truncated galactic\ndiscs following KS3. For their sample of 31 nearby edge-on spiral\ngalaxies, they found $R_{\\rm max}/h_R = 2.9 \\pm 0.7$, significantly\nlower than the ratio found by KS3. This may reflect a selection bias\ntowards large galaxies and/or small-number statistics in the KS3 sample. \n\nWhile the discs of ESO 201-G22, ESO 416-G25, and ESO 446-G18 are\ntruncated at comparable radii as found by KS and Bottema (1995), expressed\nin units of their disc scalelengths, ESO 446-G44 is clearly truncated at\nmuch smaller radii. This is the sample galaxy with the greatest\nscalelength and the only one without a bulge. ESO 446-G44, as well as\nESO 416-G25, exhibit truncated discs well within the range found by\nPohlen et al. (2000a).\n\n\\begin{figure}\n\\psfig{figure=cutoffs.ps,width=9cm}\n\\caption[]{\\label{cutoffs.fig}Radial surface brightness profiles of our\nsample galaxies taken parallel to their major axes (Sect. \n\\ref{radlum.sect}). Overplotted are the model exponential profiles\n(dashed lines), based on our full surface brightness modelling. The\narrows indicate the measured truncation radii; for ESO 446-G18 the\ncrosses indicate the side where we cannot determine the maximum radius\ndue to the presence of foreground stars.}\n\\end{figure}\n\n\\begin{figure}\n\\psfig{figure=Ibandprofs.ps,width=9cm}\n\\caption[]{\\label{Iband.fig}{\\it I}-band radial surface brightness\nprofiles of our sample galaxies taken parallel to their major axes, as\nin Fig. \\ref{cutoffs.fig}. Overplotted are the model exponential\nprofiles (dashed lines), based on our full surface brightness modelling. \nThe arrows indicate the measured truncation radii; for ESO 446-G18 the\ncross indicates the side where we cannot determine the maximum radius\ndue to the presence of foreground stars. The dotted lines correspond to\n{\\it infinitely sharply truncated} models, used for a comparison with\nprevious work (Sect. \\ref{asymm.sect}); the thin solid lines are our\nmodel fits using a slightly modified Casertano model (Sect. \n\\ref{3Dmodel.sect}).}\n\\end{figure}\n\n{\n\\begin{table*}\n\n\\caption[]{\\label{Rmax.tab}{\\bf Disc truncations in our sample galaxies}\\\\ \nColumns: (1) Galaxy name; (2) Passband observed in; (3) Side of the\ngalactic centre; (4) and (5) Cut-off radius (lower limit) and\nobservational error (arcsec); (6) and (7) Cut-off radius (in units of\nthe {\\it I-}band scalelength) and observational error; (8) Truncation\nlength (arcsec; typical observational uncertainties are of order $\\ge\n10''$); (9) and (10) Scalelength in the truncation region and\nobservational error, in arcsec; (11) and (12) Scalelength in the\ntruncation region and observational error, in kpc (based on the\nheliocentric velocities obtained by Mathewson, Ford \\& Buchhorn 1992\n[see de Grijs 1998])}\n\n\\begin{center}\n\\tabcolsep=1mm\n\n\\begin{tabular}{cccrrccccrcccc}\n\\hline\nGalaxy & Band & Side & $R_{\\rm max}$ & $\\pm$ & & $R_{\\rm max}$ &\n$\\pm$ & $\\delta$ & $h_{R,\\delta}$ & $\\pm$ & & $h_{R,\\delta}$ & $\\pm$ \\\\\n\\cline{4-5}\\cline{7-8}\\cline{10-11}\\cline{13-14}\n\\noalign{\\vspace{2pt}}\n(ESO-LV) & & & \\multicolumn{2}{c}{$('')$} & & \\multicolumn{2}{c}{$(h_{R,I})$} &\n$('')$ & \\multicolumn{2}{c}{$('')$} & & \\multicolumn{2}{c}{($h^{-1}$ kpc)} \\\\\n(1) & (2) & (3) & (4) & (5) & & (6) & (7) & (8) & (9) & (10) & & (11) & (12) \\\\\n201-G22 & {\\it B} & E & 114 & 10 & & 4.9 & 0.4 & 44 & 14.2 & 1.0 & & 3.0 & 0.2 \\\\\n & & W & 99 & 4 & & 4.2 & 0.2 & 50 & 16.8 & 0.9 & & 3.6 & 0.2 \\\\\n & {\\it V} & E & 109 & 3 & & 4.7 & 0.1 & 26 & 16.4 & 0.3 & & 3.5 & 0.1 \\\\\n & & W & 99 & 10 & & 4.2 & 0.4 & 50 & 15.7 & 0.4 & & 3.4 & 0.1 \\\\\n & {\\it I} & E & 109 & 10 & & 4.7 & 0.4 & 28 & 17.6 & 0.3 & & 3.8 & 0.1 \\\\\n & & W & 89 & 8 & & 3.8 & 0.3 & 31 & 12.9 & 0.2 & & 2.8 & 0.1 \\\\\n416-G25 & {\\it B} & N & 73 & 5 & & 3.3 & 0.2 & 13 & 6.2 & 1.6 & & 1.7 & 0.4 \\\\\n & & S & 75 & 10 & & 3.3 & 0.5 & 30 & 10.6 & 1.4 & & 2.9 & 0.4 \\\\\n & {\\it V} & N & 82 & 10 & & 3.7 & 0.5 & 24 & 7.2 & 1.0 & & 2.0 & 0.3 \\\\\n & & S & 100 & 15 & & 4.5 & 0.7 & 60 & 14.9 & 0.3 & & 4.1 & 0.1 \\\\\n & {\\it I} & N & 80 & 15 & & 3.6 & 0.7 & 28 & 9.2 & 1.7 & & 2.6 & 0.4 \\\\\n & & S & 76 & 10 & & 3.4 & 0.5 & 53 & 14.3 & 1.3 & & 4.0 & 0.4 \\\\\n446-G18 & {\\it B} & S & 83 & 8 & & 4.6 & 0.4 & 35 & 11.6 & 0.6 & & 2.8 & 0.1 \\\\\n & {\\it V} & S & 79 & 8 & & 4.4 & 0.4 & 31 & 10.3 & 0.3 & & 2.5 & 0.1 \\\\\n & {\\it I} & S & 83 & 8 & & 4.6 & 0.4 & 35 & 9.8 & 1.0 & & 2.4 & 0.2 \\\\\n446-G44 & {\\it B} & E & 66 & 8 & & 2.3 & 0.3 & 25 & 10.2 & 0.2 & & 1.6 & 0.1 \\\\\n & & W & 63 & 8 & & 2.1 & 0.3 & 22 & 7.6 & 0.8 & & 1.2 & 0.1 \\\\\n & {\\it V} & E & 67 & 8 & & 2.3 & 0.3 & 27 & 8.6 & 1.3 & & 1.3 & 0.3 \\\\\n & & W & 62 & 8 & & 2.1 & 0.3 & 22 & 7.6 & 0.8 & & 1.2 & 0.2 \\\\\n & {\\it I} & E & 72 & 7 & & 2.4 & 0.3 & 33 & 9.8 & 0.3 & & 1.5 & 0.1 \\\\\n & & W & 72 & 8 & & 2.4 & 0.3 & 32 & 8.6 & 0.3 & & 1.3 & 0.1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n}\n\n\\subsubsection{Asymmetry and sharpness}\n\\label{asymm.sect}\n\nDisc truncations do not necessarily occur at the same galactocentric\ndistances or with the same abruptness on either side (e.g., KS1, Jensen\n\\& Thuan 1982, N\\\"aslund \\& J\\\"ors\\\"ater 1997, Abe et al. 1999, Fry et\nal. 1999). In most cases, however, the truncation radii on either side\nof the galactic disc occur within $\\sim 10-15$\\% of each other. The\nprofiles in Figs. \\ref{cutoffs.fig} and \\ref{Iband.fig}, and our\nestimates for $R_{\\rm max}$ in Table \\ref{Rmax.tab} show that in\ngeneral, the discs of our sample galaxies are truncated at similar\nradii, within their observational uncertainties, with the possible\nexception of ESO 201-G22. \n\nTable \\ref{Rmax.tab} also contains our best estimates for the\nexponential scalelength in the truncation region, $h_{R,\\delta}$. To\nmeasure this scalelength, we defined the inner fitting radius as the\nradius where the radial surface brightness profile starts to deviate\nsignificantly from the model radial exponential light distribution (cf. \nBarteldrees \\& Dettmar 1994); the uncertainty in this radius is\ngenerally $\\le 15$\\%, depending on the particular galaxy considered. As\nouter fitting boundary we used $R_{\\rm max}$. The errors associated\nwith $h_{R,\\delta}$ are observational uncertainties, obtained from the\ncomparison of several similar fits in which we adjusted the inner\nboundary of the radial fitting range by 10--20\\%. The uncertainties in\n$R_{\\rm max}$ are included in the observational errors given; the formal\nerrors were $\\le 1$\\%. The effects on disc truncations and scale\nparameters of an inclination $i \\neq 90^\\circ$ are negligible for our\ngalaxy sample of $i \\ge 87^\\circ$ inclined galaxies, as was convincingly\nshown by Barteldrees \\& Dettmar (1994), in particular in view of the\nsystematic and observational uncertainties involved. The combination of\n$h_{R,\\delta}$ and $\\delta$ gives us an indication of the asymmetry and\nsharpness of the actual disc truncations. \n\nConsidering the relatively large systematic and observational errors\nthat cannot be avoided at this point, we cannot claim that we detect any\nsystematic asymmetries, perhaps with the exception of ESO 416-G25. The\nnorthern edge of the disc of ESO 416-G25 is very sharply truncated\ncompared to its southern edge. Upon close examination of the actual CCD\nimages, we believe that this may be explained by the fact that we likely\nobserve the outer stellar envelope of a spiral arm, whereas on the\nsouthern side we are looking into the inside of a spiral arm. Note,\nhowever, that also at the southern edge of the disc a clear truncation\nsignature is observed (Fig. \\ref{cutoffs.fig}). \n\nThe last two columns of Table \\ref{Rmax.tab} show that in none of our\nsample galaxies the radial scalelength in the truncation region\ndecreases to values of order or less than 1 kpc. Using $H_0 = 50$ km\ns$^{-1}$ Mpc$^{-1}$, or other values closer to the current best\nestimate, will further increase the truncation scalelengths measured in\nour galaxies. Although two of our sample galaxies may not be exactly\nedge-on, their inclinations are sufficiently close to 90$^\\circ$ so as\nnot to increase the measurements of $h_{R,\\delta}$ by more than their\nobservational uncertainties (cf. Barteldrees \\& Dettmar 1994). We are\ntherefore forced to conclude that, although our discs are clearly\ntruncated, the truncation occurs over a larger region and not as\nabruptly as found in previous studies. \n\nAs a comparison between our approach and that used in previous work, in\nFig. \\ref{Iband.fig} we present enlarged versions of the {\\it I}-band\nprofiles of Fig. \\ref{cutoffs.fig}, in which we show both our best-fit\nradial profiles using a slightly modified Casertano model (Sect. \n\\ref{3Dmodel.sect}; thin solid lines) and the corresponding profiles for\na symmetric, projected, sharply truncated exponential disc, with the\ndisc truncations occurring at $R_{\\rm max}$ (dotted lines). These\nprofiles were obtained by applying the same method used to extract the\nobserved profiles used for Fig. \\ref{cutoffs.fig} to model images of\nour sample galaxies. From a comparison between these dotted lines and\nthe actual, observed profiles, it is clear that the discs of our sample\ngalaxies are generally {\\it not infinitely sharply truncated}, but show\na more gradual decrease of their radial luminosity density. \n\nFinally, from a galaxy-by-galaxy inspection of the values for $R_{\\rm\nmax}$, no apparent trend with wavelength can be discerned, within the\nobservational errors. However, a detailed comparison of the radial $(B\n- I)$ colour profiles, our longest colour baseline, reveals that the\ndisc colour tends to get bluer in the truncation region compared to the\ncolours in the main disc (Fig. \\ref{truncolors.fig}). A similar result\nwas obtained by Sasaki (1987) for the truncation region of NGC 5907. \nAlthough this may be indicative of more recent star formation at the\nedge of the discs (e.g., Larson 1976, Seiden et al. 1984), the opposite\nbehaviour is exhibited both on the northern side of the disc of ESO\n416-G25, and on the western side of the disc of ESO 446-G44, where the\ncolour reddens in the truncation region. This is confirmed by the fact\nthat at these edges, these discs appear to be more sharply truncated in\nthe {\\it B} band compared to the {\\it I-}band observations, which may be\nthe signature of a spiral arm. \n\n\\begin{figure}\n\\psfig{figure=truncolors.ps,width=9cm}\n\\caption[]{\\label{truncolors.fig}Radial $(B-I)$ colour profiles of our\nsample galaxies parallel to their major axes, constructed from the\nprofiles shown in Fig. \\ref{cutoffs.fig}. The dashed lines indicate the\napproximate start of the truncation regions. All data points with\ncolour uncertainties $\\sigma_{(B-I)} \\le 0.4$ mag have been included.}\n\\end{figure}\n\n\\section{Dynamical consequences of disc truncations: Outlook}\n\\label{dynamics.sect}\n\nAlthough the fact that many spiral discs seem to have truncated stellar\ndiscs is an interesting observation in the context of galactic\nstructure, the physical implications of a similar truncation in the mass\ndistribution of galactic discs has far-reaching consequences making the\nstudy of disc truncations fundamental to our understanding of galactic\ndisc maintenance and evolution. This is therefore the main science\ndriver of our attempts to define a unique and objective method to\nmeasure disc truncations. \n\n\\subsection{Edge smearing and disc asymmetries}\n\nThe persistence of sharp disc cut-offs places a strong upper limit on\nthe stellar velocity dispersion at the disc edge (KS1). Adopting a\nrotational velocity of 250 km s$^{-1}$ at 20 kpc for NGC 4565, the\nradial stellar velocity dispersion, $\\langle v^2_R \\rangle^{1/2}$, must\nbe $\\le 10$ km s$^{-1}$ so that random motions do not wash out the sharp\ncut-off within one revolution time (Jensen \\& Thuan 1982; see also KS1,\nMay \\& James 1984), or the sharp cut-off must be a transient feature\n(e.g., Sasaki 1987). This low upper limit for $\\langle v^2_R\n\\rangle^{1/2}$ is close to the minimum value of $\\sim 2$ km s$^{-1}$\nneeded to satisfy Toomre's (1964) criterion of local stability for disc\ngalaxies, and thus for star formation.\n\nAlternatively, in the case of a disc formation scenario in which the\ndisc grows from the inside outward (e.g., Larson 1976, Gunn 1982, Seiden\n1983, Seiden et al. 1984) a sharp edge can be maintained if this\noutward growth is sufficiently rapid, so that the random motion of the\nstars does not smear out the edge. Note, however, that the disc\ntruncations in our sample galaxies are not as sharp as those found by\nKS1--4, among others, which will relax these requirements. \n\nThe situation becomes more complicated if the galactic disc is lopsided\nor if the truncations occur at different radii. Following the epicyclic\ndescription of Baldwin, Lynden-Bell \\& Sancisi (1980), van der Kruit\n(1988) estimates a smearing time of $1.7 \\times 10^{10}$ yr for the\nGalactic disc, and he concludes that a variation in the truncation radii\nof order 10\\% may just survive a Hubble time. With the possible\nexception of ESO 416-G25, our sample galaxies appear to comfortably meet\nthis requirement. \n\n\\subsection{Rotation curves as diagnostic tools}\n\nCasertano (1983) has shown that a truncated stellar disc leaves a\nsignature on the rotation curve in the form of a region of slowly\nvarying velocity followed by a steep decline just outside the truncation\nradius (see also Hunter, Ball \\& Gottesman 1984). The amount of this\ndecrease is a measure of the disc mass. The effect of a truncation is a\n{\\it flattening} of the rotation curve inside the truncation itself,\nfrom some radius $R_0$ to $R_{\\rm max}$, and a steep decrease of the\nvelocity outside. \n\nThe well-known warped edge-on galaxy NGC 4013, for which Bottema (1995)\nsuspected a sudden decrease in the mass density corresponding to the\ntruncation radius, has indeed been shown to exhibit a sudden drop in the\nrotational velocity of about 20 km s$^{-1}$ just at the optical edge\n(Bottema, Shostak \\& van der Kruit 1987, Bottema 1995, 1996). This drop\ncan be understood if one realises that near the edge of the galactic\ndisc the mass distribution will be irregular: there is no smooth,\ncircular end to the disc, but it likely ends in spiral arms. Bottema\n(1996) argues that therefore gas moving in the potential of such patches\nof stellar matter will not be in precise circular motion and hence the\nradial velocity along the line of sight is somewhat lower than the true\nrotation. \n\nFinally, Bahcall (1983) showed that, for Sb or Sc galaxies like NGC 891\nor the Galaxy, the feature in the rotation curve due to the truncated\nstellar disc is observable only if $R_{\\rm max} \\le 4 h_R$ (smaller for\ngalaxies with more prominent bulges), if the truncation length is small\ncompared to $h_R$, and if the halo mass inside $R_{\\rm max}$ is smaller\nthan the disc mass (Casertano 1983). \n\nUnfortunately, the currently available velocity information for the four\ngalaxies in our pilot sample does not allow us to confirm the presence\nof sharp truncations in the disc mass based on the shape of the rotation\ncurves: only for ESO 446-G18 and ESO 446-G44 rotation curves have been\npublished, for the H$\\alpha$ emission (Mathewson et al. 1992) and the\nH{\\sc i} component (Persic \\& Salucci 1995, based on the raw Mathewson\net al. 1992 data), but these rotation curves do not or just barely\nreach those radii where we expect to be able to see a truncation\nsignature.\n\n\\section{Summary and Conclusions}\n\nIn this paper we have presented the first results of a systematic\nanalysis of galactic disc structure in general and of radially truncated\nexponential discs in particular for a pilot sample of four ``normal''\ndisc-dominated edge-on spiral galaxies. We have carefully considered\nthe importance of (residual) dust, deviations from 90$^\\circ$\ninclinations, and spiral arms, and concluded that these effects do not\naffect our results significantly. We have also shown that the truncated\ndiscs in our sample galaxies are not caused artificially by inaccurate\nsky subtraction, but are real deviations from the radial exponential\nlight profiles. \n\nAn independent approach to obtain the statistics of truncated galactic\ndiscs, using a sample of galaxies selected in a uniform way, is needed\nin order to better understand their overall properties and physical\nimplications. If the truncations seen in the stellar light are also\npresent in the mass distribution, they would have important dynamical\nconsequences at the disc's outer edges. We have shown that the\ntruncated luminosity distributions of our pilot sample galaxies, if also\npresent in the mass distributions, comfortably meet the requirements for\nlongevity. \n\nThe truncation radii, expressed in units of $h_R$, for the discs of ESO\n201-G22, ESO 416-G25, and ESO 446-G18 are comparable to those found by\nKS1--4 and Bottema (1995), while ESO 446-G44 is truncated at much\nsmaller radii. In fact, the truncations of the discs of ESO 416-G25 and\nESO 446-G44 are within the range found by Pohlen et al. (2000a) for\ntheir sample of 31 nearby edge-on spiral galaxies. In general, the\ndiscs of our sample galaxies are truncated at similar radii on either\nside of their centres, within the observational uncertainties, with the\nexception of ESO 201-G22. \n\nWith the possible exception of the disc of ESO 416-G25, it appears that\nour sample galaxies are fairly symmetric, in terms of both the sharpness\nof their disc truncations and the truncation length, although the\ntruncations occur over a larger region and not as abruptly as found in\nprevious studies. The northern edge of the disc of ESO 416-G25 is very\nsharply truncated compared to its southern edge. We believe that this\nmay be explained by the fact that we likely observe the outer stellar\nenvelope of a spiral arm, whereas on the southern side we are looking\ninto the inside of a spiral arm. \n\n\\section*{Acknowledgments} We thank Piet van der Kruit for stimulating\ndiscussions and acknowledge useful suggestions by the referee, M. \nPohlen. This work is partially based on the undergraduate senior thesis\nof KHW at the University of Virginia. RdeG acknowledges partial funding\nfrom NASA grants NAG 5-3428 and NAG 5-6403 and hospitality at the\nUniversity of Groningen. This research has made use of NASA's\nAstrophysics Data System Abstract Service and of the NASA/IPAC\nExtragalactic Database (NED). \n\n\\begin{thebibliography}{}\n\n\\bibitem[]{} Abe F., Bond I.A., Carter B.S., et al., 1999, AJ, 118, 261\n\\bibitem[]{} Bahcall J.N., 1983, ApJ, 267, 52\n\\bibitem[]{} Baldwin J.E., Lynden-Bell D., Sancisi R., 1980, MNRAS, 193,\n313 \n\\bibitem[]{} Barnaby D., Thronson Jr. H.A., 1992, AJ, 103, 41\n\\bibitem[]{} Barteldrees A., Dettmar R.-J., 1994, A\\&AS, 103, 475\n\\bibitem[]{} Bottema R., 1995, A\\&A, 295, 605\n\\bibitem[]{} Bottema R., 1996, A\\&A, 306, 345\n\\bibitem[]{} Bottema R., Shostak G.S., van der Kruit P.C., 1987, Nat,\n328, 401\n\\bibitem[]{} Casertano S., 1983, MNRAS, 203, 735\n\\bibitem[]{} de Grijs R., 1997, PhD Thesis, Univ. of Groningen, the\nNetherlands\n\\bibitem[]{} de Grijs R., 1998, MNRAS, 299, 595\n\\bibitem[]{} de Grijs R., Peletier R.F., 1997, A\\&A, 320, L21\n\\bibitem[]{} de Grijs R., Peletier R.F., van der Kruit P.C., 1997, A\\&A,\n327, 966\n\\bibitem[]{} de Grijs R., van der Kruit P.C., 1996, A\\&AS, 117, 19\n\\bibitem[]{} de Jong R.S., 1995, PhD Thesis, Univ. of Groningen, the\nNetherlands \n\\bibitem[]{} de Vaucouleurs G., 1948, Ann. Astrophys., 11, 247\n\\bibitem[]{} de Vaucouleurs G., Capaccioli M., 1979, ApJS, 40, 699\n\\bibitem[]{} Freeman K.C., 1970, ApJ, 160, 811 \n\\bibitem[]{} Fry A.M., Morrison H.L., Harding P., Boroson T.A., 1999,\nAJ 118, 1209\n\\bibitem[]{} Giovanelli R., Haynes M.P., Salzer J.J., Wegner G., Da\nCosta L.N., Freudling W., 1994, AJ, 107, 2036\n\\bibitem[]{} Gunn J.E., 1982, in: Astrophysical cosmology, Proceedings of the\nStudy Week on Cosmology and Fundamental Physics, Vatican City State, p. 233\n\\bibitem[]{} Hunter Jr. J.H., Ball R., Gottesman S.T., 1984, MNRAS,\n208, 1\n\\bibitem[]{} Jensen E.B., Thuan T.X., 1982, ApJS, 50, 421\n\\bibitem[]{} Kent S.M., Dame T.M., Fazio G., 1991, ApJ, 378, 131\n\\bibitem[]{} Knapen J.H., van der Kruit P.C., 1991, A\\&A, 248, 57\n\\bibitem[]{} Kregel M., van der Kruit P.C., de Grijs R., 2001, MNRAS, \nsubmitted\n\\bibitem[]{} Kylafis N.D., Bahcall J.N., 1987, ApJ, 317, 637 \n\\bibitem[]{} Larson R.B., 1976, MNRAS, 176, 31\n\\bibitem[]{} Lauberts A., Valentijn E.A., 1989, The Surface Photometry\nCatalogue of the ESO-Uppsala Galaxies, Garching bei M\\\"unchen: ESO\n(ESO-LV)\n\\bibitem[]{} Lequeux J., Dantel-Fort M., Fort B., 1995, A\\&A, 296, L13\n\\bibitem[]{} Mathewson D.S., Ford V.L., Buchhorn M., 1992, ApJS, 81, 413\n\\bibitem[]{} May A., James R.A., 1984, MNRAS, 206, 691\n\\bibitem[]{} Morrison H.L., Boroson T.A., Harding P., 1994, AJ, 108, 1191\n\\bibitem[]{} N\\\"aslund M., J\\\"ors\\\"ater S., 1997, A\\&A, 325, 915\n\\bibitem[]{} Persic M., Salucci P., 1995, ApJS, 99, 501\n\\bibitem[]{} Pohlen M., Dettmar R.-J., L\\\"utticke R., 2000a, A\\&A, 357, L1\n\\bibitem[]{} Pohlen M., Dettmar R.-J., L\\\"utticke R., Schwarzkopf U.,\n2000b, A\\&AS, 144, 405\n\\bibitem[]{} Sasaki T., 1987, PASJ, 39, 849\n\\bibitem[]{} Seiden P.E., 1983, ApJ, 266, 555\n\\bibitem[]{} Seiden P.E., Schulman L.S., Elmegreen B.G., 1984, ApJ, 282, 95\n\\bibitem[]{} Shaw M.A., Gilmore G., 1990, MNRAS, 242, 59\n\\bibitem[]{} Toomre A., 1964, ApJ, 139, 1217 \n\\bibitem[]{} van der Kruit P.C., 1988, A\\&A, 192, 117\n\\bibitem[]{} van der Kruit P.C., Searle L., 1981a, A\\&A, 95, 105 (KS1)\n\\bibitem[]{} van der Kruit P.C., Searle L., 1981b, A\\&A, 95, 116 (KS2)\n\\bibitem[]{} van der Kruit P.C., Searle L., 1981a, A\\&A, 110, 61 (KS3)\n\\bibitem[]{} van der Kruit P.C., Searle L., 1981b, A\\&A, 110, 79 (KS4)\n\\bibitem[]{} van Dokkum P.G., Peletier R.F., de Grijs R., Balcells M.,\n1994, A\\&A, 286, 415\n\\bibitem[]{} Xilouris E.M., Kylafis N.D., Papamastorakis J., Paleologou\nE.V., Haerendel G., 1997, A\\&A, 325, 135\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n"
}
] |
[
{
"name": "astro-ph0002523.extracted_bib",
"string": "\\begin{thebibliography}{}\n\n\\bibitem[]{} Abe F., Bond I.A., Carter B.S., et al., 1999, AJ, 118, 261\n\\bibitem[]{} Bahcall J.N., 1983, ApJ, 267, 52\n\\bibitem[]{} Baldwin J.E., Lynden-Bell D., Sancisi R., 1980, MNRAS, 193,\n313 \n\\bibitem[]{} Barnaby D., Thronson Jr. H.A., 1992, AJ, 103, 41\n\\bibitem[]{} Barteldrees A., Dettmar R.-J., 1994, A\\&AS, 103, 475\n\\bibitem[]{} Bottema R., 1995, A\\&A, 295, 605\n\\bibitem[]{} Bottema R., 1996, A\\&A, 306, 345\n\\bibitem[]{} Bottema R., Shostak G.S., van der Kruit P.C., 1987, Nat,\n328, 401\n\\bibitem[]{} Casertano S., 1983, MNRAS, 203, 735\n\\bibitem[]{} de Grijs R., 1997, PhD Thesis, Univ. of Groningen, the\nNetherlands\n\\bibitem[]{} de Grijs R., 1998, MNRAS, 299, 595\n\\bibitem[]{} de Grijs R., Peletier R.F., 1997, A\\&A, 320, L21\n\\bibitem[]{} de Grijs R., Peletier R.F., van der Kruit P.C., 1997, A\\&A,\n327, 966\n\\bibitem[]{} de Grijs R., van der Kruit P.C., 1996, A\\&AS, 117, 19\n\\bibitem[]{} de Jong R.S., 1995, PhD Thesis, Univ. of Groningen, the\nNetherlands \n\\bibitem[]{} de Vaucouleurs G., 1948, Ann. Astrophys., 11, 247\n\\bibitem[]{} de Vaucouleurs G., Capaccioli M., 1979, ApJS, 40, 699\n\\bibitem[]{} Freeman K.C., 1970, ApJ, 160, 811 \n\\bibitem[]{} Fry A.M., Morrison H.L., Harding P., Boroson T.A., 1999,\nAJ 118, 1209\n\\bibitem[]{} Giovanelli R., Haynes M.P., Salzer J.J., Wegner G., Da\nCosta L.N., Freudling W., 1994, AJ, 107, 2036\n\\bibitem[]{} Gunn J.E., 1982, in: Astrophysical cosmology, Proceedings of the\nStudy Week on Cosmology and Fundamental Physics, Vatican City State, p. 233\n\\bibitem[]{} Hunter Jr. J.H., Ball R., Gottesman S.T., 1984, MNRAS,\n208, 1\n\\bibitem[]{} Jensen E.B., Thuan T.X., 1982, ApJS, 50, 421\n\\bibitem[]{} Kent S.M., Dame T.M., Fazio G., 1991, ApJ, 378, 131\n\\bibitem[]{} Knapen J.H., van der Kruit P.C., 1991, A\\&A, 248, 57\n\\bibitem[]{} Kregel M., van der Kruit P.C., de Grijs R., 2001, MNRAS, \nsubmitted\n\\bibitem[]{} Kylafis N.D., Bahcall J.N., 1987, ApJ, 317, 637 \n\\bibitem[]{} Larson R.B., 1976, MNRAS, 176, 31\n\\bibitem[]{} Lauberts A., Valentijn E.A., 1989, The Surface Photometry\nCatalogue of the ESO-Uppsala Galaxies, Garching bei M\\\"unchen: ESO\n(ESO-LV)\n\\bibitem[]{} Lequeux J., Dantel-Fort M., Fort B., 1995, A\\&A, 296, L13\n\\bibitem[]{} Mathewson D.S., Ford V.L., Buchhorn M., 1992, ApJS, 81, 413\n\\bibitem[]{} May A., James R.A., 1984, MNRAS, 206, 691\n\\bibitem[]{} Morrison H.L., Boroson T.A., Harding P., 1994, AJ, 108, 1191\n\\bibitem[]{} N\\\"aslund M., J\\\"ors\\\"ater S., 1997, A\\&A, 325, 915\n\\bibitem[]{} Persic M., Salucci P., 1995, ApJS, 99, 501\n\\bibitem[]{} Pohlen M., Dettmar R.-J., L\\\"utticke R., 2000a, A\\&A, 357, L1\n\\bibitem[]{} Pohlen M., Dettmar R.-J., L\\\"utticke R., Schwarzkopf U.,\n2000b, A\\&AS, 144, 405\n\\bibitem[]{} Sasaki T., 1987, PASJ, 39, 849\n\\bibitem[]{} Seiden P.E., 1983, ApJ, 266, 555\n\\bibitem[]{} Seiden P.E., Schulman L.S., Elmegreen B.G., 1984, ApJ, 282, 95\n\\bibitem[]{} Shaw M.A., Gilmore G., 1990, MNRAS, 242, 59\n\\bibitem[]{} Toomre A., 1964, ApJ, 139, 1217 \n\\bibitem[]{} van der Kruit P.C., 1988, A\\&A, 192, 117\n\\bibitem[]{} van der Kruit P.C., Searle L., 1981a, A\\&A, 95, 105 (KS1)\n\\bibitem[]{} van der Kruit P.C., Searle L., 1981b, A\\&A, 95, 116 (KS2)\n\\bibitem[]{} van der Kruit P.C., Searle L., 1981a, A\\&A, 110, 61 (KS3)\n\\bibitem[]{} van der Kruit P.C., Searle L., 1981b, A\\&A, 110, 79 (KS4)\n\\bibitem[]{} van Dokkum P.G., Peletier R.F., de Grijs R., Balcells M.,\n1994, A\\&A, 286, 415\n\\bibitem[]{} Xilouris E.M., Kylafis N.D., Papamastorakis J., Paleologou\nE.V., Haerendel G., 1997, A\\&A, 325, 135\n\n\\end{thebibliography}"
}
] |
astro-ph0002524
|
ARE THERE STRANGE STARS IN THE UNIVERSE ?
|
[
{
"author": "IGNAZIO BOMBACI"
}
] |
Definitely, an affirmative answer to this question would have implications of fundamental importance for astrophysics (a new class of compact stars), and for the physics of strong interactions (deconfined phase of quark matter, and strange matter hypothesis). In the present work, we use observational data for the newly discovered millisecond X-ray pulsar SAX J1808.4-3658 and for the atoll source 4U~1728-34 to constrain the radius of the underlying compact stars. Comparing the mass--radius relation of these two compact stars with theoretical models for both neutron stars and strange stars, we argue that a strange star model is more consistent with SAX J1808.4-3658 and 4U~1728-34, and suggest that they are likely strange star candidates.
|
[
{
"name": "hong-kong_1999.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Talk Hong Kong conference 1999\n% 1999 Pacific Rim Conference on Stellar Astrophysics\n% Hong Kong, China, August 3-6, 1999\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%\\documentstyle[11pt,newpasp,twoside]{article}\n\\documentstyle[11pt,newpasp,twoside,epsf]{article}\n%%%%%%%%%\\markboth{Bombaci}{Are there strange stars in the Universe?}\n\\markboth{Bombaci}{Do strange stars exist in the Universe?}\n\\pagestyle{myheadings}\n\\nofiles\n% Some definitions I use in these instructions.\n\\def\\sax{\\mbox{SAX J1808.4-3658}}\n\\def\\be{\\begin{equation}}\n\\def\\ee{\\end{equation}}\n\n\\def\\lsim{\\lower0.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}}\n\\def\\ergs{\\mbox{erg\\,s$^{-1}$}}\n\\def\\gcm{\\mbox{G\\,cm$^3$}}\n\\def\\ergcms{\\mbox{erg\\,cm$^{-2}$\\,s$^{-1}$}}\n\\def\\dm{\\mbox{$\\dot{M}$}}\n\\def\\msun{\\mbox{$M_{\\odot}$}}\n\\def\\ra{\\mbox{$R_{\\rm A}$}}\n\\def\\rco{\\mbox{$R_{\\rm c}$}}\n\\def\\ro{\\mbox{$R_0$}}\n\\def\\rs{\\mbox{$R_{\\rm s}$}}\n\\def\\rms{\\mbox{$R_{\\rm ms}$}}\n\\def\\dmmax{\\mbox{$\\dm_{\\rm max}$}}\n\\def\\dmmin{\\mbox{$\\dm_{\\rm min}$}}\n\\def\\fmax{\\mbox{$F_{\\rm max}$}}\n\\def\\fmin{\\mbox{$F_{\\rm min}$}}\n\\def\\etal{\\mbox{\\it et al.}}\n\\def\\apj{Astrophys. J.\\ }\n\\def\\aap{Astron. Astrophys.\\ }\n\\def\\mn{Mon. Not. R. Astron. Soc.\\ }\n\\def\\nat{\\mbox{Nature\\ }}\n\\def\\sci{\\mbox{Science\\ }}\n\\let\\prog=\\arg\n\\def\\edcomment#1{\\iffalse\\marginpar{\\raggedright\\sl#1\\/}\\else\\relax\\fi}\n\\marginparwidth 1.25in\n\\marginparsep .125in\n\\marginparpush .25in\n\\reversemarginpar\n\n\\begin{document}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n%%\\title{ARE THERE STRANGE STARS IN THE UNIVERSE ?} \n\\title{DO STRANGE STARS EXIST IN THE UNIVERSE?}\n%\n% **** change running heads if title is changed ******\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\author{IGNAZIO BOMBACI}\n\\affil{Dipartimento di Fisica, Universit\\`a di Pisa, and INFN Sez. di Pisa, \n via Buonarroti, 2, I-56127 Pisa, Italy}\n%%\\author{Ima Co-Author}\n%%\\affil{The Name of My Institution, The Full Address of My Institution}\n\n\\begin{abstract}\nDefinitely, an affirmative answer to this question would have implications of \nfundamental importance for astrophysics (a new class of compact stars), \nand for the physics of strong interactions (deconfined phase of quark matter, \nand strange matter hypothesis). \nIn the present work, we use observational data for the newly discovered \nmillisecond X-ray pulsar SAX J1808.4-3658 and for the atoll source \n4U~1728-34 to constrain the radius of the underlying compact stars. \nComparing the mass--radius relation of these two compact stars \nwith theoretical models for both neutron stars and strange stars, \nwe argue that a strange star model is more consistent with \nSAX J1808.4-3658 and 4U~1728-34, and suggest that they are likely \nstrange star candidates. \n\\end{abstract}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Introduction}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe possible existence of a new class of compact stars, which are made \nentirely of deconfined {\\it u,d,s} quark matter ({\\it strange quark matter} \n(SQM)), is one of the most intriguing aspects of modern astrophysics. \nThese compact objects are called strange stars. \nThey differ from neutron stars, where quarks are confined within neutrons, \nprotons, and eventually within other hadrons ({\\it e.g. hyperons}). \nThe investigation of such a possibility is relevant not only for \nastrophysics, but for high energy physics too. \nIn fact, the search for a deconfined phase of quark matter is one of the main \ngoals in heavy ion physics. Experiments at Brookhaven National Lab's \nRelativistic Heavy Ion Collider (RHIC) and at CERN's Large Hadron \nCollider (LHC), will hopefully clarify this issue in the near future. \n\nThe possibility that strange stars do exist is based on the so called \n{\\it strange matter hypothesis}, formulated by Witten~(1984) (see also \nBodmer, 1971). \nAccording to this hypothesis, strange quark matter, in equilibrium with \nrespect to the weak interactions, could be the true ground state of strongly \ninteracting matter rather than $^{56}Fe$, {\\it i.e.} the energy per baryon \nof SQM must fulfil the inequality \n\\be\n\\bigg( {{E}\\over{A}} \\bigg)_{SQM} \\leq {{E(^{56}Fe)}\\over{56}} \\simeq 930~MeV, \n\\label{eq:stableSQM}\n\\ee\nat the baryon density where the pressure is equal to zero. \n\nIf the strange matter hypothesis is true, then a nucleus with A nucleons, \ncould in principle lower its energy by converting to a strangelet \n(a drop of SQM). However, this process requires a very high-order \nsimultaneous weak interactions to convert about a number A of {\\it u} and \n{\\it d} quarks of the nucleus into strange quarks. The probability for such \na process is extremely low \n%%%%%%%%%%%%%\n{\\footnote{~It is proportional to $G_F^{2A}$, being $G_F$ the Fermi constant, \nand assuming a number $A$ of simultaneous weak processes.}}, \n%%%%%%%%%%%%% \nand the mean life time for an atomic nucleus to decay to a strangelet is \nmuch higher than the age of the Universe. \nOn the other hand, a step by step production of {\\it s} quarks, at different \ntimes, will produce hyperons in the nucleus, {\\it i.e.} a system (hypernucleus) \nwith a higher energy per baryon with respect to the original nucleus. \nIn addition, finite size effects (surface and shell effects) place a \nlower limit (A $\\sim$ 10--100) on the baryon number of a stable \nstrangelet even if bulk SQM is stable (Farhi \\& Jaffe, 1984). \nThus, according to the strange matter hypothesis, the ordinary state \nof matter, in which quarks are confined within hadrons, is a \nmetastable state. \n\n The success of traditional nuclear physics, in explaining an astonishing \namount of experimental data, provides a clear indication that quarks in \na nucleus are confined within protons and neutrons. Thus, the energy per \nbaryon $(E/A)_{ud}$ of {\\it u,d} quark matter (nonstrange quark matter) \nmust be higher than the energy per baryon of nuclei \n\\be\n\\bigg( {{E}\\over{A}} \\bigg)_{ud} \\geq 930~MeV + \\Delta , \n\\label{eq:stableNSQM}\n\\ee\nbeing $\\Delta \\sim 4$~MeV a quantity which accounts for the lower energy \nper baryon of a finite chunk ($A \\sim 250$) of nonstrange quark matter with \nrespect to the bulk ($A \\rightarrow \\infty$) case (Farhi \\& Jaffe, 1984). \nThese stability conditions (eq.s (1) and (2)) in turn may be used to \nconstrain the parameters entering in models for the equation of state (EOS) \nof SQM. As we show below, the existence of strange stars is allowable \nwithin the uncertainties inherent in perturbative Quantum Chromo-Dynamics \n(QCD). Thus {\\it strange stars may exist in the Universe}. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{The equation of state for strange quark matter}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nFrom a basic point of view the equation of state for SQM should be calculated \nsolving the equations of QCD. As we know, such a fundamental approach is \npresently not doable. Therefore one has to rely on phenomenological models. \nIn this work, we discuss two phenomenological models for the EOS \nof strange quark matter. \nThe first one is a well known model related to the MIT bag model \n(Chodos \\etal\\ 1974) for hadrons. \nThe second one is a new model developed by Dey \\etal\\ (1998). \n%%Both these models contains the basic features of QCD, namely \n%%asymptotic freedom, and quark confinement. \n\n%%%------------------------------\n%%%\\subsection{The EOS based on the MIT bag model for hadrons}\n%%%------------------------------\nAt very high density SQM behaves as a relativistic gas of weakly interacting \nfermions. This is a consequence of one of the basic features of QCD, namely \nasymptotic freedom. \n%%The starting point to derive an EOS for SQM is the relativistic extension of \n%%Landau theory of Fermi liquids (Baym \\& Chin 1976). \nTo begin with consider the case of massless quarks, and \nconsider gluon exchange interactions to the first order in the \nQCD structure constant $\\alpha_c$. Under these circumstances the EOS of \n$\\beta$--stable SQM can be written in the parametrical form: \n\\be \n \\varepsilon = K n_B^{4/3} + B, \\qquad\n P = {1\\over{3}} K n_B^{4/3} - B, ~\\qquad\n K \\equiv {9\\over{4}} \\pi^{2/3} \n \\bigg(1 + {{2\\alpha_c}\\over{3\\pi}}\\bigg) \\hbar c \n\\label{eq:eosB1}\n\\ee \n$\\varepsilon$ being the energy density, and $P$ the pressure. \nEliminating the baryon number density $n_B$ one gets: \n\\be\n P = {1\\over{3}} (\\varepsilon - 4 B) \n\\label{eq:eosBag}\n\\ee \nHere $B$ is a phenomenological parameter which represents the difference \nbetween the energy density of ``perturbative vacuum'' and true QCD vacuum. \n$B$ is related to the ``bag constant'' which in the MIT bag model for \nhadrons (Chodos \\etal\\ 1974) gives the confinement of quarks within \nthe hadronic bag. \nThe density of zero pressure SQM is just $\\rho_s = 4B/c^2$. \nThis is the value of the surface density of a bare strange star. \nTaking a non-vanishing value for the mass $m_s$ of the strange quark, \nthe EOS becomes more involved (see {\\it e.g.} Farhi \\& Jaffe, 1984) \nwith respect to the simple expression (4). \nHowever, for $m_s = 100$--300~MeV, equation (4) is less \nthan 5\\% different from the ``exact'' case for $m_s\\neq 0$. \nIn summary, in this model for the equation of state for SQM there are \nthree phenomenological parameters, namely: $B$, $m_s$, and $\\alpha_c$. \nIt is possible to determine ranges in the values of these parameters in which \nSQM is stable, and nonstrange quark matter is not (Farhi \\& Jaffe, 1984). \nFor example, in the case of non--interacting quarks ($\\alpha_c=0$) one has \n$B \\simeq$ 57--91~MeV/fm$^3$ for $m_s = 0$, and \n$B \\simeq$ 57--75~MeV/fm$^3$ for $m_s = 150$~MeV. \n\nThe schematic model outlined above becomes less and less trustworthy \ngoing from very high density region (asymptotic freedom regime) to lower \ndensities, where confinement (hadrons formation) takes place. \nRecently, Dey {\\it et al.} (1998) derived a new EOS for SQM using a \n``dynamical'' density-dependent approach to confinement. \nThe EOS by Dey {\\it et al.} has asymptotic freedom built in, shows \nconfinement at zero baryon density, deconfinement at high density. \nIn this model, the quark interaction is described by a colour-Debye-screened \ninter-quark vector potential originating from gluon exchange, \nand by a density-dependent scalar potential which restores chiral \nsymmetry at high density (in the limit of massless quarks). \nThe density-dependent scalar potential arises from the density dependence of \nthe in-medium effective quark masses $M_q$, which, in the model by Dey \\etal \n(1998), are taken to depend upon the baryon number density according to\n\\be \nM_q = m_q + 310 \\cdot sech\\bigg(\\nu {{n_B}\\over{n_0}}\\bigg)\n \\qquad \\qquad ({\\rm MeV}), \n\\ee\nwhere $n_0 = 0.16$~fm$^{-3}$ is the normal nuclear matter density, \n$q (= u,d,s)$ is the flavor index, and $\\nu$ is a parameter. \nThe effective quark mass $M_q(n_B)$ goes from its constituent masses \nat zero density, to its current mass $m_q$, as $n_B$ goes to infinity. \nHere we consider two different parameterizations of the EOS by Dey \n{\\it et al.}, which correspond to a different choice for the parameter $\\nu$. \nThe equation of state SS1 (SS2) corresponds to $\\nu = 0.333$ ($\\nu = 0.286$). \nThese two models for the EOS give absolutely stable SQM according to the \nstrange matter hypothesis. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Strange star candidates}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nTo distinguish whether a compact star is a neutron star or \na strange star, one has to find a clear observational signature.\nThere is a striking qualitative difference in the mass--radius (MR) \nrelation of strange stars with respect to that of neutron stars (see Fig.~1). \nFor strange stars with ``small'' ($M << M_{max}$) gravitational mass, \n$M$ is proportional to $R^3$. \nIn contrast, neutron stars have radii that decrease with increasing mass. \nThis is a consequence of the underlying interaction between the stellar \nconstituents which makes ``low'' mass strange stars self-bound objects\n(see {\\it e.g.} Bombaci 1999) contrary to the case of neutron stars \nwhich are bound by gravity \n%%%%%%%%%%%%%\n{\\footnote{~As an idealized example, remember that pure neutron matter is \nnot bound by nuclear forces.}}.\n%%%%%%%%%%%%% \nAs we know, there is a minimum mass for a neutron star \n($M_{min} \\sim 0.1~M_\\odot$). In the case of a strange star, there is \nessentially no minimum mass. \nAs the central density $\\rho_c \\to \\rho_s$ (surface density), a \nstrange star (or better a strangelet for very low baryon number) is a \nself--bound system, until the baryon number becomes so low that finite size \neffects destabilize it. \n% A strange star has a very sharp boundary. In fact, the density drops abruptly \n%from $\\rho_s \\sim $ 4--10$\\times 10^{14} g/cm^3$ to zero on a length scale \n%typical of the strong interaction, {\\it i.e.} the thickness of the \n%``quark surface'' is of the order of $10^{-13}cm$. This is of the same \n%order of the thickness of the surface of an atomic nucleus. \n%In the case of the simple EOS ~({\\ref{eq:eosBag}) the surface density is: \n%\\be\n% \\rho_s = {{4B}\\over{c^2}}, \\qquad \\qquad\\qquad \n% n_s = \\bigg({{3B}\\over{K}}\\bigg)^{3/4}. \n%\\ee\n%with $K$ given in eq.({\\ref{eq:eosB1}) \n%\n\n%%%%%%%%%%%%% SAX J1808.4-3658 \n\n\\subsection{SAX J1808.4-3658}\n\nThe transient X-ray burst source \\sax\\ was discovered in September 1996 \nby the BeppoSAX satellite. \nTwo bright type-I X-ray bursts were detected, each lasting less than \n30 seconds. Analysis of the bursts in \\sax\\ indicates that it is 4~kpc \ndistant and has a peak X-ray luminosity of $6\\times 10^{36}~$erg/s in its \nbright state, and a X-ray luminosity lower than $10^{35}~$erg/s in \nquiescence (in't Zand 1998). \nThe object is nearly certainly the same as the transient X-ray source \ndetected with the Proportional Counter Array (PCA) on board the Rossi \nX-ray Timing Explorer (RXTE) (Marshall, 1998). \nCoherent pulsations at a period of 2.49 milliseconds were discovered \n(Wijnands \\& van der Klis 1998). \nThe star's surface dipolar magnetic moment was derived \nto be less than $10^{26}$~G~cm$^3$ from detection of X-ray pulsations at \na luminosity of $10^{36}$~erg/s (Wijnands \\& van der Klis 1998), \nconsistent with the weak fields expected for type-I X-ray bursters and \nmillisecond radio pulsars (MS PSRs) (Bhattacharya \\& van den Heuvel 1991). \nThe binary nature of \\sax\\ was firmly established with the detection of a \n2 hour orbital period (Chakrabarty \\& Morgan 1998) as well as with the \noptical identification of the companion star (Roche \\etal\\ 1998). \n\\sax\\ is the first pulsar to show both coherent pulsations in its persistent \nemission and X-ray bursts, and by far the fastest-rotating, lowest-field \naccretion-driven pulsar known.\nIt presents direct evidence for the evolutionary link between low-mass \nX-ray binaries (LMXBs) and MS PSRs. \n\\sax\\ is the only known LMXB with an MS PSR. \n \nA mass--radius (MR) relation for the compact star in \\sax\\ has been \nrecently obtained by Li \\etal\\ (1999a)\n{\\footnote{~see also Burderi \\& King (1998), Psaltis \\& Chakrabarty (1999).}} \nusing the following two requirements. \n({\\it i}) Detection of X-ray pulsations requires that the inner radius $\\ro$ \nof the accretion flow should be larger than the stellar radius $R$. \nIn other words, the stellar magnetic field must be strong enough to \ndisrupt the disk flow above the stellar surface. \n({\\it ii}) The radius $\\ro$ must be less than the so-called co-rotation \nradius $\\rco$, {\\it i.e.} the stellar magnetic field must be weak enough \nthat accretion is not centrifugally inhibited: \n\\be \n \\ro \\lsim \\rco = [GM P^2/(4\\pi^2)]^{1/3}. \n\\ee\nHere $G$ is the gravitation constant, $M$ is the mass of the star, \nand $P$ is the pulse period. The inner disk radius $\\ro$ is generally \nevaluated in terms of the Alfv\\'en radius $\\ra$, at which the magnetic \nand material stresses balance (Bhattacharya \\& van den Heuvel 1991): \n$\\ro=\\xi\\ra=\\xi[B^2R^6/\\dm(2GM)^{1/2}]^{2/7}$, where $B$ and $\\dm$ \nare respectively the surface magnetic field and the mass accretion \nrate of the pulsar, and $\\xi$ is a parameter of order of unity \nalmost independent of $\\dm$ (Li 1997, Burderi \\& King 1998). \nSince X-ray pulsations in \\sax\\ were detected over a wide range of mass \naccretion rate (say, from $\\dmmin$ to $\\dmmax$), the two conditions \n({\\it i}) and ({\\it ii}) give $R\\lsim \\ro(\\dmmax)< \\ro(\\dmmin)\\lsim \\rco$. \nNext, we assume that the mass accretion rate $\\dm$ is proportional to the \nX-ray flux $F$ observed with RXTE. This is guaranteed by the fact that the \nX-ray spectrum of \\sax\\ was remarkably stable and there was only slight \nincrease in the pulse amplitude when the X-ray luminosity varied by a factor \nof $\\sim 100$ during the 1998 April/May outburst (Gilfanov \\etal\\ 1998, \nCui \\etal\\ 1998, Psaltis \\& Chakrabarty 1999). Therefore, Li \\etal\\ (1999a) \nget the following upper limit of the stellar radius: \n$ R < (F_{min}/F_{max})^{2/7} \\rco$, or \n\\be \n R < 27.5 \\bigg({{F_{min}}\\over{F_{max}}}\\bigg)^{2/7} \n \\bigg({{P}\\over{2.49~ms}}\\bigg)^{2/3} \n \\bigg({{M}\\over{M_\\odot}}\\bigg)^{1/3} ~{\\rm km}, \n\\label{eq:MR-sax}\n\\ee\nwhere $\\fmax$ and $\\fmin$ denote the X-ray fluxes measured \nduring X-ray high- and low-state, respectively, $\\msun$ is the \nsolar mass. \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%% Figure 1 %%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure} \n%%\\plotfiddle{saxfig.ps}{7.5cm}{-90}{60}{60}{-210}{300}\n\\plotfiddle{hong-kong_1999_fig1.ps}{7.0cm}{90}{50}{50}{200}{-60}\n\\caption{Comparison of the mass--radius relation of SAX J1808.4 -3658 \ndetermined from RXTE observations with theoretical models of neutron \nstars and of strange stars. See text for more details.} \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nNote that in writing inequality (7) it is assumed that the pulsar's \nmagnetic field is basically dipolar (see Li \\etal\\ 1999a for \narguments to support this hypothesis)\n%%%%%%%%%%%%%\n{\\footnote{~see also Psaltis \\& Chakrabarty (1999) for a study of the \ninfluence on the MR relation for \\sax\\ of a quadrupole magnetic moment, \nand of a {\\it non-standard} disk--magnetosphere interaction model.}}. \n%%%%%%%%%%%%% \n\nGiven the range of X-ray flux at which coherent pulsations were \ndetected, inequality (7) defines a limiting curve in the \nmass--radius plane for SAX J1808.4-3658, as plotted in the dashed curve \nin Fig.~1. The authors of ref. (Li \\etal\\ 1999a) adopted the flux ratio \n$\\fmax/\\fmin\\simeq 100$ from the observations that during the 1998 \nApril/May outburst, the maximum $2-30$ keV flux of \\sax\\ at the \npeak of the outburst was $\\fmax\\simeq 3\\times 10^{-9}\\,\\ergcms$, \nwhile the pulse signal became barely detectable when the flux \ndropped below $\\fmin\\simeq 2\\times 10^{-11}\\,\\ergcms$ \n(Cui \\etal\\ 1998, Psaltis \\& Chakrabarty 1999). \nThe dashed line $R = \\rs \\equiv 2GM/c^2$ represents the Schwartzschild \nradius - the lower limit of the stellar radius to prevent the star collapsing \ninto a black hole. Thus the allowed range of the mass and radius of \n\\sax\\ is the region confined by these two dashed curves in Fig.~1. \n\nIn the same figure, we report the theoretical MR relations (solid curves) \nfor neutron stars given by some recent realistic models for the EOS of \ndense matter (see Li \\etal\\ 1999a for references to the EOS models). \nModels BBB1 and BBB2 are relative to ``conventional'' neutron stars \n({\\it i.e} the core of the star is assumed to be composed by an \nuncharged mixture of neutrons, protons, electrons and muons in \nequilibrium with respect to the weak interaction). \nThe curve labeled Hyp depicts the MR relation for a neutron \nstar in which hyperons are considered in addition to nucleons as hadronic \nconstituents. The MR curve labeled $K^-$ is relative to neutron stars \nwith a Bose-Einstein condensate of negative kaons in their cores. \nIt is clearly seen in Fig.~1 that none of the neutron star MR curves \nis consistent with \\sax. Including rotational effects will shift \nthe $MR$ curves to up-right in Fig.~1 (Datta \\etal\\ 1998), and does not \nhelp improve the consistency between the theoretical neutron star models \nand observations of \\sax. \n%%Additionally, it is unlikely that the actual mass and radius of \\sax\\ lie \n%%very close to the dashed curve, since the minimum flux $F_{min}$ at which \n%%X-ray pulsations were detected by RXTE was determined by the instrumental\n%%sensitivity, and the actual value could be even lower than adopted\n%%in inequality (7). \nTherefore \\sax\\ is not well described by a neutron star model. \nThe curve B90 in Fig.~1 gives the MR relation for strange stars described \nby the schematic EOS (4) with B = 90 MeV/fm$^3$. \nThe two curves SS1 and SS2 give the MR relation for strange stars calculated \nwith the EOS by Dey et al. (1998). \nFigure~1 clearly demonstrates that a strange star model is more \ncompatible with \\sax\\ than a neutron star one. \n\n\\subsection{4U 1728-34}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nRecently, Li et al. (1999b) investigated possible signatures for the \nexistence of strange stars in connection with the newly discovered \nphenomenon of kilohertz quasi--periodic oscillations (kHz QPOs) in \nthe X-ray flux from LMXB (for a review see van der Klis 2000). \nInitially, kHz QPO data from various sources were interpreted assuming a \nsimple {\\it beat--frequency model} (see {\\it e.g.} Kaaret \\& Ford 1997). \nIn many cases, two simultaneous kHz QPO peaks (``twin peaks'') are observed. \nThe QPO frequencies vary and are strongly correlated with source flux. \nIn the beat--frequency model the highest observed QPO frequency \n$\\nu_u$ is interpreted as the Keplerian orbital frequency $\\nu_K$ \nat the inner edge of the accretion disk. \n%%of the innermost marginally stable orbit (as predicted by general \n%%relativity) of the central compact object. \nThe frequency $\\nu_l$ of the lower QPO peak is instead interpreted as the \nbeat frequency between $\\nu_K$ and the neutron star spin frequency $\\nu_0$, \nwhich within this model is equal to the separation frequency \n$\\Delta\\nu \\equiv \\nu_u - \\nu_l$ of the two peaks. \nThus $\\Delta\\nu$ is predicted to be constant. \nNevertheless, novel observations for different \nkHz QPO sources have challenged this simple beat--frequency model. \nThe most striking case is the source 4U 1728-34, where it was found that \n$\\Delta\\nu$ decreases significantly, from $349.3\\pm1.7$ Hz to \n$278.7\\pm11.6$ Hz, as the frequency of the lower kHz QPO increases \n(M\\'endez \\& van der Klis 1999). \n%%%\nFurthermore, in the spectra observed by the RXTE for 4U 1728-34, \nFord \\& van der Klis (1998) found low-frequency Lorentian \noscillations with frequencies between 10 and 50 Hz. \nThese frequencies as well as the break frequency ($\\nu_{break}$) of the \npower spectrum density for the same source were shown to be correlated with \n$\\nu_u$ and $\\nu_l$. \n\nA different model was recently developed by Osherovich \\& Titarchuk (1999) \n(see also Titarchuk \\& Osherovich 1999), who proposed a unified \nclassification of kHz QPOs and the related observed low frequency \nphenomena. \nIn this model, kHz QPOs are modeled as Keplerian oscillations under the \ninfluence of the Coriolis force in a rotating frame of reference \n(magnetosphere). The frequency $\\nu_l$ of the lower kHz QPO peak is \nthe Keplerian frequency at the outer edge of a viscous transition layer \nbetween the Keplerian disk and the surface of the compact star. \nThe frequency $\\nu_u$ is a hybrid frequency related to the \nrotational frequency $\\nu_m$ of the star's magnetosphere by: \n$\\nu_u^2=\\nu_K^2+(2\\nu_m)^2$. \nThe observed low Lorentzian frequency in 4U 1728-34 is\nsuggested to be associated with radial oscillations in the viscous transition \nlayer of the disk, whereas the observed break frequency is determined by \nthe characteristic diffusion time of the inward motion of the matter in the \naccretion flow (Titarchuk \\& Osherovich 1999). Predictions of this model \nregarding relations between the QPO frequencies mentioned above compare \nfavorably with recent observations for 4U 1728-34, Sco X-1, 4U 1608-52, \nand 4U 1702-429. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%% Figure 2 %%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure} \n%%%\\plotfiddle{4U1728_fig1.ps}{7.5cm}{-90}{60}{60}{-210}{300}\n%\\plotfiddle{hong-kong_1999_fig2.ps}{7.5cm}{180}{50}{50}{-210}{300}\n\\plotfiddle{hong-kong_1999_fig2.ps}{7.0cm}{90}{50}{50}{200}{-60}\n\\caption{Comparison of the $MR$ relation of 4U 1728-34 determined from RXTE\nobservations with theoretical models of neutron stars and of strange\nstars. The range of mass and radius of 4U 1728-34 is allowed in the\nregion outlined by the dashed curve $R=\\ro$, the horizontal dashed line, \nand the dashed line $R=R_s$. The solid curves represents theoretical MR \nrelations for neutron stars and strange stars.} \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe presence of the break frequency and the correlated Lorentzian frequency \nsuggests the introduction of a new scale in the phenomenon. \nOne attractive feature of the model by Titarchuk \\& Osherovich (1999)\nis the introduction of such a scale in the model through the Reynolds number \nfor the accretion flow. \nThe best fit for the observed data was obtained by Titarchuk \\& Osherovich \n(1999) when \n\\begin{equation} \na_k=(M/M_{\\odot})(\\ro/3\\rs)^{3/2}(\\nu_0/364\\,{\\rm Hz})=1.03,\n\\end{equation}\nwhere $M$ is the stellar mass, \n$\\ro$ is the inner edge of the accretion disk\n%%%%%%%%%%%%%\n{\\footnote{~In the expression for $a_k$ reported in \nTitarchuk \\& Osherovich (1999), one has $x_0 = \\ro/\\rs$, where $\\ro$ is \nerroneously indicated as the neutron star radius \n(Titarchuk, private communication).}}, \n%%%%%%%%%%%%% \n$\\rs$ is the Schwarzschild radius, and $\\nu_0$ is the spin\nfrequency of the star. \nGiven the 364 Hz spin frequency of 4U 1728-34 (Strohmayer et al. 1996), \nthe inner disk radius can be derived from the previous equation. \nSince the innermost radius of the disk must be larger than the radius $R$ \nof the star itself, this leads to a mass-dependent upper bound on the \nstellar radius,\n\\begin{equation}\n R \\leq \\ro \\simeq 8.86~ a_k^{2/3} (M/M_{\\odot})^{1/3}\\,{\\rm km},\n\\end{equation}\nwhich is plotted by dashed curve in Fig.~2. \n\nA second constraint on the mass and radius of 4U 1728-34 results from \nthe requirement that the inner radius $\\ro$ of the disk must be larger than\nthe radius of the last stable circular orbit $\\rms$ around the star: \n\\be \n \\ro \\geq \\rms. \n\\ee\nTo make our discussion more transparent, neglect for a moment the \nrotation of the compact star. \nFor a non-rotating star $\\rms = 3 \\rs$, then the second condition gives:\n\\be \n \\ro \\geq 3 \\rs = 8.86~ \\big(M/M_\\odot\\big)~{\\rm km}. \n\\ee\nTherefore, the allowed range of the mass and radius for \n4U1728-34 is the region in the lower left corner of the MR plane confined \nby the dashed curve ($R=\\ro$), by the horizontal dashed line, and by \nthe Schwartzschild radius (dashed line $R=R_s$). \nIn the same figure, we compare with the theoretical MR relations \nfor non-rotating neutron stars and strange stars, for the same models \nfor the EOS considered in Fig. 1. \nIt is clear that a strange star model is more compatible with 4U 1728-34 \nthan a neutron star one. \nIncluding the effects of rotation ($\\nu_0 = $364 Hz) in the calculation \nof the theoretical MR relations and $\\rms$, does not change the \nprevious conclusion (Li \\etal ~1999b). \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Final remarks}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe main result of the present work ({\\it i.e.} the likely existence \nof strange stars) is based on the analysis of observational data \nfor the X-ray sources SAX J1808.4-3658 and 4U~1728-34. \nThe interpretation of these data is done using {\\it standard} models for \nthe accretion mechanism, which is responsible for the observed phenomena. \nThe present uncertainties in our knowledge of the accretion mechanism, \nand the disk--magnetosphere interaction, do not allow us to definitely \nrule out the possibility of a neutron star for the two X-ray sources \nwe discussed. \nFor example, making {\\it a priori} the {\\it conservative} assumption that \nthe compact object in \\sax\\ is a neutron star, and using a MR relation similar \nto our eq. (7) Psaltis \\& Chakrabarty (1999) try to constrain \ndisk--magnetosphere interaction models or to infer the presence of a \nquadrupole magnetic moment in the compact star. \n\n \\sax\\ and 4U~1728-34 are not the only LMXBs which could harbour a \nstrange star. Recent studies have shown that the compact objects \nassociated with the X-ray burster 4U 1820-30 (Bombaci 1997), the bursting \nX-ray pulsar GRO J1744-28 (Cheng \\etal\\ 1998b) and the X-ray pulsar \nHer X-1 (Dey \\etal\\ 1998) are likely strange star candidates. \nFor each of these X-ray sources (strange star candidates) the conservative \nassumption of a neutron star as the central accretor would require \nsome particular (possibly {\\it ad hoc}) assumption about the nature of the \nplasma accretion flow and/or the structure of the stellar magnetic field. \nOn the other hand, the possibility of a strange star gives a simple \nand unifying picture for all the systems mentioned above. \nFinally, strange stars have also been speculated to model $\\gamma$-ray \nbursters (Haensel \\etal\\ 1991, Bombaci \\& Datta 2000)\nand soft $\\gamma$-ray repeaters (Cheng \\& Dai 1998a). \n\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\\vskip 0.6cm\n\\leftline{{\\bf Acknowledgements}}\n\\vskip 0.2cm\n\\noindent\nI thank my colleagues J. Dey, M. Dey, E.P.J. van den Heuvel, X.D. Li, \nand S. Ray with whom the ideas presented in this talk were developed. \nI am grateful to the Organizing Committee of the Pacific Rim Conference \non Stellar Astrophysics for inviting me and for financial support. \nParticularly, I thank Prof. K.S. Cheng for the warm hospitality, \nand for many stimulating discussions during the conference. \nIt is a pleasure to acknowledge fruitful and stimulating discussions with \nProf. G. Ripka during the workshop Quark Condensates in Nuclear Matter, \nheld at the ECT* in Trento. \n\n%%%%%%%%%%%%%\n\\vskip 0.6cm\n\\leftline{{\\bf In memory of Bhaskar Datta}}\n\\vskip 0.2cm\n\\noindent\nI dedicate this paper to my great friend and colleague Bhaskar Datta, \nwho passed away on december $3^{rd}$ 1999 in Bangalore. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{references}\n%%%\\reference \n%%%Baym, G. \\& Chin, S.A. 1976, Nucl. Phys. A 262, 527\n\\reference \nBhattacharya, D., \\& van den Heuvel, E. P. J. 1991, Phys. Rep., 203, 1 \n\\reference \nBodmer, A. R. 1971, Phys. Rev. D, 4, 1601\n\\reference \nBombaci, I., 1997, Phys. Rev. C, 55, 1587\n\\reference \nBombaci, I. 1999, Neutron stars' structure and nuclear equation of state, \nin M. Baldo (ed.), {\\it Nuclear methods and the nuclear equation of state}, \nWorld Scientific, Singapore, pp. 381-457\n\\reference\nBombaci, I., \\& Datta, B, 2000, \\apj 530, L69 \n\\reference\nBurderi, L., \\& and King, A.R. 1998, \\apj 505, L135\n\\reference\nChakrabarty, D., \\& Morgan, E. H. 1998, \\nat 394, 346\n\\reference\nCheng, K.S, \\& Dai, Z.G., 1998a, Phys. Rev. Lett. 80, 1998\n\\reference\nCheng, K.S, Dai, Z.G., Wai, D.M. \\& Lu, T. 1998b, \\sci 280, 407 \n\\reference\nChodos, A. \\etal, 1974, Phys. Rev. D, 9, 3471\n\\reference \nCui, W., Morgan, E.H., \\& Titarchuk, L. 1998, \\apj 504, L27 \n\\reference \nDatta, B., Thampan, A.V., \\& Bombaci, I. 1998, \\aap 334, 943 \n\\reference \nDey, M., Bombaci, I., Dey, J., Ray, S., \\& Samanta, B. C. 1998, Phys.\nLett. B, 438, 123; erratum, 1999 Phys. Lett. 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No. 6885\n\\reference \nStrohmayer, T. E., Zhang, W., Swank, J. H., Smale, A., Titarchuk, L., \nDay, C., \\& Lee, U. 1996, \\apj, 469, L9\n\\reference \nTitarchuk, L. \\& Osherovich, V. 1999, \\apj, 518, L95 \n\\reference \nvan der Klis, M., 2000, Ann. Rev. Astr. Astrophys. (to appear Sept. 2000).\narXiv:astro-ph/0001167 \n\\reference \nWijnands, R., \\& M. van der Klis, M., 1998, \\nat 394, 344 \n\\reference \nWitten, E. 1984, Phys. Rev. D, 30, 272\n\n\\end{references}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\end{document}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n"
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astro-ph0002525
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On the particle acceleration near the light surface of radio pulsars
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[
{
"author": "V.S.~Beskin$^{1,2}$ and R.R.~Rafikov$^3$"
},
{
"author": "$^1$ National Astronomical Observatory"
},
{
"author": "Osawa 2--21--1"
},
{
"author": "Mitaka"
},
{
"author": "Tokyo 181--8588"
},
{
"author": "Japan"
},
{
"author": "Leninsky prosp."
},
{
"author": "53"
},
{
"author": "Moscow"
},
{
"author": "117924"
},
{
"author": "Russia"
},
{
"author": "Princeton"
},
{
"author": "NJ"
},
{
"author": "08544"
},
{
"author": "USA"
}
] |
The two--fluid effects on the radial outflow of relativistic electron--positron plasma are considered. It is shown that for large enough Michel (1969) magnetization parameter $\sigma \gg 1$ and multiplication parameter $\lambda=n/n_{GJ} \gg 1$ one--fluid MHD approximation remains correct in the whole region $|{\bmath E}| < |{\bmath B}|$. In the case when the longitudinal electric current is smaller than the Goldreich--Julian one, %$I < I_{GJ}$ %(which may be realized in the pulsar magnetosphere) the acceleration of particles near the light surface $|{\bmath E}| = |{\bmath B}|$ is determined. It is shown that, as in the previously considered (Beskin Gurevich \& Istomin 1983) cylindrical geometry, almost all electromagnetic energy is transformed into the energy of particles in the narrow boundary layer $\Delta\varpi/\varpi \sim \lambda^{-1}$.
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[
{
"name": "paper.tex",
"string": "\\documentstyle[onecolumn]{mn}\n\n\\voffset=-1.5cm\n\n\\title{ On the particle acceleration near the light surface of radio pulsars}\n\n\\author[V.S.~Beskin and R.R.~Rafikov]\n {V.S.~Beskin$^{1,2}$ and R.R.~Rafikov$^3$ \\\\\n $^1$ National Astronomical Observatory, Osawa 2--21--1,\n Mitaka, Tokyo 181--8588, Japan \\\\\n $^2$ P.N.Lebedev Physical Institute, Leninsky prosp., 53,\nMoscow, 117924, Russia\\\\\n $^3$ Princeton University Observatory, Princeton, NJ, 08544, USA}\n\\date{Accepted 1999 .\n Received 1999 ;\n in original form 1999}\n\n\\pagerange{\\pageref{firstpage}--\\pageref{lastpage}}\n\n\\pubyear{1999}\n\n\\begin{document}\n\n\\maketitle\n\n\\label{firstpage}\n\n\\begin{abstract}\n\nThe two--fluid effects on the radial outflow of relativistic\nelectron--positron plasma are considered. It is shown that for\nlarge enough Michel (1969) magnetization parameter $\\sigma \\gg 1$\nand multiplication parameter $\\lambda=n/n_{GJ} \\gg 1$ one--fluid\nMHD approximation remains correct in the whole region\n$|{\\bmath E}| < |{\\bmath B}|$. In the case when the\nlongitudinal electric current is smaller than the Goldreich--Julian\none, %$I < I_{GJ}$\n%(which may be realized in the pulsar magnetosphere)\nthe acceleration of\nparticles near the light surface $|{\\bmath E}| = |{\\bmath B}|$\nis determined. It is shown that, as in the previously considered\n(Beskin Gurevich \\& Istomin 1983) cylindrical geometry,\nalmost all electromagnetic energy is transformed into the energy\nof particles in the narrow boundary layer\n$\\Delta\\varpi/\\varpi \\sim \\lambda^{-1}$.\n\n\\end{abstract}\n\n\\begin{keywords}\ntwo--fluid relativistic MHD:radio pulsars---particle acceleration\n\\end{keywords}\n\n\\section{Introduction}\n\nDespite the fact that the structure of the magnetosphere of radio\npulsars remains\none of the fundamental astrophysical problems, the common view on the key\ntheoretical question -- what is the physical nature of the neutron star\nbraking -- is absent (Michel 1991, Beskin Gurevich \\& Istomin 1993,\nMestel 1999). Nevertheless, very extensive theoretical studies in the\nseventies and the eighties allowed to obtain some model-independent\nresults. One of them is the absence of magnetodipole energy loss.\nThis result was first obtained theoretically (Henriksen \\& Norton\n1975, Beskin et al 1983). It was shown that the electric charges\nfilling the magnetosphere screen fully the magnetodipole radiation\nof a neutron star for an arbitrary inclination angle $\\chi$ between\nthe rotational and magnetic axes if there are no longitudinal currents\nflowing in the magnetosphere. Later this result was also confirmed\nby observations. The direct detections of the interaction\nof the pulsar wind with a companion star in close binaries (see e.g.\nDjorgovsky \\& Evans 1988, Kulkarni \\& Hester 1988) have shown that\nit is impossible to explain the heating of the companion by\na low--frequency magnetodipole wave.\n\nOn the other hand, the detailed mechanism of particle acceleration\nremains unclear. Indeed, a very high magnetization parameter $\\sigma$\n(Michel 1969) in the pulsar magnetosphere\ndemonstrates that within the light cylinder $r < R_{\\rm L} = c/\\Omega$\nthe main part of the energy is transported\nby the Poynting flux. It means that\nthe additional mechanism of particle acceleration must work\nin the vicinity of the light cylinder.\nIt is necessary to stress that %, as was already mentioned above,\nan effective particle acceleration can only take place for small enough\nlongitudinal electric currents $I < I_{GJ}$ when the plasma has no\npossibility to pass smoothly through the fast magnetosonic surface\nand when the light surface $|{\\bmath E}| = |{\\bmath B}|$ is located at a\nfinite distance.\n%It is this case can be realized in the pulsar magnetosphere\n%(Beskin et al 1983, Beskin \\& Malyshkin 1998).\nAs to the case of the large longitudinal currents $I > I_{GJ}$,\nboth analytical (Tomimatsu 1994, Begelman \\& Li 1994, Beskin et al 1998)\nand numerical (Bogovalov 1997)\nconsiderations demonstrate that the\n acceleration becomes ineffective\noutside the fast magnetosonic surface, and the particle-to-Poynting\nflux ratio remains small: $\\sim \\sigma^{-2/3}$ (Michel 1969, Okamoto 1978).\n\nThe acceleration of an electron--positron plasma near the light surface\nwas considered by Beskin Gurevich and Istomin (1983) in the simple\n$1D$ cylindrical geometry for $I \\ll I_{GJ}$.\nIt was shown that in a narrow boundary layer\n$\\Delta\\varpi/\\varpi \\sim 1/\\lambda$ almost all electromagnetic energy\nis actually converted to the particles energy. Nevertheless, cylindrical\ngeometry does not provide the complete picture of particle acceleration.\nIn particular, it was impossible to include self--consistently\nthe disturbance of a poloidal magnetic field and an electric potential,\nthe later playing the main role in the problem of the plasma acceleration\n(for more details see e.g. Mestel \\& Shibata 1994).\nHence, a more careful $2D$ consideration is necessary.\n\nIn Sect. 2 we formulate a complete system of $2D$ two--fluid MHD equations\ndescribing the electron--positron outflow from a magnetized body with\na monopole magnetic field. The presence of an exact analytical force--free\nsolution (Michel 1973) allows us to linearize this system\nwhich results in the existence of invariants (energy and angular momentum)\nalong unperturbed monopole field lines similar to the ideal one--fluid\nMHD flow.\nIn Sect. 3 it is shown that for $\\sigma \\gg 1$ and $\\lambda \\gg 1$\n($\\lambda=n/n_{GJ}$ is the multiplication factor)\nthe one--fluid MHD approximation remains true in the entire region within the\nlight surface. Finally, in Sect. 4 the acceleration of\nparticles near the light surface $|{\\bmath E}| = |{\\bmath B}|$\nis considered. It is shown that, as in the case of cylindrical\ngeometry, in a narrow boundary layer\n$\\Delta\\varpi/\\varpi \\sim \\lambda^{-1}$ almost all the electromagnetic\nenergy is converted into the energy of particles.\n\n\\section{Basic Equations}\n\nLet us consider a stationary axisymmetric outflow of a two--component\nplasma in the vicinity of an active object with a monopole magnetic field.\nIt is necessary to stress that, of course, the monopole magnetic field\nis a rather crude approximation for a pulsar magnetosphere. Nevertheless,\neven for a dipole magnetic field near the origin, at large distances\n$r \\gg R_{\\rm L}$ in the wind zone the magnetic field can have a monopole--like\nstructure. For this reason the disturbance of a monopole magnetic field\ncan give us an important information concerning particle acceleration\nfar from the neutron star.\n\nThe structure of the flow is described by the set of\nMaxwell`s equations and the equations of motion\n\\begin{eqnarray}\n\\nabla{\\bmath E}=4\\pi \\rho_e,~~~~~\n\\nabla\\times{\\bmath E} =0,\\nonumber \\\\\n\\nabla{\\bmath B}=0,~~~~~\n\\nabla\\times{\\bmath B} =\\frac{4 \\pi}{c}{\\bmath j}, \\label{cc1} \\\\\n({\\bmath v}^{\\pm}\\nabla){\\bmath p}^{\\pm}=\n\\pm e\\left( {\\bmath E}+ \\frac{{\\bmath v}^{\\pm}}{c}\\times{\\bmath B}\\right).\n\\label{eq:2}\n\\end{eqnarray}\nHere $\\bmath E$ and $\\bmath B$ are the electric and magnetic\nfields, $\\rho_e$ and $\\bf j$ are the charge and\ncurrent densities, and ${\\bmath v}^{\\pm}$ and ${\\bmath p}^{\\pm}$ are\nthe speed and momentum of particles.\nIn the limit of infinite particle energy\n\\begin{equation}\n\\gamma=\\infty, \\quad\nv_r^0=c, \\quad\nv_{\\varphi}^0=0, \\quad\nv_{\\theta}^0=0,\n\\end{equation}\nand for charge and current density\n\\begin{equation}\n\\rho_e^0=\\rho_{\\rm s}\\frac{R_{\\rm s}^2}{r^2}\\cos\\theta, \\quad\nj_r = \\rho_{\\rm s}c\\frac{R_{\\rm s}^2}{r^2}\\cos\\theta,\n\\label{gj}\n\\end{equation}\nthe monopole poloidal magnetic field\n\\begin{equation}\nB_r^0 = B_{\\rm s}\\frac{R_{\\rm s}^2}{r^2}, \\quad\nB_{\\theta}^0=0,~\n\\end{equation}\nis the exact solution of Maxwell's equations. In this case\n\\begin{equation}\nB_{\\varphi}^0 = E_{\\theta}^0 =-B_{\\rm s}\\frac{R_{\\rm s}\\Omega}{c}\\frac{R_{\\rm s}}{r}\\sin\\theta,\n\\quad E_r^0 = E_{\\varphi}^0=0,\n\\end{equation}\nwhich coincides with the well--known Michel (1973) solution.\nHere $\\gamma$ is the Lorentz-factor of particles,\n$B_{\\rm s}$ is the magnetic field on the surface of the sphere\n$r = R_{\\rm s}$, and $\\rho_{\\rm s} = $ const. As a result,\nthe angular velocity can be rewritten in a form\n$\\Omega = 2\\pi c |\\rho_{\\rm s}|/B_{\\rm s}$.\nThe limit $\\gamma \\rightarrow \\infty$ just corresponds to zero particle\nmass in the force--free approximation.\n\nIt is convenient to introduce the electric field potential $\\Phi(r,\\theta)$,\nso that ${\\bmath E}=-\\nabla \\Phi$ and\n\\begin{equation}\n\\Phi^0=-\\frac{\\Omega R_{\\rm s}^2 B_{\\rm s}}{c}\\cos\\theta,\n\\end{equation}\nand the flux function $\\Psi(r,\\theta)$, so that\n\\begin{equation}\n{\\bmath B}_{\\rm p}^0 = \\frac{\\nabla \\Psi\\times {\\bmath e}_{\\varphi}}\n{2 \\pi r\\sin\\theta},\n\\end{equation}\nand $\\Psi^0=2\\pi B_{\\rm s} R_{\\rm s}^2(1-\\cos\\theta)$. Then one can seek the\nfirst--order corrections for the case $v\\ne c$ in the following form:\n\\begin{eqnarray}\nn^{+} & = & \\frac{\\Omega B_{\\rm s}}{2\\pi c e}\\frac{R_{\\rm s}^2}{r^2}\n\\left[\\lambda-\\frac{1}{2}\\cos\\theta+\\eta^{+}(r,\\theta)\\right],\n\\label{cc8}\\\\\nn^{-} & = & \\frac{\\Omega B_{\\rm s}}{2\\pi c e}\\frac{R_{\\rm s}^2}{r^2}\n\\left[\\lambda+\\frac{1}{2}\\cos\\theta+\\eta^{-}(r,\\theta)\\right],\\\\\n\\Phi(r,\\theta) & = & \\frac{\\Omega R_{\\rm s}^2\nB_{\\rm s}}{c}\\left[-\\cos\\theta+\\delta(r,\\theta)\\right],\\\\\n\\Psi(r,\\theta) & = & 2\\pi B_{\\rm s} R_{\\rm s}^2\\left[1-\\cos\\theta+\\varepsilon\nf(r,\\theta)\\right],\\\\\nv_r^{\\pm} & = & c\\left[1-\\xi_r^{\\pm}(r,\\theta)\\right],~~~v_{\\theta}^{\\pm}=\nc\\xi_{\\theta}^{\\pm}(r,\\theta),~~~\nv_{\\varphi}^{\\pm}=\nc\\xi_{\\varphi}^{\\pm}(r,\\theta),\\\\\nB_r & = & B_{\\rm s}\\frac{R_{\\rm s}^2}{r^2}\\left(\n1+\\frac{\\varepsilon}{\\sin\\theta}\n\\frac{\\partial f}{\\partial \\theta}\\right),\\\\\nB_{\\theta} & = & -\\varepsilon \\frac{B_{\\rm s} R_{\\rm s}^2}{r \\sin\\theta}\n\\frac{\\partial f}{\\partial r},\\\\\nB_{\\varphi} & = & B_{\\rm s}\\frac{\\Omega R_{\\rm s}}{c}\\frac{R_{\\rm s}}{r}\\left[-\\sin\\theta\n-\\zeta(r,\\theta)\\right],\\\\\nE_r & = & -\\frac{\\Omega B_{\\rm s} R_{\\rm s}^2}{c}\\frac{\\partial \\delta}{\\partial r},\\\\\nE_{\\theta} & = & \\frac{\\Omega R_{\\rm s}^2 B_{\\rm s}}{c r}\\left(\n-\\sin\\theta - \\frac{\\partial \\delta}{\\partial \\theta}\\right).\n\\label{cc17}\n\\end{eqnarray}\nSuch a choice corresponds to a constant particle-to-magnetic flux ratio.\nHere $\\lambda \\gg 1$ is the multiplication parameter\n($\\lambda=en_{\\rm s}/|\\rho_{\\rm s}|$, where $n_{\\rm s}$ is the concentration\nof particles on the surface $r = R_{\\rm s}$) which is $10^3 - 10^5$ for\nradio pulsars. In what follows we consider for simplicity the case\n$\\lambda =$ const.\n\nSubstituting (\\ref{cc8})--(\\ref{cc17}) into equations \n(\\ref{cc1})--(\\ref{eq:2}), we obtain\nto the first-order approximation the following system of equations:\n\\begin{eqnarray}\n-\\frac{1}{\\sin\\theta}\\frac{\\partial}\n{\\partial \\theta}(\\zeta\\sin\\theta)= %\\nonumber \\\\\n2(\\eta^+-\\eta^-)\n-2\\left[\\left(\\lambda-\\frac{1}{2}\\cos\\theta\\right)\\xi_r^+\n-\\left(\\lambda+\\frac{1}{2}\\cos\\theta\\right)\\xi_r^-\\right],\n\\label{b1}\\\\\n2(\\eta^+-\\eta^-)+\\frac{\\partial}{\\partial r}\\left(r^2\n\\frac{\\partial \\delta}{\\partial r}\\right)+\n\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial \\theta}\n\\left(\\sin\\theta \\frac{\\partial\\delta}{\n\\partial \\theta}\\right)=0,\n\\label{k1} \\\\\n\\frac{\\partial\\zeta}{\\partial\nr}=\\frac{2}{r}\n\\left[\\left(\\lambda-\\frac{1}{2}\\cos\\theta\\right)\\xi_{\\theta}^+\n-\\left(\\lambda+\\frac{1}{2}\\cos\\theta\\right)\\xi_{\\theta}^-\\right],\n\\label{z1} \\\\\n-\\frac{\\varepsilon}{\\sin\\theta}\\frac{\\partial^2 f}{\\partial r^2}\n-\\frac{\\varepsilon}{r^2}\\frac{\\partial}{\\partial\\theta}\n\\left(\\frac{1}{\\sin\\theta}\\frac{\\partial f}\n{\\partial \\theta}\\right)= %\\nonumber \\\\\n2\\frac{\\Omega}{r\nc}\\left[\\left(\\lambda-\\frac{1}{2}\\cos\\theta\\right)\\xi_{\\varphi}^+\n-\\left(\\lambda+\\frac{1}{2}\\cos\\theta\\right)\\xi_{\\varphi}^-\\right],\n\\label{n1}\\\\\n\\frac{\\partial}{\\partial r}\\left(\\xi_{\\theta}^+\\gamma^+\\right)+\n\\frac{\\xi_{\\theta}^+\\gamma^+}{r}=\n4\\lambda\\sigma\\left(\n-\\frac{1}{r}\\frac{\\partial\\delta}{\\partial\\theta}+\n\\frac{\\zeta}{r}-\\frac{\\sin\\theta}{r}\\xi_r^++\n\\frac{c}{\\Omega r^2}\\xi_{\\varphi}^+\\right),\n\\label{s1}\\\\\n\\frac{\\partial}{\\partial r}\\left(\\xi_{\\theta}^-\\gamma^-\\right)+\n\\frac{\\xi_{\\theta}^-\\gamma^-}{r}=\n-4\\lambda\\sigma\\left(\n-\\frac{1}{r}\\frac{\\partial\\delta}{\\partial\\theta}+\n\\frac{\\zeta}{r}-\\frac{\\sin\\theta}{r}\\xi_r^-+\n\\frac{c}{\\Omega r^2}\\xi_{\\varphi}^-\\right),\n\\label{z2}\\\\\n\\frac{\\partial}{\\partial r}\\left(\\gamma^+\\right)=\n4\\lambda\\sigma\\left(\n-\\frac{\\partial\\delta}{\\partial r}-\n\\frac{\\sin\\theta}{r}\\xi_{\\theta}^+\\right),\n\\label{s2}\\\\\n\\frac{\\partial}{\\partial r}\\left(\\gamma^-\\right)\n=-4\\lambda\\sigma\\left(\n-\\frac{\\partial\\delta}{\\partial r}-\n\\frac{\\sin\\theta}{r}\\xi_{\\theta}^-\\right),\n\\label{z3} \\\\\n\\frac{\\partial}{\\partial r}\\left(\\xi_{\\varphi}^+\\gamma^+\\right)+\n\\frac{\\xi_{\\varphi}^+\\gamma^+}{r}=\n4\\lambda\\sigma\\left(\n-\\varepsilon\\frac{c}{\\Omega r \\sin\\theta}\\frac{\\partial f}\n{\\partial r}-\\frac{c}{\\Omega r^2}\\xi_{\\theta}^+\\right), \\\\\n\\frac{\\partial}{\\partial r}\\left(\\xi_{\\varphi}^-\\gamma^-\\right)+\n\\frac{\\xi_{\\varphi}^-\\gamma^-}{r}=\n-4\\lambda\\sigma\\left(\n-\\varepsilon\\frac{c}{\\Omega r \\sin\\theta}\\frac{\\partial f}\n{\\partial r}-\\frac{c}{\\Omega r^2}\\xi_{\\theta}^-\\right).\n\\label{b2}\n\\end{eqnarray}\nHere\n\\begin{equation}\n\\sigma=\\frac{\\Omega e B_{\\rm s} R_{\\rm s}^2}{4\\lambda m c^3}\n\\end{equation}\nis the Michel`s (1969) magnetization parameter,\n$m$ is the electron mass,\nand all deflecting functions are supposed to be $\\ll 1$.\nIt is necessary to stress that for applications the magnetic field\n$B_{\\rm s}$ is to be taken near the light cylinder\n$R_{\\rm s} \\approx R_{\\rm L}$ because in the internal region\nof the pulsar magnetosphere $B \\propto r^{-3}$.\nAs it has already been mentioned, only outside\nthe light cylinder the poloidal magnetic field may have quasi\nmonopole structure. As a result,\n\\begin{equation}\n\\sigma=\\frac{\\Omega^2 e B_0 R^3}{4\\lambda m c^4}\n\\approx 10^{4}B_{12}\\lambda_{3}^{-1}P^{-2},\n\\end{equation}\nwhere $B_0$ -- magnetic field on the neutron star surface $r = R$.\nHence, for ordinary pulsars ($P \\sim 1$s, $B_0 \\sim 10^{12}$G) we have\n$\\sigma \\sim 10^{4} - 10^{5}$, and only for fast ones\n($P \\sim 0.1 - 0.01$s, $B_0 \\sim 10^{13}$G) we have\n$\\sigma \\sim 10^{6} - 10^{7}$.\n\nFormally, this system of equations requires twelve boundary conditions.\nWe consider for simplicity the case $\\Omega R/c \\ll 1$ when the star\nradius $R$ is much smaller than the light cylinder.\nAs a result, one writes down\nthe first six boundary conditions as\n\\begin{eqnarray}\n\\xi_{\\theta}^{\\pm}(R_{\\rm s},\\theta) & = & 0, \\\\\n\\xi_{\\varphi}^{\\pm}(R_{\\rm s},\\theta) & = & 0, \\\\\n\\gamma^{\\pm}(R_{\\rm s}, \\theta) & = & \\gamma_{\\rm in},\n\\end{eqnarray}\ni.e. $\\xi_r^{\\pm}(R_{\\rm s},\\theta) = 1/(2\\gamma_{\\rm in}^2)$.\nAccording to all theories of particle generation near the neutron star\nsurface (Ruderman Sutherland 1975, Arons Scharlemann 1979),\n$\\gamma_{\\rm in} \\leq 10^2$ for secondary plasma. For this reason in what\nfollows we consider in more details the case\n\\begin{equation}\n\\gamma_{\\rm in}^3 \\ll \\sigma, %\\nonumber\n\\label{sgm}\n\\end{equation}\nwhen the additional acceleration of particles\ninside the fast magnetosonic surface takes place\n(see e.g. Beskin Kuznetsova Rafikov 1998).\nIt is this case that can be realized for fast pulsars. Moreover, it has\nmore general interest because the relation (\\ref{sgm}) may be true also\nfor AGNs.\nAs to the case $\\gamma_{\\rm in}^3 \\gg \\sigma$ corresponding to ordinary\npulsars, the particle energy remains constant ($\\gamma = \\gamma_{\\rm in}$)\nat any way up to the fast magnetosonic surface (see Bogovalov 1997 for details).\n\nFurther, one can put\n \\begin{eqnarray}\n\\delta(R_{\\rm s},\\theta) & = & 0,\n\\label{bc1}\\\\\n\\varepsilon f(R_{\\rm s},\\theta) & = & 0,\n\\label{bc2}\\\\\n\\eta^+(R_{\\rm s},\\theta)-\\eta^-(R_{\\rm s},\\theta) & = & 0.\n\\end{eqnarray}\nThese conditions result from the relation\n$c{\\bmath E}_{\\rm s}\n+ \\Omega R_{\\rm s}{\\bmath e}_{\\varphi} \\times {\\bmath B}_{\\rm s} = 0$\ncorresponding rigid rotation and perfect conductivity of the surface\nof a star. Finally, as will be shown in Sect. 3.2, the derivative\n$\\partial\\delta/\\partial r$ actually determines the phase of plasma\noscillations only and plays no role in the global structure. Finally,\nthe determination of the electric current and, say, the derivative\n$\\partial f/\\partial r$ depend on the problem under consideration.\nIndeed, as is well--known, the cold one--fluid MHD outflow contains\ntwo singular surfaces, Alfv\\'enic and fast magnetosonic ones. It means that\nfor the transonic flow two latter functions are to be determined from\ncritical conditions (Heyvaerts 1996). In particular, the longitudinal\nelectric current within this approach is not a free parameter. On the\nother hand, if the electric current is restricted by some physical\nreason, the flow cannot pass smoothly through the fast magnetosonic surface.\nIn this case, which can be realized in the magnetosphere of radio\npulsars (Beskin et al 1983, Beskin \\& Malyshkin 1998), near the\nlight surface $|{\\bmath E}| = |{\\bmath B}|$ an effective particle\nacceleration may take place. Such an acceleration will be considered\nin Sect. 4.\n\n\\section{The electron--positron outflow}\n\n\\subsection{The MHD Limit}\n\nIn the general case Eqns. (\\ref{b1}) -- (\\ref{b2}) have several\nintegrals. Firstly, Eqns.(\\ref{z1}), (\\ref{s2}), and (\\ref{z3}) result in\n\\begin{equation}\n\\zeta-\\frac{2}{\\tan\\theta}\\delta\n+\\frac{(\\lambda-1/2\\cos\\theta)\\gamma^++(\\lambda+1/2\\cos\\theta)\\gamma^-}\n{2\\sigma\\lambda\\sin\\theta}=\n\\frac{1}{\\sigma\\sin\\theta}\\gamma_{\\rm in}+\\frac{l(\\theta)}{\\sin\\theta},\n\\label{1}\n\\end{equation}\nwhere $l(\\theta)$ describe the disturbance of the electric current\nat the star surface by the equation\n$I(R,\\theta)=I_A\\left[\\sin^2\\theta+l(\\theta)\\right]$.\nExpression (\\ref{1}) corresponds\nto conservation of the total energy flux along a magnetic field line.\nFurthermore, Eqns. (\\ref{s2}) -- (\\ref{b2}) together with the boundary\nconditions (\\ref{bc1}), (\\ref{bc2}) result in\n\\begin{eqnarray}\n\\delta & = & \\varepsilon f-\\frac{1}{4\\lambda\\sigma}\\gamma^+\n\\left(1-\\frac{\\Omega r\\sin\\theta}{c}\\xi_{\\varphi}^+\\right)\n+\\frac{1}{4\\lambda\\sigma}\\gamma_{\\rm in};\n\\label{7}\\\\\n\\delta & = & \\varepsilon f+\\frac{1}{4\\lambda\\sigma}\\gamma^-\n\\left(1-\\frac{\\Omega r\\sin\\theta}{c}\\xi_{\\varphi}^-\\right)\n-\\frac{1}{4\\lambda\\sigma}\\gamma_{\\rm in}.\n\\label{8}\n\\end{eqnarray}\nThey correspond to conservation of the $z$--component of the angular momentum\nfor both types of particles. It is necessary to stress that the\ncomplete nonlinearized\nsystem of equations contains no such simple invariants.\n%because in general case magnetic surfaces are not conical.\n\nAs $\\sigma\\lambda \\gg 1$, we can neglect in Eqns. (\\ref{s1})--(\\ref{b2})\ntheir left-hand sides. In this approximation we have $\\xi^+ = \\xi^-$\ni.e. $\\gamma^- = \\gamma^+ = \\gamma$, so that\n\\begin{eqnarray}\n-\\frac{1}{r}\\frac{\\partial\\delta}{\\partial\\theta}+\n\\frac{\\zeta}{r}-\\frac{\\sin\\theta}{r}\\xi_r+\n\\frac{c}{\\Omega r^2}\\xi_{\\varphi}=0,\n\\label{6} \\\\\n\\varepsilon\\frac{c}{\\Omega r \\sin\\theta}\\frac{\\partial f}\n{\\partial r}+ \\frac{c}{\\Omega r^2}\\xi_{\\theta}=0,\n\\label{6a}\n\\end{eqnarray}\nand\n\\begin{equation}\n\\gamma\\left(1-\\frac{\\Omega r\\sin\\theta}{c}\\xi_{\\varphi}\\right)=\\gamma_{\\rm in}.\n\\label{5}\n\\end{equation}\nHence, within this approximation\n\\begin{eqnarray}\n\\delta & = & \\varepsilon f, \\\\\n\\zeta & = & \\frac{2}{\\tan\\theta}\\varepsilon f+\\frac{l(\\theta)}{\\sin\\theta}-\n\\frac{1}{\\sigma\\sin\\theta}(\\gamma-\\gamma_{\\rm in}).\n\\end{eqnarray}\nSubstituting these expressions into (\\ref{6}) and using Eqns.\n(\\ref{b1})--(\\ref{n1}), we obtain the following equation describing\nthe disturbance of the magnetic surfaces\n\\begin{eqnarray}\n\\varepsilon (1-x^{2}\\sin^{2}\\theta)\n\\frac{\\partial^{2}f}{\\partial x^2}\n+\\varepsilon(1-x^{2}\\sin^{2}\\theta)\n\\frac{\\sin\\theta}{x^2}\\frac{\\partial}{\\partial\\theta}\n\\left(\\frac{1}{\\sin\\theta}\\frac{\\partial f}{\\partial\\theta}\\right)\n%\\nonumber \\\\\n- 2\\varepsilon x\\sin^{2}\\theta\\frac{\\partial f}{\\partial x}\n-2\\varepsilon\\sin\\theta\\cos\\theta\\frac{\\partial f}{\\partial\\theta}\n+2\\varepsilon(3\\cos^{2}\\theta-1)f \\\\\n+\\frac{1}{\\sin\\theta}\\frac{{\\rm d}}{{\\rm d}\\theta}(l\\sin^2\\theta)\n-2\\frac{\\cos\\theta}{\\sigma}\\left(\\gamma-\\gamma_{\\rm in}\\right)\n-\\frac{\\sin\\theta}{\\sigma}\\frac{\\partial\\gamma}{\\partial\\theta}\n%\\nonumber \\\\\n-2\\lambda\\sin^2\\theta(\\xi_{r}^+-\\xi_{r}^-)\n+\\frac{2\\lambda}{x}\\sin\\theta(\\xi_{\\varphi}^+-\\xi_{\\varphi}^-)=0,\n\\nonumber\n\\label{mnq}\n\\end{eqnarray}\nwhere $x=\\Omega r/c$. One can see\nthat it actually coincides with the one--fluid MHD\nEqns.(32), (52) from Beskin et al (1998), but contains the two last\nadditional nonhydrodynamical terms. Nevertheless, as will be shown\nin the next subsection, at small distances $r \\ll r_{\\rm F}$\nwhere $r_{\\rm F}$ is the radius of the fast magnetosonic surface\nwe have\n\\begin{equation}\n-\\lambda\\sin^2\\theta(\\xi_{r}^+-\\xi_{r}^-)\n+\\frac{\\lambda}{x}\\sin\\theta(\\xi_{\\varphi}^+-\\xi_{\\varphi}^-) \\approx 0,\n\\label{add}\n\\end{equation}\nso actually there is perfect agreement with the MHD approximation\n\\begin{eqnarray}\n\\varepsilon (1-x^{2}\\sin^{2}\\theta)\n\\frac{\\partial^{2}f}{\\partial x^2}\n+\\varepsilon(1-x^{2}\\sin^{2}\\theta)\n\\frac{\\sin\\theta}{x^2}\\frac{\\partial}{\\partial\\theta}\n\\left(\\frac{1}{\\sin\\theta}\\frac{\\partial f}{\\partial\\theta}\\right)\n%\\nonumber \\\\\n- 2\\varepsilon x\\sin^{2}\\theta\\frac{\\partial f}{\\partial x}\n-2\\varepsilon\\sin\\theta\\cos\\theta\\frac{\\partial f}{\\partial\\theta} \n\\label{mn1}\\\\\n+2\\varepsilon(3\\cos^{2}\\theta-1)f\n%\\\\\n+\\frac{1}{\\sin\\theta}\\frac{{\\rm d}}{{\\rm d}\\theta}(l\\sin^2\\theta)\n-2\\frac{\\cos\\theta}{\\sigma}\\left(\\gamma-\\gamma_{\\rm in}\\right)\n-\\frac{\\sin\\theta}{\\sigma}\\frac{\\partial\\gamma}{\\partial\\theta}\n= 0.\n\\nonumber\n%\\label{mn1}\n\\end{eqnarray}\n\nAs was shown earlier (Beskin et al 1998), to pass through\nthe fast magnetosonic surface it's necessary to have\n\\begin{equation}\n|l| < \\sigma^{-4/3}.\n\\label{nn}\n\\end{equation}\nHence, within the fast magnetosonic surface $r \\ll r_{\\rm F}$\none can neglect terms containing $\\delta = \\varepsilon f$ and $\\zeta$.\nThen, relations (\\ref{6}) and (\\ref{6a}) result in\n\\begin{eqnarray}\n\\gamma(1-x\\sin\\theta\\xi_{\\varphi}) & = & \\gamma_{\\rm in}, \\\\\n\\xi_r & = & \\frac{\\xi_{\\varphi}}{x\\sin\\theta}, \\\\\n\\xi_{\\theta}& = & 0.\n\\end{eqnarray}\nFinally, using the definition\n\\begin{equation}\n\\gamma^2=\\frac{1}{2\\xi_r-\\xi_{\\varphi}^2},\n\\label{3}\n\\end{equation}\nwe obtain for $\\sigma \\gg \\gamma_{\\rm in}^3$ for $r \\ll r_{\\rm F}$\n\\begin{eqnarray}\n\\gamma^2 & = & \\gamma_{\\rm in}^2+x^2\\sin^2\\theta,\n\\label{v0}\\\\\n\\xi_{\\varphi} & = &\n\\frac{\\sqrt{\\gamma_{\\rm in}^2+x^2\\sin^2\\theta}-\\gamma_{\\rm in}}\n{x\\sin\\theta\\sqrt{\\gamma_{\\rm in}^2+x^2\\sin^2\\theta}}\n\\sim \\frac{1}{x\\sin\\theta},\n\\label{v1} \\\\\n\\xi_r & = &\n\\frac{\\sqrt{\\gamma_{\\rm in}^2+x^2\\sin^2\\theta}-\\gamma_{\\rm in}}\n{x^2\\sin^2\\theta\\sqrt{\\gamma_{\\rm in}^2+x^2\\sin^2\\theta}}\n\\sim \\frac{1}{x^2\\sin^2\\theta},\n\\label{v2}\n\\end{eqnarray}\nin full agreement with the MHD results.\n\nNext, to determine the position of the fast magnetosonic surface $r_{\\rm F}$,\none can analyze the algebraic equations (\\ref{1}) and (\\ref{6}) which\ngive\n\\begin{equation}\n-\\frac{\\partial\\delta}{\\partial\\theta}+\\frac{2}{\\tan\\theta}\\delta\n-\\frac{1}{\\sigma\\sin\\theta}\\gamma-\\sin\\theta\\xi_r\n+\\frac{1}{x}\\xi_{\\varphi}=0.\n\\end{equation}\nUsing now expressions (\\ref{5}) and (\\ref{3}), one can find\n\\begin{equation}\n2\\gamma^3-2\\sigma\\left[K+\\frac{1}{2x^2}\\right]\\gamma^2\n+\\sigma\\sin^2\\theta = 0,\n\\label{9}\n\\end{equation}\nwhere\n\\begin{equation}\nK(r,\\theta)=2\\cos\\theta\\delta-\\sin\\theta\\frac{\\partial\\delta}{\\partial\\theta}.\n\\end{equation}\n\nEquation (\\ref{9}) allows us to determine the position of the fast magnetosonic\nsurface and the energy of particles. Indeed, determining the derivative\n$r\\partial\\gamma/\\partial r$, one can obtain\n\\begin{equation}\nr\\frac{\\partial\\gamma}{\\partial r}=\n\\frac{\\gamma\\sigma\\left(r\\partial K/\\partial r-x^{-2}\\right)}\n{3\\gamma-\\sigma\\left(2K+x^{-2}\\right)}.\n\\end{equation}\nAs the fast magnetosonic surface is the $X$--point, both the nominator and\ndenominator are to be equal to zero here.\nAs a result, evaluating $r\\partial K/\\partial r$ as $K$, we obtain\n\\begin{eqnarray}\n\\delta & \\sim & \\sigma^{-2/3}; \\\\\nr_{\\rm F} & \\sim & \\sigma^{1/3}R_{\\rm L};\n\\label{51}\\\\\n\\gamma(r_{\\rm F}) & = & \\sigma^{1/3}\\sin^{2/3}\\theta,\n\\end{eqnarray}\nwhere the last expression is exact. These equations\n coincide with those obtained\nwithin the MHD consideration. It is the self--consistent analysis when\n$\\delta=\\varepsilon f$, and hence $K$ depends on the radius $r$ that results in\nthe finite value for the fast magnetosonic radius $r_{\\rm F}$. On the other hand, in\na given monopole magnetic field, when $\\varepsilon f$\ndoes not depend on the radius,\nthe critical conditions result in $r_{\\rm F}\\rightarrow \\infty$ for a cold outflow\n(Michel, 1969, Li et al 1992).\n\nNear the fast magnetosonic surface $r \\sim \\sigma^{1/3}R_{\\rm L}$ the MHD\nsolution gives\n\\begin{eqnarray}\n\\gamma & \\sim & \\sigma^{1/3}, \\\\\n\\varepsilon f & \\sim & \\sigma^{-2/3}.\n\\end{eqnarray}\nHence, Eqns. (\\ref{3}), (\\ref{v1}), and\n (\\ref{v2}) result in\n\\begin{eqnarray}\n\\xi_r & \\sim & \\sigma^{-2/3}, \\\\\n\\xi_{\\theta} & \\sim & \\sigma^{-2/3}, \\\\\n\\xi_{\\varphi} & \\sim & \\sigma^{-1/3}.\n\\end{eqnarray}\nAs we see, the $\\theta$--component of the velocity plays no role\nin the determination of the $\\gamma$.\n\nHowever, analyzing the\nleft-hand sides of the Eqns. (\\ref{s1})--(\\ref{b2})\none can evaluate the additional (nonhydrodynamic) variations of the velocity\ncomponents\n\\begin{eqnarray}\n\\Delta\\xi_r^{\\pm} & \\sim & \\lambda^{-1}\\sigma^{-4/3}, \\\\\n\\Delta\\xi_{\\theta}^{\\pm} & \\sim & \\lambda^{-1}\\sigma^{-2/3}, \\\\\n\\Delta\\xi_{\\varphi}^{\\pm} & \\sim & \\lambda^{-1}\\sigma^{-1}.\n\\end{eqnarray}\nHence, for nonhydrodynamic velocities $\\Delta\\xi_r^{\\pm} \\ll \\xi_r$ and\n$\\Delta\\xi_{\\varphi}^{\\pm} \\ll \\xi_{\\varphi}$ to be small,\nit is necessary to have a large magnetization parameter $\\sigma\n\\gg 1$ only. On the other hand,\n$\\Delta\\xi_{\\theta}^{\\pm}/\\xi_{\\theta}\\sim \\lambda^{-1}$.\nIn other words, for a highly magnetized plasma $\\sigma \\gg 1$\neven outside the fast magnetosonic surface\nthe velocity components (and, hence, the\nparticle energy) can be considered hydrodynamically.\nThe difference $\\sim \\lambda^{-1}$ appears in the $\\theta$ component\nonly, but it does not affect the particle energy. Finally,\none can obtain from (\\ref{7}), (\\ref{8}) that\n\\begin{equation}\n\\frac{\\delta-\\varepsilon f}{\\varepsilon f} \\sim \\lambda^{-2}\\sigma^{-2/3}.\n\\end{equation}\nTo put it differently, at large distances the nonhydrodynamical terms are much\nsmaller than hydrodynamical ones.\n\nAs a result, at large distances where, according to (\\ref{7})--(\\ref{8}),\none can neglect the toroidal component $\\xi_{\\varphi}$, we obtain\n\\begin{eqnarray}\n\\delta & = & \\varepsilon f, \\\\\n\\zeta & = & \\frac{2}{\\tan\\theta}\\delta\n-\\sigma^{-1}\\frac{1}{\\sin\\theta}\\gamma.\n\\label{z1a}\n\\end{eqnarray}\nOn the other hand, Eqn. (\\ref{s1}) gives\n\\begin{equation}\n\\zeta=\\frac{\\partial\\delta}{\\partial\\theta}+\\sin\\theta\\xi_r.\n\\end{equation}\nTogether with (\\ref{z1}) one can obtain for $r \\gg r_{\\rm F}$\n\\begin{equation}\n\\gamma=\\sigma\\left(2\\cos\\theta\\varepsilon f\n-\\varepsilon\\sin\\theta\\frac{\\partial\nf}{\\partial\\theta}\\right),\n\\label{k2}\n\\end{equation}\nwhich coincides with the MHD solution. Finally,\nusing Eqns. (\\ref{b1}), (\\ref{k1}), and neglecting the nonhydrodynamic\nterm $4\\lambda(\\xi_r^+-\\xi_r^-)$, one can find\n\\begin{equation}\n\\varepsilon\\frac{\\partial}{\\partial r}\\left(r^2\\frac{\\partial f}\n{\\partial r}\\right)\n-4\\cos\\theta\\xi_r\n-\\sin\\theta\\frac{\\partial}{\\partial\\theta}\\xi_r+\n\\frac{1}{x\\sin\\theta}\\frac{\\partial}{\\partial\\theta}(\\xi_{\\varphi}\\sin\\theta)\n=0.\n\\label{pp}\n\\end{equation}\nTogether with (\\ref{k2}) this equation in the limit $r \\gg r_{\\rm F}$\ncoincides with the asymptotic version of the trans--field equation\n(Tomimatsu 1994, Beskin et al 1998)\n\\begin{equation}\n\\varepsilon\\frac{\\partial^2 f}{\\partial r^2}+\n2\\varepsilon r\\frac{\\partial f}{\\partial r}\n-\\sin\\theta\\frac{D+1}{D}\\frac{\\partial g}{\\partial\\theta} = 0,\n\\end{equation}\nwhere $g(\\theta)=K(\\theta)/\\sin^2\\theta$, and\n\\begin{equation}\nD+1 = \\frac{1}{\\sigma^2\\sin^4\\theta}g^{-3}(\\theta) \\ll 1.\n\\end{equation}\nIn this limit, none of the terms containing $\\xi_r^{\\pm}$ and $\\xi_{\\varphi}$\nplays role in the asymptotic trans--field equation. Hence, it\nis not necessary to consider the effect of the nonhydrodynamical term\n$4\\lambda(\\xi_r^+-\\xi_r^-)$ either.\n\n\\subsection{Plasma Oscillations}\n\nIn the intermediate region $r \\ll r_{\\rm F}$\nEqn. (\\ref{pp}) cannot be used. The point is that in the limit\n$\\lambda \\gg 1$ the important role in Eqns. (\\ref{b1}) and (\\ref{n1})\nis played by the nonhydrodynamic terms (\\ref{add})\ncorresponding to different velocities of two\ncomponents. As a result, the full version of Eqn. (\\ref{pp}) has the form\n\\begin{equation}\n\\frac{\\partial}{\\partial r}\\left(r^2\\frac{\\partial\\delta}\n{\\partial r}\\right)\n-4\\cos\\theta\\xi_r\n-\\sin\\theta\\frac{\\partial\\xi_r}{\\partial\\theta}+\n\\frac{1}{x\\sin\\theta}\\frac{\\partial}{\\partial\\theta}(\\xi_{\\varphi}\\sin\\theta)\n+2\\lambda(\\xi_r^+-\\xi_r^-)=0.\n\\label{pk}\n\\end{equation}\nIndeed, one can see from equations (\\ref{b1}) and (\\ref{k1}) that near\nthe origin $x=R_{\\rm s}$ in the case $\\gamma_{\\rm in}^+=\\gamma_{\\rm in}^-$\n(and for the small variation of the current $\\zeta \\sim \\sigma^{-4/3}$\nwhich is necessary, as was already stressed, to pass through a fast\nmagnetosonic surface) the density variation on the surface is large enough:\n$(\\eta^+-\\eta^-) \\sim \\gamma_{\\rm in}^{-2} \\gg \\zeta$. Hence, the derivative\n$\\partial^2\\delta/\\partial r^2$ here is of the order of $\\gamma_{\\rm in}^{-2}$.\nOn the other hand, according to (\\ref{n1}), the derivative\n$\\varepsilon\\partial^2 f/\\partial r^2$ is $x^2$ times smaller.\nThis means that in the two--component system the longitudinal electric field\nis to appear resulting in a redistribution of the particle energy.\n%As a result, the variation of charge density can be diminished.\nClearly, such a redistribution is impossible for the charge--separated outflow.\nIn other words, for a finite particle energy a one--component plasma cannot\nmaintain simultaneously both the Goldreich charge and Goldreich\ncurrent density (\\ref{gj}).\nIn a two--component system with $\\lambda \\gg 1$ it can be realized by a small\nredistribution of particle energy (Ruderman \\& Sutherland 1975,\nArons \\& Scharlemann 1989).\n\nFor simplicity, let us consider only small distances $x \\ll 1$.\nIn this case one can neglect the changes of the magnetic surfaces.\nUsing now (\\ref{s2}) and (\\ref{z3}), we have\n\\begin{eqnarray}\n\\gamma^+ & = & \\gamma_{\\rm in}-4\\lambda\\sigma\\delta; \\\\\n\\gamma^- & = & \\gamma_{\\rm in}+4\\lambda\\sigma\\delta.\n\\end{eqnarray}\nFinally, taking into account that $\\xi_{\\theta}$ and $\\xi_{\\varphi}$ are\nsmall here, one can obtain from (\\ref{k1}) %and (\\ref{z1})\n\\begin{equation}\nr^2\\frac{\\partial^2\\delta}{\\partial r^2}+2r\\frac{\\partial\\delta}{\\partial r}\n+\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial\\theta}\\left(\\sin\\theta\n\\frac{\\partial\\delta}{\\partial\\theta}\\right)\n+A\\delta=\\frac{\\cos\\theta}{\\gamma_{\\rm in}^2},\n\\label{p3}\n\\end{equation}\nwhere\n\\begin{equation}\nA=16\\frac{\\lambda^2\\sigma}{\\gamma_{\\rm in}^3} \\gg 1.\n\\end{equation}\nEqn. (\\ref{p3}) has a solution\n\\begin{equation}\n\\delta=\n\\delta_0+r^{-1/2}\n\\left[C_1\\sin(\\mu\\ln r)+C_2\\cos(\\mu\\ln r)\\right]\\cos\\theta,\n\\label{p4}\n\\end{equation}\nwhere\n\\begin{equation}\n\\delta_0 \\approx\n\\frac{\\gamma_{\\rm in}\\cos\\theta}{16\\lambda^2\\sigma},\n\\end{equation}\nand $\\mu\\approx \\sqrt{A}$. As we see, Eqn. (\\ref{p4})\ndescribes plasma oscillations similar to those considered by\nShibata (1997) for charge--separated flow. The decrease of\noscillations results from a more accurate consideration of\nthe Laplace operator in a 3D space.\n\nOne can easily check that the additional potential $\\delta_0$ is small,\nand it is not necessary to add it in (\\ref{1}) and (\\ref{7})--(\\ref{8}).\nMoreover, the nonhydrodynamic disturbance $\\Delta\\xi_r$ (as well as\n$\\Delta\\gamma$) is also small, $\\Delta\\xi_r/\\xi_r \\approx \\lambda^{-1}$.\nHence, as was already stressed, the boundary condition $\\partial\\delta/\\partial\nr$ (determining together with (\\ref{bc1}) the coefficients $C_1$ and $C_2$)\ndoes not affect the general structure of the flow.\nOn the other hand, the presence of an additional electric potential $\\delta_0$\nresults in a full compensation of the last term in (\\ref{b1})\n\\begin{equation}\n2\\lambda(\\xi_r^+-\\xi_r^-)-\\cos\\theta\\xi_r \\approx 0.\n\\label{u1}\n\\end{equation}\nNext, as $\\varepsilon f \\ll \\sigma^{-2/3}$ for $r \\ll r_{\\rm F}$, a similar\nexpression can be written for the $\\varphi$--components as well\n\\begin{equation}\n2\\lambda(\\xi_{\\varphi}^+-\\xi_{\\varphi}^-)-\\cos\\theta\\xi_{\\varphi} \\approx 0.\n\\label{u2}\n\\end{equation}\nExpressions (\\ref{u1}) -- (\\ref{u2}) must hold for the whole region $r < r_{\\rm F}$.\nIn this case, the final version\nof Eqn. (\\ref{pk}) in the internal region $r \\ll r_{\\rm F}$ can be\nrewritten as\n\\begin{equation}\n\\frac{\\partial}{\\partial r}\\left(r^2\\frac{\\partial\\delta}\n{\\partial r}\\right)\n-2\\cos\\theta\\xi_r\n-\\sin\\theta\\frac{\\partial}{\\partial\\theta}\\xi_r+\n\\frac{1}{x\\sin\\theta}\\frac{\\partial}{\\partial\\theta}(\\xi_{\\varphi}\\sin\\theta)\n=0.\n\\label{ps}\n\\end{equation}\n\n\nAs $\\delta \\sim \\varepsilon f \\ll \\sigma^{-2/3}$ for $r \\ll r_{\\rm F}$,\nand $\\xi_r \\sim \\gamma_0^{-2} \\gg \\delta$, the first\nterm in (\\ref{ps}) can be omitted. As a result, the solution\nof Eqn. (\\ref{ps}) coincides exactly with the MHD expression, i.e.\n$\\gamma^2=\\gamma_{\\rm in}^2+x^2\\sin^2\\theta$ (\\ref{v0}).\nFinally, using (\\ref{u1}), (\\ref{u2}), and (\\ref{v1})--(\\ref{v2}),\none can easily check that the nonhydrodynamical terms (\\ref{add}) in\nthe trans--field equation (\\ref{mn1}) do actually vanish.\n\n\\section{The Boundary Layer}\n\nLet us now consider the case when the longitudinal electric current\n$I(R,\\theta)$ in the magnetosphere of radio pulsars is too small\n(i.e. the disturbance $l(\\theta)$ is too large) for the flow to\npass smoothly through the fast magnetosonic surface. First of all,\nit can be realized when the electric current is much smaller\nthan the Goldreich one. This possibility was already discussed within\nthe Ruderman--Sutherland model of the internal gap (Beskin et al 1983,\nBeskin \\& Malyshkin 1998). But it may take place in the Arons model\n(Arons \\& Scharlemann 1979) as well. Indeed, within this model the\nelectric current is determined by the gap structure. Hence, in general\ncase this current does not correspond to the critical condition at the\nfast magnetosonic surface. In particular, it may be smaller than the\ncritical current (of course, the separate consideration is necessary to\ncheck this statement).\n\nFor simplicity let us consider the case $l(\\theta)=h\\sin^2\\theta$.\nNeglecting now the last terms $\\propto \\sigma^{-1}$ in the trans--field\nequation (\\ref{mn1}), we obtain\n\\begin{eqnarray}\n\\varepsilon (1-x^{2}\\sin^{2}\\theta)\n\\frac{\\partial^{2}f}{\\partial x^2}\n+\\varepsilon(1-x^{2}\\sin^{2}\\theta)\n\\frac{\\sin\\theta}{x^2}\\frac{\\partial}{\\partial\\theta}\n\\left(\\frac{1}{\\sin\\theta}\\frac{\\partial f}{\\partial\\theta}\\right)\n\\nonumber \\\\\n- 2\\varepsilon x\\sin^{2}\\theta\\frac{\\partial f}{\\partial x}\n-2\\varepsilon\\sin\\theta\\cos\\theta\\frac{\\partial f}{\\partial\\theta}\n+2\\varepsilon(3\\cos^{2}\\theta-1)f\n+4h\\sin^2\\theta\\cos\\theta = 0,\n\\end{eqnarray}\nwhich actually coincides with the force--free equation (Beskin et al 1998).\nThis equation has an exact analytical solution\n\\begin{equation}\n\\varepsilon f=hx^2\\sin^2\\theta\\cos\\theta.\n\\label{qa}\n\\end{equation}\nFor $h < 0$ (when the electric current is smaller than the\nGoldreich one) this solution results in the appearance of the light\nsurface $|{\\bmath E}|=|{\\bmath B}|$ at the finite distance\n\\begin{equation}\n\\varpi_c = \\frac{R_{\\rm L}}{(2|h|)^{1/4}}.\n\\label{qq}\n\\end{equation}\nAs we see, for $l(\\theta)=h\\sin^2\\theta$ this surface has the form\nof a cylinder. It is important that the disturbance of magnetic\nsurfaces $\\varepsilon f \\sim\n(|h|)^{1/2}$ remains small here.\n\nComparing now the position of the light surface (\\ref{qq}) with that of the\nfast magnetosonic surface (\\ref{51}), one can find that the light surface\nis located inside the fast magnetosonic one if\n\\begin{equation}\n\\sigma^{-4/3} \\ll |h| \\ll 1,\n\\label{qs}\n\\end{equation}\nwhich is opposite to (\\ref{nn}).\nOne can check that the condition (\\ref{qs}) just allows to neglect\nthe non force--free term in Eqn. (\\ref{mn1}).\n\nUsing now the solution (\\ref{qa}) and the MHD condition\n$\\delta=\\varepsilon f$, one can find from (\\ref{9})\n\\begin{equation}\n2\\gamma^3-2\\sigma\\left(hx^2\\sin^4\\theta+\\frac{1}{2x^2}\\right)\\gamma^2\n+\\sigma\\sin^2\\theta=0.\n\\end{equation}\nThis equation shows that near the force--free boundary\n$x_{ff} = (2|h|)^{-1/4}$ (\\ref{qq})\n\\begin{equation}\n\\gamma = \\sigma^{1/3}\\sin^{2/3}\\theta -\n\\frac{2|2h|^{3/8}}{\\sqrt{3}}\\sigma^{1/3}\\sin^{4/3}\\theta\n\\sqrt{x_0-x\\sin\\theta},\n\\label{fig}\n\\end{equation}\nwhere\n\\begin{equation}\nx_0 = \\frac{1}{(2|h|)^{1/4}}\\left[1-\\frac{3}{4(2|h|)^{1/2}}\n\\frac{1}{(\\sigma\\sin^2\\theta)^{2/3}}\\right],\n\\label{qw}\n\\end{equation}\n(see Fig. \\ref{figure}).\nHence, the real solution is absent for $x\\sin\\theta > x_0$.\nHere $\\gamma(x_0)=\\sigma^{1/3}\\sin^{2/3}\\theta$,\nthe condition $\\sigma^{-4/3} \\ll |h|$ resulting in $\\varpi_c < r_{\\rm F}$,\nand the last term in (\\ref{qw}) being small.\n\n\\begin{figure}\n\\vspace{10.cm}\n\\special{psfile=figure.ps voffset=30 hoffset=400 vscale=40\nhscale=42 angle=91}\n\\caption{\nThe behavior of the Lorentz factor in the case $\\sigma^{-4/3} \\ll |h| \\ll 1$.\nOne can see that the one--fluid MHD solution (\\ref{fig}) exists for\n$\\gamma < \\sigma^{1/3}$ only. But in the two--fluid approximation\nin the narrow layer $\\Delta\\varpi = \\varpi_c/\\lambda$\nthe particle energy increases up to the value $\\sim \\sigma$\ncorresponding to the full conversion of the electromagnetic\nenergy to the energy of particles.}\n\\label{figure}\n\\end{figure}\n\n\nAlthough the energy of particles at the limiting point is finite,\nthe derivative $d\\gamma/d\\varpi$ moves to infinity.\nHence, near the light surface the left-hand sides in the Eqns.\n(\\ref{s1}) -- (\\ref{b2}) are to be taken into consideration.\nSince in our case the light surface has the form of a cylinder,\none can move to derivatives perpendicular to the boundary layer\nonly by\n\\begin{eqnarray}\n\\partial/\\partial r \\rightarrow \\sin\\theta\\partial/\\partial\\varpi; \\\\\n\\partial/\\partial\\theta \\rightarrow \\varpi\\cos\\theta\\partial/\\partial\\varpi.\n\\end{eqnarray}\nAs a result, $\\zeta$ can be eliminated from (\\ref{b1}) and\n(\\ref{z1}). Together with (\\ref{k1}) they give the equation for\n$\\delta$ (see (\\ref{p6})). Next, the invariants (\\ref{7}\n) and\n(\\ref{8}) can be used to define $\\xi_{\\varphi}^{\\pm}$:\n\\begin{eqnarray}\n\\xi_{\\varphi}^+ & = & \\frac{1}{x\\sin\\theta}\n\\left[1+\\frac{4\\lambda\\sigma(\\delta-\\varepsilon f)}{\\gamma^+}\\right];\n\\label{p1}\\\\\n\\xi_{\\varphi}^- & = & \\frac{1}{x\\sin\\theta}\n\\left[1-\\frac{4\\lambda\\sigma(\\delta-\\varepsilon f)}{\\gamma^-}\\right].\n\\label{p2}\n\\end{eqnarray}\nFurthermore, one can define\n\\begin{eqnarray}\n2\\xi_r^+ & = & \\frac{1}{(\\gamma^+)^2}+(\\xi_{\\varphi}^+)^2+(\\xi_{\\theta}^+)^2;\n\\label{p3a}\\\\\n2\\xi_r^- & = & \\frac{1}{(\\gamma^-)^2}+(\\xi_{\\varphi}^-)^2+(\\xi_{\\theta}^-)^2.\n\\label{p4a}\n\\end{eqnarray}\nAs to the energy integral (\\ref{1}), it determines the variation of\nthe current $\\zeta$. Now it can be rewritten as\n\\begin{equation}\n\\zeta=\\frac{2}{\\tan\\theta}\\delta\n-\\frac{(\\gamma^++\\gamma^-)}{2\\sigma\\sin\\theta}.\n\\label{p5}\n\\end{equation}\nFinally, Eqns. (\\ref{b1}) -- (\\ref{b2}) look like\n\\begin{eqnarray}\n\\varpi_c^2\\frac{d^2\\delta}{d\\varpi^2}=\n2\\sin\\theta\\cos\\theta\n\\left[\\left(\\lambda-\\frac{1}{2}\\cos\\theta\\right)\\xi_{\\theta}^+\n-\\left(\\lambda+\\frac{1}{2}\\cos\\theta\\right)\\xi_{\\theta}^-\\right]\n\\nonumber \\\\\n-2\\sin^2\\theta\\left[\\left(\\lambda-\\frac{1}{2}\\cos\\theta\\right)\\xi_r^+\n-\\left(\\lambda+\\frac{1}{2}\\cos\\theta\\right)\\xi_r^-\\right],\n\\label{p6} \\\\\n\\varpi_cR_{\\rm L}\\varepsilon\\frac{d^2 f}{d\\varpi^2}=\n-2\\sin^2\\theta\\left[\\left(\\lambda-\\frac{1}{2}\\cos\\theta\\right)\\xi_{\\varphi}^+\n-\\left(\\lambda+\\frac{1}{2}\\cos\\theta\\right)\\xi_{\\varphi}^-\\right],\n\\label{p7}\\\\\n\\varpi_c\\frac{d}{d\\varpi}\\left(\\xi_{\\theta}^+\\gamma^+\\right) =\n%\\nonumber \\\\\n4\\lambda\\sigma\\left(-\\frac{\\gamma^++\\gamma^-}{2\\sigma\\sin\\theta}\n-\\varpi_c\\frac{\\cos\\theta}{\\sin\\theta}\\frac{d\\delta}{d\\varpi}\n-\\sin\\theta\\xi_r^++\\frac{\\sin\\theta}{x_0}\\xi_{\\varphi}^+\\right),\n\\label{p8}\\\\\n\\varpi_c\\frac{d}{d\\varpi}\\left(\\xi_{\\theta}^-\\gamma^-\\right) =\n%\\nonumber \\\\\n-4\\lambda\\sigma\\left(-\\frac{\\gamma^++\\gamma^-}{2\\sigma\\sin\\theta}\n-\\varpi_c\\frac{\\cos\\theta}{\\sin\\theta}\\frac{d\\delta}{d\\varpi}\n-\\sin\\theta\\xi_r^-+\\frac{\\sin\\theta}{x_0}\\xi_{\\varphi}^-\\right),\n\\label{p9}\\\\\n\\varpi_c\\frac{d}{d\\varpi}\\gamma^+ =\n4\\lambda\\sigma\\left(\n-\\varpi_c\\frac{d\\delta}{d\\varpi}-\\sin\\theta\\xi_{\\theta}^+\\right),\n\\label{p10}\\\\\n\\varpi_c\\frac{d}{d\\varpi}\\gamma^- =\n-4\\lambda\\sigma\\left(\n-\\varpi_c\\frac{d\\delta}{d\\varpi}-\\sin\\theta\\xi_{\\theta}^-\\right),\n\\label{p11}\n\\end{eqnarray}\nwhere we neglected the terms $\\propto\\delta/r$ in (\\ref{p8}) and (\\ref{p9}).\n\nComparing the leading terms, we have inside the\nlayer $\\Delta\\varpi/R_{\\rm L} \\sim \\lambda^{-1}$\n\\begin{eqnarray}\n\\gamma^{\\pm} & \\sim & h_c^{1/2}\\sigma, \\\\\n\\xi_{\\theta}^{\\pm} & \\sim & h_c^{1/4}, \\\\\n\\xi_r^{\\pm} & \\sim & h_c^{1/2}, \\\\\n\\Delta\\delta & \\sim & h_c^{3/4}/\\lambda,\n\\end{eqnarray}\nwhere $h_c=|h|$. Then the leading terms in\n(\\ref{p1}) -- (\\ref{p5}) for $\\Delta\\varpi > \\lambda^{-1}R_{\\rm L}$ are\n\\begin{eqnarray}\n\\xi_{\\varphi}^+ & = & \\frac{1}{x\\sin\\theta} \\approx \\frac{1}{x_0},\n\\label{p12}\\\\\n\\xi_{\\varphi}^- & = & \\frac{1}{x\\sin\\theta} \\approx \\frac{1}{x_0},\n\\label{p13}\n\\end{eqnarray}\n\\begin{eqnarray}\n2\\xi_r^+ & = & (\\xi_{\\varphi}^+)^2+(\\xi_{\\theta}^+)^2,\n\\label{p14}\\\\\n2\\xi_r^- & = & (\\xi_{\\varphi}^-)^2+(\\xi_{\\theta}^-)^2,\n\\label{p15}\n\\end{eqnarray}\n\\begin{equation}\n\\zeta=-\\frac{(\\gamma^++\\gamma^-)}{2\\sigma\\sin\\theta},\n\\label{p16}\n\\end{equation}\nwhere $x_0=\\varpi_c/R_{\\rm L}=(2|h|)^{-1/4}$.\nHence, one can totally neglect $\\varepsilon f$ and $\\delta$ in\n(\\ref{p1}) -- (\\ref{p2}), so Eqns. (\\ref{p6}) -- (\\ref{p11})\nin the region $\\Delta\\varpi > \\lambda^{-1}R_{\\rm L}$ can be rewritten as\n%\n%\\newpage\n%\n\\begin{eqnarray}\n\\varpi_c^2\\frac{d^2\\delta}{d\\varpi^2}=\n2\\lambda\\sin\\theta\\cos\\theta(\\xi_{\\theta}^+-\\xi_{\\theta}^-),\n\\label{p17} \\\\\n\\varpi_c\\frac{d}{d\\varpi}\\left(\\xi_{\\theta}^+\\gamma^+\\right) =\n%\\nonumber \\\\\n4\\lambda\\sigma\\left(-\\frac{\\gamma^++\\gamma^-}{2\\sigma\\sin\\theta}\n-\\varpi_c\\frac{\\cos\\theta}{\\sin\\theta}\\frac{d\\delta}{d\\varpi}\n-\\sin\\theta\\xi_r^+\\right),\n\\label{p18}\\\\\n\\varpi_c\\frac{d}{d\\varpi}\\left(\\xi_{\\theta}^-\\gamma^-\\right) =\n%\\nonumber \\\\\n-4\\lambda\\sigma\\left(-\\frac{\\gamma^++\\gamma^-}{2\\sigma\\sin\\theta}\n-\\varpi_c\\frac{\\cos\\theta}{\\sin\\theta}\\frac{d\\delta}{d\\varpi}\n-\\sin\\theta\\xi_r^\n-\\right),\n\\label{p19}\\\\\n\\varpi_c\\frac{d}{d\\varpi}\\left(\\gamma^+\\right)=\n-4\\lambda\\sigma\\sin\\theta\\xi_{\\theta}^+,\n\\label{p20}\\\\\n\\varpi_c\\frac{d}{d\\varpi}\\left(\\gamma^-\\right)=\n4\\lambda\\sigma\\sin\\theta\\xi_{\\theta}^-,\n\\label{p21}\n\\end{eqnarray}\nwith all the terms in the right--hand sides of (\\ref{p18}) and\n(\\ref{p19}) being of the same order of magnitude.\n\nAs a result, the nonlinear equations (\\ref{p17}) -- (\\ref{p21}) and\n(\\ref{p7}) give the following simple asymptotic solution\n\\begin{eqnarray}\n\\gamma^{\\pm} & = & 4\\sin^2\\theta\\sigma(\\lambda l)^2, \\\\\n\\xi_{\\theta}^{\\pm} & = & \\mp 2\\sin\\theta\\lambda l, \\\\\n%\\xi_r^{\\pm} & = & , \\\\\n\\Delta\\delta & = &\n-\\frac{4}{3}\\sin^2\\theta\\cos\\theta\\lambda^{-1}(\\lambda l)^3,\n\\label{o2}\\\\\n\\Delta(\\varepsilon f) & = & \\sin^2\\theta\\cos\\theta\\lambda^{-2}(\\lambda l)^2,\n\\label{o3} \\\\\n\\zeta & = & -4\\sin\\theta(\\lambda l)^2,\n\\label{o4}\n\\end{eqnarray}\nwhere now $l=\\Delta\\varpi/\\varpi_c$. It is important that the last expressions\nare correct for arbitrary relations between $\\gamma_{\\rm in}^3$ and $\\sigma$.\nAs we can see, in the narrow layer $\\Delta\\varpi = \\varpi_c/\\lambda$\nthe particle energy increases up to the value $\\sim \\sigma$ which\ncorresponds to the full conversion of the electromagnetic\nenergy to the energy of particles. For this reason we have\nhere $|\\zeta| \\sim 1$, which just means the diminishing of the\ntoroidal magnetic field determining the flux of the electromagnetic\nenergy. On the other hand, the variation of the electric potential\nremains small $\\delta \\sim \\lambda^{-1}$, to say nothing about\nthe variation of the magnetic surfaces $\\Delta\\varepsilon f \\sim\n\\lambda^{-2}$. These results coincide exactly with our previous\nevaluations (Beskin et al 1983) allowing us to neglect\nvariations of the electric potential and the poloidal magnetic\nstructure in the 1D cylindrical case.\n\nThe latter result has a simple physical explanation.\nIndeed, the diminishing of the toroidal magnetic field is connected\nwith the $\\theta$--component of the electric current which\nis produced by all the particles moving in opposite directions.\nOn the other hand, the change of the electric potential\ndepends on the small difference between the electron and positron\ndensities. As a result, according to (\\ref{o3}) and (\\ref{o4}),\nthe change of the toroidal magnetic field is just $\\lambda$ times\nlarger than the change of the electric potential. Unfortunately,\nit is impossible to consider this region more thoroughly because\nfor $\\lambda l \\sim 1$ we have $\\xi_{\\theta}^{\\pm} \\sim 1$ and\n$\\xi_{r}^{\\pm} \\sim 1$, i.e. the linear approximation\n(\\ref{b1}) -- (\\ref{b2}) itself becomes incorrect.\n\nIt is necessary to stress as well that we do not include into consideration\nthe radiation reaction force\n\\begin{equation}\nF_x^{\\rm (rad)} = -\\frac{2}{3}\\frac{e^4}{m^2c^4}\\gamma^2\n\\left[(E_y - B_z)^2 + (E_z - B_y)^2\\right],\n\\label{frad}\n\\end{equation}\nwhich can be important for large enough particle energy.\nComparing (\\ref{frad}) with appropriate terms in\n(\\ref{p18}) -- (\\ref{p21}) one can conclude that the radiation\nforce can be neglected for $\\sigma < \\sigma_{\\rm cr}$, where\n\\begin{equation}\n\\sigma_{\\rm cr} = \\left(\\frac{c}{\\lambda r_e \\Omega}\\right)^{1/3}\n\\approx 10^6,\n\\end{equation}\nand $r_e = e^2/mc^2$ -- classical electron radius.\nThis relation can be rewritten in the form\n\\begin{equation}\n\\frac{\\Omega R}{c} < 3 \\times 10^{-3}B_{12}^{-3/7} \\lambda_4^{2/7}\n\\end{equation}\nwhich gives\n\\begin{equation}\nP > 0.06B_{12}^{3/7} \\lambda_4^{-2/7} s.\n\\end{equation}\nHence, for most radio pulsars the radiation force indeed can be neglected.\nAs to the pulsars with $\\sigma > \\sigma_{\\rm cr}$,\nit is clear that for $\\gamma > \\sigma_{\\rm cr}$ the radiation force\nbecomes larger than the electromagnetic one and strongly inhibits\nany further acceleration.\nAs a result, we can evaluate the maximum gamma--factor which can\nbe reached during the acceleration as\n\\begin{equation}\n\\gamma_{\\rm max} \\approx \\sigma_{\\rm cr} \\approx 10^6.\n\\end{equation}\n\n\\section{Discussion}\n\nThus, on a simple example it was demonstrated that for real\nphysical parameters of the magnetosphere of radio pulsars\n($\\sigma \\gg 1$ and $\\lambda \\gg 1$)\nthe one--fluid MHD approximation remains true in the whole region\nwithin the light surface $|{\\bmath E}| = |{\\bmath B}|$.\nOn the other hand, it was shown that in a more realistic 2D case\nthe main properties of the boundary layer near the light surface\nexisting for small enough longitudinal currents $I < I_{GJ}$\n(effective energy transformation from electromagnetic field to particles,\ncurrent closure in this region, smallness of the disturbance of\nelectric potential and poloidal magnetic field)\nremain the same as in the 1D case considered previously (Beskin et\nal 1983).\n\nIt is necessary to stress the main astrophysical consequences of our\nresults. First of all, the presence of such a boundary layer explains\nthe effective energy transformation of electromagnetic energy\ninto the energy of particles. As was already stressed, now the\nexistence of such an acceleration is confirmed by observations of\nclose binaries containing radio pulsars (as to the particle\nacceleration far from a neutron star, see e.g. Kennel \\& Coroniti\n1984, Hoshino et al 1992, Gallant \\& Arons 1994). Simultaneously,\nit allows us to understand the current closure in the pulsar\nmagnetosphere.\n Finally, particle acceleration results in the\nadditional mechanism of high--energy radiation from the boundary of\nthe magnetosphere (for more details see Beskin et al 1993).\n\nNevertheless, it is clear that the results obtained do not solve\nthe whole pulsar wind problem. Indeed, as in the cylindrical\ncase, it is impossible to describe the particle motion outside\nthe light surface. The point is that, as one can see directly from\nEqn. (\\ref{o2}), for a complete conversion of electromagnetic\nenergy into the energy of particles it is enough for them\nto pass only $\\lambda^{-1}$ of the total potential drop between\npulsar magnetosphere and infinity. It means that\n%(at any way in the axisymmetric case)\nthe electron--positron wind propagating\nto infinity has to pass the potential drop which is much larger\nthan their energy. It is possible only in the presence of\nelectromagnetic waves even in an axisymmetric magnetosphere which\nis stationary near the origin. Clearly, such a flow cannot be\nconsidered even within the two--fluid approximation. In our opinion,\nit is only a numerical consideration that can solve the problem\ncompletely and determine, in particular, the energy spectrum of\nparticles and the structure of the pulsar wind. Unfortunately, up\nto now such numerical calculations are absent.\n\n\\section*{Acknowledgments}\n\nThe authors are grateful to I.~Okamoto and H.~Sol for fruitful discussions.\nVSB thanks National Astronomical Observatory, Japan for hospitality.\nThis work was supported by INTAS Grant 96--154 and by Russian Foundation\nfor Basic Research (Grant 96--02--18203).\n%\n%\\newpage\n\n\\begin{thebibliography}{}\n\\bibitem{ar} Arons J., Scharlemann E.T., 1979, ApJ, 231, 854\n\\bibitem{bl} Begelman M.C., Li Z.-Y., 1994, ApJ, 426, 269\n\\bibitem{bgi1} Beskin V.S., Gurevich A.V., Istomin Ya.N., 1983, Soviet\nPhys. JETP, 58, 235\n\\bibitem{bgi} Beskin V.S., Gurevich A.V., Istomin Ya.N., 1993, Physics\nof the Pulsar Magnetosphere, Cambridge Univ. Press, Cambridge\n\\bibitem{bkr} Beskin V.S., Kuznetsova I.V., Rafikov R.R., 1998, MNRAS\n299, 341\n\\bibitem{b3} Beskin V.S., Malyshkin L.M., 1998, MNRAS 298, 847\n\\bibitem{bg7} Bogovalov S.V., 1997, A\\&A, 327, 662\n\\bibitem{d} Djorgovsky S., Evans C.R., 1988, ApJ, 335, L61\n%\\bibitem{b10} Fendt C., Camenzind M., Appl S., 1995, A\\&A, 300, 791\n\\bibitem{ga} Gallant Y.A., Arons J., 1994, ApJ, 435, 230\n\\bibitem{gj68} Goldreich P., Julian, W.H., 1969, ApJ, 157, 869\n\\bibitem{hn} Henriksen R.N., Norton J.A., 1975, ApJ, 201, 719\n\\bibitem{h1} Heyvaerts J., 1996, in Chiuderi C., Einaudi G., ed, Plasma\nAstrophysics, Springer, Berlin, p.31\n\\bibitem{h} Hoshino M., Arons J., Gallant Y.A., Langdon A.B., 1992, ApJ,\n390, 454\n\\bibitem{kc1} Kennel C.F., Coroniti, F.V., 1984, ApJ, 283, 694\n\\bibitem{k} Kulkarni S.R., Hester J., 1988, Nature, 335, 801\n\\bibitem{b13}Li Zh.--Yu., Chiueh T., Begelman M.C., 1992, ApJ, 394, 459\n\\bibitem{lm} Mestel L., 1999, Cosmical Magnetism, Clarendon Press, Oxford\n\\bibitem{ms} Mestel L., Shibata S., 1994, MNRAS, 271, 621\n\\bibitem{b16} Michel F.C., 1969, ApJ, 158, 727\n\\bibitem{mch73} Michel F.C., 1973, ApJ, 180, L133\n\\bibitem{mch91} Michel F.C., 1991, Theory of Neutron Star Magnetosphere, The\nUniv. of Chicago Press, Chicago\n\\bibitem{b19} Okamoto I., 1978, MNRAS, 185, 69\n\\bibitem{rs} Ruderman M.A., Sutherland P.G., 1975, ApJ, 196, 51\n\\bibitem{shb} Shibata S., 1997, MNRAS, 287, 262\n\\bibitem{t1} Tomimatsu A., 1994, Proc. Astron. Soc. Japan, 46, 123\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n"
}
] |
[
{
"name": "astro-ph0002525.extracted_bib",
"string": "\\begin{thebibliography}{}\n\\bibitem{ar} Arons J., Scharlemann E.T., 1979, ApJ, 231, 854\n\\bibitem{bl} Begelman M.C., Li Z.-Y., 1994, ApJ, 426, 269\n\\bibitem{bgi1} Beskin V.S., Gurevich A.V., Istomin Ya.N., 1983, Soviet\nPhys. JETP, 58, 235\n\\bibitem{bgi} Beskin V.S., Gurevich A.V., Istomin Ya.N., 1993, Physics\nof the Pulsar Magnetosphere, Cambridge Univ. Press, Cambridge\n\\bibitem{bkr} Beskin V.S., Kuznetsova I.V., Rafikov R.R., 1998, MNRAS\n299, 341\n\\bibitem{b3} Beskin V.S., Malyshkin L.M., 1998, MNRAS 298, 847\n\\bibitem{bg7} Bogovalov S.V., 1997, A\\&A, 327, 662\n\\bibitem{d} Djorgovsky S., Evans C.R., 1988, ApJ, 335, L61\n%\\bibitem{b10} Fendt C., Camenzind M., Appl S., 1995, A\\&A, 300, 791\n\\bibitem{ga} Gallant Y.A., Arons J., 1994, ApJ, 435, 230\n\\bibitem{gj68} Goldreich P., Julian, W.H., 1969, ApJ, 157, 869\n\\bibitem{hn} Henriksen R.N., Norton J.A., 1975, ApJ, 201, 719\n\\bibitem{h1} Heyvaerts J., 1996, in Chiuderi C., Einaudi G., ed, Plasma\nAstrophysics, Springer, Berlin, p.31\n\\bibitem{h} Hoshino M., Arons J., Gallant Y.A., Langdon A.B., 1992, ApJ,\n390, 454\n\\bibitem{kc1} Kennel C.F., Coroniti, F.V., 1984, ApJ, 283, 694\n\\bibitem{k} Kulkarni S.R., Hester J., 1988, Nature, 335, 801\n\\bibitem{b13}Li Zh.--Yu., Chiueh T., Begelman M.C., 1992, ApJ, 394, 459\n\\bibitem{lm} Mestel L., 1999, Cosmical Magnetism, Clarendon Press, Oxford\n\\bibitem{ms} Mestel L., Shibata S., 1994, MNRAS, 271, 621\n\\bibitem{b16} Michel F.C., 1969, ApJ, 158, 727\n\\bibitem{mch73} Michel F.C., 1973, ApJ, 180, L133\n\\bibitem{mch91} Michel F.C., 1991, Theory of Neutron Star Magnetosphere, The\nUniv. of Chicago Press, Chicago\n\\bibitem{b19} Okamoto I., 1978, MNRAS, 185, 69\n\\bibitem{rs} Ruderman M.A., Sutherland P.G., 1975, ApJ, 196, 51\n\\bibitem{shb} Shibata S., 1997, MNRAS, 287, 262\n\\bibitem{t1} Tomimatsu A., 1994, Proc. Astron. Soc. Japan, 46, 123\n\n\\end{thebibliography}"
}
] |
astro-ph0002526
|
Absorption-Line Probes of Gas and Dust in Galactic Superwinds
|
[
{
"author": "Timothy M. Heckman$^1$"
}
] |
We have obtained moderate resolution ($R$ = a few thousand) spectra of the $NaI\lambda\lambda$5890,5896 ($NaD$) absorption-line in a sample of 32 far-IR-bright starburst galaxies. In 18 cases, the $NaD$ line in the nucleus is produced primarily by interstellar gas, while cool stars contribute significantly in the others. In 12 of the 18 ``interstellar-dominated'' cases the $NaD$ line is blueshifted by over 100 km s$^{-1}$ relative to the galaxy systemic velocity (the ``outflow sources''), while no case shows a net {redshift} of more than 100 km s$^{-1}$. The absorption-line profiles in these outflow sources span the range from near the galaxy systemic velocity to a maximum blueshift of $\sim$ 400 to 600 km s$^{-1}$. The outflow sources are galaxies systematically viewed more nearly face-on than the others. We therefore argue that the absorbing material consists of ambient interstellar material that has been entrained and accelerated along the minor axis of the galaxy by a hot starburst-driven superwind. The $NaD$ lines are optically-thick, but indirect arguments imply total Hydrogen column densities of $N_H \sim$ few $\times 10^{21}$ cm$^{-2}$. This implies that the superwind is expelling matter at a rate comparable to the star-formation rate. This outflowing material is evidently very dusty: we find a strong correlation between the depth of the $NaD$ profile and the line-of-sight reddening. Typical implied values are $E(B-V)$ = 0.3 to 1 over regions several-to-ten kpc in size. We briefly consider some of the potential implications of these observations. The estimated terminal velocities of superwinds inferred from the present data and extant X-ray data are typically 400 to 800 km s$^{-1}$, are independent of the galaxy rotation speed, and are comparable to (substantially exceed) the escape velocities for $L_*$ (dwarf) galaxies. The resulting selective loss of metals from shallower potential wells can establish the mass-metallicity relation in spheroids, produce the observed metallicity in the intra-cluster medium, and enrich a general IGM to of-order 10$^{-1}$ solar metallicity. If the outflowing dust grains can survive their journey into the IGM, their effect on observations of cosmologically-distant objects would be significant.
|
[
{
"name": "ms.tex",
"string": "%\\documentstyle[aasms4,12pt,epsf]{article}\n%\\documentstyle[12pt,aasms4]{article}\n\\documentstyle[12pt,aaspp4,psfig]{article}\n\\def\\fnsiz{\\footnotesize}\n%\\tighten\n%\\eqsecnum\n \n\\lefthead{T. Heckman et al.}\n\\righthead{Absorption-Lines in Galactic Winds}\n \n\\slugcomment{.}\n \n\\begin{document}\n \n\\title{Absorption-Line Probes of Gas and Dust in Galactic Superwinds}\n \n\\author{Timothy M. Heckman$^1$}\n\\affil{Department of Physics and Astronomy, Johns Hopkins\nUniversity, Homewood Campus, 3400 North Charles Street, Baltimore, MD 21218}\n\\author{Matthew D. Lehnert$^1$}\n\\affil{Max-Plank-Institut f\\\"ur extraterrestrische Physik, Postfach 1603,\nD-85740 Garching, Germany}\n\\author{David K. Strickland}\n\\affil{Department of Physics and Astronomy, Johns Hopkins\nUniversity, Homewood Campus, 3400 North Charles Street, Baltimore, MD 21218}\n\\and\n\\author{Lee Armus$^1$}\n\\affil{SIRTF Science Center, 310-6, Caltech, Pasadena, CA 91125}\n \n\\parindent=0em\n\\vspace{4cm}\n \n1. Visiting astronomers, Kitt Peak National Observatory and Cerro Tololo\nInteramerican Observatory, NOAO,\noperated by AURA, Inc. under cooperative agreement with the\nNational Science Foundation.\n\n\\newpage\n \n\\parindent=2em\n \n\\begin{abstract}\nWe have \nobtained moderate resolution ($R$ = a few thousand) spectra of\nthe $NaI\\lambda\\lambda$5890,5896 ($NaD$) absorption-line in\na sample of 32 far-IR-bright starburst galaxies.\nIn 18 cases,\nthe $NaD$ line in the nucleus is produced primarily by interstellar gas,\nwhile cool stars\ncontribute significantly in the others.\nIn 12 of the 18 ``interstellar-dominated'' cases\nthe $NaD$ line is blueshifted by over 100\nkm s$^{-1}$ relative to the galaxy systemic velocity (the\n``outflow sources''), while no case shows a net\n{\\it redshift} of more than 100 km s$^{-1}$. \nThe absorption-line profiles in these outflow sources\nspan the range from near the galaxy systemic velocity\nto a maximum blueshift\nof $\\sim$ 400 to 600 km s$^{-1}$. The outflow\nsources are galaxies systematically viewed more nearly face-on than\nthe others. We therefore argue that the absorbing\nmaterial consists of ambient interstellar material that has been \nentrained and accelerated along the minor axis of the galaxy by \na hot starburst-driven superwind. The\n$NaD$ lines are optically-thick, but indirect arguments imply\ntotal Hydrogen column densities of $N_H \\sim$ few\n$\\times 10^{21}$ cm$^{-2}$. This implies that\nthe superwind is expelling matter\nat a rate comparable to the star-formation rate. \nThis outflowing material\nis evidently very dusty: we\nfind a strong correlation between the depth of the $NaD$ profile\nand the line-of-sight reddening.\nTypical implied values\nare $E(B-V)$ = 0.3 to 1 over regions several-to-ten kpc in size.\nWe briefly consider some of the potential implications of these observations.\nThe estimated terminal velocities of superwinds inferred from the\npresent data and extant X-ray data are typically 400 to 800 km s$^{-1}$,\nare independent of the galaxy rotation speed, and are comparable to \n(substantially exceed) the escape\nvelocities for $L_*$ (dwarf) galaxies. The resulting selective loss of metals\nfrom shallower potential wells can\nestablish the mass-metallicity relation in spheroids,\nproduce the observed metallicity in the intra-cluster medium,\nand enrich a general IGM to of-order 10$^{-1}$ solar metallicity.\nIf the outflowing dust grains can survive their journey into the IGM, \ntheir effect on observations\nof cosmologically-distant objects would be significant. \n\\end{abstract}\n \n\\keywords{\ngalaxies: starburst -- galaxies: nuclei --\ngalaxies: active --\ngalaxies: ISM -- galaxies: kinematics and dynamics -- galaxies: halos\n-- galaxies: intergalactic medium -- galaxies: evolution --\ninfrared: galaxies --\nISM: dust, extinction}\n \n\\newpage\n \n%\\twocolumn\n \n\\section{Introduction}\n\nBy now, it is well-established that galactic-scale outflows of gas (sometimes\ncalled `superwinds') are a ubiquitous phenomenon in the most actively star-\nforming galaxies in the local universe (Heckman, Lehnert,\n\\& Armus 1993; Dahlem 1997;\nBland-Hawthorn 1995). They are powered by the energy deposited in the\ninterstellar medium by massive stars via supernovae and stellar winds.\nOver the history of the universe, outflows like these may\nhave polluted the intergalactic medium with metals (e.g. Giroux \\& Shull\n1997) and dust (Alton, Davis, \\& Bianchi 1999; Aguirre 1999a,b), heated and\npolluted the intracluster medium\n(e.g. Gibson, Loewenstein, \\& \nMushotzky 1997; Ponman, Cannon, \\& Navarro 1999),\nand may have established the mass-metallicity relation and radial metallicity\ngradients in galactic spheroids (e.g. Carollo \\& Danziger 1994).\nHowever,\nthe astrophysical relevance of superwinds can not be\nreliably assessed without first understanding their physical, dynamical, and\nchemical properties. To date, most of the pertinent information has come\nfrom observations of the X-ray emission produced by the hot gas (e.g.\nDahlem, Weaver, \\& Heckman 1998) or the optical line-emission produced\nby the warm gas (e.g. Lehnert \\& Heckman 1996a).\n\nIn the present paper, we take a complementary approach, and discuss an\nextensive body of new data that probes the outflowing gas via its interstellar\nabsorption-lines. This technique has some important advantages. First, since\nthe gas is seen in absorption against the background starlight, there is no\npossible ambiguity as to the sign (inwards or outwards) of any radial flow\nthat is detected. Second, the strength of the absorption will be related to the\ncolumn density of the gas. In contrast, the X-ray or optical surface-brightness\nof the emitting gas is proportional to the emission-measure. Thus, the\nabsorption-lines more fully probe the whole range of gas densities in the\noutflow, rather than being strongly weighted in favor of the densest material\n(which may contain relatively little mass). Finally, provided that suitably-\nbright background sources can be found, interstellar absorption-lines can be\nused to study outflows in high-redshift galaxies where the associated X-ray\nor optical {\\it emission} may be undetectably faint. This promise has already\nbeen realized in the case of the 'Lyman Dropout' galaxies, where the\nkinematic signature of outflows is clear in their rest-frame UV spectra\n(Franx et al 1997; Pettini et al 1998, 1999).\n\nA few pioneering studies have already detected interstellar absorption-lines\nfrom superwinds in local starburst galaxies. Phillips (1993) discussed \nspatially-resolved optical spectroscopy of the $NaI$ ``D'' doublet in NGC 1808,\nshowing that an outflow of gas at velocities of up to 700 km s$^{-1}$ could\nbe traced over a region several kpc in size, coincident with a region of extra-\nplanar dust plumes. Several recent papers (Lequeux et al. 1995; Heckman \\&\nLeitherer 1997; Sahu \\& Blades 1997; Kunth et al. 1998; Gonzalez-Delgado\net al 1998a) have detected blueshifted interstellar absorption-lines in $HST$\nand $HUT$\nUV spectra of a handful of starburst galaxies, implying outflows of metal-\nbearing gas at velocities of 10$^2$ to 10$^3$ km s$^{-1}$.\n\nMost of the strong resonance lines of cosmically-abundant ions are found in\nthe UV (e.g. Morton 1991; Savage \\& Sembach 1996),\nand so must be studied with $HST$ or $FUSE$\nin local starbursts. In the present program, we have instead exploited the\nrelatively greater sensitivity and availability of ground-based telescopes at\nvisible wavelengths to study a large sample of starbursts/superwinds using\nthe $NaI$ doublet at $\\lambda$$\\lambda$5890,5896 \\AA. In a few cases,\nwe have also observed the $KI$ $\\lambda$$\\lambda$7665,7699 \\AA\\ doublet, since\nit probes gas with nearly the same ionization state as $NaI$, but is more\nlikely to be optically thin (the $K$ abundance is down from $Na$ by a factor\nof 15, while the two doublets have similar oscillator strengths). The\nionization potentials of $NaI$ and $KI$ are only 5.1 eV and 4.3 eV\nrespectively, so these species should primarily probe the $HI$ and $H_2$\nISM phases. Observations in vacuum-UV will be required to study the hotter\nand more highly ionized gas in absorption.\n\n\\section{The Data}\n\n\\subsection{Sample Selection}\n\nTable 1 lists the salient properties of the 32 objects in our sample. These\nobjects have been drawn from the larger samples of infrared-selected\ngalaxies studied by Armus, Heckman, \\& Miley (1989 - hereafter AHM) and\nby Lehnert \\& Heckman (1995 - hereafter LH95). The specific selection\ncriteria used in these in two programs are described in detail in these\nreferences. Briefly, AHM selected on the basis of far-IR flux and very warm\nfar-IR color-temperatures. LH95 selected on the basis of far-IR flux,\nmoderately warm far-IR color-temperatures, and high galaxy inclination (\ndisk galaxies seen within $\\sim$30$^{\\circ}$ of edge-on). The AHM and LH95\nsamples overlap in galaxy properties, but the former preferentially selects\nmore powerful and more distant objects.\n\nAHM measured the equivalent widths of the $NaD$ lines in their sample\nusing low-resolution spectra. The galaxies from AHM were selected for the\npresent program on the basis of the brightness of their nucleus at $\\sim$\n5900\\AA\\ and the equivalent width of $NaD$. The galaxies from LH95 had no\nprior measures of the $NaD$ line, and were selected based primarily on their\nproximity and availability at the time of the observations. \n\nOf the 32 objects, only 3 are classified as {\\it bona fide} AGN of the basis\nof their optical spectra: the type 2 Seyferts NGC7582 and Mrk273 and the\nhighly peculiar AGN IRAS11119+3257. No published classification exists\nfor IRAS10502-1843. The remaining objects are\noptically classified as $HII nuclei$ or $LINER's$, and are presumed to be\nprimarily powered by dusty starbursts (Lutz, Veilleux, \\& Genzel 1999). \nEven in the Seyfert galaxies NGC7582 and Mrk273, optical spectra show the\npresence of a young stellar population in the nucleus (Schmitt, Storchi-\nBergmann, \\& Cid-Fernadez 1999; Gonalzez-Delgado, Heckman, \\&\nLeitherer 2000). Thus, with the possible exception of IRAS11119+3257\nand IRAS10502-1843, the\nobjects in our sample all contain powerful starbursts that can drive superwinds.\n\n\\subsection{Observations}\n\nThe observations were undertaken during the period from 1988 through 1994\nusing three different facilities: the 4-meter Blanco Telescope with the\nCassegrain Spectrograph at $CTIO$, the 4-meter Mayall Telescope with the\nRC Spectrograph at $KPNO$, and the 2.5-meter Dupont Telescope with the\nModular Spectrograph at the Las Campanas Observatory. Various\nspectrograph configurations were used at each observatory, the details of\nwhich are listed in Table 2. The spectral resolution used to study the \n$NaD$ lines ranged from 55 to\n170 km s$^{-1}$. While low by the standards of interstellar absorption-line\nstudies, the resolution was good enough to cleanly resolve the $NaD$ lines\nin most cases (deconvolved line widths of 100 to 600 km s$^{-1}$ - see\nbelow).\n\n\\subsection{Data Reduction \\& Analysis}\n\nThe spectra were all processed using the standard {\\it LONGSLIT} package\nin $IRAF$. All the data were bias-subtracted using the overscan region of\nthe chip and then flat-fielded using observations of either a quartz-lamp-\nilluminated screen inside the dome or of a quartz lamp inside the\nspectrograph. The spectra were then rectified using observations of bright\nstars to determine and remove the distortion perpendicular to the dispersion\ndirection and observations of a $HeNeAr$ arc lamp to determine the two-\ndimensional dispersion solution. The zero-points in the wavelength scale\nwere verified by measuring the wavelengths of strong night-sky emission-\nlines. Corrections to the heliocentric reference-frame were computed for the\nspectra. \nThe spectra were then sky-subtracted by\ninteractively fitting a low-order polynomial along the spatial direction,\ncolumn-by-column. For a few of the galaxies we obtained relatively\nlow-dispersion spectra\nthat will be used to measure the reddening (Balmer decrement and continuum\ncolor). \nThese\ndata were flux-calibrated using observations of spectrophotometric\nstandard stars, and otherwise were reduced in the same way as the other data.\n\nThe spectra were analyzed using the interactive {\\it SPLOT} spectral fitting\npackage in $IRAF$. In all cases, a one-dimensional `nuclear' spectrum was\nextracted, covering a region with a size set by the slit width and summed over\n3 to 5 pixels\nin the spatial direction (typically 2 by 3 arcsec). The\ncorresponding linear size of the projected aperture is generally a few hundred\nto a few thousand parsecs in these galaxies (median diameter 700 pc). \nThis is a reasonable match to the typical sizes of powerful starbursts\nlike these (e.g. Meurer et al 1997; Lehnert \\& Heckman 1996b).\nPrior to further analysis, each 1-D spectrum was normalized to unit intensity by\nfitting it with, and then dividing it by, a low-order polynomial. These nuclear\nspectra are shown in Figure 1. Similar one-dimensional spectra for off-\nnuclear regions were extracted over the spatial region with adequate signal-\nto-noise in the continuum for each galaxy. The primary focus of the present\npaper is on the nuclear spectra, but we will describe the results obtained in\nthe off-nuclear bins when these are particularly illuminating or interesting.\n\nGiven the relatively low resolution of our spectra, the likelihood that the\nobserved $NaD$ line profile contains many unresolved and/or blended kinematic\nsub-components, and the saturated nature of the $NaD$ lines, we have\nchosen to parameterize the lines as simply as possible. Thus, for each\nextracted spectrum, we have fit the $NaD$ doublet with a single pair of\nGaussians, constrained to have the same line width and a wavelength\nseparation appropriate to the redshifted doublet. In a few objects, the\nadjacent \n$HeI\\lambda$5876 nebular emission-line was strong and broad enough to\nslightly contaminate the blue half of the $NaD\\lambda$5890 profile. For\nthese cases, we first fit and subtracted the $HeI$ \nemission line.\nOnly the parameters of the stronger member of $KI$ doublet at 7665 \\AA\\\nwere measured. The weaker member at 7699 \\AA\\ was detected, with a\nstrength consistent with the doublet being optically-thin (i.e. an equivalent\nwidth ratio of $\\sim$2:1).\n\nWe have not attempted a rigorous determination of the measurement uncertainties\nassociated with these data. The relatively high signal-to-noise in\nthe nuclear spectra (typically better than 30:1 per pixel) means\nthat the uncertainties in the measured quantities will be\ndominated in most cases by systematic effects due to the contamination of the\n$NaD$ line by weaker stellar photospheric features\n(whose ubiquity likewise\nmakes it difficult to determine the true continuum level to use\nin the line-fitting) and by the mismatch in profile shape between the\nactual data and the single Gaussian\ncomponent used to fit each member of the doublet (see section 3.2 below). \nThe most straightforward way to estimate the measurement\nuncertainties is to compare\nthe measurements for the 11 galaxies in the sample for which we have more\nthan one independent spectrum (taken at a different position angle).\nWe have done so, and the results are reported in the Notes to Table 3.\n\n\\section{Results}\n\n\\subsection{The Stellar {\\it vs.} Interstellar Contribution}\n\nBefore using the $NaD$ line to diagnose conditions in the starburst galaxies,\nit is imperative to establish that the line is primarily interstellar\nin origin in\nthese galaxies. The $NaD$ line is strong in the spectra of cool stars, reaching\na peak strength in the range from K3 through M0 (see Jacoby, Hunter, \\&\nChristian 1984). These stellar types can make a significant contribution to the\noptical spectrum of a starburst galaxy. First, the oldest underlying population\nin the galaxy bulge will have a dominant contribution from K -type giants\n(indeed the $NaD$ line is one of the strongest stellar absorption-lines in\noptical spectra of early-type galaxies and bulges - e.g. Heckman 1980; Bica\net al 1991). Second, for starbursts with ages greater than about 10 Myr, cool\nsupergiants make a significant contribution to the optical and near-IR light\n(Bruzual \\& Charlot 1993; Leitherer et al 1999).\n\nWe have therefore tried to estimate empirically what fraction of the measured\nequivalent width of the $NaD$ doublet is contributed by late-type stars in\nour sample galaxies. To do so, we have considered other absorption-lines that\nare conspicuous in the spectra of late-type stars and galactic nuclei, but which\narise from highly-excited states and are therefore of purely stellar origin\n(i.e. they are not resonance lines like $NaD$). The best-studied example is the\n$MgI$ b-band at 5174\\AA. The strength of this line is well-correlated with\nthe strength of the $NaD$ line in spectra of the nuclei of normal galaxies (\nBica et al 1991; Heckman 1980) and in stars (Jacoby, Hunter, \\& Christian\n1984). We have used the latter two data sets to determine a best-fit to the\ncorrelation: $W_{NaD}$ $\\sim$ 0.75 $W_{Mg-b}$. The measured strength\nof $Mg-b$ in our galaxies (from AHM, Veilleux et al 1995, or\nour own unpublished spectra) was then used to predict the equivalent width\nof the {\\it stellar} contribution to the observed $NaD$ line. We have also\ncompared our data to spectra of K giant stars obtained during the same\nobserving runs listed in Table 2. Rather than measuring the strengths of a few\nparticular stellar features, we have used the entire ensemble of features in the\nrange between about 5750 and 6450 \\AA\\ to estimate by-eye the fractional\nstellar contribution to the $NaD$ line. The agreement between these two\nmethods is generally satisfactory (the predicted $NaD$ stellar equivalent\nwidths agree on-average to $\\sim$ 0.1 dex). Heckman \\& Lehnert (2000)\nhave measured the fraction of the red continuum contributed by cool\nstars for the seven nuclei in the present sample\nhaving the highest quality detection of the {\\it interstellar}\ncomponent of the $NaD$ line. They find that this fraction is 20 to 30\\%,\nconsistent with the rough estimates reported here.\n\nAs listed in Table 3, the estimated stellar contribution to the observed $NaD$\nline in our sample galaxies ranges from negligible ($<$10\\%) to substantial\n($>$70\\%), with hints of a bimodal distribution. Thus, rather than attempting\na very uncertain direct correction for the effects of the stellar contribution,\nwe have taken the simpler approach of dividing our sample into two bins: the\nstrong-stellar-contamination objects (`SSC') in which stars produce $\\geq$40\\% \nof the measured $NaD$ equivalent-width, and the interstellar-dominated\nobjects (`ISD') in which the stellar contribution is $\\leq$ 30\\%. In the\ndiscussion to follow, we will see that the $NaD$ lines in the two sub-samples\nhave significantly different properties, which can be readily understood as\nreflecting the relative importance of the stellar and interstellar components.\n\n\\subsection{Kinematics}\n\nThe most robust indicator of an outflow is the presence of interstellar\nabsorption-lines that are significantly blueshifted with respect the the\nsystemic velocity of the galaxy ($v_{sys}$). Thus, we have first compiled the\nbest available measures of $v_{sys}$ for our galaxies\nThe velocities of the\n{\\it nuclear} emission-lines are potentially affected by radial gas flows and\nare not always reliable indicators of $v_{sys}$ (Mirabel \\& Sanders 1988;\nLehnert \\& Heckman 1996a). We have therefore determined $v_{sys}$ from \n(in order of preference) spatially-resolved galactic rotation curves, global\nmm-wave $CO$ line profiles, nuclear stellar velocities, global\n$HI\\lambda$21cm {\\it emission-line} profiles, and optical nuclear emission-\nline velocities (only used for 4 objects). See Table 1 for details and\nthe estimated uncertainties.\n\nThe results are shown in Figure 2. For the $ISD$ subsample there is strong\ntrend for the centroid of the $NaD$ feature to be blueshifted with respect to\n$v_{sys}$. Specifically, 11 of the 18 $ISD$ nuclei have $NaD$ blueshifts\n$\\Delta$$v$ greater than 100 km s$^{-1}$ (hereafter the `outflow sources'). \nIn addition, while the nuclear $NaD$ absorption-line in NGC1808 lies close\nto $v_{sys}$, the galaxy exhibits strongly blueshifted absorption over a\nseveral-kpc-scale region along its minor axis (Phillips 1993). We therefore\ninclude it as a 12th member of the outflow sample. The net blueshifts in the\n$ISD$ nuclei are in the range $\\Delta$$v$ $\\sim$ 100 to 300 km s$^{-1}$,\n(with the exception of IRAS11119+3257).\n\\footnote{IRAS11119+3257 has perhaps the most peculiar optical spectrum of\nany ultra-luminous system. It shows very broad (1500 km s$^{-1}$)\nBalmer, [OIII]$\\lambda$$\\lambda$4959,5007, FeII, HeI, and\n[OI]$\\lambda$6300 emission-lines.\nIt appears to be a member of the ``I~Zw~1'' class of quasars (e.g. Phillips\n1976), or possibly related to Mrk 231.\nIt is very compact (barely resolved) in optical images\n(Armus, Heckman, \\& Miley 1987). The $NaD$ absorption profile is complex,\nwith a strong narrow system that is blueshifted by 934 km s$^{-1}$,\nand a weaker system blueshifted by 1410 km s$^{-1}$. See\nTable 3 and Figure 1.} \nIn contrast to these large blueshifts, no net {\\it redshifts}\ngreater\nthan 100 km s$^{-1}$ are observed in the $ISD$ sample. Moreover, none of\nthe 14 members of the $SSC$ sample show a net $NaD$ blueshift or redshift\nthat is greater than 70 km s$^{-1}$. This is consistent with expectations that\nthe velocity of the nuclear stellar $NaD$ component will be very close to\n$v_{sys}$.\n\nThe $NaD$ linewidths in the $ISD$ and $SSC$ subsamples are also\nsignificantly different (Figure 3). The lines are relatively narrow \nin the $SSC$ subsample ($W\n\\sim$ 100 to 300 km s$^{-1}$, with a median of 180 km s$^{-1}$),\nand much broader\nin the $ISD$ nuclei ($W$ $\\sim$ 150 to 600 km s$^{-1}$, with a median\nof 425 km s$^{-1}$). The lines are\nespecially broad (typically 400 to 600 km s$^{-1}$) in the outflow sources.\nAs shown in Figure 4, the net blueshift in these sources is typically about\nhalf the line width ($\\Delta$$v \\sim$ 1/2 $W$). The peculiar AGN\nIRAS11119+3257, with $W << \\Delta$$v$, is the\nnotable exception. Thus, in a typical outflow, the redmost absorption occurs\nclose to $v_{sys}$. This is strongly suggestive of a flow in which\nmatter is injected at roughly zero velocity and then accelerates outward. The\napproximate implied terminal velocity of the flow is then $v_{term} \\sim$\n$\\Delta$$v + 0.5W$, which ranges from 220 to 1450 km s$^{-1}$ in our\nsample (Table 4). This picture is quite different\nfrom the standard one of a simple expanding `superbubble' in which\nthe absorption is due to a thin layer of cooled post-shock gas, and\nfor which $W << \\Delta$$v$ would be expected (e.g. Weaver et al. 1977).\n\nIt is instructive to compare the observed velocities in the absorbing material\nto the velocities expected from purely gravitational forces in the starburst\ngalaxy. This is shown in Figure 5, where we plot $W$ {\\it vs.}\nthe galaxy rotation\nspeed ($v_{rot}$ - \nsee Table 1 for details). This figure has several interesting\nimplications. First, neither the sample as-a-whole nor any of the above\nsubsamples show any correlation between the velocity dispersion in the\nabsorbing material and the galaxy rotation speed. This suggests that gravity\ndoes not play a dominant role in determining the dynamics of the\nabsorbing gas.\n\nFigure 5 also shows that the $NaD$ lines are surprisingly narrow in the\n$SSC$ sources compared with expectations for either stars or gas in the\nbulge of the starburst `host' galaxy. The lines are exceptionally \nnarrow if they are stellar in origin, since in this case the observed\nline broadening ($W_{obs}$) will be produced by both the intrinsic stellar line\nbroadening ($W_*$) and that produced by galactic dynamics ($W_{gal}$):\n$W_{gal} = \\sqrt{W_{obs}^2 - W_*^2}$. The observed equivalent\nwidths of the $Na\\lambda$5890 line are in the range 2.45$\\pm$0.4 \\AA\\\nin the $SSC$ nuclei. If the absorption were purely stellar,\nthe {\\it minimum} required\nvalues for $W_*$ would be 125$\\pm$20 km s$^{-1}$ (corresponding to\ncompletely black stellar lines). The typical implied values for\n$W_{gal}$ in the $SSC$ sample would then be\n60 to 200 km s$^{-1}$, with a median value of 130 km s$^{-1}$.\n\nTo emphasize how narrow the lines are in the $SSC$ sample, we show in\nFigure 5 the empirical relation (Whittle 1992; Franx 1993) between the galaxy\nrotation speed and the bulge velocity dispersion as a function of\nHubble type for a sample of normal disk\ngalaxies. The values for $W_{obs}$ in the $SSC$ objects are on-average\n$\\sim$ 0.2 dex below this relation for normal galaxies of the same rotation\nspeed and\nHubble type (typically Sa to Sc), while the implied values for $W_{gal}$\nwould be even more discrepant (see above).\nPut another way, based on the Hubble\ntypes and the galaxy absolute magnitudes ($M_B$\n$\\sim$ -19 to -21) for the $SSC$ subsample, the Faber-\nJackson relation for normal galactic bulges would predict typical values\nof $W_{gal}$ $\\sim$ 200 to 300 km s$^{-1}$ (e.g. Nelson \\& Whittle\n1996), while the observed widths are typically only 140 to 200 km s$^{-1}$,\neven without a correction for the line broadening due to $W_*$.\n\nThe nebular {\\it emission} lines are also narrow in the $SSC$ nuclei, as\nhas been shown to be more generally true for starbursts by Weedman (1983). \nIn this case, Lehnert \\& Heckman (1996b) showed that the narrowness of the\nnuclear emission-lines could be understood because the ionized gas was\nrotationally supported and did not fairly sample the galaxy rotation curve (it\nlies within the region of the galaxy with solid-body rotation). If this\nexplanation applies to the $NaD$ lines in the $SSC$ nuclei, it implies that\na significant fraction of\nthe stellar contribution comes from a dynamically-cold (disk/starburst)\ncomponent rather than from the bulge.\n\nFinally, Figure 5 shows that the $NaD$ linewidths are relatively large in the\noutflow sources ($W$ $\\sim$ 1 to 3 $v_{rot}$). As we have argued above,\nthe kinematic properties of the $NaD$ profiles suggest that gas is `loaded'\ninto the outflow at $v \\sim v_{sys}$ and is then accelerated up to some\nterminal velocity that corresponds to the most-blueshifted part of the $NaD$\nline profile. We plot $v_{term}$ {\\it vs.} $v_{rot}$ in Figure 6, from which\nit is clear that $v_{term}$ is significantly larger than $v_{rot}$,\nbut is uncorrelated with it. This suggests that the outflows {\\it may} be able\nto selectively escape the shallower galactic potential wells, as we will \ndiscuss in section 4.2 below.\n\nNeither the $SSC$ nor the $ISD$ subsamples show a significant correlation\nbetween the widths of the $NaD$ absorption-line and the $H\\alpha$\nemission-line. In particular, the outflow sources with very broad (400 to 600\nkm s$^{-1}$) $NaD$ absorption-lines have $H\\alpha$ emission-line widths\nranging from 145 km\ns$^{-1}$ (NGC7552) to 1500 km s$^{-1}$ (IRAS11119+3257). This\npresumably means that the dynamics of the more tenuous outflowing absorbing \ngas is\nlargely decoupled from that of the dense (high emission-measure) gas that\nprovides most of the nuclear line-emission.\n\nAs described above, we have fit the profile of the $NaD$ doublet with a single\npair of Gaussians constrained to have the same widths and a fixed\nseparation. Inspection of Figure 1 clearly shows that the observed profiles\nof many of the $ISD$ sample are more complex than this. The $ISD$ profiles \ngenerally have a larger kurtosis than a Gaussian (i.e. narrower core and\nbroader wings) and are sometimes asymmetric with a weak blueward\nwing on the $\\lambda$5890 profile (e.g. NGC 1808, IRAS 10565+2448,\nIRAS 11119+3257, NGC 6240), and/or definite substructure \n(e.g. NGC 1614, NGC 3256, IRAS 11119+3257). Observations at higher\nspectral resolution should prove instructive.\n\n\\subsection{The Roles of Luminosity and Geometry}\n\nOf the 32 galaxies in our sample, 14 show relatively weak {\\it interstellar}\n$NaD$\nabsorption-lines (the SSC sample, in which the {\\it stellar} contribution to the\nline is strong), 6 have predominantly interstellar $NaD$ lines lying\nclose to the\nsystemic velocity of the galaxy, and 12 have interstellar lines that\nare blueshifted by more than 100 km s$^{-1}$ relative to $v_{sys}$.\nThese 12 outflow sources differ systematically\nfrom the other objects in two striking\nrespects: they are \nmore luminous starbursts and they are preferentially located in galaxies\nseen relatively face-on.\n\nSpecifically, 64\\% (9/14) of the \ngalaxies with $L_{IR} >$ 10$^{11}$ L$_{\\odot}$ show outflows, compared to only\n28\\% (5/18) of the less luminous galaxies. The mean values for $logL_{IR}$ are\n11.44$\\pm$0.18 and 10.86$\\pm$0.13 for the outflow and other sources\nrespectively, a difference that is significant at the 2.6 $\\sigma$ level.\nThe relationship to galaxy inclination is stronger:\n69\\% (11/16) of the galaxies\nwith a ratio of semi-major to semi-minor axes $a/b \\leq$ 2.0 show outflows,\nwhile this is true for only 6\\% (1/16) of the flatter (more highly inclined)\ngalaxies. The mean values for $log(a/b)$ are\n0.20$\\pm$0.03 and 0.42$\\pm$0.03 for the outflow and other sources\nrespectively, a difference that is significant at the 4.6 $\\sigma$ level.\n\nIt is likely that the primary correlation is between an observed outflow\nand low galaxy inclination (small $a/b$). The weaker apparent correlation\nwith $L_{IR}$ is probably induced by the \nloose anti-correlation in our sample between $L_{IR}$ and $a/b$. This anti-\ncorrelation \nreflects our selection of galaxies from both the LH95 `edge-on' galaxy \nsample (large $a/b$ and moderate $L_{IR}$) and the AHM `FIR-warm' sample\n(broad range in $a/b$ and large $L_{IR}$).\n\nTaken at face value, the correlation with galaxy inclination implies\nthat there is a high probability ($\\sim$70\\%) that an observer located\nwithin $\\sim$60$^{\\circ}$ of the rotation axis of a starburst galaxy will\nsee outflowing gas in absorption. This geometrical constraint is \nconsistent with the observed loosely-collimated outflows seen in emission\nalong the minor axes of edge-on starburst galaxies (e.g. Dahlem, Weaver,\n\\& Heckman 1998).\n\n\\subsection{Column Densities and Optical Depths}\n\nThe $NaD$ line is clearly optically-thick in these galaxies. The ratio of the\nequivalent widths of the $\\lambda$5890 and $\\lambda$5896 members of the\ndoublet ($R$) can be used to estimate the optical depth (e.g. Spitzer 1968). \nThe distribution of $R$ is markedly different in the $SSC$ and $ISD$\nsubsamples. In the former, there is a very narrow observed range ($R$\n$\\sim$ 1.1 to 1.3). This is consistent with a strong stellar contribution to\nthe\n$NaD$ line, since $R$ $\\sim$ 1.0 to 1.3 (indicative of large optical depths)\nis characteristic of cool stars. The\nrange is much broader for the $ISD$ sample, from $R$ = 1.1 to 1.7. This\nrange corresponds to central optical depths in the $\\lambda$5896 line of\n$\\tau \\sim$ 20 to 0.5.\n\nAt first sight, it might appear odd that the $NaD$ line is optically-thick,\nyet is\nnot black at line center. This can be seen for the $ISD$ sample in Figure 7,\nwhere we have plotted $R$ {\\it vs.} the normalized residual intensity at the\ncenter of the\n$\\lambda$5890 feature: $I_{5890} = F_{5890}/ F_{cont}$ (with the respective\nfluxes measured at line center and in the adjacent continuum). There is a\nbroad range in $I_{5890}$ from 0.14 (nearly black) to 0.7. More tellingly,\nthere is no correlation between $R$ and $I_{5890}$ . This implies that the\nabsorbing gas does not fully cover the background continuum light, and that\n$I_{5890}$ is determined more by this covering factor ($C_f$) than by the\noptical depth. A covering factor less than unity is natural in these galaxies.\nFirst,\nthe continuum light may arise in part from stars in the galaxy that are\nlocated in\nfront of most of the absorbing gas (i.e. this is not the idealized case of a\npurely foreground absorbing screen: the gas and stars are likely to be mixed).\nSecondly, the gas is likely to be quite clumpy and inhomogeneous (e.g.\nCalzetti 1997; Gordon, Calzetti, \\& Witt 1997).\n\nIn the limit of large optical depth,\n$C_f = (1 - I_{5890})$, but for low or moderate optical\ndepth $C_f > (1 - I_{5890}$). For the typical optical depths in this sample,\nwe can approximate $C_f$ by (1 - $I_{5890}$).\nThis can be demonstrated quantitatively for those members of\nthe $ISD$ sample in which the $NaD$ lines are well-resolved, narrow enough\nso that the two doublet members are cleanly separated from one-another\n($W <$ 300 km s$^{-1}$), and that have high signal-to-noise spectra. These \nconstraints leave us with only three objects: NGC1808, NGC2146, and M82.\nFollowing Hamann et al. (1997) and Barlow \\& Sargent (1997), we have:\n\n\\begin{equation}\nC_f = (I_{5896}^2 - 2I_{5896} + 1)/(I_{5890} - 2I_{5896} + 1)\n\\end{equation}\n\nwhere $I_{5896}$ is the normalized intensity at the center of the \n$\\lambda$5896\nline. The measured values of $C_f$ are 0.83, 0.84 and 0.84 for\nNGC1808, NGC2146, and M82 respectively, while the corresponding values for\n$(1 - I_{5890})$ are 0.83, 0.82, and 0.82.\n\nIn this circumstance - in which optically-thick gas only partially covers the\ncontinuum source - the measured equivalent width of the $NaD$ doublet\n($EQ$) will be insensitive to the $NaI$ column density, and will instead be\nprimarily determined by the product of $C_f$ and the line-of-sight\nvelocity dispersion in the gas.\nWe plot the separate dependences of $EQ$ on\n$I_{5890}$ and $W$ in Figures 8 and 9 respectively for the $ISD$ sample.\nIt is clear from these two figures that $EQ$ is determined largely by the\ncovering factor (Figure 8), since there is no correlation between $W$ and\n$EQ$ (Figure 9).\n\nGiven that the $NaD$ doublet is moderately optically-thick in these\ngalaxies, it is not straightforward to estimate a $NaI$ column density\n($N_{NaI}$). We have taken three approaches, and emphasize that these are\ndesigned to give us only a rough (order-of-magnitude) estimate. Our\ntechniques can potentially underestimate $N_{NaI}$, because they are\ninsensitive to any $NaI$ sub-component that is highly optically-thick, yet\nkinematically quiescent.\n\nThe first is the classical doublet ratio method (e.g. Spitzer 1968), which\nrelates $R$ directly to the optical depth at line center, and thereby allows the\ncolumn density to be deduced from the equivalent width. In the spirit of this\nanalysis, we will not attempt to measure columns for all the individual cases,\nbut will instead estimate a characteristic value based on the typical observed\nparameters. The median value observed in the $ISD$ sample is $R \\sim$ 1.2,\nimplying that the corresponding median optical depth at the center of the\n$NaD\\lambda$5896 line is $\\tau_{5896}$ $\\sim$ 4 \n(see Table 2.1 in Spitzer 1968).\nThe median\nobserved value $EQ \\sim$ 6 \\AA\\ for the doublet, equation 2-41 and Table\n2.1 in Spitzer (1968), and the oscillator strength from Morton (1991),\ntogether imply $N_{NaI}$ $\\sim$ 10$^{14}$ cm$^{-2}$. Note that this assumes\n$C_f$ = 1, and should be increased by $C_f^{-1}$, or a typical\nfactor of $\\sim$ 1.6. \n\nA variant of the doublet-ratio technique can be applied to the three cases\ndiscussed above\nin which the two members of the $NaD$ doublet are cleanly separated\nand well-resolved (NGC1808, NGC2146, and M82). Again, following \nHamann et al (1997) we have:\n\n\\begin{equation}\n\\tau_{5896} = ln[C_f/(I_{5896} + C_f -1)]\n\\end{equation}\n\nThe resulting values for $\\tau_{5896}$ are 2.3, 2.1, and 1.9 for\nNGC1808, NGC2146, and M82 respectively. These are smaller than\nthe values implied by $R$ by a factor of $\\sim$ 2 in these cases. \nThe implied values for\n$N_{NaI}$ are 1.0 $\\times$ 10$^{14}$, 6 $\\times$ 10$^{13}$,\nand 6 $\\times 10^{13}$ cm$^{-2}$ after\ncorrection by $C_f^{-1}$.\n\nWe have also measured the equivalent width of the $KI\\lambda$7665 line\nin three of the nuclei (NGC1614, NGC1808, and NGC3256). Since $KI$ and\n$NaI$ have very similar ionization potentials, and since $K$ and $Na$ show\nsimilar grain depletion patterns (Savage \\& Sembach 1996), the expected\nratio of the $NaI$ and $KI$ column densities should be 15 for gas with a\nsolar Na/K ratio. \nThe measured values for the $NaD$ doublet ratio imply optical depths at\nthe center of the $NaD\\lambda$5890 line of 8, 16, and 1.6 for NGC1614,\nNGC1808, and NGC3256 respectively. The implied optical depths for the\n$KI\\lambda$7665 line would then be 0.5, 1.1, and 0.1 respectively. Using the\noscillator strength tabulated by Morton (1991), the measured equivalent\nwidths of the line imply that $N_{KI}$ = 3 $\\times$ 10$^{12}$, 4 $\\times$\n10$^{12}$, and 1.3 $\\times$ 10$^{12}$ cm$^{-2}$ respectively. Assuming\n$N_{NaI}$ = 15 $N_{KI}$, the corresponding $NaI$ columns are 4.5 $\\times$\n10$^{13}$, 6 $\\times$ 10$^{13}$, and 2 $\\times$ 10$^{13}$\ncm$^{-2}$. These values are about a factor of two or three smaller than\nwould have been deduced for these three cases using the $NaD$ doublet\nratio alone. The value for NGC1808 is in good agreement with that derived\nfrom Equation 2.\nUnder the circumstances, we regard the agreement between the three methods\nas satisfactory, and\nconclude that the typical value in the $ISD$ sample is $logN_{NaI}$ =\n13.5 to 14. A final indirect indication that these $NaI$\ncolumn densities are roughly correct comes from the detections\nof the ``Diffuse Interstellar Bands'' in the seven highest-quality\nspectra of the $ISD$ sample (Heckman \\& Lehnert 2000). The observed\nstrengths of these features agree with the strengths seen in Galactic\nsight-lines with $logN_{NaI} \\sim$ 13.5 to 14. \n\nWhat is the {\\it total} gas column density associated with the outflow? To\ncalculate this directly from the (already uncertain) $NaI$ column requires\nknowing the metallicity of the gas, the fractional depletion of $Na$ onto\ngrains (typically a factor of $\\sim 10$ in diffuse clouds in the Milky Way)\nand the potentially substantial ionization correction to account for ionized\n$Na$. Assuming solar $Na$ abundances and a factor of ten correction for\ndepletion onto grains (e.g Savage \\& Sembach 1996),\n$N_{NaI}$ = 10$^{14}$ cm$^{-2}$ implies a typical\nvalue for $N_H$ of 5 $\\times$ 10$^{20}$ ($N_{Na}/N_{NaI}$) cm$^{-2}$.\nWe can also take an empirical approach suggested by the correlation between\n$N_{NaI}$ and the total gas column towards stars in our own Galaxy. Using\nthe data in Herbig (1993), values for $N_{NaI}$ in the range we estimate\n($logN_{NaI} \\sim$ 13.5 to 14) correspond to sight-lines with\n$N_H$ $\\sim$ 1.5 to 4 $\\times$ 10$^{21}$ cm$^{-2}$. Interestingly,\nthis is just the range of values for $N_H$ deduced from the amount\nof reddening along the line-of-sight to these nuclei based on either the\nBalmer decrement or the colors of the optical continuum, assuming\na normal Galactic extinction-curve and dust-to gas ratio (\nsection 3.5, and see also AHM; Veilleux et\nal 1995).\n\nThese estimates suggest that the ionization correction factor is significant but\nnot huge (i.e. $N_{Na}/N_{NaI} \\sim$ 3 to 10). Since its ionization potential\nis only 5.1 eV, the presence of relatively significant amounts of $NaI$ implies that it is\nassociated with gas having a significant dust optical depth in the near-UV:\nfor a Galactic extinction curve and dust-to-gas ratio, a Hydrogen column\ndensity of $N_H = 8 \\times 10^{20}$ cm$^{-2}$ is required to produce\n$\\tau_{dust} = 1$ at 5.1 eV ($\\lambda \\sim$ 2420 \\AA).\n\nIt is instructive to compare the total column densities we infer for the\noutflows of a few $\\times$ 10$^{21}$ cm$^{-2}$ to the column densities\nin the other components of the ISM in these galaxies. Column densities to the\nnucleus for the hot X-ray-emitting gas are estimated to be of-order\n10$^{21}$ cm$^{-2}$ in superwinds (e.g. Suchkov et al 1994; Heckman et\nal 1999; Strickland 1998). In the nuclei themselves, the dominant ISM component\nis molecular, and the inferred columns range from $\\sim$ 10$^{23}$ to\n10$^{25}$ cm$^{-2}$ (e.g. Sanders \\& Mirabel 1996).\n\nThe $HI\\lambda$21cm line is observed in absorption against the bright\nnonthermal radio sources in starburst nuclei (e.g. Koribalski 1996; Heckman\net al 1983; Mirabel \\& Sanders 1988). The implied column densities are\ntypically a few $\\times$ 10$^{21}$ to 10$^{22} (T_{spin}/100K)$ cm$^{-\n2}$. The absorption is centered close to $v_{sys}$ and spans a velocity\nrange similar to that of the molecular gas. This strongly suggests that this gas\nis a trace atomic component in the starbursting molecular disk or ring. \nThe kinematics of the gas responsible for the $\\lambda$21cm absorption are\ntherefore quite distinct from the gas that produces the blueshifted $NaD$\nabsorption. This has several plausible explanations. First, the outflowing\n$HI$ is probably too hot ($T > 10^3$ K) to produce strong absorption at\n$\\lambda$21cm. Second, the background radio continuum source against which\nthe gas that produces the $\\lambda$ 21cm absorption is observed will almost\ncertainly be invisible in the optical: it lies behind a total column density\n(overwhelmingly $H_2$) of $\\sim$ 10$^{23}$ to 10$^{25}$ cm$^{-2}$,\ncorresponding to $A_V$ = 60 to 6000! Clearly, such material will not\ncontribute to the observed $NaD$ absorption-lines.\n\n\\subsection{Dust Associated with the Absorbing Gas}\n\nWe have argued in section 3.4 that the $NaD$ lines are\nare optically thick, and that $EQ$ is set primarily by \nthe covering fraction for the absorbing gas ($C_f$) \nrather than by the line width ($W$ - see Figures 8 and 9).\nThis inference helps explain the otherwise puzzling correlations found by\nAHM and Veilleux et al (1995) between $EQ$ and the reddening inferred\nfrom either the Balmer decrement or the color of the optical continuum\nin the nuclear spectra of large samples of starbursts.\nFor $\\tau_{NaD}$ $>>$ 1 and $C_f$ = 1, $EQ$ would be set\nby $W$, and so no correlation with the reddening would be expected. If\ninstead $EQ$ is principally determined by the fraction of the starburst that is\ncovered by gas containing $NaI$ (and dust grains), then this correlation is\nmore reasonable. \n\nVeilleux et al (1995) have also shown (via the Balmer decrement)\nthat the region of significant reddening extends far beyond the\nnucleus in many far-IR-bright galaxies.\nThus, to gain further insight into the relationship between the\n$NaD$ absorption\nand dust-reddening, we have mapped out the spatial variation in the\ndepth of the $NaD$ line ($I_{5890}$) and the reddening\nin the six galaxies in our $ISD$ sub-sample for which we have the relevant\ndata on the reddening (M82, NGC3256, NGC6240, Mrk273, IRAS03514+1546,\nand IRAS10565+2448). The size of the region mapped was set by the\ndetectability of the $NaD$ line, and ranges from\n3 to 9 kpc (except for M 82, where the mapped region is only 500 pc\nin diameter).\nIn each case, we have corrected\nthe H$\\alpha$ and H$\\beta$ emission-line fluxes for the effects of stellar\nabsorption-lines\n(using measures of the equivalent widths of the high-order\nstellar Balmer absorption-lines in NGC 3256 and NGC 6240 and an assumed value\nof 2 \\AA\\ for the other galaxies).\nWe have also corrected the data for foreground reddening\nusing the measured Galactic $HI$ column\ndensity and assuming a standard extinction curve.\n\nFigure 10 shows that not only do the extra-nuclear data points for these six\ngalaxies define a good correlation between\nthe amount of reddening and the depth of the $NaD$ line\nalong a given line-of-sight through the starburst and its outflow,\nthey define the {\\it same} correlation\nas that defined by the ensemble of all the $ISD$ nuclei in our sample.\nThe nuclear and off-nuclear points are\npretty well-mixed in Figure 10, although there is some tendency for the nuclear\nlines-of-sight to have the larger values of reddening and deeper $NaD$\nabsorption-lines.\nThe correlation of $I_{5890}$ is better with the color of the stellar continuum\nthan with the Balmer decrement. This is reasonable because the \n$NaD$ line is observed in absorption against the background stellar continuum\n(rather than against the emission-line gas)\nand because the Balmer decrement is likely to be significantly affected\nby dust directly associated with the emission-line gas itself\n(in addition to the dust in the foreground material responsible for the $NaD$\nabsorption).\n\nThe observed Balmer decrements imply extinctions of $A_V$ $\\sim$ 1 to 5\nfor a standard Galactic extinction curve.\nSimilar values are implied by the continuum colors: a typical starburst\nis predicted (in the absence of reddening) to have a color of\n$log[C_{65}/C_{48}] \\sim$ -0.3 (Leitherer \\& Heckman 1995), while the observed\ncolors in Figure 10 range from $log[C_{65}/C_{48}] \\sim$ -0.2 to +0.3\n(corresponding to $A_V$ = 0.7 to 4.2). Note also that in both Figure 10a and\n10b, the extrapolation of the correlation to $I_{5890}$ = 1.0 (no\nabsorption, $C_f$ = 0) has an x-intercept at the intrinsic values expected\nfor an unreddened starburst\n($log[C_{65}/C_{48}] \\sim$ -0.3 and $log[H\\alpha/H\\beta]$\n= 0.46).\n\nIn summary, the data imply that over regions with sizes of several or many kpc,\nthe outflows contain inhomogeneous {\\it highly dusty} material. For a\nstandard Galactic extinction law and dust-to-gas ratio, the typical implied \n$HI$ columns are a few $\\times$ 10$^{21}$ cm$^{-2}$. These $HI$ column densities\nagree well with the estimates in section 3.4 above based upon the $NaI$\ncolumn density.\n\n\\subsection{Sizes, Masses, and Energies}\n\nWe have measured the size of the region over which\nsignificantly blueshifted $NaD$ absorption is detected ($\\Delta$$v$ $>$ 100\nkm s$^{-1}$) for the 12 outflow sources.\nThe sizes are listed in Table 4, and range from 1 to 10 kpc\nin diameter. They\nmust be regarded as lower limits (since the background starlight usually\nbecomes too faint to detect the absorption at larger radii). Tracing the full\nextent of the absorbing material farther out into the galactic halos will\nprobably require observing suitably bright background QSO's (see\nNorman et al 1996).\n\nThese lower limits to the size of absorbing region can be used to estimate the\n(minimum) mass and kinetic energy in the outflow. That is, for a region with\na surface area $A$, a column density $N_H$, and an outflow velocity\n$\\Delta$$v$:\n\n\\begin{equation}\nM > 5 \\times 10^8 (A/10 kpc^2) (N_H/3 \\times 10^{21} cm^{-2})\nM_{\\odot}\n\\end{equation}\n\n\\begin{equation}\nE > 2 \\times 10^{56} (A/10 kpc^2) (N_H/3 \\times 10^{21} cm^{-2}) (\\Delta\nv/200 km/s) erg\n\\end{equation}\n\nWe have scaled these relations using values for $A$, $N_H$, and\n$\\Delta$$v$ that are typical, and have assumed an equal contribution to $M$\nand $E$ from the front (observed) and back sides of the outflow.\n\nIf we adopt a simple model of a constant-velocity, mass-conserving superwind\nflowing\ninto a solid angle $\\Omega_w$, extending to arbitrarily large radii from some\nminimum radius ($r_*$ - taken to be the radius of the starburst within which\nthe flow originates), we obtain:\n\n\\begin{equation}\n\\dot{M} \\sim 60 (r_*/kpc) (N_H/3 \\times 10^{21} cm^{-2}) (\\Delta v/200 km/s)\n (\\Omega_w/4\\pi) M_{\\odot}/yr\n\\end{equation}\n\n\\begin{equation}\n\\dot{E} \\sim 8 \\times 10^{41} (r_*/kpc) (N_H/3 \\times 10^{21} cm^{-2})\n(\\Delta v/200 km/s)^3 (\\Omega_w/4\\pi) erg/s\n\\end{equation}\n\nThe statistics of the $ISD$ subsample in the present paper imply that\noutflows are commonly observed in absorption in IR-selected starbursts\n(12/18 cases). On the other hand, many of the outflow galaxies in the present\nsample were selected from AHM on the basis of the strength of their $NaD$\nline (objects above the $\\sim$ 70th percentile in $EQ$). If the presence of\nobservable blueshifted absorption is determined by viewing angle\n(see section 3.3), this\nsuggests that $\\Omega_w$/4$\\pi$ lies in the range $\\sim$ 0.2 to 0.6 (consistent\nwith the weakly-collimated bipolar outflows seen in well-studied\nsuperwinds).\n\nTo put the above estimates into context, we can consider the rate at which\nmass and energy are returned by massive stars. The median bolometric\nluminosity of the 12 outflow galaxies in our sample is $L_{bol} \\sim$ 2\n$\\times$ 10$^{11}$ L$_{\\odot}$. The implied median rates of mass and kinetic\nenergy returned from supernovae and stellar winds are roughly\n$\\dot{M_{ret}}$ = 5\nM$_{\\odot}$ per year and\n$\\dot{E_{ret}}$ = 10$^{43}$ erg s$^{-1}$ respectively (e.g.\nLeitherer \\& Heckman 1995). Since $(\\dot{M}/\\dot{M_{ret}}) \\sim$ 3 to 10, the\nabsorption-line gas in the outflow must be primarily ambient gas that has\nbeen loaded into the flow. This inference agrees with similar conclusions\nabout the hot X-ray-emitting gas in superwinds (section 4.2 below, and see\ne.g. Strickland 1998;\nSuchkov et al 1996; Heckman et al 1999).\nSince $(\\dot{E}/\\dot{E_{ret}}) <$ 10\\%, the absorbing gas does not\ncarry the bulk of the energy supplied by the starburst. Most of this energy\nprobably resides in the form of the thermal and kinetic energy of the much\nhotter ($T > 10^{5.5}$ K) X-ray-emitting gas.\n\nA bolometric luminosity of 2 $\\times$ 10$^{11}$ L$_{\\odot}$ corresponds\nto a star-formation rate of about 12 M$_{\\odot}$ per year (for a Salpeter IMF\nextending from 1 to 100 M$_{\\odot}$). Thus, the outflow rates estimated\nfrom the $NaD$ lines are comparable to the star-formation rate: {\\it the\nfeedback from massive stars drives the ejection of as much gas as is\nbeing converted\ninto stars.} Similar inferences for starbursts have been made using the X-ray\nand optical emission-line data (e.g. Suchkov et al 1996;\nHeckman et al 1999; Della Ceca et al\n1996,1999; Martin 1999).\n\n\\section{Discussion \\& Implications}\n\n\\subsection{The Origin \\& Dynamics of the Absorbing Material}\n\nAs discussed above, the red-most part of absorption-line profile in the\noutflow objects is close to $v_{sys}$, suggesting that absorbing material is\ninjected from quiescent material at or near $v_{sys}$, and is then\naccelerated up to some terminal velocity as it flows outward. This is\nphysically plausible, as the hot (X-ray emitting) outflowing gas interacts\nhydrodynamically with colder denser material that is located either inside the\nstarburst, or in the inner portions of the galactic halo (see for example\nHartquist, Dyson, \\& Williams 1997; Suchkov et al 1994; Strickland 1998).\n\nLet us assume that a cloud of gas with a column density $N$, originally\nlocated a distance $r_0$ from the starburst, is accelerated by a constant-\nvelocity superwind that carries an outward momentum flux $\\dot{p}$ into a solid\nangle $\\Omega_w$. Ignoring the effects of gravity for the moment, the clump's\nterminal velocity will be (Strel'nitskii \\& Sunyaev 1973):\n\n\\begin{equation}\nv_{term} = 420 (\\dot{p}/7 \\times 10^{34} dynes)^{1/2}(\\Omega_w/1.6\\pi)^{-\n1/2} (r_{0}/kpc)^{-1/2}(N/3 \\times 10^{21} cm^{-2})^{-1/2} km/s\n\\end{equation}\n\nIn this expression, we have used the momentum flux supplied by stellar\nwinds and supernovae (Leitherer \\& Heckman 1995) in a starburst having a\nbolometric luminosity equal to the median value for the outflow sample (2\n$\\times 10^{11}$ L$_{\\odot}$) and have adopted the estimates in section 3\nabove for $N$ and $\\Omega_w$. Starbursts with this luminosity have typical\nestimated radii of roughly 1 kpc (see for example Heckman, Armus, \\& Miley\n1990; Meurer et al 1997).\nFrom these elementary considerations, we conclude that the observed\nterminal velocities (typically 400 to 600 km s$^{-1}$) are easily\naccommodated.\n\nEquation 7 also predicts that the outflow speeds will be larger in more\nluminous starbursts. This trend is mitigated to some degree by the fact that\nmore powerful starbursts tend to be larger. Lehnert \\& Heckman (1996b) and\nMeurer et al (1997) argue that starbursts have a maximum characteristic\nsurface-brightness, which then implies $\\dot{p} \\propto L_{bol} \\propto r^2$.\nTogether with Equation 7, this implies that such `maximum starbursts' will\nhave $v_{term} \\propto L_{bol}^{1/4}$ (although the clouds can not be\naccelerated to velocities larger than that of the flow that accelerates them!).\nOur sample shows no convincing evidence of a trend for larger $v_{term}$\nin the more luminous systems, but this sample covers a rather small range in\nstarburst luminosity. It will be instructive to extend\nthis study to dwarf starbursts with $L_{bol} < 10^9$ $L_{\\odot}$.\n\nAssume now that the cloud immersed in the outflow is subjected to a\ngravitational force imposed by an isothermal galaxy potential whose depth\ncorresponds to a circular rotation speed $v_{rot}$. In order that the\noutwardly-directed force due to the superwind exceed the inwardly-directed force\nof gravity, the value for the cloud column density must satisfy the condition:\n\n\\begin{equation}\nN < 7 \\times 10^{21} (\\dot{p}/ 7 \\times 10^{34})(\\Omega_w/1.6 \\pi)^{-\n1}(r/kpc)^{-1} (v_{rot}/200 km/s)^{-2} cm^{-2}\n\\end{equation}\n\nThus, the typical column densities estimated for the outflows ($\\sim$ few\n$\\times$ 10$^{21}$ cm$^{-2}$) lie near the upper bound for material that\nwill flow out (rather than falling in). This may not be a coincidence: given a\nrange of cloud column densities, the blueshifted absorption-line will be\ndominated by the largest-column-density clouds that can be expelled.\nAlternatively, the observed column densities may simply arise because $N_H\n\\sim 2 \\times$ 10$^{21}$ cm$^{-2}$ corresponds to a dust optical depth of \nunity in the continuum\nat the wavelength of the $NaD$ doublet (for a standard Galactic dust-\ngas ratio). That is, continuum-emitting regions in these nuclei lying behind\nsight-lines with much higher columns are invisible in optical light and\nsources behind sight-lines with much lower columns contain little $NaI$. It\nwould be interesting to test Equation 8 by measuring values of $N$ for\noutflows in `low-intensity'\nstarbursts (small $\\dot{p}/r$) and starbursts occurring in\ndwarf galaxies (small $v_{rot}$). \n\n\\subsection{Insights from Numerical Simulations}\n\nThe above elementary considerations give some simple physical insights\ninto the origin and dynamics of the absorbing material. More detailed insight\ncomes from hydrodynamical simulations of starburst-driven superwinds\n(e.g. Tomisaka \\& Bregman 1993, Suchkov et al 1994, Strickland\n1998, Tenorio-Tagle \\& Munoz-Tunon 1998). In these simulations the coolest\ndensest gas that has been\nhydrodynamically disturbed\nby the starburst is associated with the swept-up shell of\nISM that propagates laterally in the plane of the galaxy,\nand fragments of the \ncap of the original superbubble shell now being carried vertically \nout of the disk by faster, more tenuous, wind material.\n \nShear between the hot shocked starburst ejecta\nand the cool dense shell in the disk of the galaxy leads to entrainment\nand stripping of cool dense gas into the wind flowing out of the\ndisk (through Kelvin-Helmholtz instabilities, and presumably additional\ninterchange processes such as thermal conduction and turbulent mixing\nlayers that can not be included in current simulations). Dense\ngas already in the wind interior, for example the superbubble shell fragments,\nis accelerated outward via the ram pressure of the\nwind.\n\nThis process can be seen in Figure 11, in which the outward trajectories\nof four typical entrained clouds are traced over an interval of 1.5 Myr\nfrom a\n2-D hydrodynamic simulation of M82's galactic wind (Strickland 1998). This is\nbased on the thick-disk ISM distribution of Tomisaka \\& Bregman (1993).\nIn this model a mass of 10$^8$ M$_{\\odot}$ is turned into stars\nin an instantaneous burst (Salpeter IMF over the mass range 1 to 100\nM$_{\\odot}$). At a time 7 Myr after the burst, the resulting wind\nhas properties that are a reasonable match to M82. By this time, supernovae\nand stellar winds have returned 1.3 $\\times$ 10$^7$ M$_{\\odot}$\nand 6 $\\times$ 10$^{56}$ ergs to the ISM.\n\nWithin $|z| \\leq 1.5$ kpc of the disk there is $M = 1.9 \n\\times 10^{8} M_{\\odot}$ of gas cooler than $T = 3 \\times 10^{5}$ K. \nThe majority of this gas is at the minimum temperature allowed \nin these simulations of $T \\sim 6 \\times 10^{4}$ K. This material\noccupies a projected area of $\\sim 2$ kpc$^{2}$, so the\naverage hydrogen column density of this gas is $N_{\\rm H} \\sim\n9 \\times 10^{21}$ cm$^{-2}$. This cool gas has a broad range of velocities,\nfrom $v \\sim 10$ -- $10^{3}$ km s$^{-1}$,\nwith a mode of $\\sim 60$ km s$^{-1}$\n(which is associated with the slow expansion of the outer shock in the\nplane of the galaxy). The associated kinetic energy is 1.3 $\\times$ 10$^{55}$\nergs. \n \nAt higher distances above the disk there is much less cool gas.\nFor gas above $|z| = 1.5$ kpc the mass of cool gas, associated\nprimarily with the fragments of superbubble shell cap, is\n$M = 1.5 \\times 10^{7} M_{\\odot}$. For a projected area of $\\sim$\n8.5 kpc$^{2}$, the average column density\nis $N_{\\rm H} \\sim 1.6 \\times 10^{20}$ cm$^{-2}$.\nThis cool gas high above the plane typically has higher\nvelocities than the gas within the plane of the galaxy, the\ndistribution of mass with velocity being approximately flat\nbetween velocities of $v = 10^{2}$ -- $10^{3}$ km s$^{-1}$. \nThe kinetic\nenergy associated with this gas is 1.7 $\\times$ 10$^{55}$ ergs.\n\nThus, the total mass of cool gas in the wind is 2 $\\times$ 10$^8$\nM$_{\\odot}$. This is twice as large as the mass of stars formed in\nthe burst, and is 16 times larger than the mass directly returned\nby massive stars. In contrast, the total kinetic energy in the cool\ngas (3 $\\times$ 10$^{55}$) is only 5\\% of the kinetic energy\nreturned by massive stars. The lion's share of this energy is the\nform of thermal and kinetic energy of the hot ($T > 10^{5.5}$ K) \ngas in the wind.\n\nIt is worth noting several of the limitations of these simulations\nwith respect to their treatment of the cool dense ISM:\nnone of these simulations can explicitly include the cool dense clouds\nof material within the ISM and starburst region that are known to\nexist, and are thought to play a key role in ``mass-loading'' the outflow\n(e.g. Hartquist, Dyson, \\& Williams 1997). As a result, the cool dense\ngas in these simulations is\nconfined to larger radii near the outer shock and to the shell fragments\n(e.g. there is no way of explicitly treating the entrainment of clouds from \nwithin the starburst region itself). \n\\footnote{The entrainment\nof material into the hotter, more tenuous phases, from the hydrodynamical\ndestruction of such clouds can and has been simulated, but these\n``mass-loaded'' simulations (Suchkov et al 1996, Strickland 1998)\ndo not consistently treat the properties of the clouds themselves.}\nThese simulations also have a minimum allowed gas temperature\nof $T \\sim 6 \\times 10^{4} {\\thinspace \\rm K}$, due to the method of simulating\nISM turbulent pressure support by an enhanced thermal pressure.\nHence all the gas that would in reality have lower temperature is forced\nto have this minimum temperature, which in turn affects the density\nof this gas, and prevents us from knowing the exact distribution of this\nmass between the gas phases cooler than this minimum temperature.\nSimilarly the processes of entrainment and acceleration of cool gas\ninto the wind are both uncertain and occur at (or below) the scale of\nthe physical resolution of these simulations.\n \nNevertheless, the results of these simulations are encouragingly\nsimilar to the observed \nparameters in our sample of starburst outflows: the cool gas is predicted\nto have column densities of several $\\times 10^{20}$ to\n$\\sim 10^{22} \\thinspace {\\rm cm}^{-2}$, a mass that is comparable to\nthat of the stars formed in the burst, outflow velocities\nin the range $v \\sim 10^{2}$ -- $10^{3} {\\thinspace \\rm km} \n{\\thinspace \\rm s}^{-1}$, and a kinetic energy that is of-order 10$^{-1}$ of\nthe total kinetic energy returned to the ISM by the starburst. Since\nthe cool gas was originally cold dense material entrained and accelerated\nby the hot outflow, the presence of substantial\namounts of dust\n(section 3.5) is perhaps not too surprising.\n\n\\subsection{The Fate of the Outflow \\& the Chemical Evolution of Galaxies}\n\nAs Figure 6 shows, the inferred terminal velocity in the outflows is typically\n400 to 600 km s$^{-1}$, or about two to three times larger than the rotation\nspeed of the starburst's host galaxy. Are these velocities sufficient to expel\nthe gas from the galaxy altogether, or will the gas return to the galactic disk\nas a fountain flow?\n\nFor an isothermal gravitational potential that extends to a maximum radius\n$r_{max}$, and has a virial velocity $v_{rot}$, the escape velocity at a\nradius $r$ is given by:\n\n\\begin{equation}\nv_{esc} = [2v_{rot}(1 + ln(r_{max}/r))]^{1/2}\n\\end{equation}\n\nThus, $v_{esc}$ = 3.0 $v_{rot}$ for $(r_{max}/r)$ = 33 (e.g. $r$ = 3 kpc\nand $r_{max}$ = 100 kpc). As shown in Figure 6, the estimated terminal\nvelocities in the outflows are typically $v_{term} \\sim$ 2 $v_{rot}$,\nbut $v_{term}$ is uncorrelated with $v_{rot}$. \n\nSimilar results have been obtained for the hot X-ray-emitting gas in starburst\ngalaxies. This gas has temperatures of a few to ten million K in dwarf\ngalaxies (e.g. Della Ceca et al 1996; Strickland, Ponman, \\& Stevens 1997),\n$L_*$ disk\ngalaxies (e.g. Dahlem, Weaver, \\& Heckman 1998; Read, Ponman, \\& Strickland\n1997),\nand extremely powerful\nstarbursts in galactic mergers (e.g. Heckman et al 1999; Moran, Lehnert, \\&\nHelfand 1999; Read \\& Ponman 1998). Martin (1999) has used these X-ray data to\nestimate that the gas will escape from galaxies with\n$v_{rot} <$ 130 km s$^{-1}$.\n\nWe can place these disparate data on common ground by comparing the\nkinetic ($NaD$) and thermal (X-ray) energy per particle to the energy\nneeded for escape. For convenience, we do so by defining an energetically-\nequivalent velocity for the X-ray gas. The terminal velocity in an adiabatic\nsuperwind fed by gas at a temperature $T_X$ will be \n$v_X \\sim (5kT_X/\\mu)^{1/2}$,\nwhere $\\mu$ is the mean mass per particle (Chevalier \\& Clegg 1985).\n\\footnote{\nThis is a conservative approach as it ignores any kinetic energy the\nX-ray-emitting gas may already have. Currently the velocity \nand kinetic energy of the X-ray-emitting material \nin superwinds can not be measured directly, but \nnumerical simulations suggest that the kinetic\nenergy of the hot gas is typically 2 to 3 times\nits thermal energy (Strickland 1998).}\n\nThese results are shown in Figure\n12, which includes the data from Fig. 6 plus 14\nfar-IR-bright galaxies for which analyses of broad-band ($\\sim$ 0.1 to 10 keV)\nX-ray data have been published. These are:\nM82, NGC253, NGC3628, NGC3079, NGC4631 (Dahlem, Weaver, \\& Heckman\n1998), NGC1569 (Della Ceca et al. 1996), NGC1808 (Awaki et al. 1996), \nNGC2146 (Della Ceca et al. 1999), NGC3256 (Moran, Helfand, \\& Lehnert\n1999), NGC3310 (Zezas, Georgantopoulos, \\& Ward 1998), NGC4038/4039 (\nSansom et al. 1996),\nNGC4449 (Della Ceca, Griffiths, \\& Heckman 1997),\nNGC6240 (Iwasawa \\& Comastri 1998), and Arp299 (Heckman et al 1999). \nIn the cases where two-temperature plasma models were fit to the X-ray data,\nwe have plotted\nboth the corresponding outflow velocities.\nThe agreement between the two data sets is satisfactory.\nThere are three members of the $NaD$ outflow sample with X-ray data in\nFigure 12, and the agreement between the $NaD$ terminal velocity and the X-\nray temperatures is reasonably good: $v_{term}$ = 700 km s$^{-1}$ {\\it vs.}\n$v_X$ = 520 and 780 km s$^{-1}$ for NGC1808, $v_{term}$ = 580 km s$^{-\n1}$ {\\it vs.} $v_X$ = 490 and 800 km s$^{-1}$ for NGC3256, and $v_{term}$ =\n580 km s$^{-1}$ {\\it vs.} $v_X$ = 700 and 940 km s$^{-1}$ for NGC6240.\nThis suggests that\nthe fastest-moving $NaD$ absorbers are roughly co-moving with the hot\nsuperwind fluid.\n\nFigure 12 strongly suggests that shallower galaxy potential wells\nwill be less able to retain the newly-synthesized metals that are\nreturned to the ISM in the aftermath of a starburst. As has been\nsuggested many times (e.g. Wyse \\& Silk 1985; Lynden-Bell 1992;\nKauffmann \\& Charlot 1998) the\nselective loss of metal-enriched gas from shallower potential\nwells could explain both the mass-metallicity relation and radial\nmetallicity gradients in elliptical galaxies \nand galaxy bulges (Bender, Burstein,\\& Faber\n1993; Franx\n\\& Illingworth 1990; Carollo \\& Danziger 1994;\nJablonka, Martin, \\& Arimoto 1996; Pahre, de\nCarvallo, \\& Djorgovski 1998; Trager et al 1998).\n \nA simple prediction of this idea would be that the relationship\nbetween metallicity and escape velocity should saturate\n(flatten) for the deepest potential wells - i.e. locations where\nthe local escape velocity exceeds the velocity of the outflowing\nmetal-enriched gas. Lynden-Bell (1992) has parameterized this in\na simple physically-motivated fashion by positing that the fraction of\nmetals produced by massive stars that\nare retained by the galaxy ($f_{retained}$) is proportional to the\ndepth of the galaxy`s potential well ($\\Phi$) for low-mass galaxies, but\nasymtotes to $f_{retained}$ = 1 for the most massive galaxies. We\nchose to cast his formulation as follows:\n\n\\begin{equation}\nf_{retained} = v_{esc}^2/(v_{esc}^2 + v_{term}^2)\n\\end{equation}\n\nHere $v_{term}$ is some characteristic velocity associated with the mixture\nof supernova (and stellar wind) debris and entrained gas that is ejected\nfrom the starburst.\nIt is assumed\nto be a constant. For low-mass galaxies with $v_{esc} << v_{term}$,\n$f_{retained} \\propto v_{esc}^2 \\propto \\Phi$, \nor $f_{retained} \\propto L_{gal}^{1/2}$\nvia the Faber-Jackson relation. Lynden-Bell shows that this simple\nformula can reproduce the observed mass-metallicity relation for elliptical\ngalaxies over a range of $\\sim$ 10$^6$ in galaxy mass,\nand finds that the characteristic mass at which $f_{retained} = 1/2$\n(e.g. a galaxy in which $v_{esc} = v_{term}$) corresponds to an elliptical\nwith $M_B$ $\\sim$ -18 (adjusted to our assumed value of $H_0$ = 70). Such a\ngalaxy would have a line-of-sight velocity dispersion\n$\\sigma \\sim$ 140 km s$^{-1}$,\ncorresponding to $v_{rot} = \\sqrt{2} \\sigma \\sim$ 200 km s$^{-1}$\n(Binney \\& Tremaine 1987). Using equation 9 above, this would correspond\nto $v_{esc} \\sim$ 600 km s$^{-1}$. This in turn is a reassuringly good\nmatch to the characteristic superwind outflow speeds implied by\nFigure 12 ($\\sim$ 400 to 800 km s$^{-1}$).\n\nThus, starburst-driven outflows might imprint a\nrelationship between metallicity and mass in ellipticals (and\nbulges) over most of observed ranges for these two parameters.\nWhile the loss of metal-enriched gas has the most\nsevere impact on dwarf elliptical galaxies, it may nevertheless have general\nsignificance in galaxy chemical evolution.\n\n\\subsection{The Metal-Enrichment of the Intergalactic Medium}\n\nThe data discussed in this paper directly establish the flow of {\\it metals}\nout of highly-actively-star-forming galaxies\nin the local universe, and the process is observed at high-redshift\nas well (Franx et al 1997; Pettini et al 1998, 1999).\nSuch data allow us to estimate the column densities, outflow rates,\nand outflow speeds of this material as a function of the rate of\nstar-formation. Meanwhile, over the past few years, the rate of high-mass\nstar-formation over the history of the universe has been measured\nfor the first time (e.g. Madau et al 1996; Steidel et al 1999;\nBarger et al 1999).\nThis emboldens us to attempt to estimate\nthe amount of metals that have flowed out of galaxies,\nthereby polluting the inter-galactic medium, over the course of cosmic time.\n\nThe discussion in section 3.4 above implies that gas is flowing out of\nstarbursts at a rate that is proportional to the rate of star-formation:\n$\\dot{M} = \\alpha\\dot{M_*}$ where $\\alpha$ is one-to-a-few (see also\nMartin 1999). The present-day mass in stars will be smaller than the total\nmass turned into stars, since mass has been subsequently \nreturned from these stars:\n$M_{*,0} = \\beta M_{*}$ where $\\beta \\sim$ 0.7 is reasonable for an old\npresent-day system like an elliptical or bulge. The discussion in section\n4.3 implies that the outflowing gas will be mostly retained by galaxies with\nthe deepest potential wells, but mostly lost by the less massive systems.\nIntegrating equation 10 above over a Schechter luminosity function\nimplies that $\\dot{M_{lost}} = \\gamma\\dot{M}$ with $\\gamma \\sim$ 0.5\n(depending on the value of $v_{term}$ and the `mapping' of\n$M_B$ to $v_{esc}$ in spheroids).\n\nWe then assume that over\ncosmic time, we can attribute the construction of spheroidal systems\n(elliptical and bulges) to starbursts (see Kormendy \\& Sanders 1992;\nElmegreen 1999; Renzini 1999; Lilly et al. 1999).\nIf we further assume\nthat {\\it all} star-formation in spheroids over the history of the universe\nejected\ngas at the relative rate seen in local starbursts, then the ratio of the mass\nof lost-gas to present-day stars in spheroids\nwould be $\\sim \\alpha\\gamma/\\beta$ (of-order unity). Fukugita, Hogan, \\& Peebles\n(1998 - hereafter FHP) estimate that the stars\nin spheroidal systems today comprise $\\Omega_{*,sph}$ = 2.6 $\\times$ 10$^{-3}$\n(for $H_0$ = 70). Since the\nimplied value for the gas expelled from forming spheroids is\ncomparable to this, this gas is therefore a significant repository\nof baryons, but only of-order 10$^{-1}$ of the total estimated baryonic content\nof the universe (FHP). In rich clusters,\nnearly the entire stellar mass resides in spheroidal systems, while the\ncluster potential well is deep enough to have retained all the mass\nexpelled by superwinds (e.g. Renzini 1997). The {\\it observed} average ratio\nof the mass of the intra-cluster medium to the stellar mass is $\\sim$ 6\n(FHP), so gas ejected by superwinds during spheroid formation would\ncomprise a significant, but minority share of this.\n\nEquation 10 also \nimplies that - integrated over the spheroid luminosity function - roughly\nhalf of the metals produced by the stars will have been lost from the galaxies\nand reside in the intergalactic medium or intracluster medium. Once\nmixed with the metal-poor ``primordial'' baryons, the net metallicity\nwould be $\\sim$ 1/6th solar in both the intracluster medium\nand the general inter-galactic medium (assuming the\nFHP global value \n$\\Omega_{*,sph}/\\Omega_{IGM} \\sim$ 6, and assuming a mass-weighted mean\nmetallicity\nequal to solar for stars in spheroids). The estimated metallicity\nagrees reasonably well with the measured value of 0.3 solar in rich clusters \n(e.g. Renzini 1997). A measure of\nthe metal content of the present-day\ngeneral IGM may be possible with the next generation of UV and X-ray\nspace spectrographs\n(e.g. Cen \\& Ostriker 1999). \n\nThese are not new arguments by any means (e.g. Gibson, Loewenstein, \\&\nMushotzky 1997; Renzini 1997). What {\\it is} new is that we are now\nin a position to {\\it observationally verify} that intense starbursts\nof the kind that plausibly built galactic spheroids do indeed drive\nmass and metals out at a rate and velocity perhaps high enough to\naccount for the observed inter-galactic metals.\nThe presence of such substantial amounts of inter-galactic\nmetals does not violate constraints imposed by the ``Madau diagram''\n(star-formation rate {\\it vs.} redshift), once reasonable corrections\nfor the effects of dust-extinction are made, nor does it violate\nthe limits set by the far-IR/sub-mm cosmic background (see for \nexample Calzetti \\& Heckman 1999; Renzini 1997).\n\n\\subsection{The Outflow of Dust}\n\nThe strong correlation\nbetween reddening and the strength of the $NaD$ line in starbursts\n(AHM; Veilleux et al 1995; Figure 10) implies that there is an intimate\nrelationship between the dust and gas, especially given the\nclose way in which the two track one another spatially\nthroughout the outflow (section 3.5, Figure 10, and see Phillips 1993\nfor the spectacular case of NGC 1808).\nMoreover, we have argued above that\nsignificant dust column densities in the absorbing matter are needed to shield\nthe $NaI$ from photoionization by the starburst's intense UV radiation field.\n\nWe therfore conclude that\n{\\it dust is being\nexpelled from starbursts at a significant rate.}\nMore quantitatively, for normal Galactic dust,\nthe observed reddening implies a dust surface mass density in the outflow\nregion of $\\sim$10$^{-4}$ gm cm$^{-2}$, an outflowing dust mass\nof $\\sim$ 10$^6$ to 10$^7$ M$_{\\odot}$ (see equation 3), and a dust\noutflow rate of 0.1 to 1 M$_{\\odot}$ yr$^{-1}$ (see equation 5).\n\nAdditional evidence for dusty galactic outflows comes from\na variety of observations. Spectroscopy with $HST$ and $HUT$ has\nestablished that - just as in the case of the $NaD$ lines - the strong\n$UV$ interstellar absorption-lines are frequently\nblueshifted by several hundred km s$^{-1}$ in local starbursts\n(Lequeux et al 1995; Heckman \\& Leitherer 1997; Kunth et al 1998;\nGonzalez-Delgado et al 1998a).\nMoreover, as discussed by Heckman et al (1998), there is\na strong correlation\nbetween the equivalent widths of these $UV$ absorption-lines\nand the reddening in the $UV$ that is analogous to the correlation between\nreddening in the optical and the $NaD$ equivalents widths.\nThe $IUE$ spectra discussed by Heckman et al (1998)\ndo not resolve the $UV$ absorption-lines, and so can not verify\nthat the correlation is primarily driven by the covering fraction of the\nabsorbing dusty material (as in the case of the $NaD$ line). As\nthe archive of $HST$ $UV$ spectra of starbursts grows, it will be\npossible to test this.\n\nImages of several edge-on starburst and star-forming galaxies show\nfar-IR and/or sub-mm\nemission extending one or two kpc along the galaxy minor axis\n(Alton, Davies, \\& Bianchi 1999; Alton et al 1998).\nMulti-color optical images show that kpc-scale extraplanar dust filaments \nare common in star-forming edge-on\ngalaxies (Howk \\& Savage 1997, 1999; Sofue, Wakamatsu, \\& Malin 1994;\nPhillips 1993; Ichikawa et al 1994). Imaging polarimetry\nreveals light scattered by extraplanar dust in \nstarburst galaxies (Scarrott, Eaton, \\& Axon 1991; Scarrott et al. 1993;\nScarrott, Draper, \\& Stockdale 1996;\nAlton et al 1994; Draper et al 1995).\nZaritsky (1994) finds evidence\nfor very extended dust in the halos of spiral galaxies\nbased on the possible detection of reddening in background field galaxies.\n\nAs discussed by Howk \\& Savage (1997) and Aguirre (1999b), there are a variety \nof mechanisms\nby which an episode of intense star-formation could lead to the outflow\nof dust grains. Radiation pressure can ``photo-levitate'' the grains\n(Ferrara et al 1991; Ferrara 1998; Davies et al 1997),\nthe Parker instability could help loft\nmaterial out of the starburst disk (e.g. Kamaya et al 1996), or cold, dusty\ngas in and around the starburst could be entrained and accelerated outward\nby the hot outflowing X-ray gas in the superwind (Suchkov et al 1994, and\nsee section 4.2 above). \n\nThe superwind mechanism is of\nthe most direct relevance to the present paper, so we briefly evaluate\nits plausibility. First, we can show that\nthe outward force of the wind on even the largest grains will exceed\nthe inward force of gravity on the grain. For an isothermal galactic potential\nthis force-ratio at a distance $r$ from the starburst is given by:\n\n\\begin{equation}\nF_w/F_g = 3 \\dot{M}v_{term}/4\\Omega_{w} r v_{rot}^2 a \\rho\n\\end{equation}\n\nwhere $a$ and $\\rho$ are the radius and density of the grain (we take\n$\\rho$ = 2 gm cm$^{-3}$ as representative).\nFor the estimated properties of typical outflow sources in our sample\n($\\dot{M} \\sim$ 25 M$_{\\odot}$\nper year, $v_{term} \\sim$ 600 km s$^{-1}$, and $\\Omega_{w}/4\\pi \\sim$ 0.4),\n$F_w/F_g$ will be greater than unity for grains smaller than\n7 $\\mu$$m$ ($r$/10 kpc)$^{-1}$.\n\nNext, we follow\nAguirre (1999b) and estimate the ratio of the sputtering\nand outflow times for graphite grains\nimmersed in a hot galactic wind ($\\tau_{sp}$/$\\tau_{out}$).\nFor the typical parameters\nwe deduce for the outflows in our sample (see above),\nthis ratio is $\\tau_{sp}$/$\\tau_{out}$ = 4 ($a$/0.1$\\mu$$m$)($r$/10 kpc).\nThus, {\\it large} grains could in fact survive the journey\nto the galactic halo and beyond. The survivability of grains may actually be\nhigher than the above simple estimate if the grains are imbedded inside\ncold gas clouds propelled by the hot outflow (so that the grains\nare not directly exposed\nto the hot gas).\n\n{\\it If} starburst and star-forming galaxies are indeed capable of ejecting\nsubstantial quantities of dust, this could have a profound impact\non observational cosmology (e.g. Heisler \\& Ostriker 1988; Davies et al 1997;\nFerrara 1998;\nFerrara et al 1999; Aguirre 1999a,b). However, to date, the\ndirect evidence for the existence of intergalactic dust is very sparse.\nThermal far-IR emission has been detected from the\nICM of the Coma cluster (Stickel et al 1998), and a\npossible deficit of background QSO's seen through foreground\ngalaxy clusters has been reported (Romani \\& Maoz 1992; but see Maoz 1995).\n\nAguirre (1999a,b) has recently calculated that a dusty inter-galactic medium\nwith $\\Omega_{dust}$ = few $\\times$ 10$^{-5}$ would have a visual\nextinction ($\\sim$ 0.5 magnitudes out to z = 0.7) that would be sufficient\nto reconcile the Type Ia supernova Hubble diagram (Reiss et al 1998;\nPerlmutter et al 1999) with a standard $\\Omega_{M}$ = 1, $\\Omega_{\\Lambda}$\n= 0 cosmology. Data on the optical colors of high-redshift supernovae\nshow no evidence for {\\it reddening},\nbut Aguirre argues that intergalactic dust will have a much greyer\nextinction curve than standard Galactic dust. This is plausible because\nsmall grains will be more easily destroyed by sputtering \nduring and after their journey into the IGM (see above).\n\nIn this context, it is instructive to estimate the cosmic mass\ndensity of dust grains by the type of outflows investigated\nin this paper. Aguirre (1999b) has considered this in more detail\nfrom a somewhat different perspective, but comes to rather similar\nconclusions.\nLet us assume that superwinds associated with the\nformation of galactic spheroids propelled dust and gas-phase metals\ninto the ICM and IGM, with an amount proportional to the mass\nin the present-day stars in such systems. We further assume that the \nmass fractions\nof the metals locked into grains in the ICM and IGM are $f_{g,icm}$\nand $f_{g,igm}$ respectively. These assumptions imply:\n\n\\begin{equation}\n\\Omega_{dust,igm} = f_{g,igm}(1 - f_{g,icm})^{-1}\\Omega_{spheroids}\\Omega_{icm}\nZ_{icm}\\Omega_{stars,cl}^{-1}\n\\end{equation}\n\nFollowing FHP and Renzini (1997), we take $Z_{icm}$ = 6.7 $\\times$ 10$^{-3}$\n(1/3 solar metallicity), $\\Omega_{spheroids}$ = 0.0026 $h_{70}^{-1}$,\n$\\Omega_{icm}$ = 0.0026 $h_{70}^{-1.5}$, and $\\Omega_{stars,cl}$ = 0.00043\n$h_{70}^{-1}$. This implies $\\Omega_{dust,igm}$ = 1.0$\\times$ 10$^{-4}$\n$f_{g,igm}(1 - f_{g,icm})^{-1}h_{70}^{-1.5}$. For a normal Galactic\ndust/metals ratio ($f_g \\sim$ 0.5), the implied value for $\\Omega_{dust,igm}$\nis twice as large as the value needed to explain the Type Ia\nsupernova-dimming (Aguirre 1999a).\nGiven the higher densities (and thus, faster grain sputtering times)\nin the ICM compared to the IGM, we might expect\nthat $f_{g,icm} < f_{g,igm}$. More importantly, it\nis also possible that $f_{g,igm} <<$ 0.5 due to\nthe destruction of dust in\nsuperwinds and/or the \nIGM (but see Aguirre 1999b for an optimistic assessment).\nWhile the above estimate for $\\Omega_{dust,igm}$\nshould therefore probably be regarded as an absolute upper\nbound, it is an intriguingly large one from a cosmological perspective.\n\nFinally, we note that since intergalactic dust will emit as well\nas absorb, its amount is constrained by\nthe cosmic background measured by $COBE$\n(Ferrara et al 1999). Indeed, Aguirre \\& Haiman (2000) argue that a significant\nfraction of the detected cosmic far-IR and sub-mm background must have an\nintergalactic origin if this dust is abundant enough to strongly\naffect the Type Ia supernova Hubble Diagram.\n\n\\subsection{Relationship to ``Associated Absorption'' in AGN}\n\nOver the past few years, it has become increasingly clear that a young\nstellar population is present in the circumnuclear region of\na significant fraction of type 2 Seyfert galaxies (e.g. Heckman et al 1995,1997;\nGonzalez-Delgado et al 1998b;\n Schmitt et al 1999; Oliva et al. 1999). Most\nrecently, a near-UV spectroscopic survey of a complete sample of the brightest\ntype-2 Seyfert nuclei by Gonzalez-Delgado, Heckman, \\& Leitherer\n(2000) finds direct evidence for hot, young stars in roughly half\nof the nuclei. \nIn this paper we have established that starbursts drive outflows of cool\nor warm gas with total column densities of a few $\\times$ 10$^{21}$\ncm$^{-2}$, velocities of a few hundred km s$^{-1}$, and covering factors\nalong the line-of-sight of typically 50\\%. The implication then is that\nthis absorbing material should be detectable in those Seyfert nuclei\nthat also contain a circumnuclear starburst. \n\nIn the standard ``unified''\nscenario, type 1 and type 2 Seyfert nuclei are drawn from the same parent\npopulation, with the former viewed from a direction near the polar axis\nof an optically and geometrically-thick ``obscuring torus'' and the latter\nfrom a direction near the equatorial plane of the torus (e.g. Antonucci\n1993 and references therein). Thus, in type 1 Seyferts, any starburst-driven\noutflow could be observed in absorption against the bright nuclear\ncontinuum source. While the total column density of the\noutflowing gas\nshould be similar to the flows studied in this paper, the\ngas would be exposed to the intense ionizing continuum from the central\nnucleus, and therefore would be significantly more highly-ionized. \n\nThis can\nbe quantified as follows. The ionization state of photoionized gas is\ndetermined by the ionization parameter:\n\\begin{equation}\nU = Q/4\\pi r^2 n c\n\\end{equation}\nwhere $Q$ is the production rate of ionizing photons and \n$n$ is the electron density in the photoionized\nmaterial located a distance $r$ from the source. \nThe radial density gradients observed in starburst-driven outflows\nare consistent with predictions for clouds subjected to the\nram pressure associated with the superwind (Heckman, Armus, \\&\nMiley 1990): \n\\begin{equation}\n2n(r)kT \\sim P(r) = \\dot{p}/\\Omega_w r^2\n\\end{equation}\nwhere $\\dot{p}$ is the rate at which the starburst feeds momentum into\nthe superwind.\nFor photoionized gas, $T \\sim$ 10$^4$ K (e.g Osterbrock 1989), so\nequations 13 and 14 together\nimply that the magnitude of $U$ is set by\n$Q/\\dot{p}$,\nand that $U$ will be independent of $r$ (neglecting\nradiative transfer effects).\nWe adopt a generic Leitherer \\& Heckman (1995) starburst model\n(Salpeter IMF extending\nup to 100 M$_{\\odot}$ and a starburst lifetime of a few $\\times$ 10$^7$\nyears),\ninclude sources of ionization due\nto both a starburst ($Q_*$) and the type 1 Seyfert nucleus ($Q_{sy1}$).\nFor a starburst and type 1 Seyfert nucleus of the same bolometric\nluminosity, $Q_{sy1}/Q_*$ would be a factor of several.\nWe then obtain the following estimate for $U$:\n\\begin{equation}\nU = 2.6 \\times 10^{-3}(\\Omega_w/4\\pi)(1 + Q_{sy1}/Q_*)\n\\end{equation}\nThe predicted properties of the absorbing material then overlap\nsignificantly with the ``associated absorbers'' seen in $UV$ spectra\nof type 1 Seyfert nuclei (e.g. Crenshaw et al 1999; Kraemer et al 1999):\nan incidence rate of roughly 50\\%, a high line-of-sight covering\nfraction, outflow velocities of 10$^2$ to 10$^3$ km s$^{-1}$, and\ninferred ionization parameters of $\\sim$ 10$^{-2}$. Crenshaw et al\n(1999) find an essentially one-to-one correspondence between\nthe presence of $UV$ absorption-lines and soft X-ray absorption\nby hotter and more highly ionized material (the ``warm absorber'').\nWe speculate that \nthe hotter and more tenuous phases of the starburst superwind could\ncontribute to the warm absorber.\n\nWe emphasize that we are not proposing that all of the ``associated\nabsorption'' seen in type 1 Seyfert nuclei is produced by gas\nin a starburst-driven outflow. In some cases, rapid variability\nor the presence of absorption out of highly-excited lower levels\nimply densities that are orders-of-magnitude higher than would be\ntenable for material in a starburst-driven outflow (Crenshaw et al 1999\nand references therein). However, it appears that the \nabsorbing material in type 1 Seyfert nuclei can span a broad range\nin physical and dynamical conditions (Kriss et al 2000).\nGiven important roles\nfor starbursts in the Seyfert phenomenon and for superwinds in the starburst\nphenomenon, significant absorption due to the superwind material seems\nunavoidable in some Seyfert nuclei.\n\n\\section{Conclusions}\n\n%Galactic superwinds - outflows driven by the mechanical energy supplied\n%by supernovae and stellar winds - are a well-established phenomenon\n%in local starburst galaxies. Recent spectroscopy of the interstellar\n%absorption-lines in high-redshift star-forming galaxies imply that\n%similar outflows occurred in the early universe. Such outflows\n%over a Hubble time may well have chemically-enriched and heated the\n%inter-galactic medium and played a key role in regulating the chemical\n%and structural properties of galaxies (particularly ellipticals and the bulges\n%of disk galaxies).\n\n%It is therefore vital to elucidate the physics and document the basic\n%properties of the nearest superwinds. The study of such flows using\n%interstellar absorption-lines offers several key advantages over the\n%past approaches (which have emphasized the analysis of optical and X-ray\n%emission). The interstellar lines can probe the regions of low-density\n%that may dominate the mass of the outflow, they offer unique and\n%unambiguous information about the kinematics, dynamics, and energetics\n%of the outflow, and allow a direct comparison to be made between\n%superwinds at low- and high-redshift.\n\nWe have discussed the results of moderate-resolution\n($R$ = a few thousand) spectroscopy of \nthe $NaI\\lambda\\lambda$5890,5896 ($NaD$) absorption-line in\na sample of 32 far-IR-selected starburst galaxies. These galaxies were\nselected from either the far-IR-warm sample of Armus, Heckman, \\&\nMiley (1989) or the edge-on sample of Lehnert \\& Heckman (1995),\nand together span a range from 10$^{10}$ to few $\\times$ 10$^{12}$\nL$_{\\odot}$ in IR luminosity. \nWe found that the stellar contribution to the\n$NaD$ absorption-line is negligible ($<$10\\%) in some objects, but significant\n($\\sim$ 70\\%) in others. We have thus divided our sample into\n18 interstellar-dominated (``ISD'') objects ($<$ 30\\% stellar\ncontribution) and 14 strong-stellar-contamination (``SSC'')\nobjects ($>$ 40\\% stellar contribution).\n\nThe $NaD$ line lies within\n70 km s$^{-1}$ of $v_{sys}$ in all the SSC objects\n(consistent with a predominantly\nstellar origin).\nThe $NaD$ lines in the SSC nuclei are about\n0.2 dex narrower than expected for dynamics of the old stellar population\nin the bulges of normal galaxies of similar disk rotation speed\nand Hubble type. Thus,\ndynamically ``cold'' material (red supergiants and/or\ninterstellar gas) in the inner part of the starburst makes\na significant contribution to the observed $NaD$ line in these\nnuclei.\n\nThe kinematics of $NaD$ line are markedly different in the ISD objects.\nThe $NaD$ line is blueshifted by $\\Delta$$v >$ 100\nkm s$^{-1}$ relative to the galaxy systemic velocity in 12 of the 18\ncases (the\n``outflow sources''), and the outflow can be mapped over a region\nof a few-to-ten kpc in size.\nIn contrast, no objects in our sample\nshowed a net\n{\\it redshift} in $NaD$ of more than 100 km s$^{-1}$.\nThe outflow\nsources are galaxies systematically viewed more nearly face-on than\nthe other galaxies in our sample:\n69\\% of the galaxies\nwith a ratio of semi-major to semi-minor axes $a/b \\leq$ 2.0 show \n$NaD$ outflows,\nwhile this is true for only 6\\% of the flatter (more highly inclined)\ngalaxies. This is consistent with the absorbing\nmaterial being\naccelerated out along the galaxy minor axis by a bipolar\nsuperwind.\nThe absorbing\nmaterial typically spans the velocity range from near\nthe galaxy systemic velocity ($v_{sys}$) to a maximum blueshift\nof 300 to 700 km s$^{-1}$. We therefore suggest that the outflowing\nsuperwind ablates the\nabsorbing gas from ambient clouds at $\\sim v_{sys}$, and then\naccelerates it up to a terminal velocity similar to the wind speed. We\nfound no correlation between the widths of the H$\\alpha$\nemission-line and the $NaD$ absorption-line\nsubsamples. Evidently, the dynamics of the more tenuous absorbing\ngas is largely decoupled from that of the dense\n(high emission-measure) gas that\nprovides most of the nuclear line-emission.\n\nThe ratio of the equivalent widths of the two members of the $NaD$\ndoublet ($R$) ranges from 1.1 to 1.7 in the ISD sample, implying\nthat the doublet is optically-thick. However,\n$R$ does not correlate with the residual relative intensity\nat the ``bottom'' of the stronger $\\lambda$5890 line profile \n($I_{5890}$), which ranges from 0.14 (nearly black) to 0.7.\nThus, the optically-thick gas does not fully cover\nthe emitting stars (covering factor $\\sim$ 1 - $I_{5890}$). The observed\nequivalent width of the $NaD$ line is then set by the product\nof velocity dispersion and covering factor for the\nabsorbing gas, and we showed that the\nlatter quantity is the dominant one.\nUsing two variants of the classic doublet-ratio technique, we estimated that\nthe $NaI$ column densities are $logN_{NaI}$ = 13.5 to 14 cm$^{-2}$. This\nis roughly consistent with column densities measured in a few cases for $KI$\nusing the optically-thin $\\lambda$$\\lambda$7665,7699 \\AA\\ doublet\n(assuming a solar $NaI/KI$ ratio). The total gas columns are uncertain,\nbut the empirical correlation between $N_{NaI}$ and $N_H$ in the\nISM of the Milky Way implies $N_H \\sim$ few $\\times$ 10$^{21}$ cm$^{-2}$.\n\nWe found a strong correlation in the ISD sample between the reddening of the\nobserved stellar\ncontinuum and the depth of the $NaD$ absorption-line, and a significant\nbut weaker correlation of the line-depth with the reddening of the\nBalmer emission-lines. Evidently,\nthe gas responsible for the $NaD$ absorption is very dusty. The typical\nimplied reddening is $E(B-V) \\sim$ 0.3 to 1 magnitudes over regions\nseveral-to-ten\nkpc in size. For a normal dust-to-gas ratio, the corresponding column\ndensities are $N_H \\sim$ few $\\times$ 10$^{21}$ cm$^{-2}$ (in agreement\nwith the above estimate).\n\nThe inferred column densities and measured outflow velocities and\nsizes imply that the typical mass and kinetic energy associated\nwith the absorbing gas is of-order 10$^9$ M$_{\\odot}$ and\n10$^{56}$ erg, respectively. The estimated outflow rates of mass and energy\nare typically 10 to 100 M$_{\\odot}$ per year and\n10$^{41}$ to 10$^{42}$ erg s$^{-1}$. The mass outflow rates are comparable\nto the estimated star-formation rate, and much larger than the rate at\nwhich massive stars are returning mass to the ISM. Thus, powerful\nstarbursts can eject as much gas as is being converted into stars,\nand most of this gas is ambient material that has been ``mass-loaded''\ninto the hot gas returned directly by supernovae and stellar winds.\nThe energy outflow rates in the absorption-line gas are of-order 10$^{-1}$ of\nthe rate at which\nmassive stars supply mechanical energy. Most of the energy\nreturned by massive stars probably\nresides in the kinetic and thermal energy of the much hotter X-ray-emitting\ngas. We showed\nthat the overall properties of the absorbing gas in the outflow sources\ncan be easily reproduced in the context of simple analytic\nestimates for the properties of interstellar clouds accelerated\nby the ram pressure of the hot high-speed wind seen via its X-ray emission.\nDetailed hydrodynamical simulations of galactic winds, while still missing some\nessential physics, also predict the observed properties of the\ncool absorbing gas.\n\nWe have discussed the implications of our results for the chemical\nevolution of galaxies and the intergalactic medium.\nThe estimates derived for $v_{term}$ using the $NaD$ line in the outflow\nsources agree reasonably well with the outflow speeds implied for\nan adiabatic wind ``fed'' by hot gas whose temperature is measured\nby the observed X-ray-emitting gas. The typical implied values\nare 300 to 800 km s$^{-1}$, and are independent of the rotation speed\nof the ``host galaxy'' over the range $v_{rot}$ = 30 to 300 km s$^{-1}$,\nconfirming and extending the result in Martin (1999) based on X-ray data alone.\nThis strongly suggests that the outflows selectively escape the potential\nwells of the less massive galaxies.\nWe considered a simple model based on Lynden-Bell (1992) in which the fraction\nof starburst-produced metals\nthat are retained by a galaxy experiencing an outflow is proportional to the\ngalaxy potential-well\ndepth for galaxies with $v_{esc} < v_{term}$, and asymtotes to\nfull retention for the most massive galaxies ($v_{esc} > v_{term}$).\nFor $v_{term}$ in the range we measure, such a simple prescription\ncan reproduce the observed mass-metallicity relation for elliptical\ngalaxies and deposit the required amount of observed metals in the\nintra-cluster medium. If the ratio of ejected metals to stellar\nspheroid mass is the same globally as in clusters of galaxies,\nwe predicted that the present-day\nmass-weighted metallicity of an intergalactic medium\nwith $\\Omega_{igm}$ = 0.015 will be $\\sim$ 1/6 solar (see also\nRenzini 1997).\n\nWe have summarized the evidence that starbursts are ejecting significant\nquantities\nof dust, emphasizing the results from the present paper. {\\it If}\nthis dust can survive a trip into the intergalactic medium and remain\nintact for a Hubble time,\nwe estimated that the upper bound on the global amount of intergalactic\ndust is $\\Omega_{dust}$ $\\sim$ 10$^{-4}$. While this is clearly an\nupper limit, it is a cosmologically interesting one: \nAguirre (1999a,b) argues that dust this abundant\ncould in principle obviate the need for a positive cosmological\nconstant, based on the Type Ia supernova Hubble diagram.\n\nFinally, given the mounting evidence for a connection between starbursts and\nthe Seyfert phenomenon, we have suggested that outflows like those studied\nhere may account for some (but not all) aspects of the ``associated\nabsorption'' seen in type 1 Seyfert nuclei.\n\n\n\\acknowledgements\n\nWe would like to thank Ken Sembach for useful on-going discussions and advice.\nDiscussions with David Neufeld, Mark Voit, Don York, and Donna Womble were\nhelpful during the formative stages of the project. 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Col. (2) --- Galaxy systemic velocity (km s$^{-1}$) in the heliocentric\nframe. In order of preference, these are determined from: galaxy\nrotation curves (r), global $CO$ 115 GHz emission-line profiles (c), nuclear\nstellar velocities (s), global $HI\\lambda$21cm emission-line profiles (h),\nand nuclear optical emission-line profiles (e). The rotation curve velocities\n(r) are taken from LH95 except for NGC2146 from Prada et al (1994).\nItems marked `n' come from NED. Items marked `e' or `s' are based on\ndata obtained during the observing runs discussed in the present paper.\nItems marked `c' are: NGC 660 (Elfhag et al 1996; Young et al 1995),\nNGC 1614 (Elfhag et al 1996; Young et al 1995; Aalto et al 1991;\nSanders, Scoville, \\& Soifer 1991; Casoli et al 1991), M 82 (Lo et al\n1987), IRAS10173+0828 (Planesas, Mirabel, \\& Sanders 1991), NGC 3256\n(Aalto et al 1991; Casoli et al 1991; Mirabel et al 1990),\nIRAS10565+2448 (Downes \\& Solomon 1998), and Arp 220 (Young et al 1995;\nSolomon, Downes, \\& Radford 1992). Based on the intercomparison\nof independent measurements for a given galaxy, the typical uncertainties\nin $v_{sys}$ range from 10 km s$^{-1}$ for the nearby, relatively\nnormal galaxies\nto as much as 100 km s$^{-1}$ for the most distant systems (generally,\nhighly disturbed mergers).\nCol. (3) --- Total infrared luminosity from 8 to 100 microns, based on IRAS\ndata and the definition of L$_{IR}$ given in Sanders \\& Mirabel (1996). We\nassume throughout that $H_0$ = 70 km s$^{-1}$ Mpc$^{-1}$.\nCol. (4) --- Blue absolute magnitude for the galaxy, corrected for\nforeground (Galactic) extinction, but not for internal extinction.\nTaken from LH95 when available (adjusted to $H_0$ = 70 km s$^{-1}$\nMpc$^{-1}$) or based on the data in NED or Armus, Heckman, \\& Miley\n(1987).\nCol. (5) --- The ratio of the optical semi-major to semi-minor axes. These\nare taken (in order of preference) from LH95, the images published in\nArmus, Heckman, \\& Miley (1987; 1990), or NED. For the highly disturbed\nmerging systems, we have measured this ratio at intermediate radii\n(excluding both faint tidal tails and the inner regions where dust\nobscuration is most significant).\nCol. (6) --- The amplitude of the rotation speed of the galaxy. In order\nof preference, we have based these on rotation curves (``r'' from LH95),\nglobal $HI\\lambda$21cm profiles corrected for inclination and turbulence\n(``h'' - see LH95 for details), and global $CO$ 115 GHz emission-line\nprofiles using the half-width at 20\\% of the peak intensity and then\ncorrecting for inclination (``c'' - using the same data as in Column 2).\nFor M 82 we have replaced the value listed in LH95 by the more recent\ndetermination by Sofue (1998).\nThe uncertainties in most cases are dominated by the inclination correction.\nWe estimate the resulting uncertainties to be $<$ 0.1 dex for all but the\ncases of mergers and strongly interacting galaxies, where the\ninclination corrections lead to an uncertainty of roughly 0.2 dex\n(denoted by :).\nCol. (7) --- Sample from which the galaxy was drawn (Armus, Heckman, \\& Miley\n1989; Lehnert \\& Heckman 1995).\nCol. (8) --- Observing runs used in this paper (see Table 2).\n\n% table2 = observing runs\n\\include{tab2}\n\n% table 3 = measured properties\n\\include{tab3}\nNote. Col. (2) --- The estimated contribution to the observed $NaD$ line by\ncool stars (the remainder is interstellar in origin). See text for\ndetails. Galaxies with $f_* \\geq$ 40\\% are members of the\nstrong-stellar-contamination sample (SSC), while those with\n$f_* \\leq$ 30\\% are members of the interstellar-dominated\n(ISD) sample. Based on the agreement between $f_*$ determined by the two\nindependent techniques discussed in the text, the uncertainty is\ntypically $\\pm$10\\% .\nCol. (3) --- Heliocentric velocity of the $NaD$ absorption-line (km s$^{-1}$)\nmeasured by fitting the profile with a pair of Gaussians constrained\nto have the separation in wavelength appropriate for the red-shifted\n$NaD$ doublet. Based on comparison of independent measurements of this\nquantity in the cases for which we have multiple spectra,\nwe estimate that the typical measurement uncertainty\nis $\\pm$ 20 km s$^{-1}$. \nCol. (4) --- The velocity difference between the $NaD$ absorption-line\nand the galaxy systemic velocity in the galaxy rest-frame:\n$\\Delta$v = (v$_{NaD}$ - v$_{sys}$)/(1 + v$_{sys}$/c). The relevant\nquantities are given in Col.2 of Table 1 and Col. 3 of this Table.\nA typical uncertainty in this velocity difference is $\\pm$ 20 km s$^{-1}$,\nfor the relatively bright nearby galaxies with well-determined values\nfor $v_{sys}$ up to $\\pm$ 100 km s$^{-1}$ for the most distant and\nfar-IR-luminous\ngalaxies (highly disturbed mergers with uncertain $v_{sys}$).\nCol. (5) --- The full-width at half-maximum of each of the two Gaussians\nfit to the doublet (km s$^{-1}$).\n$W$ was constrained to be the same for\nthe two doublet members. The listed value has had the instrumental\ncontribution to the measured value removed by assuming the intrinsic\nand instrumental widths add in quadrature: W = [W$_{obs}^2$ -\nW$_{instr}^2$]$^{1/2}$. Based on comparison of independent measurements of this\nquantity in the cases for which we have multiple spectra,\nwe estimate that the typical measurement uncertainty\nis $\\pm$ 20 km s$^{-1}$. \nCol. (6) --- The rest-frame equivalent width (\\AA) for the $NaD$ doublet.\nBased on comparison of independent measurements of this\nquantity in the cases for which we have multiple spectra,\nwe estimate that the typical measurement uncertainty\nis $\\pm$ 0.2 \\AA.\nCol. (7) --- The normalized residual intensity at the center of the $NaD$\n$\\lambda$5890 line profile (I$_{5890}$ = 0 corresponds to a totally black\nline center). This has been corrected for the effect of the spectral\nresolution assuming Gaussian profiles: (1-I$_{5890}$) = (W$_{obs}$/W)(1 -\nI$_{5890,obs}$). Based on comparison of independent measurements of this\nquantity in the cases for which we have multiple spectra,\nwe estimate that the typical measurement uncertainty\nis $\\pm$ 0.02.\nCol. (8) --- The ratio of the equivalent widths of the\n$NaD\\lambda\\lambda$5890,5896 transitions. A ratio $R$ = 2 (1) corresponds\nto an optical depth of 0 (infinity). Based on comparison of independent\nmeasurements of this\nquantity in the cases for which we have multiple spectra,\nwe estimate that the typical measurement uncertainty\nis $\\pm$ 0.1.\nCol. (9) --- The full-width at half-maximum of a Gaussian fit to the nuclear\n$H\\alpha$ emission-line. These are taken from AHM, LH95, or our own\nunpublished spectra. The listed value has had the instrumental\ncontribution to the measured value removed by assuming the intrinsic\nand instrumental widths add in quadrature. Typical uncertainties\nare $\\pm$ 20 km s$^{-1}$.\n\n%table 4 = misc properties of the ISM dominated sample\n\\include{tab4}\nNote. Col. (2) --- The ratio of the nuclear H$\\alpha$ and H$\\beta$ emission-line\nfluxes. These have been corrected for the effects of underlying stellar\nabsorption-lines\n(assuming a stellar equivalent width of 1.5 \\AA) and for foreground\nGalactic extinction (using a standard Galactic extinction curve and\nthe measured Galactic $HI$ column density. The data come from AHM,\nVeilluex et al (1995), data from runs 3 or 6 (Table 2), Vaceli et al\n(1997), or Dahari \\& DeRobertis (1988). Typical uncertainties are\n$\\pm$5\\%. Unreddened ionized gas\nwould have a flux ratio of 2.86 for standard Case B conditions.\nCol. (3) --- The ratio of the flux densities (F$_{\\lambda}$) in the nuclear\ncontinuum\nnear the wavelengths of H$\\alpha$ and H$\\beta$. The values have\nbeen corrected for foreground Galactic extinction (see above).\nThe data come from AHM, Veilluex et al (1995), or our runs 3 or 6 (Table 2).\nTypical uncertainties are $\\pm$5\\%.\nNote that an unreddened starburst corresponding to constant star-formation\nfor 30 Myr would have an intrinsic color in these units of 0.5.\nCol. (4) --- The projected size (in kpc) of the region along the spectrograph\nslit exhibiting strongly blueshifted (by $>$ 100 km/s) $NaD$\nabsorption. In NGC1572, NGC1614, NGC3256, NGC7552, and NGC7582\nwe have measured this along two position angles.\nCol. (5) --- An estimate of the terminal velocity implied by the $NaD$\nabsorption-line profile (v$_{term}$ = $\\Delta$ v + 0.5 W). See Table 3.\nCol. (6) --- The rotation speed of the starburst galaxy. See Table 1.\n\n\\newpage\n\n\\figcaption [] {a) through d)\nNormalized spectra of the nuclear $NaD$ absorption-line\nprofile in the 32 galaxies in our sample. The region displayed is\ntypically 2 by 4 arcsec in size\ncentered on the region of peak brightness in the red continuum.\nEach displayed spectrum covers an observed range of 60 \\AA\\\n($\\sim$ 3000 km s$^{-1}$). Note the weak foreground Galactic $NaD$\nabsorption at 5890,5896 \\AA\\ in NGC660, NGC2146, and NGC4945.}\n\n\\figcaption [] {Histogram of the difference between the mean radial velocity\nof the $NaD$ line and the galaxy systemic velocity corrected to the\ngalaxy rest-frame - see Table 3. The $ISD$ (interstellar-dominated)\nlines are indicated by vertical hashing and the $SSC$ (strong-stellar\ncontamination) lines by horizontal hashing. The majority of the $ISD$\nlines are blueshifted by at least 100 km s$^{-1}$, while the $SSC$\nlines are all close to $v_{sys}$. Uncertainties in $\\Delta$$v$\nrange from $\\pm$20 to 100 km s$^{-1}$ - see text and Table 3.\nA typical uncertainty is represented by the plotted error-bar.}\n\n\\figcaption [] {Histogram of the full-width at half-maximum of each\nof the two members of the $NaD$ doublet, corrected for the effects\nof instrumental resolution and in the galaxy rest-frame. See Table 3.\nThe $ISD$ (interstellar-dominated)\nlines are indicated by vertical hashing and the $SSC$ (strong-stellar\ncontamination) lines by horizontal hashing. The $ISD$ lines are much\nbroader than the $SSC$ lines. Typical uncertainties in $W$ are\n$\\pm$20 km s$^{-1}$, as indicated by the plotted error-bar.}\n\n\\figcaption [] {Plot of the full-width at half-maximum ($W$) {\\it vs.}\nthe blueshift of the $NaD$ doublet ($\\Delta$$v$) for the sources\nshowing nuclear outflows ($\\Delta$$v \\leq$ 100 km s$^{-1}$). This\nFigure omits IRAS11119+3257, which has a blueshift of $\\sim$ 10$^3$\nkm s$^{-1}$. The diagonal line shows the relation\n$W$ = 2 $\\Delta$$v$, expected in the case that gas is injected\ninto the outflow at $v \\sim v_{sys}$, and accelerated up to\na terminal velocity $v_{term} \\sim \\Delta$$v$ + 0.5$W$.\nUncertainties in $\\Delta$$v$\nrange from $\\pm$20 to 100 km s$^{-1}$, and typical uncertainties in $W$ are\n$\\pm$20 km s$^{-1}$ (as indicated by the plotted error-bar).}\n\n\\figcaption [] {Plot of the galaxy rotation speed (Table 1)\n{\\it vs.} the full-width at half maximum of the $NaD$ line\n(Table 3). The $ISD$ (interstellar-dominated)\nlines are indicated by solid dots and the $SSC$ (strong-stellar\ncontamination) lines by hollow dots. The empirical relations between\nbulge stellar velocity dispersion and disk rotation speed\nfound by Whittle (1992) for normal Sa (Sc) spiral galaxies are indicated\nby the lower (upper) diagonal line. The $ISD$ lines are nearly all broader\nthan these relations, while the $SSC$ lines are nearly all narrower.\nTypical uncertainties are $\\pm$20 km s$^{-1}$ for $W$ and range\nfrom $<$ 0.1 dex to 0.2 dex for $v_{rot}$ (dominated by the uncertain\ninclination correction in highly disturbed systems). Typical uncertainties\nare indicated by the plotted error-bar.}\n\n\\figcaption [] {Plot of the galaxy rotation speed (Tables 1 and 4)\n{\\it vs.} the inferred terminal velocity of the outflow\n($v_{term} = \\Delta$$v$ + 0.5$W$; see Table 4) for the sources\nshowing outflows in $NaD$ line ($\\Delta$$v \\leq$ -100 km s$^{-1}$).\nThe diagonal line shows the relation\n$v_{term}$ = 2$v_{rot}$. The terminal velocity shows no dependence\non the rotation speed. Uncertainties in $v_{term}$\nrange from $\\pm$20 to 100 km s$^{-1}$, and \nfrom $<$ 0.1 dex to 0.2 dex for $v_{rot}$ (dominated by the uncertain\ninclination correction in highly disturbed systems). The typical \nuncertainties are indicated by the plotted error-bar.}\n\n\\figcaption [] {Plot of the\nnormalized residual intensity at the center of the $\\lambda$5890\ntransition ($I_{5890}$) {\\it vs.}\nthe ratio of the \nequivalent widths of the $\\lambda$5890\nand $\\lambda$5896 members of the $NaD$ doublet ($R$),\nfor the $ISD$ (interstellar-dominated)\n$NaD$ lines. See Table 3. $R$ = 1 (2) corresponds to the limit\nof optically-thick (-thin) conditions. There is no correlation,\nimplying that $I_{5890}$ is determined primarily by covering factor\nrather than optical depth. Typical uncertainties are $\\pm$ \n0.02 in $I_{5890}$ and $\\pm$ 0.1 in $R$, as shown by the plotted\nerror-bar.}\n\n\\figcaption [] {Plot of the normalized residual intensity at the \ncenter of the $\\lambda$5890\ntransition ($I_{5890}$) {\\it vs.} the rest-frame equivalent width of the\n$NaD$ doublet for the $ISD$ (interstellar-dominated)\nlines. There is a significant correlation, implying that\nthe equivalent width depends on the covering fraction of the\noptically-thick absorbing gas. Typical uncertainties are $\\pm$\n0.02 in $I_{5890}$ and $\\pm$ 0.2 \\AA\\ in $EQW_{NaD}$, and are shown\nby the plotted error-bar.}\n\n\\figcaption [] {Plot of the full-width-at-half-maximum {\\it vs.}\nthe rest-frame equivalent width of the\n$NaD$ doublet for the $ISD$ (interstellar-dominated)\nlines. There is no correlation, implying that\nthe equivalent width does not depend on the velocity dispersion of the\noptically-thick absorbing gas. Typical uncertainties are $\\pm$\n20 km s$^{-1}$ for $W$ and $\\pm$ 0.2 \\AA\\ in $EQW_{NaD}$, as shown\nby the plotted error-bar.}\n\n\\figcaption [] {a) Plot of the normalized residual intensity at the \ncenter of the $\\lambda$5890\ntransition ($I_{5890}$) {\\it vs.} the log of the color of the optical continuum\n(the ratio of $F_{\\lambda}$ at rest wavelengths of 6560 and 4860\n\\AA). Points plotted as solid dots are the nuclei of the $ISD$ \n(interstellar-dominated) sample members (Table 4). Other points\nare off-nuclear locations in M82, NGC3256, NGC6240, Mrk273, IRAS03514+1546,\nand IR10565+2448.\nThe deeper the\n$NaD$ line (higher covering factor), the more-reddened the background starlight.\nThe correlation is obeyed by both the nuclear and off-nuclear regions.\nAn unreddened starburst population should have $log(C_{65}/C_{48})$\n= -0.3. For a standard Galactic reddening curve, the implied\n$A_V$ ranges up to roughly 4 magnitudes for the most-reddened sight-lines.\nTypical uncertainties for the nuclear (extra-nuclear) data are indicated\nby the error-bar in the lower-left (upper-right) of the plot.\nb) As in a), except that $I_{5890}$ is plotted {\\it vs.}\nthe log of the Balmer decrement (H$\\alpha$/H$\\beta$ flux ratio). Again, the\nmore-reddened sight-lines correspond to the deepest $NaD$ line profiles\n(highest covering factors). The log of the intrinsic H$\\alpha$/H$\\beta$ flux\nratio is 0.46, and the implied values of $A_V$ range up to $\\sim$ 5 magnitudes\nfor the most-reddened sight-lines.} \n\n\\figcaption [] {Logarithm of the gas number density (in units of \n${\\rm cm}^{-3}$)\nin the hydrodynamic simulation described in the text, at four\ndifferent epochs. The figure shows dense cool gas, either entrained\ninto the flow at the walls of the cavity or remnants of the fragmented\nsuperbubble shell, being swept up and carried out of the galaxy by the wind. \nAs material is locally in pressure equilibrium,\nthe densest material visible is also the coolest, and might be\nconsidered analogous to the dense cool gas responsible for the\noptical absorption lines. \nThe four clumps or clouds shown have average velocities (over the 1.5 Myr\nperiod shown) of 177 km s$^{-1}$ (cloud A), \n540 km s$^{-1}$ (cloud B), 348 km s$^{-1}$ (cloud C) and\n858 km s$^{-1}$ (cloud D) respectively. }\n\n\\figcaption [] {As in Figure 6, except that we have added galaxies\nin which we have estimated wind outflow velocities from the\nobserved temperature of the hot X-ray-emitting gas via the\nrelation from Chevalier \\& Clegg (1985): $v_{term} \\sim (5kT_X/\\mu)^{1/2}$.\nThe X-ray temperatures are taken from the references listed in the text.\nThe data points\nbased on the $NaD$\nprofile are indicated by solid dots and the points based on the X-ray data\nare indicated by hollow dots. Note that the two data sets are consistent with\neach other,\nimply that the outflow speed is independent of the host galaxy\npotential well depth, and thus suggest that outflows will\npreferentially escape from the least massive galaxies. The two diagonal\nlines indicate the galaxy escape velocity under the assumption\nthat $v_{esc}$ = 2 $v_{rot}$ and $v_{esc}$ = 3 $v_{rot}$ respectively\n(see equation 9 in the text). Typical uncertainties in the\nX-ray ($NaD$) estimates of $v_{term}$ are shown by the error-bar on the\nbottom right (upper center).}\n\n\\end{document}\n"
},
{
"name": "tab1.tex",
"string": "\\begin{deluxetable}{lrccccccc}\n\\tablecolumns{9}\n\\tablewidth{0pt}\n\\tablenum{1}\n\\tablecaption{Sample}\n\\tablehead{\n\\colhead{Galaxy}&\\colhead{v$_{sys}$}&\\colhead{}&\n\\colhead{log L$_{IR}$}&\\colhead{M$_{B_T}$}&\n\\colhead{${a \\over b}$}&\\colhead{v$_{rot}$}&\n\\colhead{Sample}&\\colhead{Run} \\\\\n\\colhead{(1)}&\\colhead{(2)}&\\colhead{}&\n\\colhead{(3)}&\\colhead{(4)}&\n\\colhead{(5)}&\\colhead{(6)}&\n\\colhead{(7)}&\\colhead{(8)}}\n\\startdata\nNGC 253 & 245&h,n & 10.5 &$-$19.8 & 4.1 & 202 h & AHM,LH & 2 \\nl\nNGC 660 & 885&c & 10.0 &$-$17.7 & 2.6 & 175 h & AHM,LH & 5 \\nl\nIIIZw035 & 8294&e & 11.5 &$-$19.5 & 2.0 & 87 h & LH & 1 \\nl\nIRAS02021--2104& 34630&e & 12.0 &$-$21.1 & 2.5 & ? & AHM & 4 \\nl\nNGC 1134 & 3620&r & 10.7 &$-$21.2 & 2.9 & 211 r & LH & 1 \\nl\nIRAS03514+1546& 6662&h,n & 11.1 &$-$20.8 & 1.1 & 270:h & AHM & 5,6 \\nl\nNGC1572 & 6142&r & 11.2 &$-$21.4 & 2.0 & 312 r & LH & 1 \\nl\nNGC1614 & 4760&c & 11.3 &$-$20.8 & 1.8 & 210:c & AHM & 4 \\nl\nIRAS04370--2416& 4537&r & 11.0 &$-$20.0 & 2.5 & 172 r & LH & 1 \\nl\nNGC 1808 & 1001&r & 10.6 &$-$20.1 & 1.7 & 160 r & LH & 3,4 \\nl\nIRAS05447--2114& 11977&e & 11.0 &$-$18.7 & 2.0 & ? & AHM,LH & 5 \\nl\nNGC 2146 & 916&r & 10.7 &$-$19.3 & 1.8 & 272 h & AHM,LH & 5 \\nl\nNGC 2966 & 2045&h,n & 10.2 &$-$19.2 & 2.6 & 124 r & LH & 4 \\nl\nM 82 & 214&c & 10.5 &$-$18.5 & 2.6 & 50 h & AHM.LH & 5,6 \\nl\nNGC 3094 & 2409&h,n & 10.4 &$-$19.5 & 1.4 & 150 h & AHM & 5 \\nl\nIRAS10173+0828& 14669&c & 11.8 &$-$19.1 & 2.5 & 140 c & AHM,LH & 5 \\nl\nNGC 3256 & 2801&c & 11.5 &$-$21.3 & 1.8 & 170:c & AHM & 3,4 \\nl\nIRAS10502--1843& 16131&s & 11.8 &$-$19.0 & 1.2 & ? & AHM & 4 \\nl\nIRAS10565+2448& 12923&c & 12.0 &$-$20.7 & 1.3 & 300:c & AHM & 5,6 \\nl\nIRAS11119+3257& 56866&e & 12.5 & ? & 1.2 & ? & AHM & 5 \\nl\nNGC 3628 & 843&h,n & 10.2 &$-$19.8 & 5.0 & 218 h & LH & 3 \\nl\nNGC 3885 & 1938&r & 10.3 &$-$20.9 & 2.5 & 195 r & LH & 4 \\nl\nNGC 4666 & 1511&r & 10.9 &$-$20.7 & 3.5 & 186 r & LH & 3 \\nl\nNGC 4945 & 560&h,n & 10.6 &$-$19.8 & 5.2 & 172 h & LH & 3 \\nl\nNGC 5104 & 5615&r & 11.1 &$-$20.2 & 2.8 & 231 r & LH & 4 \\nl\nMrk 273 & 11326&c,n & 12.2 &$-$21.0 & 2.0 & 260:c & AHM & 5,6 \\nl\nArp 220 & 5441&c & 12.2 &$-$20.7 & 1.3 & 330:c & AHM & 5 \\nl\nNGC 6240 & 7339&c,n & 11.7 &$-$21.6 & 1.9 & 290:c & AHM,LH & 3 \\nl\nIC 5179 & 3424&r & 11.0 &$-$20.9 & 2.1 & 194 r & LH & 1 \\nl\nNGC 7541 & 2714&r & 10.8 &$-$20.3 & 2.8 & 221 r & LH & 1 \\nl\nNGC 7552 & 1585&n & 10.8 &$-$20.2 & 1.2 & 230 h & LH & 1 \\nl\nNGC 7582 & 1575&n & 10.6 &$-$20.2 & 2.4 & 180 r & LH & 1 \\nl\n\\enddata\n\\end{deluxetable}\n"
},
{
"name": "tab2.tex",
"string": "\\begin{deluxetable}{ccccccccc}\n\\tablecolumns{9}\n\\tablewidth{0pt}\n\\tablenum{2}\n\\tablecaption{Observing Runs}\n\\tablehead{\n\\colhead{Run Num}&\\colhead{Date}&\n\\colhead{Obs}&\\colhead{Tel}&\n\\colhead{Spec}&\\colhead{Detector}&\n\\colhead{pixels}&\\colhead{Slit}&\n\\colhead{Res} \\\\\n\\colhead{(1)}&\\colhead{(2)}&\n\\colhead{(3)}&\\colhead{(4)}&\n\\colhead{(5)}&\\colhead{(6)}&\n\\colhead{(7)}&\\colhead{(8)}&\n\\colhead{(9)}}\n\\startdata\n1 & 11/90& LCO &2.5m& ModSpec &TEK1024 &0.68$\\times$1.2 &2 &3.4 \\nl\n2 & 10/92& CTIO& 4m&Blue Air Schmidt&Reticon1 &0.77$\\times$0.93 &2 &2.1 \\nl\n3 & 3/93 & CTIO& 4m&Folded Schmidt &TEK1024 &0.79$\\times$0.60 &2.2&1.8 \\nl\n & & & & & &0.79$\\times$1.98 &2.2&5.9 \\nl\n4 & 1/94 & CTIO& 4m&Folded Schmidt &TEK1024 &0.82$\\times$0.6 &2 &1.8 \\nl\n5 & 1/94 & KPNO& 4m&RC Spec &T2KB &0.69$\\times$0.5 &2 &1.1 \\nl\n6 & 1/88 & KPNO& 4m&RC Spec &TI2 &0.90$\\times$3.4 &2 &13.5 \\nl\n\\enddata\n\\tablecomments{\nCol. (7) --- Pixel size in arcsec by \\AA.\nCol. (8) --- Slit width in arcsec.\nCol. (9) --- Spectral resolution (FWHM) in \\AA\\ for $NaD$.}\n\\end{deluxetable}\n\n\n"
},
{
"name": "tab3.tex",
"string": "\\begin{deluxetable}{lrccccccc}\n\\tablecolumns{9}\n\\tablewidth{0pt}\n\\tablenum{3}\n\\tablecaption{Measured Properties}\n\\tablehead{\n\\colhead{Galaxy}&\\colhead{f$_{\\star}$}&\n\\colhead{v$_{NaD}$}&\\colhead{$\\Delta$v}&\n\\colhead{W}&\\colhead{EQW}&\n\\colhead{I$_{5890}$}&\\colhead{R}&\n\\colhead{W$_{H\\alpha}$} \\\\\n\\colhead{(1)}&\\colhead{(2)}&\n\\colhead{(3)}&\\colhead{(4)}&\n\\colhead{(5)}&\\colhead{(6)}&\n\\colhead{(7)}&\\colhead{(8)}&\n\\colhead{(9)}}\n\\startdata\nNGC253 & 50\\% & 193& $-$52& 150&6.0 & 0.04 &1.1 &280 \\nl\nNGC660 & 50\\% & 878& $-$7& 140&4.6 & 0.22 &1.2 &190 \\nl\nIIZw035 & 50\\% & 8365& 69& 100&2.5 & 0.37 &1.4 &310 \\nl\nIRAS02021--2104&$<$20\\% &34692& 56& 340& 8.1& 0.28& 1.5& 420 \\nl\nNGC1134 & 70\\% & 3620& 0& 320 &4.4 &0.57 &1.2 &220 \\nl\nIRAS03514+1546 & 30\\% & 6461& $-$197& 430 &4.4 &0.60 &1.1 &170 \\nl\nNGC1572 & 30\\% & 6011& $-$128& 360 &4.3 &0.65 &1.2 &330 \\nl\nNGC1614 &$<$10\\% & 4636& $-$122& 420& 8.3& 0.35& 1.2& 300 \\nl\nIRAS04370--2416& 40\\% & 4525& $-$12& 200 &2.4 &0.67 &1.3 &170 \\nl\nNGC1808 & 20\\% & 1013& 12& 300 &9.2 &0.18 &1.1 &260 \\nl\nIRAS05447--2114&$<$30\\% &12072& 91& 200& 4.2& 0.47& 1.2& 290 \\nl\nNGC2146 & 30\\% & 930& 14& 140 &4.7 &0.17 &1.2 &120 \\nl\nNGC2966 & 50\\% & 2043& $-$2& 190 &3.8 &0.45 &1.2 &210 \\nl\nM 82 &$<$20\\% & 204& $-$10& 170& 5.8& 0.18& 1.2& 100 \\nl\nNGC3094 & 50\\% & 2394& $-$15& 190 &3.9 &0.48 &1.3 &120 \\nl\nIRAS10173+0828 &$<$30\\% &14708& 37& 150& 4.3& 0.44& 1.2& 200 \\nl\nNGC3256 & 20\\% & 2489& $-$309& 550&5.5 &0.59 &1.6 &210 \\nl\nIRAS10502--1843& $<$20\\%&16022& $-$103& 240& 6.8& 0.31& 1.2& 310 \\nl\nIRAS10565+2448 &$<$20\\% &12717& $-$197& 500& 8.5& 0.34& 1.3& 210 \\nl\nIRAS11119+3257 &$<$10\\% &55755& $-$934& 170& 6.1& 0.14& 1.2& 1500 \\nl\n & &55189&$-$1410& 80& 1.8& 0.45& ...& \\nl\nNGC3628 & 70\\% & 812& $-$31& 170& 4.5 & 0.29 & 1.2 & 120 \\nl\nNGC3885 & 70\\% & 1962& 24& 200& 4.1 & 0.46 & 1.2 & 300 \\nl\nNGC4666 & 70\\% & 1521& 10& 150& 4.3 & 0.25 & 1.2 & 200 \\nl\nNGC4945 & 50\\% & 622& 62& 180& 5.5 & 0.33 & 1.1 & 390 \\nl\nNGC5104 & 50\\% & 5603& $-$12& 240& 5.3 & 0.41 & 1.1 & 470 \\nl\nMrk273 & 30\\% &11145& $-$174& 560& 4.6 & 0.68 & 1.2 & 520 \\nl\nArp 220 & 30\\% & 5422& $-$19& 500& 6.5 & 0.45 & 1.4 & 610 \\nl\nNGC6240 & 20\\% & 7049& $-$283& 600&10.3 & 0.31 & 1.6 & 890 \\nl\nIC5179 & 40\\% & 3431& 7& 130& 4.5 & 0.28 & 1.2 & 180 \\nl\nNGC7541 & 50\\% & 2695& $-$19& 160& 3.8 & 0.42 & 1.2 & 260 \\nl\nNGC7552 & 30\\% & 1323& $-$261& 480& 4.3 & 0.70 & 1.1 & 140 \\nl\nNGC7582 & 30\\% & 1344& $-$230& 510& 4.7 & 0.65 & 1.7 & 190 \\nl\n\\enddata\n\\end{deluxetable}\n"
},
{
"name": "tab4.tex",
"string": "\\begin{deluxetable}{lccccccc}\n\\tablecolumns{6}\n\\tablewidth{0pt}\n\\tablenum{4}\n\\tablecaption{Miscellaneous Properties of ``ISD'' Sub-Sample}\n\\tablehead{\n\\colhead{Galaxy}&\\colhead{${H\\alpha \\over H\\beta}$}&\n\\colhead{${C65 \\over C48}$}&\\colhead{Size}&\n\\colhead{v$_{term}$}&\\colhead{v$_{rot}$} \\\\\n\\colhead{(1)}&\\colhead{(2)}&\n\\colhead{(3)}&\\colhead{(4)}&\n\\colhead{(5)}&\\colhead{(6)}}\n\\startdata\nIRAS02021--2104& ... & 1.77 & ... & ... & ... \\nl\nIRAS03514+1546&7.6 & 1.21 & 4.4 & 410 & 270: \\nl\nNGC1572 & ... & ... & 2.4x3.2 & 310 & 312 \\nl\nNGC1614 &7.4 & 1.32 & 3.2x3.2 & 330 & 210: \\nl\nIRAS04370--2416&5.4 & 0.86 & ... & ... & 172 \\nl\nNGC 1808 &8.0 & 1.39 & 3.7 & 700* & 160 \\nl\nIRAS05447--2114& ...& ... & ... & ... & ... \\nl\nNGC 2146 &10.0& 1.39 & ... & ... & 272 \\nl\nM 82 &8.4& 1.44 & ... & ... & 50 \\nl\nIRAS10173+0828&...& 1.61 & ... & ... & 140 \\nl\nNGC 3256 &5.3& 0.94 & 5.6x1.8 & 580 & 170: \\nl\nIRAS10502--1843&...& ... & 7.7 & 220 & ... \\nl\nIRAS10565+2448&9.7& 1.14 & 8.5 & 450 & 300: \\nl\nIRAS11119+3257&7.3& 1.89 & $<$6 & 1450 & ... \\nl\nMrk 273 &8.7& 1.02 & 3 & 450 & 260: \\nl\nArp 220 &21 & 1.23 & ... & ... & 330: \\nl\nNGC 6240 &13.6& 1.98 & 9 & 580 & 290: \\nl\nNGC 7552 &9.8& ... & 1.2x1.2 & 500 & 230 \\nl\nNGC 7582 &8.2& ... & 1.0x1.0 & 490 & 180 \\nl\n\\enddata\n\\end{deluxetable}\n"
}
] |
[
{
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}
] |
astro-ph0002527
|
Prompt and afterglow emission from the X-ray rich GRB981226 observed with BeppoSAX
|
[
{
"author": "F. Frontera\\altaffilmark{1,2}"
},
{
"author": "L.A.~Antonelli\\altaffilmark{3}"
},
{
"author": "L.~Amati\\altaffilmark{2}"
},
{
"author": "E.~Montanari\\altaffilmark{1}"
},
{
"author": "E.~Costa\\altaffilmark{4}"
},
{
"author": "D.~Dal~Fiume\\altaffilmark{2}"
},
{
"author": "P.~Giommi\\altaffilmark{6}"
},
{
"author": "M.~Feroci\\altaffilmark{4}"
},
{
"author": "G.~Gennaro\\altaffilmark{11}"
},
{
"author": "J.~Heise\\altaffilmark{5}"
},
{
"author": "N.~Masetti\\altaffilmark{2}"
},
{
"author": "J.M.~Muller\\altaffilmark{5,6}"
},
{
"author": "L.~Nicastro\\altaffilmark{7}"
},
{
"author": "M.~Orlandini\\altaffilmark{2}"
},
{
"author": "E.~Palazzi\\altaffilmark{2}"
},
{
"author": "E.~Pian\\altaffilmark{2}"
},
{
"author": "L.~Piro\\altaffilmark{4}"
},
{
"author": "P.~Soffitta\\altaffilmark{4}"
},
{
"author": "S.~Stornelli\\altaffilmark{11} J.J.M.~in 't Zand\\altaffilmark{5}"
},
{
"author": "D.A.~ Frail\\altaffilmark{8}"
},
{
"author": "S.R.~Kulkarni\\altaffilmark{9}"
},
{
"author": "and M. Vietri\\altaffilmark{10}"
}
] |
We discuss observations of the prompt X-- and $\gamma$--ray emission and X--ray afterglow from GRB981226. This event has the weakest gamma-ray peak flux detected with the \S\ Gamma-Ray Burst Monitor. It shows an isolated X-ray precursor and the highest X-ray to gamma-ray fluence ratio measured thus far with the \S\ Wide Field Cameras. The event was followed up with the \S\ Narrow Field Instruments, and the X-ray afterglow was detected up to 10 keV. The afterglow flux is observed to rise from a level below the sensitivity of the MECS/LECS telescopes up to a peak flux of (5$\pm 1) \times 10^{-13}\, \ergcms$ in the 2-10~keV energy band. This rise is followed by a decline according to a power law with index of 1.31$^{+0.44}_{-0.39}$. We discuss these results in the light of the current GRB models.
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"string": "%\\documentstyle[12pt,epsf,aasms4]{article}\n\\documentstyle[11pt,epsf,aaspp4]{article}\n\\lefthead{F. Frontera et~al.}\n\\righthead{GRB981226 with BeppoSAX}\n\n\\def\\S{{\\em BeppoSAX\\/}} \n\\def\\etal{{\\it et al. }}\n\\def\\eg{{\\em e.g.,\\ }}\n\\def\\ergcm{\\mbox{ erg cm$^{-2}$}}\n\\def\\apj{{\\it Astrophys. J. }}\n\\def\\apjs{{\\it Astrophys. J. Suppl. Ser. }}\n\\def\\pasj{{\\it Publs. Astron. Soc. Japan }}\n\\def\\nature{{\\it Nature }}\n\\def\\aa{{\\it Astron. Astrophys. }}\n\\def\\aas{{\\it Astron. Astrophys. Suppl. Ser. }}\n\\def\\mnras{{\\it Mon. Not. R. Astr. Soc. }}\n\\def\\ergcms{\\mbox{ erg cm$^{-2}$ s$^{-1}$}}\n\\makeatletter\n\\def\\@cite#1#2{(#1\\if@tempswa , #2\\fi)}\n\\def\\preprint{preprint} \\newif\\ifPreprintMode\n\\ifx\\preprint\\revtex@genre\\PreprintModetrue\\else\\PreprintModefalse\\fi\n\\makeatother\n\n\\begin{document}\n\n\\title{Prompt and afterglow emission from the X-ray rich GRB981226 \nobserved with BeppoSAX}\n\n\\author{F. Frontera\\altaffilmark{1,2},\nL.A.~Antonelli\\altaffilmark{3},\nL.~Amati\\altaffilmark{2},\nE.~Montanari\\altaffilmark{1},\nE.~Costa\\altaffilmark{4},\nD.~Dal~Fiume\\altaffilmark{2},\nP.~Giommi\\altaffilmark{6},\nM.~Feroci\\altaffilmark{4},\nG.~Gennaro\\altaffilmark{11},\nJ.~Heise\\altaffilmark{5},\nN.~Masetti\\altaffilmark{2},\nJ.M.~Muller\\altaffilmark{5,6},\nL.~Nicastro\\altaffilmark{7},\nM.~Orlandini\\altaffilmark{2},\nE.~Palazzi\\altaffilmark{2},\nE.~Pian\\altaffilmark{2},\nL.~Piro\\altaffilmark{4},\nP.~Soffitta\\altaffilmark{4},\nS.~Stornelli\\altaffilmark{11}\nJ.J.M.~in 't Zand\\altaffilmark{5},\nD.A.~ Frail\\altaffilmark{8},\nS.R.~Kulkarni\\altaffilmark{9},\nand M. Vietri\\altaffilmark{10}\n}\n\n\\altaffilmark{1}{Dipartimento di Fisica, Universit\\`a di Ferrara, Via Paradiso\n 12, 44100 Ferrara, Italy}\n\n\\altaffilmark{2}{Istituto Tecnologie e Studio Radiazioni Extraterrestri, \nCNR, Via Gobetti 101, 40129 Bologna, Italy}\n\n\\altaffilmark{3}{Osservatorio Astronomico di Roma, Via Frascati, 33, 00040\nMonteporzio Catone (RM), Italy}\n\n\\altaffilmark{4}{Istituto Astrofisica Spaziale, C.N.R., Via Fosso del \nCavaliere, 00133 Roma, Italy}\n\n\\altaffilmark{5}{Space Research Organization in the Netherlands,\n Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands}\n\n\\altaffilmark{6}{\\S\\ Scientific Data Center, Via Corcolle 19, 00131 Roma,\n Italy} \n\n\\altaffilmark{7}{Istituto Fisica Cosmica e Applicazioni all'Informatica, \nC.N.R., Via U. La Malfa 153, 90146 Palermo, Italy}\n\n\\altaffilmark{8}{National Radio Astronomy Observatory, P.O. Box O, Socorro,\nNM 87801, USA}\n\n\\altaffilmark{9}{Palomar Observatory 105-24, Caltech, Pasadena, CA 91125, USA}\n\n\\altaffilmark{10}{Dipartimento di Fisica, Universit\\`a Roma Tre, Via della \nVasca Navale, 84, 00146 Roma, Italy}\n\n\\altaffilmark{11}{\\S\\ Operative Control Center, Via Corcolle 19, 00131 Roma,\n Italy} \n\n\n\\begin{abstract}\nWe discuss observations of the prompt X-- and $\\gamma$--ray emission and X--ray afterglow\n from GRB981226. This event has the weakest gamma-ray peak flux\ndetected with the \\S\\ Gamma-Ray Burst Monitor. It shows an isolated X-ray precursor\nand the highest X-ray to gamma-ray fluence ratio measured thus far with the \\S\\\nWide Field Cameras. The event was followed up \nwith the \\S\\ Narrow Field Instruments, and the X-ray afterglow was detected up to 10 keV. \nThe afterglow flux is observed to rise from a level below the sensitivity of the\nMECS/LECS telescopes up to a peak flux of (5$\\pm 1) \\times 10^{-13}\\, \n\\ergcms$ in the 2-10~keV energy band. This rise is followed by a decline according to a \npower law with index of 1.31$^{+0.44}_{-0.39}$. We discuss these results \nin the light of the current GRB models.\n\\end{abstract}\n\n\\keywords{gamma rays: bursts --- gamma rays: observations --- X--rays:\ngeneral ---shock waves}\n\n\n\\section{Introduction}\nFollow-up observations of arcminute positions of Gamma-Ray Bursts (GRBs) \nprovided by provided by the Wide Field Cameras on the \\S\\ satellite \\cite{Jager97,Boella97a} \nhave shown that in most\ncases a fading X-ray counterpart identified as the GRB X-ray afterglow is\ndetected \\cite{Frontera98}. The fading law is generally a smooth power law \n(e.g., Frontera et~al. 1999\\nocite{Frontera99}) except in two cases: GRB970508 \n\\cite{Piro98}, in which a late-time outburst of about 10$^5$~s duration \nstarted about 6$\\times 10^4$s after the main event, and GRB970828 \n\\cite{Yoshida99}, in which a peak structure of 4000~s duration appeared \n1.25 $\\times 10^5$~s after the main event. Generally the \nafterglow light curves extrapolated back to the time of the bursts, are in \nagreement with the tail of the GRB time profiles \n\\cite{Costa97,Piro98,Frontera99}. This fact is\nconsidered as evidence that late afterglow emission and tail of the prompt \nGRB emission have the same origin \\cite{Frontera99}. In the fireball model \nscenario (see, e.g., the recent review by Piran 1999 \\nocite{Piran99}), \nthis means that both the tail of the prompt emission \nand the late afterglow emission can be due to an external shock produced \nby the interaction of a relativistically expanding fireball \nwith the Interstellar Medium (ISM). \n\\\\\nOf the promptly localized GRBs 80\\% show X-ray afterglow,\nabout 50\\% exhibit also optical emission, and 30\\% \nshow radio emission. Radio emission is generally accompanied by optical \nemission, except in two cases: GRB990506 \\cite{Taylor99}\nfrom which also X-ray afterglow emission was observed \nwith the RXTE/PCA experiment (Bacodine trigger 7549, Hurley 1999 \\nocite{Hurley99}), \nand GRB981226. For this event, in spite of several attempts to find the \noptical counterpart \n\\cite{Galama98,Rhoads98,Bloom98,Schaefer98,Castro-Tirado98,Wozniak98,Wozniak98,Lindgren99}\nnone was identified. The best upper limit (R$\\sim$23~mag) to the optical flux of the \nGRB counterpart was reported by Lindgren \\etal (1999) \\nocite{Lindgren99}. \nFrail \\etal (1999a) \\nocite{Frail99a} reported the detection of a \nradio counterpart (VLA 232937.2--235553). \nThe radio source peaked to 173$\\pm27$~$\\mu$Jy at 8.46~GHz at about 10 days after \nthe burst. This time delay is tipical of \nall previously studied radio afterglows. However, the source declined relatively fast, \nfollowing a power law decay ($\\propto t^{-\\delta_R}$) with \n$\\delta_R \\, = \\, 2.0 \\pm 0.4$ \\cite{Frail99b}. Two interpretations of this\nrapid decay have been proposed by Frail \\etal (1999b) \\nocite{Frail99b}: it is \neither the consequence of a jetted GRB or the result of a fireball shock in an ambient \nmedium with variable density.\nAn optical galaxy (R~=~24.85 mag), consistent with the radio transient\nposition, has been proposed as the host galaxy of GRB981226 \\cite{Frail99b}.\n\nGRB981226 was also followed up with the \\S\\ Narrow Field Instruments. A transient X-ray \nsource was observed, which was proposed as the X-ray afterglow of GRB 981226 \n\\cite{Frontera98}.\nHere we report the properties of this afterglow emission along \nwith those of the prompt X-- and gamma--ray emission. We will discuss these \nproperties in the light of the current models of GRBs.\n\n\n\\section{Observations} \n\\label{obs}\nGRB981226 was detected with the \\S\\ Gamma--Ray Burst Monitor (GRBM, \n40--700~keV; Frontera et~al. 1997, Amati et~al. 1997\\nocite{Frontera97,Amati97}) and \nWFC unit 1 (1.5--26.1~keV; Jager et~al. 1997\\nocite{Jager97}) on December 26 starting \nat 09:47:25 UT \\cite{Diciolo98}. Its \nposition was determined with an error radius of $6'$ (99\\% confidence level) \nand was centered at $\\alpha_{2000} = 23^{\\rm h}29^{\\rm m}40^{\\rm s}$, $\\delta_{2000} =\n-23^\\circ55'30''$. A precursor is detected only in the WFC on 09:44:20 UT.\n \nAbout eleven hours after the burst, the Narrow Field Instruments \non-board \\S\\ were pointed at the burst location for a first\ntarget of opportunity (TOO1) observation, from December 26.8785 UT to December\n28.2986~UT. A new X--ray source was detected\n\\cite{Frontera98} in the GRB error box with the Low Energy (LECS, 0.1-10~keV, \nParmar et~al. 1997 \\nocite{Parmar97}) and Medium Energy (MECS, 2--10~keV,\nBoella et~al. 1997b\\nocite{Boella97b}) Concentrators/Spectrometers. The net exposure \ntime on the source for MECS\nand LECS was 58.4~ks and 25.6~ks, respectively. The same field was\nagain observed about 7 days after the main event (TOO2) from 1999\nJanuary 2.7604~UT to\nJanuary 3.5590~UT (total net exposure time of 25.1~ks for MECS and 8.0~ks for\nLECS). During this observation, the source was no more detected.\n\nData available from GRBM include two 1~s ratemeters in two energy\nchannels (40--700~keV and $>$100~keV), 128~s count spectra (40--700~keV,\n225 channels) and high time resolution data (up to 0.5~ms) in the\n40--700~keV energy band.\n%The energy resolution of the GRBM unit 1, co-aligned \n%with the WFC No. 1, is 20\\% at 280 keV \\cite{Amati97}. \nWFCs (energy resolution $\\approx$ 20\\% at 6~keV) were operated in normal\nmode with 31 channels in 1.5--26~keV and 0.5~ms time resolution \n\\cite{Jager97}. \nThe burst direction was offset by 7$^\\circ$ with respect to\nthe WFC axis. With this offset, the\neffective area exposed to the GRB was\n$\\approx$~420~cm$^2$ in the 40-700~keV band and 91~cm$^2$\nin the 2--26~keV energy band.\nThe background in the WFC and GRBM energy bands was fairly stable\nduring the burst, with a slight increase with time in the $>$100~keV\nchannel (about 2\\% in 350~s). The GRBM background was estimated by linear \ninterpolation using the 250~s count rate data before and after the burst.\nThe WFC spectra were extracted through the Iterative Removal Of Sources procedure\n(IROS \\footnote{WFC software version 105.108}, e.g. Jager et al. 1997 \\nocite{Jager97})\nwhich implicitly subtracts the contribution of the background and of other point sources \nin the field of view.\n\\\\\nThe MECS source count rates and\nspectra for TOO1 were extracted, using the XSELECT package, from a $\\sim 3'$ radius region\naround the source centroid, while the background level was estimated\nfrom an annulus centered on the source with inner and outer radii of 4 and \n8.5 arcmin, respectively. The spectra from MECS 2 and 3 were equalized and co-added.\nGiven the much lower exposure time, the source was much less visible in the LECS. We used\nfor the source extraction from the LECS image the XIMAGE package \\cite{Giommi91}, \nthat permits a more\nrefined choice of the background region. The source counts were extracted \nfrom a square box centered on the source centroid consistent with the MECS centroid \nposition and with side of 3.5 arcmin, while the corresponding background was \nextracted from a square annulus of\ninner side 3.5 arcmin and outer side 12 arcmin, centered on the source.\nThe uncertainties will be given as single parameter errors at 90\\% confidence level.\n\n\\section{Results}\n\n\\subsection{Prompt emission}\n\\label{prompt}\nFigure~1 shows the measured time profiles of GRB981226 in three energy channels after the \nbackground subtraction.\nIn the $\\gamma$--ray band (40--700 keV; fig.~1, middle panel), the GRB shows a single peak \nof about 5~s duration. Some marginal evidence of the peak appears in the high energy range \n($>$100 keV; fig.~1, bottom panel).\nIn the X--ray energy band (2--26~keV; fig~1, top panel), the prompt emission starts about 180~s \nearlier with a precursor-like event of about 50~s duration. The X--ray main event\nexhibits two peaks, the first of which is coincident with that detected by the GRBM.\nThe total duration of X-ray main event is about 80~s. From the WFC images both precursor \nand main event are consistent with the same direction in the sky.\n\nThe spectral evolution of precursor and main event was studied\nby subdividing the GRB time profile into five temporal slices and \nperforming an analysis on the average spectrum of each\nslice (see Fig.~1). \nWe fit the spectra with a power law (N(E)$\\propto \\ E^{\\alpha}$) and a smoothly \nbroken power law \n\\cite{Band93}, both photoelectrically absorbed by a neutral hydrogen column\ndensity N$_H$ \\cite{Morrison83}. The count statistics \ndo not permit to constrain N$_H$, that was thus fixed to the \nGalactic value along the GRB direction (1.8 $ \\times 10^{20}\\,{\\rm cm^{-2}}$). \nIn Table 1 we show the results. Both laws fit the data. The Band law permits\nto determine the value of the peak energy E$_p$ of the logarithmic power per photon energy\ndecade (the $\\nu F_\\nu$ spectrum). For the time slices A, C, D, where only upper\nlimits to the gamma-ray flux were available, the upper limits of E$_p$ were derived\nby freezing the value of the high energy index $\\beta$ to $-2$.\nIn Fig.~2 we show the $\\nu F(\\nu)$ spectra of the chosen temporal slices. \nThe continuous line is the best fit of the Band law for slice B and of the power\nlaw for the other slices.\nA spectral softening is observed from slice B (onset of the main \nevent) to the following slices. Also the precursor spectrum appears softer \nthan that in the slice B. The spectral evolution is better shown by the behavior\nof the peak energy E$_p$ (see Table~1): it is low of our energy passband during the\nprecurson, it achieves a value of about 60 keV at the onset of the main event \nand then goes down below our passband at the end of the burst.\n\\\\\nThe $\\gamma$--ray (40--700~keV) fluence of the burst is S$_\\gamma\\,=\\, \n(4\\pm 1)\\times 10^{-7}$ \\ergcm, while the corresponding value found\nin the 2--10 keV band is S$_X\\,=\\,(5.7\\pm 1.0)\\times 10^{-7}$~\\ergcm, with\na ratio S$_X$/S$_\\gamma$ \\,=\\,(1.4$\\pm$0.4).\n%In the 50--300~keV range the fluence of GRB981226 is 2.8$\\times 10^{-7} \\ergcm$.\nThe $\\gamma$--ray peak flux, derived from the 1~s ratemeters, is P$_\\gamma\\,=\\, \n0.33$$\\pm$0.13~photons/cm$^2$~s corresponding to \n$(6.5\\pm2.6)\\times 10^{-8}$ \\ergcms, while the\ncorresponding 2--10 keV peak flux is P$_X\\,=\\,2.7$$\\pm$0.3~photons/cm$^2$~s,\ncorresponding to $(1.7\\pm0.2)\\times 10^{-8} \\ergcms$.\nThe peak flux in gamma-rays is the lowest observed thus far with \\S\\ . \n\\\\\n\\subsection{Afterglow emission}\n\\label{after}\nDuring TOO1, a previously unknown X-ray source, 1SAX J2329.6-2356, was detected \nin the MECS, almost at the center of the GRB error box, at celestial coordinates\n$\\alpha_{2000} = 23^{\\rm h}29^{\\rm m}36.1^{\\rm s}$,\n$\\delta_{2000} = -23^\\circ55'58.3''$, with an error radius of 1 arcmin \\cite{Frontera98}.\nThe MECS image obtained in the first part of TOO1 (exposure time of 26950~s), when\nthe source was stronger, is shown in fig.\\ref{fig:mecs_image_1_2}, left.\nThe source is also visible in the corresponding 0.1--2~keV LECS image (6880 exposure \ntime). \n%The MECS and LECS source exposure times during TOO1 were 26950~s and 6880~s, \n%respectively. \nThe source was not detected in the TOO2, when the MECS exposure time was 25080~s \n(see fig.~3, right).\n\\\\\nOther count excesses, compatible with very weak celestial \nsources, are present in the TOO1 image. They are likely field sources, \nthe number of which\nis in agreement with the log~N-log~S distribution of the 5-10~keV X-ray \nsources found by Fiore et al. (1999b) \\nocite{Fiore99b} with \\S\\ . Given its \ntransient behavior, 1SAX J2329.6-2356 is likely the X-ray aftergow \nof GRB981226. From the above log~N-log~S distribution, the chance probability for \nits coincidence with a background source is about 5$\\times 10^{-3}$. \n\n\\subsubsection{Spectrum}\n\\label{spectrum}\nWe derived the average 0.1-10~keV count spectrum of 1SAX J2329.6-2356 over the \nfirst 15~hrs of TOO1, when the source was brightest. We fit it both with a power\nlaw (N(E)$\\propto \\ E^{\\alpha}$) and a blackbody, photoelectrically absorbed by the Galactic \ncolumn density along the GRB direction (see Section \\ref{prompt}). In the fits a\nnormalization of a factor 0.8 was applied to the LECS spectra following the\ncross-calibration tests between the LECS and MECS \\cite{Fiore99a}. \nBoth laws are acceptable descriptions of the data: $\\chi^2_\\nu\\,=\\,0.5$ (5 degrees of \nfreedom, dof) \nfor a power-law and $\\chi^2_\\nu\\,=\\,1.3$ (5 dof) for a blackbody. However, \nin the case of the\nblackbody, for energies above 5~keV, the best fit curve is constantly below \nthe measured bins. This suggests that the power law provides a better description \nof the data.\nThe best fit power-law index is $\\alpha \\,= \\, -1.92 \\pm 0.47 $.\nWe do not find evidence of a spectral evolution of the emission: the time\nbehavior of the C(4--10~keV)/ C(1.4--4~keV) hardness ratio is statistically consistent \nwith a constant. \n\\\\\nNo evidence of the source is found in the 15--300~keV energy range: the \n\\S\\ PDS instrument \\cite{Frontera97} during TOO1 does not show any statistically \nsignificant count excess over the background level. Assuming the above power-law \nindex, the 2~$\\sigma$ upper limit in the 15--60~keV energy band is \n$5.5\\times 10^{-12} \\ergcms$, which is \na factor $\\sim$30 higher than the extrapolated flux from the LECS+MECS spectrum.\n\\\\\nAs above mentioned, no source excess is apparent in the MECS~+~LECS data\nduring TOO2. Assuming the best fit power law index obtained from the\nTOO1 spectrum and the Galactic column density,\nwe derived the following 2~$\\sigma$ upper limits to the source flux during\nTOO2: $1.3\\times 10^{-13} \\ergcms$ and $8.2 \\times 10^{-14} \\ergcms$ in the 0.1--2~keV\nand 2--10~keV ranges, respectively. \n\n\n\\subsubsection{Light curve}\n\\label{lc}\n\nThe 2--10~keV MECS light curve of the afterglow in bins of \n7000~s elapsed time is shown in fig.~\\ref{fig:lc} (top).\nIts main feature is the weakness of the source in the first 7000~s bin, where its flux\nis below the MECS sensitivity limit (2$\\sigma$ upper limit of $1.5 \\times 10^{-13} \n\\ergcms$ in the 2-10~keV energy band).\nChecks were done to verify whether this non detection could be due to attitude \nmalfunctions of the satellite. However we found that the source was correctly \npointed since the beginning of the NFI TOO1. Afterwards the source flux increases by\nmore than a factor 3 in about 10000~s ((5$\\pm 1)\\times 10^{-13} \\ergcms$). After this \npeak flux the source starts fading.\nThe light curve is also reported in the bottom panel of fig.~\\ref{fig:lc} along\nwith the WFC data points. The later fading of 1SAX J2329.6-2356 is apparent, that\nis well described by a power law, F(t)$\\propto$~t$^{-\\delta}$, \nwith index $\\delta \\,= \\, 1.31^{+0.44}_{-0.39}$. In fig.~\\ref{fig:lc}, bottom \nwe show the best fit power law along with the slope uncertainty region (90\\% \nconfidence level).\nFrom the best fit light curve, the 2--10~keV afterglow fluence\nintegrated over the time interval from the GRB end (80~s) to 10$^6$~s is\n$S_a \\, = \\, (4.3\\pm 3.8)\\times 10^{-7} \\ergcm$, with\na ratio between X--ray afterglow fluence and the prompt $\\gamma$--ray \nfluence of 1.1$\\pm$0.9, vs. a corresponding value of (1.4$\\pm$0.4) for the \nprompt emission (see Sect. \\ref{prompt}).\n\n\n\\section{Discussion}\n%\nGRB981226 shows the lowest gamma-ray peak flux among the bursts localized thus \nfar with \\S\\ . In the log~N--log~P distribution of the GRBs observed with BATSE \n\\cite{Paciesas99}, it is located near the faint end of the distribution.\nThe burst is marked by three peculiarities, two of which have been\nseen for the first time:\n\n\\begin{enumerate}\n\n\\item\nOf the bursts localized by BeppoSAX, this burst is the richest in the\nX-ray band: the X-ray to gamma-ray fluence ratio of $1.4\\pm 0.4$ is\nthe highest of all SAX bursts \\cite{Frontera99}. The peak energy\n$E_p\\sim 60$ keV is softer than that measured in other \\S\\ bursts.\n%\n\\item\nAn isolated X-ray precursor occurs about 180 s before the main event (Figures\n1--2). Onset of X-ray emission before the gamma-rays has been observed\nin other GRBs \\cite{Laros84,Murakami91,Zand99,Feroci99}, but only another isolated X-ray \nprecursor, started about 25~s before the start of $\\gamma$--rays, has been\nreported thus far \\cite{Laros84}.\n%\n\\item\nThe X-ray afterglow light curve is peculiar. As can be seen from\nFigure 4 (top), the afterglow emission is undetectable during first \n2 hours of the start of the NFI observations (2$\\sigma$ upper\nlimit of $1.5 \\times 10^{-13} \\ergcms$ in 2-10~keV), after which it rises\nrapidly to (5$\\pm 1) \\times 10^{-13} \\ergcms$, then undergoes a decline in the \ntypical power law fashion (index $\\delta \\,= \\, 1.31^{+0.44}_{-0.39}$).\nThus at least during epoch 11--13 hrs the afterglow is undetectable.\n%\n\\end{enumerate}\n\nItem (2) is an uncommon feature of GRBs. Assuming the synchrotron shock\nmodel (e.g., Piran 1999 \\nocite{Piran99}), it implies a \nan initial fireball expansion Lorentz factor smaller than that found in other GRBs\n(e.g., Frontera et~al. 1999\\nocite{Frontera99}).\n\\\\\nItem (3) is the most interesting and mysterious aspect of this burst.\nIn all SAX observed bursts to date, the afterglow emission at X-ray\nwavelengths begins more or less as soon as the main gamma-ray event\nends. Also in the case of GRB981226 we have evidence that the afterglow starts\nduring the main event. Indeed the peak flux of the X-ray prompt emission \n($\\gtrsim 300$~$\\mu$Jy) has the same order of magnitude as that\n(173$\\pm27$~$\\mu$Jy at 8.46~GHz) observed in the radio band $\\sim$10 days after the \nburst \\cite{Frail99b}. On the contrary, the X-ray flux measured by us about 13~hrs \nafter the \nmain event is about 4 orders of magnitude lower than the radio peak flux. Now\nsimilar values of peak fluxes in X-rays and in the radio band are \nexpected in quasi-adiabatic cooling shocks of relativistically expanding \nfireballs \\cite{Sari98}. \nThus a simplest conclusion is that the afterglow emission began immediately after \nthe burst \n(phase E in fig.~4), rather than 13~hrs later, and then it was strongly reduced or\ncompletely ceased, and eventually restarted 13 hours after the burst.\nIf we accept this explanation, then we must explain the physical cause for this\nrather extended gap.\n\\\\\nThe afterglow emission is attributed to the forward shock, i.e. shock\nof the ambient gas particles swept up by the advancing blast wave.\nA cessation of X-ray afterglow would require that there be no ambient\ngas (or very reduced density) and the resurgence of the afterglow at\n13 hr would then require increased density -- in short, a cavity\nsurrounding the explosion. The rapid decline of the X-ray and radio\nafterglow require that the ambient density not be a constant but\ndecrease with the radius \\cite{Chevalier99}.\n\nIndeed, one expects such a circumburst medium around massive stars to\nhave a complicated geometry. For example, a star which exploded as a\nblue supergiant would have suffered two episodes of mass loss. During\nthe first phase, the red supergiant phase, the wind speed is low\nleading to a rich circumstellar medium. During the next phase, the blue\nsupergiant phase, the fast blue supergiant wind sweeps up the\ncircumburst medium shaped by the red supergiant wind with the\nnet result of a low-density cavity surrounded by a dense\nshell at the outskirts. \n\nIf one accepts this explanation then we have found an additional evidence\nlinking GRBs to massive stars. In this scenario, we find a satisfying\nexplanation also for item (1) and (2). The X-ray richness is because the\nblast wave picks up matter (baryons) as it tunnels through the envelope\nof the massive star. The resulting large baryon content leads to lower\n$E_p$ and hence more X-ray emission. We attribute the percursor\nemission to emission from shock breakout that is a natural consequence\nof models in which GRBs arise from the death of massive stars \\cite{MacFadyen99}.\n\nThe rapid decline in the radio and X-ray afterglow is because of\nthe radial density gradient in the circumburst medium. Finally, the\nabsence of an optical afterglow could well be due to extinction towards\nthe GRB. If this GRB arises from death of a massive star then most\nlikely the progenitor was in a dusty region and hence the extinction.\n\nWe tend to exclude that the X-ray increase of the afterglow observed $\\sim$13~hrs \nafter the burst is a signature of a supernova emission. Given that the X-ray\nflux is similar to that measured for the X-ray counterpart of SN1998bw \\cite{Pian99}, one would\nexpect a moderate distance for GRB981226 (z$\\lesssim$~0.01) and therefore a significant \ndetection in the optical band also in the case of a large extinction.\n\n\n\\acknowledgements\n\nThis research is supported by the Italian Space Agency (ASI). \nWe thank the \\S\\\nmission director R.C.\\ Butler and the teams of the \\S\\ Operative Control\nCenter and Scientific Data Center\nfor their efficient and enthusiastic support to the GRB alert program. \n\n\\newpage\n\n\\begin{thebibliography}{99}\n\n\\bibitem[Amati et~al.\\ 1997]{Amati97}\nAmati, L. \\etal 1997, SPIE Proceedings, 3114, 176\n%\n\\bibitem[Band et~al.\\ 1993]{Band93}\nBand, D. \\etal 1993, \\apj, 413, 281\n%\n\\bibitem[Bloom et~al.\\ 1998]{Bloom98}\nBloom, J.S. \\etal 1998, GCN 182\n%\n%\\bibitem[Bloom et~al.\\ 1999]{Bloom99}\n%Bloom, J.S. et~al. 1999, Nature, submitted (astro-ph/9905301)\n%\n\\bibitem[Boella et~al.\\ 1997a]{Boella97a}\nBoella, G. \\etal 1997a, \\aaps, 122, 299\n%\n\\bibitem[Boella et~al.\\ 1997b]{Boella97b}\nBoella, G. \\etal 1997b, \\aaps, 122, 327\n%\n\\bibitem[Castro-Tirado \\etal 1998]{Castro-Tirado98}\nCastro-Tirado, A.J. \\etal 1998, GCN 172\n%\n\\bibitem[Chevalier and Li 1999]{Chevalier99}\nChevalier, R.~A. \\& Li, Z.~Y. 1999, submitted to ApJ (astro-ph/9908272)\n%\n\\bibitem[Costa et~al.\\ 1997]{Costa97}\nCosta, E. \\etal 1997, Nature, 387, 783\n%\n\\bibitem[Di Ciolo et~al.\\ 1998]{Diciolo98}\nDi Ciolo, L. et~al. 1998, IAU Circ. 7074\n%\n\\bibitem[Feroci et~al.\\ 1999]{Feroci99}\nFeroci, M. \\etal 1999, in preparation.\n%\n\\bibitem[Fiore et~al. \\ 1999a]{Fiore99a}\nFiore, F. \\etal 1999a, at the site $http://www.sdc.asi.it/software/cookbook$\n%\n\\bibitem[Fiore et~al. \\ 1999b]{Fiore99b}\nFiore, F. \\etal 1999b, in Proceedings of the first XMM workshop \n\"Science with XMM\" (ed. M. Dahlem), at the site\n$http://astro.estec.esa.nl/XMM/news/ws1$\\_$top.html$ (astro-ph/9811149)\n%\n\\bibitem[Frail et~al. 1999a]{Frail99a}\nFrail, D.A. \\etal 1999a, GCN 195\n%\n\\bibitem[Frail et~al. \\ 1999b]{Frail99b}\nFrail, D.A. \\etal 1999b, \\apj, 525, L81 \n%\n\\bibitem[Frontera et~al.\\ 1997]{Frontera97}\nFrontera, F., Costa, E., Dal~Fiume, D., Feroci, M., Nicastro, L., Orlandini,\n M., Palazzi, E., and Zavattini, G. 1997, \\aaps, 122, 357\n%\n\\bibitem[Frontera et~al.\\ 1998]{Frontera98}\nFrontera, F. \\etal 1998, IAU Circ. 7078\n%\n\\bibitem[Frontera et~al.\\ 1999]{Frontera99}\nFrontera, F. \\etal 1999, \\apjs, accepted (astro-ph/9911228) \n%\n\\bibitem[Galama et~al.\\ 1999a]{Galama98} \nGalama, T. \\etal 1998, GCN 172\n%\n\\bibitem[Giommi et~al.\\ 1991]{Giommi91} \nGiommi, P. \\etal 1991, ASP Conf. Ser., 25, 100\n%\n\\bibitem[Hurley 1999]{Hurley99}\nHurley, K. 1999, GCN 290\n%\n\\bibitem[Jager et~al.\\ 1997]{Jager97}\nJager, R., \\etal 1997, \\aaps, 125, 557\n%\n\\bibitem[Laros et~al.\\ 1984]{Laros84}\nLaros, J.G. \\etal 1984, ApJ, 286, 681\n%\n\\bibitem[Lindgren \\etal 1998]{Lindgren99}\nLindgren, B. \\etal 1999, GCN 190\n%\n\\bibitem[MacFadyen et~al. 1999]{MacFadyen99}\nMacFadyen A.~I. \\etal 1999, submitted to \\apj (astro-ph/9910034)\n%\n\\bibitem[M\\'esz\\'aros and Rees 1997]{Meszaros97}\nM\\'esz\\'aros, P. and Rees, M.J. 1997, \\apj, 476, 232\n\n\\bibitem[Morrison and McCammon 1983]{Morrison83}\nMorrison R. and McCammon, D. 1983, \\apj, 270, 119\n%\n\\bibitem[Murakami et~al. 1991]{Murakami91}\nMurakami, T., \\etal 1991, Nature 350, 592\n%\n\\bibitem[Paciesas et~al. 1999]{Paciesas99}\nPaciesas, W.S. et~al. 1999, \\apjs, 122, 465\n%\n\\bibitem[Parmar et~al.\\ 1997]{Parmar97}\nParmar, A.N. \\etal 1997, \\aaps, 122, 309\n%\n\\bibitem[Pian et~al. 1999]{Pian99}\nPian, E. \\etal 1999, \\aas, 138, 463\n%\n\\bibitem[Piran 1999]{Piran99}\nPiran, T. 1999, Phys. Rep., 314, 575\n%\n\\bibitem[Piro et~al. 1998]{Piro98}\nPiro, L., \\etal 1998, \\aa, 331, L41\n%\n\\bibitem[Rhoads et~al. 1998]{Rhoads98}\nRhoads, J.E. 1998, GCN 181\n%\n%\\bibitem[Rhoads 1999]{Rhoads99}\n%Rhoads, J.E. 1999, \\apj, 525, 737\n%\n\\bibitem[Sari, Piran and Narayan 1998]{Sari98}\nSari, R., Piran, T., and Narayan, R. 1998, \\apj, 497, L17\n%\n%\\bibitem[Sari, Piran and Halpern 1999]{Sari99}\n%Sari, R., Piran, T. and Halpern, J.P. 1999, \\apj, 519, L17\n%\n%\\bibitem[Sari and Piran 1999]{Sari99}\n%Sari, R. and Piran, T. 1999, \\aas, 138, 537\n%\n\\bibitem[Schaefer et~al. 1998]{Schaefer98}\nSchaefer, B.E. \\etal 1998, GCN 185\n%\n%\\bibitem[Tavani 1996]{Tavani96}\n%Tavani, M. 1996, \\apj, 466, 768\n%\n%\\bibitem[Tavani 1997]{Tavani97}\n%Tavani, M. 1997, \\apj, 483, L87\n%\n\\bibitem[Taylor, Frail and Kulkarni 1999]{Taylor99}\nTaylor, G.B., Frail, D.A., and Kulkarni, S.R. 1999, GCN 350\n%\n%\\bibitem[van~Paradijs et~al.\\ 1997]{vanparadijs}\n%van~Paradijs, J. \\etal 1997, \\nat, 386, 686\n%\n%\\bibitem[Wijers et~al.\\ 1997]{Wijers97}\n%Wijers, R.A.M.J., Rees, M.J., and M\\'esz\\'aros, P. 1997, \\mnras, \n%\n\\bibitem[Wozniak 1998]{Wozniak98}\nWozniak, P.R. 1998, GCN 177\n%\n\\bibitem[Yoshida et~al.\\ 1999]{Yoshida99}\nYoshida, A. \\etal 1999, \\aas, 138, 433\n%\n\\bibitem[in 't Zand at~al.\\ 1999]{Zand99}\nin 't Zand \\etal 1999, \\apj, 516, L57\n%\n\\end{thebibliography}\n\n\\clearpage\n\n\\begin{deluxetable}{cccccc}\n%\\small\n%\\footnotesize\n%\\scriptsize\n\\tablewidth{0pt}\n\\tablenum{1}\n\\tablecaption{Spectral evolution of GRB981226 prompt emission}\n\\label{table1}\n\\tablehead{\nSect. & Model \\tablenotemark{(a)} &\n$\\alpha$ & \n$\\beta$ & \nE$_{p}$(keV) &\n$\\chi^{2}_{\\nu}$ (dof) \n} \n\\startdata\nA & Power-law & $-$1.97 $\\pm$ 0.41 & & & 1.32 (7) \\nl\n & Band law& $-$1.88 $\\pm$ 0.21 & $<$ $-$2.0 & $<$ 4 & 1.16 (5) \\nl\n & & & & & \\nl\nB & Power-law & $-$1.66 $\\pm$ 0.07 & & & 1.35 (7) \\nl\n & Band law & $-$1.25 $\\pm$ 0.25 & $-$2.6 $\\pm$ 0.7 & 61 $\\pm$ 15 & 0.91 (5) \\nl\n & & & & & \\nl\nC & Power-law & $-$2.14 $\\pm$ 0.14 & & & 0.85 (7) \\nl\n & Band law& $-$1.76 $\\pm$ 0.38 & $<$ $-$2.0 & $<$ 7 & 1.2 (5) \\nl\n & & & & & \\nl\nD & Power-law & $-$2.17 $\\pm$ 0.13 & & & 1.4 (7) \\nl\n & Band law& $-$1.83 $\\pm$ 0.28 & $<$ $-$2.0 & $<$ 6 & 2.1 (5) \\nl\n & & & & & \\nl\nE & Power-law & $-$2.06 $\\pm$ 0.16 & & & 1.36 (7) \\nl\n\\enddata\n\\tablenotetext{(a)}{Band law refers to \nthe smoothed broken power-law proposed by Band et al. (1993): $\\alpha$\nand $\\beta$ are the power--law photon \nindices below and above the break energy E$_{0}$, respectively. \nE$_{p}$=E$_{0}$(2+$\\alpha$)} \n\\end{deluxetable}\n\n\n\n\n\n\\clearpage\n\n% Figure 1\n\\figcaption[fig1.ps]{Light curves of GRB981226 in three energy bands,\nafter background subtraction (see text). The zero abscissa corresponds to 1998 Dec. 26,\n09:47:25 UT. The time slices on which the spectral analysis was performed \nare indicated by vertical dashed lines. The X-ray precursor start time corresponds to\n$-185$~s.}\n\n% Figure 2\n\\figcaption[fig2.ps]{$\\nu F_\\nu$ spectrum of the 5 time slices in which we\ndivided the burst time profile ($\\nu$ is the photon\nenergy in keV and $F_\\nu$ is the specific energy flux in\nkeV~cm$^{-2}$~s$^{-1}$~keV$^{-1}$). Continuous curves for slices A, C, D and E \nrepresent the power-law fits to WFC data, while that for slice B represents the\nfit with a Band law \\cite{Band93} to WFC+GRBM data (see also Table~1). The dashed lines \nrepresent the extrapolations of the spectral fits to higher energies.}\n\n% Figure 3\n\\figcaption[fig3.ps]{False colour MECS images (2-10~keV) of the field\nof GRB981226 obtained during TOO1 (left) and TOO~2 (right). }\n\n\n%Figure 4\n\\figcaption[fig4.ps] {{\\it Top:} 2-10 keV MECS light curve of \n1SAX J2329.6-2356 in time bins of 7000~s each. Vertical bars represent \n$\\pm$1~$\\sigma$ errors.\\\\\n{\\it Bottom:} 2-10 keV WFC and MECS data. Superposed to the data points is the \npower law curve (dashed line) obtained from the best fit to the late afterglow data, \nexcept the first bin with the upper limit. Also shown is the uncertainty region in the \nslope (1.31) derived from the best fit at 90\\% confidence level. \n}\n\n\n%\\fi\n\n\\clearpage\n\n\\begin{figure}\n\\label{fig:timeprofile}\n\\plotone{fig1.ps}\n\\ifPreprintMode\n\\vspace{2cm}\n\n\n\\fi\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\label{fig:power}\n\\epsscale{0.5}\n\\plotone{fig2.ps}\n\\ifPreprintMode\n\\vspace{2cm}\n\n\\fi\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\label{fig:mecs_image_1_2}\n\\epsscale{1.0}\n\\vspace{-1cm}\n\\plotone{fig3.ps}\n\\ifPreprintMode\n\n\\fi\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\label{fig:lc}\n\\epsscale{0.9}\n\\plotone{fig4.ps}\n\\ifPreprintMode\n\n\\fi\n\\end{figure}\n\n\\end{document}\n\n\n"
}
] |
[
{
"name": "astro-ph0002527.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem[Amati et~al.\\ 1997]{Amati97}\nAmati, L. \\etal 1997, SPIE Proceedings, 3114, 176\n%\n\\bibitem[Band et~al.\\ 1993]{Band93}\nBand, D. \\etal 1993, \\apj, 413, 281\n%\n\\bibitem[Bloom et~al.\\ 1998]{Bloom98}\nBloom, J.S. \\etal 1998, GCN 182\n%\n%\\bibitem[Bloom et~al.\\ 1999]{Bloom99}\n%Bloom, J.S. et~al. 1999, Nature, submitted (astro-ph/9905301)\n%\n\\bibitem[Boella et~al.\\ 1997a]{Boella97a}\nBoella, G. \\etal 1997a, \\aaps, 122, 299\n%\n\\bibitem[Boella et~al.\\ 1997b]{Boella97b}\nBoella, G. \\etal 1997b, \\aaps, 122, 327\n%\n\\bibitem[Castro-Tirado \\etal 1998]{Castro-Tirado98}\nCastro-Tirado, A.J. \\etal 1998, GCN 172\n%\n\\bibitem[Chevalier and Li 1999]{Chevalier99}\nChevalier, R.~A. \\& Li, Z.~Y. 1999, submitted to ApJ (astro-ph/9908272)\n%\n\\bibitem[Costa et~al.\\ 1997]{Costa97}\nCosta, E. \\etal 1997, Nature, 387, 783\n%\n\\bibitem[Di Ciolo et~al.\\ 1998]{Diciolo98}\nDi Ciolo, L. et~al. 1998, IAU Circ. 7074\n%\n\\bibitem[Feroci et~al.\\ 1999]{Feroci99}\nFeroci, M. \\etal 1999, in preparation.\n%\n\\bibitem[Fiore et~al. \\ 1999a]{Fiore99a}\nFiore, F. \\etal 1999a, at the site $http://www.sdc.asi.it/software/cookbook$\n%\n\\bibitem[Fiore et~al. \\ 1999b]{Fiore99b}\nFiore, F. \\etal 1999b, in Proceedings of the first XMM workshop \n\"Science with XMM\" (ed. M. Dahlem), at the site\n$http://astro.estec.esa.nl/XMM/news/ws1$\\_$top.html$ (astro-ph/9811149)\n%\n\\bibitem[Frail et~al. 1999a]{Frail99a}\nFrail, D.A. \\etal 1999a, GCN 195\n%\n\\bibitem[Frail et~al. \\ 1999b]{Frail99b}\nFrail, D.A. \\etal 1999b, \\apj, 525, L81 \n%\n\\bibitem[Frontera et~al.\\ 1997]{Frontera97}\nFrontera, F., Costa, E., Dal~Fiume, D., Feroci, M., Nicastro, L., Orlandini,\n M., Palazzi, E., and Zavattini, G. 1997, \\aaps, 122, 357\n%\n\\bibitem[Frontera et~al.\\ 1998]{Frontera98}\nFrontera, F. \\etal 1998, IAU Circ. 7078\n%\n\\bibitem[Frontera et~al.\\ 1999]{Frontera99}\nFrontera, F. \\etal 1999, \\apjs, accepted (astro-ph/9911228) \n%\n\\bibitem[Galama et~al.\\ 1999a]{Galama98} \nGalama, T. \\etal 1998, GCN 172\n%\n\\bibitem[Giommi et~al.\\ 1991]{Giommi91} \nGiommi, P. \\etal 1991, ASP Conf. Ser., 25, 100\n%\n\\bibitem[Hurley 1999]{Hurley99}\nHurley, K. 1999, GCN 290\n%\n\\bibitem[Jager et~al.\\ 1997]{Jager97}\nJager, R., \\etal 1997, \\aaps, 125, 557\n%\n\\bibitem[Laros et~al.\\ 1984]{Laros84}\nLaros, J.G. \\etal 1984, ApJ, 286, 681\n%\n\\bibitem[Lindgren \\etal 1998]{Lindgren99}\nLindgren, B. \\etal 1999, GCN 190\n%\n\\bibitem[MacFadyen et~al. 1999]{MacFadyen99}\nMacFadyen A.~I. \\etal 1999, submitted to \\apj (astro-ph/9910034)\n%\n\\bibitem[M\\'esz\\'aros and Rees 1997]{Meszaros97}\nM\\'esz\\'aros, P. and Rees, M.J. 1997, \\apj, 476, 232\n\n\\bibitem[Morrison and McCammon 1983]{Morrison83}\nMorrison R. and McCammon, D. 1983, \\apj, 270, 119\n%\n\\bibitem[Murakami et~al. 1991]{Murakami91}\nMurakami, T., \\etal 1991, Nature 350, 592\n%\n\\bibitem[Paciesas et~al. 1999]{Paciesas99}\nPaciesas, W.S. et~al. 1999, \\apjs, 122, 465\n%\n\\bibitem[Parmar et~al.\\ 1997]{Parmar97}\nParmar, A.N. \\etal 1997, \\aaps, 122, 309\n%\n\\bibitem[Pian et~al. 1999]{Pian99}\nPian, E. \\etal 1999, \\aas, 138, 463\n%\n\\bibitem[Piran 1999]{Piran99}\nPiran, T. 1999, Phys. Rep., 314, 575\n%\n\\bibitem[Piro et~al. 1998]{Piro98}\nPiro, L., \\etal 1998, \\aa, 331, L41\n%\n\\bibitem[Rhoads et~al. 1998]{Rhoads98}\nRhoads, J.E. 1998, GCN 181\n%\n%\\bibitem[Rhoads 1999]{Rhoads99}\n%Rhoads, J.E. 1999, \\apj, 525, 737\n%\n\\bibitem[Sari, Piran and Narayan 1998]{Sari98}\nSari, R., Piran, T., and Narayan, R. 1998, \\apj, 497, L17\n%\n%\\bibitem[Sari, Piran and Halpern 1999]{Sari99}\n%Sari, R., Piran, T. and Halpern, J.P. 1999, \\apj, 519, L17\n%\n%\\bibitem[Sari and Piran 1999]{Sari99}\n%Sari, R. and Piran, T. 1999, \\aas, 138, 537\n%\n\\bibitem[Schaefer et~al. 1998]{Schaefer98}\nSchaefer, B.E. \\etal 1998, GCN 185\n%\n%\\bibitem[Tavani 1996]{Tavani96}\n%Tavani, M. 1996, \\apj, 466, 768\n%\n%\\bibitem[Tavani 1997]{Tavani97}\n%Tavani, M. 1997, \\apj, 483, L87\n%\n\\bibitem[Taylor, Frail and Kulkarni 1999]{Taylor99}\nTaylor, G.B., Frail, D.A., and Kulkarni, S.R. 1999, GCN 350\n%\n%\\bibitem[van~Paradijs et~al.\\ 1997]{vanparadijs}\n%van~Paradijs, J. \\etal 1997, \\nat, 386, 686\n%\n%\\bibitem[Wijers et~al.\\ 1997]{Wijers97}\n%Wijers, R.A.M.J., Rees, M.J., and M\\'esz\\'aros, P. 1997, \\mnras, \n%\n\\bibitem[Wozniak 1998]{Wozniak98}\nWozniak, P.R. 1998, GCN 177\n%\n\\bibitem[Yoshida et~al.\\ 1999]{Yoshida99}\nYoshida, A. \\etal 1999, \\aas, 138, 433\n%\n\\bibitem[in 't Zand at~al.\\ 1999]{Zand99}\nin 't Zand \\etal 1999, \\apj, 516, L57\n%\n\\end{thebibliography}"
}
] |
astro-ph0002528
|
The Final Fate of Coalescing Compact Binaries:\protect\\ From Black Hole to Planet Formation
|
[
{
"author": "Frederic A.~Rasio"
}
] |
Coalescing compact binaries are thought to be involved in a wide variety of astrophysical phenomena. In particular, they are important sources of gravitational radiation for both ground-based and space-based laser-interferometer detectors, and they may be sources of supernova explosions or gamma-ray bursts. Mergers of two white dwarfs may produce neutron stars with peculiar properties, including perhaps millisecond radio pulsars sometimes accompanied by planets (as observed in PSR 1257+12). According to a widely held belief, the coalescence of two neutron stars should produce a rapidly rotating black hole surrounded by an accretion disk or torus, but this is by no means certain. This review paper focuses on the final hydrodynamic coalescence and merger of double neutron stars and double white dwarfs, and addresses the question of the nature of the final merger products.
|
[
{
"name": "paper.tex",
"string": "\\documentstyle[nato,epsf]{crckapb}\n\n\\def\\go{\\mathrel{\\raise.3ex\\hbox{$>$}\\mkern-14mu\\lower0.6ex\\hbox{$\\sim$}}}\n\\def\\lo{\\mathrel{\\raise.3ex\\hbox{$<$}\\mkern-14mu\\lower0.6ex\\hbox{$\\sim$}}}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{opening}\n\n\\title{The Final Fate of Coalescing Compact Binaries:\\protect\\\\\nFrom Black Hole to Planet Formation}\n\n\\author{Frederic A.~Rasio}\n\n\\institute{Department of Physics,\\\\\n Massachusetts Institute of Technology,\\\\\n Cambridge, MA 02139, USA}\n\n\\end{opening}\n\n\\runningtitle{Coalescing Compact Binaries}\n\n\\begin{document}\n\n\\begin{abstract}\nCoalescing compact binaries are thought to be involved in a wide variety \nof astrophysical phenomena. In particular, they are important sources of \ngravitational radiation for both ground-based and space-based laser-interferometer\ndetectors, and they may be sources of supernova explosions or gamma-ray bursts. \nMergers of two white dwarfs may produce neutron stars with peculiar\nproperties, including perhaps millisecond radio pulsars sometimes\naccompanied by planets (as observed in PSR 1257+12). According to a widely\nheld belief, the coalescence of two neutron stars should produce a rapidly \nrotating black hole surrounded by an accretion disk or torus, but this is by \nno means certain. This review paper focuses on the final hydrodynamic \ncoalescence and merger of double neutron stars and double white dwarfs, \nand addresses the question of the nature of the final merger products.\n\\end{abstract}\n\n\n\\section{Introduction}\n\nThe coalescence and merging of two compact stars into a single object\nis a very common end-point of close binary evolution. \nDissipation mechanisms such as friction in common envelopes, tidal dissipation,\nor the emission of gravitational radiation, are always present and cause \nthe orbits of close binary systems to decay. \nThis review will concentrate on the coalescence of compact binaries\ncontaining either two neutron stars (hereafter NS) or two white dwarfs (WD). \n\n\\subsection{Double Neutron Stars}\n\nMany theoretical models of gamma-ray bursts (GRBs) \nrely on coalescing NS \nbinaries to provide the energy of GRBs at\ncosmological distances (e.g., Eichler et al.\\ 1989; Narayan,\nPaczy\\'nski, \\& Piran 1992; M\\'esz\\'aros \\& Rees 1992;\nfor recent reviews see M\\'esz\\'aros 1999 and Piran 1999). \nThe close spatial association of some GRB afterglows \nwith faint galaxies at high redshifts may not be inconsistent\nwith a NS binary merger origin, in spite of the large recoil\nvelocities acquired by NS binaries at birth (Bloom,\nSigurdsson, \\& Pols 1999; but see also Bulik \\& Belczynski 2000).\nCurrently the most popular models all assume that the coalescence of two\nNS leads\nto the formation of a rapidly rotating black hole (BH) \nsurrounded by a torus of debris. \nEnergy can then be extracted either from the rotation of the \nKerr BH or from\nthe material in the torus so that, with sufficient beaming, the\ngamma-ray fluxes observed from even the most distant GRBs can be\nexplained (M\\'esz\\'aros, Rees, \\& Wijers 1999). However, it is important to\nunderstand the hydrodynamic processes taking place during the final \ncoalescence before making assumptions about its outcome. In particular,\nas will be argued below (\\S2.2), it is not clear that the coalescence of\ntwo $1.4\\,M_\\odot$ NS forms an object that will collapse to a BH\non a dynamical timescale,\nand it is not certain either that a significant amount of \nmatter will be ejected\nduring the merger to form an outer torus around the central object\n(Faber \\& Rasio 2000).\n\nCoalescing NS binaries are also important sources of gravitational\nwaves that may be directly detectable by the large laser interferometers \ncurrently under construction,\nsuch as LIGO (Abramovici et al.\\ 1992; see Barish \\& Weiss 1999 for a\nrecent pedagogical introduction) and VIRGO (Bradaschia et al.\\ 1990). \nIn addition to providing a major new confirmation of\nEinstein's theory of general relativity (GR), including the first direct\nproof of the existence of black holes (see, e.g., Flanagan \\& Hughes 1998;\nLipunov, Postnov, \\& Prokhorov 1997), the detection of gravitational\nwaves from coalescing binaries at cosmological distances could provide \naccurate independent measurements of the Hubble constant\nand mean density of the Universe (Schutz 1986; Chernoff \\& Finn 1993; \nMarkovi\\'c 1993). \nExpected rates of NS binary coalescence in the Universe, \nas well as expected event rates in laser interferometers, have \nnow been calculated by many groups. Although there is some disparity \nbetween various published results, the estimated rates are generally \nencouraging (see Kalogera 2000 for a recent review).\n\nMany calculations of gravitational wave emission from coalescing binaries \nhave focused on the waveforms emitted during the last few thousand orbits, \nas the frequency sweeps upward from $\\sim10\\,$Hz to $\\sim300\\,$Hz.\nThe waveforms in this frequency range, where the sensitivity of\nground-based interferometers \nis highest, can be calculated very accurately by \nperforming high-order post-Newtonian (PN)\nexpansions of the equations of \nmotion for two {\\it point masses\\/} (see, e.g., Owen \\& Sathyaprakash 1999\nand references therein). However, at the end of the inspiral, \nwhen the binary separation becomes comparable \nto the stellar radii (and the frequency is $\\go1\\,$kHz), \nhydrodynamics becomes important and the character \nof the waveforms must change. \nSpecial purpose narrow-band detectors that can sweep up frequency in real \ntime will be used to try to catch the last $\\sim10$ cycles of the gravitational\nwaves during the final coalescence\n(Meers 1988; Strain \\& Meers 1991). These ``dual recycling''\ntechniques are being tested right now on the German-British interferometer\nGEO 600 (Danzmann 1998). In this terminal phase of the coalescence,\nwhen the two stars merge together into a single object, \nthe waveforms contain information not just about the \neffects of GR, but also about the interior structure \nof a NS and the nuclear equation of state \n(EOS) at high density. \nExtracting this information from observed waveforms, \nhowever, requires detailed theoretical knowledge about all relevant\nhydrodynamic processes. \nIf the NS merger is followed by the formation \nof a BH, the corresponding gravitational radiation waveforms will also \nprovide direct information on the dynamics of rotating core collapse\nand the BH ``ringdown'' (see, e.g., Flanagan \\& Hughes 1998).\n\n\\subsection{Double White Dwarfs}\n\nCoalescing WD binaries have long been discussed as possible progenitors\nof Type~Ia supernovae (Iben \\& Tutukov 1984; Webbink 1984; Paczy\\'nski 1985;\nsee Branch et al.\\ 1995 for a recent review). To produce a supernova,\nthe total mass of the system must be above the Chandrasekhar mass. Given\nevolutionary considerations, this requires two C-O or O-Ne-Mg WD.\nYungelson et al.\\ (1994) showed that the expected merger rate for close WD\npairs with total mass exceeding the Chandrasekhar mass is consistent\nwith the rate of Type~Ia supernovae deduced from observations.\nAlternatively, a massive enough merger may collapse to form a rapidly\nrotating NS (Nomoto \\& Iben 1985; Colgate 1990).\nChen \\& Leonard (1993) speculated that most\nmillisecond pulsars in globular clusters might have formed in this way.\nIn some cases planets may also form in the disk of material ejected during\nthe coalescence and left in orbit around the central pulsar \n(Podsiadlowski, Pringle, \\& Rees 1991). Indeed the very\nfirst extrasolar planets were discovered\nin orbit around a millisecond pulsar, PSR B1257$+$12 (Wolszczan \\& Frail 1992).\nA merger of two magnetized WD might lead to the formation of \na NS with extremely high magnetic field, and this scenario has been \nproposed as a source of GRBs (Usov 1992). \n\nClose WD binaries are expected to be extremely abundant in our Galaxy,\neven though their direct detection remains very challenging (Han 1998;\nSaffer, Livio, \\& Yungelson 1999).\nIben \\& Tutukov (1984, 1986) predicted that $\\sim20$\\% of all binary stars \nproduce close WD pairs at the end of their stellar evolution.\nMore recently, theoretical estimates of the double WD formation rate\nin the Galaxy have converged to a value \n$\\simeq0.1\\,{\\rm yr}^{-1}$, with an uncertainty that may be only a factor of \ntwo (Han 1998; Kalogera 2000). \nThe most common systems should be those containing two low-mass helium WD.\nTheir final coalescence can produce an object massive enough\nto start helium burning. Bailyn (1993 and references therein) and others \nhave suggested that some\n``extreme horizontal branch'' stars in globular clusters \nmay be such helium-burning stars formed by the coalescence of two WD.\nPlanets in orbit around a massive WD may also form following\nthe binary coalescence (Livio, Pringle, \\& Saffer 1992).\n\nCoalescing WD binaries are also important sources of low-frequency \ngravitational waves\nthat should be easily detectable by future space-based laser interferometers.\nThe currently planned LISA (Laser Interferometer Space Antenna; see Folkner 1998)\nshould have an\nextremely high sensitivity (down to a characteristic strain $h\\sim10^{-23}$)\nto sources with frequencies in the range $\\sim10^{-4}\\,-\\,1\\,$Hz.\nHan (1998) estimated a WD merger rate \n$\\sim0.03\\,{\\rm yr}^{-1}$ in our own Galaxy. Individual coalescing systems \nand mergers may be detectable in the frequency range $\\sim10$--$100\\,$mHz. \nIn addition, the total number ($\\sim10^4$) of close WD binaries \nin our Galaxy emitting at lower frequencies $\\sim0.1$--$10\\,$mHz (the emission \nlasting for $\\sim10^2$--$10^4\\,$yr before final merging) should \nprovide a continuum background signal of amplitude \n$h\\sim10^{-20}$--$10^{-21}$ (Hils et al.\\ 1990).\nThe detection of the final burst of gravitational waves emitted during \nan actual merger would provide a unique opportunity to observe in ``real time'' \nthe hydrodynamic interaction between the two degenerate stars, possibly followed\nimmediately by a supernova explosion, nuclear outburst, or some other type of \nelectromagnetic signal.\n\n\n\\section{Coalescing Binary Neutron Stars}\n\n\\subsection{Hydrodynamics of Neutron Star Mergers}\n\nThe final hydrodynamic merger of two NS is driven by a combination\nof relativistic and fluid effects. Even in Newtonian gravity,\nan innermost stable circular orbit (ISCO) is imposed by\n{\\it global hydrodynamic instabilities\\/}, which can drive \na close binary system to rapid coalescence once the tidal interaction \nbetween the two stars becomes sufficiently strong.\nThe existence of these global instabilities \nfor close binary equilibrium configurations containing a compressible fluid, \nand their particular importance for binary NS systems, \nwere demonstrated for the first time by \nRasio \\& Shapiro (1992, 1994, 1995; hereafter RS1--3) \nusing numerical hydrodynamic calculations.\nThese instabilities can also be studied using analytic methods.\nThe classical analytic work for close binaries containing an\nincompressible fluid (e.g., Chandrasekhar 1969) was\nextended to compressible fluids in the work of Lai, Rasio, \\& Shapiro \n(1993a,b, 1994a,b,c, hereafter LRS1--5).\nThis analytic study confirmed the existence of dynamical \ninstabilities for sufficiently close binaries.\nAlthough these simplified analytic studies can give much physical\ninsight into difficult questions of global fluid instabilities, \nfully numerical calculations remain essential for establishing\nthe stability limits of close binaries accurately and for following \nthe nonlinear evolution of unstable systems all the way to complete \ncoalescence. \n\nA number of different groups have now performed such calculations, using\na variety of numerical methods and focusing on different aspects of the\n problem. Nakamura and collaborators (see Nakamura \\& Oohara 1998 and \nreferences therein)\nwere the first to perform 3D hydrodynamic calculations of binary \nNS coalescence, using a traditional Eulerian finite-difference code. \nInstead, RS used the \nLagrangian method SPH (Smoothed Particle Hydrodynamics). They focused\non determining the ISCO for initial binary models in strict\nhydrostatic equilibrium and calculating the emission of gravitational waves\nfrom the coalescence of unstable binaries. Many of the results of RS were\nlater independently confirmed by New \\& Tohline (1997) and Swesty,\nWang, \\& Calder (1999), who used completely\ndifferent numerical methods but also focused on stability questions, and \nby Zhuge, Centrella, \\& McMillan (1994, 1996), who also \nused SPH. Zhuge et al.\\ (1996) also explored in detail the dependence of\nthe gravitational wave signals on the initial NS spins. \nDavies et al.\\ (1994) and Ruffert et al.\\ (1996, 1997) have\nincorporated a treatment of the nuclear physics in their hydrodynamic\ncalculations (done using SPH and PPM codes, respectively), motivated\nby models of GRBs at cosmological distances.\nAll these calculations were performed in {\\it Newtonian gravity\\/}, with\nsome of the more recent studies adding an approximate treatment of\nenergy and angular momentum dissipation through the gravitational \nradiation reaction (e.g., Janka et al.\\ 1999; Rosswog et al.\\ 1999),\nor even a full treatment of PN gravity to lowest order (Ayal et al.\\ 2000;\nFaber \\& Rasio 2000).\n\nAll recent hydrodynamic calculations agree on\nthe basic qualitative picture that emerges for the final coalescence\n(see Fig.~1). As the ISCO is approached, the secular orbital\ndecay driven by gravitational wave emission is dramatically accelerated\n(see also LRS2, LRS3).\nThe two stars then plunge rapidly toward each other, and merge together \ninto a single object in just a few rotation periods. In the corotating \nframe of the binary, the relative radial velocity of the two stars always \nremains very subsonic, so that the evolution is nearly adiabatic.\nThis is in sharp contrast to the case of a head-on collision between\ntwo stars on a free-fall, radial orbit, where\nshock heating is very important for the dynamics (RS1; Shapiro 1998).\nHere the stars are constantly being held back by a (slowly receding)\ncentrifugal barrier, and the merging, although dynamical, is much more gentle. \nAfter typically $1-2$ orbital periods following first contact,\n the innermost cores of the \ntwo stars have merged and \na secondary instability occurs: {\\it mass shedding\\/} \nsets in rather abruptly. Material (typically $\\sim10$\\% of the total mass) \nis ejected through the outer Lagrange\npoints of the effective potential and spirals out rapidly.\nIn the final stage, the spiral arms widen and merge together, \nforming a nearly axisymmetric thick disk or torus around the inner, \nmaximally rotating dense core. \n\n\\begin{figure}\n\\epsfxsize=5.truein\n\\epsfbox{fig1.eps}\n\\caption{Post-Newtonian SPH calculation of the coalescence of two identical\nneutron stars modeled as simple $\\Gamma=3$ polytropes. Projections of a subset\nof all SPH particles onto the orbital ($x-y$) plane are shown at various times.\nUnits are such that $G=M=R=1$ where $M$ and $R$ are the mass and radius\nof each star initially. The orbital rotation is counter-clockwise. From\nFaber \\& Rasio (2000).}\n\\end{figure}\n\n\nIn GR, strong-field gravity between the masses in\na binary system is alone sufficient to drive a close circular \norbit unstable. In close NS binaries, GR effects combine nonlinearly\nwith Newtonian tidal effects so that the ISCO is encountered\nat larger binary separations and lower orbital frequency than \npredicted by Newtonian hydrodynamics alone, or GR alone for two point\nmasses. The combined effects\nof relativity and hydrodynamics on the stability of close compact\nbinaries have only very recently begun to be studied,\nusing both analytic approximations\n(basically, PN generalizations of LRS; see, e.g., \nLai \\& Wiseman 1997; Lombardi, Rasio, \\& Shapiro 1997; \nShibata \\& Taniguchi 1997), as well as numerical \ncalculations in 3D incorporating simplified treatments of \nrelativistic effects \n(e.g., Baumgarte et al.\\ 1998; Marronetti, Mathews \\& Wilson 1998; \nWang, Swesty, \\& Calder 1998; Faber \\& Rasio 2000).\n\nSeveral groups have been working on a fully general relativistic\ncalculation of the final coalescence, combining the techniques of \nnumerical relativity and numerical hydrodynamics in 3D\n(Baumgarte, Hughes, \\& Shapiro 1999;\nLandry \\& Teukolsky 2000; Seidel 1998; Shibata \\& Uryu 2000). \nHowever this work is still in its infancy, and only very preliminary results\nof test calculations have been reported so far.\n\n\\subsection{Black Hole Formation}\n\nThe final fate of a NS--NS merger depends crucially on the NS EOS,\nand on the extraction of angular momentum from the system during the \nfinal merger. For a stiff NS EOS, it is by no means\ncertain that the core of the final merged configuration will collapse\non a dynamical timescale to form a BH. One reason is that the Kerr\nparameter $J/M^2$ of the core may exceed unity for extremely stiff\nEOS (Baumgarte et al.\\ 1998), although Newtonian and PN \nhydrodynamic calculations suggest that this is never the case\n(see, e.g., Faber \\& Rasio 2000). \nMore importantly, the rapidly rotating core may in fact be \ndynamically stable. \n\nTake the obvious example of a system containing two \nidentical $1.35\\,M_\\odot$ NS. The total baryonic mass of the system\nfor a stiff NS EOS is then about $3\\,M_\\odot$. Almost independent of \nthe spins of the NS, all hydrodynamic calculations suggest that about\n$10\\%$ of this mass will be ejected into the outer torus, leaving at\nthe center a {\\it maximally rotating\\/} object with baryonic mass \n$\\simeq2.7\\,M_\\odot$ (Any hydrodynamic merger process that leads to mass\nshedding will produce a maximally rotating object since the system will\nhave ejected just enough mass and angular momentum to reach its new,\nstable quasi-equilibrium state). Most stiff NS EOS (including the\nwell-known ``AU'' and ``UU'' EOS of Wiringa et al.\\ 1988; see Akmal et al.\\\n1998 for a recent update) allow stable,\nmaximally rotating NS with baryonic masses exceeding\n$3\\,M_\\odot$ (Cook, Shapiro, \\& Teukolsky 1994), i.e., well above the mass\nof the final merger core. Differential rotation (not taken into account in the\ncalculations of Cook et al.\\ 1994) can further increase this maximum stable \nmass very significantly (see Baumgarte, Shapiro, \\& Shibata 2000).\nThus the hydrodynamic merger of two NS with stiff EOS and realistic\nmasses is not expected to produce a BH. This expectation is confirmed by\nthe preliminary full-GR calculations of Shibata \\& Uryu (2000), for\npolytropes with $\\Gamma=2$, which indicate collapse to a BH only\nwhen the two NS are initially very close to the maximum stable\nmass. \n\nFor {\\it slowly rotating\\/} stars, the same stiff NS EOS give\nmaximum stable baryonic masses in the range $2.5-3\\,M_\\odot$, which may\nor may not exceed the total merger core mass. Therefore, collapse to\na BH could still occur on a timescale longer than the dynamical timescale,\nfollowing a significant loss of angular momentum.\nIndeed, processes such as \nelectromagnetic radiation, neutrino emission, and the development of\nvarious secular instabilities (e.g., r-modes), which may lead to angular\nmomentum losses, take place on timescales much longer than the dynamical\ntimescale (see, e.g., Baumgarte \\& Shapiro 1998, who show that\nneutrino emission is probably negligible). These processes are\ntherefore decoupled from the hydrodynamics of the coalescence.\nUnfortunately their\nstudy is plagued by many fundamental uncertainties in the microphysics.\n\n\n\\subsection{The Importance of the Neutron Star Spins}\n\nThe question of the final fate of the merger could also depend crucially\non the NS spins and on the \nevolution of the fluid vorticity during the final coalescence.\nClose NS binaries are likely to be {\\it nonsynchronized\\/}. Indeed,\nthe tidal synchronization time \nis almost certainly much longer than the orbital decay\ntime (Kochanek 1992; Bildsten \\& Cutler 1992).\nFor NS binaries that are far from synchronized,\nthe final coalescence involves\nsome new, complex hydrodynamic processes (Rasio \\& Shapiro 1999).\n\nConsider for example the case of an irrotational system (containing\ntwo nonspinning stars at large separation; see LRS3).\nBecause the two stars appear to be counter-spinning in the corotating\nframe of the binary, a {\\it vortex sheet\\/} (where the tangential velocity\njumps discontinuously by $\\Delta v\\sim 0.1\\,c$) appears when the stellar \nsurfaces come into contact.\nSuch a vortex sheet is Kelvin-Helmholtz unstable on all \nwavelengths and the hydrodynamics is therefore extremely\ndifficult to model accurately given the limited spatial\nresolution of 3D calculations.\nThe breaking of the vortex sheet generates some turbulent\nviscosity so that the final configuration may no longer be\nirrotational. In numerical simulations, however, vorticity is\nquickly generated through spurious shear viscosity, and the \nmerger remnant is observed to evolve rapidly (in just a few\nrotation periods) toward uniform rotation.\n\nThe final fate of the merger could be affected drastically by these\nprocesses. In particular, the shear flow inside the merging stars \n(which supports a highly triaxial shape; see Rasio \\& Shapiro 1999) may\nin reality persist long enough to allow a large fraction of the total\nangular momentum\nin the system to be radiated away in gravitational waves during the\nhydrodynamic phase of the coalescence. In this case\nthe final merged core may resemble a Dedekind ellipsoid, i.e., it will have\na triaxial shape supported entirely by internal fluid motions, but with\na stationary shape in the inertial frame (so that it no longer\nradiates gravitational waves). \nThis state will be reached on the gravitational radiation reaction\ntimescale, which is no more than a few tens of rotation periods.\nOn the (much longer) {\\it viscous timescale\\/}, the core will\nthen evolve to a uniform, slowly rotating state and will probably collapse \nto a BH.\nIn contrast, in all 3D numerical simulations performed to date,\nthe shear is quickly dissipated, so that gravitational radiation\nnever gets a chance to extract more than a small fraction ($\\lo10$\\%)\nof the angular momentum, and the final core appears to be a uniform,\nmaximally rotating object (stable to collapse) exactly as in calculations starting\nfrom synchronized binaries. However this behavior is most likely\nan artefact of the large spurious shear viscosity present in the\n3D simulations. \n\nIn addition to their obvious significance for gravitational wave emission,\nthese issues are also of great importance for models \nof GRBs that depend on energy extraction from a torus of material around\nthe central BH. Indeed, if a large fraction of the total angular momentum\nis removed by the gravitational waves, rotationally-induced mass shedding \nmay not occur at all during the merger, eventually\nleaving a BH with no surrounding \nmatter, and no way of extracting energy from the system.\nNote also that, even without any additional loss of angular momentum\nthrough gravitational radiation, PN effects tend to reduce drastically the\namount of matter ejected during the merger (Faber \\& Rasio 2000).\n\n\n\\section{Coalescing White Dwarf Binaries}\n\n\\subsection{Hydrodynamics of White Dwarf Mergers}\n\nThe results of RS3 for polytropes with $\\Gamma=5/3$ show that\nhydrodynamics also plays an important role in the coalescence of two\nWD, either because of dynamical instabilities of the\nequilibrium configuration, or following the onset of dynamically unstable\nmass transfer. Systems with mass ratios $q\\approx1$ must evolve into deep contact\nbefore they become dynamically unstable and merge. Instead, equilibrium\nconfigurations for binaries with $q$ \nsufficiently far from unity never become dynamically unstable. However,\nonce these binaries reach their Roche limit, dynamically unstable mass\ntransfer occurs and the less massive star is completely disrupted after\na small number ($<10$) of orbital periods (see also Benz et al.\\ 1990). In both\ncases, the final merged configuration is an axisymmetric, rapidly rotating \nobject with a core -- thick disk structure similar to that obtained for coalescing \nNS (RS2, RS3; see also Mochkovitch \\& Livio 1989). \n\n\n\\subsection{The Final Fate: Collapse to a Neutron Star? Planets?}\n\nFor two massive enough WD, the merger product may be well above the \nChandrasekhar mass $M_{Ch}$.\nThe object may therefore explode as a (Type~Ia) supernova, or perhaps collapse \nto a NS. The rapid rotation and possibly high mass (up to $\\sim2M_{Ch}$) \nof the object must be taken into account for determining its final fate.\nUnfortunately, rapid rotation and the possibility of starting from an object\nwell above the Chandrasekhar limit have not been taken into account in \nmost previous theoretical calculations of ``accretion-induced collapse''\n (AIC), which consider a \nnonrotating WD just below the Chandrasekhar limit \naccreting matter slowly and quasi-spherically (e.g., Canal et al.\\ 1990; \nNomoto \\& Kondo 1991; see Fryer et al.\\ 1999 for a\nrecent 2-D SPH calculation including rotation). \nUnder these assumptions it is found that collapse to a NS\nis possible only for a narrow range of initial conditions.\nIn most cases, a supernova explosion follows the ignition of the\nnuclear fuel in the degenerate core.\nHowever, the fate of a much more massive object with substantial \nrotational support and large deviations from spherical symmetry \n(as would be formed by dynamical coalescence) may be very different.\n\nIf a NS does indeed form, and later accretes some of the \nmaterial ejected during the coalescence, a millisecond radio pulsar \nmay emerge. Planets around this millisecond pulsar may be formed\nat large distances $\\sim1\\,$AU following the viscous evolution of the remaining\nmaterial in the outer disk \n(Podsiadlowski, Pringle \\& Rees 1991; Phinney \\& Hansen 1993).\nThis is one of the possible formation scenarios for the extraordinary\nplanetary system discovered around the millisecond pulsar PSR~B1257$+$12\n(see Wolszczan 1999 for a recent update; Podsiadlowski 1993 for\nalternative planet formation scenarios). \nThis system contains three \nconfirmed Earth-mass planets in quasi-circular orbits \n(Wolszczan \\& Frail 1992; Wolszczan 1994). \nThe planets have masses of $0.015/\\sin i_1\\,\\rm M_\\oplus$, \n$3.4/\\sin i_2\\,\\rm M_\\oplus$, and $2.8/\\sin i_3\\,\\rm M_\\oplus$, where $i_1$, \n$i_2$ and $i_3$ are the inclinations of the orbits with respect to the \nline of sight, \nand are at distances of 0.19\\,AU, 0.36\\,AU, and 0.47\\,AU, respectively, \nfrom the pulsar. In addition, the unusually large second and third frequency \nderivatives of the pulsar suggest the existence of a fourth, more distant \nand massive planet in the system (Wolszczan 1999).\nThe simplest interpretation of the \npresent best-fit values of the frequency derivatives implies \na mass of about $100/\\sin i_4 \\,\\rm M_\\oplus$ (i.e., comparable to \nSaturn's mass) for the fourth planet, at a distance of about\n$38\\,\\rm AU$ (i.e., comparable to Pluto's distance from the Sun), and\nwith a period of about $170\\,\\rm yr$ in a circular, coplanar orbit \n(Wolszczan 1996; Joshi \\& Rasio 1997). However, if, as may well be the case,\nthe first pulse frequency derivative is not entirely acceleration-induced, \nthen the fourth planet can have a wide range of masses\n(Joshi \\& Rasio 1997).\nIn particular, it can have a mass comparable to that of \nMars (at a distance of $9\\,$AU), Uranus (at a distance of $25\\,$AU) or \nNeptune (at a distance of $26\\,$AU).\nThe presence of this fourth planet, if confirmed,\nwould place strong additional constraints on possible formation\nscenarios, as both the minimum mass and minimum angular momentum\nrequired in the protoplanetary disk would increase considerably\n(see Phinney \\& Hansen 1993 for a general discussion).\n\n\n\\section*{Acknowledgements}\n\nThis work was supported by NSF Grant AST-9618116, NASA ATP Grant NAG5-8460, \nand by an Alfred P.\\ Sloan Research Fellowship. 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D.Q.~Lamb \\& J.~Patterson (Dordrecht: Reidel), 1\n\n\\bibitem{}\nPhinney, E.S., \\& Hansen, B.M.S. 1993, in Planets around Pulsars,\ned. J.A.~Phillips, S.E.~Thorsett, \\& S.R.~Kulkarni (ASP Conf.\\ Series\nVol. 36), 371\n\n\\bibitem{}\nPiran, T. 1999, Phys.\\ Rep., 314, 575 [astro-ph/9907392]\n\n\\bibitem{} \nPodsiadlowski, P., Pringle, J.E., \\& Rees, M.J. 1991, \n Nature, 352, 783\n\n\\bibitem{} \nPodsiadlowski, P. 1993, in Planets around Pulsars,\ned. 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}
] |
astro-ph0002529
|
Spherical, Oscillatory $\mbox{\boldmath$\alpha^2$}$-Dynamo Induced by Magnetic Coupling \\ Between a Fluid Shell and an Inner Electrically Conducting Core:\\ Relevance to the Solar Dynamo
|
[
{
"author": "G. Schubert"
}
] |
A two-layer spherical $\alpha^2$-dynamo model consisting of an inner electrically conducting core (magnetic diffusivity $\lambda_i$ and radius $r_i$) with $\alpha = 0$ surrounded by an electrically conducting spherical shell (magnetic diffusivity $\lambda_o$ and radius $r_o$) with a constant $\alpha$ is shown to exhibit oscillatory behavior for values of $\beta = \lambda_i/\lambda_o$ and $r_i/r_o$ relevant to the solar dynamo. Time-dependent dynamo solutions require $r_i/r_o \geq 0.55$ and $\beta \leq O(1)$. For the Sun, $r_i/r_o$ is about 0.8 and $\beta\approx 10^{-3}$. The time scale of the oscillations matches the 22 year period of the sunspot cycle for $\lambda_0 = O(10^2\mbox{\thinspace km}^2\mbox{\thinspace s}^{-1}$). It is unnecessary to hypothesize an $\alpha\omega$-dynamo to obtain oscillatory dynamo solutions; an $\alpha^2$-dynamo suffices provided the spherical shell region of dynamo action lies above a large, less magnetically diffusive core, as is the case for the solar dynamo.
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The right\n%% head is a modified title of up to roughly 44 characters. Running heads\n%% will not print in the manuscript style.\n\n\\shorttitle{???}\n\\shortauthors{Schubert and Zhang}\n\n\n%% This is the end of the preamble. Indicate the beginning of the\n%% paper itself with \\begin{document}.\n\n\\begin{document}\n\n%% LaTeX will automatically break titles if they run longer than\n%% one line. However, you may use \\\\ to force a line break if\n%% you desire.\n\n\\title{Spherical, Oscillatory $\\mbox{\\boldmath$\\alpha^2$}$-Dynamo Induced by Magnetic Coupling \\\\\n Between a Fluid Shell and an Inner Electrically Conducting Core:\\\\\n Relevance to the Solar Dynamo}\n\n\n%% Use \\author, \\affil, and the \\and command to format\n%% author and affiliation information.\n%% Note that \\email has replaced the old \\authoremail command\n%% from AASTeX v4.0. You can use \\email to mark an email address\n%% anywhere in the paper, not just in the front matter.\n%% As in the title, you can use \\\\ to force line breaks.\n\n\\author{G. Schubert}\n\\affil{Department of Earth and Space Sciences,\\\\\n Institute of Geophysics and Planetary Physics,\\\\\n University of California,\n Los Angeles, CA 90095}\n\\email{schubert@ucla.edu}\n\n\\and\n\n\\author{K. Zhang}\n\\affil{School of Mathematical Sciences\\\\\nUniversity of Exeter, Exeter EX4 4QE\\\\\nUnited Kingdom}\n\\email{KZhang@maths.ex.ac.uk}\n\n%%\\and\n%%\n%%\\author{R. J. Hanisch\\altaffilmark{5}}\n%%\\affil{Space Telescope Science Institute, Baltimore, MD 21218}\n\n%% Notice that each of these authors has alternate affiliations, which\n%% are identified by the \\altaffilmark after each name. Specify alternate\n%% affiliation information with \\altaffiltext, with one command per each\n%% affiliation.\n\n%%\\altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American Observatory.\n%%CTIO is operated by AURA, Inc.\\ under contract to the National Science\n%%Foundation.}\n%%\\altaffiltext{2}{Society of Fellows, Harvard University.}\n%%\\altaffiltext{3}{present address: Center for Astrophysics,\n%% 60 Garden Street, Cambridge, MA 02138}\n%%\\altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute}\n%%\\altaffiltext{5}{Patron, Alonso's Bar and Grill}\n\n\n%% Mark off your abstract in the ``abstract'' environment. In the manuscript\n%% style, abstract will output a Received/Accepted line after the\n%% title and affiliation information. No date will appear since the author\n%% does not have this information. The dates will be filled in by the\n%% editorial office after submission.\n\n\\begin{abstract}\nA two-layer spherical $\\alpha^2$-dynamo model consisting of an inner electrically conducting core (magnetic diffusivity $\\lambda_i$ and radius\n$r_i$) with $\\alpha = 0$ surrounded by an electrically conducting spherical shell (magnetic diffusivity $\\lambda_o$ and radius $r_o$) with\na constant $\\alpha$ is shown to exhibit oscillatory behavior for values of $\\beta = \\lambda_i/\\lambda_o$ and $r_i/r_o$ relevant to the solar dynamo.\nTime-dependent dynamo solutions require $r_i/r_o \\geq 0.55$ and $\\beta \\leq O(1)$. For the Sun, $r_i/r_o$ is about 0.8 and $\\beta\\approx 10^{-3}$. The time scale of the oscillations matches the 22 year period of the\nsunspot cycle for $\\lambda_0 = O(10^2\\mbox{\\thinspace km}^2\\mbox{\\thinspace s}^{-1}$). It is unnecessary to hypothesize an $\\alpha\\omega$-dynamo to obtain oscillatory dynamo solutions; an $\\alpha^2$-dynamo suffices provided the spherical\nshell region of dynamo action lies above a large, less magnetically diffusive core, as is the case for the solar dynamo.\n\\end{abstract}\n\n%% Keywords should appear after the \\end{abstract} command. The uncommented\n%% example has been keyed in ApJ style. See the instructions to authors\n%% for the journal to which you are submitting your paper to determine\n%% what keyword punctuation is appropriate.\n\n\\keywords{convection --- hydrodynamics --- instabilities --- magnetic fields ---Sun: magnetic fields --- stars: magnetic fields}\n\n\n%% From the front matter, we move on to the body of the paper.\n%% In the first two sections, notice the use of the natbib \\citep\n%% and \\citet commands to identify citations. The citations are\n%% tied to the reference list via symbolic KEYs. The KEY corresponds\n%% to the KEY in the \\bibitem in the reference list below. We have\n%% chosen the first three characters of the first author's name plus\n%% the last two numeral of the year of publication as our KEY for\n%% each reference.\n\n\\section{Introduction}\n\n$\\alpha^2$- and $\\alpha\\omega$-dynamo models have\nbeen studied for decades\n%%(Braginsky, 1964; Steenbeck and Krause, 1966;\n%%Roberts, 1972, Moffatt, 1978;\n%%Gubbins and Roberts, 1987;Barysnikova and Shukurov, 1987).\n\\citep{Bra64,Ste66,Rob72,Mof78,Gub87,Bar87}.\nThe $\\alpha^2$-dynamo is usually stationary\n%%(e.g., Roberts, 1972; Gubbins and Roberts, 1987; Hollerbach, 1996)\n\\citep[e.g.,][]{Rob72,Gub87,Hol96},\nalthough oscillatory $\\alpha^2$-dynamos have been found to occur\nin the special circumstance wherein $\\alpha$ changes rapidly in\nboundary layers\n%%(Radler and Brauer, 1987; Baryshnikova and Shukurov, 1987).\n\\citep{Rad87,Bar87}.\nIn such cases, the period of the $\\alpha^2$-dynamo depends strongly\non the location of the $\\alpha$-boundary layer and is typically\nan order of magnitude or more smaller than the magnetic diffusion\ntime across the dynamo generation region. In general, oscillatory\ndynamo behavior has been produced by combination of the\n$\\alpha$- and $\\omega$-effects. Kinematic models of the solar dynamo, which is \ninherently oscillatory, have been of the $\\alpha\\omega$-type\n%%(e.g., Roald and Thomas, 1997).\n\\citep[e.g.,][]{Roa97}.\n\nIn this paper, we report that spherical oscillatory $\\alpha^2$-dynamos\ncan be simply induced by the magnetic coupling between an\nelectrically conducting outer fluid shell and a conducting\ninner spherical core even when $\\alpha$ in the outer shell is\na constant. The period of oscillation is of the same order\nof magnitude as the magnetic diffusion time across the outer\nshell and depends largely on the electrical conductivity\nof the inner core. Oscillatory behavior occurs when\nthe outer region of dynamo action surrounds a large, less\nmagnetically diffusive core.\nThe radiative interior of the Sun\nis a large region with a smaller magnetic diffusivity than\nthe overlying convection zone wherein dynamo action occurs.\nThe oscillatory character of the Sun's magnetic field, as expressed in\nthe 22 year periodicity of the sunspot cycle, could then be related\nto the electromagnetic coupling of the region of magnetic field generation\nin the convection zone with the radiative core of the Sun. The importance\nof an inner electrically conducting core to the problem of magnetic\nfield generation in an overlying spherical shell has been emphasized\nin the $\\alpha^2$-type models of the geodynamo by \n%%Hollerbach and Jones (1993, 1995),\n%%Hollerbach (1996), and Gubbins~(1999).\n\\citet{Hol93,Hol95},\n\\citet{Hol96},\nand\n\\cite{Gub99}.\nNevertheless, the effects of an inner core, with magnetic diffusivity\ndifferent from that of the overlying fluid convecting shell in\nwhich dynamo action takes place, have not been fully elucidated.\n\n\nThe problem of inner core-fluid shell coupling is made\ndifficult by complicated electromagnetic matching conditions at\nthe interface between the regions. For this reason, and to facilitate understanding\nof the physical effects, we consider the simplest type of \n$\\alpha^2$-dynamo model consisting of a spherical shell with\n$\\alpha = \\mbox{constant}$ surrounding a core with $\\alpha = 0$.\nWe derive the appropriate matching conditions for a core and shell\nof arbitrary magnetic diffusivity. These matching conditions do not\nappear to have been considered in previous studies of spherical\n$\\alpha^2$-dynamos and they result in oscillatory dynamo solutions.\n\n\\section{Model, Equations and Boundary Conditions}\n\nThe model consists of a turbulent fluid spherical shell of inner radius\n$r_i$ and outer radius $r_o$ with constant (turbulent) magnetic diffusivity\n$\\lambda_o$. A magnetic field is generated in the shell by the\n$\\alpha$-effect\n%(Steenbeck and Krause, 1966;\n%Krause and Steenbeck, 1967; Roberts, 1972).\n\\citep{Ste66,Rob72}.\nFor $r > r_o$, we assume there is a non-conductor; for\n$r < r_i$ we assume that there is a conductor with magnetic\ndiffusivity $\\lambda_i$. The kinematics of the $\\alpha^2$-dynamo\nin the spherical shell is governed by the non-dimensional linear\nequations for the magnetic field $\\mathbf{B}_o$\n\\begin{equation}\n \\frac{\\partial\\mathbf{B}_o}{\\partial t} = R\\left(1-\\eta\\right)\n \\nabla\\times\\alpha\\mathbf{B}_o\n +\\nabla^2\\mathbf{B}_o\n\\end{equation}\n\\begin{equation}\n \\nabla\\cdot \\mathbf{B}_o = 0\n\\end{equation}\nIn the inner sphere the magnetic field $\\mathbf{B}_i$ is governed by\n\\begin{equation}\n \\frac{\\partial\\mathbf{B}_i}{\\partial t} = \\beta\\nabla^2\\mathbf{B}_i\n\\end{equation}\n\\begin{equation}\n \\nabla \\cdot \\mathbf{B}_i = 0\n\\end{equation}\nEquations (1)--(4) are scaled by the thickness of the shell\n$\\left(r_o - r_i\\right)$ and by the magnetic diffusion timescale\n$\\left(r_o - r_i\\right)^2/\\lambda_o$. The scaling of the linear system\nof equations for the magnetic field is arbitrary. The non-dimensional\nparameters in the above equations, $\\beta$, $\\eta$, and the\nmagnetic Reynolds number $R$ are defined as\n\\begin{equation}\n \\beta = \\frac{\\lambda_i}{\\lambda_o},\\quad \\eta = \\frac{r_i}{r_o},\\quad\n R = \\frac{r_o\\alpha}{\\lambda_o}\n\\end{equation}\nSince the main purpose of this paper is to understand the effect\nof an electrically conducting inner core, we adopt the simplest possible\nmodel and take $\\alpha$ constant in the spherical shell\n$r_i < r < r_o$; $\\alpha$ is zero outside the shell.\nWith this assumption, spherical harmonics are decoupled and the problem\nis reduced to a one-dimensional problem with complicated boundary conditions.\n\nAt the interface between the shell and the perfectly insulating exterior,\ni.e., at $r = r_o$, the magnetic field must be continuous\n\\begin{equation}\n \\mathbf{B}_o = \\mathbf{B}^{(e)}\\quad \\mbox{at}\\quad r = r_o\n\\end{equation}\nwhere $\\mathbf{B}^{(e)} = -\\nabla\\phi$ is the magnetic field\nin the insulating exterior\n$r > r_o$, and $\\nabla^2\\phi = 0$. At the interface\nbetween the shell and the conducting inner sphere, i.e., \nat $r = r_i$, both the magnetic field $\\mathbf{B}$\nand tangential components of the electric field\n$\\mathbf{E}$ must be continuous\n\\begin{equation}\n {\\bf B}_o = \\mathbf{B}_i,\\quad \\mbox{\\boldmath$\\hat r$} \\times\n \\mathbf{E}_o = \\mbox{\\boldmath$\\hat r$} \\times\n \\mathbf{E}_i\\quad \\mbox{at}\\quad r = r_i\n\\end{equation}\nwhere $\\mbox{\\boldmath$\\hat r$}$ is the unit radial vector,\n$\\mathbf{E}_o$ is the electric field in the outer shell, and\n$\\mathbf{E}_i$ is the electric field in the inner core.\n\nConditions (2) and (4) allow us to express the magnetic fields as\na sum of poloidal and toroidal vectors\n\\begin{eqnarray}\n \\mbox{\\boldmath{$B$}}_o &=& \\nabla \\times \\nabla \\times \\mbox{\\boldmath{$r$}}h_o + \\nabla\n \\times \\mbox{\\boldmath{$r$}}g_o\\\\\n \\mbox{\\boldmath{$B$}}_i &=& \\nabla \\times \\nabla \\times \\mbox{\\boldmath{$r$}}h_i + \\nabla\n \\times \\mbox{\\boldmath{$r$}}g_i\n\\end{eqnarray}\nwhere $\\mbox{\\boldmath$r$}$ is the position vector. Use of Equation (8) in boundary condition\n(6) and expansion of $h_o$ and $g_o$ in terms of spherical harmonics give\n\\begin{equation}\n g_o = 0,\\quad \\frac{\\partial h_o}{\\partial r} + \\frac{(l + 1) h_o}{r}\n = 0\\quad \\mbox{at}\\ r = r_o\n\\end{equation}\nwhere $l$ is the degree of the spherical harmonic $Y_l^m$.\n\nExtra care must be taken for the magnetic boundary conditions\nat the interface $r = r_i$. There are four different cases that we\nhave studied:\n\n\\noindent (I). The limit $\\beta\\rightarrow \\infty$ for both\n stationary and oscillatory dynamos. In this case, the boundary\ncondition for the magnetic field is simply\n\\begin{equation}\n g_o = 0,\\quad \\frac{\\partial h_o}{\\partial r} \n - \\frac{lh_o}{r} = 0\\quad\\mbox{at}\\ r = r_i\n\\end{equation}\n\\noindent(II). The limit $\\beta\\rightarrow 0$ for a stationary dynamo.\nIn this case, boundary conditions (7) require\n\\begin{equation}\n h_o = 0,\\quad R \\left(1-\\eta\\right)\\, r\\,\\frac{\\partial h_o}{\\partial r}\n - \\frac{\\partial (rg_o)}{\\partial r} = 0\\quad\n \\mbox{at}\\quad r = r_i\n\\end{equation}\n\\noindent(III). The limit $\\beta\\rightarrow 0$ for an oscillatory dynamo.\nIn this case, boundary conditions (7) require\n\\begin{equation}\n \\frac{\\partial h_o}{\\partial r} -\\frac{l h_o}{r} = 0,\\quad\n R\\left(1-\\eta\\right) \\left(l + 1\\right) h_o\n -\\frac{\\partial (rg_o)}{\\partial r} = 0\n \\qquad \\mbox{at}\\ r = r_i\n\\end{equation}\n\\noindent (IV). The general case for $\\beta$ not tending toward \n0 or $\\infty$, for both stationary and oscillatory dynamos.\nIn this case, boundary conditions (7) require\n\\begin{equation}\n g_o = g_i,\\quad h_o = h_i,\\quad \\frac{\\partial h_o}{\\partial r}\n = \\frac{\\partial h_i}{\\partial r},\\quad R\\left(1-\\eta\\right)\\,\n \\frac{\\partial (rh_o)}{\\partial r} -\\frac{\\partial(rg_o)}{\\partial r}\n + \\beta\\,\\frac{\\partial (rg_i)}{\\partial r} = 0\n \\quad \\mbox{at}\\quad r = r_i\n\\end{equation}\nThe last case is evidently the most complicated one. The solutions\npresented below show that for $\\beta \\geq 10$, case I provides\na good approximation to case IV, while for $\\beta \\leq 0.1$, cases\nII and III provide a good approximation to case IV. In cases II through\nIV, $g$ and $h$ are coupled by boundary conditions (7). The solutions\nare invariant to a change in the sign of $R$.\n\n\\section{Solution Method}\n\nIn all cases, solutions are expanded in terms of spherical harmonics,\nimplicit in the forms of the boundary and interface conditions\ngiven above. The spherical harmonics are decoupled and only the lowest one\n$\\left(\\ell = 1\\right)$ is used in the analysis. The $\\ell = 2$ mode,\nnot discussed here, behaves similarly to the $\\ell = 1$ mode.\n\nThe time dependence of the solutions is written as\n$\\exp \\left(\\sigma_r + i\\omega t\\right)$ and onset of dynamo action\n$\\left(\\sigma_r = 0\\right)$ is sought. As discussed below,\ndynamos are either stationary $\\left(\\omega = 0\\right)$ or\noscillatory $\\left(\\omega\\neq 0\\right)$. The frequency $\\omega$\nis dimensionless with respect to the timescale\n$\\left(r_o - r_i\\right)^2/\\lambda_o$.\n\nIn case I, a perfectly insulating core, and in cases II and III,\na perfectly conducting core, it is only necessary to\nsolve for $g_o$ and $h_o$ subject to the above boundary conditions\nat $r_i$ and $r_o$. Exact analytic solutions for $g_o$ and $h_o$\nin these cases can be found in terms of the spherical Bessel\nfunctions of the first and second kind. The solutions reduce\nto finding the eigenvalues of a $4\\times 4$ matrix. The eigenvalues\ngive critical values of the magnetic Reynolds number $R$\nas a function of $\\eta = r_i/r_o$, for which steady or\noscillatory dynamos are possible (i.e., $\\sigma_r = 0$).\n\nIn case IV, a core of arbitrary $\\beta$, solutions must be obtained\nin both the shell and the core subject to the above matching conditions,\ni.e., $g_o$, $h_o$, $g_i$, and $h_i$ must be determined.\nIn principle, analytic solutions are possible, but it is computationally\nmore efficient to seek numerical solutions. We do this by employing\na spectral-Tau method which expands solutions in terms of Chebyshev\npolynomials. The numerical solutions for arbitrary $\\beta$, and the\nanalytic solutions determined independently in cases I, II, and III,\nprovide a mutual validation of the separate methods. For appropriate\nvalues of $\\beta$, the solutions of the separate methods agree\nessentially exactly.\n\n\\section{Results}\n\nThe principal results of this study are summarized in Figure~1\n\\begin{figure}[b!]\n%%\\resizebox{3.0in}{4.0in}{\n%%\\includegraphics[0in,0in][.02in,0in]{schubertfigure1.eps}}\n\\plotone{.6}{schubertfigure1.eps}\n\\caption{Critical magnetic Reynolds number $R_{cr}$\nand dimensionless frequency $\\omega$ of dipolar dynamo solutions\nat the onset of dynamo action vs.\\ $\\eta$. \\label{fig1}}\n\\end{figure}\nwhich gives the critical magnetic Reynolds number $R_{cr}$\nfor the onset of dynamo action in the dipole $(l = 1)$ mode\nas a function of $\\eta$, the ratio of the inner radius\nof the shell to its outer radius. The critical value of $R$\nincreases with increasing $\\eta$. For an insulating core\n($\\beta > > 1$, dashed curve) dynamo solutions are always steady\n$\\left(\\omega\\equiv 0\\right)$. For a perfectly conducting core\n($\\beta < < 1$, solid curve) there are two branches of dynamo\nsolutions depending on $\\eta$; for $\\eta\\leq 0.55$\nthe dynamo is steady, but for $\\eta \\geq 0.55$ the dynamo\nis oscillatory. There is a jump in $R_{cr}$ at the transition\nfrom steady dynamos to oscillatory dynamos near $\\eta = 0.55$.\nThe dimensionless frequency $\\omega$ of the oscillatory dynamos,\nalso shown in Figure 1 as a function of $\\eta$, varies between\nabout 2.5 and 3 for all values of $\\eta$ considered.\n\nThe values of $R_{cr}$ for arbitrary $\\beta$ lie in the narrow space\nbetween the solid and dashed curves of Figure 1. Importantly,\nit is found that $\\beta$ need not in fact be very small compared with\nunity for oscillatory dynamos to exist. For example, when\n$\\eta = 0.8$, the value appropriate to the solar dynamo,\noscillatory dynamo solutions are found for $\\beta$ less\nthan 2 to 3.\n\n\\section{Discussion}\n\nThe problem solved above is a classically simple one of the type\nconsidered by \n%%Steenbeck and Krause~(1966, 1969)\n\\citet{Ste66}\nand \n%%Roberts~(1972)\n\\citet{Rob72}\ndecades ago. Yet the effects\nof an inner electrically conducting core on the\n$\\alpha^2$-dynamo are subtle and not heretofore appreciated.\nThey enter through the complicated electromagnetic matching conditions\nat the interface between the core and the surrounding shell in which\ndynamo action occurs. The main effect of the core is to introduce\ntime dependence into the dynamo solutions for cores whose radii\nare greater than about 0.55 of the outer radius of the shell.\nAn additional requirement for time dependence is that the core\nbe a reasonably good electrical conductor; in terms of the magnetic\ndiffusivity ratio $\\beta = \\lambda_i/\\lambda_o$,\n$\\beta\\leq O(1)$ suffices for oscillatory dynamo behavior.\n\n\nThe importance of all this to the solar dynamo is that the parameters\nof the solar dynamo satisfy the requirements of oscillatory\n$\\alpha^2$-dynamo solutions. For the solar dynamo\n$\\eta$ is about 0.8 and $\\beta$ is about $10^{-3}$\n%%(Moffatt, 1978).\n\\citep{Mof78}.\nIn addition, if $\\omega$ (from Figure 1) is made dimensional using\nthe time scale $\\left(r_o - r_i\\right)^2/\\lambda_o$ with\n$r_o - r_i = 1.4 \\times 10^5~\\mbox{km}$ and\n$\\lambda_o = O\\left(10^2\\mbox{\\thinspace km}^2\\mbox{\\thinspace s}^{-1}\\right)$\n(eddy magnetic diffusivity), then the period of the\noscillatory dynamo solution is comparable to the 22 year\nperiod of the sunspot cycle. Thus, $\\alpha^2$-dynamo action\nalone could be responsible for the observed time dependence\nof the large scale solar magnetic field. \nIt is not our intent to suggest that the $\\omega$-effect\nis not significant in dynamo action in general, or in the solar\ndynamo in particular, because it represents a physically important\nprocess. Our purpose is only to clarify some physics and demonstrate\nthe potential importance of a hitherto overlooked effect, that of\nthe oscillatory $\\alpha^2$-dynamo. A detailed analysis of the\ncases $\\alpha = \\alpha\\/(r)$ and $\\alpha\\propto \\cos \\theta\n(\\theta = \\mbox{polar\\ angle})$ is in progress.\n\n\\acknowledgments\n\nG.\\ S.\\ acknowledges support from NASA's Planetary Atmospheres Program.\nK.\\ Z.\\ is supported by PPARC and NATO grants.\n\n\n\n%% The reference list follows the main body and any appendices.\n%% Use LaTeX's thebibliography environment to mark up your reference list.\n%% Note \\begin{thebibliography} is followed by an empty set of\n%% curly braces. If you forget this, LaTeX will generate the error\n%% \"Perhaps a missing \\item?\".\n%%\n%% thebibliography produces citations in the text using \\bibitem-\\cite\n%% cross-referencing. Each reference is preceded by a\n%% \\bibitem command that defines in curly braces the KEY that corresponds\n%% to the KEY in the \\cite commands (see the first section above).\n%% Make sure that you provide a unique KEY for every \\bibitem or else the\n%% paper will not LaTeX. The square brackets should contain\n%% the citation text that LaTeX will insert in\n%% place of the \\cite commands.\n\n%% We have used macros to produce journal name abbreviations.\n%% AASTeX provides a number of these for the more frequently-cited journals.\n%% See the Author Guide for a list of them.\n\n%% Note that the style of the \\bibitem labels (in []) is slightly\n%% different from previous examples. The natbib system solves a host\n%% of citation expression problems, but it is necessary to clearly\n%% delimit the year from the author name used in the citation.\n%% See the natbib documentation for more details and options.\n\n\\begin{thebibliography}{}\n\\bibitem[Baryshnikova and Shukurov(1987)]{Bar87} Baryshnikova, Y.,\n and Shukurov, A. 1987, Astronomische Nachrichten, 308, 89\n\\bibitem[Braginsky (1964)]{Bra64} Braginsky, S. I.,\n 1964, Geomag. and Aeronomy, 4, 572\n\\bibitem[Gubbins(1999)]{Gub99} Gubbins, D. 1999, Geophys. J. Int., 137, F1\n\\bibitem[Gubbins and Roberts(1987)]{Gub87} Gubbins, D.,\n and Roberts, P. H. 1987, in Geomagnetism, Vol. 2,\n ed. J. A. Jacobs, London: Academic Press, 1\n\\bibitem[Hollerbach(1996)]{Hol96} Hollerbach, R. 1996, Phys. Earth Planet. Int., 98, 163\n\\bibitem[Hollerbach and Jones(1993)]{Hol93} Hollerbach, R., and Jones, \n C. A. 1993, Nature, 365, 541\n\\bibitem[Hollerbach and Jones(1995)]{Hol95} Hollerbach, R.,\n and Jones, C. A. 1995, Phys. Earth Planet. Int., 87, 171\n\\bibitem[Moffatt(1978)]{Mof78} Moffatt, H. K. 1978, Magnetic Field\n Generation in Electrically Conducting Fluids, Cambridge, England: Cambridge\n University Press\n\\bibitem[Radler and Brauer(1987)]{Rad87} Radler, K. H., and\n Brauer, H. J. 1987, Astronomische Nachrichten, 308, 101\n\\bibitem[Roald and Thomas(1997)]{Roa97} Roald, C. B., and Thomas, J. H.\n1997, Mon. Not. Roy. Astron. Soc., 288, 551\n\\bibitem[Roberts(1972)] {Rob72} Roberts, P. H. 1972, Phil. Trans.\nR. Soc. London, Ser. A, 272, 663\n\\bibitem[Steenbeck and Krause(1966)]{Ste66} Steenbeck, M.,\nand Krause, F. 1966, Z. Naturforsch., 21a, 1285\n(Eng. transl. Roberts and Stix, 1971, 49)\n\\end{thebibliography}\n\n\n%% Generally speaking, only the figure captions, and not the figures\n%% themselves, are included in electronic manuscript submissions.\n%% Use \\figcaption to format your figure captions. They should begin on a\n%% new page.\n\n\\clearpage\n\n%% No more than seven \\figcaption commands are allowed per page,\n%% so if you have more than seven captions, insert a \\clearpage\n%% after every seventh one.\n\n%% There must be a \\figcaption command for each legend. Key the text of the\n%% legend and the optional \\label in curly braces. If you wish, you may\n%% include the name of the corresponding figure file in square brackets.\n%% The label is for identification purposes only. It will not insert the\n%% figures themselves into the document.\n%% If you want to include your art in the paper, use \\plotone.\n%% Refer to the on-line documentation for details.\n\n\n%%\\figcaption[schubzhangfig1.eps]{Critical magnetic Reynolds number $R_{cr}$\n%%and dimensionless frequency $\\omega$ of dipolar dynamo solutions\n%%at the onset of dynamo action vs.\\ $\\eta$ \\label{fig1}}\n\n%%\\figcaption[sgi9279.eps]{This is an example of a long figure caption that\n%%must be set as a paragraph. The processor has to buffer the text of the\n%%caption, so it is good not to be too wordy, but that would make for\n%%poor communication as well. \\label{fig2}}\n%%\n%%\\figcaption{This figure has no associated EPS file, so the optional\n%%parameter is omitted. \\label{fig3}}\n\n\n\n\n\n\n\\end{document}\n\n%%\n%% End of file `sample.tex'.\n"
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"string": "\\begin{thebibliography} is followed by an empty set of\n%% curly braces. If you forget this, LaTeX will generate the error\n%% \"Perhaps a missing \\item?\".\n%%\n%% thebibliography produces citations in the text using \\bibitem-\\cite\n%% cross-referencing. Each reference is preceded by a\n%% \\bibitem command that defines in curly braces the KEY that corresponds\n%% to the KEY in the \\cite commands (see the first section above).\n%% Make sure that you provide a unique KEY for every \\bibitem or else the\n%% paper will not LaTeX. The square brackets should contain\n%% the citation text that LaTeX will insert in\n%% place of the \\cite commands.\n\n%% We have used macros to produce journal name abbreviations.\n%% AASTeX provides a number of these for the more frequently-cited journals.\n%% See the Author Guide for a list of them.\n\n%% Note that the style of the \\bibitem labels (in []) is slightly\n%% different from previous examples. The natbib system solves a host\n%% of citation expression problems, but it is necessary to clearly\n%% delimit the year from the author name used in the citation.\n%% See the natbib documentation for more details and options.\n\n\\begin{thebibliography}{}\n\\bibitem[Baryshnikova and Shukurov(1987)]{Bar87} Baryshnikova, Y.,\n and Shukurov, A. 1987, Astronomische Nachrichten, 308, 89\n\\bibitem[Braginsky (1964)]{Bra64} Braginsky, S. I.,\n 1964, Geomag. and Aeronomy, 4, 572\n\\bibitem[Gubbins(1999)]{Gub99} Gubbins, D. 1999, Geophys. J. Int., 137, F1\n\\bibitem[Gubbins and Roberts(1987)]{Gub87} Gubbins, D.,\n and Roberts, P. H. 1987, in Geomagnetism, Vol. 2,\n ed. J. A. Jacobs, London: Academic Press, 1\n\\bibitem[Hollerbach(1996)]{Hol96} Hollerbach, R. 1996, Phys. Earth Planet. Int., 98, 163\n\\bibitem[Hollerbach and Jones(1993)]{Hol93} Hollerbach, R., and Jones, \n C. A. 1993, Nature, 365, 541\n\\bibitem[Hollerbach and Jones(1995)]{Hol95} Hollerbach, R.,\n and Jones, C. A. 1995, Phys. Earth Planet. Int., 87, 171\n\\bibitem[Moffatt(1978)]{Mof78} Moffatt, H. K. 1978, Magnetic Field\n Generation in Electrically Conducting Fluids, Cambridge, England: Cambridge\n University Press\n\\bibitem[Radler and Brauer(1987)]{Rad87} Radler, K. H., and\n Brauer, H. J. 1987, Astronomische Nachrichten, 308, 101\n\\bibitem[Roald and Thomas(1997)]{Roa97} Roald, C. B., and Thomas, J. H.\n1997, Mon. Not. Roy. Astron. Soc., 288, 551\n\\bibitem[Roberts(1972)] {Rob72} Roberts, P. H. 1972, Phil. Trans.\nR. Soc. London, Ser. A, 272, 663\n\\bibitem[Steenbeck and Krause(1966)]{Ste66} Steenbeck, M.,\nand Krause, F. 1966, Z. Naturforsch., 21a, 1285\n(Eng. transl. Roberts and Stix, 1971, 49)\n\\end{thebibliography}"
}
] |
astro-ph0002530
|
The Discovery of an \\ Embedded Cluster of High-Mass Stars Near SGR 1900+14
|
[
{
"author": "Frederick J. Vrba"
},
{
"author": "Arne A. Henden\\altaffilmark{1}"
},
{
"author": "Christian B. Luginbuhl"
},
{
"author": "and Harry H. Guetter"
}
] |
Deep I-band imaging to I~$\approx$~26.5 of the soft gamma--ray repeater SGR 1900+14 region has revealed a compact cluster of massive stars located only a few arcseconds from the fading radio source thought to be the location of the SGR \citep{fra99}. This cluster was previously hidden in the glare of the pair of M5 supergiant stars (whose light was removed by PSF subtraction) proposed by \citet{vrb96} as likely associated with the SGR 1900+14. The cluster has at least 13 members within a cluster radius of $\approx$~0.6~pc based on an estimated distance of 12--15 kpc. It is remarkably similar to a cluster found associated with SGR 1806--20 \citep{fuc99}. That similar clusters have now been found at or near the positions of the two best--studied SGRs suggests that young neutron stars, thought to be responsible for the SGR phenomenon, have their origins in proximate compact clusters of massive stars.
|
[
{
"name": "sgr1900.tex",
"string": "%%\n%% Beginning of file 'sample.tex'\n%%\n%% This is a sample manuscript marked up using the\n%% AASTeX v5.0 LaTeX 2e macros.\n\n%% The first piece of markup in an AASTeX v5.0 document\n%% is the \\documentclass command. LaTeX will ignore\n%% any data that comes before this command.\n\n%% The command below calls the default manuscript style,\n%% which will produce a double-spaced document on one column.\n%% Examples of commands for other substyles follow. Use\n%% whichever is most appropriate for your purposes.\n\n%% \\documentclass{aastex}\n\n%% preprint produces a one-column, single-spaced document:\n\n% \\documentclass[preprint]{aastex}\n\n%% preprint2 produces a double-column, single-spaced document:\n\n\\documentclass[preprint2]{aastex}\n\n%% If you want to create your own macros, you can do so\n%% using \\newcommand. Your macros should appear before\n%% the \\begin{document} command.\n%%\n%% If you are submitting to a journal that translates manuscripts\n%% into SGML, you need to follow certain guidelines when preparing\n%% your macros. See the AASTeX v5.0 Author Guide\n%% for information.\n\n\\newcommand{\\vdag}{(v)^\\dagger}\n\\newcommand{\\myemail}{skywalker@galaxy.far.far.away}\n\n%% You can insert a short comment on the title page using the command below.\n\n\n%% If you wish, you may supply running head information, although\n%% this information may be modified by the editorial offices.\n%% The left head contains a list of authors,\n%% usually a maximum of three (otherwise use et al.). The right\n%% head is a modified title of up to roughly 44 characters. Running heads\n%% will not print in the manuscript style.\n\n\\shorttitle{Vrba et al.}\n\\shortauthors{Embedded Cluster Near SGR 1900+14}\n\n\n%% This is the end of the preamble. Indicate the beginning of the\n%% paper itself with \\begin{document}.\n\n\\begin{document}\n\n%% LaTeX will automatically break titles if they run longer than\n%% one line. However, you may use \\\\ to force a line break if\n%% you desire.\n\n\\title{The Discovery of an \\\\\n Embedded Cluster of High-Mass Stars Near SGR 1900+14 }\n%% Use \\author, \\affil, and the \\and command to format\n%% author and affiliation information.\n%% Note that \\email has replaced the old \\authoremail command\n%% from AASTeX v4.0. You can use \\email to mark an email address\n%% anywhere in the paper, not just in the front matter.\n%% As in the title, you can use \\\\ to force line breaks.\n\n\\author{Frederick J. Vrba, Arne A. Henden\\altaffilmark{1}, Christian B. Luginbuhl, and Harry H. Guetter}\n\\affil{U.S. Naval Observatory, Flagstaff Station,\n Flagstaff, AZ 86002-1149}\n\n\\author{Dieter H. Hartmann}\n\\affil{Dept. of Physics and Astronomy,\n Clemson, University, Clemson, SC 29634-0978}\n\n\\and\n\n\\author{Sylvio Klose}\n\\affil{Th\\\"uringer Landessternwarte Tautenburg, D--07778 Tautenburg, Germany}\n\n%% Notice that each of these authors has alternate affiliations, which\n%% are identified by the \\altaffilmark after each name. Specify alternate\n%% affiliation information with \\altaffiltext, with one command per each\n%% affiliation.\n\n\\altaffiltext{1}{Universities Space Research Association}\n\n\n%% Mark off your abstract in the ``abstract'' environment. In the manuscript\n%% style, abstract will output a Received/Accepted line after the\n%% title and affiliation information. No date will appear since the author\n%% does not have this information. The dates will be filled in by the\n%% editorial office after submission.\n\n\\begin{abstract}\n\nDeep I-band imaging to I~$\\approx$~26.5 of the soft gamma--ray\nrepeater SGR 1900+14 region has revealed a compact cluster of massive\nstars located only a few arcseconds from the fading radio\nsource thought to be the location of the SGR \\citep{fra99}. This\ncluster was previously hidden in the glare of the pair of M5 supergiant \nstars (whose light was removed by PSF subtraction) proposed by \n\\citet{vrb96} as likely associated with the SGR 1900+14. The cluster\nhas at least 13 members within a cluster radius of $\\approx$~0.6~pc\nbased on an estimated distance of 12--15 kpc. It is remarkably\nsimilar to a cluster found associated with SGR 1806--20 \\citep{fuc99}.\nThat similar clusters have now been found at or near the\npositions of the two best--studied SGRs suggests that young neutron\nstars, thought to be responsible for the SGR phenomenon, have their\norigins in proximate compact clusters of massive stars.\n\n\\end{abstract}\n\n%% Keywords should appear after the \\end{abstract} command. The uncommented\n%% example has been keyed in ApJ style. See the instructions to authors\n%% for the journal to which you are submitting your paper to determine\n%% what keyword punctuation is appropriate.\n\n\\keywords{gamma rays: bursts -- gamma rays: observations-- star clusters}\n\n\n%% From the front matter, we move on to the body of the paper.\n%% In the first two sections, notice the use of the natbib \\citep\n%% and \\citet commands to identify citations. The citations are\n%% tied to the reference list via symbolic KEYs. The KEY corresponds\n%% to the KEY in the \\bibitem in the reference list below. We have\n%% chosen the first three characters of the first author's name plus\n%% the last two numeral of the year of publication as our KEY for\n%% each reference.\n\n\\section{Introduction}\n\nThe \\citet[V96]{har96,vrb96} survey of the original Network \nSynthesis Localization (NSL) of SGR 1900+14 \\citep{hur94} found a pair \nof nearly identical M5 supergiant stars, separated by 3.3 arcsec, and \nat an estimated distance of 12-15 kpc. While just outside of the original \nNSL, they lie within the ROSAT HRI localization of the quiescent\nX--ray source RX~J190717+0919.3 thought to be associated with SGR 1900+14 \n\\citep{hur96}. On the basis of the small probability that even one supergiant \nwould lie within the ROSAT error circle and that at least one other \nsupergiant had been associated with an SGR (1806--20; \\citet{van95,kul95}), \nV96 proposed that the M star pair may be associated with the SGR 1900+14 \nsource. The position of the M star pair has continued to be consistent with more\nrecent X--ray and gamma--ray observations which, taken together, have narrowed\nconsiderably the actual location of SGR 1900+14 from the original NSL \narea of 5 arcmin$^2$. These recent X--ray and gamma--ray observations\nhave also detected variations with a period of 5.16 sec\n\\citep{hur99a,mur99,kou99} and a deceleration of $\\dot P \\approx\n10^{-10}$~sec/sec. Taken together, these are interpreted as\nevidence that the SGR source is a magnetar, though there remains\nsome uncertainty in this interpretation \\citep{mar99}.\n\nAdditionally, a variable and fading radio source was detected \nshortly after the 27 August SGR 1900+14 superburst by \\citet{fra99}, providing \nstrong evidence that it was the radio counterpart to the SGR. Its\nsubarcsec accurate position is located only a few arcseconds from the M\nstars. These positional coincidences, the lack of a plerionic radio source, \nand, despite arguments for SNR G42.8+0.6 in the literature, the lack of a \ncoincident supernova remnant, suggest that the system of proximate, \nhigh--mass M stars should not yet be dismissed as an evolutionary companion \nto the pulsating X--ray source associated with the SGR. \n\nFinding direct evidence that the M star pair may be associated with SGR\n1900+14 has proven elusive as summarized by \\citet{gue20}. Also difficult\nis a theoretical understanding of how isolated, albeit high mass, stars\ncould play a role in the formation of a pulsating X--ray source, despite the\npresence of a high mass luminous blue variable (LBV) very near the SGR 1806--20\nlocalization position, a remarkably similar situation to that for SGR\n1900+14. Recent near-- and mid--infrared observations of SGR 1806--20\n\\citep[F99]{fuc99}, however, have revealed the LBV to be only the most luminous \nmember of a compact cluster of massive stars. Such proximate regions \nof recent star formation provide a natural location for the birth of \nsuch pulsating X--ray sources,\nwhich cannot be very old, without the need for invoking\nenormous space velocities from the nearest supernova remnants.\n\nIn this paper we present evidence for a similar compact cluster of\nhigh-mass stars which has heretofore been hidden in the\nglare of its brightest components, the pair of M5 supergiant stars.\n\n\\section{Observations}\n\nThe 1998 outburst season of SGR 1900+14 presented an opportunity\nto search for optical and near-infrared\nvariability of the double M stars, or other sources\nwithin the ROSAT HRI error circle, which might be correlated to the \nSGR outbursts via some process such as mass transfer to a compact object. \nBeginning in early May and continuing through mid--July 1998 we\ncarried out an I-- and J--band monitoring campaign at the U.S. Naval\nObservatory,\nFlagstaff Station (NOFS) which eventually comprised 2025 short exposure frames \nof data with 54,460 seconds of open shutter time during 16 nights,\nintended to sample variability timescales down to a few seconds. The\nresults of this work found no variablity for any object within\nthe ROSAT HRI error circle and are presented more fully in \\citet{vrb20}.\n \nHowever, it was recognized that the numerous short I--band exposures\nconstituted several hours of total exposure time, which could be\nstacked to form a deep I--band image to search for a counterpart at\nthe position of the \\citet{fra99} variable radio\nsource. To the 1998 data were added additional short exposure frames\nfrom 1995 and 1999. In all, 217 frames of individual exposure time\nbetween 1 and 10 minutes were coadded to form a net image of about\n6.5 hours total exposure. All frames were obtained with one\nof two Tektronix 2K CCDs on the 1.55--m Strand Astrometric Telescope at the\nUSNOFS. It was additionally\nrecognized that, since the exposures used were short enough not to\nsaturate the three bright M stars (A, B,and C of V96), their\nlight could largely be removed by PSF subtraction. \n\nFigure~1 is an approximately 45 x 45 arcsec portion of the median--filtered\ncomposite I--band image centered on the V96 M stars, with a limiting \ndetection magnitude of I~$\\approx$~26.5. In this image the M\nstars ABC have been removed, although their positions are still\napparent due to imperfect subtraction. \nThe position of the variable radio source is shown, but\nno counterpart is visible to I~$\\approx$~26.5, which is consistent with the\nnon-detections in the near infrared of \\citet{eik99}. Unfortunately,\nnone of the nearly 200 frames from 1998 were obtained simultaneously\nwith a gamma--ray burst from SGR 1900+14.\n\nOf greater interest is that the subtracted I--band image shows what appears \nto be a cluster of stars, and possibly nebulosity, centered on the position \nof the M stars. The IRAS source found at this location by\n\\citet{van96} shows a steeply rising energy spectrum that can be\ninterpreted as warm dust in the cluster region.\nFigure~1 also shows\nidentification numbers of the possible cluster stars. On UT 1999 October 28 \nwe used the ASTROCAM IR imager, which employs an SBRC 1024$^2$ InSb \ndetector, at the 1.55--m telescope to obtain a 1600 second net exposure \nJ--band image of this region. An approximately 45 x 45 arcsec region\nof this image is shown in Figure~2, where again the M stars were\nsomewhat successfully PSF--subtracted.\n\nWe obtained photometry for the cluster stars from the I-- and J--band frames,\ncalibrated with several I-- and J--band local standards which had previously\nbeen set up for our variability monitoring program. The photometric\nresults are presented in Table~1 where the results for stars 5 and 6\nare presented together as they could not be separated in the J--band\nobservations. The observed (I--J)~$\\approx$~7 colors are far larger\nthan for any unreddened star and indicate that they suffer extremely high \nextinction.\n\n\\section{Nature of the Cluster}\n\nAssuming that the cluster stars are at the\nsame distance and suffer the same extinction as the M supergiant stars \n(12-15 kpc; A$_V$~=~19.2$\\pm$1.0; V96) we placed all stars in an\n$M_I$ vs. (I--J) CM diagram (Figure~3), assuming normal interstellar\nextinction \\citep{bes88}, and where the error bars include the\nranges in distance and A$_V$ values given above. The solid curves \nshow the approximate loci for supergiants and dwarfs later than A0 and for\ngiants later than G0 while the dashed lines show the M0 (I--J) colors,\nfor reference. The large uncertainty of the intrinsic (I--J) colors of the\nstars after subtracting a huge baseline of extinction renders them\nessentially useless in estimating their spectral types. However, at\nthis assumed distance and extinction\nthe stars have luminosities far greater than\nthat of main sequence stars. We note that even assuming the stars\nare at a much closer distance (for instance d~$\\approx$~5 kpc as has\noften been quoted by association with the SNR G42.8+0.6) has little affect\non the conclusion that these are highly luminous stars.\n\nSeveral examples of compact high mass young clusters serve as templates for\nthese objects: NGC 3603 \\citep{mof94}, W43 \\citep{blu99}, and several\nclusters summarized in \\citet{fig99}. These clusters are characterized\nby 10 -- 30 cluster members, radii of 0.2 -- 1.0 pc, and ages \nof 1 -- 10 Myr. The SGR 1900+14 cluster has at least 13 members\n(including stars A\nand B) and an approximate 7 arcsec radius which,\nat a distance of 12 -- 15 kpc, corresponds to a cluster radius of\n$\\approx$ 0.4 pc. A remarkably similar example to that of the SGR 1900+14\ncluster is described by \\citet{mof76} in which a group of 12 luminous stars\nsurround the M3 I supergiant star HD143183 within a cluster radius of\n0.6~pc. These examples support the idea that the small cluster of stars\nnear SGR 1900+14 and dominated by the M5 supergiants is likely a real\nassociation. A formal astrometric solution, not previously presented, for \nthe positions of the M supergiants based on 21 USNO-A2.0 stars gives the \nresult ($\\pm$ 0.1 arcsec):\n\n\\centerline{Star A: $\\alpha$~=~19$^h$~07$^m$~15.35$^s$, $\\delta$~=~+09$^d$~19'~21.4\" (J2000)}\n\n\\centerline{Star B: $\\alpha$~=~19$^h$~07$^m$~15.13$^s$, $\\delta$~=~+09$^d$~19'~20.7\" (J2000)}\n\n\\section{Discussion}\n\nIf the cluster was the birthplace of SGR 1900+14, this essentially excludes\nSNR G42.8+0.6 as playing any role in the SGR. Although one can envision\nscenarios in which the SNR progenitor was ejected from the cluster by\ndynamical interaction or a much earlier supernova, this leaves the\nnecessity of the neutron star having been kicked back to almost exactly \nits place of origin by the supernova that formed SNR G42.8+0.6 (since the \ncluster and SGR localizations are coincident), an unlikely coincidence both \nin space and timing. However, despite the association of G42.8+0.6 with\nSGR 1900+14 in the literature, there has been no evidence supporting this \nassociation offered, such as the probablity of finding any SNR within a\ngiven distance, based on the number density of SNRs in the Galactic plane.\n\nA more plausable scenario is one in which the cluster and associated\ndense gas/dust cloud hides a recent supernova. Evidence for this\ncloud comes from Figures 1 and 2 and the coincident extended strong \nfar--infrared source indicating compact warm and extended cool dust\n(see V96). Optical extinction from this cloud combined with a 12--15~kpc\ndistance explains why the supernova would not have been noticed\nhistorically. A very young SNR expanding into the dense wind--blown\nbubble due to mass loss from the supergiant stars in the cluster would\nbe consistent with the otherwise unexplained persistent X--ray source at \nthis position, RX J190717+0919.3 \\citep{hur96}. While no quiescent radio source\nis known at this position, a combination of self--absorption within\nthe dense medium and rapid decay \\citep{rey88} could account for this.\nThe supernova remnant evolutionary calculations of \\citet{tru99} indicate that \nfor an ejecta mass of 1 M$_\\sun$, and an external density medium of\n10 cm$^{-3}$, one finds characteristic sizes of $\\approx$ 1~pc at\nt~=~1000 yr; similar to that of the cluster dimensions at the estimated\nM supergiant distances.\n\nThe most likely position for the SGR itself is the \\citet{fra99} \nfading radio source located at\n$\\alpha$~=~19$^h$~07$^m$~14.33$^s$, $\\delta$~=~+09$^d$~19'~21.1\"\n(J2000), with positional accuracy of $\\pm$ 0.15 arcsec in each coordinate. \nWith these astrometric positions we estimate the approximate distances from \nthe center and edge of the cluster to the radio position as 12 arcsec\n(0.7--0.9~pc) and 5 arcsec (0.3--0.4~pc), respectively, based on the 12--15~kpc \ndistance estimate.\nThus, even at the extreme minimum age of the SGR based on the simplest \nmagnetar physics ($\\approx$~700 yr; Kouveliotou et al. 1999) this implies \na tangential velocity of $\\approx$ 420 km~s$^{-1}$ from the near edge of the\ncluster. While still an ample velocity for the runaway neutron star,\nit obviates the enormous space velocities implied by associating it with \nG42.8+0.6 \\citep{kou99}, which is about 12 arcmin away \\citep{hur99b}.\n\nWhile an isolated instance of the compact, high mass cluster found\nat/near SGR 1900+14 would be dismissed as a chance superposition,\nits striking similarity to the cluster found near SGR 1806-20 by \\citet{fuc99}\nmust be recognized. In that case,\nan LBV supergiant is found associated with a cluster of at least another \nfour massive young stars enshrouded in a bright dust cloud as imaged by\nISO and located only 7 arcsec from the SGR gamma--ray localization. \nWith an approximate cluster radius of 8 arcsec and an estimated\ndistance of 14.5 kpc, this implies a cluster radius of $\\approx$~0.6~pc.\nNow that similar compact clusters have been found near the \npositions of the two best studied SGRs (1806-20 and 1900+14)\nthe possiblity that young SGR neutron stars have their origins in\ncompact clusters should be considered seriously.\n\n\n\n%% If you wish to include an acknowledgments section in your paper,\n%% separate it off from the body of the text using the \\acknowledgments\n%% command.\n\n\n%% The reference list follows the main body and any appendices.\n%% Use LaTeX's thebibliography environment to mark up your reference list.\n%% Note \\begin{thebibliography} is followed by an empty set of\n%% curly braces. If you forget this, LaTeX will generate the error\n%% \"Perhaps a missing \\item?\".\n%%\n%% thebibliography produces citations in the text using \\bibitem-\\cite\n%% cross-referencing. Each reference is preceded by a\n%% \\bibitem command that defines in curly braces the KEY that corresponds\n%% to the KEY in the \\cite commands (see the first section above).\n%% Make sure that you provide a unique KEY for every \\bibitem or else the\n%% paper will not LaTeX. The square brackets should contain\n%% the citation text that LaTeX will insert in\n%% place of the \\cite commands.\n\n%% We have used macros to produce journal name abbreviations.\n%% AASTeX provides a number of these for the more frequently-cited journals.\n%% See the Author Guide for a list of them.\n\n%% Note that the style of the \\bibitem labels (in []) is slightly\n%% different from previous examples. The natbib system solves a host\n%% of citation expression problems, but it is necessary to clearly\n%% delimit the year from the author name used in the citation.\n%% See the natbib documentation for more details and options.\n\n\\begin{thebibliography}{}\n\\bibitem[Bessel \\& Brett (1988)]{bes88} Bessell, M. S. \\& Brett, J. M.\n 1988, \\pasp, 100, 1134\n\\bibitem[Blum, Damineli, \\& Conti(1999)]{blu99} Blum, R. D., Damineli,\n A., \\& Conti, P. S. 1999, \\aj, 117, 1392\n\\bibitem[Eikenberry \\& Dror (1999)]{eik99} Eikenberry, S. S. \\& Dror,\n D. H. 1999, \\apj, in press\n\\bibitem[Figer, McLean, \\& Morris(1999)]{fig99} Figer, D. F.,\n McLean, I. S., \\& Morris, M. 1999, \\apj, 514, 202\n\\bibitem[Frail, Kulkarni, \\& Bloom(1999)]{fra99} Frail, D. A., Kulkarni, S. R.,\n \\& Bloom, J. S. 1999, \\nat, 398, 127 \n\\bibitem[Fuchs et al.(1999)]{fuc99} Fuchs, Y., Mirabel, F., Chaty, S.,\n Claret, A., Cesarsky, C. J., \\& Cesarsky, D. A. 1999, \\aap, 358, 891\n (F99)\n\\bibitem[Guenther, Klose, \\& Vrba(2000)]{gue20} Guenther, E. W.,\n Klose, S., \\& Vrba, F. 2000, AIP Conf. Proc. ???, Gamma--Ray\n Bursts; 5th Huntsville Symposium, ed. M. Kippen, (New York, AIP),\n in press \n\\bibitem[Hartmann et al.(1996)]{har96} Hartmann, D. H., et al. 1996, in\n AIP Conf. Proc. 366, Workshop on High Velocity Neutron Stars, ed.\n R. E. Rothschild \\& R. E. Lingenfelter (New York: AIP), 84\n\\bibitem[Hurley et al.(1999b)]{hur99b} Hurley, K. et al. 1999b, \\apj,\n 510, L107\n\\bibitem[Hurley et al.(1999a)]{hur99a} Hurley, K, et al. 1999a, \\apj,\n 510, L111\n\\bibitem[Hurley et al.(1996)]{hur96} Hurley, K., et al. 1996, \\apj, 463, L13\n\\bibitem[Hurley et al.(1994)]{hur94} Hurley, K., Sommer, M.,\n Kouveliotou, C., Fishman, G., Meegan, C., Cline, T., Boer, M.,\n \\& Niel, N. 1994, \\apj, 431, L31\n\\bibitem[Kouveliotou et al.(1999)]{kou99} Kouveliotou, C., et al.\n 1999, \\apj, 510, L115\n\\bibitem[Kulkarni et al.(1995)]{kul95} Kulkarni, S. R., Matthews, K.,\n Neugebauer, G., Reid, I. N., van Kerkwijk, M. H., \\& Vasisht, G.\n 1995, \\apj, 440, L61\n\\bibitem[Marsden, Rothschild, \\& Lingenfelter(1999)]{mar99} Marsden,\n D., Rothschild, R. E., \\& Lingenfelter, R. E. 1999, \\apj, 520, L107\n\\bibitem[Moffat(1976)]{mof76} Moffat, A. F. J. 1976, \\aap, 50, 429\n\\bibitem[Moffat, Drissen, \\& Shara(1994)]{mof94} Moffat, A. F. J.,\n Drissen, L., \\& Shara, M. M. 1994, \\apj, 436, 183\n\\bibitem[Murakami et al.(1999)]{mur99} Murakami, T., Kubo, S.,\n Shibazaki, N., Takeshima, T., Yoshida, A., \\& Kawai, N. 1999, \\apj,\n 510, L119 \n\\bibitem[Reynolds(1988)]{rey88} Reynolds, S. P. 1988, in Galactic and\n Extragalactic Radio Astronomy, ed. G. L. Verschuur \\& K. I.\n Kellermann (Berlin: Springer--Verlag), 460\n\\bibitem[Truelove \\& McKee(1999)]{tru99} Truelove, J. K., \\& McKee,\n C. F. 1999, \\apjs, 120, 299\n\\bibitem[van Kerkwijk et al.(1995)]{van95} van Kerkwijk, M. H.,\n Kulkarni, S. R., Matthews, K., \\& Neugebauer, G. 1995, \\apj, 444, L33\n\\bibitem[van Paradijs et al.(1996)]{van96} van Paradijs, J., et al.\n 1996, \\aap, 314, 146\n\\bibitem[Vrba et al.(1996)]{vrb96} Vrba, F. J., et al. 1996, \\apj,\n 468, 225 (V96)\n\\bibitem[Vrba et al.(2000)]{vrb20} Vrba, F. J., Luginbuhl, C. B., Henden, \n A. A., Guetter, H. H, \\& Hartmann, D. H. 2000, AIP Conf. Proc. ???, \n Gamma--Ray Bursts; 5th Huntsville Symposium, ed. M. Kippen, (New York, AIP),\n in press\n\n\\end{thebibliography}\n\n\n%% Generally speaking, only the figure captions, and not the figures\n%% themselves, are included in electronic manuscript submissions.\n%% Use \\figcaption to format your figure captions. They should begin on a\n%% new page.\n\n\\clearpage\n\n%% No more than seven \\figcaption commands are allowed per page,\n%% so if you have more than seven captions, insert a \\clearpage\n%% after every seventh one.\n\n%% There must be a \\figcaption command for each legend. Key the text of the\n%% legend and the optional \\label in curly braces. If you wish, you may\n%% include the name of the corresponding figure file in square brackets.\n%% The label is for identification purposes only. It will not insert the\n%% figures themselves into the document.\n%% If you want to include your art in the paper, use \\plotone.\n%% Refer to the on-line documentation for details.\n\n\n\\figcaption[fig1.ps]{An approximately 45 x 45 arcsec portion of the\n6.5 hour exposure I--band image of the SGR 1900+14 region formed from\nnumerous short exposures as explained in the text. North is at the\ntop, East to the left. In this image the bright M stars, discussed in\n\\citet{vrb96}, have been subtracted revealing the cluster of faint\nstars. The position of the \\citet{fra99} fading radio source is\nindicated by the circle. The 11 stars forming the cluster are\nnumbered for identification. \\label{fig1}}\n\n\\figcaption[fig2.ps]{An approximately 45 x 45 arcsec portion of a\nJ-band image of the SGR 1900+14 region. North is at the top and East\nto the left. In this image the bright M stars, discussed in \\citet{vrb96}, \nhave been subtracted, showing the nebulosity associated with the\ncluster. \\label{fig2}}\n\n\\figcaption[fig3.ps]{The M$_I$ vs I--J color--magnitude diagram for\nthe 11 newly discovered cluster stars. The I and J band photometry\nhas been dereddened by A$_V$ = 19.2 $\\pm$ 1.0 magnitudes and the\ncluster stars have been assumed to be at a distance range of 12 to\n15~kpc as explained in the text. The approximate loci for luminosity \nclass I, III, and V stars are also shown by the solid curves, with the\npositions for M0 spectral type stars shown for reference by the dashed\nlines. \\label{fig3}}\n\n%% Tables should be submitted one per page, so put a \\clearpage before\n%% each one.\n\n%% Two options are available to the author for producing tables: the\n%% deluxetable environment provided by the AASTeX package or the LaTeX\n%% table environment. Use of deluxetable is preferred.\n%%\n\n%% Three table samples follow, two marked up in the deluxetable environment,\n%% one marked up as a LaTeX table.\n\n\n%% In this first example, note that the \\footnotesize command has been\n%% used to shrink the table so it will fit on one page. Note also that\n%% the \\label command needs to be placed inside the \\tablecaption.\n\n\\clearpage\n\n\\begin{deluxetable}{cccc}\n\\footnotesize\n\\tablecaption{I-- and J--Band Photometry of Cluster Stars \\label{tbl-1}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Star} & \\colhead{I $\\pm$ $\\sigma$(I)} & \\colhead{J $\\pm$ $\\sigma$(J)} & \\colhead{(I -- J) $\\pm$ $\\sigma$(I -- J)} \n}\n\\startdata\n1 & 20.00 $\\pm$ 0.02 & 13.17 $\\pm$ 0.02 & 6.83 $\\pm$ 0.03 \\\\\n2 & 21.10 $\\pm$ 0.02 & 14.26 $\\pm$ 0.02 & 6.84 $\\pm$ 0.03 \\\\\n3 & 21.85 $\\pm$ 0.02 & 14.51 $\\pm$ 0.02 & 7.34 $\\pm$ 0.03 \\\\\n4 & 22.01 $\\pm$ 0.02 & 15.06 $\\pm$ 0.04 & 6.95 $\\pm$ 0.05 \\\\\n5 & 22.77 $\\pm$ 0.03 & \\nodata & \\nodata \\\\\n6 & 23.18 $\\pm$ 0.04 & \\nodata & \\nodata \\\\\n5+6& 22.20 $\\pm$ 0.04 & 15.47 $\\pm$ 0.06 & 6.73 $\\pm$ 0.07 \\\\\n7 & 23.16 $\\pm$ 0.05 & 15.70 $\\pm$ 0.09 & 7.46 $\\pm$ 0.10 \\\\\n8 & 22.85 $\\pm$ 0.04 & 15.62 $\\pm$ 0.06 & 7.23 $\\pm$ 0.07 \\\\\n9 & 23.01 $\\pm$ 0.05 & 15.53 $\\pm$ 0.05 & 7.48 $\\pm$ 0.07 \\\\\n10& 23.60 $\\pm$ 0.06 & 16.41 $\\pm$ 0.07 & 7.19 $\\pm$ 0.09 \\\\\n11& 23.78 $\\pm$ 0.06 & 16.12 $\\pm$ 0.04 & 7.66 $\\pm$ 0.07 \\\\\n \\enddata\n\\end{deluxetable}\n\n\\end{document}\n\n"
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[
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"name": "astro-ph0002530.extracted_bib",
"string": "\\begin{thebibliography} is followed by an empty set of\n%% curly braces. If you forget this, LaTeX will generate the error\n%% \"Perhaps a missing \\item?\".\n%%\n%% thebibliography produces citations in the text using \\bibitem-\\cite\n%% cross-referencing. Each reference is preceded by a\n%% \\bibitem command that defines in curly braces the KEY that corresponds\n%% to the KEY in the \\cite commands (see the first section above).\n%% Make sure that you provide a unique KEY for every \\bibitem or else the\n%% paper will not LaTeX. The square brackets should contain\n%% the citation text that LaTeX will insert in\n%% place of the \\cite commands.\n\n%% We have used macros to produce journal name abbreviations.\n%% AASTeX provides a number of these for the more frequently-cited journals.\n%% See the Author Guide for a list of them.\n\n%% Note that the style of the \\bibitem labels (in []) is slightly\n%% different from previous examples. The natbib system solves a host\n%% of citation expression problems, but it is necessary to clearly\n%% delimit the year from the author name used in the citation.\n%% See the natbib documentation for more details and options.\n\n\\begin{thebibliography}{}\n\\bibitem[Bessel \\& Brett (1988)]{bes88} Bessell, M. S. \\& Brett, J. M.\n 1988, \\pasp, 100, 1134\n\\bibitem[Blum, Damineli, \\& Conti(1999)]{blu99} Blum, R. D., Damineli,\n A., \\& Conti, P. S. 1999, \\aj, 117, 1392\n\\bibitem[Eikenberry \\& Dror (1999)]{eik99} Eikenberry, S. S. \\& Dror,\n D. H. 1999, \\apj, in press\n\\bibitem[Figer, McLean, \\& Morris(1999)]{fig99} Figer, D. F.,\n McLean, I. S., \\& Morris, M. 1999, \\apj, 514, 202\n\\bibitem[Frail, Kulkarni, \\& Bloom(1999)]{fra99} Frail, D. A., Kulkarni, S. R.,\n \\& Bloom, J. S. 1999, \\nat, 398, 127 \n\\bibitem[Fuchs et al.(1999)]{fuc99} Fuchs, Y., Mirabel, F., Chaty, S.,\n Claret, A., Cesarsky, C. J., \\& Cesarsky, D. A. 1999, \\aap, 358, 891\n (F99)\n\\bibitem[Guenther, Klose, \\& Vrba(2000)]{gue20} Guenther, E. W.,\n Klose, S., \\& Vrba, F. 2000, AIP Conf. Proc. ???, Gamma--Ray\n Bursts; 5th Huntsville Symposium, ed. M. Kippen, (New York, AIP),\n in press \n\\bibitem[Hartmann et al.(1996)]{har96} Hartmann, D. H., et al. 1996, in\n AIP Conf. Proc. 366, Workshop on High Velocity Neutron Stars, ed.\n R. E. Rothschild \\& R. E. Lingenfelter (New York: AIP), 84\n\\bibitem[Hurley et al.(1999b)]{hur99b} Hurley, K. et al. 1999b, \\apj,\n 510, L107\n\\bibitem[Hurley et al.(1999a)]{hur99a} Hurley, K, et al. 1999a, \\apj,\n 510, L111\n\\bibitem[Hurley et al.(1996)]{hur96} Hurley, K., et al. 1996, \\apj, 463, L13\n\\bibitem[Hurley et al.(1994)]{hur94} Hurley, K., Sommer, M.,\n Kouveliotou, C., Fishman, G., Meegan, C., Cline, T., Boer, M.,\n \\& Niel, N. 1994, \\apj, 431, L31\n\\bibitem[Kouveliotou et al.(1999)]{kou99} Kouveliotou, C., et al.\n 1999, \\apj, 510, L115\n\\bibitem[Kulkarni et al.(1995)]{kul95} Kulkarni, S. R., Matthews, K.,\n Neugebauer, G., Reid, I. N., van Kerkwijk, M. H., \\& Vasisht, G.\n 1995, \\apj, 440, L61\n\\bibitem[Marsden, Rothschild, \\& Lingenfelter(1999)]{mar99} Marsden,\n D., Rothschild, R. E., \\& Lingenfelter, R. E. 1999, \\apj, 520, L107\n\\bibitem[Moffat(1976)]{mof76} Moffat, A. F. J. 1976, \\aap, 50, 429\n\\bibitem[Moffat, Drissen, \\& Shara(1994)]{mof94} Moffat, A. F. J.,\n Drissen, L., \\& Shara, M. M. 1994, \\apj, 436, 183\n\\bibitem[Murakami et al.(1999)]{mur99} Murakami, T., Kubo, S.,\n Shibazaki, N., Takeshima, T., Yoshida, A., \\& Kawai, N. 1999, \\apj,\n 510, L119 \n\\bibitem[Reynolds(1988)]{rey88} Reynolds, S. P. 1988, in Galactic and\n Extragalactic Radio Astronomy, ed. G. L. Verschuur \\& K. I.\n Kellermann (Berlin: Springer--Verlag), 460\n\\bibitem[Truelove \\& McKee(1999)]{tru99} Truelove, J. K., \\& McKee,\n C. F. 1999, \\apjs, 120, 299\n\\bibitem[van Kerkwijk et al.(1995)]{van95} van Kerkwijk, M. H.,\n Kulkarni, S. R., Matthews, K., \\& Neugebauer, G. 1995, \\apj, 444, L33\n\\bibitem[van Paradijs et al.(1996)]{van96} van Paradijs, J., et al.\n 1996, \\aap, 314, 146\n\\bibitem[Vrba et al.(1996)]{vrb96} Vrba, F. J., et al. 1996, \\apj,\n 468, 225 (V96)\n\\bibitem[Vrba et al.(2000)]{vrb20} Vrba, F. J., Luginbuhl, C. B., Henden, \n A. A., Guetter, H. H, \\& Hartmann, D. H. 2000, AIP Conf. Proc. ???, \n Gamma--Ray Bursts; 5th Huntsville Symposium, ed. M. Kippen, (New York, AIP),\n in press\n\n\\end{thebibliography}"
}
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astro-ph0002531
|
Models of Wave Supported Clumps in Giant Molecular Clouds
|
[
{
"author": "R.F. Coker"
},
{
"author": "J.G.L. Rae"
},
{
"author": "T.W. Hartquist"
}
] |
We present plane-parallel equilibrium models of molecular clumps that are supported by Alfv\'en waves damped by the linear process of ion-neutral friction. We used a WKB approximation to treat the inward propagation of waves and adopted a realistic ionization structure influenced by dissociation and ionization due to photons of external origin. The model clumps are larger and less centrally condensed than those obtained for an assumed ionization structure, used in some previous studies, that is more appropriate for dark regions. \keywords{ISM: magnetic fields -- ISM: clouds -- Turbulence -- Waves }
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[
{
"name": "paper.tex",
"string": "\\documentclass[]{aa}\n\\usepackage{epsfig}\n\\def\\simlt{\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}}\n\\def\\simgt{\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$}}\n\\def\\H2{\\element[][][][2]{H}}\n\n\\begin{document}\n\n \\msnr{H2063}\n\n \\thesaurus{08 % A&A Sect. 9: ISM\n (09.03.1; % ISM: clouds\n 09.13.1; % ISM: magnetic fields\n 09.19.1; % ISM: structure\n 02.20.1; % Turbulence\n 02.23.1 % Waves\n )}\n\n \\title{Models of Wave Supported Clumps \n in Giant Molecular Clouds}\n\n \\author{R.F. Coker, J.G.L. Rae, \\and T.W. Hartquist}\n\n \\offprints{R.F. Coker}\n\n \\institute{Department of Physics and Astronomy, \\\\ University of Leeds, \n Leeds LS2 9JT UK \\\\\n email: robc@ast.leeds.ac.uk,jglr@ast.leeds.ac.uk,twh@ast.leeds.ac.uk\n }\n\n \\date{Received 2000; accepted 2000}\n\n \\titlerunning{Turbulent Wave Support of Clumps}\n \\authorrunning{Coker, Rae, \\& Hartquist}\n\n \\maketitle\n\n \\begin{abstract}\n\nWe present plane-parallel equilibrium models of molecular clumps that are supported\nby Alfv\\'en waves damped by the linear process of ion-neutral friction.\nWe used a WKB approximation to treat the inward propagation of waves and \nadopted a realistic ionization structure influenced by dissociation and\nionization due to photons of external origin.\nThe model clumps are larger and less centrally condensed than those\nobtained for an assumed ionization structure, used in some previous studies,\nthat is more appropriate for dark regions.\n\n \\keywords{ISM: magnetic fields -- ISM: clouds -- Turbulence\n -- Waves\n }\n \\end{abstract}\n%________________________________________________________________\n\n\\section{Introduction}\n\nGiant molecular cloud complexes (GMCs), the birth places of stars,\nare typically many tens of parsecs in linear extent and have masses\nfrom $10^4$ to $10^6 M_{\\sun}$ and temperatures of 10--30 K (see\n\\cite{HCRRW98} for a recent review).\nObservations of CO emission from GMCs (\\cite{BT80}; \\cite{WBS95})\nshow them to be composed of many smaller clumps that are a few parsecs\nin extent and contain $\\simlt 10^3 M_{\\sun}$. \n\nThe widths of CO emission lines originating in individual clumps\nare supersonic and have been attributed to the presence of Alfv\\'en waves\nhaving subAlfv\\'enic velocity amplitudes (\\cite{AM75}). The\nAlfv\\'en waves contribute to the support of a clump along the direction\nof the large-scale magnetic field; the damping of the waves affects the degree\nof support that they provide. An important and well understood mechanism\nfor the damping of linear Alfv\\'en waves in a partially ionized medium is\nthat due to ion-neutral friction which depends on the ionization structure\n(\\cite{KP69}). Ruffle et al. (1998, hereafter R98) and Hartquist et al. (1993)\nhave emphasised that the dependence of the ionization structure\non total visual extinction, $A_\\mathrm{V}$, should greatly influence the density\nprofiles of clumps if Alfv\\'en waves contribute to their support. To quantify\nthe assertion of Hartquist et al. (1993) and R98, we present in this\npaper models of plane-parallel, wave-supported GMC clumps like those identified\nin the work of Williams et al. (1995), who made a detailed\nanalysis of the CO maps of the Rosette Molecular Cloud (RMC), identifying\nmore than 70 clumps. The models that\nwe have constructed are for RMC-type clumps in equilibrium, a restriction justified\nby the fact that clear spectral signatures of collapse have been found only\nwhen much smaller scale features have been resolved (see, e.g., \\cite{HCRRW98}).\n\nWe have adopted a WKB description of the wave propagation as did Martin et al. (1997)\nin their work on wave-supported clumps. Their work differs substantially from ours\nin that they used an ionization structure appropriate for dark regions. Also, we have\nconsidered inwardly rather than outwardly propagating waves, as many of the clumps\nmapped by Williams et al. (1995) do not contain detected stars and\nmay have no internal means of generating waves. Indeed, the waves may be produced\nat the surface of a clump by its interaction with an interclump medium.\n\nOther authors have addressed the importance of photoabsorption for the effects\nthat the ionization structure will have upon a clump's dynamics. These authors\nhave been concerned primarily with dense cores and/or envelopes around them;\ncores are much smaller-scale objects than the clumps identified in Williams et al. (1995).\nMcKee (1989) addressed the possibility that collapse in a system of dense\ncores is a self-regulating process due to the ionization of metals such as Magnesium and Sodium\nby photons emitted by stars formed in the collapse; he was concerned with infall due to\nambipolar diffusion of a large-scale magnetic field. Ciolek \\& Mouschovias (1995)\nhave shown that the large-scale magnetic field can support a photoionized envelope\naround a dense core for a time that is very long compared to the ambipolar diffusion\ntimescale in the center of the dense core. In contrast to McKee (1989) and \nCiolek \\& Mouschovias (1995), Myers \\& Lazarian (1998) addressed the effect\nof photoabsorption on support by waves rather than by the large-scale magnetic\nfield. They stressed that observed infall of dense core envelopes is slower than\nthat expected due to gravitational free-fall and more rapid than collapse due to the\nreduction by ambipolar diffusion of support by an ordered large-scale magnetic field.\nThey considered collapse of material supported primarily by\nwaves and subjected to an external radiation field. While they\nmade clear comments about the importance of the $A_\\mathrm{V}$ dependence of the ionization structure for their\nmodel, they did not perform any detailed calculations in which a realistic dependence\nof the ionization fraction on $A_\\mathrm{V}$ was used.\n\nSeveral sets of authors have considered\nnonlinear effects in the dissipation of waves supporting a clump. Gammie \\& \nOstriker (1996)\ninvestigated models of plane-parallel clumps and from their\n``1 2/3-dimensional'' models found dissipation times due to nonlinear effects to be\nlonger than the Alfv\\'en crossing times for a fairly large range of parameters.\nThe three dimensional investigations of Mac Low et al. (1998) and \nStone et al. (1998) suggest the more restrictive condition that the\nangular frequency of the longest waves be no more than a few times\n$2\\pi/t_\\mathrm{A}$ (where $t_\\mathrm{A}$ is the Alfv\\'en crossing time) in order for the\ndissipation timescale due to nonlinear damping to be roughly the Alfv\\'en crossing time\nor more. In most cases addressed in this paper we have restricted our attention to\nsuch angular frequencies so that we are justified to lowest order in focusing\non only the damping due to ion-neutral friction. It should be noted\nthat the above three dimensional studies of nonlinear effects concerned homogeneous\nturbulence and did not include ion-neutral damping for a realistic ionization structure.\nIf we are correct in supposing that the waves in clumps are driven externally,\nthen the turbulence is not homogeneous \nand its nature depends on both the viscous scale set by ion-neutral damping\nand the exact boundary conditions. The effects of nonlinear damping \nand multiple dimensions will be considered in\nsubsequent work.\n\nIn Sect.\n2 we present the equations for the wave energy, the static equilibrium clump structure,\nand the gravitational field. In Sect. 3 we give a description of the calculations of the\nionization structure for various values of the clump density and $A_\\mathrm{V}$ while Sect. 4\ncontains details of the models considered here. Finally, in Sect. 5, we present\nconclusions.\n\n%__________________________________________________________________\n\n\\section{Equations of Wave Propagation and Static Equilibrium}\n\nWe consider plane-parallel clumps with $z = 0$ corresponding to the clump midplane and\n$z = z_\\mathrm{b}$ (with $z_\\mathrm{b}$ defined as positive) corresponding to boundary between\nthe clump and the interclump medium.\nThe large-scale magnetic field is taken to be $B_0 \\hat{z}$ with $\\hat{z}$ normal to\nthe surface of a plane-parallel clump. We study\nwaves of angular frequency $\\omega$ propagating from $z=z_\\mathrm{b}$ in the $-\\hat{z}$\ndirection. \n\nThe ion\nvelocity can be expressed as\n\\begin{equation}\\label{eq:vieq}\nv_\\mathrm{i} = V e^{-i \\int{k_\\mathrm{r} {\\rm d}z}} \\;,\n\\end{equation}\nwhere $V$ is defined below and $k_\\mathrm{r}$ is the real component of the complex wave vector.\nHartquist \\& Morfill (1984) used a two-fluid\ntreatment to examine a related problem and showed that for an\ninwardly propagating linear Alfv\\'en wave\n\\begin{equation}\\label{eq:conseq}\n{{\\rm d}\\over{\\rm d}z} \\left(k_\\mathrm{r} v_\\mathrm{i} v_\\mathrm{i}^*\\right) = {{V_2}\\over{v_{A_\\mathrm{i}}^2}} v_\\mathrm{i} v_\\mathrm{i}^* \\;,\n\\end{equation}\nwhere $v_\\mathrm{i}^*$ is the complex conjugate of the ion velocity perturbation and\n\\begin{equation}\\label{eq:V2eq}\nV_2 \\equiv { {\\nu_0 \\omega^3 \\rho_\\mathrm{n}^2} \\over {\\nu_0^2 \\rho_\\mathrm{i}^2 + \\omega^2 \\rho_\\mathrm{n}^2} } \\;,\n\\end{equation}\nwhere $\\rho_\\mathrm{i}$ is ion mass density and $\\rho_\\mathrm{n}$ is the neutral mass density.\nThe ion Alfv\\'en velocity is given by\n\\begin{equation}\\label{eq:alfeq}\nv_{A_\\mathrm{i}}^2 = {{B_0}\\over{\\sqrt{4 \\pi \\rho_\\mathrm{i}}}} \\;.\n\\end{equation}\nThe ion-neutral coupling frequency, $\\nu_0 \\rho_\\mathrm{i}$, is such that the momentum transfer\nper unit volume per unit time from ions to neutrals is given by $\\nu_0 \\rho_\\mathrm{n} \\rho_\\mathrm{i} (v_\\mathrm{i} - v_\\mathrm{n})$\nwhere $v_\\mathrm{n}$ is the velocity of the neutrals. To a reasonably\ngood approximation, C$^+$ is the dominant ion and we may take \n\\begin{equation}\\label{eq:nu0}\n\\nu_0 \\rho_\\mathrm{i} \\simeq 2.1\\times10^{-9}\\mathrm{sec}^{-1} \n\\left({{n_\\mathrm{i}}\\over{1~\\mathrm{cm}^{-3}}}\\right)\n\\end{equation}\nwhere $n_\\mathrm{i}$ is the ion number density (\\cite{O61}).\nIf the dominant ion species are very massive, as occurs at large $A_\\mathrm{V}$ and densities,\nthe constant would approach $2.3\\times10^{-9}$s$^{-1}$. However, in this work we\nignore this dependence and use $12~m_\\mathrm{H}$ as the mass per ion.\n\nIf $V$ in Eq.~\\ref{eq:vieq} is written as $e^{\\int{k_\\mathrm{i} {\\rm d}z}}$,\nwhere $k_\\mathrm{i}$ is the imaginary component of the wave vector, and the linearized version\nof the induction equation (cf. Eq. 3 of Hartquist \\& Morfill 1984) is used, Eq.~\\ref{eq:conseq}\nyields\n\\begin{equation}\\label{eq:conseq2}\n{{\\rm d}\\over{\\rm d}z} \\left({{k_\\mathrm{r}}\\over{k_\\mathrm{r}^2 + k_\\mathrm{i}^2}} b b^*\\right) = {{V_2}\\over{v_{A_\\mathrm{i}}^2}} \n{{b b^*}\\over{k_\\mathrm{r}^2 + k_\\mathrm{i}^2}} \\;,\n\\end{equation}\nwhere $b$ is the perturbation magnetic field and $b^*$ is its complex\nconjugate. In the WKB approximation, d$^2V/$d$z^2$ is taken\nto be equal to zero, which is equivalent to assuming\n\\begin{equation}\\label{eq:ass}\nk_\\mathrm{i}^2 + {\\rm d}k_\\mathrm{i}/{\\rm d}z \\ll k_\\mathrm{r}^2\n\\end{equation} \nThen\nit follows (cf. Eq. 7b of Hartquist \\& Morfill 1984) that in the WKB approximation\n\\begin{equation}\\label{eq:WKBeq}\nk_\\mathrm{r}^2 \\simeq {{\\omega^2 + V_1} \\over{ v_{A_\\mathrm{i}}^2}} \\;,\n\\end{equation}\nwith \n\\begin{equation}\\label{eq:V1eq2}\nV_1 \\equiv {{ \\nu_0^2 \\omega^2 \\rho_\\mathrm{n} \\rho_\\mathrm{i}} \\over {\\nu_0^2 \\rho_\\mathrm{i}^2 + \\omega^2 \\rho_\\mathrm{n}^2} }\\;.\n\\end{equation}\nAs is consistent with the WKB approximation, we assume $k_\\mathrm{i}^2 \\ll k_\\mathrm{r}^2$ and substitute \nEq.~\\ref{eq:WKBeq} into Eq.~\\ref{eq:conseq2} to find\n\\begin{equation}\\label{eq:conseq3}\n{{\\rm d}\\over{\\rm d}z} \\left({{v_{A_\\mathrm{i}} U}\\over{\\sqrt{\\omega^2 + V_1}}} \\right) = \n{{V_2}\\over{\\omega^2 + V_1}} U \\;,\n\\end{equation}\nwhere $U = bb^*/16\\pi$ is the time-averaged energy density of the perturbation\nmagnetic field.\n\nWe solve Eq.~\\ref{eq:conseq3} along with the static equilibrium equation\n\\begin{equation}\\label{eq:eqlbrm}\nc_s^2 {{ {\\rm d} (\\rho_\\mathrm{n} + \\rho_\\mathrm{i})}\\over{\\rm d}z} + {{ {\\rm d}U}\\over{ {\\rm d}z}} =\n-(\\rho_\\mathrm{n} + \\rho_\\mathrm{i}) g \\;\n\\end{equation}\n(\\cite{MHP97})\nand the gravitational equation\n\\begin{equation}\\label{eq:grav}\n{{ {\\rm d}g} \\over{\\rm d}z} = 4 \\pi G ( \\rho_\\mathrm{n} + \\rho_\\mathrm{i}) \\;,\n\\end{equation}\nwhere $c_s$, $g$, and $G$ are the isothermal sound speed, the strength of the\ngravitational field, and the gravitational constant, respectively.\n\nWe verify the assumption that $k_\\mathrm{i}^2 \\ll k_\\mathrm{r}^2$ {\\sl a posteri} by checking that \n\\begin{equation}\\label{eq:test}\nk_\\mathrm{i} = {{1}\\over{V}}{{{\\rm d}V}\\over{{\\rm d}z}} \\simeq {{V_2}\\over{v_{A_\\mathrm{i}}\\sqrt{V_1}}} \\ll {{V_1}\\over{v_{A_\\mathrm{i}}^2}} \\;.\n\\end{equation}\nNote that\nif Eq.~\\ref{eq:test} is satisfied, Eq.~\\ref{eq:ass} is as well.\n\n%________________________________________________________________\n\n\\section{Calculations of the Ionization Structure}\n\nThe ionization structure determined by R98 and presented in their Fig. 1 \nwas calculated on the assumption that, due to shielding of the CO by itself and by H$_2$, the rate\nof CO dissociation by photons of external origin is negligible. \nFor a plane-parallel semi-infinite cloud with constant Hydrogen nucleus number density,\n$n_\\mathrm{H} =10^3~$cm$^{-3}$, we\nassume an $A_\\mathrm{V}$-dependent dissociation rate that results\nin a CO abundance relative to $n_\\mathrm{H}$, x(CO), that is\nin harmony with the measurements shown in Fig. 6 of van Dishoeck (1998).\nFor the $n_\\mathrm{H} = 10^3~$cm$^{-3}$ model, Table~\\ref{tab:rates} gives x(CO) and\nthe photodissociation rate as a function of $A_\\mathrm{V}$. Note that the total abundance\nof carbon nuclei relative to $n_\\mathrm{H}$ is fixed at $10^{-4}$. \nIn the work reported here, we used an ionization fraction that depends\non both $A_\\mathrm{V}$ and $n_\\mathrm{H}$. \nUsing the $A_\\mathrm{V}$-dependent CO photodissociation rate from Table~\\ref{tab:rates},\nwe calculated the fractional ionization as a function of $A_\\mathrm{V}$ \nfor $n_\\mathrm{H} = 3\\times10^2, 3\\times10^3, $~and~$ 10^4~$cm$^{-3}$ at various $A_\\mathrm{V}$ values. A bilinear\ninterpolation in $A_\\mathrm{V}$ and $n_\\mathrm{H}$ is used to find the actual ionization\nfraction used for a given point in the clump. For densities above $10^4~$cm$^{-3}$\nwe assume that the total ionization fraction, $\\xi \\equiv \\rho_\\mathrm{i}/\\rho_\\mathrm{n}$, goes as $n_\\mathrm{H}^{-1/2}$,\nas expected in a dark region. Note that in all models, $A_\\mathrm{V}$ is always\ngreater than 4 whenever $n_\\mathrm{H} > 10^4~$cm$^{-3}$.\n\nSince the depletions in RMC-type clumps are very uncertain,\nwe present results for both depletion cases given in R98. As discussed therein\n(also see Shalabiea \\& Greenberg 1995), case A abundances resemble those seen\nin dark cores with $A_\\mathrm{V}\\simgt 5$ while case B, with higher fractional abundances of lower\nionization potential elements, is more appropriate for more diffuse clouds.\n\n\\begin{table}\n \\caption{Assumed CO Photodissociation Rates}\n \\label{tab:rates}\n \\begin{tabular}{lll}\n \\hline\n & & \\\\ [-11pt]\n $A_\\mathrm{V}$ &Rate (sec$^{-1}$) &x(CO) \\\\\n & & \\\\ [-11pt]\n \\hline\n & & \\\\ [-10pt]\n 0.5 &$7.685\\times10^{-13}$&$9.132\\times10^{-7}$ \\\\\n 1.0 &$4.279\\times10^{-13}$&$2.248\\times10^{-6}$ \\\\\n 1.5 &$2.132\\times10^{-13}$&$5.248\\times10^{-6}$ \\\\\n 1.75 &$1.440\\times10^{-13}$&$7.538\\times10^{-6}$ \\\\\n 2.0 &$7.465\\times10^{-14}$&$1.226\\times10^{-5}$ \\\\\n 2.25 &$4.864\\times10^{-14}$&$1.503\\times10^{-5}$ \\\\\n 2.5 &$2.245\\times10^{-14}$&$2.166\\times10^{-5}$ \\\\\n 2.75 &$1.405\\times10^{-14}$&$2.381\\times10^{-5}$ \\\\\n 3.0 &$4.291\\times10^{-15}$&$3.621\\times10^{-5}$ \\\\\n 3.25 &$4.112\\times10^{-15}$&$3.620\\times10^{-5}$ \\\\\n 3.5 &$3.669\\times10^{-16}$&$6.336\\times10^{-5}$ \\\\\n 3.75 &$1.200\\times10^{-16}$&$8.976\\times10^{-5}$ \\\\\n 4.0 &$ 0 $&$9.767\\times10^{-5}$ \\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\n%________________________________________________________________\n\n\\section{Details of the Models}\n\nMany RMC-type clumps are not bound by their own gravity and must be confined\nby interclump media\n(\\cite{BM92}).\nWe shall assume that an interclump medium is sufficiently tenuous that\nit does not shield the clump from the standard interstellar background\nradiation field used in the calculation of $\\xi$ so that $A_\\mathrm{V} = 0$ at\n$z = z_\\mathrm{b}$. The material on either side of the interface between a clump\nand the interclump medium is in two distinct phases, and we may assume\nthat at its outer boundary a clump has a \nsubstantial density; we take $n(\\H2) = n_\\mathrm{H} / 2$ everywhere in the clump\nand at $z = z_\\mathrm{b}$ set $n(\\H2) = n_\\mathrm{b}$. \nWaves may exist in the interclump medium and be partially\ntransmitted into the clump, or, as mentioned earlier, may be generated\nnear the interface by the interaction between the clump and the interclump medium.\nConsequently, we assume that the magnitude of the amplitude of the perturbation\nmagnetic field, $\\delta B = \\sqrt{bb^*}$, is, at $z = z_\\mathrm{b}$, a substantial fraction, $f_\\mathrm{b}$,\nof the large-scale field, $B_0$.\n\nIn all models we have taken the mean mass per neutral particle, $\\mu_\\mathrm{n}$, to be\n2.3 amu, corresponding to $14\\%$ of the neutral particles being He and $86\\%$ being $\\H2$.\nAs is consistent with data given by Savage \\& Mathis (1979), we have assumed \n\\begin{equation}\nA_\\mathrm{V} = {{N_\\mathrm{H}}\\over{1.9\\times10^{21}{\\rm cm}^{-2}}} \\;,\n\\end{equation} where\n$N_\\mathrm{H}$ is the column density of Hydrogen nuclei. Also, the temperature throughout a clump was\ntaken to be 20 K.\n\nFor a given setup, an initial boundary value of $g(z=z_\\mathrm{b})\\equiv g_\\mathrm{b}$, was selected and \nEqs.~\\ref{eq:conseq3},~\\ref{eq:eqlbrm},\nand ~\\ref{eq:grav} were numerically integrated using an adaptive Gear\nalgorithm (\\cite{G71}). The value of $g_\\mathrm{b}$ was changed by iteration\nuntil the inner boundary condition $g(z=0)=0$ was satisfied. \nNote that as the total visual extinction (or,\nequivalently for a plane-parallel cloud, the column density) for a cloud\nis increased, the clump reaches a maximum size, $z_\\mathrm{max}$, and\nstarts shrinking as a larger and larger thermal pressure (and\ntherefore density) is required to balance gravity. Thus, as\nlong as $z_\\mathrm{b} < z_\\mathrm{max}$, there are\ntwo values of $g_\\mathrm{b}$ for each setup that satisfy the boundary conditions.\nOne solution has a relatively flat density profile and a small total\nvisual extinction while the\nother solution is more centrally condensed. We deal here exclusively\nwith the latter solutions.\n\nWe have considered other models but present full results only for\nmodels which have a total edge-to-center visual extinction of\n5 magnitudes since, as discussed in R98, it is in the region of $A_\\mathrm{V}$ of a few that \nclumps appear to begin to contain detected stars, while many dense cores may\nhave $A_\\mathrm{V} \\simlt 5$ (\\cite{M89}). For our canonical model (Model 1) we require\na velocity amplitude, $V$, of 2 km sec$^{-1}$ and an Alfv\\'en speed, $v_\\mathrm{A}$, of\n3 km sec$^{-1}$ at $A_\\mathrm{V} = 2$. Since the concentration of CO at $A_\\mathrm{V} \\simlt 2$ is\nvery low and measurements of\nGMC clump CO profiles have a width of $\\sim 2~$ km sec$^{-1}$ \n(\\cite{WBS95}),\nobservations require\nsuch velocities to exist well within the cloud. Also, we use a wave frequency, $\\omega$,\nof $2\\times10^{-12}$~sec$^{-1}$; this results in relatively strong neutral-ion coupling\nwhile keeping the wavelength of the perturbing wave less than the size of the\ncloud. In other words,\n\\begin{equation}\\label{eq:omega}\n\\nu_0 \\xi \\ll \\omega \\simlt {{2 \\pi}\\over{\\int{\\rm d}z/v_\\mathrm{A}}} \\;.\n\\end{equation} \nWithin the above constraints, we find for our canonical model the solution requiring the smallest\nvalue of $B_0$, and, since the velocity amplitude of\na linear Alfv\\'en wave is thought to be comparable to but less than the Alfv\\'en speed, the largest\nvalue of $f_\\mathrm{b}$. However, throughout the clump, $V \\leq v_\\mathrm{A}$,\nconsistent with our assumption of linear Alfv\\'en waves.\n\nWe present the results of 5 models. Model 1 is for the above canonical parameters\nwith R98's depletion case A while Model 2 is for case B. Model 3 is the same as\nModel 1 but with $\\omega = 1\\times10^{-12}$ to illustrate the effect of a scenario\nwith roughly maximum ion-neutral coupling. Model 4 is as Model 1 but\nwith an ionization profile given by $\\xi = 3\\times10^{-16} {\\rho_\\mathrm{n}}^{-1/2}$\n(see, e.g., McKee 1989 and Myers \\& Lazarian 1998).\nThus, Model 4 is for a dark region surrounded by interclump material.\nFinally, for completeness, we present a model with no turbulence; Model 5 is \nidentical to Model 1 except with $f_\\mathrm{b} = 0$. A summary of the parameters of\nthe 5 models is given in Table~\\ref{tab:models}. For all models, $n_\\mathrm{b} = 375~\\H2$ cm$^{-3}$\nand $B_0 = 135~\\mu$G.\n\n\\begin{table}\n \\centering\n \\caption{Summary of Parameters used in Models}\n \\label{tab:models}\n \\begin{tabular}{lllll}\n \\hline\n & & & & \\\\ [-11pt]\n Model &$\\omega$ (sec$^{-1}$)& $f_\\mathrm{b}$ & $z_\\mathrm{b}$ (pc) & Depletions \\\\\n & & & & \\\\ [-11pt]\n \\hline & & & & \\\\ [-10pt]\n 1 &$2\\times10^{-12}$ & 0.436 & 0.455 & A \\\\\n 2 &$2\\times10^{-12}$ & 0.436 & 0.4855 & B \\\\\n 3 &$1\\times10^{-12}$ & 0.436 & 0.495 & A \\\\\n 4$^{\\mathrm{a}}$&$2\\times10^{-12}$ & 0.436 & 0.1978 & A \\\\\n 5 &$2\\times10^{-12}$ & 0.0 & 0.082 & A \\\\\n \\hline\n \\end{tabular}\n \\begin{list}{}{}\n \\item[$^{\\mathrm{a}}$] Uses $\\xi = 3\\times10^{-16} {\\rho_\\mathrm{n}}^{-1/2}$.\n \\end{list}\n\\end{table} \n\n%________________________________________________________________\n\n\\section{Results and Conclusions}\n\n \\begin{figure}\n \\epsfxsize=8.8cm \\epsfbox{fig1_rho.eps} \n \\caption[]{Plot of density (normalized to the density at the outer edge of the\n clump) versus $A_\\mathrm{V}$ for the 5 Models. The solid\n curve is for Model 1, the dotted curve for Model 2, the dashed curve for Model 3,\n the dot-dashed curve for Model 4, and the dash-chain-dot curve for Model 5.}\n \\label{fig:rho}\n \\end{figure} \n\n \\begin{figure}\n \\epsfxsize=8.8cm \\epsfbox{fig2_U.eps} \n \\caption[]{As Fig.~\\ref{fig:rho} but for the perturbing magnetic field, $\\delta B$.}\n \\label{fig:deltaB}\n \\end{figure} \n\n \\begin{figure}\n \\epsfxsize=8.8cm \\epsfbox{fig3_xi.eps} \n \\caption[]{As Fig.~\\ref{fig:rho} but for the absolute ion mass fraction, $\\xi$. The\n bumps seen on the curves for Models 1--3 are a result of the interpolation over\n $A_\\mathrm{V}$.}\n \\label{fig:xi}\n \\end{figure} \n\n \\begin{figure}\n \\epsfxsize=8.8cm \\epsfbox{fig4_flux.eps} \n \\caption[]{As Fig.~\\ref{fig:rho} but for the magnetic energy flux, $F \\equiv k_\\mathrm{r} U \\omega / |k|^2$.}\n \\label{fig:flux}\n \\end{figure} \n\n \\begin{figure}\n \\epsfxsize=8.8cm \\epsfbox{fig5_zed.eps} \n \\caption[]{As Fig.~\\ref{fig:rho} but for z, the absolute spatial extent of the plane-parallel cloud.}\n \\label{fig:size}\n \\end{figure} \n\nIn Fig.~\\ref{fig:rho} we present density as a function of visual extinction for each of the 5 Models.\nIt is clear that the newer ionization profiles used in Models 1--3 result\nin less condensed, more extended clumps than the profile used in Model 4. In fact, $n_\\mathrm{H}$ is roughly proportional to $1/z$ \nin Models 1--3 while $n_\\mathrm{H}$ goes roughly as $1/z^2$ in Model 4. In Fig.~\\ref{fig:deltaB} we show\nplots of $\\delta B$ versus $A_\\mathrm{V}$ for Models 1--4. Except for Model 4, which has $k_\\mathrm{i}/k_\\mathrm{r}$ approaching\n$0.5$ at the clump boundary so that Eq.~\\ref{eq:test} is not satisfactorily satisfied,\nthe perturbing field obeys flux conservation\nnear the surface of the clump. In Model 1, in the central region of the clump \ndissipation is rapid enough that $\\delta B$ begins to decrease. In order to compensate\nfor the loss of support, the equilibrium solution requires a complementary increase in the density,\nas can be seen in Fig.~\\ref{fig:rho}. On the other hand, in Model 4, the turbulence is dissipated\nmuch nearer to the cloud boundary, thus requiring a steeper overall density profile.\nThe higher ionization fractions of the case B depletions result in very little \ndissipation even in the center of the clump for Model 2.\nNote that even though\nobservations (\\cite{WBS95}) suggest that the\ntemperature of RMC-type\nclumps is closer to 10 K rather than the 20 K used here, thermal\nsupport is insignificant except in the centre of Model 1, so the effect of a lower\ntemperature on the models would be merely to enhance slightly any central condensations. \n\nThe ionization profiles used in the Models are shown in Fig.~\\ref{fig:xi}. The ionization profiles\ndescribed in Sec. 3 result in $\\xi$ for Models 1--3 being more than 50 times greater near the surface\nof the clump than in Model 4. However, in the center of the clump\nthe ionization fraction drops, resulting in more dissipation. Again, this leads to clumps\nthat are overall more diffuse but with small condensed cores. Clearly, clumps with $A_\\mathrm{V} \\simgt 5$\nwill have distinct central condensations with $n/n_\\mathrm{b} \\simgt 100$ and central fractional ionizations of $\\simlt\n5\\times10^{-7}$. Though dense cores may be formed during the fairly rapid collapse\n(as envisaged by Fielder \\& Mouschovias 1993) of more extended objects (i.e. RMC-like clumps)\nthat become unstable, even in our equilibrium models we find central cores having densities and fractional\nionizations similar to those measured for dense cores and their envelopes (\\cite{WBS95}; \\cite{WBCMP98}; \\cite{BPWM99}).\n\nFig.~\\ref{fig:flux} shows the flux of magnetic energy\nthrough the clumps for Models 1--4. Near the clump center, Model 1 is nearly thermally supported due to the\ndissipation of the turbulence.\nThe higher ionization fraction for the case B depletions used in Model 2 results in less\ndissipation and thus more turbulent support for the clump. Consequently, as can be\nseen in Fig.~\\ref{fig:rho}, Model 2 has no central condensation and is more extended. Unfortunately,\nwe can only speculate about how the depletions of Sulphur, metals, and some other species\nbehave in RMC-like clumps (\\cite{RHCW99}). Thus, \ncases A and B are merely representative; as can\nbe clearly seen in the figures, the clump profiles are very sensitive to the choice of\nabundances and the subsequent fractional ionizations.\nIn addition, compared to Model 1, the stronger ion-neutral coupling in Model 3 results in less dissipation\nand subsequently the clump has little central condensation, as expected. Note however that\nfor Model 3 the lower limit in Eq.~\\ref{eq:omega} is not adequately satisfied.\n\nFig.~\\ref{fig:flux} also shows the effect of external wave generation. If the fractional ionization\nis too low, as in Model 4, dissipation occurs close to the surface of the clump.\nConversely, if the fractional ionization is too large, as in Model 2, significant\ndissipation occurs only at the clump's very centre. Both extremes produce density profiles\nwhich lack a central condensation. Note that if our externally generated wave model\nis correct, one should not see turbulence within a condensed core if there is no\nturbulence in its surrounding envelope.\n\nIn Fig.~\\ref{fig:size} we present curves which map the visual extinction to the spatial extent of\nthe clumps. Clouds with larger extents which match the observed 2--3 pc size of RMC-type\nclumps (\\cite{WBS95}) cannot be reproduced within the constraints given\nin Sect. 4. However, observations generally measure the largest linear extent of\na clump. Thus, since the waves only support the model clumps parallel to the large-scale\nfield, it is not surprising that the model sizes given here are less than the observed sizes.\n\nSimilarly, the models require high boundary\ndensities and magnetic field strengths in order for the Alfv\\'en speed and wave velocity\namplitude at visual extinctions\nwhere CO is abundant to be large enough to be compatible with observed linewidths.\nFor Model 1, $n_\\mathrm{b} = 375~\\H2$ cm$^{-3}$. This is rather higher than the typical value of\n$n(\\H2) = 220$ cm$^{-3}$ given by Williams et al. (1995) for RMC-type clumps but,\ngiven the uncertainties, it is within a reasonable range of\nthe Williams et al. (1995) value. For Model 1, $B_0 = 135~\\mu$G,\nsignificantly higher than the value of $30\\, \\mu$G suggested\nby observations (\\cite{H87}) and expected from robust theoretical arguments\n(\\cite{M87}). In order to determine\nwhether the values of $n_\\mathrm{b}$ and $B_0$ could be lower and still allow model\nproperties to be consistent with observed linewidths, we constructed models for \nclumps with total edge-to-center extinctions of 3 magnitudes. The model giving $V = 2$ km sec$^{-1}$\nand $v_\\mathrm{A} = 3$ km sec$^{-1}$ at $A_\\mathrm{V} = 2$ had $n_\\mathrm{b} = 325~\\H2$ cm$^{-3}$, $B_0 = 105\\,\\mu$G, \n$f_\\mathrm{b} = 0.49$, and $z_\\mathrm{max} = 0.565$ pc; although $n_\\mathrm{b}$ and $B_0$ were\nsmaller and $z_{max}$ larger, the agreement with observations is nonetheless poor.\n\nThus, the next step in the modelling of clumps\nin which wave support is important is the inclusion of\nwave support in models analogous to the axisymmetric models\nof magnetically and thermally suported clumps described in classic papers by\nMouschovias (1976a,1976b). 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{
"name": "astro-ph0002531.extracted_bib",
"string": "\\begin{thebibliography}{}\n\n\\bibitem[Arons \\& Max 1975]{AM75} Arons J., Max, C.E.,\n1975, ApJ 196, L77 \n\n\\bibitem[Bergin et al. 1999]{BPWM99} Bergin\nE.A., Plume R., Williams J.P., Myers P.C., 1999, ApJ 512, 724 \n\n\\bibitem[Bertoldi \\& McKee 1992]{BM92} Bertoldi F.,\nMcKee C.F., 1992, ApJ 395, 140 \n\n\\bibitem[Blitz \\& Thaddeus 1980]{BT80} Blitz L., \nThaddeus P., 1980, ApJ 241, 676 \n\n\\bibitem[Ciolek \\& Mouschovias 1995]{CM95} Ciolek G. E.,\nMouschovias T.Ch., 1995, ApJ 454, 194 \n\n\\bibitem[Fiedler \\& Mouschovias 1993]{FM93} Fiedler R. A.,\nMouschovias T.Ch., 1993, ApJ 415, 680 \n\n\\bibitem[Gammie \\& Ostriker 1996]{GO96} Gammie C. F., \nOstriker E. C., 1996, ApJ 466, 814 \n\n\\bibitem[Gear 1971]{G71} Gear C. W., 1971, Prentice-Hall \nSeries in Automatic Computation, Prentice-Hall, Engelwood Cliffs\n\n\\bibitem[Hartquist et al. 1998]{HCRRW98} Hartquist T.W., \nCaselli P., Rawlings J.M.C., Ruffle D.P., Williams D.A., 1998. \nIn: Hartquist T.W., Williams, D.A. (eds) The Molecular Astrophysics of Stars and Galaxies.\nClarendon Press, Oxford, p. 101\n\n\\bibitem[Hartquist \\& Morfill 1984]{HM84} Hartquist T.W.,\nMorfill G.E., 1984, ApJ 287, 194 \n\n\\bibitem[Hartquist et al. 1993]{HRWD93} \nHartquist T.W., Rawlings J.M.C., Williams D.A., Dalgarno A., 1993, \nQJRAS 34, 213 \n\n\\bibitem[Heiles 1987]{H87} Heiles C., 1987. In: Hollenbach D.J., Thronson, H.A. (eds) Interstellar \nProcesses, Reidel, Dordrecht, p. 171 \n\n\\bibitem[Kulsrud \\& Pearce 1969]{KP69} Kulsrud R., \nPearce W.P., 1969, ApJ 156, 445 \n\n\\bibitem[Mac Low, Klessen, Burkert \\& Smith 1998]{MKBS98} Mac \nLow M.-M., Klessen R.S., Burkert A., Smith M.D., 1998, Physical \nReview Letters 80, 2754 \n\n\\bibitem[Martin et al. 1997]{MHP97} Martin C.E,.\nHeyvaerts J., Priest E.R., 1997, A\\&A 326, 1176 \n\n\\bibitem[McKee 1989]{M89} McKee C.F., 1989, ApJ 345, 782 \n\n\\bibitem[Mouschovias 1976]{M76a} Mouschovias T.Ch., 1976, \nApJ 206, 753 \n\n\\bibitem[Mouschovias 1976]{M76b} Mouschovias T.Ch., 1976, \nApJ 207, 141 \n\n\\bibitem[Mouschovias 1987]{M87} Mouschovias T.Ch., 1987. \nIn: Morfill G.E., Scholer M. (eds) Physical Processes in Interstellar Clouds, p. 453 \n\n\\bibitem[Myers \\& Lazarian 1998]{ML98} Myers P.C., \nLazarian A., 1998, ApJ 507, L157 \n\n\\bibitem[Osterbrock 1961]{O61} Osterbrock D.E., 1961, \nApJ 134, 270 \n\n\\bibitem[Ruffle et al. 1999]{RHCW99} \nRuffle D.P., Hartquist T.W., Caselli P., Williams D.A., 1999, \nMNRAS 306, 691 \n\n\\bibitem[Ruffle et al. 1998]{RHRW98} \nRuffle D.P., Hartquist T.W., Rawlings J.M.C., Williams D.A.,\n1998, A\\&A 334, 678 (R98)\n\n\\bibitem[Savage \\& Mathis 1979]{SM79} Savage B.D., \nMathis J.S., 1979, ARA\\&A 17, 73\n\n\\bibitem[Shalabiea \\& Greenberg 1995]{SG95} Shalabiea O.M.,\nGreenberg J.M., 1995, A\\&A 296, 779 \n\n\\bibitem[Stone Ostriker \\& Gammie 1998]{SOG98} Stone J.M., \nOstriker E.C., Gammie C.F., 1998, ApJ 508, L99 \n\n\\bibitem[van Dishoeck 1998]{vD98} van Dishoeck E.F., 1998. \nIn: Hartquist T.W., Williams, D.A. (eds) The Molecular Astrophysics of Stars and Galaxies.\nClarendon Press, Oxford, p. 53 \n\n\\bibitem[Williams et al. 1998]{WBCMP98} Williams J.P., \nBergin E.A., Caselli P., Myers P.C., Plume R., 1998, ApJ 503, 689 \n\n\\bibitem[Williams et al. 1995]{WBS95} Williams J.P.,\nBlitz L., Stark A.A., 1995, ApJ 451, 252 \n\n\\end{thebibliography}"
}
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cond-mat0002001
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Spin- and charge-density oscillations in spin chains and quantum wires
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[
{
"author": "Stefan Rommer"
}
] |
We analyze the spin- and charge-density oscillations near impurities in spin chains and quantum wires. These so-called Friedel oscillations give detailed information about the impurity and also about the interactions in the system. The temperature dependence of these oscillations explicitly shows the renormalization of backscattering and conductivity, which we analyze for a number of different impurity models. We are also able to analyze screening effects in one dimension. The relation to the Kondo effect and experimental consequences are discussed.
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[
{
"name": "density-osc.tex",
"string": "%\\documentstyle[epsf,aps,prb,preprint,tighten]{revtex}\n\\documentstyle[epsf,prb,twocolumn,aps]{revtex}\n\n\n\\begin{document}\n\n\\draft\n\n%\\preprint{...}\n\\wideabs{\n\n\\title{Spin- and charge-density oscillations in spin chains and quantum wires}\n\n\\author{Stefan Rommer}\n\n\\address{Department of Physics and Astronomy, University of California,\nIrvine, California 92697}\n\n\\author{Sebastian Eggert}\n\n\\address{Institute of Theoretical Physics, Chalmers University of\nTechnology and G\\\"oteborg University, 41296 G\\\"{o}teborg, Sweden}\n\\date{Submitted: December 7, 1999. Last change: \\today}\n\n\\maketitle\n\n\n\n\\begin{abstract}\nWe analyze the spin- and charge-density oscillations near \nimpurities in spin chains and quantum wires. These so-called\nFriedel oscillations give detailed information about the \nimpurity and also about the interactions in the system.\nThe temperature dependence of these oscillations \nexplicitly shows the renormalization of backscattering and\nconductivity, which we analyze for a number of different \nimpurity models. We are also able to analyze screening effects\nin one dimension. The relation to the Kondo effect \nand experimental consequences are discussed. \n\\end{abstract}\n\n\\pacs{PACS numbers: 71.10.Pm, 75.10.Jm, 72.15.Qm, 73.23.-b}\n\n% 71.10.Pm Fermions in reduced dimensions (anyons, composite fermions, Luttinger liquid, etc.)\n% 75.10.Jm Quantized spin models\n% 72.15.Qm Scattering mechanisms and Kondo effect\n% 73.23.-b Mesoscopic systems\n\n% 75.40.Mg Numerical simulation studies\n% 72.10.-d Theory of electronic transport; scattering mechanisms\n% 75.20.Hr Local moment in compounds and alloys; Kondo effect, valence\n% fluctuations, heavy fermions \n% 73.20.Dx Electron states in low-dimensional structures (superlattices,\n% quantum well structures and multilayers) \n% 73.40.Gk Tunneling \n% 71.27.+a Strongly correlated electron systems; heavy fermions\n% 72.10.Fk Scattering by point defects, dislocations, surfaces, and other\n% imperfections (including Kondo effect)\n\n\n}\\narrowtext\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Introduction} \\label{introduction}\nThere is growing interest in impurities in low-dimensional electron \nand magnetic systems spurred by high temperature superconductivity\nand experimental progress in producing ever smaller electronic\nstructures. There appears to be two central aspects that are studied\nmost in this context, namely the effect of impurities on the transport\nproperties in mesoscopic systems on the one hand,\\cite{kane1} and\nimpurity-impurity interactions in antiferromagnetic systems due to\nimpurity induced magnetic order\\cite{elbio} on the other hand.\nIn this paper we show that the charge- and spin-densities near impurities\ngive a great deal of information about both of those aspects and\nallow us to study a number of impurity models in one dimension \nin detail.\n\nInduced density fluctuations at twice the Fermi wave-vector, \nso-called Friedel oscillations,\\cite{friedel} are\na common impurity effect in fermionic systems, which\nare enhanced in lower dimensions. \n%and play an important role\n%in impurity-impurity interactions, the so-called RKKY interactions.\\cite{RKKY}\n There are two distinct physical effects that\ncan give rise to Friedel oscillations. The most common source \nis a simple {\\it interference effect}\n as considered in the original work by Friedel.\\cite{friedel}\nFermions scatter off the impurity, resulting in \na superposition of incoming and outgoing wave-functions.\nSumming up the squares of the corresponding wave-functions \nup to the sharp cutoff at the\nFermi wave-vector $k_F$ results in a characteristic interference\npattern with a $2 k_F x$ modulation, namely the Friedel oscillations.\nClearly, this pattern can give a great deal of information about the \nimpurity, in particular details about the scattering process.\nA second source for the $2 k_F x$ oscillations are {\\it interaction\neffects} due to the screening of an impurity with a net charge or a\nmagnetic moment. A typical example of this effect is the\nKondo screening cloud,\\cite{sorensen} which we also analyze in\nthis paper. The $2 k_F x$ oscillations due to screening have typically\na different characteristic amplitude as a function of $x$ than \nthose due to backscattering, as we will discuss in more detail below.\n%A related phenomenon in low dimensional antiferromagnets\n%is the enhanced antiferromagnetic order near impurities, which \n%also gives details about the impurity and the system. In this\n%case a more suitable explanation can be found in terms of\n%the valence bond basis.\\cite{elbio}\n\nWe now consider the density oscillations in one-dimensional systems\nsuch as spin-chains and interacting quantum wires (Luttinger Liquids) \nin order to understand the detailed\neffects of impurity scattering and screening as a function of temperature.\nIn the classic work by Kane and Fisher\\cite{kane1,kanereview} it was found that\na generic impurity in a spinless Luttinger Liquid results in a\nrenormalization of the conductivity with temperature, which leads to\na perfectly reflecting barrier at $T=0$ for repulsive interactions.\nInterestingly, this behavior can also be explained in terms of repeated\nscattering off the Friedel oscillations,\nwhich gives an explicit expression of the transmission coefficient \nin the weak coupling limit.\\cite{yue}\nIndependently, the analogous renormalization behavior was also found in the \nspin-1/2 chain,\\cite{eggert} where a generic \nperturbation in the chain effectively renormalizes to an open boundary \ncondition as $T\\to 0$. \nHowever, it is possible that a special symmetry in the Hamiltonian\nreverses this renormalization, which leads to resonant tunneling in quantum\nwires\\cite{kanereview,kane2} or the healing of a two-link problem in the \nspin-1/2 chain.\\cite{eggert} The renormalization behavior in \nthat case is analogous to the two-channel Kondo effect.\\cite{eggert,2CK} \n\nThe renormalization flow can easily be tested numerically \nby examining the scaling of the finite size energy gaps,\\cite{eggert,qin}\nbut we now would like to determine the reflection coefficient \ndirectly by analyzing the induced density oscillations which are \nalso interesting in their own right. In addition, we also consider \nthe density oscillations from impurity models near an edge, \nimpurities with a net charge or magnetic moment (Kondo-type impurities), \nand integrable impurities. The detailed renormalization of the \nimpurity backscattering as well as screening can\nbe studied in each case by analyzing the induced density oscillations\nas a function of temperature, which we determined numerically \nwith the Transfer Matrix \nRenormalization Group (TMRG) for impurities.\\cite{2CK,rommer,TMDMRG}\nThis allows us to make predictions for conductivity\nmeasurements in quantum wires and for Knight shift measurements\nin spin chains, e.g.~Nuclear Magnetic Resonance (NMR) experiments.\nIn all cases we find a typical renormalization to a\nfixed point of the Luttinger Liquid model, which is described\nin terms of a simple (open or periodic) boundary condition in agreement \nwith field theory calculations.\n\nThe rest of this paper is organized as follows.\nIn Sec.~\\ref{model} we present the model Hamiltonian and review the\nresults for Friedel oscillations due to an open end (i.e.~complete\nbackscattering). \nDifferent impurity models of a modified link, two modified links,\nan edge impurity, Kondo impurities, and an integrable impurity are \nthen analyzed in detail in Sec.~\\ref{impurities}. \nSection~\\ref{numerics} contains a description of the\nnumerical methods used and a critical discussion about the possible\nnumerical errors. \nWe conclude with a summary and a discussion about \nexperimental relevance in Sec.~\\ref{summary}.\n\n\\section{The Model} \\label{model}\nThe standard model we are considering here are spinless interacting fermions\non a one-dimensional lattice, described by the Hamiltonian\n\\begin{equation}\nH = \\sum_{i} \\left[-t (\\Psi_i^\\dagger \\Psi_{i+1}^{} +\n\\Psi_{i+1}^\\dagger \\Psi_{i}^{}) + U n_i n_{i+1} - \\mu n_i \\right],\n\\label{fermions}\n\\end{equation}\nwhere $n_i = \\Psi_i^\\dagger \\Psi_{i}^{}$ is the fermion density.\nAlthough this Hamiltonian neglects the spin degrees of freedom of real \nelectrons in quantum wires, it captures the essential physics in\nconductivity experiments. Moreover, this model is equivalent to \nthe spin-1/2 chain \n\\begin{equation}\nH =\\sum_i \\left[\\frac{J}{2}(S_i^+ S_{i+1}^- + S_i^- S_{i+1}^+) \n+ J_z S_i^z S_{i+1}^z - B S_i^z\\right]\n\\label{heisenberg}\n\\end{equation}\nwhere the spin operators are related to the fermion field by the\nJordan-Wigner transformation\n\\begin{equation}\nS_i^z = n_i - \\case{1}{2}, \\ \\ \\ S_i^- = (-1)^i \\Psi_i \\exp{i \\pi \\sum_j^{i-1}n_j},\n\\label{JWtrafo}\n\\end{equation}\nwith $J = 2 t, \\ J_z = U$ and $B= \\mu - U$.\n\nThe model in Eq.~(\\ref{fermions}) can be analyzed by standard bosonization\ntechniques in the low-temperature limit. For low energies we only\nconsider excitations around the Fermi-points $\\pm k_F$ and introduce\nleft- and right-moving fermion fields with a linear dispersion relation\n\\begin{equation}\n\\Psi(x) = \ne^{-ik_Fx}\\psi_{L}(x) + e^{ik_Fx}\\psi_{R}(x). \\label{linearization}\n\\end{equation}\nThe chiral fermion fields can then be bosonized using the usual \nbosonization rules\n\\begin{equation}\n\\psi_{L/R}^\\dagger \\psi_{L/R}^{} = \\case{1}{\\sqrt{4 \\pi}} \\left(\\partial_x \\phi\n\\pm \\Pi_\\phi\\right),\n\\end{equation}\nwhere $\\Pi_\\phi$ is the conjugate momenta to the boson field $\\phi$.\nThis results in the following boson Hamiltonian density\n\\begin{equation}\n{\\cal H} \\ = \\ \\case{v}{2}\n\\left[g^{-1}({\\partial_x \\phi})^2 + g \\Pi_\\phi^2\\right],\\label{boson}\n\\end{equation}\nwhich can be solved by a simple rescaling of the boson with the\ninteraction parameter $g$. The parameter $g$ and the\nvelocity $v$ can in principle be calculated for any interaction\nstrength $U$ and chemical potential $\\mu$ with Bethe ansatz \ntechniques.\\cite{korepin} To lowest order in $U$ we get \n$g = 1- 2 U/\\pi v$ and $v = \\sqrt{4 t^2-\\mu^2} + 2 U/\\pi$,\nso that $g<1$ for repulsive interactions.\n\nWe now want to analyze the density oscillations using this formalism.\nAlready from the decomposition of the fermion field in \nEq.~(\\ref{linearization}) it is clear that the fermion density\nmay contain an oscillating component with $2 k_F x$. To see this\nexplicitly we can\nwrite the charge density in quantum wires (or equivalently the \nspin density $\\langle S_z \\rangle$ in spin chains)\nin terms of left- and right-movers \n\\begin{eqnarray}\n\\langle \\Psi^{\\dagger} \\Psi^{}\\rangle & = &\n\\langle \\psi_{L}^{\\dagger}\\psi_{L}^{} \\rangle +\n\\langle \\psi_{R}^{\\dagger}\\psi_{R}^{} \\rangle \\nonumber \\\\\n& & \\ \\ \\ +\ne^{i 2k_F x} \\langle \\psi_{L}^{\\dagger}\\psi_{R}^{} \\rangle +\ne^{-i 2k_F x} \\langle \\psi_{R}^{\\dagger}\\psi_{L}^{} \\rangle.\n\\label{density}\n\\end{eqnarray}\nThe first two uniform terms just represent the overall fermion density\nin the bulk system, while the last two ``Friedel'' terms are the \ndensity oscillations $n_{\\rm osc}$ we are interested in. \nIn a system with translational invariance\nthe left- and right-moving fields are uncorrelated \n$\\langle \\psi_{L}^{\\dagger}\\psi_{R}^{} \\rangle =0$ and no \ndensity oscillations are present. An impurity, however, scatters \nleft- into right-movers and the amplitude of the oscillations \n%$\\langle \\psi_{L}^{}\\psi_{R}^{\\dagger} \\rangle$ \ngives detailed information about the backscattering.\n\nAs the simplest example of this effect, let us consider an open boundary,\ni.e.\\ an impurity with complete backscattering at the origin. \nIn this case the correlation functions can be calculated \ndirectly.\\cite{eggert,mattsson,boundcorr,fabrizio,eggopen,finiteT} \nFor the particular case of the left-right correlation function at \nequal space and time we find\n\\begin{equation}\n\\langle \\psi_{L}^{\\dagger}(x)\\psi_{R}^{}(x) \\rangle \n\\propto \\left(\\frac{\\pi T}{v \\sinh{2 \\pi x T/v}}\\right)^g,\n\\label{LR}\n\\end{equation}\n%\\begin{eqnarray}\n% & & \\langle \\psi_{L}^{\\dagger}(x,t)\\psi_{R}^{}(x,0) \\rangle \\propto \n%\\nonumber \\\\\n%& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n% \\left[\\frac{i \\pi T}{v \\sinh\\pi(v t + 2 x - i\\alpha) T/v} \\right]^{(g+1/g+2)/4}\n%\\nonumber \\\\\n%& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n% \\left[\\frac{i \\pi T}{v \\sinh\\pi(v t - 2 x - i\\alpha) T/v} \\right]^{(g+1/g-2)/4}\n%\\nonumber \\\\\n%& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n% \\left[\\frac{\\sinh 2 \\pi x T/v}{\\sinh\\pi(v t - i\\alpha) T/v} \\right]^{(1/g-g)/2},\n%\\label{LR}\n%\\end{eqnarray}\n%where $\\alpha$ is the ultraviolet cut-off which is of the order of one lattice\n%spacing. The Friedel oscillations $n_{\\rm osc}$ are immediately \n%given by the last two terms in\n%Eq.~(\\ref{density}) and taking the limit $t\\to 0$ in Eq.~(\\ref{LR})\nso that the density oscillations are given by\n\\begin{equation}\nn_{\\rm osc} \\propto \\sin(2 k_F x)\n\\left(\\frac{\\pi T}{v \\sinh{2 \\pi x T/v}}\\right)^g.\n\\label{n_osc}\n\\end{equation}\nThe Friedel oscillations are exponentially damped with \ntemperature, because the incoming and outgoing wave-functions that \nform the interference pattern lose coherence due to temperature fluctuations.\nIn the limit $T\\to 0$ we recover the result of Ref.~\\onlinecite{egger}\nwhere a power-law decay of the Friedel \noscillation $n_{\\rm osc} \\propto 1/x^{g}$ was predicted.\n\nIt is now important to realize that the fermions or spins are still\npinned to a lattice, i.e. $x = $ Integer,\nwhich gives interesting additional effects.\nIn particular, at half-filling $k_F = \\pi/2$ the Friedel oscillations\nin Eq.~(\\ref{n_osc}) are identically zero $\\sin(\\pi x) = 0$ \nfor integer $x$, which can \neasily be understood from particle-hole symmetry (or equivalently\nspin-flip symmetry). Half-filling is a natural state for \nthe spin chains in zero magnetic field, but a small magnetic field\nchanges the Fermi vector slightly $k_F = \\pi/2 + B/v$. In that case,\nEq.~(\\ref{n_osc}) becomes\n\\begin{equation}\nn_{\\rm osc} \\propto (-1)^x \\sin(2 B x/v)\n\\left(\\frac{\\pi T}{v \\sinh{2 \\pi x T/v}}\\right)^g.\n\\label{n_osc2}\n\\end{equation}\nNow, the Friedel oscillations are simply alternating on the lattice\nand for distances below the magnetic length scale $x < v/B$ we can use\n$\\sin(2 B x/v) \\to 2 B x/v$ so that remarkably the oscillations actually\n{\\it increase} with $x^{1-g}$. This effect was first observed for\nthe Heisenberg chain ($J_z = J, \\ \\ g = 1/2, \\ \\ v=J \\pi/2$), where the \nlocal susceptibilities $\\chi(x)$ can be written as\\cite{eggopen}\n\\begin{equation}\n\\chi(x) = \\chi_0 \\ - \\ c\\ (-1)^{x} \\; \\chi^{\\rm bs}(x) , \\label{susc}\n\\end{equation}\nwith the amplitude of the alternating part given by\n\\begin{equation}\n\\chi^{\\rm bs}(x) = \\frac{x \\sqrt{T}}{\\sqrt{\\sinh 4xT}}.\n\\label{chi} \n\\end{equation}\nHere $\\chi_0$ is the bulk susceptibility in the chain\\cite{suscept}\nand we measure $T$ in units of $J$.\nThe sign was chosen so that the (constant) overall amplitude $c$\nof the alternating part is positive.\nThe superscript bs indicates that the alternating\nsusceptibility is due to backscattering.\nAs shown in Fig.~\\ref{altsusc} from TMRG simulations\nthere is a characteristic maximum \nbecause the temperature damping eventually dominates over the increasing \noscillations. Clearly, the expression in Eq.~(\\ref{chi}) reproduces \nthe shape of this alternating part rather well,\nalthough we have neglected possible logarithmic corrections \n(multiplicative and additive), which may be responsible for the \napparent shift in the characteristic maximum in Fig.~\\ref{altsusc}. \nThe numerical TMRG results of the local susceptibility \nnear the open end $\\chi^{\\rm bs}(x)$\nwill be used as the reference data for a completely backscattering \nimpurity in our studies in the next section. The numerical data \nautomatically contains all corrections due to irrelevant higher order\noperators. The logarithmic corrections to Eq.~(\\ref{chi})\ndue to the leading irrelevant \noperator have a special behavior near a boundary,\\cite{brunel} which \nwe have not tried to predict for the local susceptibility, \nbut numerically we find that a possible \nmultiplicative logarithmic correction for $\\chi(x)$\nappears to have a {\\it negative} power of $\\ln(x)$.\nThe maximum in Fig.~\\ref{altsusc} occurs at $x \\propto 1/T$ with an\namplitude $\\chi_{\\rm alt} \\propto 1/\\sqrt{T}$, which results in a \ncharacteristic feature in NMR experiments, so that it was possible to\nconfirm this effect experimentally as well.\\cite{NMR}\nAt zero temperature the ground state has a staggered magnetization \nwhich has a maximum in the center of a finite chain (assuming\nan odd number of sites).\\cite{laukamp}\nThe magnetization for finite chains with impurities \nhas also recently been analyzed, which resulted in interesting patterns\nthat reveal the nature of the strong correlations in the system.\\cite{nishino}\n\nEven for a partially reflecting impurity we expect that\nthe same alternating contribution as in Eq.~(\\ref{chi}) due to backscattering\nis present, but with an amplitude $c$ that \nincreases monotonically with the reflection coefficient $R$.\nIn fact we can make a firm connection between the relative amplitudes\nand the reflection coefficients by considering free fermions $U=0$ for which \nwe can find the eigenfunctions exactly even in the presence of impurities.\nClearly the eigenfunctions are given by plane wave solutions $|k\\rangle$\nwhich contain a special mix of left- and right-moving components due\nto the impurity. Just like without impurities there are in fact always two\nsuch degenerate orthogonal solutions. We found the solutions for \ngeneric impurity models and looked at the spatial structure of the\nsquare of the wave-functions, which contains an interference pattern of\nincoming and outgoing waves. In general, we always find \n\\begin{equation}\n\\left| \\langle x | k \\rangle \\right|^2 = \\frac{1}{\\pi} \\left( 1 + \n\\sqrt{R(k)} \\cos (2 k x + 2 \\Phi) \\right), \\label{wavefcn}\n\\end{equation}\nwhere the summation over the two degenerate solutions is implied. Here\n$R(k)$ is the ordinary k-dependent reflection coefficient which has been\ndetermined \nindependently according to text-book methods. Therefore, the magnitude\nof the interference is exactly given by the square root of\nthe reflection coefficient, which is maybe not too surprising but very\nuseful in our analysis. In particular, when we consider the fermion\ndensity at half filling we are really directly looking \nat the spatial structure of the wave-function. We can write\nnear half-filling (i.e. for a small field $B$\nin the spin chain model) \n\\begin{eqnarray}\nn(x)-1/2 & = & \\int_{\\pi/2}^{\\pi/2 + B/v} \\left| \\langle x | k \\rangle\n\\right|^2 dk \\nonumber \\\\ \n& \\stackrel{B\\to 0}{=} & \\case{B}{v} \n\\left| \\langle x | \\case{\\pi}{2} \\rangle \\right|^2 \n\\nonumber \\\\\n& = & B \\left[ \\chi_0 - c_R \n(-1)^x \\chi^{\\rm bs} \\right],\n\\label{linresp}\n\\end{eqnarray}\nwhere we have used the fact that the spin density for the Heisenberg chain\nin a small field \nis just given by the susceptibility in Eq.~(\\ref{susc}), but with\na coefficient $c_R$ which now depends on the reflection coefficient $R$\nnear half filling. Together with Eq.~(\\ref{wavefcn}) we therefore\narrive at the central result that at half-filling {\\it the \nreflection coefficient is proportional to \nto the square of the alternating density amplitude}\n\\begin{equation}\nR = \\left( \\frac{c_R}{c} \\right)^2, \\label{R}\n\\end{equation}\nwhere $c=c_{R=1}$ is the coefficient corresponding to complete \nbackscattering in Eq.~(\\ref{susc}).\nWe use this formula to estimate the reflection coefficient from the density \noscillations for various impurity models in the following. \n\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig1-density-osc.eps}}\n\\caption{Local susceptibility close to the open end of a spin-1/2\nchain from TMRG data for $T=0.04J$ compared to Eqs.~(\\ref{susc}) \nand (\\ref{chi}) \nwith $c=0.51$, which was determined by matching the characteristic maxima.}\n\\label{altsusc}\n\\end{figure}}\n\nAs mentioned above there may also be $2 k_F x$ density oscillations due\nto screening, so that the alternating susceptibility is in general a sum\nof two parts\n\\begin{equation}\n\\chi^{\\rm alt}(x) \\equiv \\chi(x) - \\chi_0\n = (-1)^x \\left[\\chi^{\\rm screening}(x) - c_R \\chi^{\\rm bs}(x)\\right].\n\\label{altchi}\n\\end{equation}\nIn the case of overscreening the \nneighboring spins (or electrons) overcompensate the magnetic (or electric) \nimpurity and leave an effective impurity with opposite moment \nwhich in turn gets screened by\nthe next nearest neighbors and so on. This finally results in a\nscreening cloud. \nScreening is purely an interaction effect where a $2 k_F x$ density \noscillation is induced by an ``active'' impurity Hamiltonian \n$\\langle \\psi_L^\\dagger \\psi_R^{} H_{\\rm imp} \\rangle \\neq 0$.\nThe $2 k_F x$ oscillations due to backscattering, however, are purely an \ninterference effect and are even present in non-interacting fermion systems.\nThe special shape and the increasing nature of the alternating part in \nEq.~(\\ref{chi}) for $g=1/2$\nmakes it possible to easily identify the contribution due\nto backscattering, so that we can always separate the two possible effects\nnear half-filling. In what follows we therefore always\nuse the special choice of coupling $U=2t$ \ncorresponding to the Heisenberg model $J_z=J$. \nThis model can be used to demonstrate the generic behavior of impurity \neffects in mesoscopic systems and also gives experimental consequences for \nspin-chain compounds. The Luttinger Liquid\nparameter takes the value $g=1/2$ in this case, which is the \nstrongest possible interaction at half-filling before Umklapp\nscattering becomes relevant.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Impurity models} \\label{impurities}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{One modified link} \\label{onelink}\n\nMaybe the simplest impurity to consider is a weak link in the chain,\ni.e.\\ a modified hopping $J'$ between two sites in the chain\nas shown in Fig.~\\ref{fig:oneweak}\n\\begin{equation}\nH = -t \\sum_{i\\neq 0} (\\Psi_i^\\dagger \\Psi_{i+1}^{} + \n\\Psi_{i+1}^\\dagger \\Psi_{i}^{}) -\nJ' (\\Psi_0^\\dagger \\Psi_{1}^{} + \n\\Psi_{1}^\\dagger \\Psi_{0}^{}) .\n\\end{equation}\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig2-density-osc.eps}}\n\\caption{One modified link.}\n\\label{fig:oneweak}\n\\end{figure}}\n\nThe wave-functions and reflection coefficient $R(k)$ for this problem can \nbe calculated exactly, with the result that \n\\begin{equation}\nR(k) =\\frac{ t^4 -2 t^2 J'^2 + J'^4}{t^4 -2 t^2 J'^2 \\cos 2 k + J'^4}.\n\\label{Rfree}\n\\end{equation}\nHowever, once the interaction $U$ is introduced this problem becomes\nhighly non-trivial and the reflection coefficient renormalizes with\ntemperature $T$. The interacting system has been studied in the context of \nboth spinless fermions\\cite{kane1} and the spin-1/2 chain,\\cite{eggert}\nwhere it was found that repulsive interactions $U > 0$ make the\nperturbation of one link relevant, so that it renormalizes \nto a completely reflecting barrier as $T\\to 0$. \nA small weakening of a link $J' \\alt t$ \nproduces a relevant backscattering operator in the periodic chain of \nscaling dimension $d=g$, so that this link effectively weakens further as\nthe temperature is lowered. Below a cross-over temperature\n$T_K$ (analogous to a Kondo-temperature) the link has weakened so \nmuch that it is more useful to consider the problem of two open ends\nthat are weakly coupled, which is now described by an irrelevant operator\nof scaling dimension $d=1/g$. Therefore, this coupling weakens further\nand ultimately the open boundary condition represents the stable fixed point\nas $T\\to 0$. The same analysis is also true for a slight strengthening\nof a link $J' \\agt t$, because in this case the two ends lock into \na ``singlet'' state as the effective coupling grows,\nand the remaining ends are weakly coupled with a virtual\ncoupling of order $t^2/J'$ which is again irrelevant.\n\nWe consider the interacting system with $U = 2 t$, which we can write \nin terms of an SU(2) invariant spin Hamiltonian via the Jordan-Wigner\ntransformation in Eq.~(\\ref{JWtrafo}) with a modified\nHeisenberg coupling between two spins\n\\begin{equation}\nH = J \\sum_{i\\neq 0} {\\bf S}_i \\cdot {\\bf S}_{i+1}\n + J^\\prime {\\bf S}_0 \\cdot {\\bf S}_1.\n\\label{oneweakham}\n\\end{equation}\nWe now want to analyze the density oscillation near the impurity\nin order to extract the reflection coefficient as described above.\nIn Fig.~\\ref{fig:oneweakaltpart} we show the amplitude of the \nalternating spin density for different coupling strengths $J'$.\nClearly the shape as a function of distance $x$ remains largely the \nsame as in Fig.~\\ref{altsusc} for all $J'$ so that the functional dependence\nin Eq.~(\\ref{chi}) is still adequate, but with an overall\ncoefficient $c$ which is now related to the reflection coefficient $R$\nas postulated in Eq.~(\\ref{R}).\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig3-density-osc.eps}}\n\\caption{Envelope of the alternating susceptibility of\nthe one-link impurity at\n$T=0.04J$ for $J^\\prime/J = 0.0, 0.2, 0.4,\n0.6, 0.8$ from above. }\n\\label{fig:oneweakaltpart}\n\\end{figure}}\n\nThe reflection coefficient is directly related to the renormalization \nbehavior above. The basic idea behind renormalization is to use an \neffective Hamiltonian with renormalized parameters as a function of\n$T$. To estimate the reflection coefficient it is \ntherefore possible to make a simplified but\nintuitive analysis by using the\nfree fermion result in Eq.~(\\ref{Rfree}), but with a renormalized\ncoupling strength $\\tilde J'(T)$. \nBelow the cross-over temperature $T<T_K$, the effective potential \nis small and given by the renormalization behavior of the\nleading irrelevant operator $\\tilde J'(T) \\propto J' T^{1/g-1}$. \nThis results in\n\\begin{equation}\n1-R \\propto J'^2 T^{2/g-2}, \\label{stable}\n\\end{equation}\nwhich is the universal behavior near the stable\nfixed point as first predicted in Ref.~\\onlinecite{kane1}.\nAbove the cross-over temperature $T> T_K$ the renormalization behavior\nis better described by a relevant operator on the periodic chain\ngiving $J - \\tilde J'(T) \\propto (J-J')T^{g-1}$. \nFrom this result it would even seem that we can recover\nthe periodic chain in the high temperature limit, but it is of course \nimportant to realize that the renormalization is no longer possible above\na cutoff of order $J$. For an initial bare coupling $J' \\sim J$\nvery close to the unstable fixed point $T_K \\ll J$\nwe therefore find that the effective coupling stops renormalizing \nat its bare value $\\tilde J' \\to J'$ for large $T$. \nIn summary, the temperature dependence above $T_K$ is not as universal as\nin Eq.~(\\ref{stable}), but we may still write \n\\begin{equation}\nR \\propto (J-J')^2, \\label{unstable}\n\\end{equation}\nfor $J' \\sim J$ and $T> T_K$.\n\nIt is now straightforward to extract the relative coefficient $c_R/c$ in \nEq.~(\\ref{linresp}) from the numerical data by simply dividing\nthe amplitude of the alternating part for each coupling $J'$ in\nFig.~\\ref{fig:oneweakaltpart} by the reference data of $\\chi^{\\rm bs}$\nfor the open chain.\nAccording to Eq.~(\\ref{R}) the square of this relative coefficient then\ngives the reflection coefficient.\nFig.~\\ref{fig:Goneweak} shows the results for the temperature \ndependent reflection coefficient from our TMRG data. \nThe renormalization to a perfectly\nreflective barrier can clearly be seen as $T\\to 0$. \nThe behavior for couplings close to the periodic fixed point \n($J' \\agt 0.4 J$) is consistent with \nEq.~(\\ref{unstable}). For smaller\ncouplings the cross-over temperature $T_K$ is larger, and we see an extended\nregion where the scaling of the stable fixed point with $J'^2$ and\n $T^{2/g-2}$ in Eq.~(\\ref{stable}) holds (here $g=1/2$). \nWe can also compare our results to the findings of\nMatveev {\\it et al} in Ref.~\\onlinecite{yue} where an explicit formula\nfor the transmission coefficient was given \n$1-R \\propto [(D/T)^{2 \\alpha} R_0/(1 -R_0) \\ + \\ 1]^{-1}$\nin terms of the non-interacting reflection coefficient $R_0$ in \nEq.~(\\ref{Rfree}), a cut-off $D$, and a {\\it small} interaction parameter \n$\\alpha = 1/g-1$. Unfortunately, the interaction parameter is large in\nour case $\\alpha = 1$ so that this formula does not quantitatively agree \nwith our findings in Fig.~\\ref{fig:Goneweak}. Qualitatively,\ntheir result looks rather similar, but we observe a sharper renormalization\nat low temperatures near the unstable fixed point ($J' \\agt 0.4 J$).\nIndeed we find that the region where the famous\nscaling in Eq.~(\\ref{stable}) is valid turns out to be extremely narrow\nfor $J' \\agt 0.4J$.\n\nAnother aspect is the high temperature behavior where the \nnon-interacting reflection coefficient in Eq.~(\\ref{Rfree}) should be \napproached\\cite{yue}. This is indeed the case near the unstable fixed point\n$J' \\agt 0.4J$ where the non-interacting value is quickly reached\nwith high accuracy. However, near the stable fixed point ($J' \\alt 0.4J$)\nwe find that the reflection coefficient can renormalize even well below\nthe non-interacting value, so that the interactions actually {\\it enhance}\nthe conductivity at higher temperatures in this case. \nThe reason for this unexpected\nbehavior is that the cross-over temperature is larger than the cut-off\nnear the stable fixed point $T_K \\gg J$, so that the renormalization \nmay continue beyond the bare coupling constants at higher temperatures.\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig4-density-osc.eps}}\n\\caption{Reflection coefficient $R$ of one modified link for\n$J'/J \\ =\\ 0.1, 0.2, 0.4, 0.6,\n0.8$ from above. The lines are guides for the eye.}\n\\label{fig:Goneweak}\n\\end{figure}}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Two modified links} \\label{twolink}\n\nWe now consider the impurity of {\\it two} neighboring \nmodified links in the chain as shown in Fig.~\\ref{twoweakfig}.\nFor the interacting case $U=2 t$ we can again write this model \nin terms of a Heisenberg spin chain model\n\\begin{equation}\nH = J \\sum_{i\\neq -1,0} {\\bf S}_i \\cdot {\\bf S}_{i+1}\n + J^\\prime {\\bf S}_0 \\cdot \\left( {\\bf S}_{-1} + {\\bf\nS}_1 \\right). \\label{twoweakham}\n\\end{equation}\nThis type of impurity may correspond to a charge island that is \nweakly coupled to a mesoscopic wire or to doping in \na quasi-one dimensional compound where one atom in the chain\nhas been substituted.\nWe have recently considered this type of impurity in the context \nof doping in spin-1/2 compounds and as a simple experimental \nexample of the two channel Kondo effect.\\cite{2CK} In this section\nwe analyze the induced density oscillations in more detail, especially\nin connection with the reflection coefficient. \n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig5-density-osc.eps}}\n\\caption{Two modified links.}\n\\label{twoweakfig}\n\\end{figure}}\n\nThe model in Eq.~(\\ref{twoweakham}) is\nequally simple as the one-link impurity, but the \nrenormalization behavior is known to be quite different.\\cite{eggert}\nAlready for the non-interacting case at half filling the system shows\na resonant behavior with perfect transmission $R=0$, so that this\ncorresponds to the simplest case of resonant tunneling considered\nby Kane and Fisher\\cite{kanereview,kane2} (at half-filling\nthe impurity potential is automatically tuned to the resonant condition). \nWith interactions $U\\neq 0$ the reflection coefficient is no longer \nexactly zero, but shows nontheless a renormalization to perfect \ntransmission as $T\\to 0$ in sharp contrast to the one-link impurity.\nThis difference in renormalization behavior is easily explained\nby the different parity symmetry of the problem (namely site- instead\nof link-parity). For a small perturbation from a periodic chain $J' \\sim J$\nthe leading operator is now {\\it irrelevant} with scaling dimension of\n$d=1+g$, so that a perfectly transmitting chain is the stable fixed point.\nFor small couplings $J' \\agt 0$ on the other hand, the leading perturbing \noperator is marginally relevant, and the situation is similar \nto the two channel Kondo effect where the two ends of the chain play the\nrole of two independent channels.\\cite{eggert,2CK} \n\nApart from the renormalization behavior there is another key difference\nbetween the one- and two-link impurities: In the two-link impurity model\nthere is an ``active'' impurity site that carries a spin or charge degree\nof freedom, which in turn must be {\\it screened} by the surrounding system.\nTherefore, the density oscillations are no longer simply determined by \nthe backscattering in Eq.~(\\ref{chi}), but there is also a so-called\nscreening cloud induced in the system. From perturbation theory in the \nleading irrelevant operator the functional dependence of this screening\ncloud can be calculated\\cite{2CK} and the total alternating density\n$\\chi^{\\rm alt}$ is a sum of two contributions\n\\begin{equation}\n\\chi^{\\rm alt} (x) = c_I (-1)^{x} \\ln[\\coth(x T)] - \nc_R (-1)^{x} \\chi^{\\rm bs}(x),\n\\label{twolinkalt} \n\\end{equation}\nwhere the first term is the induced screening cloud while the second\nterm is the familiar contribution due to backscattering in Eq.~(\\ref{chi}).\nInterestingly, the two contributions have opposite sign, so that \nthe density oscillations vanish at a special distance from the\nimpurity, but then increase again due to the backscattering contribution. \nThis behavior is shown in \nFig.~\\ref{fig:altchitwoweak} together with a fit to the two contributions\nin Eq.~(\\ref{twolinkalt}).\nThe special distance at which the density oscillations vanish grows\nas we approach the stable fixed point ($J' \\to J$ or $T \\to 0$).\nAs already with the one-link problem, we use again the numerical \nopen chain data as a reference for $\\chi^{\\rm bs}$\ninstead of the more simplified analytical form of the\nbackscattering contribution in Eq.~(\\ref{chi}) since this minimizes \nthe corrections due to irrelevant operators. However, even the \nanalytical form in Eq.~(\\ref{chi}) gives very good fits, so that none of our \nour findings are affected by this choice.\n\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig6-density-osc.eps}}\n\\caption{Alternating part of the local susceptibility for the two-link\nimpurity for $T/J=0.04$ and $J^\\prime/J=0.6$. Fit to\nEq.~(\\ref{twolinkalt}).}\n\\label{fig:altchitwoweak}\n\\end{figure}}\n\nIt is now straightforward to extract the reflection coefficient\nfrom the numerical data \nwith the help of Eq.~(\\ref{R}) and Eq.~(\\ref{twolinkalt}) as shown in \nFig.~\\ref{fig:Gtwoweak}. Below a cross-over temperature $T_K$ \ndepending on $J'$\nthe reflection coefficient clearly decreases and eventually \napproaches perfect transmission as $T\\to 0$. \nAbove $T_K$ the renormalization of the reflection coefficient is\nrather weak and converges to a finite constant (never approaching \ncomplete reflection as the temperature increases).\n \n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig7-density-osc.eps}}\n\\caption{Reflection coefficient $R$ of the two-link impurity for\n $J'/J = 0.05, 0.1, 0.2, 0.4, 0.6,\n0.8$ from above. The lines are guides for the eye.}\n\\label{fig:Gtwoweak}\n\\end{figure}}\n\n\nEqually interesting is the induced screening cloud. In this case,\nthe coefficient $c_I$ approaches a constant as $T < T_K$ as it should,\nsince this contribution was determined from perturbation theory around\nthe stable fixed point. Above the cross-over temperature, however, \nthis contribution vanishes quickly. This behavior is shown in\nFig.~\\ref{fig:c1twoweak}: In general the behavior of the \ncoefficient $c_I$ vs. $J'$ is temperature dependent and $c_I$ increases as\nthe temperature is lowered. However, as $T \\ll T_K$\nall curves approach a limiting value, which gives a universal behavior \nas a function of $J'$ (thick line).\n\n \n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig8-density-osc.eps}}\n\\caption{Coefficient $c_I$ vs $J'$ of the two link impurity\nfor different temperatures $T/J =$\n%$0.2,0.1,0.067,0.04,0.025$,$0.02,0.0167,0.0133,0.01$\n$0.2,0.1,0.04,0.025,0.0167,0.01$\nfrom below. For $J' \\approx J$ and/or low temperatures\n$c_I$ approaches a universal $T$-independent curve (thick line).\nInset: $c_I$ vs $T$. The lines are guides for the eye.}\n\\label{fig:c1twoweak}\n\\end{figure}}\n \nThe competing contributions in Eq.~(\\ref{twolinkalt})\nhave the opposite renormalization behavior:\nAbove $T_K$ backscattering is constant, while the screening cloud is\nreduced which is the open chain behavior. Below $T_K$ on the other hand\nbackscattering is reduced, while the coefficient for the \ninduced screening cloud is constant, which is the behavior of the two channel \nKondo fixed point. Note, that although the coefficient $c_I$ is finite as\n$T\\to 0$, the screening cloud itself diverges logarithmically with\n$-\\ln(x T)$, which is a clear indication of the famous over-screening\nin the two channel Kondo effect. As we approach the unstable fixed point \nthe order of limits becomes crucial: \nFor zero coupling there is no screening cloud at all\n$\\lim_{T\\to 0} \\lim_{J' \\to 0} c_I = 0$, while\nfor zero temperature the\ncoefficient becomes infinite $\\lim_{J'\\to 0} \\lim_{T \\to 0} c_I = \\infty$. \nRemarkably, exactly at zero \ntemperature a minute perturbation therefore induces an infinite screening\ncloud, although this behavior occurs in an unphysical limit.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\subsection{Impurity at the edge} \\label{edgeimp}\n\nAnother category of impurities we can consider are imperfections\nnear the end of a chain. In this case the boundary always gives\ncomplete backscattering, but as we will see the impurity \ncan still give interesting effects on the density oscillations.\nThe simplest case to consider is a modified link at the edge of\na chain as depicted in Fig.~\\ref{fig:halfedge}. For the interacting\ncase $U=2 t$ it is again useful to write the Hamiltonian in terms\nof the Heisenberg spin-chain model\n\\begin{equation}\nH = J \\sum_{i=1}^\\infty {\\bf S}_i \\cdot {\\bf S}_{i+1}\n + J^\\prime {\\bf S}_0 \\cdot {\\bf S}_1 .\n\\label{edgeham}\n\\end{equation}\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig9-density-osc.eps}}\n\\caption{Edge impurity.}\n\\label{fig:halfedge}\n\\end{figure}}\n\nJust like the two-link impurity was related to the two-channel\nKondo problem, we can identify the field theory description of the\nedge impurity model with the regular\none-channel Kondo problem. There are two possible fixed points:\nThe case $J' = 0$ corresponds to the unstable fixed point of a decoupled\nspin at the end of a chain with a marginally relevant perturbation\nfor $J' \\agt 0$. The case $J' = J$ corresponds to the completely \nscreened spin, which is a stable fixed point with a \nleading irrelevant operator of scaling dimension $d=2$. \nJust like in the ordinary Kondo effect both fixed points \nare represented by the same boundary condition and differ only\nby a simple $\\pi/2$ phase shift on the fermions. (The infinite\ncoupling fixed point $J' \\to \\infty$ is also stable, but is actually \nabsolutely\nequivalent to the $J'=J$ fixed point since both cases represent a $\\pi/2$\nphase shift on the fermions by removing or adding a site, respectively).\nFor intermediate\ncouplings the phase shift $\\Phi$ takes on values between 0 and $\\pi/2$ which \nwill be reflected in the backscattering contribution of the density\noscillations as we will see below.\n\nA screening cloud for the impurity spin at the end should\nalso be present in this model, but with a different behavior than\nfor the overscreened case in Eq.~(\\ref{twolinkalt}). Instead we find that\nthe leading operator that causes the screening cloud is the same \nas that for an edge magnetic field in the xxz-chain which has been analyzed in \nRef.~\\onlinecite{affleckedgefield}, so we can use the corresponding\nresult for the shape of the induced screening cloud. \nTaking into account finite temperatures and the phase shift \non the fermions we can write for the density oscillations\n\\begin{equation}\n\\chi^{\\rm alt}(x) = c_I \\frac{(-1)^x \\sqrt{T}}{\\sqrt{\\sinh(4xT)}} -\n\\cos(\\pi x +2\\Phi) \\ c \\ \\chi^{\\rm bs}(x), \\label{edgealt}\n\\end{equation}\nwhere the first term is the induced screening cloud, while the second\nterm is the backscattering contribution in Eq.~(\\ref{chi}) but\nwith a phase shift $\\Phi$. \nHowever, the coefficient $c$ always takes the value corresponding to complete \nbackscattering in Eq.~(\\ref{susc}). There is also an implied shift of \n$2 \\Phi/\\pi$ in the argument of $\\chi^{\\rm bs}$, which we used\nfor a self-consistent fitting. The effective boundary condition in the\ncontinuum limit is therefore technically between two lattice sites (although\nit is not really that meaningful to define locations on the scale of less than\na lattice spacing in the continuum limit theory anyway).\n\nFigure~\\ref{fig:Szhalfedge} shows the envelope of the alternating part of\nthe susceptibility for temperature $T=0.04J$ and different couplings $J'$,\nwhich always fits well to the superposition in Eq.~(\\ref{edgealt}). At the\nfixed points $J'=0$ and $J'=J$ there is no screening, but the backscattering\ncontribution has opposite signs due to the $\\pi/2$ phase shift.\n\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig10-density-osc.eps}}\n\\caption{Alternating susceptibility for the edge impurity at $T/J=0.04$ for\n$J'/J = 0, 0.1, 0.2, 0.4, 0.6, 1.0$ from above. Fits to Eq.~(\\ref{edgealt}).}\n\\label{fig:Szhalfedge}\n\\end{figure}}\n\nIt is now straightforward to extract the screening cloud amplitude $c_I$\nand the phase shift $\\Phi$ from our numerical data for all temperatures\nand couplings $J'$. As expected we find that the phase shift increases with\n$J'$ and renormalizes to larger \nvalues as the temperature is lowered as shown in Fig.~\\ref{fig:phihalfedge}.\nIn the limit of very low temperatures the jump to the stable fixed point value \n$\\Phi =\\pi/2$ becomes more abrupt as a function of $J'$. \n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig11-density-osc.eps}}\n\\caption{Phase shift of alternating part of the edge impurity for\n$T/J = 0.25, 0.1, 0.04, 0.02, 0.01$ from below. The lines are guides\nfor the eye.}\n\\label{fig:phihalfedge}\n\\end{figure}}\n\nThe screening cloud coefficient $c_I$ again approaches a constant as we \nlower the temperature below $T_K$ as shown in Fig.~\\ref{fig:c1halfedge}.\nAlthough formally the behavior looks similar to the \nover-screened case of the two link problem in Fig.~\\ref{fig:c1twoweak}\nit is important to realize that now the screening cloud in \nEq.~(\\ref{edgealt}) is finite as $T\\to 0$ and drops off with $1/x$ (while\nin the two link case the screening cloud was divergent with $\\ln xT $).\n\n \n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig12-density-osc.eps}}\n\\caption{Coefficient $c_I$ vs. $J'$ of the edge impurity for\n%%%%$T/J = 0.25,0.2,0.1,0.067,0.04,0.02,0.0167,0.133,0.01$\n$T/J = 0.2,0.1,0.04,0.02,0.133,0.01$ from below.\nThe lines are a guide for the eye.}\n\\label{fig:c1halfedge}\n\\end{figure}}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Generalized two link impurity} \\label{genimp}\n\nIt is now instructive to summarize the findings of the three\nimpurity models in the previous subsections by considering one\ngeneralized two link impurity model that is {\\it not symmetric}\nas shown in Fig.~\\ref{genfig}\n\\begin{equation}\nH = J \\sum_{i\\neq -1,0} {\\bf S}_i \\cdot {\\bf S}_{i+1}\n + J_1 {\\bf S}_{-1} \\cdot {\\bf S}_{0} + J_2 {\\bf S}_0 \\cdot {\\bf\nS}_1 . \\label{genham}\n\\end{equation}\nThe three impurity cases above can be identified easily:\n\\begin{itemize}\n\\item $J_2 \\neq J_1 = J$ one modified link in Eq.~(\\ref{oneweakham})\n\\item $J_1 = J_2 \\neq J$ two modified links in Eq.~(\\ref{twoweakham})\n\\item $J_1 = 0, \\ J_2 \\neq J$ edge impurity in Eq.~(\\ref{edgeham})\n\\end{itemize}\nThe density oscillations for the more general model in Eq.~(\\ref{genham})\nare much more complex than in the special cases, so that a detailed\nanalysis of this effect is not always useful. The renormalization behavior\non the other hand is straightforward and can be read off from what we \nalready know about the special cases.\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig13-density-osc.eps}}\n\\caption{Generalized two-link impurity.}\n\\label{genfig}\n\\end{figure}}\n\nA weak coupling $J_1 \\agt 0$ and $J_2 \\agt 0$ to an additional site\nis always marginally relevant, so that the open chain with a \ndecoupled impurity site is unstable for any antiferromagnetic coupling\n(i.e.~negative hopping probability). \nThe periodic chain on the other hand is only stable for the special\nsite-parity symmetric case $J_1 = J_2$, where the \nrenormalization behavior is analogous to the two channel Kondo \neffect. In general, however, one of the two couplings is larger\nand renormalizes to unity, absorbing the spin. The smaller coupling\nis then irrelevant as in the one-weak problem, so that the stable\nfixed point is an open chain with an absorbed impurity \nsite $J_1 = J, \\ J_2 =0$ (or $J_2 = J, \\ J_1 =0$) in most cases, \nexcept for a site-parity symmetric impurity \nor two ferromagnetic coupling constants. The complete renormalization \nflow is summarized in Fig.~\\ref{renflow} where the possible \nfixed points are indicated by the black dots. In cases where the\ncoupling diverges to infinity a singlet forms, and we can therefore again\ndescribe the system by one of the four finite fixed points in\nthe figure. Interestingly, the more stable fixed points\nalways have a lower ground state degeneracy, in accordance with the \ng-theorem.\\cite{g-theorem} The phase diagram in Fig.~\\ref{renflow}\nis valid for all interaction strengths $0 < U \\leq 2 t$ as long as\nthe system is half-filled.\n \n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig14-density-osc.eps}}\n\\caption{Renormalization flow diagram.}\n\\label{renflow}\n\\end{figure}}\n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Spin-1 impurity} \\label{spin1}\n\nWe now turn to a magnetic impurity in the chain with spin $S_{\\rm imp}=1$ \ngiven by the Heisenberg Hamiltonian\n\\begin{equation}\nH = J \\sum_{i\\neq 0} {\\bf S}_i \\cdot {\\bf S}_{i+1}\n + J^\\prime {\\bf S}_{\\rm imp} \\cdot \\left( {\\bf S}_{0} + {\\bf\nS}_1 \\right). \\label{spin1ham}\n\\end{equation}\nas shown in Fig.~\\ref{S1_twoweakfig}.\nIn the previous impurity models in \nSections \\ref{onelink}-\\ref{genimp}\nit was always possible to interpret \nthe Heisenberg Hamiltonians equally well in terms of mesoscopic \nsystems and electrons \nhopping on the lattice by identifying the spin-1/2 impurity\nin terms of an extra site or charge island. However, for the \nspin-1 impurity in Eq.~(\\ref{spin1ham}) no meaningful\ninterpretation in terms of spinless fermions is possible. On the\nother hand this impurity model has important implications for doping in \nquasi one-dimensional spin-1/2 compounds, so that we find it useful to \ndiscuss it here.\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig15-density-osc.eps}}\n\\caption{The spin-1 impurity.}\n\\label{S1_twoweakfig}\n\\end{figure}}\n\nSimilar to the impurity models in Sections \\ref{twolink} and \\ref{edgeimp}\nwe find again that the field theory language is analogous to a\nKondo impurity model. The two ends of the spin-chain play the role\nof the two channels coupled to a spin-1 impurity. A small antiferromagnetic\ncoupling is therefore marginally relevant and the renormalization\nflow goes to the strong coupling limit. The stable fixed point is \ngiven by an open spin chain with two sites removed and a decoupled singlet \ncontaining the spin-1 and the two end spins ($J'\\to \\infty$).\n\nJust like the edge impurity in Sec.~\\ref{edgeimp} this Kondo-type model \nis an exactly screened impurity. The shape of the screening cloud\nis again given by that of an edge magnetic field\\cite{affleckedgefield}\njust like in Eq.~(\\ref{edgealt})\n\\begin{equation}\n\\chi^{\\rm alt}(x) = c_I \\frac{(-1)^x \\sqrt{T}}{\\sqrt{\\sinh(4xT)}} - \n c_R (-1)^{x} \\chi^{\\rm bs}(x)\n, \\label{spin1alt}\n\\end{equation}\nwhere the first term is again the induced screening cloud, while the second\nterm is the backscattering contribution in Eq.~(\\ref{chi}). As shown in\nFig.~\\ref{fig:S1_twoweakc1} the fits to this expression are \nexcellent (again using the open chain data as a reference for $\\chi^{\\rm bs}$).\nThe coefficient $c_I$ for the induced screening cloud again approaches a \nconstant for temperatures below $T_K$ which results in a universal\ncurve as $T\\to 0$ as shown in Fig.~\\ref{fig:S1_c1twoweak}.\nThe backscattering coefficient is an indication of the \neffective phase shift and changes sign depending \non the temperature and coupling strength. \nFrom Fig.~\\ref{fig:S1_twoweakc1}\nit is clear that the backscattering coefficient $c_R$ is positive\nfor small coupling strengths $J'$ (or equivalently high temperatures) \nand negative for\nlarger coupling strengths $J'$ (or equivalently lower temperatures).\nThe renormalization of $c_R$ is explicitly shown in the inset of \nFig.~\\ref{fig:S1_c1twoweak}. As $T\\to 0$ the jump of $c_R$ to negative\nvalues happens at smaller $J'$ and becomes very sharp.\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig16-density-osc.eps}}\n\\caption{Envelope of alternating part at $T/J=0.04$ for the\nspin-1 impurity. From above at $x \\protect\\agt 20$:\n$J^\\prime/J = 0, 0.05, 0.1, 0.2, 0.4, 0.8, 4.0$.\nFits to Eq.~(\\ref{spin1alt}).}\n\\label{fig:S1_twoweakc1}\n\\end{figure}}\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig17-density-osc.eps}}\n\\caption{Coefficient $c_I$ of the spin-1 impurity for\n$T/J =$\n%%$0.5,0.25,0.2,0.1,0.067,0.04$,$0.025,0.02,0.0167,0.0133,0.01$\n$0.2,0.1,0.04,0.025,0.0167,0.0133,0.01$ from below. Inset: Backscattering\ncoefficient $c_R$. The dashed lines are a guide for the eye.}\n\\label{fig:S1_c1twoweak}\n\\end{figure}}\n\nMore interesting are the experimental consequences for Knight shift \nexperiments in doped spin-1/2 chain compounds (as for example\nNi doping in CuO chains). For that case we can predict an interesting NMR\nspectrum with a characteristic feature (sharp edge) corresponding to the \nmaximum in the alternating susceptibility. Such a sharp edge has been \nobserved before in NMR experiments on spin-1/2 chain compounds with\nnon-magnetic defects.\\cite{NMR} In that case the sharp edge broadens with\na $1/\\sqrt{T}$ behavior as discussed in Sec.~\\ref{model}.\nFor the magnetic spin-1 impurities a sharp edge from the maximum in the \nbackscattering part may also be present, but it depends on if the temperature \nis above or below $T_K$ how this feature changes.\nAbove $T_K$ the backscattering part becomes weaker as the temperature \nis lowered,\nbut the induced screening cloud increases, so that the sharp kink \nmay vanish in a quickly broadening line-shape from the screening cloud\nas shown in the left part of Fig.~\\ref{fig:S1_twoweakNMR2}.\nBelow $T_K$ on the other hand, the screening has saturated and\nthe backscattering contribution dominates again (albeit with\na phase shift). Therefore, the kink feature in the NMR spectrum will\nsharpen further as the temperature is lowered and widen with the usual \n$1/\\sqrt{T}$ behavior as shown in the right part \nof Fig.~\\ref{fig:S1_twoweakNMR2}. The detailed T-dependence can be\npredicted for any particular value of $J'$ of an actual experimental\ncompound.\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig18-density-osc.eps}}\n\\caption{NMR signal of the spin-1 impurity for $J^\\prime/J=0.1$\n(left) and $J^\\prime/J=1.4$ (right). }\n\\label{fig:S1_twoweakNMR2}\n\\end{figure}}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Integrable impurity model}\n\n\nFinally, we would like to consider a more exotic impurity model \nwhich has been especially constructed to preserve the integrability \nof the entire system.\\cite{henrik} We consider here the simplest\nnon-trivial example of such an impurity model which corresponds\nto an impurity spin with $S_{\\rm imp} = 1$\nthat is coupled in a special way to two\nsites in the chain. The corresponding Hamiltonian has been set up \nin Ref.~\\onlinecite{henrik}\n\\begin{eqnarray}\nH & = &\n J \\sum_{i\\neq 0} {\\bf S}_i \\cdot {\\bf S}_{i+1} \n- \\case{7 J}{9} {\\bf S}_{0} \\cdot {\\bf S}_1 \n\\label{intham} \\\\\n& & + \\case{4 J}{9} \n\\left[ ({\\bf S}_{0} +\n {\\bf S}_1) \\cdot {\\bf S}_{\\rm imp} \n+ \\left\\{ {\\bf S}_{0} \\cdot {\\bf S}_{\\rm imp}, {\\bf S}_1 \\cdot {\\bf\nS}_{\\rm imp} \\right\\} \\right] , \n\\nonumber \n\\end{eqnarray}\nwhere ${\\bf S}_{\\rm imp}$ is the external spin-1 impurity\nand $\\{,\\}$ denotes the anticommutator. \n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig19-density-osc.eps}}\n\\caption{The integrable impurity.}\n\\label{fig:integrable}\n\\end{figure}}\n\nA closer analysis of this model\\cite{schlottmann} \nshowed that the thermodynamics\nat low temperatures were in fact described by a {\\it periodic}\nspin chain with one additional site and an asymptotically free impurity\nspin with $S=1/2$, so that it appears that the original spin-1 has\nsomehow been partially absorbed by the chain. From a field theory\npoint of view it was later shown that this type of impurity corresponds \nin fact to an unstable fixed point which can only be reached by an artificial\ntuning of the coupling parameters.\\cite{erik}\n\nWe are now interested in what kind of density oscillations might be\nobservable from such an impurity. Interestingly, we found that the\ndensity oscillations were {\\it identically zero at all temperatures} \nas if the system was translationally invariant.\nThe impurity Hamiltonian in Eq.~(\\ref{intham}) was of course constructed\nin a way to avoid all backscattering, but it is remarkable that even \nthe induced alternating part from the magnetic impurity vanishes exactly, \ni.e.~no conventional screening takes place.\n\nNonetheless, the impurity spin is somehow reduced from a spin-1 to an\neffective spin-1/2 as the temperature is lowered. This can be explicitly\nseen from the impurity susceptibility\nin small magnetic fields \n\\begin{equation}\n\\langle S^z_{\\rm imp}\\rangle = B\\frac{C_{\\rm Curie}}{T}\n\\end{equation}\nwhere we have assumed some type of Curie-law.\nAt high temperatures\nthe impurity susceptibility must follow the Curie-law for a spin-1\n$C_{\\rm Curie} = 2/3$, \nwhile at low temperatures a Curie-law for a spin-1/2 $C_{\\rm Curie} = 1/4$\nhas been predicted up to logarithmic corrections.\\cite{schlottmann} \nIn Fig.~\\ref{fig:chiintegrable}\nwe plot the temperature dependent Curie constant (i.e. the\nimpurity susceptibility times temperature). It appears that the\nasymptotic value $C_{\\rm Curie} = 1/4$ is indeed approached with \nlogarithmic corrections as $T\\to 0$. The fit in the figure is \n\\begin{equation}\nC_{\\rm Curie} = \n\\frac{1}{4} + \\frac{1}{8 \\ln(2 \\pi/T)} + a \\frac{\\ln \\left( \\ln(2 \\pi/T)/b\n\\right) }{\\ln(2 \\pi/T)^2} \\label{integrfit}\n\\end{equation}\nwith $a=1.62$ and $b=1.32$.\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig20-density-osc.eps}}\n\\caption{Susceptibility of the spin-1\nin the integrable model multiplied by temperature. Fit to\nEq.~(\\ref{integrfit}).}\n\\label{fig:chiintegrable}\n\\end{figure}}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Numerical Method} \\label{numerics}\nThe numerical method we have \nused here is based on the Density Matrix Renormalization\nGroup (DMRG)\\cite{DMRG} applied to transfer matrices. \nWhile the ordinary DMRG considers the properties of individual \neigenstates in a finite system, we are interested in the thermodynamic\nlimit, namely properties of an infinite system at finite temperatures.\nThis can be achieved by the Transfer Matrix Renormalization Group \n(TMRG),\\cite{TMDMRG} which we adapted especially for \nimpurities\\cite{2CK,rommer} as we will review briefly.\nWe consider the partition function $Z$ of the\nmodels in Eqs.~(\\ref{fermions}) and (\\ref{heisenberg}). After the \nstandard Trotter decomposition, we obtain for an infinite \nsystem $(L\\to \\infty)$\n\\begin{equation}\nZ = \\lim_{M\\to \\infty} {\\rm tr} T_M^{L/2} \\to \n\\lim_{M\\to \\infty} \\lambda_M^{L/2}, \\label{Z}\n\\end{equation}\nwhere $T_M$ is the transfer matrix with $M$ time-slices. In the limit of \ninfinite system size only the largest eigenvalue $\\lambda_M$ determines\nthe thermodynamics of the system, which we find numerically.\nWe start with small time-steps so that the Trotter-error is negligible, and\nsuccessively increase the number of time-slices $M$ to reach lower \ntemperatures. At each step the dimension of $T_M$ increases so we keep only\nthe most important states to describe the state with the highest \neigenvalue $\\lambda_M$ by using the DMRG algorithm with some modifications\nfor asymmetric matrices.\\cite{rommer}\nA measurement of the local spin-density at site $j$\nfor example is straightforward,\nsince we can just absorb the measuring operator \n$S^z_j$ into one of the transfer matrices $T_M \\to T^{sz}_M$\n\\begin{equation}\n\\langle S_j^z \\rangle = \\frac{1}{Z} \\; \\mbox{tr} \\; S_j^z e^{-\\beta H}\n\\to \\frac{\\langle \\psi_M | {T}_M^{sz}(j) | \\psi_M \\rangle}{\\lambda_M},\n\\label{sz}\n\\end{equation}\nwhere $\\langle \\psi_M |$ and $| \\psi_M \\rangle$ are the left and right \ntarget states for the eigenvalue $\\lambda_M$. So far we have considered\na translational invariant system.\n\nWe now introduce a generic impurity which modifies one of the\ntransfer matrices $T_M \\to T_{\\rm imp}$. \nEven in the presence of impurities the thermodynamics of the system\nis entirely determined by the highest eigenvalue $\\lambda_M$ and \ncorresponding eigenstate of the pure transfer matrix $T_M$ which always \nappears with an infinite power in the partition function in Eq.~(\\ref{Z}).\nThe measurement of\nthe spin (or charge) density near the impurity is again straightforward.\nFor the spin density at a distance of $j$ sites from the \nimpurity we write\n\\begin{equation}\n\\langle S^z_j \\rangle = \\frac{ \\langle \\psi_M | {T}_{M}^{sz} \n\\ ({T}_{M})^{j/2} \\ {T}_{\\rm imp} | \\psi_M \\rangle }{ \\lambda_M^{j/2+1} \\;\n\\langle \\psi_M | {T}_{\\rm imp} | \\psi_M \\rangle } . \\label{szimp}\n\\end{equation}\n\nSince we step-wise approximate the transfer matrix, it is\nimportant to make a careful error-analysis. The error due to the\nTrotter approximation is the simplest to estimate since it\nis just proportional to the square of the time-step $\\tau = 1/T M$.\nWe found that a value of $\\tau = 0.05/J$ makes this error negligible\ncompared to the DMRG truncation errors. To estimate the truncation\nerrors we can compare our results to the exact solution of the\nfree fermion Hamiltonian in Eq.~(\\ref{fermions}) with $U=0$.\nThe structure of the transfer matrix is not fundamentally changed\nby taking $U=0$ so that the truncation error will be of the same \norder as for $U\\neq 0$. Keeping 64 states we find\nfor the local response of the spins closest to\ntypical impurities a relative error of less than\n$10^{-4}$ for $T>0.04$, less than $10^{-3}$ for $0.02 < T < 0.04$ and a relative\nerror of less than $10^{-2}$ for temperatures $0.01 < T < 0.02$. However, already\nfrom Eq.~(\\ref{szimp}) it is clear \nthat the spin and charge densities far away from the impurity will\ncontain a larger error. Each transfer matrix contains a small\nerror $\\epsilon$ which then gets exponentiated in Eq.~(\\ref{szimp})\nand hence the oscillating part\nof the density $\\langle S_j^z \\rangle$ is suppressed exponentially with\ndistance $j$ \n\\begin{equation}\n\\langle S_j^z \\rangle_{\\rm osc} \\propto (1-\\epsilon)^{j} = \\exp(-j \\epsilon)\n\\label{error}\n\\end{equation}\nwhere $\\epsilon$ depends only on temperature. This exponential \nsuppression with the distance from the boundary\nis again a consequence of the fact that the incoming and\noutgoing waves lose coherence but this time \ndue to error fluctuations. However, the corresponding energy \nscale from the truncation error is always smaller than the \ntemperature in our case. We observe that the suppression error \nin Eq.~(\\ref{error}) is actually very systematic, so that we \ncan even correct our data very well using Eq.~(\\ref{error}).\nFor free fermions we find to high accuracy the following dependence\nof the error on temperature\n\\begin{equation}\n\\epsilon = 0.06 \\exp(- 58 T),\n\\label{epsilon}\n\\end{equation}\nwhere we have kept 64 states in the TMRG simulations.\nFor interacting fermions the suppression also has the exponential dependence\nin Eq.~(\\ref{error}), but the energy scale $\\epsilon$ is in general\ndependent on the interaction $U$. For the Heisenberg model an \nindependent analysis of the free energy hinted at a value of \napproximately $\\epsilon = 0.02 \\exp(-34 T)$, but the value in \nEq.~(\\ref{epsilon}) is more reliable and gives a relatively \ngood estimate of the error for all interaction strengths.\nWe chose to correct our data for the alternating \nfermion densities by dividing out the factor in Eq.~(\\ref{error})\ntogether with the estimate in Eq.~(\\ref{epsilon}) in all cases presented\nabove. However, the use of this correction or the \nparticular choice of the error $\\epsilon$ makes no qualitative difference\nin any of our findings, since the temperature suppression always \ndominates (i.e.~the energy scale in Eq.~(\\ref{epsilon}) is always\nsmaller than the temperature). \nAnother important energy scale is the finite magnetic field $B$\nthat is used in the simulations (i.e.~how close the system is to \nhalf filling). We typically used a value of $B=0.003$ which makes the \nmagnetic length scale in Eq.~(\\ref{n_osc2})\nalways negligible compared to the finite temperature correlation length.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Conclusion} \\label{summary}\nWe have considered a number of impurity models and were able to \nextract detailed information about the backscattering amplitude,\nthe backscattering phase-shift, and the impurity screening effects\nby examining the Friedel oscillations. The results for the\nvarious impurities have direct and indirect implications for\na large number of theoretical models and experimental systems\nas we will summarize below.\n\n\\subsection{Kondo-type impurities}\nKondo impurity problems are maybe the most famous examples of\nimpurity renormalization effects ever since the classic work by \nWilson.\\cite{wilson} \nMany of the impurity models we have considered here are analogous to \nKondo impurity problems in terms of the field theory language. \nIn particular, the field theory description of a Heisenberg chain \nis the same as that of the spin-channel for a spin-full electron\nfield (while the charge excitations are neglected).\nMoreover, it is known that coupling the open end of a Heisenberg chain\nto an impurity spin produces the same impurity operators as in\nthe real Kondo problem.\\cite{eggert,affleck} \nThe number of channels in the equivalent Kondo problems \nis given by the open ends that the \nimpurity spin is connected to (e.g.~the two link impurity in \nSec.~\\ref{twolink} is analogous to the two channel S=1/2 Kondo problem).\nIt is important to realize that the Heisenberg spins \nin the chains that we consider here have\ndifferent expressions in terms of the boson fields than the real \nelectron spins in the full three dimensional Kondo problems.\nNonetheless, we can still use our models to gain some insight into the \ncentral aspects of renormalization, scaling, cross-over temperature, and\nscreening clouds. \n\nWe have shown that the Kondo-type impurities indeed show the \nexpected renormalization to a screened impurity spin. In particular, \nwe have found a diverging screening cloud (and vanishing backscattering)\nfor the overscreened case in Sec.~\\ref{twolink}, while the \nexactly screened cases in Sec.~\\ref{edgeimp} and \\ref{spin1}\nare characterized by a finite screening cloud and a phase shift in\nthe backscattering as $T\\to 0$. \n\nTo analyze the renormalization process more quantitatively it is important\nto introduce the concept of scaling. It can be expected that \nthe impurity introduces a new energy scale that depends on the initial bare \ncoupling constants. Commonly this energy scale is \nreferred to as the cross-over temperature $T_K$. By making use of \nscale invariance it is then possible to describe the renormalization\nprocess universally in terms of the single parameter $T/T_K$.\nIn particular, impurity properties like the impurity susceptibility \nare described by a universal scaling function\n$\\chi_{\\rm imp} = f(T/T_K)/T$, which is valid for all $T$ and $T_K$ below\nthe cut-off. This behavior was demonstrated explicitly before \nfor the two weak link problem\\cite{2CK} and works for all\nKondo-type impurities in this paper (not shown). In fact it is \npossible to extract the cross-over \ntemperature $T_K$ up to an arbitrary overall scale explicitly\nby collapsing the data according to the scaling analysis.\\cite{2CK,rommer}\nWe have determined $T_K$ this way as a \nfunction of coupling $J'$ in each case as\nshown in Fig.~\\ref{fig:tkcollapse} (up to an arbitrary overall scale).\nThe Kondo temperature shows the same exponential dependence for small $J'$ \n\\begin{equation}\nT_K \\propto \\exp(-0.85J/J') \\label{TK}\n\\end{equation}\nas shown in Fig.~\\ref{fig:tkcollapse} (coming from the same\nmarginally relevant operator at the unstable fixed point in all cases).\nThe underscreened case of a spin-1 coupled to the end of one chain has also \nbeen included in Fig.~\\ref{fig:tkcollapse} for completeness.\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig21-density-osc.eps}}\n\\caption{Crossover temperature $T_K$ of four different Kondo-type impurities.\n$T_K$ has been multiplied by arbitrary constants in order to compare\nthe four cases.}\n\\label{fig:tkcollapse}\n\\end{figure}}\n\nMore interesting in the context of the density oscillations\nis maybe the scaling of the screening cloud. As the screening cloud\nwe define that part of the alternating density that is induced by\nthe magnetic impurity, labeled by $c_I$ in Eqs.~(\\ref{twolinkalt}), \n(\\ref{edgealt}), and (\\ref{spin1alt}). In Ref.~\\onlinecite{sorensen}\nit was postulated that the screening cloud in the real\nKondo effect should be a function of the \nscaling variables $xT$ and $T/T_K$.\nIn our cases we can make a similar argument except that we need\nto include an overall factor $T^{g-1}$ to account for the dimensionality \nof the correlation functions. We therefore obtain the \nfollowing scaling law\n\\begin{equation}\n\\chi^{\\rm screening} = T^{g-1} f(x T, T/T_K). \\label{scaling}\n\\end{equation}\nIndeed we find that \nthe shape of the screening cloud is not affected by $T_K$\nand can always be expressed as a function of the scaling variable $xT$. \nThe coefficient $c_I$\nmust therefore be a function of $T/T_K$ multiplied by \nappropriate powers of $T$. As an example we can take the\ntwo link problem at $g=1/2$ with the screening cloud given in \nEq.~(\\ref{twolinkalt}), where the coefficient can be written as\n$c_I = f(T_K/T)/\\sqrt{T}$ with some function $f$. In Fig.~\\ref{cIscale}\nwe replot the coefficient $c_I$ analogous \nto Fig.~\\ref{fig:c1twoweak} but with the argument replaced by $T_K/T$\ninstead of $J'$. The inset shows that \nthe data indeed collapses if multiplied by $\\sqrt{T}$ as implied by \nEq.~(\\ref{scaling}). The solid line in Fig.~\\ref{fig:c1twoweak}\ntherefore is proportional to $1/\\sqrt{T_K}$ and diverges exponentially\nwith $J'$ according to Eq.~(\\ref{TK}).\nSimilar arguments can be made for the coefficients $c_I$ in the screening \nclouds of the exactly screened cases in Eqs.~(\\ref{edgealt}) \nand (\\ref{spin1alt}), except that $c_I = f(T_K/T)/T$ \nand the solid line is proportional to $1/T_K$ in that case.\n\n{\\begin{figure}\n\\epsfxsize=7cm\n\\centerline{\\epsfbox{fig22-density-osc.eps}}\n\\caption{The coefficient $c_I$ for the two link impurity in\nEq.~(\\ref{twolinkalt}) as a function of $T_K/T$ for different\ntemperatures $T/J = 0.2,0.1,0.04,0.025,0.0167,0.01$ from below.}\n\\label{cIscale}\n\\end{figure}}\n\n\\subsection{Doping in spin chains}\nOur results also have immediate experimental consequences for impurities in \nspin-chain compounds such as KCuF$_3$ or $\\rm Sr_2CuO_3$.\nThe spin density oscillations are directly linked\nto the local Knight shifts (susceptibilities) close to the\ncorresponding impurities, which can be measured by standard NMR techniques\nor muon spin resonance. NMR experiments have already successfully\ndetected the sharp feature corresponding to the maximum \nin Fig.~\\ref{altsusc} from open boundaries due to \nnon-magnetic defects that were naturally present in the crystal.\\cite{NMR}\nWe now propose to use intentional doping with magnetic or non-magnetic \nimpurities to see the predicted\nrenormalization effects. Impurities of one or\ntwo modified links in the chain\ncan possibly be created by doping the surrounding \nnon-magnetic atoms in the crystal at link or site parity symmetric \nlocations. The spin-1 impurities in Sec.~\\ref{spin1} could be produced \nin a more straightforward way by \nsubstituting Cu ions by Ni ions in the corresponding compounds.\nIn Sec.~\\ref{spin1} we discussed explicitly how the renormalization\neffects for spin-1 impurities would show up in an actual experiment.\nSimilar arguments can also be made for the two link\\cite{2CK}\nor one link impurities by simply using the analytic form of \nthe corresponding alternating spin densities with the \ncoefficients $c_R$ and $c_I$ that we have calculated.\n\nIn general we find a strong enhancement of the antiferromagnetic order \nnear impurities. This enhancement can also be observed \nin higher dimensions\\cite{elbio} and may have important consequences for\nimpurity-impurity interactions. In one dimension this effect is \nstrongest, but the complex functional dependence we found here\nis often beyond the intuitive explanation in terms of valence bond \nstates.\\cite{elbio}\n\n\n\\subsection{Impurities in Mesoscopic systems}\nFinally, our analysis also allows us to draw important conclusions\nfor transport measurements in one dimensional mesoscopic structures.\nThis is probably the first time that the conductivity could be \nexplicitly extracted from numerical data for Luttinger Liquid type models.\nNot surprisingly, we found that a generic\nimpurity indeed renormalizes to complete backscattering as the temperature is \nlowered, and we also could explicitly observe the ``healing effect'' in \nthe symmetric resonant tunneling case as predicted by Kane and \nFisher.\\cite{kane1,kanereview,kane2} Our numerical results\nnot only confirm the asymptotic power-laws, but also \ngive a quantitative estimate of the conductivity for all temperatures\nand impurity strengths. For a generic impurity with little or \nintermediate backscattering we find that the asymptotic scaling region\nturns out to be extremely narrow. For impurities with strong backscattering \nwe find that the conductivity is {\\it enhanced} by interactions at \nhigher temperatures.\n\nOne obvious question is how those results can be generalized to spinful \nelectron systems and carbon nanotubes. A number of works have addressed\nthe question of impurities in spinful wires\\cite{kanereview,yue,furusaki,wong} \nand found a richer structure since renormalization takes place in both the\nspin and the charge channels. However, if realistic SU(2) invariant \ninteractions are assumed the generic behavior is very similar to the spinless\ncase, so that we expect that our results for the reflection coefficient \ncarry over in a straight forward fashion. The shape and amplitude\nof the density oscillations, however, will in general be very different\nfor spinful electron systems. For carbon nanotubes it has been shown that \nthe Friedel oscillations impose a characteristic pattern that can be \nobserved with scanning tunneling microscopy.\\cite{mele} For spinful wires\nit is expected that the Friedel oscillations from an open end\ncan reveal the nature of the spin-charge separation in real space.\\cite{stm}\nAlthough our results do not allow for quantitative predictions of the \ndensity oscillations in spinful systems, \nwe generally expect that strong, long-range density oscillations should\nbe present from backscattering in one-dimension.\nOne experimental consequence of those oscillations\nis that the measurement through a lead close\nto an impurity is very sensitive to the exact location. \nPrevious studies have shown that even \nthe distance between two leads can play a crucial role.\\cite{kinaret}\nThe current\nmay be strongly enhanced or depleted, depending on if the distance \nto the impurity is a multiple of $2 k_F x$ or not.\nEspecially interesting are therefore experiments with an adjustable \nlead such as a tunneling tip.\nThe direct observation of those oscillations could give detailed \ninformation about both the nature of the impurity and also about the\ninteractions in the system.\n\n%Not surprisingly, we found \n%that a generic impurity of a weak link indeed renormalizes to a completely\n%backscattering open boundary condition in interacting quantum wires\n%and spin chains. If a special symmetry is present this effect can be \n%reversed so that the backscattering renormalizes to zero, e.g.~by tuning \n%the potential of a quantum dot in the wire or by a simple site symmetric \n%impurity in the spin chain case. However, at the same time there is an\n%induced screening cloud in the system, which does not affect the \n%transport properties, but should be observable by probing the local\n%spin or charge density (e.g. by scanning tunneling microscopy or NMR\n%experiments).\n%\n%An impurity perturbation at the end is analogous to the ordinary Kondo \n%effect in terms of the field theory description. \n%Such an impurity always exhibits complete backscattering\n%by construction, but we were able to observe the phase shift directly,\n%which also gives detailed information about the renormalization behavior\n%and the Kondo temperature as a function of coupling strength. The\n%induced screening cloud can also be extracted explicitly as a function of \n%temperature and Kondo coupling.\n%\n%An integrable impurity does not exhibit any backscattering whatsoever\n%due to the high symmetry of the integrable model. We can confirm the\n%exotic behavior of the integrable spin-1 impurity being partially absorbed in\n%the chain and leaving a decoupled effective spin-1/2 as the only remnant\n%of the impurity.\n%\n%In our simulations we have effectively measured the transmission \n%coefficient which may be of use in interpreting \n%transport measurements in quantum wires especially \n%at intermediate temperatures. More interesting, however, is the \n%possibility of examining the density oscillations directly by tunneling\n%microscopy or NMR experiments. To some extend this has already\n%been done for the simplest impurities in spin chains\\cite{NMR}\n%and even carbon nanotubes.\\cite{STM}\n%As we have shown here more complicated impurities may give rise\n%to additional interesting effects, which\n%provide detailed information about the nature of the impurity, the\n%interactions in the system and possibly even impurity-impurity \n%interactions.\n\n\\begin{acknowledgements}\nS.R.\\ acknowledges support from the Swedish Foundation for International\nCooperation in Research and Higher Education (STINT). S.E.\\ is thankful\nfor the support from the Swedish Natural Science Research Council \nthrough the research grants F-AA/FU 12288-301 and S-AA/FO 12288-302.\n\\end{acknowledgements}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{references}\n\\bibitem{kane1} C.L.\\ Kane and M.P.A.\\ Fisher, Phys.\\ Rev.\\ Lett. {\\bf\n68}, 1220 (1992).\n\n\\bibitem{elbio}\nG.B.\\ Martins, M.\\ Markus J.\\ Riera and E.\\ Dagotto, {Phys.\\ Rev.\\ Lett.}\n{\\bf 78}, 3563 (1997).\n\n\\bibitem{friedel} J.\\ Friedel, Nuovo Cimento Suppl. {\\bf 7}, 187 (1958).\n\n%\\bibitem{RKKY} M.A.\\ Rudderman and C.\\ Kittel, Phys. Rev. {\\bf 96}, 99 (1954); \n%T.\\ Kasuya, Prog.\\ Theor.\\ Phys. {\\bf 16}, 45 (1956);\n%K.\\ Yosida, Phys.\\ Rev. {\\bf 106}, 893 (1957).\n\n\\bibitem{sorensen} E.S.~S\\o rensen and I.~Affleck, Phys.\\ Rev.\\ B {\\bf 53}, \n9153 (1996).\n\n\n\\bibitem{kanereview} C.L.\\ Kane and M.P.A.\\ Fisher, {Phys.\\ Rev.\\ B} {\\bf 46},\n15233 (1992).\n\n\\bibitem{yue} K.A.~Matveev, D.X.~Yue and L.I.~Glazman, Phys.~Rev.~Lett. {\\bf 71},\n3351 (1993); D.X.~Yue, L.I.~Glazman and K.A.~Matveev, Phys.~Rev.~B {\\bf 49},\n1966 (1994). \n\n\\bibitem{eggert} S.\\ Eggert and I.\\ Affleck, { Phys.\\ Rev.\\ B} {\\bf 46}, 10866 \n(1992). \n\n\\bibitem{kane2} C.L.\\ Kane and M.P.A.\\ Fisher, Phys.\\ Rev.\\ B {\\bf\n46}, 7268 (1992).\n\n\\bibitem{2CK} S.\\ Eggert and S.\\ Rommer, Phys.\\ Rev.\\ Lett. {\\bf 81}, 1690\n(1998); Physica B {\\bf 261}, 200 (1999).\n\n\\bibitem{qin} S.J.~Qin, M.~Fabrizio and L.~Yu, Phys.~Rev.~B {\\bf 54}, \nR9643 (1996); S.J.~Qin, M.~Fabrizio, L.~Yu, M.~Oshikawa and I.~Affleck, \nPhys.~Rev.~B {\\bf 56}, 9766 (1997). \n\n\n\\bibitem{rommer}\nS.\\ Rommer and S.\\ Eggert, Phys.\\ Rev.\\ B {\\bf 59}, 6301 (1999).\n\n\\bibitem{TMDMRG}\nR.J.\\ Bursill, T.\\ Xiang and G.A.\\ Gehring, { J.\\ Phys.\\ C} {\\bf 8}, L583 (1996); \nX.Q.\\ Wang and T.\\ Xiang, { Phys.\\ Rev.\\ B} {\\bf 56}, 5061 (1997); N.\\ Shibata,\n{ J Phys.\\ Soc.\\ Jpn.} {\\bf 66}, 2221 (1997).\n\n\n\\bibitem{korepin} N.M.\\ Bogoliubov, A.G.\\ Izergin and V.E.\\ Korepin, \nNucl.\\ Phys. B {\\bf 275}, 687 (1986).\n\n\\bibitem{mattsson} A.E.\\ Mattsson, S.\\ Eggert and H.\\ Johannesson,\nPhys.\\ Rev.\\ B {\\bf 56}, 15615 (1997).\n\n\\bibitem{boundcorr} S.\\ Eggert, H.\\ Johannesson and A.\\ Mattsson,\nPhys.\\ Rev.\\ Lett. {\\bf 76}, 1505 (1996).\n\n\\bibitem{fabrizio} M.\\ Fabrizio and A.O.\\ Gogolin, { Phys.\\ Rev.\\ B} {\\bf 51}, \n17827 (1995). \n\n\\bibitem{eggopen}\nS.\\ Eggert and I.\\ Affleck, Phys.\\ Rev.\\ Lett. {\\bf 75}, 934 (1995)\n\n\\bibitem{finiteT}\nS.\\ Eggert, A.E.\\ Mattsson and J.M.\\ Kinaret, \nPhys.\\ Rev.\\ B {\\bf 56}, R15537 (1997).\n\n\\bibitem{egger} R.\\ Egger and H.\\ Grabert, Phys.\\ Rev.\\ Lett. {\\bf 75}, 3505\n(1995).\n \n\n\\bibitem{suscept} S. Eggert, I. Affleck and M. Takahashi, Phys. Rev. Lett.\n{\\bf 73}, 332 (1994).\n\n\\bibitem{brunel} V.\\ Brunel, M.\\ Bocquet and Th.\\ Jolicoeur, \nPhys.~Rev.~Lett. {\\bf 83} 2821 (1999); I.\\ Affleck and S.J.\\ Qin, \nJ. Phys. A: Math. Gen. {\\bf 32}, 7815 (1999).\n\n\n\\bibitem{NMR} M.\\ Takigawa, N.\\ Motoyama, H.\\ Eisaki and S.\\ Uchida, {\nPhys.\\ Rev.\\ B} {\\bf 55}, 14129 (1997); { Phys.\\ Rev.\\ Lett.}\n{\\bf 76}, 4612 (1996); N.\\ Fujiwara, H.\\ Yasuoka, M.\\ Isobe and Y.\\ Ueda, \nPhys. Rev. B {\\bf 58}, 11134 (1998).\n\n\\bibitem{laukamp} M.~Laukamp, G.B.~Martins, C.~Gazza, A.L.~Malvezzi, E.~Dagotto,\n P.M.~Hansen, A.C.~Lopez and J.~Riera, Phys.~Rev.~B {\\bf 57}, 10755 (1998).\n\n\\bibitem{nishino} M.~Nishino, H.~Onishi, P.~Roos, K.~Yamaguchi and S.~Miyashita,\nPhys.~Rev.~B {\\bf 61}, 4033 (2000).\n\n\\bibitem{affleckedgefield}\nI.\\ Affleck, J.\\ Phys.\\ A {\\bf 31}, 2761 (1998).\n\n\\bibitem{g-theorem}\nI.\\ Affleck and A.W.W.~Ludwig, Phys.~Rev.~Lett. {\\bf 67}, 161 (1991).\n\n\\bibitem{henrik}\nN.\\ Andrei and H.\\ Johannesson, Phys.\\ Lett. {\\bf 100A}, 108 (1984)\n\n\\bibitem{schlottmann}\nP.\\ Schlottmann, J.\\ Phys.\\ Condens.\\ Matter {\\bf 3}, 6617 (1991)\n\n\\bibitem{erik} E.S.~S\\o rensen, S.~Eggert and I.~Affleck, \nJ.~Phys.~A {\\bf 26}, 6757 (1993).\n\n\\bibitem{DMRG}\nS.R. White, Phys.\\ Rev.\\ Lett {\\bf 69}, 2863 (1992); {Phys.\\ Rev.\\ B} {\\bf 48}, \n10345 (1993). \n\n\\bibitem{wilson} K.G.~Wilson, Rev. Mod. Phys. {\\bf 47}, 773 (1975).\n\n\\bibitem{affleck} I. Affleck, {\\it Acta Physica Polonica B}\n{\\bf 26}, 1869 (1995); preprint cond-mat/9512099.\n\n\\bibitem{furusaki} A.~Furusaki and N.~Nagaosa, \nPhys.~Rev.~B {\\bf 47}, 4631 (1993).\n \n\\bibitem{wong} E.~Wong and I.~Affleck Nucl.~Phys.~B {\\bf 417}, 403 (1994).\n\n\\bibitem{mele} C.L.~Kane and E.J.~Mele, Phys. Rev. B {\\bf 59}, 12759 (1999);\nW.~Clauss, D.J.~Bergeron, M.~Freitag, C.L.~Kane, E.J.~Mele and A.T.~Johnson, \nEurophys.~Lett. {\\bf 47}, 601 (1999).\n\n\\bibitem{stm} S.~Eggert, Phys.~Rev.~Lett.~{\\bf 84}, 4413 (2000).\n\n\\bibitem{kinaret} J.M.~Kinaret, M.~Jonson, R.I.~Shekter and S.~Eggert, \nPhys. Rev. B {\\bf 57}, 3777 (1998); Physica E {\\bf 1}, 265 (1997).\n\n\n\n\n\\end{references}\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002001.extracted_bib",
"string": "\\bibitem{kane1} C.L.\\ Kane and M.P.A.\\ Fisher, Phys.\\ Rev.\\ Lett. {\\bf\n68}, 1220 (1992).\n\n\n\\bibitem{elbio}\nG.B.\\ Martins, M.\\ Markus J.\\ Riera and E.\\ Dagotto, {Phys.\\ Rev.\\ Lett.}\n{\\bf 78}, 3563 (1997).\n\n\n\\bibitem{friedel} J.\\ Friedel, Nuovo Cimento Suppl. {\\bf 7}, 187 (1958).\n\n%\n\\bibitem{RKKY} M.A.\\ Rudderman and C.\\ Kittel, Phys. Rev. {\\bf 96}, 99 (1954); \n%T.\\ Kasuya, Prog.\\ Theor.\\ Phys. {\\bf 16}, 45 (1956);\n%K.\\ Yosida, Phys.\\ Rev. {\\bf 106}, 893 (1957).\n\n\n\\bibitem{sorensen} E.S.~S\\o rensen and I.~Affleck, Phys.\\ Rev.\\ B {\\bf 53}, \n9153 (1996).\n\n\n\n\\bibitem{kanereview} C.L.\\ Kane and M.P.A.\\ Fisher, {Phys.\\ Rev.\\ B} {\\bf 46},\n15233 (1992).\n\n\n\\bibitem{yue} K.A.~Matveev, D.X.~Yue and L.I.~Glazman, Phys.~Rev.~Lett. {\\bf 71},\n3351 (1993); D.X.~Yue, L.I.~Glazman and K.A.~Matveev, Phys.~Rev.~B {\\bf 49},\n1966 (1994). \n\n\n\\bibitem{eggert} S.\\ Eggert and I.\\ Affleck, { Phys.\\ Rev.\\ B} {\\bf 46}, 10866 \n(1992). \n\n\n\\bibitem{kane2} C.L.\\ Kane and M.P.A.\\ Fisher, Phys.\\ Rev.\\ B {\\bf\n46}, 7268 (1992).\n\n\n\\bibitem{2CK} S.\\ Eggert and S.\\ Rommer, Phys.\\ Rev.\\ Lett. {\\bf 81}, 1690\n(1998); Physica B {\\bf 261}, 200 (1999).\n\n\n\\bibitem{qin} S.J.~Qin, M.~Fabrizio and L.~Yu, Phys.~Rev.~B {\\bf 54}, \nR9643 (1996); S.J.~Qin, M.~Fabrizio, L.~Yu, M.~Oshikawa and I.~Affleck, \nPhys.~Rev.~B {\\bf 56}, 9766 (1997). \n\n\n\n\\bibitem{rommer}\nS.\\ Rommer and S.\\ Eggert, Phys.\\ Rev.\\ B {\\bf 59}, 6301 (1999).\n\n\n\\bibitem{TMDMRG}\nR.J.\\ Bursill, T.\\ Xiang and G.A.\\ Gehring, { J.\\ Phys.\\ C} {\\bf 8}, L583 (1996); \nX.Q.\\ Wang and T.\\ Xiang, { Phys.\\ Rev.\\ B} {\\bf 56}, 5061 (1997); N.\\ Shibata,\n{ J Phys.\\ Soc.\\ Jpn.} {\\bf 66}, 2221 (1997).\n\n\n\n\\bibitem{korepin} N.M.\\ Bogoliubov, A.G.\\ Izergin and V.E.\\ Korepin, \nNucl.\\ Phys. B {\\bf 275}, 687 (1986).\n\n\n\\bibitem{mattsson} A.E.\\ Mattsson, S.\\ Eggert and H.\\ Johannesson,\nPhys.\\ Rev.\\ B {\\bf 56}, 15615 (1997).\n\n\n\\bibitem{boundcorr} S.\\ Eggert, H.\\ Johannesson and A.\\ Mattsson,\nPhys.\\ Rev.\\ Lett. {\\bf 76}, 1505 (1996).\n\n\n\\bibitem{fabrizio} M.\\ Fabrizio and A.O.\\ Gogolin, { Phys.\\ Rev.\\ B} {\\bf 51}, \n17827 (1995). \n\n\n\\bibitem{eggopen}\nS.\\ Eggert and I.\\ Affleck, Phys.\\ Rev.\\ Lett. {\\bf 75}, 934 (1995)\n\n\n\\bibitem{finiteT}\nS.\\ Eggert, A.E.\\ Mattsson and J.M.\\ Kinaret, \nPhys.\\ Rev.\\ B {\\bf 56}, R15537 (1997).\n\n\n\\bibitem{egger} R.\\ Egger and H.\\ Grabert, Phys.\\ Rev.\\ Lett. {\\bf 75}, 3505\n(1995).\n \n\n\n\\bibitem{suscept} S. Eggert, I. Affleck and M. Takahashi, Phys. Rev. Lett.\n{\\bf 73}, 332 (1994).\n\n\n\\bibitem{brunel} V.\\ Brunel, M.\\ Bocquet and Th.\\ Jolicoeur, \nPhys.~Rev.~Lett. {\\bf 83} 2821 (1999); I.\\ Affleck and S.J.\\ Qin, \nJ. Phys. A: Math. Gen. {\\bf 32}, 7815 (1999).\n\n\n\n\\bibitem{NMR} M.\\ Takigawa, N.\\ Motoyama, H.\\ Eisaki and S.\\ Uchida, {\nPhys.\\ Rev.\\ B} {\\bf 55}, 14129 (1997); { Phys.\\ Rev.\\ Lett.}\n{\\bf 76}, 4612 (1996); N.\\ Fujiwara, H.\\ Yasuoka, M.\\ Isobe and Y.\\ Ueda, \nPhys. Rev. B {\\bf 58}, 11134 (1998).\n\n\n\\bibitem{laukamp} M.~Laukamp, G.B.~Martins, C.~Gazza, A.L.~Malvezzi, E.~Dagotto,\n P.M.~Hansen, A.C.~Lopez and J.~Riera, Phys.~Rev.~B {\\bf 57}, 10755 (1998).\n\n\n\\bibitem{nishino} M.~Nishino, H.~Onishi, P.~Roos, K.~Yamaguchi and S.~Miyashita,\nPhys.~Rev.~B {\\bf 61}, 4033 (2000).\n\n\n\\bibitem{affleckedgefield}\nI.\\ Affleck, J.\\ Phys.\\ A {\\bf 31}, 2761 (1998).\n\n\n\\bibitem{g-theorem}\nI.\\ Affleck and A.W.W.~Ludwig, Phys.~Rev.~Lett. {\\bf 67}, 161 (1991).\n\n\n\\bibitem{henrik}\nN.\\ Andrei and H.\\ Johannesson, Phys.\\ Lett. {\\bf 100A}, 108 (1984)\n\n\n\\bibitem{schlottmann}\nP.\\ Schlottmann, J.\\ Phys.\\ Condens.\\ Matter {\\bf 3}, 6617 (1991)\n\n\n\\bibitem{erik} E.S.~S\\o rensen, S.~Eggert and I.~Affleck, \nJ.~Phys.~A {\\bf 26}, 6757 (1993).\n\n\n\\bibitem{DMRG}\nS.R. White, Phys.\\ Rev.\\ Lett {\\bf 69}, 2863 (1992); {Phys.\\ Rev.\\ B} {\\bf 48}, \n10345 (1993). \n\n\n\\bibitem{wilson} K.G.~Wilson, Rev. Mod. Phys. {\\bf 47}, 773 (1975).\n\n\n\\bibitem{affleck} I. Affleck, {\\it Acta Physica Polonica B}\n{\\bf 26}, 1869 (1995); preprint cond-mat/9512099.\n\n\n\\bibitem{furusaki} A.~Furusaki and N.~Nagaosa, \nPhys.~Rev.~B {\\bf 47}, 4631 (1993).\n \n\n\\bibitem{wong} E.~Wong and I.~Affleck Nucl.~Phys.~B {\\bf 417}, 403 (1994).\n\n\n\\bibitem{mele} C.L.~Kane and E.J.~Mele, Phys. Rev. B {\\bf 59}, 12759 (1999);\nW.~Clauss, D.J.~Bergeron, M.~Freitag, C.L.~Kane, E.J.~Mele and A.T.~Johnson, \nEurophys.~Lett. {\\bf 47}, 601 (1999).\n\n\n\\bibitem{stm} S.~Eggert, Phys.~Rev.~Lett.~{\\bf 84}, 4413 (2000).\n\n\n\\bibitem{kinaret} J.M.~Kinaret, M.~Jonson, R.I.~Shekter and S.~Eggert, \nPhys. Rev. B {\\bf 57}, 3777 (1998); Physica E {\\bf 1}, 265 (1997).\n\n\n\n\n"
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cond-mat0002002
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Localization-delocalization transition of disordered $d$-wave superconductors in class $C$I
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[
{
"author": "Takahiro Fukui\\cite{Email}"
}
] |
A lattice model for disordered $d$-wave superconductors in class $C$I is reconsidered. Near the band-center, the lattice model can be described by Dirac fermions with several species, each of which yields WZW term for an effective action of the Goldstone mode. The WZW terms cancel out each other because of the four-fold symmetry of the model, which suggests that the quasiparticle states are localized. If the lattice model has, however, symmetry breaking terms which generate mass for any species of the Dirac fermions, remaining WZW term which avoids the cancellation can derive the system to a delocalized strong-coupling fixed point.
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[
{
"name": "cond-mat0002002.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% REVTeX file of the paper: %\n% %\n% Title: Localization-delocalization transition of %\n% disordered $d$-wave superconductors in class $C$I %\n% %\n% %\n% Author: T. Fukui %\n% %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentstyle[epsf,multicol,prl,aps]{revtex}\n%\\documentstyle[epsf,floats,multicol,prl,aps,eqsecnum]{revtex}\n%\\documentstyle[preprint,aps,12pt,epsf]{revtex}\n%---------------------------------------------------------------------\n\\newcommand{\\rmt}{{\\rm t}}\n\\newcommand{\\rmd}{{\\rm d}}\n\\newcommand{\\rmR}{{\\rm r}}\n\\newcommand{\\rmA}{{\\rm a}}\n\\newcommand{\\rmRS}{{\\rm rs}}\n\\newcommand{\\rmF}{{\\rm F}}\n\\newcommand{\\Det}{{\\rm Det}}\n\\newcommand{\\Tr}{{\\rm Tr}}\n\\newcommand{\\rmWZW}{{\\rm WZW}}\n\\newcommand{\\calH}{{\\cal H}}\n\\newcommand{\\calC}{{\\cal C}}\n\\newcommand{\\calT}{{\\cal T}}\n\\newcommand{\\calL}{{\\cal L}}\n\\newcommand{\\calZ}{{\\cal Z}}\n\\newcommand{\\calD}{{\\cal D}}\n\\newcommand{\\bmC}{ \\mbox{\\boldmath $C$} }\n\\newcommand{\\tr}{{\\rm tr}}\n\\newcommand{\\bra}[1]{\\langle #1|}\n\\newcommand{\\ket}[1]{|#1\\rangle}\n\\newcommand{\\tPsi}{{\\widetilde\\Psi}}\n\\newcommand{\\tpsi}{{\\widetilde\\psi}}\n\\newcommand{\\tH}{{\\widetilde H}}\n\\newcommand{\\bpsi}{{\\bar\\psi}}\n\\newcommand{\\bchi}{{\\bar\\chi}}\n\\newcommand{\\otimesshort}{{\\!\\!\\otimes\\!\\!}}\n\\newcommand{\\slsh}[1]{\\!\\!\\not\\! #1}\n%---------------------------------------------------------------------\n\\begin{document}\n\\draft\n\\preprint{}\n\\title{\nLocalization-delocalization transition of\ndisordered $d$-wave superconductors in class $C$I\n} \n\\author{ \nTakahiro Fukui\\cite{Email}\n} \n\\address{\nDepartment of Mathematical Sciences,\nIbaraki University, Mito 310-8512, Japan\n}\n\\date{January 24, 2000}\n\\maketitle\n%---------------------------------------------------------------------\n% Abstract\n%---------------------------------------------------------------------\n\\begin{abstract}\nA lattice model for disordered $d$-wave superconductors in class\n$C$I is reconsidered. Near the band-center, the lattice model can be \ndescribed by Dirac fermions with several species, \neach of which yields WZW term for an effective action of \nthe Goldstone mode. The WZW terms cancel out each other \nbecause of the four-fold symmetry of the model, \nwhich suggests that the quasiparticle states are localized.\nIf the lattice model has, however, \nsymmetry breaking terms which generate\nmass for any species of the Dirac fermions,\nremaining WZW term which avoids the cancellation \ncan derive the system to a delocalized strong-coupling fixed point.\n\\end{abstract}\n\n%\\pacs{PACS: 72.15.Rn, 74.20.-z, 05.10.Cc}\n\n\\begin{multicols}{2}%----------------------------------------------!!!\n%---------------------------------------------------------------------\n% Introduction\n%---------------------------------------------------------------------\n%\\section{Introduction}\\label{s:Int}\n\n\nDirty superconductors have attracted much interest,\nsince they provide wider universality classes of \ndisordered systems \\cite{AltZir,Zir,BCSZs}.\nIn particular, it is quite interesting to ask what the \nuniversality class of the disordered $d$-wave superconductors\nis \\cite{NTW,ZHH,SFBN,BCSZd}.\nA remarkable property of this unconventional superconductors\nis that near the band center, \nquasiparticle states can be described by Dirac fermions \\cite{LFSG}.\nSuch a description enables us, for example, to\nrelate $d$-wave superconductors with the quantum Hall effect\nand to predict a new spin phase called spin quantum Hall fluid \n\\cite{SQH}.\nThey should also produce the well-known effect of\nchiral anomaly \\cite{PolWie,Wit} to $d$-wave superconductors.\nRecently, Senthil {\\it et al} \\cite{SFBN} have \nstudied disordered $d$-wave superconductors with spin rotational \nsymmetry and reached the conclusion that all states are localized.\nOne knows, however, that the WZW term \ndue to chiral anomaly plays a crucial role\nin two dimensional critical phenomena \\cite{Wit}.\nTherefore, it is quite important to take the WZW term \nmissing in \\cite{SFBN} into account,\nor to answer the question why it vanishes if \nit does not exist. \n\nIn this paper, we reconsider disorder effects on \nthe quasiparticle properties of the $d$-wave superconductors\nin the class $C$I \n(those with spin rotational and time-reversal symmetries)\nusing a replica technique.\nIt is shown that each Dirac fermion associated with \nfour nodes creates the WZW term.\nIt turns out that they cancel each other and the resultant \nnonlinear sigma model suggests localization \nof the quasiparticles in dirty $d$-wave superconductors,\nas was shown by Senthil {\\it et al} \\cite{SFBN}.\nIt should be stressed, however, that \nthe WZW term is potentially realizable and\nthe cancellation is accidental:\nIt is due to the four-fold symmetry of the model.\nTherefore, if such symmetry is broken,\nthe system flows to the strong-coupling fixed point described by\nthe WZW model. \n\nLet us begin with a lattice Hamiltonian for singlet\nsuperconductors \\cite{NTW,SFBN,BCSZd},\n%---------------------------------------------------------------------\n% lattice Hamiltonian\n%---------------------------------------------------------------------\n\\begin{equation}\nH=\\sum_{i,j}\n\\left(\n t_{ij}\\sum_{\\sigma}c_{i\\sigma}^\\dagger c_{j\\sigma}\n +\\Delta_{ij} c_{i\\uparrow}^\\dagger c_{j\\downarrow}^\\dagger\n +\\Delta_{ij}^*c_{j\\downarrow} c_{i\\uparrow}\n\\right).\n\\label{LatHam}%-------------------------------------------------------\n\\end{equation}\nWe can choose real and symmetric matrices\n$t_{ij}=t_{ji}$ and $\\Delta_{ij}=\\Delta_{ji}$\ntaking account of the hermiticity as well as \nthe spin-rotational and the time-reversal symmetries.\nIn the absence of randomness,\nwe choose the following parameters for \na pure Hamiltonian $H_0$ with $d$-wave symmetry\n%---------------------------------------------------------------------\n% pure d-wave lattice Hamiltonian H_0\n%---------------------------------------------------------------------\n\\begin{equation}\nt_{j,j\\pm\\hat x}=t_{j,j\\pm\\hat y}=-t_0, \\quad\n\\Delta_{j,j\\pm\\hat x}=-\\Delta_{j,j\\pm\\hat y}=\\Delta_0,\n\\end{equation} \nwhere $\\hat x=(1,0)$ and $\\hat y=(0,1)$.\nMoreover, consider introducing small terms $H_1$ \nto break the $C$I symmetry, \n%---------------------------------------------------------------------\n% breaking term H_1\n%---------------------------------------------------------------------\n\\begin{equation}\n\\Delta_{j,j+\\hat x\\pm\\hat y}=\\Delta_{j+\\hat x\\pm\\hat y,j}=\n\\pm \\frac{i\\Delta_1}{4}, \\quad\n\\Delta_{j,j}=i\\Delta_1.\n\\label{LatHam1}%-----------------------------------------------------\n\\end{equation}\nIt will be shown momentarily that this parameter controls the\nlocalization-delocalization transition of the present model.\n\nThe pure Hamiltonian $H_0$\nhas four nodes, where gapless quasi-particle\nexcitations exist \\cite{NTW}. \nTherefore, we can firstly take the \ncontinuum limit around the nodes of $H_0$ and \nnext incorporate the continuum expression of $H_1$,\nprovided that $H_1$ is small. \nThe lattice operators are then described by the \ncontinuum slowly-varying fields near the band center as\\cite{SFBN}, \n%---------------------------------------------------------------------\n% continuum limit\n%---------------------------------------------------------------------\n\\begin{eqnarray}\nc_{j\\uparrow}/a\\sim&&\n i^{ j_x+j_y}\\psi_{\\uparrow1}^1(x)\n -i^{-j_x-j_y}\\psi_{\\downarrow2}^1(x) \n\\nonumber\\\\&& %-------------------------------------------------!!!\n +i^{-j_x+j_y}\\psi_{\\uparrow1}^2(x)\n -i^{ j_x-j_y}\\psi_{\\downarrow2}^2(x) ,\n\\nonumber\\\\\nc_{j\\downarrow}/a\\sim&&\n i^{ j_x+j_y}\\psi_{\\downarrow1}^{1\\dagger}(x)\n +i^{-j_x-j_y}\\psi_{\\uparrow2}^{1\\dagger}(x) \n\\nonumber\\\\&& %-------------------------------------------------!!!\n +i^{-j_x+j_y}\\psi_{\\downarrow1}^{2\\dagger}(x)\n +i^{ j_x-j_y}\\psi_{\\uparrow2}^{2\\dagger}(x) ,\n\\end{eqnarray}\nwhere $a$ is a lattice constant, $x=aj$, and\ntwo kinds of lower indices of the field $\\psi$ are \nreferred to as spin, left-right (LR) movers, respectively, \nand upper index as node.\nNamely, the field variable $\\psi(x)$ lives in the space\n$V=\\bmC^2\\otimes \\bmC^2\\otimes \\bmC^2$.\nThe pure Hamiltonian in the continuum limit \\cite{NTW,SFBN} is then \n%---------------------------------------------------------------------\n% pure Hamiltonian\n%---------------------------------------------------------------------\n$\nH_0=\n\\int d^2x\\psi^\\dagger(\\calH_0+\\calH_1)\\psi\n$\nwith\n\\begin{equation}\n\\calH_0=\n\\left(\n \\begin{array}{ll}\n -\\gamma_\\mu i\\partial_\\mu & \\\\\n & (x\\leftrightarrow y) \n \\end{array}\n\\right),\n\\label{ConPurHam}%----------------------------------------------------\n\\end{equation}\nwhere the coordinates have been transformed as \n$x,y\\rightarrow\\frac{\\pm x+ y}{\\sqrt{2}}$.\nThe explicit matrix in Eq. (\\ref{ConPurHam}) denotes \nthe node space, and \nmatrices $\\gamma_\\mu$ belong to the other space of $V$, \ncalculated initially as \n$\\gamma_1= v_F 1_2\\otimes\\sigma_3$ and \n$\\gamma_2=v_\\Delta 1_2\\otimes\\sigma_1$,\nwhere $v_F=2\\sqrt{2}t_0a$ and $v_\\Delta=2\\sqrt{2}\\Delta_0a$.\nIt may be more convenient to choose\n%---------------------------------------------------------------------\n% \\gamma matrices after a rotation\n%---------------------------------------------------------------------\n\\begin{equation}\n\\gamma_1= v_F 1_2\\otimes\\sigma_2,\\quad\n\\gamma_2=-v_\\Delta1_2\\otimes\\sigma_1,\n\\end{equation}\nvia suitable rotation in LR-space of $V$. In this basis,\n$\\calH_1$ is given by\n%---------------------------------------------------------------------\n% Dirac Mass term \n%---------------------------------------------------------------------\n\\begin{equation}\n\\calH_1=\n\\left(\n \\begin{array}{ll}\n 0& \\\\\n & m1_2\\otimes\\sigma_3\n \\end{array}\n\\right),\n\\label{MasTer}%-------------------------------------------------------\n\\end{equation}\nwhere $m=\\Delta_1$.\nNamely, $H_1$ yields asymmetric mass term in the continuum \nHamiltonian. For the time being we neglect it, but it will be \nshown that vanishing $m$ leads to \nlocalization whereas finite $m$ drives the system to \ndelocalization. \n\n\nThe spin-rotational and time-reversal\nsymmetries of the lattice model translate, respectively, into the \ncontinuum model as\n%---------------------------------------------------------------------\n% symmetry of the Hamiltonian\n%---------------------------------------------------------------------\n\\begin{equation}\n \\begin{array}{ll}\n \\calH=-\\calC\\calH^\\rmt\\calC^{-1} , \\qquad &\n \\calC=i\\sigma_2\\otimes\\sigma_1\\otimes1_2, \\\\\n \\calH=-\\calT\\calH\\calT^{-1} , &\n \\calT=1_2\\otimes\\sigma_3\\otimes 1_2,\n\\end{array}\n\\label{Sym}%----------------------------------------------------------\n\\end{equation}\nwhere t means the transpose.\nThe total Hamiltonian density\nis given by $\\calH=\\calH_0+\\calH_\\rmd$,\nwhere $\\calH_\\rmd$ is disorder \npotential satisfying Eq. (\\ref{Sym}).\nIt is stressed that we take account of \nall kinds of disorder potentials satisfying Eq. (\\ref{Sym}). \nThis implies that \nthe summation with respect to $i,j$\nin Eq. (\\ref{LatHam}) should be \nover on-site, nearest-neighbor, and diagonal-second-neighbor pairs.\nNamely, we can achieve ``maximum information entropy'' \nfor the Dirac Hamiltonian with the symmetries (\\ref{Sym})\nwhen we introduce \nnot only on-site disorder potentials but also \ndisordered hopping and off-diagonal pairing for the lattice model\nin Eq. (\\ref{LatHam}).\nActually Ludwig {\\it et al} have derived generic Dirac Hamiltonian\nfor the integer quantum Hall transition considering lattice model \nin a similar situation \\cite{LFSG}. \nIt may be straightforward to explicitly calculate disorder \npotentials but tedious to average over them, because\nthere are no less than twenty independent potentials satisfying\nEq. (\\ref{Sym}).\nThe key point for the ensemble average\nis that if the model has ``maximum entropy'', \nwe can use the technique developed by Zirnbauer \\cite{Zir}.\n\nTo study one-quasiparticle properties of the model,\nwe introduce the Green function\n$G_{aa'}(x,x';i\\epsilon)=\\bra{x,a}(i\\epsilon-\\calH)^{-1}\\ket{x',a'}$,\nwhere index $a$ denotes the set of spin, LR, and node \nspecies in the space $V$\nand $\\ket{x,a}=\\psi_a^\\dagger(x)\\ket{0}$. \nEspecially, we need\n%---------------------------------------------------------------------\n% Green functions\n%---------------------------------------------------------------------\n\\begin{eqnarray}\n&&\nG(x)=\\sum_a G_{aa}(x,x;i\\epsilon),\n\\nonumber\\\\\n&&\nK(x,x')=\\sum_{a,a'}G_{aa'}(x,x';i\\epsilon)G_{a'a}(x',x;-i\\epsilon)\n\\label{GreFun}%-------------------------------------------------------\n\\end{eqnarray}\nto compute the DOS and the conductance of the quasiparticle \ntransport \\cite{NLSM}. \nSome notations are convenient for this purpose.\nThe introduction of replica for the field $\\psi$ \nenables us to express the generating functional \nof these Green functions by path integrals: \n$\\psi_{a}\\rightarrow\\psi_{a\\alpha}$ and \n$\\psi_{a}^*\\rightarrow\\psi_{\\alpha a}^*$,\nwhere $a$ and $\\alpha$ are indices denoting $V$ and \nthe replica space $W_\\rmR=\\bmC^n$, respectively.\nThe fields $\\psi$ and $\\psi^*$ have been converted \ninto matrix fields, which makes it\nsimpler to define an order parameter field.\nIt should be stressed that the fields $\\psi$ and $\\psi^*$\nare completely independent variables.\nLagrangian density is then described symbolically as\n$\\calL=-\\tr_{W_\\rmR}\\psi^\\dagger\\left(i\\epsilon-\\calH\\right)\\psi$,\nwhere $\\tr_{W_\\rmR}$ is the trace in the replica space\nand the summation over the indices $a$ of the $V$ space is \nimplied according to the rule of the matrix product.\nIt should be noted again that $\\psi^\\dagger$ is independent of\n$\\psi$.\nMoreover, we introduce an auxiliary space \\cite{Zir}\nto reflect the symmetries in the $V$ space (\\ref{Sym}) \nto an auxiliary field introduced later \n[See Eq. (\\ref{SymQ})],\n$W_\\rmR\\rightarrow W=W_\\rmR\\otimes W_\\rmA$ with\n$W_\\rmA=\\bmC^2\\otimes\\bmC^2$,\nwhich are associated with the \nspin-rotational and the time-reversal symmetries, respectively.\nFermi fields are now denoted by\n$\\tPsi_{\\alpha i}$ and $\\Psi_{i\\alpha}$.\nOne of simpler choices is\n%---------------------------------------------------------------------\n% a choice for \\Psi\n%---------------------------------------------------------------------\n\\begin{eqnarray}\n&&\n\\Psi=(\\psi_+,\\psi_-), \\quad\n\\psi_\\pm=\\calT_\\pm\\tpsi, \n\\nonumber\\\\\n&&\n\\tPsi=\n\\left(\\begin{array}{c}\\bpsi_+\\\\\\bpsi_-\\end{array}\\right),\\quad\n\\bpsi_\\pm=\\tpsi^\\dagger\\calT_\\mp, \n\\label{DefPsi}%-------------------------------------------------------\n\\end{eqnarray}\nwhere \n$\\calT_\\pm=(1\\pm\\calT)/2$ \nserves as a projection operator \n( $\\calT_++\\calT_-=1_2\\otimes1_2\\otimes1_2$, \n$\\calT_\\pm^2=\\calT_\\pm$,\nand $\\calT_+\\calT_-=0$ ) \ninto each chiral component of Sp($n$)$\\times$Sp($n$) symmetry, \nas we shall see later, and\n$\\tpsi=\\frac{1}{2}\\left(\\psi_1,-i\\psi_2\\right)$\nwith \n$\\psi_{1,2}=\\psi\\pm i\\calC^{-1}\\psi^*$.\nThe newly introduced fields $\\Psi$ and $\\tPsi$\nare subject to \\cite{Zir},\n%---------------------------------------------------------------------\n% condition for \\Psi\n%---------------------------------------------------------------------\n\\begin{equation}\n\\begin{array}{ll}\n \\tPsi= \\gamma \\Psi^\\rmt \\calC^{-1}, \\qquad\n&\\Psi = \\calC \\tPsi^\\rmt \\gamma^{-1},\\\\\n \\tPsi= -\\pi \\tPsi \\calT^{-1}, \n&\\Psi = \\calT \\Psi \\pi^{-1} .\n\\end{array}\n\\label{AuxCon}%-------------------------------------------------------\n\\end{equation}\nMatrices $\\gamma$ and $\\pi$ are defined in the $W$ space by\n%---------------------------------------------------------------------\n% \\gamma and \\pi matrices\n%---------------------------------------------------------------------\n\\begin{eqnarray}\n&&\\gamma=1_n\\otimes i\\sigma_2\\otimes 1_2\\equiv\n\\gamma_0\\otimes 1_2 ,\n\\nonumber\\\\\n&&\\pi =1_n\\otimes 1_2 \\otimes \\sigma_3.\n\\end{eqnarray}\nThe identity\n$\\tr_W(i\\epsilon\\omega\\tPsi\\Psi-\\tPsi\\calH\\Psi)\n =\\tr_{W_\\rmR}\\psi^\\dagger(i\\epsilon-\\calH)\\psi$,\nwhere \n$\\omega=1_n\\otimes\\sigma_2\\otimes\\sigma_1$,\nleads to the generating functional\n$\\calZ=\\int\\!\\calD\\Psi\\calD\\tPsi e^{-S}$ with\n%--------------------------------------------------------------------\n% action by Psi field\n%--------------------------------------------------------------------\n\\begin{equation}\nS=-\\int\\!\\!d^2x\\tr_W\n \\left(\n i\\epsilon\\omega\\tPsi\\Psi-\\tPsi\\calH\\Psi+J\\tPsi\\Psi\n \\right).\n\\label{OrgAct}%------------------------------------------------------\n\\end{equation}\n\n\nAssume that disorder potential $\\calH_\\rmd$ obeys the Gaussian \ndistribution\n$P[\\calH_\\rmd]=\\exp\\left(-\\frac{1}{2g}\\tr_V\\calH_\\rmd^2\\right)$.\nThen ensemble average over\ndisorder is quite simple.\nThe procedure is as follows\\cite{Zir}:\nThe disorder potentials are integrated out by using the identity\n$-\\frac{1}{2g}\\tr_V\\calH_\\rmd^2+\\tr_V\\calH_\\rmd\\Psi\\tPsi\n=-\\frac{1}{2g}(\\tr_V\\calH_\\rmd-g\\Psi\\tPsi)^2\n+\\frac{g}{2}\\tr_V(\\Psi\\tPsi)^2$.\nIt turns out that the integration over $\\calH_\\rmd$ is automatic\nbecause $\\Psi\\tPsi$ satisfy the same symmetries \nas those of $\\calH_\\rmd$ due to Eq. (\\ref{AuxCon}). \nThis is actually a merit to consider the disorder potentials\nwith maximum entropy. If disordered hopping and off-diagonal pairing \nof the lattice model are neglected \nand on-site disorder potentials are merely taken into account, \nsome other conditions should be imposed on $\\calH_\\rmd$ and\ntherefore on the fields $\\Psi$ and $\\tPsi$.\nNow we have interaction terms of fermions due to ensemble average.\nNote the identity \n$\\tr_V(\\Psi\\tPsi)^2=-\\tr_W(\\tPsi\\Psi)^2$.\nThen, the four fermi interactions are decoupled via\nauxiliary matrix (order parameter)\nfield defined in the $W$ space.\nTo be concrete, add the following term into the action,\n$\\frac{1}{2g}\\tr_W (Q+g\\tPsi\\Psi-\\omega)^2$, which is actually\na constant after integration over $Q$.\nThen we reach an effective Lagrangian density, \n%--------------------------------------------------------------------\n% Lagrangian for \\Psi and Q\n%--------------------------------------------------------------------\n\\begin{eqnarray}\n\\calL=&&-\\tr_W\n\\left[\\frac{1}{2g}\\left(Q^2-2i\\epsilon\\omega Q\\right)\n+Q\\tPsi\\Psi-\\tPsi\\calH_0\\Psi\\right] .\n\\label{EffLag}%----------------------------------------------------\n\\end{eqnarray}\nHere we have set $J=0$ for simplicity.\nNotice that the anti-Hermitian\nauxiliary field $Q=-Q^\\dagger$ is subject to\n%--------------------------------------------------------------------\n% symmetry for Q\n%--------------------------------------------------------------------\n\\begin{equation}\nQ=-\\gamma Q^\\rmt\\gamma^{-1},\\quad\nQ=-\\pi Q \\pi^{-1} . \n\\label{SymQ}%--------------------------------------------------------\n\\end{equation}\nThe solution of these equations is\n%--------------------------------------------------------------------\n% Explicit form of Q: Introduction of matrix M\n%--------------------------------------------------------------------\n\\begin{equation}\nQ=\n\\left(\n \\begin{array}{ll}\n & -M^\\dagger\\\\ M&\n \\end{array}\n\\right)\n\\label{ExpQ}%--------------------------------------------------------\n\\end{equation}\nwith a condition \n$M= \\gamma_0 M^* \\gamma_0^{-1}$, where the explicit matrix \nin the above equation denotes the time-reversal space of $W$.\n\nThe Lagrangian (\\ref{EffLag}) has \nG=Sp($n$)$\\times$Sp($n$) symmetry.\nTo see this, let us consider the transformation \n$Q\\rightarrow gQg^{-1}$\nwhich keeps the symmetry relations (\\ref{SymQ}).\nIt turns out that $g$ should satisfy \n$\\gamma=g\\gamma g^\\rmt$ and $\\tau g\\tau^{-1}=g$\nas well as $gg^\\dagger=1$, and therefore\n$g$ is explicitly given by\n%-------------------------------------------------------------------\n% Explicit g: Sp * Sp symmetry \n%-------------------------------------------------------------------\n\\begin{equation}\ng=\n\\left(\n \\begin{array}{ll}\n g_+ & \\\\ & g_-\n \\end{array}\n\\right),\n\\quad\n\\label{Expg}%-------------------------------------------------------\n\\end{equation}\nwhere $g_\\pm\\in$ Sp($n$) is a $2n\\times2n$ matrix in the replica and \nthe spin space of $W$.\nSo far we have derived the Lagrangian as well as its symmetry group.\nTo integrate out the fermi fields, it may be convenient to \nuse the notations where the fermi fields $\\Psi$ and $\\tPsi$\nare column and row vector, respectively, as usual.\nUsing Eq. (\\ref{SymQ}), fermion part of the Lagrangian (\\ref{EffLag})\nis rewritten as\n$\\calL_\\rmF\n=\\tPsi(\\calH_0\\otimes1-1\\otimes Q^\\rmt)\\Psi\n=\\tPsi 1 \\otimes \\gamma\n (\\calH_0\\otimes1+1\\otimes Q)\n 1 \\otimes \\gamma^\\dagger\\Psi$.\n%In what follows, identity matrices in the space $V$ and $W$ \n%are suppressed for simplicity.\nTransform the fields as \n$\\tPsi \\rightarrow \\tPsi 1\\otimes\\gamma^\\dagger$\nand \n$\\Psi\\rightarrow 1\\otimes\\gamma\\Psi$. \nThen the Lagrangian is given by,\nin terms of the fields $M$ and $\\psi_\\pm$,\n%-------------------------------------------------------------------\n% Lagrangian for M and \\psi\n%-------------------------------------------------------------------\n\\begin{eqnarray}\n\\calL&&=\\frac{1}{g}\\tr_{W_\\rmRS}\n\\left[\nM^\\dagger M-\\epsilon\\gamma_0(M-M^\\dagger)\n\\right]\n+\\calL_{\\rmF1}+\\calL_{\\rmF2},\n%\\label{Expg}%-------------------------------------------------------\n\\end{eqnarray}\nwhere $W_\\rmRS$ is the replica and the spin space of $W$ and\n$\\calL_{\\rmF j}$ describes the Lagrangian of the $j$th-node \nfermion defined by\n%-------------------------------------------------------------------\n% Fermion Lagrangian for node 1\n%-------------------------------------------------------------------\n\\begin{equation}\n\\calL_{\\rmF1}=\n \\bpsi_+^1 i \\slsh{\\partial}\\psi_+^1\n+\\bpsi_-^1 i \\slsh{\\partial}\\psi_-^1\n-\\bpsi_+^1 M^\\dagger\\psi_-^1 + \\bpsi_-^1 M\\psi_+^1 ,\n\\label{LagF1}%------------------------------------------------------\n\\end{equation}\nand \n$\\calL_{\\rmF2}=\\calL_{\\rmF1}(1\\rightarrow2, x\\leftrightarrow y)$.\nHere and hereafter, the identity matrices such as those in \n$\\slsh{\\partial}\\otimes1$ and $1\\otimes M$ are suppressed.\nThe transformation laws of $M$ and $\\psi_\\pm$ fields are\n%-------------------------------------------------------------------\n% Transformation laws\n%-------------------------------------------------------------------\n\\begin{eqnarray}\n&&\nM\\rightarrow g_-Mg_+^\\dagger, \\quad\n%\\nonumber\\\\&&\n\\bpsi_\\pm \\rightarrow \\bpsi_\\pm g_\\pm^\\dagger ,\\quad\n \\psi_\\pm\\rightarrow g_\\pm \\psi_\\pm .\n\\label{TFL}%----------------------------------------------------------\n\\end{eqnarray}\n\nWe have used a bit complicated basis for $W$ in the definition of\n$\\Psi$ and $\\tPsi$ in Eq. (\\ref{DefPsi}),\nsince $Q$ and $g$ become simpler in this basis.\nOn the other hand,\nit may cause a difficulty in computing the saddle points.\nWe have known, however, from the $\\epsilon$-term in Eqs. \n(\\ref{OrgAct}) and (\\ref{EffLag}) \nthat $\\omega$ serves as a ``metric''\nin the extended auxiliary space. \nUsually we may choose a basis of the space $W$ with a diagonal metric,\nowing to \nwhich we can assume that $Q$ is also diagonal on the \nsaddle points.\nIn the present case, therefore, it is natural to\nassume that $Q$ should have the same structure\nas $\\omega$, and hence $M_0=v\\gamma_0$ with real \ndiagonal matrix $v=\\mbox{diag}(v_1,\\cdots,v_n)$.\nThen the variation with respect to $v$ \nafter the integration over fermi fields\ntells that the saddle points are given by \n$v_\\alpha=v_0\\sim \\Lambda\\exp(-\\pi v_Fv_\\Delta/g)$,\nwhere $\\Lambda$ is a ultraviolet cut-off.\nThis solution gives rise to an exponentially small density of state\nat the band-center.\nNow it is easy to identify the saddle point manifold as H=Sp($n$):\nThe chiral transformation $g\\equiv(g_+,g_-)$ in \nEqs. (\\ref{Expg}) and (\\ref{TFL}) \nis divided into two types. One is \n$g_{\\rm v}=(g_1,g_1^*)$ under which $M_0$ is still invariant, \nand the other is \n$g_{\\rm a}=(g_2,g_2^\\rmt)$ under which $M_0$ is no longer invariant.\n\n\nNonlinear sigma model is derived as small fluctuation\naround the saddle point manifold H by considering $g_{\\rm a}$ type \nlocal Sp($n$) rotation.\nLet us parameterize \n$M=\\xi^*H\\xi=\\tH U$\nwith $\\xi\\in$ Sp($n$),\n$\\tH=\\xi^*H\\xi^\\dagger$, and $U=\\xi^2$.\nThe field $\\xi$ describes the massless fluctuation around the \nsaddle point manifold H, whereas $H$ describes massive \nlongitudinal modes.\nIn what follows, we take only the leading order for the latter mode,\nsetting $H=M_0 (=\\tH)$.\nIt should be noted that nonlinear sigma model on G/H has\na global G symmetry as well as a hidden local H symmetry.\nThough the field $\\xi$ itself is not invariant \nunder local H transformation, the composite field $U$ is invariant.\n\n\nTo derive an effective action for the transverse mode, \nlet us come back to the Lagrangian (\\ref{LagF1}), \nsince we should be careful in the integration over fermi fields.\nIt is practical to firstly integrate out the fermi field of node-1. \nThen, the contribution from node-2 can be obtained by \nreplacing $x\\leftrightarrow y$. \nTo carry out the former integration,\nmake the transformation\n$\\bchi_+= \\bpsi_+U^\\dagger$ and \n$\\chi_+ = U\\psi_+$, whereas\n$\\bchi_-=\\bpsi_-1_n\\otimes\\sigma_2$ and \n$\\chi_- =1_n\\otimes\\sigma_2\\psi_-$.\nThe Lagrangian (\\ref{LagF1}) is then converted into\n%-------------------------------------------------------------------\n% Lagrangian after chiral transformation\n%-------------------------------------------------------------------\n\\begin{eqnarray}\n\\calL_{\\rmF1}=\n \\bchi_+^1 i \\slsh{D} \\chi_+^1\n+\\bchi_-^1 i \\slsh{\\partial}\\chi_-^1\n+iv_0\n\\left(\n\\bchi_+^1\\chi_-^1 + \\bchi_-^1\\chi_+^1 \n\\right),\n\\end{eqnarray}\nwhere $D_\\mu$ is defined by \n$D_\\mu=\\partial_\\mu+L_\\mu$ \nwith \n$L_\\mu=U\\partial_\\mu U^\\dagger$.\nIt is convenient to scale \n$x=v_Fx'$ and \n$y=v_\\Delta y'$ working in the node-1 sector.\nIntegration over the fermi field of node-1 yields\n%-------------------------------------------------------------------\n% Integration over Fermi fields\n%-------------------------------------------------------------------\n\\begin{eqnarray}\n&&\nZ_{\\rmF1}\n=e^{-\\frac{1}{2}\\Gamma_1(U)}\n\\nonumber\\\\&&\\times %-------------------------------------------!!!\n\\Det^{\\frac{1}{2}}_{V_1\\otimes W_\\rmRS}\n1_2\\otimes\n\\left(\n\\begin{array}{cc}\niv_0\n&\ni(-i\\partial_1-\\partial_2)\n\\\\\ni( iD_1-D_2)\n&\niv_0\n\\end{array}\n\\right),\n\\end{eqnarray}\nwhere the identity matrix $1_2$ belongs to the spin space of $V$,\n$V_1$ means the node-1 sector of $V$, \nthe derivatives are with respect to the scaled coordinates\n$x'$ and $y'$, and\n$\\Gamma_1(U)$ is the Jacobian for the chiral transformation,\nwhich can be calculated \nby using the Fujikawa method \\cite{Fuj}.\nWe simply present the final answer \n$Z_{\\rmF1}\\sim e^{-S_1}$, where\nthe effective action $S_1$ associated with the node-1 fermion\nis composed of the principal chiral action of $U$ with a coupling\nconstant $\\lambda=4\\pi$ and of the WZW term \n%---------------------------------------------------------------------\n% WZW term\n%---------------------------------------------------------------------\n\\begin{eqnarray}\n\\Gamma_{\\rmWZW}=\n\\frac{i}{12\\pi}\n\\int d^3x\\epsilon_{\\mu\\nu\\tau}\\tr_{W_\\rmRS}\n\\partial_\\mu UU^\\dagger\n\\partial_\\nu UU^\\dagger\n\\partial_\\tau UU^\\dagger .\n\\end{eqnarray}\nTherefore, \nintegration over fermion of node-1 actually yields the WZW\nterm.\n\nNext let us compute the contribution from the node-2.\nThe procedure is the rescaling $x'=x/v_F$ and $y'=y/v_\\Delta$\nand the exchange $x\\leftrightarrow y$. \nIt should be noted that the WZW term is invariant under \nthe rescaling but is odd under the exchange, \nand hence it cancels out.\nThe total effective action ends up with\n%---------------------------------------------------------------------\n% final action\n%---------------------------------------------------------------------\n\\begin{equation}\nS=\\int d^2x \\tr_{W_\\rmRS}\n\\left[\n\\frac{1}{2\\lambda}\\partial_\\mu U\\partial_\\mu U^\\dagger\n-\\epsilon\n\\left(\nU+U^\\dagger\n\\right)\n\\right]\n+k\\Gamma_{\\rmWZW}\n\\label{EffAct}%-------------------------------------------------------\n\\end{equation}\nwith \n$\\frac{1}{\\lambda}=\\frac{1}{4\\pi}\\frac{v_F^2+v_\\Delta^2}{v_Fv_\\Delta}$\nand $k=0$.\nThis is just the action derived by Senthil {\\it et al}.\nAlthough the WZW term disappears in the ordinary $d$-wave \nsuperconductors, we are tempted to expose it,\nsince the WZW term exists potentially.\nThis is indeed possible:\nIf the lattice model includes the symmetry breaking term\n(\\ref{LatHam1}) and \ntherefore the Dirac fermion for the node-2 has a mass\n(\\ref{MasTer}), we can neglect the node-2 fermion \nin the lower energy than the mass gap. \nIn this case, we have the same action (\\ref{EffAct}) but with \n$\\frac{1}{\\lambda}=\\frac{1}{4\\pi}$ and $k=1$ in the scaled \ncoordinates $x'$ and $y'$.\n\nThe renormalization group equations of the action (\\ref{EffAct})\nare calculated at the one-loop order as\n%---------------------------------------------------------------------\n% RG equations\n%---------------------------------------------------------------------\n\\begin{eqnarray}\n&&\n\\frac{d \\lambda}{d\\ln L}=\n-\\varepsilon\\lambda+\\frac{\\lambda^2}{4\\pi}\n\\left[\n1-\n\\left(\n\\frac{k\\lambda}{4\\pi}\n\\right)^2\n\\right],\n\\nonumber\\\\\n&&\n\\frac{d \\epsilon}{d\\ln L}=\n\\left(\nd-\\frac{\\lambda}{8\\pi}\n\\right)\n\\epsilon,\n\\label{RGE}%----------------------------------------------------------\n\\end{eqnarray}\nwhere $d=2$, $\\varepsilon=d-2$, and\nthe replica limit $n\\rightarrow0$ has been taken.\nIn the case where $m=0$, we reach the same conclusion as \nSenthil {\\it et al}.\nNamely, one-quasiparticle states are localized, \nsince the spin conductance against weak magnetic fields\nis related with $\\lambda$ as $\\sigma=2/(\\pi\\lambda)$, which \nis calculated \\cite{SFBN}\nvia diffusion constant in the diffusion propagator (\\ref{GreFun}).\nIt should be stressed that \nthe cancellation of \nthe WZW term is due to the four-fold symmetry of the \n$d$-wave Hamiltonian. \nActually, the latent WZW term can emerge \nvia symmetry breaking mass term for the Dirac fermions,\nand in that case the coupling constant\n$\\lambda$ flows to the strong-coupling fixed point value\n$\\lambda_{\\rm c}=4\\pi/k$. \nThis fixed-point is conformal invariant described by\nSp($n$) WZW model. From the scaling dimension of the energy\nin Eq. (\\ref{RGE}), it turns out that the density of state \nnear this fixed-point obeys the scaling law \n%---------------------------------------------------------------------\n% DOS\n%---------------------------------------------------------------------\n\\begin{equation}\n\\rho(E)=E^{\\frac{1}{4k-1}}.\n\\end{equation}\nTherefore, if the pure model has the breaking term \n(\\ref{LatHam1}), we suggest that $\\rho(E)=E^{\\frac{1}{3}}$.\n\nIn this paper, we have taken the breaking term of type \n(\\ref{LatHam1}) into account. More detailed phase diagram \nwill be published elsewhere.\nNumerical check of the present conjecture does not seem difficult.\nIt is quite interesting to expose the hidden WZW term\nwhich exists potentially but conceals itself in the \nfour-fold symmetry of the $d$-wave superconductors.\n\nThe author would like to thank Y. Hatsugai, Y. Morita,\nC. Mudry, and Y. Kato for helpful discussions and comments.\nHe is deeply indebted to M. R. Zirnbauer and A. Altland \nfor valuable discussions in the early stage of this work. \n\n%--------------------------------------------------------------------\n% references\n%--------------------------------------------------------------------\n\\begin{references}\n\\bibitem[*]{Email} Email: fukui@mito.ipc.ibaraki.ac.jp\n%\n\\bibitem{AltZir}\nA. Altland and M. R. Zirnbauer,\nPhys. Rev. {\\bf B55}, 1142 (1997).\n%\n\\bibitem{Zir}\nM. R. Zirnbauer,\nJ. Math. Phys. {\\bf 37}, 4986 (1996).\n%\n\\bibitem{BCSZs}\nR. Bundschuh, C. Cassanello, D. Serban, and M. R. Zirnbauer,\ncond-mat/9806172.\n%\n\\bibitem{NTW}\nA. A. Nersesyan, A. M. Tsvelik, and F. Wenger,\nPhys. Rev. Lett. {\\bf 72}, 2628 (1994).\n%\n\\bibitem{ZHH}\nK. Ziegler, M. H. Hettler, and P. J. Hirschfeld,\nPhys. Rev. Lett. {\\bf 77}, 3013 (1996).\n%\n\\bibitem{SFBN}\nT. Senthil, M. P. A. Fisher, L. Balents, and C. Nayak,\nPhys. Rev. Lett. {\\bf 81}, 4704 (1998):\nT. Senthil and M. P. A. Fisher,\ncond-mat/9810238:\nSee also, M. P. A. Fisher,\ncond-mat/9806164.\n%\n\\bibitem{BCSZd}\nR. Bundschuh, C. Cassanello, D. Serban, and M. R. Zirnbauer,\ncond-mat/9808297.\n%\n\\bibitem{LFSG}\nA. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein,\nPhys. Rev. {\\bf B50}, 7526 (1994).\n%\n\\bibitem{SQH}\nV. Kagalovsky, B. Horovitz, Y. Avishai, and J. T. Chalker,\ncond-mat/9812155:\nT. Senthil, J. B. Marston, and M. P. A. Fisher,\ncond-mat/9902062:\nI. A. Gruzberg, A. W. W. Ludwig, and N. Read,\ncond-mat/9902063.\n%\n\\bibitem{PolWie}\nA. Polyakov and P. B. Wiegmann,\nPhys. Lett. {\\bf 131B}, 121 (1983).\n%\n\\bibitem{Wit}\nE. Witten,\nCom. Math. Phys. {\\bf 92}, 455 (1984).\n%\n\\bibitem{NLSM}\nF. Wegner, \nZ. Phys. {\\bf B35}, 207 (1979):\nA. J. McKane and M. Stone,\nAnn. Phys. {\\bf 131}, 36 (1981):\nA. M. M. Pruisken and L. Sch\\\"afer,\nNucl. Phys. {\\bf B200}, 20 (1982).\n%\n\\bibitem{Fuj}\nK. Fujikawa,\nPhys. Rev. {\\bf D21}, 2848 (1980); \n{\\it ibid} {\\bf D22}, 1499 (1980) (E).\n\\end{references}\n\n\\end{multicols}%---------------------------------------------------!!!\n\n\\end{document}\n\n\n\n\n\n\n\n"
}
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[
{
"name": "cond-mat0002002.extracted_bib",
"string": "\\bibitem[*]{Email} Email: fukui@mito.ipc.ibaraki.ac.jp\n%\n\n\\bibitem{AltZir}\nA. Altland and M. R. Zirnbauer,\nPhys. Rev. {\\bf B55}, 1142 (1997).\n%\n\n\\bibitem{Zir}\nM. R. Zirnbauer,\nJ. Math. Phys. {\\bf 37}, 4986 (1996).\n%\n\n\\bibitem{BCSZs}\nR. Bundschuh, C. Cassanello, D. Serban, and M. R. Zirnbauer,\ncond-mat/9806172.\n%\n\n\\bibitem{NTW}\nA. A. Nersesyan, A. M. Tsvelik, and F. Wenger,\nPhys. Rev. Lett. {\\bf 72}, 2628 (1994).\n%\n\n\\bibitem{ZHH}\nK. Ziegler, M. H. Hettler, and P. J. Hirschfeld,\nPhys. Rev. Lett. {\\bf 77}, 3013 (1996).\n%\n\n\\bibitem{SFBN}\nT. Senthil, M. P. A. Fisher, L. Balents, and C. Nayak,\nPhys. Rev. Lett. {\\bf 81}, 4704 (1998):\nT. Senthil and M. P. A. Fisher,\ncond-mat/9810238:\nSee also, M. P. A. Fisher,\ncond-mat/9806164.\n%\n\n\\bibitem{BCSZd}\nR. Bundschuh, C. Cassanello, D. Serban, and M. R. Zirnbauer,\ncond-mat/9808297.\n%\n\n\\bibitem{LFSG}\nA. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein,\nPhys. Rev. {\\bf B50}, 7526 (1994).\n%\n\n\\bibitem{SQH}\nV. Kagalovsky, B. Horovitz, Y. Avishai, and J. T. Chalker,\ncond-mat/9812155:\nT. Senthil, J. B. Marston, and M. P. A. Fisher,\ncond-mat/9902062:\nI. A. Gruzberg, A. W. W. Ludwig, and N. Read,\ncond-mat/9902063.\n%\n\n\\bibitem{PolWie}\nA. Polyakov and P. B. Wiegmann,\nPhys. Lett. {\\bf 131B}, 121 (1983).\n%\n\n\\bibitem{Wit}\nE. Witten,\nCom. Math. Phys. {\\bf 92}, 455 (1984).\n%\n\n\\bibitem{NLSM}\nF. Wegner, \nZ. Phys. {\\bf B35}, 207 (1979):\nA. J. McKane and M. Stone,\nAnn. Phys. {\\bf 131}, 36 (1981):\nA. M. M. Pruisken and L. Sch\\\"afer,\nNucl. Phys. {\\bf B200}, 20 (1982).\n%\n\n\\bibitem{Fuj}\nK. Fujikawa,\nPhys. Rev. {\\bf D21}, 2848 (1980); \n{\\it ibid} {\\bf D22}, 1499 (1980) (E).\n"
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cond-mat0002003
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Rashba spin splitting in two-dimensional electron and hole systems
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"author": "R. Winkler"
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In two-dimensional (2D) hole systems the inversion asymmetry induced spin splitting differs remarkably from its familiar counterpart in the conduction band. While the so-called Rashba spin splitting of electron states increases linearly with in-plane wave vector $k_\|$ the spin splitting of heavy hole states can be of third order in $k_\|$ so that spin splitting becomes negligible in the limit of small 2D hole densities. We discuss consequences of this behavior in the context of recent arguments on the origin of the metal-insulator transition observed in 2D systems.
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"name": "rash.tex",
"string": "\\documentstyle[prb,aps,epsf,multicol]{revtex}\n\\newif\\ifmynarrow \\mynarrowfalse\n\\parskip 0pt %plus 1pt\n\\renewcommand{\\narrowtext}{%\n \\ifmynarrow\\hspace*{\\fill}\\raisebox{-1ex}[0pt][0pt]{%\n \\rule{0.3pt}{1ex}%\n \\rule[1ex]{20.5pc}{0.3pt}}\\fi\n \\mynarrowtrue\n \\vspace{-1.0ex}%\n \\begin{multicols}{2}%\n \\par\\global\\columnwidth20.5pc\n \\global\\hsize\\columnwidth\\global\\linewidth\\columnwidth\n \\global\\displaywidth\\columnwidth}\n\n\\renewcommand{\\widetext}{%\n \\end{multicols}%\n \\vspace{-2.5ex}%\n \\noindent\\raisebox{1ex}[0pt][0pt]{%\n \\rule{20.5pc}{0.3pt}%\n \\rule{0.3pt}{1ex}}%\n \\par\\global\\columnwidth42.5pc\n \\global\\hsize\\columnwidth\\global\\linewidth\\columnwidth\n \\global\\displaywidth\\columnwidth}\n\n\\textheight 245mm\n\\begin{document}\n\n\\title{Rashba spin splitting in two-dimensional electron and hole systems}\n\n\\draft\n\n\\author{R. Winkler}\n\\address{Institut f\\\"ur Technische Physik III, Universit\\\"at\nErlangen-N\\\"urnberg, Staudtstr. 7, D-91058 Erlangen, Germany}\n\n\\date{February 1, 2000}\n\\maketitle\n\\begin{abstract}\n In two-dimensional (2D) hole systems the inversion asymmetry\n induced spin splitting differs remarkably from its familiar\n counterpart in the conduction band. While the so-called Rashba\n spin splitting of electron states increases linearly with in-plane\n wave vector $k_\\|$ the spin splitting of heavy hole states can be\n of third order in $k_\\|$ so that spin splitting becomes negligible\n in the limit of small 2D hole densities. We discuss consequences\n of this behavior in the context of recent arguments on the origin\n of the metal-insulator transition observed in 2D systems.\n\\end{abstract}\n\\pacs{73.20.Dx, 71.70.Ej}\n\n\\narrowtext\n\nAt zero magnetic field $B$ spin splitting in quasi two-dimensional\n(2D) semiconductor quantum wells (QW's) can be a consequence of the\nbulk inversion asymmetry (BIA) of the underlying crystal (e.g.\\ a\nzinc blende structure) and of the structure inversion asymmetry\n(SIA) of the confinement potential. This $B=0$ spin splitting\n\\cite{kit63,chiral} is the subject of considerable interest because\nit concerns details of energy band structure that are important in\nboth fundamental research and electronic device applications (Refs.\\ \n\\onlinecite{pfe99,bych84,win93,eke85,win96,pud97,sko98,lom85,ros89,%\nwol89,luo90,nit97,hei98,eis84,las85,dre92,jus92,lu98,pap99} and references\ntherein).\n\nHere we want to focus on the SIA spin splitting which is usually the\ndominant part of $B=0$ spin splitting in 2D systems. \\cite{pfe99} To\nlowest order in $k_\\|$ SIA spin splitting in 2D electron systems is\ngiven by the so-called Rashba model, \\cite{bych84} which predicts a\nspin splitting linear in $k_\\|$. For small in-plane wave vector\n$k_\\|$ this is in good agreement with more accurate numerical\ncomputations. \\cite{win93} For 2D hole systems, on the other hand,\nthe situation is more complicated because of the fourfold degeneracy\nof the topmost valence band $\\Gamma_8^v$, and so far only numerical\ncomputations on hole spin splitting have been performed.\n\\cite{eke85,win96} In the present paper we will develop an\nanalytical model for the SIA spin splitting of 2D hole systems. We\nwill show that in contrast to the familiar Rashba model the spin\nsplitting of heavy hole (HH) states is basically proportional to\n$k_\\|^3$. This result was already implicitly contained in several\nnumerical computations. \\cite{eke85,win96} But a clear analytical\nframework was missing. We will discuss consequences of this behavior\nin the context of recent arguments on the origin of the\nmetal-insulator transition observed in 2D systems.\n\\cite{pud97,sko98}\n\nFirst we want to review the major properties of the Rashba model\n\\cite{bych84}\n%\n\\begin{equation}\n\\label{rashbafull}\nH_{6c}^{\\rm SO} = \\alpha \\, {\\bf k} \\times {\\bf E} \\cdot\n\\bbox{\\sigma} .\n\\end{equation}\n%\nIn this equation $\\bbox{\\sigma} = (\\sigma_x, \\sigma_y, \\sigma_z)$\ndenotes the Pauli spin matrices, $\\alpha$ is a material-specific\nprefactor, \\cite{lom85,ros89} and ${\\bf E}$ is an effective electric\nfield that results from the built-in or external potential $V$ as\nwell as from the position dependent valence band edge. For ${\\bf E}\n= (0,0,E_z)$ Eq.\\ (\\ref{rashbafull}) becomes (using explicit matrix\nnotation)\n%\n\\begin{equation}\n\\label{rashba}\nH_{6c}^{\\rm SO} = \\alpha \\, E_z \\left(\n \\begin{array}{cc} 0 & k_- \\\\ k_+ & 0 \\end{array}\n \\right)\n\\end{equation}\n%\nwith $k_\\pm = k_x \\pm i k_y$. By means of perturbation theory we\nobtain for the spin splitting of the energy dispersion\n%\n\\begin{equation}\n\\label{rashbaperturb}\n{\\cal E}_{6c}^{\\rm SO} ({\\bf k}_\\|) =\n\\pm \\langle \\alpha E_z \\rangle k_\\|\n\\end{equation}\n%\nwhere ${\\bf k}_\\| = (k_x, k_y, 0)$. Using this simple formula\nseveral groups determined the prefactor $\\langle \\alpha E_z \\rangle\n$ by analyzing Shubnikov-de Haas (SdH) oscillations.\n\\cite{wol89,luo90,nit97,hei98}\n\nEquation (\\ref{rashbaperturb}) predicts an SIA spin splitting which\nis linear in $k_\\|$. For small $k_\\|$ Eq.\\ (\\ref{rashbaperturb})\nthus becomes the dominant term in the energy dispersion ${\\cal\nE}_\\pm ({\\bf k_\\|})$, i.e., SIA spin splitting of electron states is\nmost important for small 2D densities. In particular, we get a\ndivergent van Hove singularity of the density-of-states (DOS) at the\nbottom of the subband \\cite{win93} which is characteristic for a $k$\nlinear spin splitting. As an example, we show in Fig.\\ \n\\ref{pic:insb} the self-consistently calculated \\cite{win93} subband\ndispersion ${\\cal E}_\\pm (k_\\|)$, DOS effective mass $m^\\ast/m_0$,\nand spin splitting ${\\cal E}_+ (k_\\|) - {\\cal E}_- (k_\\|)$ for an\nMOS inversion layer on InSb. For small $k_\\|$ the spin splitting\nincreases linearly as a function of $k_\\|$, in agreement with Eq.\\ \n(\\ref{rashbaperturb}). Due to nonparabolicity the spin splitting for\nlarger $k_\\|$ converges toward a constant. \\cite{win93}\n\nThe spin splitting results in unequal populations $N_\\pm$ of the two\nbranches ${\\cal E}_\\pm (k_\\|)$. For a given total density $N_s = N_+\n+ N_-$ and a subband dispersion ${\\cal E}_\\pm (k_\\|) = \\langle \\mu\n\\rangle k_\\|^2 \\pm \\langle \\alpha E_z \\rangle k_\\|$ with $\\mu =\n\\hbar^2/2m^\\ast$ we obtain\n%\n\\begin{equation}\n\\label{sdh_rashba}\nN_\\pm = \\frac{1}{2} N_s \\pm \\frac{\\langle \\alpha E_z \\rangle}{8\\pi\n\\langle \\mu \\rangle^2} \\sqrt{8\\pi \\langle \\mu \\rangle^2 N_s -\n\\langle \\alpha E_z \\rangle^2} .\n\\end{equation}\n%\nThis equation can be directly compared with, e.g., the results of\nSdH experiments. \\cite{wol89,luo90,nit97,hei98}\n\nThe Rashba model (\\ref{rashbafull}) can be derived by purely\ngroup-theoretical means. The electron states in the lowest\nconduction band are $s$ like (orbital angular momentum $l=0$). With\nspin-orbit (SO) interaction we have total angular momentum $j=1/2$.\nBoth ${\\bf k}$ and ${\\bf E}$ are polar vectors and ${\\bf k} \\times\n{\\bf E}$ is an axial vector (transforming according to the\nirreducible representation $\\Gamma_4$ of $T_d$). \\cite{ros89,trr79}\nLikewise, the spin matrices $\\sigma_x$, $\\sigma_y$, and $\\sigma_z$\nform an axial vector $\\bbox{\\sigma}$. The dot product\n(\\ref{rashbafull}) of ${\\bf k} \\times {\\bf E}$ and $\\bbox{\\sigma}$\ntherefore transforms according to the identity representation\n$\\Gamma_1$, in accordance with the theory of invariants of Bir and\nPikus. \\cite{bir74} In the $\\Gamma_6^c$ conduction band the scalar\ntriple product (\\ref{rashbafull}) is the only term of first order in\n${\\bf k}$ and ${\\bf E}$ that is compatible with the symmetry of the\nband.\n\nNow we want to compare the Rashba model (\\ref{rashbafull}) with the\nSIA spin splitting of hole states. The topmost valence band is $p$\nlike ($l=1$). With SO interaction we have $j=3/2$ for the HH/LH\nstates ($\\Gamma_8^v$) and $j=1/2$ for the SO states ($\\Gamma_7^v$).\nFor the $\\Gamma_8^v$ valence band there are two sets of matrices\nwhich transform like an axial vector, namely ${\\bf J} =\n(J_x,J_y,J_z)$ and $\\bbox{\\cal J} = (J_x^3,J_y^3,J_z^3)$ (Refs.\\ \n\\onlinecite{trr79,lut56}). Here $J_x$, $J_y$ and $J_z$ are the\nangular momentum matrices for $j=3/2$. Thus we get \\cite{cross}\n%\n\\begin{equation}\n\\label{lutt_spin}\nH_{8v}^{\\rm SO} = \\beta_1 \\, {\\bf k} \\times {\\bf E} \\cdot {\\bf J} +\n\\beta_2 \\, {\\bf k} \\times {\\bf E} \\cdot \\bbox{\\cal J} .\n\\end{equation}\n%\nSimilar to the Rashba model the first term has axial symmetry with\nthe symmetry axis being the direction of the electric field ${\\bf\nE}$. The second term is anisotropic, i.e., it depends on both the\ncrystallographic orientation of ${\\bf E}$ and ${\\bf k}$. Using ${\\bf\nk} \\cdot {\\bf p}$ theory we find that the prefactor $\\beta_2$ is\nalways much smaller than $\\beta_1$, i.e., the dominant term in Eq.\\ \n(\\ref{lutt_spin}) is the first term. This can be easily understood\nby noting that the ${\\bf k} \\cdot {\\bf p}$ coupling between\n$\\Gamma_8^v$ and $\\Gamma_6^c$ is isotropic, so that it contributes\nto $\\beta_1$ but not to $\\beta_2$. The prefactor $\\beta_2$ stems\nfrom ${\\bf k} \\cdot {\\bf p}$ coupling to more remote bands such as\nthe $p$ antibonding conduction bands $\\Gamma_8^c$ and $\\Gamma_7^c$.\n\nFor ${\\bf E} = (0,0,E_z)$ Eq.\\ (\\ref{lutt_spin}) becomes (using\nexplicit matrix notation with $j=3/2$ eigenstates in the order $j_z\n= +3/2,+1/2,-1/2,-3/2$)\n%\n\\widetext\n\\begin{equation}\n\\label{lutt_spin_x}\nH_{8v}^{\\rm SO} = \\beta_1 \\, E_z\n\\left(\\begin {array}{cccc}\n 0 & \\frac{1}{2} \\sqrt{3} \\, k_- & 0 & 0 \\\\[0.5ex]\n \\frac{1}{2} \\sqrt{3} \\, k_+ & 0 & k_- & 0 \\\\[0.5ex]\n 0 & k_+ & 0 & \\frac{1}{2} \\sqrt{3} \\, k_- \\\\[0.5ex]\n 0 & 0 & \\frac{1}{2} \\sqrt{3} \\, k_+ & 0 \\end {array} \\right)\n%\n+ \\beta_2 \\, E_z\n\\left(\n\\begin {array}{cccc}\n 0 & {\\frac{7}{8}} \\sqrt {3} \\, k_- & 0 & 3/4 \\, k_+ \\\\[0.5ex]\n {\\frac {7}{8}} \\sqrt {3} \\, k_+ & 0 & 5/2 \\, k_- & 0 \\\\[0.5ex]\n 0 & 5/2 \\, k_+ & 0 & {\\frac {7}{8}} \\sqrt {3} \\, k_- \\\\[0.5ex]\n 3/4 \\, k_- & 0 & {\\frac {7}{8}} \\sqrt {3} \\, k_+ & 0\n\\end {array} \\right) .\n\\end{equation}\n%\n\\narrowtext\\noindent\n%\nHere the first term couples the two LH states ($j_z = \\pm 1/2$), and\nit couples the HH states ($j_z = \\pm 3/2$) with the LH states. But\nthere is no $k$ linear splitting of the HH states proportional to\n$\\beta_1$. The second matrix in Eq.\\ (\\ref{lutt_spin_x}) contains a\n$k$ linear coupling of the HH states.\n\nWe want to emphasize that $H_{6c}^{\\rm SO}$ and $H_{8v}^{\\rm SO}$\nare {\\em effective} Hamiltonians for the spin splitting of electron\nand hole subbands, which are implicitly contained in the full\nmultiband Hamiltonian for the subband problem \\cite{win93,las85}\n%\n\\begin{equation}\n\\label{kp_sub}\nH = H_{{\\bf k} \\cdot {\\bf p}} ({\\bf k}_\\|, k_z = -i \\partial_z)\n + e E_z z \\openone .\n\\end{equation}\n%\nHere $H_{{\\bf k} \\cdot {\\bf p}}$ is a ${\\bf k} \\cdot {\\bf p}$\nHamiltonian for the bulk band structure (i.e., $H_{{\\bf k} \\cdot\n{\\bf p}}$ does not contain $H_{6c}^{\\rm SO}$ or $H_{8v}^{\\rm SO}$)\nand we have restricted ourselves to the lowest order term in a\nTaylor expansion of the confining potential $V(z) = V_0 + eE_z z +\n{\\cal O} (z^2)$ which reflects the inversion asymmetry of $V(z)$.\nThe effective Hamiltonians (\\ref{rashba}) and (\\ref{lutt_spin_x})\nstem from the combined effect of $H_{{\\bf k} \\cdot {\\bf p}}$ and the\nterm $e E_z z$. For a systematic investigation of the importance of\nthe different terms in $H$ we have developed a novel, analytical\napproach based on a perturbative diagonalization of $H$ using a\nsuitable set of trial functions and using L\\\"owdin partitioning.\n\\cite{bir74,loe51} Though we cannot expect accurate numerical\nresults from such an approach it is an instructive complement for\nnumerical methods, as we can clearly identify in the subband\ndispersion ${\\cal E} ({\\bf k}_\\|)$ the terms proportional to $E_z$\nwhich are breaking the spin degeneracy. Neglecting in $H_{{\\bf k}\n\\cdot {\\bf p}}$ remote bands like $\\Gamma_8^c$ and $\\Gamma_7^c$ we\nobtain for the SIA spin splitting of the HH states\n%\n\\begin{mathletters}\n\\label{luttperturb}\n\\begin{equation}\n\\label{luttperturbHH}\n{\\cal E}_{\\rm HH}^{\\rm SO} (k_\\|) \\propto\n\\pm \\langle \\beta_1 E_z \\rangle k_\\|^3 .\n\\end{equation}\n%\nIn particular, we have no $k$ linear splitting (and $\\beta_2 \\equiv\n0$) if we restrict ourselves to the Luttinger Hamiltonian\n\\cite{lut56} which includes $\\Gamma_8^c$ and $\\Gamma_7^c$ by means\nof second order perturbation theory. \\cite{eke85} Accurate numerical\ncomputations \\cite{win93} show that the dominant part of the $k$\nlinear splitting of the HH states is due to BIA. However, for\ntypical densities this $k$ linear splitting is rather small. For the\nLH states we have\n%\n\\begin{equation}\n\\label{luttperturbLH}\n{\\cal E}_{\\rm LH}^{\\rm SO} (k_\\|) \\propto\n\\pm \\langle \\beta_1 E_z \\rangle k_\\| .\n\\end{equation}\n\\end{mathletters}%\n%\nThus we have a qualitative difference between the spin splitting of\nelectron and LH states which is proportional to $k_\\|$ and the\nsplitting of HH states which essentially is proportional to\n$k_\\|^3$. The former is most important in the low-density regime\nwhereas the latter becomes negligible for small densities. Note that\nfor 2D hole systems the first subband is HH like so that for low\ndensities the SIA spin splitting is given by Eq.\\ \n(\\ref{luttperturbHH}). In Eq.\\ (\\ref{luttperturb}) the lengthy\nprefactors depend on the details of the geometry of the QW.\nMoreover, we have omitted a weak dependence on the direction of\n${\\bf k}_\\|$. But the order of the terms with respect to $k_\\|$ is\nindependent of these details. It is crucial that, basically, we have\n%\n\\begin{equation}\n\\label{so_bulk}\n\\alpha, \\beta_1, \\beta_2 \\propto \\Delta_0\n\\end{equation}\n%\nwith $\\Delta_0$ the SO gap between the bulk valence bands\n$\\Gamma_8^v$ and $\\Gamma_7^v$, i.e., we have no SIA spin splitting\nfor $\\Delta_0 = 0$. \\cite{ros89} This can be most easily seen if we\nexpress $H_{{\\bf k} \\cdot {\\bf p}}$ in a basis of orbital angular\nmomentum eigenstates.\n\nA more detailed analysis of our analytical model shows that both\n$H_{6c}^{\\rm SO}$ and $H_{8v}^{\\rm SO}$ stem from a third order\nperturbation theory for $k_\\pm$, $k_z = -i \\partial_z$, and $e E_z\nz$. This seems to be a rather high order. Nevertheless, the\nresulting terms are fairly large. \\cite{gap} In agreement with\nRefs.\\ \\onlinecite{pfe99,win93,las85} this is a simple argument to\nresolve the old controversy based on an argument by Ando\n\\cite{afs82} that spin splitting in 2D systems ought to be\nnegligibly small because for bound states in first order we have\n$\\langle E_z \\rangle = 0$. We note that the present ansatz for the\nprefactors $\\alpha$ and $\\beta_1, \\beta_2$ is quite different from\nthe ansatz in Ref.\\ \\onlinecite{lom85}. We obtain $H_{6c}^{\\rm SO}$\nand $H_{8v}^{\\rm SO}$ by means of L\\\"owdin partitioning of the\nHamiltonian (\\ref{kp_sub}) whereas in Ref.\\ \\onlinecite{lom85} the\nauthors explicitly introduced $H_{6c}^{\\rm SO}$ into their model.\nMoreover, we evaluate the matrix elements of $eE_z z$ with respect\nto envelope functions for the bound states whereas in\nRef.~\\onlinecite{lom85} the authors considered matrix elements of\n$eE_z z$ with respect to bulk Bloch functions. The latter quantities\nare problematic because they depend on the origin of the coordinate\nframe.\n\nAs an example, we show in Fig.\\ \\ref{pic:gaas} the self-consistently\ncalculated \\cite{win93} anisotropic subband dispersion ${\\cal E}_\\pm\n({\\bf k}_\\|)$, DOS effective mass $m^\\ast/m_0$, and spin splitting\n${\\cal E}_+ ({\\bf k}_\\|) - {\\cal E}_- ({\\bf k}_\\|)$ for a [001]\ngrown GaAs/Al$_{0.5}$Ga$_{0.5}$As heterostructure. The calculation\nwas based on a $14\\times 14$ Hamiltonian ($\\Gamma_8^c$,\n$\\Gamma_7^c$, $\\Gamma_6^c$, $\\Gamma_8^v$, and $\\Gamma_7^v$). It\nfully took into account both SIA and BIA. The weakly divergent van\nHove singularity of the DOS effective mass at the subband edge\nindicates that the $k$ linear splitting is rather small. (Its\ndominant part is due to BIA. \\cite{trr79}) Basically, the spin\nsplitting in Fig.\\ \\ref{pic:gaas} is proportional to $k_\\|^3$.\n\nOnly for the crystallographic growth directions [001] and [111] the\nhole subband states at $k_\\| = 0$ are pure HH and LH states. For\nlow-symmetry growth directions like [113] and [110] we have mixed\nHH-LH eigenstates even at $k_\\| = 0$, though often the eigenstates\ncan be labeled by their dominant spinor components. \\cite{win96a}\nThe HH-LH mixing adds a $k$ linear term to the splitting\n(\\ref{luttperturbHH}) of the HH states, which often exceeds $\\langle\n\\beta_2 E_z \\rangle k_\\|$. However, this effect is still small when\ncompared with the cubic splitting.\n\nFor a HH subband dispersion ${\\cal E}_\\pm (k_\\|) = \\langle \\mu\n\\rangle k_\\|^2 \\pm \\langle \\beta_1 E_z \\rangle k_\\|^3$ we obtain for\nthe densities $N_\\pm$ in the spin-split subbands\n%\n\\begin{mathletters}\n\\label{sdh_hh}\n\\begin{equation}\nN_\\pm = \\frac{1}{2} N_s \\pm \\frac{\\langle \\beta_1 E_z \\rangle \nN_s}{\\sqrt{2} \\, \\langle \\mu \\rangle X} \\sqrt{\\pi N_s (6 - 4/X)}\n\\end{equation}\n%\nwith\n%\n\\begin{equation}\n%\\label{}\nX = 1 + \\sqrt{1 - 4\\pi \\, N_s \\left(\\frac{\\langle \\beta_1\n E_z \\rangle}{\\langle \\mu \\rangle} \\right)^2} .\n\\end{equation}\n\\end{mathletters}%\n%\nThe spin splitting according to Eq.\\ (\\ref{sdh_hh}) is substantially\ndifferent from Eq.\\ (\\ref{sdh_rashba}). For electrons and a fixed\nelectric field $E_z$ but varying $N_s$ the difference $\\Delta N =\nN_+ - N_-$ increases like $N_s^{1/2}$ whereas for HH subbands it\nincreases like $N_s^{3/2}$. Using a fixed density $N_s$ but varying\n$E_z$ it is more difficult to detect the difference between Eqs.\\ \n(\\ref{sdh_rashba}) and (\\ref{sdh_hh}). In both cases a power\nexpansion of $\\Delta N$ gives $\\Delta N = a_1 |E_z| + a_3 |E_z|^3 +\n{\\cal O} (|E_z|^5)$ with $a_3 <0$ for electrons and $a_3>0$ for HH\nsubbands.\n\nThe proportionality (\\ref{so_bulk}) is completely analogous to the\neffective $g$ factor in bulk semiconductors. \\cite{rot59} Lassnig\n\\cite{las85} pointed out that the $B=0$ spin splitting of electrons\ncan be expressed in terms of a position dependent effective $g$\nfactor $g^\\ast (z)$. In the following we want to discuss the close\nrelationship between Zeeman splitting and $B=0$ spin splitting from\na more general point of view. Note that in the presence of an\nexternal magnetic field ${\\bf B}$ we have ${\\bf k} \\times {\\bf k} =\n(- i e / \\hbar) {\\bf B}$ and the Zeeman splitting in the\n$\\Gamma_6^c$ conduction band can be expressed as \\cite{trr79}\n%\n\\begin{equation}\n\\label{zeeman6c}\nH_{6c}^{Z} = \\frac{i\\hbar}{e} \\, \\frac{g^\\ast}{2} \\, \\mu_B \\, {\\bf\nk} \\times {\\bf k} \\cdot \\bbox{\\sigma} =\n\\frac{g^\\ast}{2} \\, \\mu_B \\, {\\bf B} \\cdot \\bbox{\\sigma}\n\\end{equation}\n%\nwith $\\mu_B$ the Bohr magneton. Thus apart from a prefactor we\nobtain the Rashba term (\\ref{rashbafull}) from Eq.\\ (\\ref{zeeman6c})\nby replacing one of the ${\\bf k}$'s with the electric field ${\\bf\nE}$. In the $\\Gamma_8^v$ valence band we have two invariants for the\nZeeman splitting \\cite{lut56,trr79}\n%\n\\begin{equation}\n\\label{lutt_zeeman}\nH_{8v}^{Z} = 2 \\kappa \\, \\mu_B \\, {\\bf B} \\cdot {\\bf J} +\n2 q \\, \\mu_B \\, {\\bf B} \\cdot \\bbox{\\cal J} .\n\\end{equation}\n%\nHere, the first term is the isotropic contribution, and the second\nterm is the anisotropic part. It is well-known that in all common\nsemiconductors for which Eq.\\ (\\ref{lutt_zeeman}) is applicable the\ndominant contribution to $H_{8v}^{Z}$ is given by the first term\nproportional to $\\kappa$ whereas the second term is rather small.\n\\cite{trr79,lut56} Analogous to $\\beta_1$ and $\\beta_2$ the\nisotropic ${\\bf k} \\cdot {\\bf p}$ coupling between $\\Gamma_8^v$ and\n$\\Gamma_6^c$ contributes to $\\kappa$ but not to $q$. The latter\nstems from ${\\bf k} \\cdot {\\bf p}$ coupling to more remote bands\nsuch as $\\Gamma_8^c$ and $\\Gamma_7^c$.\n\nSeveral authors \\cite{pud97,nit97,hei98,eis84} used an apparently\nclosely related intuitive picture for the $B=0$ spin splitting which\nwas based on the idea that the velocity $v_\\| = \\hbar k_\\| / m^\\ast$\nof the 2D electrons is perpendicular to the electric field $E_z$. In\nthe electron's rest frame $E_z$ is Lorentz transformed into a\nmagnetic field $B$ so that the $B=0$ spin splitting becomes a Zeeman\nsplitting in the electron's rest frame. However, this magnetic field\nis given by $B=(v_\\|/c^2)E_z$ (SI units) and for typical values of\n$E_z$ and $v_\\|$ we have $B \\sim 2 \\ldots 20 \\times 10^{-7}$~T which\nwould result in a spin splitting of the order of $5\\times 10^{-9}\n\\ldots 5\\times 10^{-5}$~meV. On the other hand, the experimentally\nobserved spin splitting is of the order of $0.1 \\ldots 10$~meV. The\n$B=0$ spin splitting requires the SO interaction caused by the\natomic cores. In bulk semiconductors this interaction is responsible\nfor the SO gap $\\Delta_0$ between the valence bands $\\Gamma_8^v$ and\n$\\Gamma_7^v$ which appears in Eq.\\ (\\ref{so_bulk}). The SO\ninteraction is the larger the larger the atomic number of the\nconstituting atoms. In Si we have $\\Delta_0 = 44$ meV whereas in Ge\nwe have $\\Delta_0 = 296$ meV. Therefore, SIA spin splitting in Si\nquantum structures is rather small. \\cite{win96}\n\nRecently, spin splitting in 2D systems has gained renewed interest\nbecause of an argument by Pudalov \\cite{pud97} which relates the\nmetal-insulator transition (MIT) in low-density 2D systems with the\nSIA spin splitting. Based on the Rashba model \\cite{bych84} it was\nargued that the SIA spin splitting ``results in a drastic change of\nthe internal properties of the system even without allowing for the\nCoulomb interaction.'' \\cite{sko98}. However, as we have shown\nabove, this argument is applicable only to electron and LH states.\nThe MIT has been observed also in pure HH systems in, e.g., Si/SiGe\nQW's. \\cite{lam97,col97} As noted above, SO interaction and spin\nsplitting in these systems are rather small, \\cite{win96} so that it\nappears unlikely that here the broken inversion symmetry of the\nconfining potential is responsible for the MIT. We note that in Si\n2D electron systems the effective $g$ factor is enhanced due to many\nbody effects. \\cite{afs82,oka99} It can be expected that similar\neffects are also relevant for the $B=0$ spin splitting.\n\nIn conclusion, we have analyzed the SIA spin splitting in 2D\nelectron and hole systems. In 2D hole systems the splitting differs\nremarkably from its familiar counterpart in the conduction band. For\nelectron states it increases linearly with in-plane wave vector\n$k_\\|$ whereas the spin splitting of heavy hole states can be of\nthird order in $k_\\|$. We have discussed consequences of this\nbehavior in the context of recent arguments on the origin of the\nmetal-insulator transition observed in 2D systems.\n\nThe author wants to thank O.\\ Pankratov, S.\\ J.\\ Papadakis, and M.\\\nShayegan for stimulating discussions and suggestions.\n\n%******************************************************************\n\\begin{thebibliography}{99}\n\\vspace*{-12mm}\n\n\\bibitem{kit63} C.\\ Kittel, {\\em Quantum Theory of Solids},\n (Wiley, New York, 1963).\n \n\\bibitem{chiral} The spin-split eigenstates contain equal\n contributions of up and down spinor components which reflects the\n fact that for $B=0$ the system has a vanishing magnetic moment.\n However, the eigenstates are {\\em circularly} polarized, and\n recently some authors have used the term ``chiral splitting''\n instead of ``spin splitting'' (e.g., Ref.\\ \\onlinecite{sko98}).\n\n\\bibitem{pfe99} P.\\ Pfeffer and W.\\ Zawadzki, \n Phys.\\ Rev.\\ B {\\bf 59}, R5312 (1999).\n\n\\bibitem{bych84} Y.\\ A.\\ Bychkov and E.\\ I.\\ Rashba,\n J.\\ Phys.\\ C: Solid State Phys.\\ {\\bf 17}, 6039 (1984).\n\n\\bibitem{win93} R.\\ Winkler and U.\\ R\\\"ossler,\n Phys.\\ Rev.\\ B {\\bf 48}, 8918 (1993).\n\n\\bibitem{eke85} U.\\ Ekenberg and M.\\ Altarelli, \n Phys.\\ Rev.\\ B {\\bf 32}, 3712 (1985), and references therein.\n\n\\bibitem{win96} R.\\ Winkler {\\em et al.},\n Phys.\\ Rev.\\ B {\\bf 53}, 10858 (1996).\n\n\\bibitem{pud97} V.\\ M.\\ Pudalov, \n Pis'ma Zh.\\ Eksp.\\ Teor. Fiz. {\\bf 66}, 168 (1997) \n [JETP Lett.\\ {\\bf 66}, 175 (1997)].\n\n\\bibitem{sko98} M.\\ A.\\ Skvortsov,\n Pis'ma Zh.\\ Eksp.\\ Teor. Fiz. {\\bf 67}, 118 (1998) \n [JETP Lett.\\ {\\bf 67}, 133 (1998)].\n\n\\bibitem{lom85} G.\\ Lommer, F.\\ Malcher, and U.\\ R\\\"ossler,\n Phys.\\ Rev.\\ B {\\bf 32}, 6965 (1985).\n\n\\bibitem{ros89} U.\\ R\\\"ossler, F.\\ Malcher, and G.\\ Lommer, in {\\em\n High Magnetic Fields in Semiconductor Physics II}, edited by G.\\\n Landwehr, (Springer, Berlin, 1989), p 376.\n\n\\bibitem{wol89} R.\\ Wollrab {\\em et al.},\n Semicond.\\ Sci.\\ Technol.\\ {\\bf 4}, 491 (1989).\n\n\\bibitem{luo90} J.\\ Luo {\\em et al.},\n Phys.\\ Rev.\\ B {\\bf 41}, 7685 (1990).\n\n\\bibitem{nit97} J.\\ Nitta {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 78}, 1335 (1997).\n\n\\bibitem{hei98} J.\\ P.\\ Heida {\\em et al.},\n Phys.\\ Rev.\\ B {\\bf 57}, 11911 (1998).\n\n\\bibitem{eis84} J.\\ P.\\ Eisenstein {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 53}, 2579 (1984).\n\n\\bibitem{las85} R.\\ Lassnig, \n Phys.\\ Rev.\\ B {\\bf 31}, 8076 (1985).\n\n\\bibitem{dre92} P.\\ D.\\ Dresselhaus {\\em et al.}, \n Phys.\\ Rev.\\ Lett.\\ {\\bf 68}, 106 (1992).\n\n\\bibitem{jus92} B.\\ Jusserand {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 69}, 848 (1992).\n\n\\bibitem{lu98} J.\\ P.\\ Lu {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 81}, 1282 (1998).\n\n\\bibitem{pap99} S.\\ J.\\ Papadakis {\\em et al.},\n Science {\\bf 283}, 2056 (1999).\n\n\\bibitem{trr79} H.-R.\\ Trebin, U.\\ R\\\"ossler, and R.\\ Ranvaud,\n Phys.\\ Rev.\\ B {\\bf 20}, 686 (1979), and references therein.\n\n\\bibitem{bir74} G.\\ L.\\ Bir and G.\\ E.\\ Pikus,\n {\\em Symmetry and Strain-Induced Effects in Semiconductors},\n (Wiley, New York, 1974).\n\n\\bibitem{lut56} J.\\ M.\\ Luttinger,\n Phys.\\ Rev.\\ {\\bf 102}, 1030 (1956).\n \n\\bibitem{cross} In the same way we can derive $H^{\\rm SO}$ for other\n blocks of a multiband Hamiltonian, \\cite{trr79} e.g., for\n $H_{7v}^{\\rm SO}$ we get Eq.\\ (\\ref{rashbafull}).\n\n\\bibitem{loe51} P.-O.\\ L\\\"owdin,\n J.\\ Chem.\\ Phys.\\ {\\bf 19}, 1396 (1951).\n \n\\bibitem{gap} For a third-order term such as $\\langle\n h_i|k_\\pm|c_j\\rangle \\cdot \\langle c_j|k_z|h_k\\rangle \\cdot \\langle\n h_k|E_z z|h_i\\rangle$ we have only one energy denominator of the\n order of the fundamental gap $E_g$; the second denominator is given\n by the subband spacing. Note that $1/m^\\ast \\propto 1/E_g$, i.e.,\n both spin splitting and $\\hbar^2 k_\\|^2/(2m^\\ast)$ scale like\n $E_g^{-1}$. If we restrict ourselves to the Luttinger Hamiltonian\n we get Eq.\\ (\\ref{lutt_spin_x}) from a second order perturbation\n theory for $E_z z$ and the off-diagonal terms $\\gamma_3 \\, k_\\pm\n k_z$. Here, the energy denominator $E_g$ is contained in the\n Luttinger parameter $\\gamma_3$.\n\n\\bibitem{afs82} T.\\ Ando, A.\\ B.\\ Fowler, and F.\\ Stern,\n Rev.\\ Mod.\\ Phys.\\ {\\bf 54}, 437 (1982).\n\n\\bibitem{win96a} R.\\ Winkler and A.\\ I.\\ Nesvizhskii, \n Phys.\\ Rev.\\ B {\\bf 53}, 9984 (1996), and references therein.\n\n\\bibitem{rot59} L.\\ M.\\ Roth, B.\\ Lax, and S.\\ Zwerdling,\n Phys.\\ Rev.\\ {\\bf114}, 90 (1959).\n\n\\bibitem{lam97} J.\\ Lam {\\em et al.}, \n Phys.\\ Rev.\\ B {\\bf 56}, R12741 (1997).\n\n\\bibitem{col97} P.\\ T.\\ Coleridge {\\em et al.}, \n Phys.\\ Rev.\\ B {\\bf 56}, R12764 (1997).\n\n\\bibitem{oka99} T.\\ Okamoto {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 82}, 3875 (1999).\n\n\\end{thebibliography}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{figure}\n\\centerline{\\epsfxsize=0.80\\columnwidth\\leavevmode\n \\epsffile{fig1.eps}}\\vspace{3mm}\n\\caption[]{\\label{pic:insb} Self-consistently calculated subband\ndispersion ${\\cal E}_\\pm (k_\\|)$ (lower right), DOS\neffective mass $m^\\ast/m_0$ (lower left), spin splitting ${\\cal E}_+\n(k_\\|) - {\\cal E}_- (k_\\|)$ (upper right) and subband dispersion\n${\\cal E}_\\pm (k_\\|)$ in the vicinity of $k_\\|=0$ (upper left) for\nan MOS inversion layer on InSb with $N_s = 2 \\times 10^{11}$\ncm$^{-2}$ and $|N_A - N_D| = 2\\times 10^{16}$ cm$^{-2}$. The dotted\nline indicates the Fermi energy $E_F$.}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=0.85\\columnwidth\\leavevmode\n \\epsffile{fig2.eps}} \\vspace{3mm}\n\\caption[]{\\label{pic:gaas} Self-consistently calculated anisotropic\nsubband dispersion ${\\cal E}_\\pm ({\\bf k}_\\|)$ (lower right), DOS\neffective mass $m^\\ast/m_0$ (lower left), spin splitting ${\\cal E}_+\n({\\bf k}_\\|) - {\\cal E}_- ({\\bf k}_\\|)$ (upper right) and subband\ndispersion ${\\cal E}_\\pm ({\\bf k}_\\|)$ in the vicinity of $k_\\|=0$\n(upper left) for a [001] grown GaAs/Al$_{0.5}$Ga$_{0.5}$As\nheterostructure with $N_s = 2 \\times 10^{11}$ cm$^{-2}$ and $|N_A -\nN_D| = 2\\times 10^{16}$ cm$^{-2}$. Different line styles correspond\nto different directions of the in-plane wave vector ${\\bf k}_\\|$ as\nindicated. The dotted line indicates the Fermi energy $E_F$.}\n\\end{figure}\n\n\\widetext\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002003.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\vspace*{-12mm}\n\n\\bibitem{kit63} C.\\ Kittel, {\\em Quantum Theory of Solids},\n (Wiley, New York, 1963).\n \n\\bibitem{chiral} The spin-split eigenstates contain equal\n contributions of up and down spinor components which reflects the\n fact that for $B=0$ the system has a vanishing magnetic moment.\n However, the eigenstates are {\\em circularly} polarized, and\n recently some authors have used the term ``chiral splitting''\n instead of ``spin splitting'' (e.g., Ref.\\ \\onlinecite{sko98}).\n\n\\bibitem{pfe99} P.\\ Pfeffer and W.\\ Zawadzki, \n Phys.\\ Rev.\\ B {\\bf 59}, R5312 (1999).\n\n\\bibitem{bych84} Y.\\ A.\\ Bychkov and E.\\ I.\\ Rashba,\n J.\\ Phys.\\ C: Solid State Phys.\\ {\\bf 17}, 6039 (1984).\n\n\\bibitem{win93} R.\\ Winkler and U.\\ R\\\"ossler,\n Phys.\\ Rev.\\ B {\\bf 48}, 8918 (1993).\n\n\\bibitem{eke85} U.\\ Ekenberg and M.\\ Altarelli, \n Phys.\\ Rev.\\ B {\\bf 32}, 3712 (1985), and references therein.\n\n\\bibitem{win96} R.\\ Winkler {\\em et al.},\n Phys.\\ Rev.\\ B {\\bf 53}, 10858 (1996).\n\n\\bibitem{pud97} V.\\ M.\\ Pudalov, \n Pis'ma Zh.\\ Eksp.\\ Teor. Fiz. {\\bf 66}, 168 (1997) \n [JETP Lett.\\ {\\bf 66}, 175 (1997)].\n\n\\bibitem{sko98} M.\\ A.\\ Skvortsov,\n Pis'ma Zh.\\ Eksp.\\ Teor. Fiz. {\\bf 67}, 118 (1998) \n [JETP Lett.\\ {\\bf 67}, 133 (1998)].\n\n\\bibitem{lom85} G.\\ Lommer, F.\\ Malcher, and U.\\ R\\\"ossler,\n Phys.\\ Rev.\\ B {\\bf 32}, 6965 (1985).\n\n\\bibitem{ros89} U.\\ R\\\"ossler, F.\\ Malcher, and G.\\ Lommer, in {\\em\n High Magnetic Fields in Semiconductor Physics II}, edited by G.\\\n Landwehr, (Springer, Berlin, 1989), p 376.\n\n\\bibitem{wol89} R.\\ Wollrab {\\em et al.},\n Semicond.\\ Sci.\\ Technol.\\ {\\bf 4}, 491 (1989).\n\n\\bibitem{luo90} J.\\ Luo {\\em et al.},\n Phys.\\ Rev.\\ B {\\bf 41}, 7685 (1990).\n\n\\bibitem{nit97} J.\\ Nitta {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 78}, 1335 (1997).\n\n\\bibitem{hei98} J.\\ P.\\ Heida {\\em et al.},\n Phys.\\ Rev.\\ B {\\bf 57}, 11911 (1998).\n\n\\bibitem{eis84} J.\\ P.\\ Eisenstein {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 53}, 2579 (1984).\n\n\\bibitem{las85} R.\\ Lassnig, \n Phys.\\ Rev.\\ B {\\bf 31}, 8076 (1985).\n\n\\bibitem{dre92} P.\\ D.\\ Dresselhaus {\\em et al.}, \n Phys.\\ Rev.\\ Lett.\\ {\\bf 68}, 106 (1992).\n\n\\bibitem{jus92} B.\\ Jusserand {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 69}, 848 (1992).\n\n\\bibitem{lu98} J.\\ P.\\ Lu {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 81}, 1282 (1998).\n\n\\bibitem{pap99} S.\\ J.\\ Papadakis {\\em et al.},\n Science {\\bf 283}, 2056 (1999).\n\n\\bibitem{trr79} H.-R.\\ Trebin, U.\\ R\\\"ossler, and R.\\ Ranvaud,\n Phys.\\ Rev.\\ B {\\bf 20}, 686 (1979), and references therein.\n\n\\bibitem{bir74} G.\\ L.\\ Bir and G.\\ E.\\ Pikus,\n {\\em Symmetry and Strain-Induced Effects in Semiconductors},\n (Wiley, New York, 1974).\n\n\\bibitem{lut56} J.\\ M.\\ Luttinger,\n Phys.\\ Rev.\\ {\\bf 102}, 1030 (1956).\n \n\\bibitem{cross} In the same way we can derive $H^{\\rm SO}$ for other\n blocks of a multiband Hamiltonian, \\cite{trr79} e.g., for\n $H_{7v}^{\\rm SO}$ we get Eq.\\ (\\ref{rashbafull}).\n\n\\bibitem{loe51} P.-O.\\ L\\\"owdin,\n J.\\ Chem.\\ Phys.\\ {\\bf 19}, 1396 (1951).\n \n\\bibitem{gap} For a third-order term such as $\\langle\n h_i|k_\\pm|c_j\\rangle \\cdot \\langle c_j|k_z|h_k\\rangle \\cdot \\langle\n h_k|E_z z|h_i\\rangle$ we have only one energy denominator of the\n order of the fundamental gap $E_g$; the second denominator is given\n by the subband spacing. Note that $1/m^\\ast \\propto 1/E_g$, i.e.,\n both spin splitting and $\\hbar^2 k_\\|^2/(2m^\\ast)$ scale like\n $E_g^{-1}$. If we restrict ourselves to the Luttinger Hamiltonian\n we get Eq.\\ (\\ref{lutt_spin_x}) from a second order perturbation\n theory for $E_z z$ and the off-diagonal terms $\\gamma_3 \\, k_\\pm\n k_z$. Here, the energy denominator $E_g$ is contained in the\n Luttinger parameter $\\gamma_3$.\n\n\\bibitem{afs82} T.\\ Ando, A.\\ B.\\ Fowler, and F.\\ Stern,\n Rev.\\ Mod.\\ Phys.\\ {\\bf 54}, 437 (1982).\n\n\\bibitem{win96a} R.\\ Winkler and A.\\ I.\\ Nesvizhskii, \n Phys.\\ Rev.\\ B {\\bf 53}, 9984 (1996), and references therein.\n\n\\bibitem{rot59} L.\\ M.\\ Roth, B.\\ Lax, and S.\\ Zwerdling,\n Phys.\\ Rev.\\ {\\bf114}, 90 (1959).\n\n\\bibitem{lam97} J.\\ Lam {\\em et al.}, \n Phys.\\ Rev.\\ B {\\bf 56}, R12741 (1997).\n\n\\bibitem{col97} P.\\ T.\\ Coleridge {\\em et al.}, \n Phys.\\ Rev.\\ B {\\bf 56}, R12764 (1997).\n\n\\bibitem{oka99} T.\\ Okamoto {\\em et al.},\n Phys.\\ Rev.\\ Lett.\\ {\\bf 82}, 3875 (1999).\n\n\\end{thebibliography}"
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cond-mat0002004
|
Elasticity of Thin Rods with Spontaneous Curvature and Torsion---Beyond Geometrical Lines
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[
{
"author": "Aleksey D. Drozdov$^{1}$ and Yitzhak Rabin$^{2}$"
}
] |
We study three-dimensional deformations of thin inextensible elastic rods with non-vanishing spontaneous curvature and torsion. In addition to the usual description in terms of curvature and torsion which considers only the configuration of the centerline of the rod, we allow deformations that involve the rotation of the rod's cross-section around its centerline. We derive new expressions for the mechanical energy and for the force and moment balance conditions for the equilibrium of a rod under the action of arbitrary external loads. Several illustrative examples are studied and the connection between our results and recent experiments on the stretching of supercoiled DNA molecules is discussed.
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[
{
"name": "cond-mat0002004.tex",
"string": "\\documentstyle[osa,manuscript]{revtex}\n\n\\newcommand{\\MF}{{\\large{\\manual META}\\-{\\manual FONT}}}\n\\newcommand{\\manual}{rm}\n\\newcommand\\bs{\\char '134 }\n\\newcommand{\\bsigma}{\\mbox{$\\boldmath \\sigma$}}\n\\newcommand{\\brho}{\\mbox{$\\boldmath \\rho$}}\n\\newcommand{\\bpsi}{\\mbox{$\\boldmath \\psi$}}\n\\newcommand{\\bnabla}{\\mbox{$\\boldmath \\nabla$}}\n\\newcommand{\\bmu}{\\mbox{$\\boldmath \\mu$}}\n\\newcommand{\\bepsilon}{\\mbox{$\\boldmath \\epsilon$}}\n\n\\begin{document}\n\\title{Elasticity of Thin Rods with Spontaneous Curvature\nand Torsion---Beyond Geometrical Lines }\n\\author{Aleksey D. Drozdov$^{1}$ and Yitzhak Rabin$^{2}$}\n\\address{$^{1}$ Institute for Industrial Mathematics, 4 Hanachtom Street\\\\\nBeersheba, 84311 Israel\\\\\n$^{2}$ Department of Physics, Bar-Ilan University\\\\\nRamat-Gan, 52900 Israel}\n\\maketitle\n\n\\begin{abstract}\nWe study three-dimensional deformations of thin inextensible elastic rods\nwith non-vanishing spontaneous curvature and torsion. In addition to the\nusual description in terms of curvature and torsion which considers only the\nconfiguration of the centerline of the rod, we allow deformations that\ninvolve the rotation of the rod's cross-section around its centerline. We\nderive new expressions for the mechanical energy and for the force and\nmoment balance conditions for the equilibrium of a rod under the action of\narbitrary external loads. Several illustrative examples are studied and the\nconnection between our results and recent experiments on the stretching of\nsupercoiled DNA molecules is discussed.\n\\end{abstract}\n\n\\pacs{87.15.La, 46.70.Hg}\n\n\\section{Introduction}\n\nRecent experimental advances in the art of manipulation of single DNA\nmolecules and of rigid protein assemblies such as actin filaments, etc.,\nhave led to an outbreak of theoretical activity connected with the\nelasticity of thin rods$^{1-19}$. One of the most intriguing theoretical\nquestions related to the deformation of DNA concerns the coupling between\nbending and twist in the mechanical energy of the polymer. The problem is\nusually considered in the following terms: at the first step, the thin rod\nwhich models the molecule is replaced by its centerline. With each point of\nthe line (specified by its position along the contour $\\xi $) one associates\na triad of unit vectors: the tangent to the line (${\\bf t}$), the principal\nnormal (${\\bf n}$) which lies in the plane defined by the tangents at points \n$\\xi $ and $\\xi +d\\xi ,$ and the binormal (${\\bf b}$) which is orthogonal to\nboth ${\\bf t}$ and ${\\bf n}$. As one moves along the line, the triad rotates\nand this rotation is described by the Frenet--Serret equations in which the\n``rate'' of rotation of each unit vector is determined by two parameters:\nthe local curvature $\\kappa $ and the local torsion $\\omega $ (sometimes\nreferred to as writhe)\\cite{SH94}. In order to relate this purely\ngeometrical picture to the elastic response of real rods, one has to specify\nthe physical properties of the rod in a stress--free (undeformed) reference\nstate and to write the energy as a quadratic expansion in deviations from\nthis state. In the classical theories of thin elastic rods\\cite{LL80} one\nusually assumes that the reference state corresponds to a straight untwisted\nrod (with vanishing spontaneous curvature $\\kappa _{0}$ and spontaneous\ntorsion $\\omega _{0}$) and the mechanical energy density is written as a sum\nof terms proportional to $\\kappa ^{2}$ and $\\omega ^{2}$. The generalization\nto the case of non--vanishing spontaneous curvature and torsion is then done\nby requiring that the strain energy density per unit length $U$ is minimized\nfor $\\kappa =\\kappa _{0}$ and $\\omega =\\omega _{0}$ which leads to the\nexpression \n\\begin{equation}\nU=\\frac{1}{2}\\biggl [A_{1}(\\kappa -\\kappa _{0})^{2}+A_{2}(\\omega -\\omega\n_{0})^{2}\\biggr ].\n\\end{equation}\nHere $A_{1}$ and $A_{2}$ are material parameters (products of elastic moduli\nand moments of inertia).\n\nAlthough Eq. (1) has been employed in a number of studies\\cite\n{HH91,GL92,MKJ95,MS95,GT97,GPW98,LGK98,HZ99}, its validity has been\nquestioned by several authors \\cite{KLN97,Mar97,Mar98,MN98}, who argued that\nit fails to describe, even qualitatively, the experimental data on\ntorsionally constrained DNA \\cite{MN98,SAB96}. To account for the coupling\nbetween bending and twist observed in experiment, extra terms are\nconventionally added to the mechanical energy density,\nEq. (1), {\\it by hand}.\n\nThe objective of this work is to derive an expression for the mechanical\nenergy and obtain the equations which determine the mechanical equilibrium\nof a rod subjected to arbitrary forces and moments. This is done using a new\nform of the displacement field, which accounts for both the deformation of\nthe centerline and the rotation of the cross-section around this line (i.e.,\ntwist). Instead of using ad hoc assumptions about the form of the coupling\nbetween bending and twist, we will use standard methods of the theory of\nelasticity in order to derive the correct form of the coupling.\n\nIn this work we will consider cylindrical rods with circular\ncross--sections. Although, at first sight, this case appears to be simpler\nthan that of rods with asymmetric cross--sections, the reverse is true:\nwhile in the asymmetric case one can introduce a triad of vectors associated\nwith the principal axes of inertia, which can rotate at a different rate\nthan the Frenet triad, no such natural choice is possible in the symmetric\ncase which therefore requires a more careful analysis.\n\nThe exposition is organized as follows. Section 2 deals with geometry of\ndeformation. The strain energy density of a rod is introduced in Section 3.\nStress--strain relations are developed in Section 4. In Section 5, force and\nmoment balance equations which describe the mechanical equilibrium of thin\nrods are derived. Several examples which illustrate the different aspects of\nthe interaction between elongation, torsion and twist, are discussed in\nSection 6. Finally, in Section 7 we discuss the connection between our\nresults and other theoretical and experimental works and outline directions\nfor future research.\n\n\\section{Geometry of deformation}\n\nA long chain is modeled as an elastic rod with length $L$ and a circular\ncross-section ${\\cal S}$ with radius $a\\ll L$. Denote by $\\xi $ the\narc--length of the centerline of the rod in the reference (stress-free)\nconfiguration. Let ${\\bf R}_{0}(\\xi )$ be the radius vector of the\nlongitudinal axis and ${\\bf t}_{0}(\\xi )=d{\\bf R}_{0}/d\\xi $ the unit\ntangent vector in the reference state. The unit normal vector ${\\bf n}%\n_{0}(\\xi )$ and the unit binormal vector ${\\bf b}_{0}(\\xi )$ are introduced\nby the conventional way. These vectors obey the Frenet--Serret equations\nwith given spontaneous curvature $\\kappa _{0}(\\xi )$ and torsion $\\omega\n_{0}(\\xi )$: \n\\begin{equation}\n\\frac{d{\\bf t}_{0}}{d\\xi }=\\kappa _{0}{\\bf n}_{0},\\qquad \\frac{d{\\bf n}_{0}}{%\nd\\xi }=\\omega _{0}{\\bf b}_{0}-\\kappa _{0}{\\bf t}_{0},\\qquad \\frac{d{\\bf b}%\n_{0}}{d\\xi }=-\\omega _{0}{\\bf n}_{0}.\n\\end{equation}\nPoints of the rod refer to Lagrangian coordinates $\\{\\xi _{i}\\}$, where $\\xi\n_{1},\\xi _{2}$ are Cartesian coordinates in the cross-sectional plane with\nunit vectors ${\\bf n}_{0}$ and ${\\bf b}_{0}$ and $\\xi _{3}=\\xi $, \n\\begin{equation}\n{\\bf r}_{0}(\\xi _{1},\\xi _{2},\\xi )={\\bf R}_{0}(\\xi )+\\xi _{1}{\\bf n}%\n_{0}(\\xi )+\\xi _{2}{\\bf b}_{0}(\\xi ).\n\\end{equation}\nIt follows from Eqs. (2) and (3) that the covariant base vectors in the\nreference configuration, ${\\bf g}_{0k}=\\partial {\\bf r}_{0}/\\partial \\xi\n_{k} $ are given by \n\\begin{equation}\n{\\bf g}_{01}={\\bf n}_{0},\\qquad {\\bf g}_{02}={\\bf b}_{0},\\qquad {\\bf g}%\n_{03}=(1-\\kappa _{0}\\xi _{1}){\\bf t}_{0}+\\omega _{0}(\\xi _{1}{\\bf b}_{0}-\\xi\n_{2}{\\bf n}_{0}).\n\\end{equation}\nThe position of the longitudinal axis of the rod in the actual (deformed)\nconfiguration is determined by the radius vector ${\\bf R}(\\xi )$. Following\nthe conventional theories of rods, see, e.g., \\cite{BZ79}, we assume that\nthe longitudinal axis is inextensible, which means that $\\xi $ remains the\narc--length in the actual configuration (for attempts to account for the\nextensibility of the longitudinal axis, see \\cite{Odi95,KLN97,Mar97,Mar98}).\nThe unit tangent vector in the actual configuration ${\\bf t}=d{\\bf R}/d\\xi $\ntogether with the unit normal vector ${\\bf n}$ and the unit binormal vector $%\n{\\bf b}$ satisfy the Frenet--Serret equations \n\\begin{equation}\n\\frac{d{\\bf t}}{d\\xi }=\\kappa {\\bf n},\\qquad \\frac{d{\\bf n}}{d\\xi }=\\omega \n{\\bf b}-\\kappa {\\bf t},\\qquad \\frac{d{\\bf b}}{d\\xi }=-\\omega {\\bf n}.\n\\end{equation}\nFor Kirchhoff rods \\cite{GT97}, the radius vector of an arbitrary point is\nrepresented as an expansion in the coordinates $\\xi _{1}$ and $\\xi _{2}$: \n\\begin{equation}\n{\\bf r}(\\xi _{1},\\xi _{2},\\xi )={\\bf R}(\\xi )+\\xi _{1}{\\bf n}(\\xi )+\\xi _{2}%\n{\\bf b}(\\xi ).\n\\end{equation}\nThe functional form of Eq. (6) implies that any cross-section remains planar\nand perpendicular to the centerline of the rod, even in the actual deformed\nconfiguration. Furthermore, it also implies that any cross--section rotates\nrigidly with the longitudinal axis and therefore Eq. (6) does not allow for\nthe possibility of a twist of the cross--section with respect to the\ncenterline of the rod. Since the latter assumption has no physical basis \n\\cite{GL92}, we relax it by introducing a more general displacement field \n\\begin{equation}\n{\\bf r}(\\xi _{1},\\xi _{2},\\xi )={\\bf R}(\\xi )+(\\xi _{1}\\cos \\alpha -\\xi\n_{2}\\sin \\alpha ){\\bf n}(\\xi )+(\\xi _{1}\\sin \\alpha +\\xi _{2}\\cos \\alpha )%\n{\\bf b}(\\xi ),\n\\end{equation}\nwhere $\\alpha (\\xi )$ is the rotation angle around the centerline of the\nrod. From here on we will refer to this rotation as ``twist'' and will\nreserve the terms ``torsion'' and ``writhe'' to describe the\nthree-dimensional geometry of bending of the centerline of the rod.\n\nThe covariant base vectors ${\\bf g}_{k}=\\partial {\\bf r}/\\partial \\xi _{k}$\nare given by \n\\begin{eqnarray}\n{\\bf g}_{1} &=&\\cos \\alpha {\\bf n}+\\sin \\alpha {\\bf b}, \\qquad {\\bf g}%\n_{2}=-\\sin \\alpha {\\bf n}+\\cos \\alpha {\\bf b}, \\nonumber \\\\\n{\\bf g}_{3} &=& \\Bigl [ 1 -\\kappa (\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha\n) \\Bigr]{\\bf t} \\nonumber \\\\\n&&+\\Bigl(\\omega +\\frac{d\\alpha }{d\\xi }\\Bigr) \\Bigl [ -(\\xi _{1}\\sin \\alpha\n+\\xi _{2}\\cos \\alpha ){\\bf n} +(\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha )%\n{\\bf b}\\Bigr].\n\\end{eqnarray}\nThe contravariant base vectors ${\\bf g}^{k}$ are found from Eq. (8) and the\nequality ${\\bf g}_{i}\\cdot {\\bf g}^{j}=\\delta _{i}^{j}$, where the dot\nstands for inner product and $\\delta _{i}^{j}$ is the Kronecker delta.\nSimple calculations result in \n\\begin{equation}\n{\\bf g}^{1}=\\cos \\alpha {\\bf n}+\\sin \\alpha {\\bf b}+C_{1}{\\bf t}, \\qquad \n{\\bf g}^{2}=-\\sin \\alpha {\\bf n}+\\cos \\alpha {\\bf b}+C_{2}{\\bf t}, \\qquad \n{\\bf g}^{3}=C_{3}{\\bf t},\n\\end{equation}\nwhere \n\\begin{eqnarray}\n&& A_{n}=-\\Bigl(\\frac{d\\alpha }{d\\xi } +\\omega \\Bigr)(\\xi _{1}\\sin \\alpha\n+\\xi _{2}\\cos \\alpha ), \\qquad A_{b}=\\Bigl(\\frac{d\\alpha }{d\\xi }+\\omega\n\\Bigr ) (\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha ), \\nonumber \\\\\n&& A_{t}=1-\\kappa (\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha ), \\qquad\nC_{1}=- \\frac{1}{A_{t}}(A_{n}\\cos \\alpha +A_{b}\\sin \\alpha ), \\nonumber \\\\\n&& C_{2}=\\frac{1}{A_{t}}(A_{n}\\sin \\alpha -A_{b}\\cos \\alpha ), \\qquad C_{3}= \n\\frac{1}{A_{t}}.\n\\end{eqnarray}\nOne can proceed to calculate the energy of deformation using the\ndisplacement gradient either in the deformed, $\\mbox{$\\boldmath \\nabla$}{\\bf %\nr}_{0}$, or in the reference, $\\mbox{$\\boldmath \\nabla$}_{0}{\\bf r}$, state.\nBoth approaches result in the same expression for the mechanical energy. We\nwill use the displacement gradient in the actual configuration $%\n\\mbox{$\\boldmath \\nabla$}{\\bf r}_{0}$, because the corresponding strain\ntensor is connected with the stress tensor (always defined with respect to\nthe coordinates in the deformed state) by conventional constitutive\nequations. It follows from Eqs. (4) and (9) that the tensor $%\n\\mbox{$\\boldmath \\nabla$}{\\bf r}_{0}={\\bf g}^{k}{\\bf g}_{0k}$ is given by \n\\begin{eqnarray}\n\\mbox{$\\boldmath \\nabla$}{\\bf r}_{0} &=&\\cos \\alpha ({\\bf n}{\\bf n}_{0}+{\\bf %\nb}{\\bf b}_{0}) +\\sin \\alpha ({\\bf b}{\\bf n}_{0}-{\\bf n}{\\bf b}_{0}) \n\\nonumber \\\\\n&& +(C_{1}-C_{3}\\omega _{0}\\xi _{2}){\\bf t}{\\bf n}_{0} +(C_{2}+C_{3}\\omega\n_{0}\\xi _{1}){\\bf t}{\\bf b}_{0} +C_{3}(1-\\kappa _{0}\\xi _{1}){\\bf t}{\\bf t}%\n_{0}.\n\\end{eqnarray}\nAs a measure of deformation, the Almansi tensor \\cite{Dro96} ${\\bf A}=%\n\\mbox{$\\boldmath \\nabla$}{\\bf r}_{0} \\cdot \\mbox{$\\boldmath \\nabla$}{\\bf r}%\n_{0}^{\\top }$ is employed, where $\\top $ stands for transpose. The tensor $%\n{\\bf A}$ is connected with the strain tensor $\\mbox{$\\boldmath \\epsilon$}$\nin the deformed state by the equality $\\mbox{$\\boldmath \\epsilon$}=\\frac{1}{2%\n}({\\bf I}-{\\bf A})$, where ${\\bf I}$ is the unit tensor. It follows from Eq.\n(11) that the non--zero components of $\\mbox{$\\boldmath \\epsilon$}$ are\ngiven by \n\\begin{eqnarray}\n\\epsilon _{13} &=&\\epsilon _{31}=-\\frac{1}{2} (\\xi _{1}\\sin \\alpha +\\xi\n_{2}\\cos \\alpha ) \\Bigl(\\frac{d\\alpha }{d\\xi } +\\omega -\\omega _{0}\\Bigr), \n\\nonumber \\\\\n\\epsilon _{23} &=&\\epsilon _{32} =\\frac{1}{2} (\\xi _{1}\\cos \\alpha -\\xi\n_{2}\\sin \\alpha ) \\Bigl(\\frac{d\\alpha }{d\\xi }+\\omega -\\omega _{0}\\Bigr), \n\\nonumber \\\\\n\\epsilon _{33} &=&-\\kappa (\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha )\n+\\kappa_{0}\\xi _{1},\n\\end{eqnarray}\nwhere we kept only terms up to first order in $\\xi _{1}$ and $\\xi _{2}$. The\nneglect of second and higher order terms follows from the standard small\nlocal deformation assumption, which implies that all the length scales\nassociated with bending, torsion and twist (e.g., radii of curvature) are\nmuch larger than the diameter of the rod. Note that this approximation is\nconsistent with the form of the displacement field, Eqs. (3) and (7), where\nonly terms up to linear order in the transverse coordinated $\\xi _{1}$ and $%\n\\xi _{2}$ were kept.\n\n\\section{Strain energy density}\n\nFor a linear anisotropic elastic medium, the mechanical energy of elongation\nper unit volume in the deformed state is calculated as \n\\begin{equation}\nu_{{\\rm el}}=\\frac{1}{2}E_{1}\\epsilon _{33}^{2},\n\\end{equation}\nand the mechanical energy of shear is \n\\begin{equation}\nu_{{\\rm sh}}=E_{2}(\\epsilon _{13}^{2}+\\epsilon _{31}^{2}+\\epsilon\n_{23}^{2}+\\epsilon _{32}^{2}),\n\\end{equation}\nwhere $E_{1}$ and $E_{2}$ are the appropriate elastic moduli. It follows\nfrom Eqs. (12) to (14) that the mechanical energy density \n\\begin{equation}\nu=u_{{\\rm el}}+u_{{\\rm sh}}\n\\end{equation}\nis can be written as \n\\begin{equation}\nu =\\frac{1}{2}\\biggl \\{E_{1}\\biggl [\\kappa ^{2}(\\xi _{1}\\cos \\alpha -\\xi\n_{2}\\sin \\alpha )^{2}+\\kappa _{0}^{2}\\xi _{1}{}^{2}\\Bigr] +E_{2}(\\xi\n_{1}^{2}+\\xi _{2}^{2})\\Bigl(\\frac{d\\alpha }{d\\xi }+\\omega -\\omega\n_{0}\\Bigr)^{2}\\biggr \\}.\n\\end{equation}\nThe mechanical energy per unit length is given by \n\\[\nU=\\int_{{\\cal S}}ud\\xi _{1}d\\xi _{2}, \n\\]\nwhich yields, upon integration \n\\begin{equation}\nU=\\frac{1}{2}\\biggl [ A_{1} \\Bigl(\\kappa ^{2}-2\\kappa \\kappa _{0}\\cos \\alpha\n+\\kappa _{0}^{2}\\Bigr)+A_{2}\\Bigl(\\frac{d\\alpha }{d\\xi } +\\omega -\\omega\n_{0} \\Bigr)^{2}\\biggr ]\n\\end{equation}\nwith \n\\[\nA_{1}=E_{1}I, \\qquad A_{2}=2E_{2}I, \\qquad I=\\int_{{\\cal S}}\\xi\n_{1}^{2}d\\xi_{1}d\\xi _{2} =\\int_{{\\cal S}}\\xi _{2}^{2}d\\xi _{1}d\\xi _{2},\n\\qquad \\int_{{\\cal S}}\\xi _{1}\\xi _{2}d\\xi _{1}d\\xi _{2}=0. \n\\]\nComparison of Eqs. (1) and (17) shows that the two expressions coincide in\nthe absence of rotation of the cross--section with respect to the centerline\n(no twist, $\\alpha =0$). In the general case, when $\\alpha \\neq 0$, Eq. (17)\ndiffers from Eq. (1) in several important ways:\n\n\\begin{enumerate}\n\\item The torsion $\\omega $ is replaced by $\\omega +d\\alpha /d\\xi $. This\ncorrection has a simple intuitive meaning: the rotation of a point on the\nsurface of a rod is the sum of the rotation in space of the centerline of\nthe rod and of the twist of the cross--section about this centerline. Notice\nthat this correction may always be present, independent of whether the rod\nhas a non-vanishing spontaneous curvature ($\\kappa _{0}$) and spontaneous\ntorsion ($\\omega _{0}$) or not. Such a correction was, in fact, proposed by\nprevious investigators \\cite{GL92}.\n\n\\item The term $2\\kappa \\kappa _{0}$ is replaced by $2\\kappa \\kappa\n_{0}\\cos \\alpha $, introducing a non-trivial coupling between the\nspontaneous and the actual curvatures of the rod, and the twist of its\ncross--section with respect to the centerline. Note that this term appears\nonly when the rod has a non--vanishing spontaneous curvature and therefore\nwhile it has no effect on the elasticity of straight rods ($\\kappa _{0}=0$),\nit has a dramatic effect on the elasticity of helices and other curved ($%\n\\kappa _{0}\\neq 0$) rods.\n\n\\item The usual expression for the energy, Eq. (1), is minimized when the\ncurvature ($\\kappa $) and torsion ($\\omega $) recover their spontaneous\nvalues ($\\kappa _{0}$ and $\\omega _{0}$, respectively) in the stress--free\nreference state. Although this appears to be no longer true for our energy,\nEq. (17), the difference stems from the fact that we have introduced a new\nindependent variable ($\\alpha $) that describes the twist of the\ncross--section with respect to the centerline of the rod. In the absence of\nexternally applied torques and tensile forces, minimizing the energy with\nrespect to $\\kappa $, $\\omega $ and $\\alpha $ yields their values in the\nstress--free reference state, i.e., $\\kappa _{0}$, $\\omega _{0}$ and $\\alpha\n=0$, respectively.\n\\end{enumerate}\n\n\\section{Stress--strain relations}\n\nDenote by ${\\mbox{$\\boldmath \\sigma$}}$ the Cauchy stress tensor and by $%\n\\sigma ^{ij}$ its contravariant components in the basis of the actual\nconfiguration. Substitution of Eqs. (13) to (15) into the equality \n\\[\n\\sigma ^{ij}=\\frac{\\partial u}{\\partial \\epsilon _{ij}} \n\\]\nresults in \n\\begin{eqnarray}\n\\sigma ^{13} &=&\\sigma ^{31} =-E_{2}(\\xi _{1}\\sin \\alpha +\\xi _{2}\\cos\n\\alpha ) \\Bigl(\\frac{d\\alpha }{d\\xi }+\\omega -\\omega _{0}\\Bigr), \\nonumber\n\\\\\n\\sigma ^{23} &=&\\sigma ^{32} =E_{2}(\\xi _{1}\\cos \\alpha -\\xi _{2}\\cos \\alpha\n) \\Bigl(\\frac{d\\alpha }{d\\xi }+\\omega -\\omega _{0}\\Bigr), \\\\\n\\sigma ^{33} &=& E_{1}[-\\kappa (\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha\n)+\\kappa _{0}\\xi _{1}].\n\\end{eqnarray}\nEquation (19) does not take into account the inextensibility of the\nlongitudinal axis. In order to enforce this constraint, we add an unknown\nparameter $p$ (a Lagrange multiplier analogous to pressure for\nincompressible solids) to Eq. (19): \n\\begin{equation}\n\\sigma ^{33}=-p+E_{1}[-\\kappa (\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha\n)+\\kappa _{0}\\xi _{1}].\n\\end{equation}\nSince the unit normal to a cross-section of the rod coincides with ${\\bf t}$%\n, the internal force (per unit area) ${\\bf f}$ that acts on the\ncross--section of the rod is given by \n\\begin{equation}\n{\\bf f}={\\bf t}\\cdot \\mbox{$\\boldmath \\sigma$}=\\sigma ^{13}{\\bf n}+\\sigma\n^{23}{\\bf b}+\\sigma ^{33}{\\bf t}.\n\\end{equation}\nIt follows from Eq. (7) that the radius vector $\\mbox{$\\boldmath \\rho$}$\nfrom the center point of the cross--section (its intersection with the\ncenterline) to an arbitrary point of the cross-section is \n\\begin{equation}\n\\mbox{$\\boldmath \\rho$}=(\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha ){\\bf n}%\n+(\\xi _{1}\\sin \\alpha +\\xi _{2}\\cos \\alpha ){\\bf b}.\n\\end{equation}\nThe moment (per unit area) $\\mbox{$\\boldmath \\mu$}$ of the internal force\nwith respect to the center point\\ of the cross--section is defined as \n\\begin{equation}\n\\mbox{$\\boldmath \\mu$}=\\mbox{$\\boldmath \\rho$}\\times {\\bf f},\n\\end{equation}\nwhere $\\times $ stands for vector product. Combining Eqs. (21) to (23) and\nusing the equalities \n\\begin{equation}\n{\\bf t}\\times {\\bf n}={\\bf b},\\qquad {\\bf n}\\times {\\bf b}={\\bf t},\\qquad \n{\\bf b}\\times {\\bf t}={\\bf n},\n\\end{equation}\nwe find that \n\\begin{eqnarray}\n\\mbox{$\\boldmath \\mu$} &=&\\Bigl[(\\xi _{1}\\sin \\alpha +\\xi _{2}\\cos \\alpha )%\n{\\bf n}-(\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha ){\\bf b}\\Bigr]\\sigma ^{33}\n\\nonumber \\\\\n&&+\\Bigl[(\\xi _{1}\\cos \\alpha -\\xi _{2}\\sin \\alpha )\\sigma ^{23}-(\\xi\n_{1}\\sin \\alpha +\\xi _{2}\\cos \\alpha )\\sigma ^{13}\\Bigr]{\\bf t}.\n\\end{eqnarray}\nThe internal moment ${\\bf M}$ is obtained by integrating $%\n\\mbox{$\\boldmath\\mu$}$ over the cross--section of the rod, \n\\begin{equation}\n{\\bf M}=\\int_{{\\cal S}}\\mbox{$\\boldmath \\mu$}d\\xi _{1}d\\xi _{2}=M_{n}{\\bf n}%\n+M_{b}{\\bf b}+M_{t}{\\bf t}.\n\\end{equation}\n\nIn principle, one could proceed in similar fashion and obtain the internal\nforce \n\\begin{equation}\n{\\bf F=}F_{n}{\\bf n}+F_{b}{\\bf b}+F_{t}{\\bf t}\n\\end{equation}\nby integrating ${\\bf f}$ over the cross--section of the rod. However,\ninspection of Eqs. (18)--(21) shows that since our expression for ${\\bf f}$\nis linear in the transverse coordinates $\\xi _{1}$ and $\\xi _{2}$, the\nintegral over the cross--section vanishes. The source of the problem can be\ntraced back to our choice of the displacement fields, Eqs. (3) and (7),\nwhere only linear terms in the transverse coordinates $\\xi _{1}$ and $%\n\\xi_{2} $ were taken into account. Note, however, that even if we were to\nkeep higher order terms in $\\xi _{1}$ and $\\xi _{2}$ in these equations, the\nunknown function ${\\bf F}$ would be expressed in terms of new unknown\nfunctions (coefficients of quadratic contributions in $\\xi _{1}$ and $%\n\\xi_{2} $ to the displacement fields). Instead, we will follow the standard\napproach and treat the vector ${\\bf F}$ as an additional unknown that is\nfound from the equilibrium equations (force and moment balance conditions).\n\nWe now proceed to calculate the internal moment by substituting expressions\n(18) and (21) into Eqs. (25) and (26). Upon integration we obtain the\nconstitutive relation between the parameters that characterize the\ndeformation ($\\kappa $, $\\omega $ and $\\alpha $) and the internal moment $%\n{\\bf M}$ \n\\begin{equation}\n{\\bf M}=A_{1}\\kappa _{0}\\sin \\alpha {\\bf n}+A_{1}(\\kappa -\\kappa _{0}\\cos\n\\alpha ){\\bf b}+A_{2}\\Bigl(\\omega +\\frac{d\\alpha }{d\\xi }-\\omega _{0}\\Bigr)%\n{\\bf t}.\n\\end{equation}\nEquation (28) is a new expression for the moment of internal forces which\naccounts for the twist of the cross-section with respect to the centerline\nof the rod. As expected, the internal moment vanishes in the stress--free\nreference state: $\\kappa =\\kappa _{0}$, $\\omega =\\omega _{0}$, $\\alpha =0$.\nIn the absence of twist, $\\alpha =0$, Eq. (28) reduces to the conventional\nexpression for Kirchhoff rods \n\\begin{equation}\n{\\bf M}=A_{1}(\\kappa -\\kappa _{0}){\\bf b} +A_{2}\\Bigl(\\omega -\\omega\n_{0}\\Bigr ){\\bf t}.\n\\end{equation}\n\n\\section{Equilibrium equations}\n\nConsider an element of the rod bounded by two cross-sections with\nlongitudinal coordinates $\\xi $ and $\\xi +d\\xi $. Forces acting on this\nelement consist of the internal force $-{\\bf F}(\\xi )$ applied to the\ncross-section $\\xi $, the internal force ${\\bf F}(\\xi +d\\xi )$ applied to\nthe cross-section $\\xi +d\\xi $, and the external force ${\\bf q}d\\xi $\nproportional to the length of the element $d\\xi $. Balancing the forces on\nthe element yields \n\\begin{equation}\n{\\bf F}(\\xi +d\\xi )-{\\bf F}(\\xi )+{\\bf q}(\\xi )d\\xi =0.\n\\end{equation}\nExpanding the vector function \n\\[\n{\\bf F}(\\xi +d\\xi )=F_{n}(\\xi +d\\xi ){\\bf n}(\\xi +d\\xi )+F_{b}(\\xi +d\\xi )%\n{\\bf b}(\\xi +d\\xi )+F_{t}(\\xi +d\\xi ){\\bf t}(\\xi +d\\xi ) \n\\]\ninto the Taylor series, using Eq. (5), and neglecting terms of second order\nin $d\\xi $, we find that \n\\begin{equation}\n{\\bf F}(\\xi +d\\xi )-{\\bf F}(\\xi )=\\biggl [\\Bigl(\\frac{dF_{n}}{d\\xi }+\\kappa\nF_{t}-\\omega F_{b}\\Bigr){\\bf n}+\\Bigl(\\frac{dF_{b}}{d\\xi }+\\omega F_{n}\\Bigr)%\n{\\bf b}+\\Bigl(\\frac{dF_{t}}{d\\xi }-\\kappa F_{n}\\Bigr){\\bf t}\\biggr ]d\\xi .\n\\end{equation}\nSubstitution of Eq. (31) into Eq. (30) results in the equilibrium equations \n\\begin{eqnarray}\n&&\\frac{dF_{n}}{d\\xi }+\\kappa F_{t}-\\omega F_{b}+q_{n}=0,\\qquad \\frac{dF_{b}%\n}{d\\xi }+\\omega F_{n}+q_{b}=0, \\\\\n&&\\frac{dF_{t}}{d\\xi }-\\kappa F_{n}+q_{t}=0.\n\\end{eqnarray}\nwhere $q_{n}$, $q_{b}$, and $q_{t}\\ $are the components of the external\nforce per unit length, ${\\bf q}=q_{n}{\\bf n}+q_{b}{\\bf b}+q_{t}{\\bf t}$.\n\nThe moments acting on the element of the rod consist of the internal moment $%\n-{\\bf M}(\\xi )$ applied to the cross-section $\\xi $, the internal moment $%\n{\\bf M}(\\xi +d\\xi )$ applied to the cross-section $\\xi +d\\xi $, the moments\nof internal forces $-{\\bf F}(\\xi )$ and ${\\bf F}(\\xi +d\\xi )$, and the\nexternal moment ${\\bf m}d\\xi $ proportional to the length $d\\xi $, where $%\n{\\bf m}=m_{n}{\\bf n}+m_{b}{\\bf b}+m_{t}{\\bf t}$ is the external moment per\nunit length. To first order in $d\\xi $, the moment of internal forces \n%acting on the cross-section with coordinate $\\xi+d\\xi$\nwith respect to the center of the cross-section with coordinate $\\xi $ is \n\\[\n\\Bigl[{\\bf R}(\\xi +d\\xi )-{\\bf R}(\\xi )\\Bigr]\\times {\\bf F}(\\xi +d\\xi )={\\bf %\nt}(\\xi )\\times {\\bf F}(\\xi )d\\xi . \n\\]\nThe balance equation for the moments reads \n\\begin{equation}\n{\\bf M}(\\xi +d\\xi )-{\\bf M}(\\xi )+{\\bf t}(\\xi )\\times {\\bf F}(\\xi )d\\xi +%\n{\\bf m}(\\xi )d\\xi =0.\n\\end{equation}\nIt follows from Eqs. (24) and (26) that ${\\bf t}\\times {\\bf F}=-F_{b}{\\bf n}%\n+F_{n}{\\bf b}$. By analogy with Eq. (31), one can write \n\\[\n{\\bf M}(\\xi +d\\xi )-{\\bf M}(\\xi )=\\biggl [\n\\Bigl(\\frac{dM_{n}}{d\\xi }+\\kappa M_{t}-\\omega M_{b}\\Bigr){\\bf n}+\\Bigl(%\n\\frac{dM_{b}}{d\\xi }+\\omega M_{n}\\Bigr){\\bf b}+\\Bigl(\\frac{dM_{t}}{d\\xi }%\n-\\kappa M_{n}\\Bigr){\\bf t}\\biggr ]d\\xi . \n\\]\nSubstitution of these expressions into Eq. (34) results in the equations \n\\begin{eqnarray}\n&&\\frac{dM_{n}}{d\\xi }+\\kappa M_{t}-\\omega M_{b}-F_{b}+m_{n}=0,\\qquad \\frac{%\ndM_{b}}{d\\xi }+\\omega M_{n}+F_{n}+m_{b}=0, \\\\\n&&\\frac{dM_{t}}{d\\xi }-\\kappa M_{n}+m_{t}=0.\n\\end{eqnarray}\nGiven the vectors ${\\bf M}$ and ${\\bf m}$, Eqs. (35) can be used to\ndetermine the forces $F_{n}$ and $F_{b}$. Eliminating the unknown functions $%\nF_{n}$ and $F_{b}$ from Eqs. (32) and (35), we obtain \n\\begin{eqnarray}\n&&\\frac{dF_{t}}{d\\xi }+\\kappa \\biggl (\\frac{dM_{b}}{d\\xi }+\\omega M_{n}+m_{b}%\n\\biggr )+q_{t}=0, \\\\\n&&\\frac{d}{d\\xi }\\biggl (\\frac{dM_{b}}{d\\xi }+\\omega M_{n}+m_{b}\\biggr )%\n+\\omega \\biggl (\\frac{dM_{n}}{d\\xi }+\\kappa M_{t}-\\omega M_{b}+m_{n}\\biggr )%\n-\\kappa F_{t}-q_{n}=0, \\nonumber \\\\\n&&\\frac{d}{d\\xi }\\biggl (\\frac{dM_{n}}{d\\xi }+\\kappa M_{t}-\\omega M_{b}+m_{n}%\n\\biggr )-\\omega \\biggl (\\frac{dM_{b}}{d\\xi }+\\omega M_{n}+m_{b}\\biggr )%\n+q_{b}=0.\n\\end{eqnarray}\nEquations (36) to (38) together with constitutive relation (28) are a set of\nfour nonlinear differential equations which determine the four unknown\nfunctions $F_{t}$, $\\alpha $, $\\kappa $ and $\\omega $. The neglect of $%\n\\alpha $ (that is the use of conventional formula (6) instead of Eq. (7) for\nthe displacement field ${\\bf r}$) is acceptable only when special\nrestrictions are imposed on external forces and moments. In the general\ncase, this simplification is not correct, and Eq. (7) should be employed for\nthe analysis of deformations.\n\n\\section{Examples}\n\n\\subsection{Twist of a closed loop}\n\nConsider a rod whose stress-free shape is a planar circular loop with radius \n$a_{0}$, under the action of a constant twisting moment $m_{t}$. It is\nassumed that the moments $m_{n}$ and $m_{b}$, as well as the forces $q_{n}$, \n$q_{b}$ and $q_{t}$ vanish. The solution of Eqs. (36) to (38) reads \n\\begin{eqnarray}\n&&\\kappa =\\kappa _{0}=a_{0}^{-1}, \\\\\n&&\\omega =\\omega _{0}=0,\\qquad F_{t}=0,\\qquad \\alpha =\\arcsin \\frac{%\nm_{t}a_{0}^{2}}{A_{1}}.\n\\end{eqnarray}\nAccording to these equalities, any cross-section of the rod twists around\nits centerline by a constant angle $\\alpha $. This solution is not described\nby the Kirchhoff theory of thin rods. It exists as long as the moment $m_{t}$\nsatisfies the condition $|m_{t}|\\leq A_{1}a_{0}^{-2}$. If the latter\nrestriction is not fulfilled, the planar shape of the loop becomes unstable.\n\n\\subsection{Torsion of a disconnected ring}\n\nWe analyze the deformation of a disconnected ring (no contact between the\npoints $\\xi =0$ and $\\xi =L$). The end $\\xi =0$ is fixed, and a torque $T$\nis applied to the free end $\\xi =L$. The centerline of the rod in the\nstress--free reference state describes a planar circle with radius $%\na_{0}=\\kappa _{0}^{-1}$ and no spontaneous torsion, $\\omega _{0}=0$. Similar\nproblems were recently studied in \\cite{GL92,GT97,HZ99,WT86} and their\nsolutions were applied to the analysis of kink transitions in short DNA\nrings.\n\nWe assume that in the deformed state the centerline becomes a non-planar\ncurve whose radius of curvature remains unchanged, see Eq. (39). For\nsimplicity, we confine ourselves to small displacements, and neglect terms\nof order $\\alpha ^{2}$ in the constitutive equations (28). This yields \n\\begin{equation}\nM_{n}=A_{1}\\kappa _{0}\\alpha ,\\qquad M_{b}=0,\\qquad M_{t}=A_{2}\\Bigl(\\frac{%\nd\\alpha }{d\\xi }+\\omega \\Bigr).\n\\end{equation}\nSubstitution of these expressions into the equilibrium equations (36) to\n(38) implies that the longitudinal force $F_{t}$ vanishes, whereas the\nfunctions $\\alpha $ and $\\omega $ obey the equations \n\\begin{equation}\nA_{2}\\Bigl(\\frac{d^{2}\\alpha }{d\\xi ^{2}}+\\frac{d\\omega }{d\\xi }%\n\\Bigr)-A_{1}\\kappa _{0}^{2}\\alpha =0,\\qquad \\frac{d}{d\\xi }\\biggl [A_{1}%\n\\frac{d\\alpha }{d\\xi }+A_{2}\\Bigl(\\frac{d\\alpha }{d\\xi }+\\omega \\Bigr)\\biggr\n]=0.\n\\end{equation}\nIt follows from the second equality in Eq. (42) that \n\\begin{equation}\n(A_{1}+A_{2})\\frac{d\\alpha }{d\\xi }+A_{2}\\omega =c,\n\\end{equation}\nwhere $c$ is a constant to be found. Excluding $\\omega $ from Eqs. (42) and\n(43), we obtain \n\\begin{equation}\n\\frac{d^{2}\\alpha }{d\\xi ^{2}}+\\kappa _{0}^{2}\\alpha =0.\n\\end{equation}\nThe solution of Eq. (44) is given by \n\\begin{equation}\n\\alpha =c_{1}\\sin \\kappa _{0}\\xi +c_{2}\\cos \\kappa _{0}\\xi ,\n\\end{equation}\nwhere $c_{1}$ and $c_{2}$ are arbitrary constants. Substitution of Eqs. (43)\nand (45) into the boundary conditions at the clamped end $\\xi =0$ \n\\[\n\\alpha (0)=0,\\qquad \\omega (0)=0 \n\\]\nimplies that \n\\begin{equation}\n\\alpha =c_{1}\\sin \\kappa _{0}\\xi ,\\qquad \\omega =\\frac{A_{1}+A_{2}}{A_{2}}%\n\\kappa _{0}c_{1}(1-\\cos \\kappa _{0}\\xi ).\n\\end{equation}\nEquating the moment $M_{t}$ at the end $\\xi =L$ to the external torque $T$\nand using Eqs. (41) and (46), we obtain \n\\[\nc_{1}=\\frac{T}{A_{2}\\kappa _{0}}, \n\\]\nwhich results in the formulas \n\\begin{equation}\n\\alpha =\\frac{T}{A_{2}\\kappa _{0}}\\sin (\\kappa _{0}\\xi ),\\qquad \\omega =%\n\\frac{T(A_{1}+A_{2})}{A_{2}^{2}}\\Bigl(1-\\cos (\\kappa _{0}\\xi )\\Bigr).\n\\end{equation}\nEquations (41) and (47) provide an explicit solution to the torque problem,\nwhich cannot be obtained in the framework of the Kirchhoff theory of rods.\nWhen the radius of the ring tends to infinity, i.e. for a prismatic rod, Eq.\n(47) implies that \n\\begin{equation}\n\\alpha =\\frac{T}{A_{2}}\\xi ,\\qquad \\omega =0.\n\\end{equation}\nIn this limit, the solution (48) coincides with the classical displacement\nfield for the twist of a circular cylinder \\cite{Dro96}.\n\n\\subsection{Helix under tension and torque}\n\nA helix--shaped rod whose stress--free reference state is characterized by\nspontaneous curvature $\\kappa _{0}$ and torsion $\\omega _{0}$, is deformed\nby tensile forces $P$ and torques $T$ applied to its ends. All other forces $%\n{\\bf q}$ and moments ${\\bf m}$ are assumed to vanish. We introduce Cartesian\ncoordinates $\\{x_{k}\\}$ with unit vectors ${\\bf e}_{k}$ and describe the\nconfiguration of the centerline of the rod in the stress--free reference\nstate by the vector \n\\begin{equation}\n{\\bf R}_{0}=a_{0}\\cos \\frac{\\xi }{\\sqrt{a_{0}^{2}+b_{0}^{2}}}{\\bf e}%\n_{1}+a_{0}\\sin \\frac{\\xi }{\\sqrt{a_{0}^{2}+b_{0}^{2}}}{\\bf e}_{2}+\\frac{%\nb_{0}\\xi }{\\sqrt{a_{0}^{2}+b_{0}^{2}}}{\\bf e}_{3}.\n\\end{equation}\nThe parameters $a_{0}$ and $b_{0}$ are expressed in terms of the spontaneous\ncurvature $\\kappa _{0}$ and torsion $\\omega _{0}$ by the formulas \n\\begin{equation}\n\\kappa _{0}=\\frac{a_{0}}{a_{0}^{2}+b_{0}^{2}},\\qquad \\omega _{0}=\\frac{b_{0}%\n}{a_{0}^{2}+b_{0}^{2}}.\n\\end{equation}\n\n\\subsubsection{Fixed force and torque on ends}\n\nConsider a rod whose centerline describes one complete turn of a helix (the\nangle between tangent vectors at the two ends of the undeformed rod equals $%\n2\\pi ).$ The contour length of the rod is \n\\begin{equation}\nl=2\\pi (\\kappa _{0}^{2}+\\omega _{0}^{2})^{-\\frac{1}{2}}.\n\\end{equation}\nWe assume the following boundary conditions at the ends of the rod: \n\\begin{eqnarray}\n&&M_{n}(0)=M_{n}(l)=0,\\qquad M_{b}(0)=M_{b}(l)=0, \\nonumber \\\\\n&&M_{t}(0)=M_{t}(l)=T,\\qquad F_{t}(0)=F_{t}(l)=P.\n\\end{eqnarray}\nEquations (52) imply that the torque $T$ and the tensile force $P$ are the\nonly external loads applied to the segment. Assuming the parameters $P$ and $%\nT$ to be rather small and neglecting the deviation of torsion from its value\nin the stress--free state, we look for a solution of the equilibrium\nequations in the form \n\\begin{equation}\n\\alpha =\\Delta \\alpha ,\\qquad \\kappa =\\kappa _{0}+\\Delta \\kappa ,\\qquad\n\\omega =\\omega _{0},\n\\end{equation}\nwhere $\\Delta \\alpha $ is small compared to unity, and $\\Delta \\kappa $ is\nsmall compared to $\\kappa _{0}$.\n\nNeglecting terms of the second order in the perturbations of twist angle and\ncurvature ($\\Delta \\alpha \\ $and $\\Delta \\kappa $, respectively), we find\nfrom Eq. (28) that \n\\begin{equation}\nM_{n}=A_{1}\\kappa _{0}\\Delta \\alpha ,\\qquad M_{b}=A_{1}\\Delta \\kappa ,\\qquad\nM_{t}=A_{2}\\frac{d\\Delta \\alpha }{d\\xi }.\n\\end{equation}\nWe substitute expressions (53) and (54) into Eqs. (36) to (38), neglect\nterms of the second order in $\\Delta \\alpha \\ $and $\\Delta \\kappa $, and\narrive at the equations \n\\begin{eqnarray}\n&&\\frac{dM_{t}}{d\\xi }-\\kappa _{0}M_{n}=0, \\\\\n&&\\frac{dF_{t}}{d\\xi }+\\kappa _{0}\\Bigl(\\frac{dM_{b}}{d\\xi }+\\omega\n_{0}M_{n}\\Bigr)=0, \\\\\n&&\\frac{d^{2}M_{b}}{d\\xi ^{2}}+2\\omega _{0}\\frac{dM_{n}}{d\\xi }+\\kappa\n_{0}\\omega _{0}M_{t}-\\omega _{0}^{2}M_{b}-\\kappa _{0}F_{t}=0, \\\\\n&&\\frac{d^{2}M_{n}}{d\\xi ^{2}}+\\kappa _{0}\\frac{dM_{t}}{d\\xi }-2\\omega _{0}%\n\\frac{dM_{b}}{d\\xi }-\\omega _{0}^{2}M_{n}=0,\n\\end{eqnarray}\nwhere the longitudinal force $F_{t}$ is assumed to be small as well. It\nfollows from Eqs. (55) and (58) that \n\\begin{equation}\n\\frac{dM_{b}}{d\\xi }=\\frac{1}{2\\omega _{0}}\\biggl [\\frac{d^{2}M_{n}}{d\\xi\n^{2}}+(\\kappa _{0}^{2}-\\omega _{0}^{2})M_{n}\\biggr ].\n\\end{equation}\nSubstitution of Eq. (59) into Eq. (56) results in \n\\begin{equation}\n\\frac{dF_{t}}{d\\xi }+\\frac{\\kappa _{0}}{2\\omega _{0}}\\biggl [\\frac{d^{2}M_{n}%\n}{d\\xi ^{2}}+(\\kappa _{0}^{2}+\\omega _{0}^{2})M_{n}\\biggr ]=0.\n\\end{equation}\nEquations (55), (57) and (59) imply that \n\\begin{eqnarray}\n\\frac{dF_{t}}{d\\xi } &=&\\frac{1}{\\kappa _{0}}\\biggl (\\frac{d^{3}M_{b}}{d\\xi\n^{3}}+2\\omega _{0}\\frac{d^{2}M_{n}}{d\\xi ^{2}}+\\kappa _{0}\\omega _{0}\\frac{%\ndM_{t}}{d\\xi }-\\omega _{0}^{2}\\frac{dM_{b}}{d\\xi }\\biggr ) \\nonumber \\\\\n&=&\\frac{1}{2\\kappa _{0}\\omega _{0}}\\biggl [\\frac{d^{4}M_{n}}{d\\xi ^{4}}%\n+(\\kappa _{0}^{2}+2\\omega _{0}^{2})\\frac{d^{2}M_{n}}{d\\xi ^{2}}+\\omega\n_{0}^{2}(\\kappa _{0}^{2}+\\omega _{0}^{2})M_{n}\\biggr ].\n\\end{eqnarray}\nExcluding the function $F_{t}$ from Eqs. (60) and (61), we obtain a closed\nequation for the internal moment $M_{n}$ \n\\begin{equation}\n\\frac{d^{4}M_{n}}{d\\xi ^{4}}+2(\\kappa _{0}^{2}+\\omega _{0}^{2})\\frac{%\nd^{2}M_{n}}{d\\xi ^{2}}+(\\kappa _{0}^{2}+\\omega _{0}^{2})^{2}M_{n}=0.\n\\end{equation}\nThe solution of Eq. (62) reads \n\\begin{equation}\nM_{n}=(c_{1}+c_{1}^{\\prime }\\xi )\\sin \\Bigl(\\sqrt{\\kappa _{0}^{2}+\\omega\n_{0}^{2}}\\xi \\Bigr)+(c_{2}+c_{2}^{\\prime }\\xi )\\cos \\Bigl(\\sqrt{\\kappa\n_{0}^{2}+\\omega _{0}^{2}}\\xi \\Bigr),\n\\end{equation}\nwhere $c_{k}$, $c_{k}^{\\prime }$ are constants to be found. It follows from\nthe boundary conditions (52) for the function $M_{n}$ and Eq. (63) that \n\\begin{equation}\nc_{2}=c_{2}^{\\prime }=0.\n\\end{equation}\nIntegrating Eq. (59) from 0 to $l$ and using boundary conditions (52) for\nthe function $M_{b}$, we obtain \n\\[\n\\int_{0}^{l}\\biggl [\\frac{d^{2}M_{n}}{d\\xi ^{2}}+(\\kappa _{0}^{2}-\\omega\n_{0}^{2})M_{n}\\biggr ]d\\xi =0. \n\\]\nSubstitution of expressions (63) and (64) into this equality results in \n\\begin{equation}\nc_{1}^{\\prime }=0.\n\\end{equation}\nCombining Eqs. (54) and (63) to (65), we find that \n\\begin{equation}\n\\Delta \\alpha (\\xi )=\\frac{c_{1}}{A_{1}\\kappa _{0}}\\sin \\Bigl(\\sqrt{\\kappa\n_{0}^{2}+\\omega _{0}^{2}}\\xi \\Bigr),\n\\end{equation}\nNote that although the twist angle vanishes at the ends and in the middle of\nthe rod ($\\Delta \\alpha (0)=\\Delta \\alpha (l)=\\Delta \\alpha (l/2)=0$), it\ndoes not vanish elsewhere. Differentiating Eq. (66) and using Eq. (54) and\nthe boundary conditions (52) for $M_{t}$, we arrive at the equality \n\\begin{equation}\nc_{1}=\\frac{A_{1}T\\kappa _{0}}{A_{2}\\sqrt{\\kappa _{0}^{2}+\\omega _{0}^{2}}}.\n\\end{equation}\nSubstitution of Eqs. (64), (65) and (67) into Eqs. (54), (63) and (66)\nimplies that \n\\begin{equation}\nM_{n}=\\frac{A_{1}T\\kappa _{0}}{A_{2}\\sqrt{\\kappa _{0}^{2}+\\omega _{0}^{2}}}%\n\\sin \\Bigl(\\sqrt{\\kappa _{0}^{2}+\\omega _{0}^{2}}\\xi \\Bigr),\\qquad\nM_{t}=T\\cos \\Bigl(\\sqrt{\\kappa _{0}^{2}+\\omega _{0}^{2}}\\xi \\Bigr).\n\\end{equation}\nIt follows from Eqs. (59) and (68) that \n\\[\n\\frac{dM_{b}}{d\\xi }=-\\frac{A_{1}T\\kappa _{0}\\omega _{0}}{A_{2}\\sqrt{\\kappa\n_{0}^{2}+\\omega _{0}^{2}}}\\sin \\Bigl(\\sqrt{\\kappa _{0}^{2}+\\omega _{0}^{2}}%\n\\xi \\Bigr). \n\\]\nIntegrating this equality with the boundary conditions (52) and substituting\nin Eq. (54) yields \n\\begin{equation}\n\\Delta \\kappa (\\xi )=\\frac{M_{b}}{A_{1}}=-\\frac{T\\kappa _{0}\\omega _{0}}{%\nA_{2}(\\kappa _{0}^{2}+\\omega _{0}^{2})}\\biggl [1-\\cos \\Bigl(\\sqrt{\\kappa\n_{0}^{2}+\\omega _{0}^{2}}\\xi \\Bigr)\\biggr ]\n\\end{equation}\nNote that the sign of $\\Delta \\kappa $ vanishes at the ends of the rod;\ninside it, its sign is opposite to that of the torque $T$ (positive torque\nmeans overtwisting). Substitution of Eqs. (68) and (69) into Eq. (57) gives\nthe internal tensile force \n\\begin{equation}\nF_{t}=T\\omega _{0}\\biggl [\\biggl (1+\\frac{A_{1}}{A_{2}}\\Bigl(1+\\frac{\\omega\n_{0}^{2}}{\\kappa _{0}^{2}+\\omega _{0}^{2}}\\Bigr)\\biggr )\\cos \\Bigl(\\sqrt{%\n\\kappa _{0}^{2}+\\omega _{0}^{2}}\\xi \\Bigr)-\\frac{A_{1}\\omega _{0}^{2}}{%\nA_{2}(\\kappa _{0}^{2}+\\omega _{0}^{2})}\\biggr ].\n\\end{equation}\nIt follows from Eq. (70) that our solution $\\kappa (\\xi )$, $\\omega $ and $%\n\\alpha (\\xi )$ under boundary conditions (52) is valid if the tensile force $%\nP$ and the torque $T$ applied to the ends of the rod satisfy the relation \n\\begin{equation}\nP=T\\omega _{0}\\Bigl(1+\\frac{A_{1}}{A_{2}}\\Bigr).\n\\end{equation}\nEquations (68) to (70) provide an explicit solution to the problem of\ncombined tension and torsion of a helical segment. The main results are as\nfollows:\n\n\\begin{enumerate}\n\\item The application of positive torque $T$ at the ends (overtwist) leads\nto axial compression of the helix which is maximal at the center and\nvanishes at the ends of the rod;\n\n\\item The ratio of the tensile force $P$ and the torque $T$ is independent\nof the initial curvature $\\kappa _{0}$ (and, therefore, of the length of the\nrod) and depends only on the initial torsion $\\omega _{0}$ and the ratio of\nelastic moduli $A_{1}/A_{2}=E_{1}/(2E_{2})$;\n\n\\item The force $P$ is proportional to the torque $T.$ This result is\nmarkedly different from that obtained for a similar deformation of a\ncircular incompressible cylinder, where $P$ can be shown to be proportional\nto $T^{2}$ (the Poynting effect\\cite{Dro96}).\n\\end{enumerate}\n\nNote that our solution corresponds to a helical rod (with constant $\\kappa\n_{0}$ and $\\omega _{0}$) which, upon application of external forces and\ntorques, is deformed into a new, non-helical shape. It is natural to ask\nunder which boundary conditions a helix will deform into another helix (with\nconstant $\\kappa $ and $\\omega $), and derive the corresponding\nforce--elongation relation. This is done in the following.\n\n\\subsubsection{Elongation and winding of a helix}\n\nConsider a helix made of an arbitrary number of repetitive units ${\\cal L}%\n_{0}$ (${\\cal L}_{0}$ is the smallest segment for which the angle between\ntangent vectors at its ends is $2\\pi $) such that its stress--free reference\nstate is characterized by the parameters $\\kappa _{0}$ and $\\omega _{0}$. We\nallow deformations that satisfy the following conditions: (i) the rod\nbecomes a helix with constant curvature $\\kappa $ and constant torsion $%\n\\omega $, and (ii) the twist $\\alpha $ vanishes. Under the action of\ncombined tensile force $P$ and torque $T$, any repetitive unit ${\\cal L}_{0}$\nof the helix in the reference state is transformed into an element with the\nangle between tangent vectors at the ends $2\\pi (1+\\varphi )$, where the\nangle $2\\pi \\varphi $ can take positive or negative values. The radius\nvector of the centerline of the rod in the deformed state can be written in\nthe form \n\\begin{equation}\n{\\bf R}=a\\cos (S\\xi ){\\bf e}_{1}+a\\sin (S\\xi ){\\bf e}_{2}+S_{1}\\xi {\\bf e}%\n_{3},\n\\end{equation}\nwhere $a$, $S$ and $S_{1}$ are constants which will be calculated in the\nfollowing. Differentiating Eq. (72) with respect to $\\xi $ and bearing in\nmind that $|{\\bf t}|=1$, we obtain \n\\begin{equation}\na^{2}S^{2}+S_{1}^{2}=1.\n\\end{equation}\nAccording to the definition of $\\varphi $, \n\\begin{equation}\nSl=2\\pi (1+\\varphi ).\n\\end{equation}\nThe projected distances (along the $x_{3}$-axis) between the ends of the\nrepetitive unit in the reference and deformed states are \n\\begin{equation}\n\\Pi _{0}=2\\pi b_{0},\\qquad \\Pi =S_{1}l,\n\\end{equation}\nrespectively. The axial elongation $\\eta $ is defined as the ratio of these\ndistances, \n\\begin{equation}\n\\eta =\\frac{\\Pi }{\\Pi _{0}}=\\frac{S_{1}l}{2\\pi b_{0}}=\\frac{l}{2\\pi b_{0}}%\n\\sqrt{1-a^{2}S^{2}}.\n\\end{equation}\nSimple calculations result in the formulas \n\\begin{equation}\n\\kappa =aS^{2},\\qquad \\omega =S\\sqrt{1-a^{2}S^{2}}.\n\\end{equation}\nIt follows from Eq. (72) that the projection of the force $F_{t}$ on the\naxis $x_{3}$ is $F_{t}{\\bf t}\\cdot {\\bf e}_{3}=F_{t}S_{1}$. Equating this\nexpression to the tensile force $P$ and using Eq. (73), we arrive at the\nrelation \n\\begin{equation}\nF_{t}=\\frac{P}{\\sqrt{1-a^{2}S^{2}}},\n\\end{equation}\nwhich means that $F_{t}$ is independent of $\\xi $. Equation (28) implies\nthat components of the moment ${\\bf M}$ are independent of $\\xi $ as well, \n\\begin{equation}\nM_{n}=0,\\qquad M_{b}=A_{1}(\\kappa -\\kappa _{0}),\\qquad M_{t}=A_{2}(\\omega\n-\\omega _{0}).\n\\end{equation}\nThe only equilibrium equation reads \n\\[\n\\omega (\\kappa M_{t}-\\omega M_{b})=\\kappa F_{t}. \n\\]\nSubstitution of expressions (78) and (79) into this equality yields \n\\begin{equation}\n\\omega \\Bigl[A_{2}\\kappa (\\omega -\\omega _{0})-A_{1}\\omega (\\kappa -\\kappa\n_{0})\\Bigr]=\\frac{\\kappa P}{\\sqrt{1-a^{2}S^{2}}}.\n\\end{equation}\nExcluding the parameters $a$, $S$, $\\kappa $ and $\\omega $ from Eqs. (73),\n(74), (76), (77) and (80), we express the tensile force $P$ in terms of the\naxial elongation of the helix $\\eta $: \n\\begin{equation}\nP_{0}=\\frac{\\lambda (1+\\varphi )\\eta ^{2}}{\\sqrt{1+\\lambda ^{2}}}\\biggl [%\n(1+\\varphi )\\eta -1-A\\eta \\Bigl((1+\\varphi )-\\frac{1}{\\sqrt{1+\\lambda\n^{2}(1-\\eta ^{2})}}\\Bigr)\\biggr ],\n\\end{equation}\nwhere \n\\[\nA=\\frac{A_{1}}{A_{2}},\\qquad \\lambda =\\frac{\\omega _{0}}{\\kappa _{0}},\\qquad\nP_{0}=\\frac{P}{A_{2}\\omega _{0}^{2}}. \n\\]\n\nComparing Eq. (52) with Eqs. (78) and (79) we find that the only difference\nbetween the boundary conditions for the two problems is that in the former\ncase we have neglected the moment $M_{b}$. The fact that a minor change of\nboundary conditions can drastically change the character of deformation is\nquite remarkable and indicates that these conditions should be chosen with\ncare.\n\nSince various variants of the theory of elastic rods were applied to\ninterpret the experimental force--elongation curves for stretched DNA\nmolecules at large deformations, we will present plots of some of the\nresults of this section and comment on their qualitative features. The graph \n$P_{0}=P_{0}(\\eta )$ for extension without torsion, $\\varphi =0$, is plotted\nin Figure~1. In the calculation we used $A=0.67$, in agreement with\nconventional data on DNA $\\tilde{A}_{1}=50$ nm, $\\tilde{A}_{2}=75$ nm \\cite\n{Mar98}, where $\\tilde{A}_{k}=A_{k}/(k_{B}T)$, $k_{B}$ is Boltzmann's\nconstant and $T$ is temperature. No detailed comparison with experiment is\nattempted here, but Figure~1 captures rather well the qualitative features\nof the experimental data for DNA molecules \\cite{SFB92,SCB96}.\n\nIn order to check whether our theory captures the qualitative features of\nexperimental data on the elasticity of supercoiled DNA, the dependence $\\eta\n=\\eta (\\varphi )$ is depicted in Figure~2 for various tensile forces $P_{0}$%\n. This figure also shows qualitative agreement with observations on the DNA\nchains \\cite{MN98,SAB96}: for small tensile forces, there is pronounced\nasymmetry with regard to the sign of $\\varphi$, but the $\\eta =\\eta (\\varphi\n)$ curve becomes nearly flat at large tensile forces. Throughout the\nparameter range, the elongation decreases nearly linearly with degree of\nsupercoiling. All these features were observed experimentally and were\ninterpreted as a proof for the existence of a new type of twist--stretch\ncoupling\\cite{KLN97,Mar97}. Note, however, that in the analysis that led to\nFigure 2 we assumed that the deformation of the helix takes place with no\ntwist of its cross--section around the centerline of the rod ($\\alpha =0$).\nTherefore, our solution can be derived using the standard theory of elastic\nrods, based on the elastic energy of Eq. (1), in which no such coupling\nappears. Inspection of the derivation of Eq. (81) leads to the conclusion\nthat the strong dependence of elongation on the degree of supercoiling has a\nsimple physical meaning: when an inextensible helical rod is subjected to\ntorque that produces supercoiling, each new turn has non-vanishing\nprojection on the $x_{1}-x_{2}$ plane and the projection of the deformed\nhelix on the $x_{3}$ axis (i.e., its elongation) decreases as the result.\nThis can be fully described by Eq. (1) and does not require the introduction\nof new coupling into the mechanical energy of elastic rods.\n\n\\section{Concluding remarks}\n\nIn this work we have extended the theory of elasticity of thin inextensible\nrods beyond that of three--dimensional space curves which can be completely\ndescribed by local curvature $\\kappa $ and geometric torsion $\\omega $. We\nhave shown that in order to describe the displacement of a point in a rod of\narbitrarily small but non--vanishing thickness, one has to account for\ndeformations that produce a rotation of the cross--section of the rod about\nits centerline. The modified displacement field was then used to calculate\nthe strain tensor. The resulting expression for the mechanical energy of\nrods with non--vanishing spontaneous curvature contains a new coupling term\nbetween the curvature of the rod and the twist of its cross--section with\nrespect to the centerline, which does not appear in any of the previous\ntheories. We derived the complete set of non-linear differential equations\nwhich describe the conditions of mechanical equilibrium and which can be\nsolved for the parameters of deformation $\\kappa $, $\\omega $ and $\\alpha $\nfor arbitrary external forces and moments acting on the rod. In order to\nillustrate the physical consequences of our theory, we proceeded to analyze\nseveral illustrative examples. In particular, we have analyzed the\ndeformation of a helical rod subjected to a combination of tension and\ntorque and showed that the theory captures the qualitative features of the\nrecent observations on the connection between supercoiling and elongation of\nstrongly stretched DNA molecules.\n\nNote that we have described the deformation of thin rods by three\nindependent functions $\\alpha $, $\\kappa $ and $\\omega $. This is\nreminiscent of the conventional approach\\cite{GT97} where the deformation is\ndescribed in terms of the three components of the, so called, ``twist''\nvector, $\\kappa _{1},$ $\\kappa _{2}$ and $\\kappa _{3}$. Although this was\nnot mentioned by the above authors, such an approach goes beyond the purely\ngeometric description of an elastic line in which only two functions are\nnecessary\\cite{SH94} \\ and describes a line with some ``internal\nstructure''. With each point of this line one can associate a ``physical''\ntriad of vectors that differs, in general, from the ``geometric'' (Frenet)\ntriad. While the two triads have one common vector (the tangent to the\nline), the other two pairs of vectors rotate at different rates as one moves\nalong the line contour and therefore the rotation of the physical triad can\nnot be completely described by the two Frenet parameters $\\kappa $ and $%\n\\omega $. It is important to realize that the introduction of a physical\ntriad is necessary whenever some asymmetry of the cross--section, either\ngeometric or physical\\cite{LGK98}, is present. However, even though the\nprocedure is not unique, one may also introduce the physical triad by hand\neven for a rod with a circular cross--section. For example, we may draw a\nline on the surface of a rod which describes the intersection of the normal\nvector with this surface. When the rod is deformed, the deformation of this\nline will, in general, be different from that of the centerline. We can now\nconnect the corresponding points of the two lines (having the same contour\nparameter $\\xi $) \\ and define the resulting vector as one of the vectors of\nthe physical triad. The remaining vector is then defined as the normal to\nthe plane formed by the above vector and the tangent to the centerline. This\nprocedure is completely equivalent to what we have done here, by introducing\nthe rotation $\\alpha (\\xi )$ and explains the appearance of an $\\alpha -$%\ndependent term ($\\kappa $ $\\kappa _{0}\\cos \\alpha $) in the expression for\nthe mechanical energy, that couples the curvatures in the stress--free and\nthe deformed states of the rod. Note that while for rods with asymmetric\ncross--sections, three independent parameters are needed in order to\ncharacterize the stress--free reference state, only two such parameters\n(e.g., $\\kappa _{0}$ and $\\omega _{0}$) are necessary in the degenerate case\nof rods with circular cross--sections.\n\nThere are several possible directions in which the work presented here can\nbe extended. For example, throughout this work we assumed that the\nconditions of mechanical equilibrium can be satisfied and considered only\nstable configurations of the deformed rods. However, the introduction of a\nnew type of deformations is expected to have a profound effect on various\ninstabilities (e.g., buckling under torsion and twist, plectoneme formation,\netc.) and we are now studying these questions. Another direction for future\nresearch involves the extension of the present, purely mechanical, analysis\nto include the effects of thermal fluctuations. This leads naturally to a\nnew class of physical models for rigid biopolymers and protein assemblies\nwhich can account for the spontaneous curvature\\cite{TTH87} of these objects.\n\n\\acknowledgments\nWe would like to thank M. Elbaum and D. Kessler for helpful discussions and\nsuggestions. AD gratefully acknowledges financial support by the Israeli\nMinistry of Science through grant 1202--1--98. 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Curve~1: $%\nP_{0}=0.1$; curve~2: $P_{0}=1.0$; curve~3: $P_{0}=10.0$}\n\\end{figure}\n\\newpage\n\n\\setcounter{figure}{0}\n\\setlength{\\unitlength}{1.0 mm}\n\\begin{figure}[t]\n\\begin{center}\n\\begin{picture}(100,100)\n\\put(0,0){\\framebox(100,100)}\n\\multiput(10,0)(10,0){9}{\\line(0,1){2}}\n\\multiput(0,10)(0,10){9}{\\line(1,0){2}}\n\\put(0,-7){1.0}\n\\put(96,-7){2.0}\n\\put(70,-7){$\\eta$}\n\\put(-10,0){0.0}\n\\put(-10,97){10.0}\n\\put(-10,70){$P_{0}$}\n\\put(102,43){1}\n\\put(90,102){2}\n\\put(71,102){3}\n\n\\put( 0.40, 0.02){\\circle*{0.7}} \n\\put( 0.80, 0.04){\\circle*{0.7}} \n\\put( 1.20, 0.06){\\circle*{0.7}} \n\\put( 1.60, 0.09){\\circle*{0.7}} \n\\put( 2.00, 0.11){\\circle*{0.7}} \n\\put( 2.40, 0.13){\\circle*{0.7}} \n\\put( 2.80, 0.16){\\circle*{0.7}} \n\\put( 3.20, 0.18){\\circle*{0.7}} \n\\put( 3.60, 0.20){\\circle*{0.7}} \n\\put( 4.00, 0.23){\\circle*{0.7}} \n\\put( 4.40, 0.25){\\circle*{0.7}} \n\\put( 4.80, 0.28){\\circle*{0.7}} \n\\put( 5.20, 0.30){\\circle*{0.7}} 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\n\\put( 95.50, 87.85){\\circle*{0.7}} \n\\put( 96.00, 87.80){\\circle*{0.7}} \n\\put( 96.50, 87.75){\\circle*{0.7}} \n\\put( 97.00, 87.75){\\circle*{0.7}} \n\\put( 97.50, 87.70){\\circle*{0.7}} \n\\put( 98.00, 87.65){\\circle*{0.7}} \n\\put( 98.50, 87.60){\\circle*{0.7}} \n\\put( 99.00, 87.60){\\circle*{0.7}} \n\\put( 99.50, 87.55){\\circle*{0.7}} \n\\put( 100.00, 87.50){\\circle*{0.7}} \n\n\\end{picture}\n\\end{center}\n\\vspace*{20 mm}\n\n\\caption{The axial elongation $\\eta$ versus the\novertwist $\\varphi$ for a helix with $A=0.67$ and $\\lambda=0.6$.\nCurve~1: $P_{0}=0.1$;\ncurve~2: $P_{0}=1.0$;\ncurve~3: $P_{0}=10.0$}\n\\end{figure}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002004.extracted_bib",
"string": "\\bibitem{HH91} J.E. Hearst and N.G. Hunt, {\\it J. Chem. Phys.} {\\bf 95,}\n9322 (1991).\n\n\n\\bibitem{GL92} E. Guitter and S. Leibler, {\\it Europhys. Lett.} {\\bf 17,}\n643 (1992).\n\n\n\\bibitem{MS94} J.F. Marko and E.D. Siggia, {\\it Macromolecules} {\\bf 27,}\n981 (1994).\n\n\n\\bibitem{Vol94} A. Vologodskii, {\\it Macromolecules} {\\bf 27,} 5623 (1994).\n\n\n\\bibitem{MKJ95} F.C. MacKintosh, J. K\\\"{a}s and P.A. Janmey, {\\it Phys.\nRev. Lett.} {\\bf 75,} 4425 (1995).\n\n\n\\bibitem{MS95} J.F. Marko and E.D. Siggia, {\\it Phys. Rev. E} {\\bf 52,}\n2912 (1995).\n\n\n\\bibitem{Odi95} T. Odijk, {\\it Macromolecules} {\\bf 28,} 7016 (1995).\n\n\n\\bibitem{Elb} D.K. Fygenson, M. Elbaum, B. Shraiman and A. Libchaber, {\\it %\nPhys. Rev. E} {\\bf 55,} 850 (1997).\n\n\n\\bibitem{GT97} A. Goriely and M. Tabor, {\\it Physica D} {\\bf 105,} 20\n(1997).\n\n\n\\bibitem{KLN97} R.D. Kamien, T.C. Lubensky, P. Nelson, and C.S. O'Hern, \n{\\it Europhys. Lett.} {\\bf 38,} 237 (1997).\n\n\n\\bibitem{Mar97} J.F. Marko, {\\it Europhys. Lett.} {\\bf 38,} 183 (1997).\n\n\n\\bibitem{BM98} C. Bouchiat and M. Mezard, {\\it Phys. Rev. Lett.} {\\bf 80,}\n1556 (1998).\n\n\n\\bibitem{GPW98} R.E. Goldstein, T.R. Powers, and C.H. Wiggins, {\\it Phys.\nRev. Lett.} {\\bf 80,} 5232 (1998).\n\n\n\\bibitem{LGK98} T.B. Liverpool, R. Golestanian and K. Kremer, {\\it Phys.\nRev. Lett.} {\\bf 80,} 405 (1998).\n\n\n\\bibitem{Mar98} J.F. Marko, {\\it Phys. Rev. E} {\\bf 57,} 2134 (1998).\n\n\n\\bibitem{ajdari} R. Everaers, F. Julicher, A. Ajdari and A.C. Maggs, {\\it %\nPhys. Rev. Lett.} {\\bf 82,} 3717 (1999).\n\n\n\\bibitem{BMS99} A. Balaeff, L. Mahadevan, and K. Schulten, {\\it Phys. Rev.\nLett.} {\\bf 83,} 4900 (1999).\n\n\n\\bibitem{HR99} L. Harnau and P. Reineker, {\\it Phys. Rev. E} {\\bf 60,} 4671\n(1999).\n\n\n\\bibitem{Nel99} P. Nelson, {\\it Proc. Natl. Acad. Sci.} {\\bf 16}, 14342\n(1999).\n\n\n\\bibitem{SH94} Y. Shi and J.E. Hearst, {\\it J. Chem. Phys.} {\\bf 101,} 5186\n(1994).\n\n\n\\bibitem{LL80} L.D. Landau and E.M. Lifshitz, {\\it Theory of Elasticity}\n(Pergamon Press, Oxford, 1986).\n\n\n\\bibitem{HZ99} Z. Haijun and O.-Y. Zhong-can, {\\it J. Chem. Phys.} {\\bf 110,%\n} 1247 (1999).\n\n\n\\bibitem{MN98} J.D. Moroz and P. Nelson, {\\it Macromolecules} {\\bf 31,}\n6333 (1998).\n\n\n\\bibitem{SAB96} T.R. Strick, J.-F. Allemand, D. Bensimon and V. Croquette, \n{\\it Science} {\\bf 271,} 1835 (1996).\n\n\n\\bibitem{BZ79} M.D. Barkley and B.H. Zimm, {\\it J. Chem. Phys.} {\\bf 70,}\n2991 (1979).\n\n\n\\bibitem{Dro96} A.D. Drozdov, {\\it Finite Elasticity and Viscoelasticity}\n(World Scientific, Singapore, 1996).\n\n\n\\bibitem{WT86} M. Wadati and H. Tsuru, {\\it Physica D} {\\bf 21,} 213 (1986).\n\n\n\\bibitem{SFB92} S.B. Smith, L. Finzi and C. Bustamante, {\\it Science} {\\bf %\n258,} 1122 (1992).\n\n\n\\bibitem{SCB96} S.B. Smith, Y. Cui, Y. and C. Bustamante, {\\it Science} \n{\\bf 271,} 795 (1996).\n\n\n\\bibitem{TTH87} E.N. Trifonov, R.K.-Z. Tan and S.C. Harvey, in {\\it DNA\nBending and Curvature}, W.K. Olson, M.H. Sarma and M. Sundaralingam eds.\n(Adenine Press, Schenectady, 1987), p. 243.\n"
}
] |
cond-mat0002005
|
Violation of ensemble equivalence in the antiferromagnetic mean-field XY model
|
[
{
"author": "Peter Holdsworth\\inst{1}"
},
{
"author": "Stefano Ruffo\\inst{1,2}\\thanks{INFM and INFN, Firenze (Italy)}"
}
] |
It is well known that long-range interactions pose serious problems for the formulation of statistical mechanics. We show in this paper that ensemble equivalence is violated in a simple mean-field model of $N$ fully coupled classical rotators with repulsive interaction (antiferromagnetic XY model). While in the {canonical} ensemble the rotators are randomly dispersed over all angles, in the {microcanonical} ensemble a bi-cluster of rotators separated by angle $\pi$, forms in the low energy limit. We attribute this behavior to the extreme degeneracy of the ground state: only one harmonic mode is present, together with $N-1$ zero modes. We obtain empirically an analytical formula for the probability density function for the angle made by the rotator, which compares extremely well with numerical data and should become exact in the zero energy limit. At low energy, in the presence of the bi-cluster, an extensive amount of energy is located in the single harmonic mode, with the result that the energy temperature relation is modified. Although still linear, $T = \alpha U$, it has the slope $\alpha \approx 1.3$, instead of the canonical value $\alpha =2$. % \PACS{ {05.20.-y,}{ Classical statistical mechanics } \and {05.20.Gg}{Classical ensemble theory} \and {05.45.-a}{Nonlinear dynamics and nonlinear dynamical systems} } % end of PACS codes
|
[
{
"name": "versionsoumise.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%% file template.tex %%%%%%%%%%%%%%%%%%%%%%%%%\n%\n% This is a template file for The European Physical Journal\n%\n% Copy it to a new file with a new name and use it as the basis\n% for your article\n%\n%%%%%%%%%%%%%%%%%%%%%%%% Springer-Verlag %%%%%%%%%%%%%%%%%%%%%%%%%%\n%\n\\begin{filecontents}{leer.eps}\n%!PS-Adobe-2.0 EPSF-2.0\n%%CreationDate: Mon Jul 13 16:51:17 1992\n%%DocumentFonts: (atend)\n%%Pages: 0 1\n%%BoundingBox: 72 31 601 342\n%%EndComments\n\ngsave\n72 31 moveto\n72 342 lineto\n601 342 lineto\n601 31 lineto\n72 31 lineto\nshowpage\ngrestore\n%%Trailer\n%%DocumentFonts: Helvetica\n\\end{filecontents}\n%\n%\\documentclass[epj,referee]{svjour}\n\\documentclass[epj]{svjour}\n% Remove option referee for final version\n%\n% Remove any % below to load the required packages\n%\\usepackage{latexsym}\n\\usepackage{graphics}\n%\\usepackage{/usr/local/TeX/texmf/tex/latex/localinputs/CRAL/psfig}\n%\\usepackage{/usr/local/TeX/texmf/tex/latex/localinputs/CRAL/psfig.tex}\n% etc\n%\n\\begin{document}\n\\title{Violation of ensemble equivalence in the \nantiferromagnetic mean-field XY model}\n\n\\author{\nThierry Dauxois\\inst{1}\\thanks{Thierry.Dauxois@ens-lyon.fr},\nPeter Holdsworth\\inst{1}\n\\and\nStefano Ruffo\\inst{1,2}\\thanks{INFM and INFN, Firenze (Italy)}\n} % Do not remove\n%\n%\\offprints{Thierry Dauxois} % Insert a name or remove this line\n%\n\\institute{Laboratoire de Physique, UMR-CNRS 5672,\nENS Lyon, 46 All\\'{e}e d'Italie, 69364 Lyon C\\'{e}dex 07, France\n\\and \nDipartimento di Energetica \"S. Stecco\",\nUniversit\\`a di Firenze, via S. Marta, 3, I-50139 Firenze, Italy}\n%\n\\date{Received: date / Revised version: \\today}\n% The correct dates will be entered by Springer\n%\n\\abstract{ It is well known that long-range interactions pose serious\n problems for the formulation of statistical mechanics. We show in\n this paper that ensemble equivalence is violated in a simple\n mean-field model of $N$ fully coupled classical rotators with\n repulsive interaction (antiferromagnetic XY model). While in the\n {\\it canonical} ensemble the rotators are randomly dispersed over\n all angles, in the {\\it microcanonical} ensemble a bi-cluster of\n rotators separated by angle $\\pi$, forms in the low energy limit. We\n attribute this behavior to the extreme degeneracy of the ground\n state: only one harmonic mode is present, together with $N-1$ zero\n modes. We obtain empirically an analytical formula for the\n probability density function for the angle made by the rotator,\n which compares extremely well with numerical data and should become\n exact in the zero energy limit. At low energy, in the presence of\n the bi-cluster, an extensive amount of energy is located in the\n single harmonic mode, with the result that the energy temperature\n relation is modified. Although still linear, $T = \\alpha U$, it has\n the slope $\\alpha \\approx 1.3$, instead of the canonical value\n $\\alpha =2$.\n%\n\\PACS{\n {05.20.-y,}{ Classical statistical mechanics } \\and\n {05.20.Gg}{Classical ensemble theory} \\and\n {05.45.-a}{Nonlinear dynamics and nonlinear dynamical systems}\n } % end of PACS codes\n} %end of abstract\n%\n%{\\bf Keywords}: \\\\\n%Statistical ensembles. Long-range interactions. \n%Mean-field models. Classical rotators. XY model.\n\\authorrunning{Dauxois, Holdsworth and Ruffo}\n\\titlerunning{Violation of ensemble equivalence in the \nantiferromagnetic mean-field XY model}\n\\maketitle\n%\n\n\n\\section{Introduction}\n\\label{Introduction}\n\nThe relation between microscopic dynamics and macroscopic\nthermodynamic behaviour can nowadays be explored in full detail in\ncomputer simulations~\\cite{Chaos}. This allows one to test important\nhypotheses of statistical mechanics. Formost among these is\nequivalence of ensembles.\n\nIt is widely accepted that the constant energy {\\it microcanonical}\nensemble gives the same results for average values as the constant\ntemperature {\\it canonical} ensemble. This is true under certain\nconditions~\\cite{Rue}, among them the most important is that\ninteractions must be short-ranged. If interactions are long-range and\nattractive, as for gravitating systems, all of thermodynamics breaks\ndown due to the non extensivity of thermodynamic\npotentials~\\cite{Pad90}. For systems of Coulomb charges of opposite\nsigns, screening effects may help in the construction of\nthermodynamics, but the problem is not trivially solved, even for\nclassical systems~\\cite{plasmas}, and each case must be examined\nseparately.\n\nIn this context, mean-field models occupy a special status. Here,\nthermodynamic potentials can be made extensive if the thermodynamic\nlimit is performed by rescaling the coupling with system size and\nletting the range of the interaction go to infinity~\\cite{Hem}.\nMean-field models are quite well studied in the {\\it canonical}\nensemble and serve as a zero-order characterization of phase\ntransitions. Although their solution is often trivial for systems\nwithout disorder, they may hide important subtleties for disordered\nand frustrated systems~\\cite{Par}. Far fewer studies of mean-field\nmodels exist in the {\\it microcanonical} ensemble. This may be the\nreason why, until now, ensemble equivalence has not been questioned\nfor these models. In fact, one might have already expected some\nsurprises on the basis of the exact solution of a model by Hertel and\nThirring~\\cite{Thi70}, where in the mean-field limit in the presence\nof extensive thermodynamic potentials ensemble inequivalence was\nexplicitly shown.\n\nIn this paper we present a detailed study of the low\ntemperature/energy phase of a model of classical rotators, whose\npotential energy is that of the mean-field antiferromagnetic XY model.\nThe ground state of this model is highly degenerate, and while in the\ncanonical ensemble equilibrium states are disordered at all\ntemperatures (i.e. rotators do not display any directional\norganization), microcanonical ensemble simulations show the presence\nof a ``bimodal\" state where rotators are mainly grouped in two\n``clusters\", pointing in directions separated by an angle $\\pi$. This\ndynamical effect also has thermodynamical consequences: the order\nparameter measuring the degree of clustering is non-zero in the limit\nof zero energy and the energy temperature relation is not that\npredicted by the canonical ensemble. We find that, although the\nenergy temperature relation $T=\\alpha U$ is linear, as in the\ncanonical case, the coefficient is $\\alpha \\approx 1.3$ and not the\ncanonical value $\\alpha=2$. From the point of view of dynamics, the\nrotators can be separated into two groups: a slow group which\noscillates around the bi-cluster and a group of almost freely rotating\nrotators, which we call a ``gas'', following an analogy with particle\nmotion.\n\nIn Section~\\ref{Model} we introduce the model. In Section~\\ref{Ground}\nwe discuss the unusual properties of its ground state and in the\nfollowing Section~\\ref{Statistical} we present the main controversial\npoints related to ensemble inequivalence. In Section~\\ref{Structural}\nwe present an analytical expression for the probability density\nfunction (PDF) for the orientation of a rotator, while the system is\nin the bi-cluster state, and we explain the consequences for the\nstatistical properties in the system in the microcanonical ensemble.\nIn Section~\\ref{Momentdy} we discuss the main features of the dynamics\nof the moments of the PDF. The paper ends with some conclusions and\nperspectives.\n\n\\section{The model}\n\\label{Model}\n\nWe consider classical rotators denoted by the angle $\\theta_i$, $i=1,\\dots,N$,\nwhich all interact with each other, with an antiferromagnetic coupling $1/N$ and an\nexternal field $h$\n\\begin{equation}\nV=\\frac{1}{2N} \\sum_{i,j} \\cos (\\theta_i-\\theta_j)- h \\sum_i \\sin \\theta_i~.\n\\label{potential}\n\\end{equation}\nWe will mostly restrict ourselves to $h=0$.\n\nAfter defining the complex order parameters (with $i$ the imaginary unit)\n\\begin{equation}\nM_k= \\frac{1}{N} \\sum_n \\exp (i k \\theta_n)=|M_k| \\exp (i \\psi_k)~,\n\\label{order}\n\\end{equation}\nthe potential can be rewritten as\n\\begin{equation}\nV= N \\left( \\frac{|M_1|^2}{2} - h |M_1| \\sin \\psi_1 \\right)\n\\label{newpot}\n\\end{equation}\nThe microcanonical ensemble is obtained by adding a kinetic energy term to the\nabove potential~\\cite{Kogut,Antoni}\n\\begin{equation}\nH=K+V~,\n\\end{equation}\nwith\n\\begin{equation}\nK=\\sum_{n=1}^N \\frac{p_n^2}{2}~.\n\\end{equation}\nIn this formulation the model can also be thought of as describing a system\nof particles with unitary mass, interacting through the mean-field\ncoupling $V$ (the names rotator and particle will be equivalently used\nthroughout the paper). This model has been first introduced in Ref.~\\cite{Kogut}\nfor 2D nearest-neighbour couplings and then studied in the antiferromagnetic\nmean-field context in Ref.~\\cite{Antoni}.\n\nThe total energy $E= U N$ ($U$ being the energy density) is fixed by\nthe initial conditions and is conserved in time, while temperature is\ndefined through the time averaged kinetic energy (see Ref.~\\cite{temp}\nfor alternative definitions of temperature), $T=2<K>/N$, where\n$<\\cdot>=\\displaystyle\\lim_{t \\to \\infty}\\displaystyle {1\\over t}\n\\int_0^t$. If $h=0$, as is usually the case, then the total momentum\n$P=\\displaystyle \\sum_n p_n$ is also conserved. We set $P=0$ in order\nto avoid ballistic center of mass motion when $h=0$.\n\nThe equations of motion\n\\begin{equation}\n\\ddot{\\theta}_n= |M_1| \\left[ \\sin (\\theta_n - \\psi_1) + h \\cos \\theta_n \\right]\n\\end{equation}\nhave been integrated using an improved fourth-order symplectic scheme~\\cite{Mac}.\nThe algorithm is ${\\cal O}(N)$, provided one first computes $M_1$ in the central loop.\nDuring the time evolution, we sample the instantaneous values of the order parameters\n(\\ref{order}) up to $k=20$ and we compute their running time averages.\nSystem size was varied from $N=100$ to $N=10^4$.\nWe have also performed canonical Monte-Carlo simulations, using the Metropolis algorithm,\nfor comparison.\n\nThe initial conditions were of two classes: {\\it i)} homogeneous state\n$\\theta_n=(2 \\pi n)/N$, which can be shown to be marginally stable\n(see below), to which we add either a small spatially random\nperturbation $\\theta_n \\to \\theta_n+ r_n$ and/or a small momentum\n$p_n=r_n$ with zero average; {\\it ii)} homogeneous state $\\theta_n =\n(2 \\pi n)/N$ with $p_n = A \\sin \\theta_n$, which leads to a faster\ninduction of the bimodal state we want to study. These initial\nconditions lead to the same bimodal state, discussed below, at low\nenergy.\n\nThe canonical solution of this model for $h=0$ is sketched in Ref.~\\cite{Antoni}. It uses\nthe Hubbard-Stratonovich trick to decouple the rotators and the thermodynamic limit\nis performed by a saddle-point technique. The result is that all moments (\\ref{order})\nvanish in the $N \\to \\infty$ limit, including the magnetization $M_1$, which\nimplies, on the one hand, that\n\\begin{equation}\nT=2U~,\n\\label{equi}\n\\end{equation}\non the other, that since in this limit\n\\begin{equation}\nM_k=\\int_0^{2\\pi} {\\cal P}(\\theta) \\exp (ik\\theta) d\\theta~,\n\\end{equation}\nthe PDF ${\\cal P} (\\theta)$ is flat. Hence, the bimodal state is absent in the canonical\nensemble, which we have confirmed by Monte-Carlo simulations.\n\nMost of the discussions below will concentrate on the reasons for the different\nfindings in the microcanonical and canonical statistical ensembles.\n \n\\section{Ground State and Statistical Properties in the Canonical Ensemble}\n\\label{Ground}\n\nThe long ranged interactions mean that it is impossible to satisfy all \nthe antiferromagnetic bonds at once and the model is highly frustrated.\nIn zero field, the frustration is minimized for configurations with \n$M_1 = 0$, giving a ground state energy of $U=E/N=0$. \n\n\nThe ground state is infinitely degenerate, as there is an \ninfinity of ways of minimizing the frustration. For example, grouping\nthe rotators into pairs with angles $\\theta_i$ and $\\theta_i +\\pi$\nensures that $M_1=0$ for all configurations of the pairs. The\npairs do not have to be arranged in an ordered way however, as \nany disordered arrangement will equally give \n$M_1=0$. Neither is the ground state manifold restricted to pairs: one\ncan equally construct ground states from groups of three, four, five ...\nrotators separated by angles of $2\\pi/3, 2\\pi/4, 2\\pi/5...$. By moving\nrotators in clusters whose total angle is zero the system can evolve from \none ground state to another remaining on the constant energy hypersurface.\n\nThe high dimensional ground state manifold follows from the fact that\nthe ground state condition requires the two constraints $M_{1x} =M_{1y}=0$\nonly.\nOne can therefore expect a ground state to possess $N-2$ unconstrained\ndegrees of freedom, which is easily confirmed by calculating\nthe Hessian\n$J_{i,j}=- \\partial^2 V/\\partial \\theta_i \\partial \\theta_j$. For example, \nfor the perfectly homogeneous ground state at $h=0$, with $\\theta_i=(2 \\pi i)/N$\n\\begin{equation}\nJ_{i,j}= - \\frac{1}{2N} \\cos \\left( \\frac{2 \\pi (i-j)}{N} \\right)~.\n\\label{Hess}\n\\end{equation}\nThe matrix indeed has two non-zero equal eigenvalues, $-1/4$ and $N-2$\nzero eigenvalues.\n\nCollective organization of the particles into reduced symmetry states\ncan reduce the number of constraints even further. For example, in\nour system, if the rotational symmetry is broken and the spins lie\nalong a single axis the ground state condition is reduced to a single\nconstraint $M_x=0$, leading, at the harmonic level to $N-1$ free\ndegrees of freedom~\\cite{Moessner}. This result is confirmed by\ncalculating the Hessian (\\ref{Hess}), on a perfect bi-cluster\ngroundstate with $N/2$ rotators at $\\theta = 0$ and $N/2$ at\n$\\theta=\\pi$. One finds only one non-zero eigenvalue, $-1/2$,\ncorresponding to the counter vibrating motion on the circle of the two\ngroups of particles; all the other eigenvalues are zero.\n\n\nThe model we study is an extreme case of a collective\nparamagnet~\\cite{Villain}, or classical spin\nliquid~\\cite{Moessner,CHS}; a system that remains disordered with no\nevidence of spin freezing down to the limit of zero temperature. The\nspecial points on the ground state manifold with a reduced number of\nconstraints can dominate the partition function, in the canonical\nensemble, leading to an ``Order by Disorder''\ntransition~\\cite{Villain2} to a reduced symmetry state. However, the\nmode counting arguments of Moessner and Chalker~\\cite{Moessner}\npredict this to happen only if the number of liberated modes at the\nspecial points exceeds the number of zero-modes in a state with full\nsymmetry. This is certainly not the case here and as we have confirmed\nby Monte Carlo simulation, Order by Disorder does not occur. We have\nsimulated between $N=10$ and $N=10^4$ rotators down to \ntemperatures $T < 10^{-4}$. At all temperatures, the system remains perfectly\ndisordered, with no evidence of bi-cluster formation. The number of\nzero modes can be directly verified by measuring the specific heat at\nconstant field, $C_h$, \nat low temperature, as each quadratic mode makes a contribution\n$1/2$ (in units of $k_B$), while each zero mode makes a contribution\nzero. We find $C_h = 1.0$, for the $N$ rotator system, as expected for\ntwo regular modes. There is therefore no evidence of the system\npreferring states with a single quadratic mode.\n\n\\section{Statistical Properties in the Microcanonical Ensemble}\n\\label{Statistical}\n\nThe system reserves a surprise, when studied in the microcanonical\nensemble, as we do not observe ensemble equivalence. For the classes\nof initial conditions cited above, the low energy state of the system\nis not one with a homogeneous distribution of angles, rather we\nobserve the formation of a bimodal structure, with enhanced\nprobability for two angles separated by a distance\n$\\pi$~\\cite{Antoni}. As the energy goes to zero, the asymptotically\nreached state has broken symmetry, but is not perfectly bi-modal, as\nwe discuss below. Nevertheless, when the Hessian for such a state\nis calculated numerically, we find that it also has only one non-zero\neigenvalue. It seems therefore that the formation of the bicluster in\nthe microcanonical ensemble, realises the condition of minimizing the\nnumber of non-zero modes.\n\n\nAfter a transient time $\\tau$, the formation of a stable bi-cluster is\nrevealed by a non-zero value of the second moment $|M_2|$ of ${\\cal P}\n(\\theta)$. The early time evolution of $|M_2|$ shows the typical\nbehavior observed for the growth of the mean-field in self-consistent\ndynamical models~\\cite{Tenny} with an initial exponential growth,\nfollowed by a saturation reached after a damped oscillatory motion has\ndied away, see Figs.~\\ref{dynamics}(a) and \\ref{dynamics}(b). In\nFigs.~\\ref{dynamics}(c) and \\ref{dynamics}(d) we show the space-time\nplot of ${\\cal P} (\\theta,t)$, which reveals the origin of the\noscillations in the density waves originated periodically from the\nbi-cluster, preceding stabilization. The darkest colours correspond\nto the highest densities. A very sharply peaked but unstable\nbi-cluster forms over a short time period. The structure disperses\nwith well defined instability edges that propagate from the cluster\ncenter and re-appears in a quasi-periodic manner with slowly\nlengthening period. The dispersing cluster appears to interact with\npropagating fronts from previous incarnations. The result is that the\ndispersion is successively slower and the fronts less well defined for\nfollowing quasi-periods until finally the bi-cluster stabilizes. In\naddition Fig.~\\ref{dynamics}(d) shows that once this coherent\nstructures has emerged and is stabilized, it propagates around the\ncircle with apparently ballistic dynamics.\n \n\n\\begin{figure}\n\\null\n\\vskip 1truecm\n\\resizebox{0.6\\textwidth}{!}{ \\includegraphics{dynamics.ps}}\n\\vskip 1truecm\n\\caption{(a) (resp. (b)) shows the instantaneous value of $|M_2|$ as\n a function of time\nat early (resp. later) times. (c) (resp. (d)) shows the evolution in\ngrey scales of the distribution ${\\cal P} (\\theta,t)$ at early\n(resp. later) times. The $\\theta$ axis is horizontal, the time axis\nvertical pointing downwards.}\n\\label{dynamics}\n\\end{figure}\n\n\nIn Fig.~\\ref{um2t}(a), we show $|M_2|$, averaged over times later than\n$\\tau$, as a function of temperature $T$, for various system sizes.\nFor low temperatures ($T<10^{-4}$), $|M_2|$ approaches the value\n$|M_2| \\sim 0.5$ and never decays, even in extremely long simulations\n(times as long as $10^8$ in proper units\\footnote{The appropriate\nlinear time scale of the system is $2 \\pi={\\cal O}(1)$}). On the contrary, in the\nhigh temperature regime ($T>10^{-2}$) the bi-cluster never forms,\n$|M_2| \\sim 0$ for all times and the PDF remains flat, as in Monte-Carlo\nsimulations. In the intermediate regime, $ 10^{-4}< T < 10^{-2}$, the\nbi-cluster forms but is less well-defined, corresponding\nto progressively smaller values of $|M_2|$. As the temperature\nis increased there is a smooth transition to the homogeneous state\nobserved in the canonical ensemble. In the next Section we will\nfurther discuss the internal structure of the bi-cluster.\n\n\nThe choice of initial conditions discussed above is dictated by the\nneed to progressively increase the energy starting from a ground\nstate. One may wonder what happens for more general initial\nconditions. \nWe therefore looked at statistics of the final configuration, starting from many different realizations of a uniformly random\ndistribution of angles and zero momenta. For $N=100$, at temperatures \naround $T \\sim 10^{-5}-10^{-4}$, the bimodal state was always reached\nover $500$ initial states. For $N=1000$ and $T \\sim 10^{-6}$,\nthe totality of 250 random initial states went to the\nbimodal state. It means that the bi-cluster is fully attractive for\nthis class of initial states, which is quite generic.\n\nAs the formation of the bi-cluster is a unique property of the\ndeterministic microcanonical system, it is an example of violation of\nthe equivalence hypothesis between microcanonical and canonical\nbehaviour. One might therefore expect this inequivalence to show up in\nother measurable quantities. This is indeed the case, for example we\nfind that the standard relationship between energy and temperature is\nmodified. In the canonical ensemble, as there are only two harmonic\nmodes, equipartition of energy at low temperature leads to the energy\ntemperature relation $E=NT/2+T$ ($N$ quadratic modes for the kinetic\nenergy and 2 for the potential energy), which implies that $T=2U+{\\cal\n O}(N^{-1})$, see (\\ref{equi}). The eventual Order by Disorder \naveraging over parts of\nthe phase space with only one quadratic modes (or any finite number of\nthem) produces only $1/N$ corrections to this relation. In the\nmicrocanonical simulation, in the presence of the bi-cluster we find\nthe anomalous relation $T \\sim 1.3U$ as shown in Fig.~\\ref{um2t}(b).\nOnce the temperature exceeds $T>10^{-2}$, the bi-cluster does not form\nand the energy-temperature relation crosses over to the canonical\nexpression. In the presence of the bi-cluster, the kinetic energy is\ntherefore much smaller than one should expect and the mean potential\nenergy far in excess of that predicted by equipartition for the single\nnon-zero mode associated with the bicluster. In fact, from this\nresult, we find that the configurational contribution to the heat\ncapacity $C_h \\approx <V>/T \\sim (0.35 N)/1.3$ is an extensive\nquantity, in complete contradiction to the predictions of mode\ncounting in the canonical ensemble.\n\nThe system manages to put a macroscopic amount of energy in the single\nnon-zero mode corresponding to the vibrating motion of the two groups\nof rotators and gives the system a low-dimensional aspect with the\nbi-cluster taking on many characteristics of a two-particle system.\nFluctuations in the magnetization $|M_1|$, whose mean value is\nidentically zero in the ground state illustrate this point. In zero\nfield the potential energy is $V = N|M_1|^2/2$, hence the extensive\nnature of the configurational heat capacity implies that mean value\nof the magnetization $<|M_1|>$ should be of order unity and not\nof order $1/\\sqrt{N}$ as one might expect in an uncorrelated\nparamagnetic system. We have indeed observed ordered features in the\ntime dependence of $M_1$ which are consistent with the\nnon-fluctuational value of this quantity.\n\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{ \\includegraphics{um2t.ps}}\n\\caption{(a) $|M_2|$ as a function of the temperature $T$ \n for $N=200$ (triangles), $N=500$ (circles), $N=10^3$\n (diamonds), $N=10^4$ (squares). (b) $T$ vs. $U$ for the same values\n of $N$. The solid line is the modified microcanonical relation $T =\n 1.3\\ U$, whereas the dashed one corresponds to the canonical\n relation $T=2\\ U$ and is valid in the high energy regime, where the\n cluster has disappeared ($|M_2|=0$).}\n\\label{um2t}\n\\end{figure}\n\n\n\\section{Structural and Dynamical Details of the Bi-Cluster}\n\\label{Structural}\n\nThe bi-cluster never becomes perfectly formed, with $|M_2| = 1$, even in \nthe limit $U \\to 0$. Rather, the angular distribution of rotators retains a width, with\na certain population of rotators homogeneously distributed. In Fig.~\\ref{histogram}\nwe show the angular PDF ${\\cal P}(\\theta)$ for \na system of $N=10^4$ particles and three different temperatures: one very low \n(Fig.~\\ref{histogram}(a)) and two in the transition region \n(Fig.~\\ref{histogram}b) and (c)). \nThe distribution is peaked for two angles, separated\nby $\\pi$, but there remains non-zero probability of finding angles between\nthe two peaks.\nThe degradation of the bicluster is progressive as the temperature is increased.\nWe have made the experimental observation that the moments of the PDF assume the\nfollowing values as the energy is decreased\n\\begin{eqnarray}\n<|M_k|>&\\approx&0 \\; \\mbox{for odd k} \\cr\n<|M_k|>&=&1/|k| \\; \\mbox{for even k with} \\, <|M_0|>=1~,\n\\label{moments}\n\\end{eqnarray}\nas shown in the insets of Fig.~\\ref{histogram} for the first $k=20$ modes. This observation\noffers no contradiction to our finding that $<|M_1|>$ is independent of system size. As the\nbi-cluster is stable at low temperatures only, the numerical value of $<|M_1|>$ remains small,\neven though it is an intensive quantity. By summing\nthe Fourier series for ${\\cal P}(\\theta)$,\n\\begin{equation}\n{\\cal P}(\\theta) = \\sum_{k=-\\infty}^{\\infty} <|M_k|> \\exp (-i k\\theta)~,\n\\end{equation} \none gets~\\cite{Grad}\n\\begin{equation}\n{\\cal P}(\\theta) = \\frac{1}{2\\pi} \\left( 1 - \\log (2 |\\sin \\theta| ) \\right)~.\n\\label{distr}\n\\end{equation} \nThis analytical formula is superimposed on to the numerical data in Fig.~\\ref{histogram}(a)\nwith no free parameter apart from a shift of $\\theta$, due to\nthe motion of the bi-cluster (see below). The agreement is impressive, and although\nwe have no theory for this result, we may well say we have a solution.\nWe expect the analytical formula (\\ref{distr}) to become exact in the $U \\to 0$\nlimit.\n\n\\begin{figure}\n\\resizebox{0.45\\textwidth}{!}{ \\includegraphics{histo.ps}}\n\\vskip 1truecm\n\\caption{${\\cal P} (\\theta)$ for $N=10^4$ and three different values of \n$T$: $1.46 \\cdot 10^{-5}$ (a), $2.6 \\cdot 10^{-4}$ (b), $2.8 \\cdot 10^{-3}$ (c). \nThe full line in (a) is the analytical formula\n(\\ref{distr}), the diamonds are the numerical histograms (in \n(b) we join them for convenience with a dashed line). In the insets we show the\ncorresponding momenta of the PDF (diamonds), the full line is $1/n$.\nIn (a) the moments respect the $1/n$ law for even $n$, while in (c)\nonly the mode $n=2$ is non-zero. Odd modes are much smaller.}\n\\label{histogram}\n\\end{figure}\n\nThe internal structure of the bi-cluster is further shown in\nFig.~\\ref{thetaimp}(a), where the state at time $t=2.5 \\cdot 10^5$,\nwith $N=10^4$ and $T=1.43 \\cdot 10^{-5}$, is displayed in the\n$(\\theta,p)$ single-particle phase-space (so-called Boltzmann\n$\\mu$-space). A large fraction of particles get stuck in the\nbi-cluster and have small values of $|\\theta|$, while others\ndevelop much larger $\\theta$ values. This illustrates why we speak\nof a ``gas-cluster'' coexistence: particles appear to belong to\ntwo distinct groups; those in the bi-cluster, which perform\noscillations around the two centers separated by angle $\\pi$, and those \nin the gas, which have ballistic dynamics and travel over large\ndistances. Once projected onto the $p$-axis the distribution is\nsymmetric (giving, for instance, zero average momentum), but as we see\nit in the $(\\theta,p)$ space it is skewed, with, on average, bigger\nmomenta for those particles which have traveled the furthest.\n \n\n\n\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{ \\includegraphics{thetaimp.ps}}\n\\caption{Single particle phase-space for $N=10^4$, $U=1.13 \\cdot 10^{-5}$\nand $t=2.5 \\cdot 10^{5}$. In (b) the angle $\\theta$ is folded onto $[0,2\\pi[$\nand the inset emphasizes the structure of the spirals arm around one of the bi-cluster center. }\n\\label{thetaimp}\n\\end{figure}\n\n\nThis picture is further clarified if we fold the $\\theta$-axis onto $[0, 2\\pi]$\n(Fig.~\\ref{thetaimp}b). The oscillating motion of the particles of the cluster is now\nrevealed by the spiral arms visible in the bi-cluster centers (see the inset). The particles in the cluster appear now to be distributed \nin a sinusoidal band of lesser density. \nIndeed, if we look inside the region around $\\theta=0$\n(Fig.~\\ref{serpent}a), we see that the particles are quite well distributed along the\nphase line \n\\begin{equation}\np=A \\sin (\\theta+\\phi)~,\n\\label{sin}\n\\end{equation} where $\\phi$ is a sliding phase related to the\nbi-cluster motion and $A$ an energy related amplitude. This gives us a hint \nas to how to discover the dynamical mechanism at work in the process of particle \nevaporation from the bi-cluster. In fact, differentiating (\\ref{sin}) with respect \nto time and using $\\dot{\\theta}=p$, one gets for $u=2\\theta$ the equation\nof motion\n\\begin{equation}\n\\ddot{u}= A^2 \\sin u~,\n\\label{invpend}\n\\end{equation}\nwhich describes a pendulum with gravity pointing upwards, $u=0$ being a saddle-point.\nThis system is integrable and cannot produce the phase-distribution in Fig.~\\ref{serpent}(a).\nWe must, therefore introduce some sort of perturbation capable of producing the sinusoidal\nlayer observed in Fig.~\\ref{thetaimp}(b). A well known mechanism for producing a stochastic\nlayer in pendulum motion is the introduction of a finite time step, as done\nin the Chirikov standard map. We have therefore decided to iterate the map\n\\begin{eqnarray}\nu'&=& u + \\Delta t\\ p_u \\cr\np'_u&=& p_u + A^2 \\Delta t\\ u'~,\n\\label{mod}\n\\end{eqnarray}\nwith $\\Delta t=1$, taking an ensemble of initial points homogeneously\ndistributed in a small square around the unstable point $u=p_u=0$. A\nsnapshot of these points at iteration $3000$ is shown in\nFig.~\\ref{serpent}(b); the particles have been stretched along the\nseparatrix layer and distributed along it in a similar way to that shown in \nFig.~\\ref{serpent}(a). Two features are responsible for the stretching:\nthe unstable manifold distributes the particles away from the saddles,\nwhile the stable manifold attracts them to the saddles. The\ninhomogeneous distribution of the points is a result of these two\nmechanisms. One feature of the evaporation in the full model, which is\nnot at all reproduced by the map (\\ref{mod}), is the fact that time\nevolution seems to select only one of the two lobes of the separatrix;\nin other words, the symmetry $p \\to -p$ seems to be broken in the full\nmodel, while stretching in both the lobes is present in the map\n(\\ref{mod}).\n\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{ \\includegraphics{serpent.ps}}\n\\caption{(a) Zoom of the central region of Fig.~\\ref{thetaimp}a, showing the phenomenological\nlaw $p=A \\sin (\\theta + \\phi)$ (\\ref{sin}) with $A=0.01$ and $\\phi=0.3$. (b) Phase-space of\n$10^4$ initial particles in the square $(u,p_u)=[-\\pi*0.0005,\\pi*0.0005]^2$\nof the toy model (\\ref{mod}) (with $\\Delta t=1$) at iteration $3000$. \n}\n\\label{serpent}\n\\end{figure}\n\n\nThis map is of course unable to explain why the bi-cluster forms and\nwhy it is stable. That is, if the evaporation mechanism were the only\nprocess present then the bi-cluster would be progressively depleted.\nThis is in contrast with numerical simulations for low temperature\nvalues, which show the temporal stability of this collective state.\nHence, a condensation mechanism on the bi-cluster should also exist,\nin order to establish a steady-state. Indeed, Fig.~\\ref{thetaimp}(b)\nshows that in the single-particle phase space two elliptic points are\npresent at the centers of the bi-cluster, therefore in the bordering\nchaotic region of these two points a ``trapping'' effect could be\npresent.\n\nIf formula (\\ref{sin}) is approximately verified, then the knowledge\nof the PDF at low temperatures (\\ref{distr}) would allow us to compute\nthe relationship between the temperature and the constant $A$.\nPerforming the integral, it turns out that\n\\begin{equation}\nT=<p^2>=\\frac{A^2}{4}~.\n\\label{ptoA}\n\\end{equation}\nThis relation is very well verified numerically for low temperature\nvalues. Moreover, particle momenta appear to follow a distribution\nwhich, although non-Gaussian, has a variance given by (\\ref{ptoA}).\nThe distribution shows a sharp peak in the center, due to the core of\nthe bi-cluster, which is far from Gaussian. However, the wings of the\ndistribution, which are due to the ``gas'' do follow a Gaussian\ndistribution.\n\nOne might propose that total momentum conservation plays a role in the\nobserved phenomenology. An easy way to test this idea is by adding a\nsmall external magnetic field $h$, which removes the constraint. We\nobserve that, for small $h$, the bi-cluster still forms and is stable\nat low temperatures. The conservation law is not\ntherefore relevant for cluster formation. On increasing $h$, the\ndistribution ${\\cal P}(\\theta)$ is modified, as shown in Fig. 6. The\nbi-cluster lies along the field axis and becomes asymmetric, with the\nnumber of rotators lying parallel and antiparallel to the field\ndirection becoming unequal. The antiparallel cluster is continuously\ndepleted until a single cluster eventually forms. This route towards\na single cluster is somewhat counter intuitive if one thinks of a\nN\\'eel ordered, unfrustrated antiferromagnet. Such a system would\nminimize its energy by aligning the bi-cluster perpendicular to the\nfield and relaxing the two halves continuously out of the plane in a\nsymmetric way. The bi-cluster would be destroyed by the two clusters,\nof equal size, rotating continuously onto the field direction.\n\n\n\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{ \\includegraphics{field.ps}}\n\\caption{${\\cal P}$($\\theta$) in the presence of an external field for a \nchain of $N=200$ particles. The diamonds correspond to $h=0.5$ whereas the triangles to $h=0.1$\nand the squares to $h=0.01$.}\n\\label{field}\n\\end{figure}\n\n\\section{Moment dynamics}\n\\label{Momentdy}\n\nAs we have already remarked, there is no implicit contradiction\nbetween the proposed formula for the moments of the PDF\n(\\ref{moments}) and the presence of a modified energy temperature\nrelation. The latter means that $<|M_1^2|>$ is an intensive quantity\nrather than being ${\\cal O} (1/N)$ as in the in the canonical\nensemble. Hence, although all the other odd modes of the PDF vanish\nin the $N \\to \\infty$ limit, this is not true for $<|M_1|>$. Rather,\nit remains finite in the low temperature regime, increasing linearly\n$U$. Solution (\\ref{moments}) is therefore exact, only when $U \\to 0$.\nHowever, because of the smallness of $U$ in this regime, $<|M_1|>$ is\na small quantity with respect to $<|M_2|>$ and equation\n(\\ref{moments}) is perfectly valid.\n\n\n\nWe expect that $M_1$ will display interesting dynamical behaviour,\ngiven its rather unexpected extensive nature, when the bi-cluster is\nformed. The $($Re$(M_{1})$,Im$(M_{1}))$ phase-plane is shown in\nFig.~\\ref{mm2af}(a) for $N=10^4$ and $U=1.13 \\cdot 10^{-5}(T=1.43\n\\cdot 10^{-5})$. Successive points are joined by lines to show the\nrelevant properties of the motion. The phase-point has a fast\noscillatory motion through zero and a much slower rotatory motion\ncentered on zero. The fast motion is due to the vibrations of the\nbi-cluster around the equilibrium positions (the two components of\n$M_1$ cross zero in phase). The slow rotational motion is due to the\nrigid rotation of the bi-cluster. This latter motion is further\nrevealed by the dynamics of $M_2$ in Fig.~\\ref{mm2af}(b). Again we\njoin successive phase-points with a line, showing that the phase point\nin the $M_2$ plane is rotating around the center, maintaining a fixed\nradius $|M_2| \\approx 0.5$. The time dependence of the phase\n$\\psi_2$ of $M_2$ is such that, over the long time span, the average\nis zero, but preliminary measurements of the variance $\\sigma^2$ show\nthat the motion is ballistic, rather than diffusive, $\\sigma^2 \\sim\nt^2$.\n\n\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{%\n \\includegraphics{herisson.ps}}\n\\caption{(a) Phase-points of $M_1$ for $N=10^4$ particles and \n$U=1.13 \\cdot 10^{-5}$.\n(b) Same for $M_2$.}\n\\label{mm2af}\n\\end{figure}\n\n\nThis picture is not only qualitative, but also quantitative. Indeed,\nif we average the fast oscillatory motion of $|M_1|$ in\nFig.~\\ref{mm2af}(a), we get an estimate of $<|M_1|^2>$. Taking\n$<|M_1|^2>/U \\approx 0.7$, we get the correction to the energy\ntemperature relation $T \\approx (2 - 0.7) U = 1.3 U$ as shown in\nFig.~\\ref{um2t}(b).\n\nAt the transition temperature where $|M_2|$ is decreasing, both the\nmotion of $M_1$ and that of $M_2$ become more erratic, revealing the\nbeginning of the region where the bi-cluster is progressively\ndepleting in time.\n\nThe other odd moments of the PDF always show an erratic motion around\nzero, with the variance of the cloud of points decreasing as $N$ is\nincreased; i.e. higher odd modes are not intensive. The higher even\nmoments of the PDF show a pattern similar to that of\nFig.~\\ref{mm2af}(b), with a progressively reduced radius; the motion\nof the higher phases has also ballistic features.\n\n\n\n\\section{Conclusions and perspectives}\n\\label{Conclusions}\n\nThe antiferromagnetic mean-field classical rotator system is an ideal\nlaboratory to study the relation between microcanonical and canonical\nensembles. Although it has a trivial canonical solution, the randomly\nuniform state at all temperatures, the high degeneracy of its ground\nstate induces nontrivial dynamical effects, which are revealed in the\nmicrocanonical ensemble. Instead of maintaining a random distribution\nof the rotators, the Hamiltonian dynamics selects a bimodal state,\nwhere the rotators are oriented along angles at distance $\\pi$ with\nsome spread, a bi-cluster. We have introduced an order parameter which\nreveals the formation of this state in the low energy phase and we\nhave empirically obtained an analytical formula for the probability\ndistribution function in angle, which perfectly fits the numerical\ndata.\n\nIn addition to this first remarkable difference between the two ensembles, we\nhave also shown that the energy temperature relation is modified and,\nalthough still linear, it has a different slope in the two\nensembles. The origin of this behavior lies in the extensive amount of\nenergy which Hamiltonian dynamics puts into the oscillatory vibrating\nmotion of the bi-cluster.\n\nMean-field models are characterized by self-consistency. Indeed, a\nfeature of our model is that rotators generate themselves the\nmean-field in which they move. Therefore, in the large $N$ limit, the\ndynamics of our model should depend only on the interaction of the\nsingle rotator with the mean-field. As already claimed for other\nmodels (e.g. beam-plasma instability and vorticity defect\nmodel~\\cite{Tenny}) self-consistency effectively reduces the number of\ndegrees of freedom. This is why many properties of the dynamics of\nsuch a complex $N$-body system as ours resemble those of a ``simple''\nperturbed pendulum. This is also the origin of the ordering of the\nrotators in a bi-cluster. There should be entropic reasons why the rotators\nprefer this state rather than choosing the disordered state, which is\ninstead selected in the canonical ensemble.\n\nMany questions remain to be explored, but the most\ncompelling one concerns the careful description and explanation of the\ndynamics. The perturbed pendulum analogy must be further investigated\nand a thorough study of the time evolution of the moments of the\nprobability distribution function in angle should allow a better\nunderstanding of the low-dimensional properties of the dynamics.\n\n\\acknowledgement We thank M. Droz, J. Farago, M-C Firpo, M. Paliy and\nZ. Racz for useful discussions. S.R. thanks ENS-Lyon, INFN and INFM\nfor financial support. P.H. thanks INFN for financial support. This\nwork was performed using the computing resources of the P\\^ole\nScientifique de Mod\\'elisation Num\\'erique (PSMN) of ENS Lyon and of\nthe DOCS-INFM group in Florence.\n\n\n\\begin{thebibliography}{}\n\\bibitem{Chaos} Chaos, {\\it Focus issue on chaos and irreversibility}, T. T\\`el,\nP. Gaspard and G. Nicolis Eds., (1998).\n\n\\bibitem{Rue} D. Ruelle, {\\it Statistical Mechanics: Rigorous Results},\nAddison-Wesley, 1989.\n\n\\bibitem{Pad90} T. Padmanabhan, Phys. Rep., {\\bf 188}, 285 (1990) and Refs.\ntherein.\n\n\\bibitem{plasmas} D.C. Brydges and Ph. A. Martin, J. Stat. Phys., {\\bf 96},\n1163 (1999) and Refs. therein.\n\n\\bibitem{Hem} M. Kac, G.E. Uhlenbeck and P.L. Hemmer, J. Math. Phys., {\\bf 4}, 216 (1963);\n{\\it ibidem}, {\\bf 5}, 60 (1964). \n\n\\bibitem{Par} M. Mezard, G. Parisi and M.A. Virasoro, \n{\\it Spin glass theory and beyond}, World Scientific, Singapore (1987).\n\n\\bibitem{Thi70} P. Hertel and W. Thirring, Ann. of Physics {\\bf 63}, 520 (1971).\n\n\\bibitem{Kogut} J. Kogut and J. Polonyi, Nuclear Physics {\\bf B265} [FS15], 313 (1986).\n\n\\bibitem{Antoni} M. Antoni and S. Ruffo, Phys. Rev. E {\\bf 52}, 2361 (1995);\nK. Kaneko and T. Konishi, Physica D {\\bf 71}, 146 (1994); M. Antoni, \nY. Elskens and C. Sandoz, Phys. Rev. E {\\bf 57}, 5347 (1998).\n\n\n\\bibitem{temp} H.H. Rugh, Phys. Rev. Lett {\\bf 78} 772 (1997); \nC. Giardin\\'a and R. Livi, J. Stat. Phys. {\\bf 91}, 1027 (1998).\n\n\\bibitem{Mac} I. Mac-Laghlan and P. Atela, Nonlinearity {\\bf 5}, 541 (1992).\n\n\\bibitem{Moessner} R. Moessner and J.T. Chalker, Phys. Rev. Lett. {\\bf 80}, 2929, (1998).\n\n\\bibitem{Villain} J. Villain, Z. Phys. B {\\bf 33}, 31, (1979).\n\n\\bibitem{CHS} J.T. Chalker, P.C.W. Holdsworth and E.F. Shender, Phys. Rev. Lett. {\\bf 68}, 855, (1992). \nJ.N. Reimers and A.J. Berlinksy, Phys. Rev. B {\\bf 48}, 9539, (1993).\n\n\\bibitem{Villain2} J. Villain, R. Bidaux, J.P. Carton and R. Coute, J. Phys. (Paris), {\\bf 41}, 1263, (1980).\n\n\\bibitem{Tenny} J.L. Tennyson, J.D. Meiss and P.J. Morrison, Physica D\n{\\bf 71}, 1 (1994); D. del-Castillo-Negrete, Phys. Lett. A {\\bf 241}, 99 (1998).\n\n\\bibitem{Grad} I.S. Gradshtein and I.M. Ryzhik, \n{\\it Tables of Integrals, Series, and Products}, Academic Press (1994), formula 1.441(2.).\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002005.extracted_bib",
"string": "\\begin{thebibliography}{}\n\\bibitem{Chaos} Chaos, {\\it Focus issue on chaos and irreversibility}, T. T\\`el,\nP. Gaspard and G. Nicolis Eds., (1998).\n\n\\bibitem{Rue} D. Ruelle, {\\it Statistical Mechanics: Rigorous Results},\nAddison-Wesley, 1989.\n\n\\bibitem{Pad90} T. Padmanabhan, Phys. Rep., {\\bf 188}, 285 (1990) and Refs.\ntherein.\n\n\\bibitem{plasmas} D.C. Brydges and Ph. A. Martin, J. Stat. Phys., {\\bf 96},\n1163 (1999) and Refs. therein.\n\n\\bibitem{Hem} M. Kac, G.E. Uhlenbeck and P.L. Hemmer, J. Math. Phys., {\\bf 4}, 216 (1963);\n{\\it ibidem}, {\\bf 5}, 60 (1964). \n\n\\bibitem{Par} M. Mezard, G. Parisi and M.A. Virasoro, \n{\\it Spin glass theory and beyond}, World Scientific, Singapore (1987).\n\n\\bibitem{Thi70} P. Hertel and W. Thirring, Ann. of Physics {\\bf 63}, 520 (1971).\n\n\\bibitem{Kogut} J. Kogut and J. Polonyi, Nuclear Physics {\\bf B265} [FS15], 313 (1986).\n\n\\bibitem{Antoni} M. Antoni and S. Ruffo, Phys. Rev. E {\\bf 52}, 2361 (1995);\nK. Kaneko and T. Konishi, Physica D {\\bf 71}, 146 (1994); M. Antoni, \nY. Elskens and C. Sandoz, Phys. Rev. E {\\bf 57}, 5347 (1998).\n\n\n\\bibitem{temp} H.H. Rugh, Phys. Rev. Lett {\\bf 78} 772 (1997); \nC. Giardin\\'a and R. Livi, J. Stat. Phys. {\\bf 91}, 1027 (1998).\n\n\\bibitem{Mac} I. Mac-Laghlan and P. Atela, Nonlinearity {\\bf 5}, 541 (1992).\n\n\\bibitem{Moessner} R. Moessner and J.T. Chalker, Phys. Rev. Lett. {\\bf 80}, 2929, (1998).\n\n\\bibitem{Villain} J. Villain, Z. Phys. B {\\bf 33}, 31, (1979).\n\n\\bibitem{CHS} J.T. Chalker, P.C.W. Holdsworth and E.F. Shender, Phys. Rev. Lett. {\\bf 68}, 855, (1992). \nJ.N. Reimers and A.J. Berlinksy, Phys. Rev. B {\\bf 48}, 9539, (1993).\n\n\\bibitem{Villain2} J. Villain, R. Bidaux, J.P. Carton and R. Coute, J. Phys. (Paris), {\\bf 41}, 1263, (1980).\n\n\\bibitem{Tenny} J.L. Tennyson, J.D. Meiss and P.J. Morrison, Physica D\n{\\bf 71}, 1 (1994); D. del-Castillo-Negrete, Phys. Lett. A {\\bf 241}, 99 (1998).\n\n\\bibitem{Grad} I.S. Gradshtein and I.M. Ryzhik, \n{\\it Tables of Integrals, Series, and Products}, Academic Press (1994), formula 1.441(2.).\n\n\\end{thebibliography}"
}
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cond-mat0002006
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Onset of sliding friction in incommensurate systems
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[
{
"author": "L. Consoli"
},
{
"author": "H. J. F. Knops"
},
{
"author": "and A. Fasolino"
}
] |
We study the dynamics of an incommensurate chain sliding on a periodic lattice, modeled by the Frenkel Kontorova hamiltonian with initial kinetic energy, without damping and driving terms. We show that the onset of friction is due to a novel kind of dissipative parametric resonances, involving several resonant phonons which are driven by the (dissipationless) coupling of the center of mass motion to the phonons with wavevector related to the modulating potential. We establish quantitative estimates for their existence in finite systems and point out the analogy with the induction phenomenon in Fermi-Ulam-Pasta lattices.
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[
{
"name": "e_print.tex",
"string": "\\documentstyle[prl,aps, epsfig, multicol]{revtex}\n\\begin{document}\n\\draft\n\\title{Onset of sliding friction in \nincommensurate systems}\n\\author{L. Consoli, H. J. F. Knops, and A. Fasolino}\n\\address{Institute for Theoretical Physics, University of Nijmegen, \nToernooiveld 1, 6525 ED Nijmegen, The Netherlands}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nWe study the dynamics of an incommensurate chain sliding on a \nperiodic lattice, \nmodeled by the Frenkel Kontorova hamiltonian with initial kinetic energy,\nwithout damping and driving terms. \nWe show that the onset of friction is due to a novel kind of\ndissipative parametric resonances, \ninvolving several resonant phonons which are driven by \nthe (dissipationless) coupling of the center of mass \nmotion to the phonons with wavevector related to the \nmodulating potential. We establish quantitative estimates for \ntheir existence in finite systems and\npoint out the analogy with the induction phenomenon in Fermi-Ulam-Pasta \nlattices. \n\n\\end{abstract}\n\\pacs{PACS numbers: 05.45.-a, 45.05.+x, 46.55.+d, 46.40.Ff}\n\n\\begin{multicols}{2}\nThe possibility of measuring friction at the atomic level provided by \nthe Lateral Force Microscopes\\cite{Mate} and Quartz Crystal\nMicrobalance\\cite{Krim} has stimulated intense research on \nthis topic\\cite{book}. Phonon excitations are the dominant cause \nof friction in many cases\\cite{Tomm}.\nMost studies are carried out for one-dimensional \nnon-linear lattices\\cite{shinjo,Aubry2,Braun,StrElm,Bambi,Sokoloff,Braim,Hent} \nand in\nparticular for the Frenkel-Kontorova (FK) model\\cite{FK},\nwhere the surface layer is modeled by a harmonic chain and the substrate is \nreplaced by a rigid periodic modulation potential. \nThe majority\\cite{Aubry2,Braun,StrElm,Bambi,Sokoloff,Braim,Hent} examines the \nsteady state of the dynamical FK model in presence of dissipation representing\nthe coupling of phonons to other, undescribed degrees of freedom.\\linebreak\n\\hspace*{0.3cm}We study the dynamics of an undriven incommensurate FK chain.\nOur aim is to ascertain whether the experimentally observed \nsuperlubricity \\cite{Hirexp} can be due to the blocking of the phonon\nchannels caused by an incommensurate contact of the two sliding surfaces.\nTherefore we do not include any explicit damping of the phonon modes, \nsince we wish to find out if they can be excited at all by the\nmotion of the center of mass (CM). In an earlier study, Shinjo and Hirano\\cite{shinjo} \nfound a superlubric regime for this model, where the chain would \nslide indefinitely without dynamic friction but with a recurrent exchange of\nkinetic energy between CM and a single internal mode. We will show that their\nfinding is oversimplified by either too short simulation times or too small\nsystem sizes. \nThe inherent non-linear coupling of the CM to the phonons\nleads to an irreversible decay of the CM velocity, albeit with very long time \nscales in some windows.\nThe dissipative mechanism is driven by the coupling of the CM to the modes \nwith modulation wavevector $q$ or its harmonics, $\\omega_{nq}$, \nand consists in a novel \nkind of parametric resonances with much wider\nwindows of instabilities than those deriving from the standard Mathieu \nequation\\cite{handbook}. \nThe importance of harmonic resonances at $\\omega_{nq}$ has been \npointed out before\\cite{Aubry2,StrElm,Sokoloff}, with the \nsuggestion\\cite{Sokoloff} that they \ncould be absent in finite systems due to the \ndiscreteness of the phonon spectrum. However, \nit has not been realized that they act as a {\\it driving} term for the\nonset of dissipation via subsequent complex parametric \nexcitations which we shall describe, establishing quantitative \nestimates for their existence in finite systems. \nA related mechanism has recently been identified in \nthe resonant energy transfer in the induction phenomenon in Fermi-Ulam-Pasta \nlattices\\cite{christie}. \n\nWe start with the FK hamiltonian\n\\begin{equation}\n{\\cal H}=\\sum_{n=1}^N\\left[\\frac{p_n^2}{2} + \\frac{1}{2}\\left(u_{n+1}-u_n-l\n\\right)^2 \n+\\frac{\\lambda}{2\\pi}\\sin{ (\\frac{2\\pi u_n}{m} )}\\right]\n\\label{Ham}\n\\end{equation}\nwhere $u_n$ are the lattice positions and $l$ is the equilibrium spacing of \nthe chain for $\\lambda=0$, \n$\\lambda$ being the strength of the coupling scaled to the \nelastic spring constant. We take an incommensurable ratio of \n$l$ to the period $m$ of the periodic potential, namely $m=1$, \n$l=(1+\\sqrt 5)/2$. We consider chains of $N$ atoms with periodic boundary conditions\n$u_{N+1}=Nl+u_1$. Hence, in the numerical implementation, \nwe have to choose commensurate approximations for $l$ so that\n$l\\times N=M\\times 1$ with $N$ and $M$ integer, i.e. \nwe express $l$ as ratio of consecutive Fibonacci numbers.\nThe ground state of this model displays the \nso-called Aubry transition\\cite{Aubry}\nfrom a modulated to a pinned configuration above a critical value \n$\\lambda_c=0.14$.\nHere we just note that in the limit of weak coupling \n($\\lambda<<\\lambda_c$), \ndeviations from equidistant spacing $l$ in the ground state \nare modulated with the substrate modulation wavevector \n$q=2\\pi l/m$\\cite{modfun} as due to the frozen-in phonon $\\omega_q$. \nHigher harmonics $nq$ have amplitudes which scale with $\\lambda^n$.\n\nWe define the CM position and velocity as $Q= \\frac {1}{N}\\sum_nu_n$, \n$P= \\frac {1}{N}\\sum_n p_n$. By writing\n$u_n=nl+x_n+Q$, the equations of motion for the deviations from a rigid \ndisplacement $x_n$ read\n\\begin{equation}\n\\ddot{x}_n=x_{n+1}+x_{n-1}-2x_n+\\lambda\\cos{(qn+2\\pi x_n+2\\pi Q)}\n\\label{eom1}\n\\end{equation}\nWe integrate by a Runge Kutta algorithm the $N$ Eqs. (\\ref{eom1})\nwith initial momenta $p_n=P_0$ and \n$x_n(t=0)$ \ncorresponding to the ground state. For a given velocity $P$, \nparticles pass over maxima of the potential with frequency \n$\\Omega=2\\pi P$, the so-called washboard frequency\\cite{StrElm,Sokoloff}.\n\\begin{figure}\n\\epsfig{file=fig1.ps, angle=-90}\n\\caption{Time dependence of the CM velocity for several values of $P_0$ for $N=144, \n\\lambda=0.05\\sim\\frac{\\lambda_c}{3}$. The dashed line corresponds to\n $P_0=\\frac{\\omega_q}{2\\pi}=0.2966$. Close to higher resonances (solid dots) \na similar oscillatory behavior is observed, accompanied by a slower decay which\nis not apparent on the timescale of the figure.\n}\n\\label{fig1}\n\\end{figure}\nIn Fig. 1 we show the time evolution of the CM momentum for \n$\\lambda \\sim \\lambda_c/3$ and several values of $P_0$. \nAccording to the phase diagram\nof Ref.\\onlinecite{shinjo} a superlubric behavior should be observed for \nthis value of $\\lambda$ and $P_0\\geq 0.1$.\nWe find instead a non trivial time evolution with oscillations of varying \nperiod and amplitude and, remarkably, a very fast decay \nof the CM velocity for $P_0 \\sim \\omega_{q}/(2\\pi)$ despite the absence of a \ndamping term in Eq.~(\\ref{eom1}). A similar, but much slower, decay\nis found for $nP_0\\sim \\omega_{nq}/(2\\pi)$. In the study of \nthe driven underdamped FK\\cite{StrElm} it is shown that, at\nthese superharmonic resonances, the differential mobility is extremely low.\nHere, we work out an analytical description in terms of the phonon spectrum \nwhich explains this complex time evolution and \nidentifies the dissipative mechanism which \nis triggered by these resonances. \nIn the limit of weak coupling \n$\\lambda$ it is convenient to go\nfrom real to reciprocal space by defining Fourier transformed coordinates\n$x_k=\\frac{1}{N}\\sum_n{e^{-ik n}x_n}$ and $x_n=\\sum_k{e^{ik n}x_k}$, where \n$k=2\\pi n/N$ and the normalization is chosen to remove the explicit\n$N$-dependence in the equations of motion, which become:\n\\begin{mathletters}\n\\label{eom2}\n\\begin{eqnarray}\n\\ddot{x}_k&=&-\\omega^2_k x_k + \\frac{\\lambda}{2N}\\sum_n e^{-ikn}\\left[e^{iqn} \ne^{i2\\pi Q}e^{i2\\pi x_n} +c.c.\\right]\\label{xqgen}\\\\\n\\ddot{Q}&=&\\frac{\\lambda}{2N}\\sum_n \\left[e^{i2\\pi Q}e^{iqn}e^{i2\\pi x_n} \n+c.c.\\right]\n\\end{eqnarray}\n\\end{mathletters}\nwith $\\omega_k=2 \\vert \\sin(k/2)\\vert$.\nWe expand Eq. (\\ref{xqgen}) in $x_n$ as:\n\\end{multicols}\n\\begin{equation}\n\\ddot{x}_k=-\\omega^2_k x_k + \\frac{\\lambda}{2}\\sum_{m=0}^\\infty \n\\frac{(i2\\pi)^m}{m!}\\sum_{k_1\\ldots k_m}\\left[e^{i2\\pi Q}x_{k_1}\\cdots \nx_{k_m}\\delta_{k_1+\\cdots\n+k_m,-q+k} + (-1)^m\ne^{-i2\\pi Q} x_{k_1} \\cdots x_{k_m}\\delta_{k_1+\\cdots +k_m,q+k} \\right]\n\\label{expans1}\n\\end{equation}\n\n\\begin{multicols}{2}\nSince in the ground state the only modes present in \norder $\\lambda$ are $x_q = x_{-q} = \\lambda / 2\\omega_q^2$\nthe CM is coupled only to these modes up to \nsecond order in $\\lambda$:\n\\begin{mathletters}\n\\label{Qxq}\n\\begin{eqnarray}\n\\ddot{Q}&=&i\\lambda\\pi \\left(e^{i2\\pi Q}x_{-q} - \ne^{-i2\\pi Q}x_{q}\\right) \\label{Q}\\\\\n\\ddot{x}_q &=&-\\omega^2_q x_q + \\frac{\\lambda}{2} e^{i2\\pi Q} \n\\label{xq+}\\\\\n\\ddot{x}_{-q}&=& -\\omega^2_q x_{-q}+ \\frac{\\lambda}{2} e^{-i2\\pi Q} \n\\label{xq-}\n\\end{eqnarray}\n\\end{mathletters}\n\nIn Fig.~2 we compare the behavior of \n$P(t)=\\dot Q(t)$, obtained by solving the minimal set \nof Eqs. (\\ref{Qxq}) with the appropriate initial \nconditions $Q(t=0)=0$, $P(t=0)=P_0$, $x_q(t=0)=\\lambda/(2\\omega_q^2)$, \n$\\dot x_q(t=0)=0$, with \nthe one obtained from the full system of Eqs.~(\\ref{eom1}).\nEqs.~(\\ref{Qxq}) reproduce very well the initial behavior of the CM velocity \nwhich displays oscillations of frequency $\\Delta$ around the value \n$\\Omega/2\\pi$ but do not predict\nthe decay occurring at later times because, as we show next, this is \ndue to coupling to other modes. To this aim, we analyze the relation \nbetween the initial CM velocity $P_0$ and $\\Omega/2\\pi$, respectively \n$\\Delta_{\\pm}$.\n\\begin{figure}\n\\epsfig{file=fig2.ps, angle=-90}\n\\caption{Simulation of the full FK system according to \nEq. (\\ref{eom1}) (solid lines) and numerical solution of Eqs. (\\ref{Qxq}) (dashed lines) \nfor $N=144$, $\\lambda=0.015$, and several values of $P_0$ around $P_0=0.29$.\nThe differences between the two approaches are negligible. The average value of the CM velocity \n$\\Omega/2\\pi$ (horizontal dashed line) and the period of the oscillation for $P_0=0.29$ are also shown.\n}\n\\label{fig2}\n\\end{figure}\nTake as an ansatz for the CM motion:\n\\begin{equation}\nQ(t)=\\frac{\\Omega}{2\\pi}t+\\alpha_+\\sin(\\Delta_+t)+\\alpha_-\\sin(\\Delta_-t)\n\\label{ansatz}\n\\end{equation}\nInserting the ansatz (\\ref{ansatz}) in the coupled set of Eqs. (\\ref{Qxq}) keeping\nonly terms linear in $\\alpha_{\\pm}$, we find that both $\\Delta_{\\pm}$ are roots of:\n\\begin{equation} \n\\Delta^2=\\lambda^2 \\pi^2 \\left(2Z(0)-Z(\\Delta)-Z(-\\Delta)\\right)\n\\label{eqDelta}\n\\end{equation}\n$ Z(\\Delta)$ being the impedance\n\\begin{equation} \n Z(\\Delta) = \\frac{1}{\\omega_{q}^2-(\\Omega+\\Delta)^2} \\textrm{.}\n\\label{Delta}\n\\end{equation}\nIn general Eq. (\\ref{eqDelta}) has (besides the trivial solution $\\Delta=0$) \nindeed two solutions, related to the sum \nand difference of the two basic \nfrequencies in the system, $\\omega_q$ and $\\Omega$ :\n\\begin{equation}\n\\Delta_\\pm\\cong|\\omega_q\\pm\\Omega+\\frac {\\lambda^2 \\pi^2} \n{2\\omega_q(\\Omega\\pm\\omega_q)^2} + \\cdots|\\label{deltam}\n\\end{equation}\nClose to resonance, $\\Omega\\sim\\omega_q$, the amplitude $\\alpha_-$\ndominates (see below) and the CM oscillates with a single frequency \n$\\Delta=\\Delta_-$ (see Fig.~2).\nVery close to resonance (more precisely \n$\\omega_q<\\Omega<\\omega_q+(2\\lambda^2\\pi^2/\\omega_q)^{\\frac{1}{3}}$),\nthe root $\\Delta_-$ becomes imaginary, signaling an instability. \nIn fact the system turns out to be bistable as it can be seen in Fig. 3 by the \njump in $\\Omega(P_0)$ as $P_0$ passes through $\\omega_q/(2\\pi)$.\n\\begin{figure}\n\\epsfig{file=fig3.ps, angle=-90}\n\\caption{Closed and open dots, frequencies $\\Delta$ (left axis) and $\\Omega$ (right axis) \nversus $P_0$ for simulations for $N=144$, $\\lambda=0.015$. \nDashed and solid lines: solutions of Eqs. (\\ref{deltam}) and (\\ref{p_0}) respectively.\n}\n\\label{fig3}\n\\end{figure}\nAnalytically, the relation between $\\Omega$ and $P_0$ and the\namplitudes $\\alpha_{\\pm}$, is determined\nby matching the ansatz (\\ref{ansatz}) with the initial condition:\n\\begin{equation}\n\\alpha_{\\pm}= \\frac{\\lambda^2 \\pi\\Omega}{2\\omega_q^2}\\frac{1}{(\\omega_q \n\\pm\\Omega)^3}\n\\label{alfa}\n\\end{equation}\n\\begin{equation}\nP_0 = \\frac{\\Omega}{2\\pi}+ \\frac {\\lambda^2\\pi\\Omega}{2\\omega_q^2}\\left[ \n\\frac{1}{(\\Omega+\\omega_q)^2}+\\frac{1}{(\\Omega-\\omega_q)^2}\\right] + \\cdots\n\\label{p_0}\n\\end{equation}\n\nThe fact that Eq. (\\ref{p_0}) has multiple solutions for $\\Omega$ when $P_0 \n\\sim \\omega_q/2\\pi$ is in accordance with the jump seen in Fig. 3. However, \nEq. (\\ref{p_0}), which is derived by keeping only linear terms in $\\alpha_-$,\nis not accurate enough to describe in detail the instability in the above \nrange around $\\omega_q$ where $\\alpha_-$ diverges.\n\nAn initial behavior similar to that for $P_0\\simeq\\omega_q/2\\pi$ is \nobserved in Fig. 1 for $nP_0\\simeq \\omega_{nq}/n2\\pi$. We examine\nthe case $n=2$.\nEq. (\\ref{expans1}) shows\nthat $x_{2q}$ is driven in next order in $\\lambda$ by $x_q$:\n\\begin{equation}\n\\ddot{x}_{2q}=-\\omega_{2q}^2x_{2q} + i 2\\lambda\\pi e^{i2\\pi Q}x_q\n\\label{x2q}\n\\end{equation}\nWhen $2\\pi Q\\simeq \\Omega t$, $x_q$ will be $\\simeq \\lambda e^{i\\Omega t}$, \nso that\n$x_{2q}$ is forced with amplitude $\\lambda^2$ and frequency $2\\Omega$, \nyielding \nresonance for $2\\Omega=\\omega_{2q}$. Since $x_{2q}$ couples back to \n$x_q$, we have\na set of equations similar to Eqs. (\\ref{Qxq}), but at order $\\lambda^2$.\n\nWe now come to the key issue, namely the onset of friction causing the decay \nof the CM velocity seen in Fig. 1 at later times, which\ncannot be explained by the coupling of the CM to the main harmonics $nq$. \nSince $x_q$ is by far the largest\nmode in the early stage, we consider second order terms involving $x_q$ \nin Eq. (\\ref{expans1}): \n\\begin{equation}\n\\ddot{x}_k=- \\left[ \\omega^2_k +2 \\lambda \\pi^2 \\left( e^{i2\\pi Q} x_{-q} +\n e^{-i2\\pi Q} x_{q} \\right)\n \\right] x_k\n\\label{parres}\n\\end{equation}\nInsertion of the solution obtained above for $x_q$ (Eq. (\\ref{xq+})) and $Q$ \n(Eq. (\\ref{ansatz})) yields\n\\begin{equation}\n\\ddot{x}_k=-\\left[\\omega^2_k + A + B\\cos{(\\Delta t)}\\right]x_k\n\\label{ficata}\n\\end{equation}\nwith $A=2(\\lambda\\pi)^2/Z(0)$ and $B\\sim\\alpha_-$.\nClearly, Eq. (\\ref{ficata}) is a Mathieu parametric resonance for mode $x_k$.\nThe relevance of parametric resonances has been recently stressed\\cite{Hent}. \nHowever here, due to the coupling of the CM \nto the modulation mode $q$, resonances are not with the washboard frequency\n$\\Omega$ but with $\\Delta\\sim\\Omega-\\omega_q$. Hence, we find \ninstability windows around $\\omega_k^2+A=(n\\Delta/2)^2$.\nSince $\\Delta$ is small close to resonance, one expects to find instabilities \nfor acoustic modes with $k$ small. Indeed, as shown in Fig. 4a, we find by \nsolving Eq. (\\ref{eom1}) that the decay of the CM is accompanied by the \nexponential increase of the modes $k=2, 3, 4$ and, \nwith a longer rise time, $k=1$. However, the instability windows \nresulting from Eq. (\\ref{ficata}), shown in Fig. 4b, cannot \nexplain the numerical results of Fig. 4a, i.e. the Mathieu formalism cannot \nexplain the observed instability.\nIn Eq. (\\ref{expans1}), the only linear terms left \nout in Eq. (\\ref{parres})\nare couplings with $x_{k\\pm q}$, which are much higher order in $\\lambda$. \nNevertheless, these terms are crucial since they may cause new instabilities \ndue to the fact that, for $k$ small, they are \nalso close to resonance.\nWe have solved the coupled set of equations for mode $x_{\\pm k}$\nand $x_{k\\pm q}$ :\n\\end{multicols}\n\\begin{mathletters}\n\\label{cfk}\n\\begin{eqnarray}\n\\ddot{x}_k & = & -\\left[\\omega^2_k +2\\lambda\\pi^2\\left(e^{i2\\pi Q}x_{-q}+\ne^{-i2\\pi Q}x_q\\right)\\right]x_k + i \\lambda\\pi\\left(e^{i2\\pi Q}\nx_{k-q} - e^{-i2\\pi Q}x_{k+q}\\right) \\label{cfk1} \\\\\n\\ddot{x}_{k\\pm q} & = & -\\left[\\omega^2_{k \\pm q} + 2\\lambda\\pi^2\\left(\ne^{i2\\pi Q}x_{-q}+e^{-i2\\pi Q}x_q\\right)\\right]x_{k\\pm q} \\pm i\\lambda\\pi\ne^{\\pm i2\\pi Q}x_k \\label{cfk2}\n\\end{eqnarray}\n\\end{mathletters}\n\\begin{multicols}{2}\ntogether with Eqs. (\\ref{Qxq}) for continuous $k$. Indeed, we find a wider \nrange of instabilities, giving \na detailed account of the\nnumerical result as shown in Fig. 4b. This mechanism where a parametric \nresonance is enhanced by coupling to near resonant modes \nis quite general in systems with a quasi continuous\nspectrum of excitations and is \nrelated to the one proposed~\\cite{christie} in explaining \ninstabilities in the FPU chain in a different physical context. \\linebreak\n\\hspace*{0.3cm}The number of particles in the chain is an important parameter. When \nthis number is very small, the chain is in fact commensurate and the phase\nof the CM is locked (the gap scales as $\\lambda^N$ due to Umklapp terms). Next, one\nenters a stage of apparent superlubric behavior due to the fact that the\nspectrum is still discrete on the scale of the size of the instability windows \ndiscussed above. For $N=144$ and $\\lambda=\\frac{1}{3}\\lambda_c$ (Fig. \\ref{fig1})\nwe only begin to see the decay for values of $P_0$ close to resonances.\nThe experimentally observed superlubricity in\\cite{Hirexp} could then be due either\nto the finiteness of the system or to the low sliding velocities.\\linebreak\n\\hspace*{0.3cm}The above described multiple parametric excitation gives rise to an \neffective damping for the system via a cascade of couplings\nto more and more modes via the non-linear terms in Eq. (\\ref{expans1}).\nIt remains an open question if this mechanism will eventually lead to \na full or partial equilibrium distribution of energy over the \nnormal\nmodes\\cite{Ruffo} although our preliminary results support the former \nhypothesis even at weak couplings. \\linebreak\n\\hspace*{0.3cm}In summary, we have described in detail the mechanism which gives rise to \nfriction during the sliding of a harmonic system onto an incommensurate \nsubstrate. The onset of friction occurs in two steps: the resonant coupling \nof the CM to modes with wavevector related to the substrate \nmodulation leads to long wavelength oscillations which in turn drive a complex\nparametric resonance involving several resonant modes. This mechanism is \nrobust in that it leads to wide instability windows and represents a quite \ngeneral mechanism for the onset of energy transfer in systems with a quasi continuous\nspectrum of excitations. \n\nWe are grateful to Ted Janssen for many constructive discussions and \nfor his support.\n\\begin{figure}\n\\epsfig{file=fig4.ps, angle=-90}\n\\caption{(a) $|x_k(t)|^2$ of the first 4 modes from Eq. (\\ref{eom1}) with \n$N=144$, $\\lambda=0.015$ and $P_0=0.29$.\nNote that the first mode has a longer rise time and that the third mode is the \nmost unstable. (b) Dispersion relation for a chain of $144$ atoms\n($k$-values in units of $(\\frac{2\\pi}{144}$)). \nUnstable modes resulting from the full simulation are represented by solid \ndots. The shaded $k$-ranges give the instability windows resulting from the \nMathieu-type Eq.(\\ref{parres}) and cannot explain the simulation. \nConversely the wiggled ranges of the phonon dispersion (WU=weakly unstable, \nSU=strongly unstable) are the instability windows predicted by Eqs. (\\ref{Qxq}) and \n(\\ref{cfk}).\nThey explain all instabilities as well as the long rise time of the \nfirst mode (WU) and the shortest one of the third mode which falls in the \nmiddle of the SU range.}\n\\label{fig4}\n\\end{figure}\n\\begin{references}\n\\bibitem{Mate} C. M. Mate, G. M. McClelland, R. Erlandsson, and S. Chiang, \nPhys. Rev. Lett. {\\bf 59}, 1942 (1987).\n\\bibitem{Krim} J. Krim, D. H. Solina, and R. Chiarello, Phys. Rev. Lett. \n{\\bf 66}, 181 (1991).\nJ. B. Sokoloff, J. Krim, and A. Widom, Phys. Rev. B {\\bf 48}, 9134 (1993).\n\\bibitem{book} {\\it Physics of Sliding Friction}, edited by B. N.J. Persson and\nE. Tosatti (Kluwer, Dordrecht, 1996); B. N. J. Persson, Surf. Sci. Rep. {\\bf \n33}, 83 (1999).\n\\bibitem{Tomm} M. S. Tomassone, J. B. Sokoloff, A. Widom, and J. Krim, Phys. Rev. \nLett. {\\bf 79}, 4798 (1997).\n\\bibitem{shinjo} K. Shinjo and M. Hirano, Surf. Sci. {\\bf 283}, 473 (1993).\n\\bibitem{Aubry2} S. Aubry and L. de Seze, Festk\\\"{o}rperprobleme {\\bf XXV}, 59 \n(1985).\n\\bibitem{Braun} O. M. Braun, T. Dauxois, M. V. Paliy, and M. Peyrard, Phys. \nRev. Lett. {\\bf 78}, 1295 (1997); Phys. Rev. E {\\bf 55}, 3598 (1997).\n\\bibitem{StrElm} T. Strunz and F.-J. Elmer, Phys. Rev. E {\\bf 58}, 1601 \n(1998); {\\bf 58}, 1612 (1998).\n\\bibitem{Bambi} Z. Zheng, B. Hu, and G. Hu, Phys. Rev. B {\\bf 58}, 5453 \n(1998).\n\\bibitem{Sokoloff} J. B. Sokoloff, Phys. Rev. Lett. {\\bf 71}, 3450 (1993);\nJ. Phys. Cond. Matt. {\\bf 10}, 9991 (1998).\n\\bibitem{Braim} Y. Braiman, F. Family, and H. G. E. Hentschel, Phys. Rev. B \n{\\bf 55}, 5491 (1997).\n\\bibitem{Hent} H. G. E. Hentschel, F. Family, and Y. Braiman, Phys. Rev. Lett. \n{\\bf 83}, 104 (1999).\n\\bibitem{FK} Ya. I. Frenkel and T.A. Kontorova, Zh. Eksp. Teor. Fiz. {\\bf 8}, \n89 (1938).\n\\bibitem{Hirexp} M. Hirano, K. Shinjo, R. Kaneko, and Y. Murata, Phys. Rev. Lett. {\\bf 78}, \n1448 (1997).\n\\bibitem{handbook} {\\sl Handbook of mathematical functions}, edited by\nM. Abramowitz and I. A. Stegun (Dover, New York, 1970).\n\\bibitem{christie} G. Christie and B. I. Henry, Phys. Rev. E {\\bf 58}, 3045 \n(1998).\n\\bibitem{Aubry} M. Peyrard and S. Aubry, J. Phys. C {\\bf 16}, 1593 (1983), and refereces\n therein.\n\\bibitem{modfun} See, e.g., Eq. (3) in\nT.S. van Erp. A. Fasolino, O. Radulescu, and T. Janssen, Phys. Rev. B {\\bf \n60}, 6522 (1999).\n\\bibitem{Ruffo} J. De Luca, A. J. Lichtenberg, and S. Ruffo, Phys. Rev. {\\bf E} 60, 3781 (1999).\n\\end{references}\n\n%\\begin{figure}\n%\\epsfig{file=fig1.ps, angle=-90}\n%\\caption{Time dependence of the CM velocity for several values of $P_0$ for $N=144, \n%\\lambda=0.05\\sim\\frac{\\lambda_c}{3}$. The dashed line corresponds to\n% $P_0=\\frac{\\omega_q}{2\\pi}=0.2966$. Close to higher resonances (solid dots) \n%a similar oscillatory behavior is observed, accompanied by a slower decay which\n%is not apparent on the timescale of the figure.\n%}\n%\\label{fig1}\n%\\end{figure}\n%\\begin{figure}\n%\\epsfig{file=fig2.ps, angle=-90}\n%\\caption{Simulation of the full FK system according to \n%Eq. (\\ref{eom1}) (solid lines) and numerical solution of Eqs. (\\ref{Qxq}) (dashed lines) \n%for $N=144$, $\\lambda=0.015$, and several values of $P_0$ around $P_0=0.29$.\n%The differences between the two approaches are negligible. The average value of the CM velocity \n%$\\Omega/2\\pi$ (horizontal dashed line) and the period of the oscillation for $P_0=0.29$ are also shown.\n%}\n%\\label{fig2}\n%\\end{figure}\n%\\begin{figure}\n%\\epsfig{file=fig3.ps, angle=-90}\n%\\caption{Closed and open dots, frequencies $\\Delta$ (left axis) and $\\Omega$ (right axis) \n%versus $P_0$ for simulations for $N=144$, $\\lambda=0.015$. \n%Dashed and solid lines: solutions of Eqs. (\\ref{deltam}) and (\\ref{p_0}) respectively.\n%}\n%\\label{fig3}\n%\\end{figure}\n%\\begin{figure}\n%\\epsfig{file=fig4.ps, angle=-90}\n%\\caption{(a) $|x_k(t)|^2$ of the first 4 modes from Eq. (\\ref{eom1}) with \n%$N=144$, $\\lambda=0.015$ and $P_0=0.29$.\n%Note that the first mode has a longer rise time and that the third mode is the \n%most unstable. (b) Dispersion relation for a chain of $144$ atoms\n%($k$-values in units of $(\\frac{2\\pi}{144}$)). \n%Unstable modes resulting from the full simulation are represented by solid \n%dots. The shaded $k$-ranges give the instability windows resulting from the \n%Mathieu-type Eq.(\\ref{parres}) and cannot explain the simulation. \n%Conversely the wiggled ranges of the phonon dispersion (WU=weakly unstable, \n%SU=strongly unstable) are the instability windows predicted by Eqs. (\\ref{Qxq}) and \n%(\\ref{cfk}).\n%They explain all instabilities as well as the long rise time of the \n%first mode (WU) and the shortest one of the third mode which falls in the \n%middle of the SU range.}\n%\\label{fig4}\n%\\end{figure}\n\\end{multicols}\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002006.extracted_bib",
"string": "\\bibitem{Mate} C. M. Mate, G. M. McClelland, R. Erlandsson, and S. Chiang, \nPhys. Rev. Lett. {\\bf 59}, 1942 (1987).\n\n\\bibitem{Krim} J. Krim, D. H. Solina, and R. Chiarello, Phys. Rev. Lett. \n{\\bf 66}, 181 (1991).\nJ. B. Sokoloff, J. Krim, and A. Widom, Phys. Rev. B {\\bf 48}, 9134 (1993).\n\n\\bibitem{book} {\\it Physics of Sliding Friction}, edited by B. N.J. Persson and\nE. Tosatti (Kluwer, Dordrecht, 1996); B. N. J. Persson, Surf. Sci. Rep. {\\bf \n33}, 83 (1999).\n\n\\bibitem{Tomm} M. S. Tomassone, J. B. Sokoloff, A. Widom, and J. Krim, Phys. Rev. \nLett. {\\bf 79}, 4798 (1997).\n\n\\bibitem{shinjo} K. Shinjo and M. Hirano, Surf. Sci. {\\bf 283}, 473 (1993).\n\n\\bibitem{Aubry2} S. Aubry and L. de Seze, Festk\\\"{o}rperprobleme {\\bf XXV}, 59 \n(1985).\n\n\\bibitem{Braun} O. M. Braun, T. Dauxois, M. V. Paliy, and M. Peyrard, Phys. \nRev. Lett. {\\bf 78}, 1295 (1997); Phys. Rev. E {\\bf 55}, 3598 (1997).\n\n\\bibitem{StrElm} T. Strunz and F.-J. Elmer, Phys. Rev. E {\\bf 58}, 1601 \n(1998); {\\bf 58}, 1612 (1998).\n\n\\bibitem{Bambi} Z. Zheng, B. Hu, and G. Hu, Phys. Rev. B {\\bf 58}, 5453 \n(1998).\n\n\\bibitem{Sokoloff} J. B. Sokoloff, Phys. Rev. Lett. {\\bf 71}, 3450 (1993);\nJ. Phys. Cond. Matt. {\\bf 10}, 9991 (1998).\n\n\\bibitem{Braim} Y. Braiman, F. Family, and H. G. E. Hentschel, Phys. Rev. B \n{\\bf 55}, 5491 (1997).\n\n\\bibitem{Hent} H. G. E. Hentschel, F. Family, and Y. Braiman, Phys. Rev. Lett. \n{\\bf 83}, 104 (1999).\n\n\\bibitem{FK} Ya. I. Frenkel and T.A. Kontorova, Zh. Eksp. Teor. Fiz. {\\bf 8}, \n89 (1938).\n\n\\bibitem{Hirexp} M. Hirano, K. Shinjo, R. Kaneko, and Y. Murata, Phys. Rev. Lett. {\\bf 78}, \n1448 (1997).\n\n\\bibitem{handbook} {\\sl Handbook of mathematical functions}, edited by\nM. Abramowitz and I. A. Stegun (Dover, New York, 1970).\n\n\\bibitem{christie} G. Christie and B. I. Henry, Phys. Rev. E {\\bf 58}, 3045 \n(1998).\n\n\\bibitem{Aubry} M. Peyrard and S. Aubry, J. Phys. C {\\bf 16}, 1593 (1983), and refereces\n therein.\n\n\\bibitem{modfun} See, e.g., Eq. (3) in\nT.S. van Erp. A. Fasolino, O. Radulescu, and T. Janssen, Phys. Rev. B {\\bf \n60}, 6522 (1999).\n\n\\bibitem{Ruffo} J. De Luca, A. J. Lichtenberg, and S. Ruffo, Phys. Rev. {\\bf E} 60, 3781 (1999).\n"
}
] |
cond-mat0002007
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Adsorption of benzene on Si(100) from first principles
|
[
{
"author": "Pier Luigi Silvestrelli"
},
{
"author": "Francesco Ancilotto"
},
{
"author": "and Flavio Toigo"
}
] |
Adsorption of benzene on the Si(100) surface is studied from first principles. We find that the most stable configuration is a tetra-$\sigma$-bonded structure characterized by one C-C double bond and four C-Si bonds. A similar structure, obtained by rotating the benzene molecule by 90$^{\circ}$, lies slightly higher in energy. However, rather narrow wells on the potential energy surface characterize these adsorption configurations. A benzene molecule impinging on the Si surface is most likely to be adsorbed in one of three different di-$\sigma$-bonded, metastable structures, characterized by two C-Si bonds, and eventually converts into the lowest-energy configurations. These results are consistent with recent experiments.
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[
{
"name": "benzene.tex",
"string": "\\documentstyle[aps,prl]{revtex}\n\\begin{document}\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname\n\n\\title{Adsorption of benzene on Si(100) from first principles}\n\\author{Pier Luigi Silvestrelli, Francesco Ancilotto, and Flavio Toigo}\n\\address{Istituto Nazionale per la Fisica della Materia and\nDipartimento di Fisica ``G. Galilei'',\nUniversit\\`a di Padova, via Marzolo 8, I-35131 Padova, Italy\\\\}\n\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nAdsorption of benzene on the Si(100) surface\nis studied from first principles.\nWe find that the most stable configuration is a\ntetra-$\\sigma$-bonded structure characterized by \none C-C double bond and four C-Si bonds.\nA similar structure, obtained\nby rotating the benzene molecule by 90$^{\\circ}$, lies slightly higher \nin energy.\nHowever, rather narrow wells on the potential energy surface \ncharacterize these adsorption configurations. \nA benzene molecule impinging on the \nSi surface is most likely to be adsorbed in one of \nthree different di-$\\sigma$-bonded, metastable structures, \ncharacterized by \ntwo C-Si bonds, and eventually converts into the \nlowest-energy configurations.\nThese results are consistent with recent experiments.\n\n\\end{abstract}\n\\vspace{0.5cm}\n]\n\\pacs{PACS numbers: 82.65.My, 68.35.Bs, 68.45.Da, 71.15.Pd}\n\n\\narrowtext\n\nAdsorption of benzene on the Si(100) surface \nis a topic of great current interest\n\\cite{Taguchi,Craig,Jeong,Konecny,Gokhale,Borovsky,Self,Lopinski,Kong}\nboth because it represents a prototype system for the study\nof molecular adsorption (and desorption) of hydrocarbons\non semiconductor surfaces, and \nbecause it is considered a promising precursor for \ntechnologically relevant processes, such as\nthe growth of Si-C and CVD diamond thin films on Si surfaces.\nHowever, despite many experimental and theoretical investigations,\nthe adsorption mechanism is not yet well understood.\nIn particular, at present there is no consensus \nabout the lowest-energy structure of benzene on Si(100):\nresults obtained from surface science \nexperimental techniques, semiempirical methods, and first-principles\napproaches provide a number of different predictions.\n\nBenzene is known from experiments to \nadsorb exclusively on top of the Si(100) surface dimer rows, \nthus avoiding energetically disfavored structures \nwith unsaturated, isolated Si \ndangling bonds.\nEven so, since the size of the benzene molecule is comparable to the spacing\nbetween two adjacent dimers on the same row, many different bonding\nconfigurations are possible. \nAmong the structures proposed in the literature \nas the lowest-energy configurations,\nthe 1,4-cyclohexadiene-like (``butterfly'') configuration,\nin which the benzene molecule is di-$\\sigma$ bonded to\nthe two dangling bonds of the same Si surface dimer, \nis supported by thermal desorption and angle-resolved photoelectron\nspectroscopy\\cite{Gokhale}, STM\\cite{Self}, vibrational IR spectroscopy \nand near-edge X-ray absorption fine structure techniques\\cite{Kong}, and\nfirst-principles cluster calculations\\cite{Gokhale}.\nInstead, other STM experiments\\cite{Borovsky} suggest the\n1,3-cyclohexadiene-like (``tilted'') structure.\nFinally, semiempirical calculations\\cite{Jeong,Lopinski}, STM and\nIR spectroscopy experiments \\cite{Lopinski} favor\na tetra-$\\sigma$-bonded configuration where benzene is\nbonded to two adjacent surface dimers.\n\nAnother open issue concerns the occurrence and nature of metastable\nadsorption states. In fact, the results of STM and \nIR spectroscopy\\cite{Borovsky,Lopinski} \nsupport the hypothesis that benzene is initially \nchemisorbed in a metastable, ``butterfly''-like state, and then slowly \nconverts (within minutes) to a lower-energy final state, which is \na ``tilted'' structure according to Ref. \\onlinecite{Borovsky}, \nor a tetra-$\\sigma$-bonded one according to Ref. \\onlinecite{Lopinski}.\nMoreover, recent IR experiments \\cite{Kong} \nsuggest that, at room temperature, benzene\nis predominantly adsorbed in the ``butterfly'' configuration, \nwhile the existence of a less stable structure,\nconsistent with a tetra-$\\sigma$-bonded configuration, is proposed.\n\nPrevious theoretical calculations on benzene on Si(100) \nhave been restricted to semiempirical or {\\em ab initio} \ncluster-model methods. \nIn the latter approach\nthe Si surface is modeled with \na cluster of Si atoms, thus considerably reducing the\ncost of a first-principles calculation. However, the effects of such an \napproximation can be relevant. It is well known, for instance, that\nthe characteristic buckling of the Si dimers on the\nclean Si(100) surface can only be obtained by using models with\na slab geometry and periodic boundary conditions.\nAs shown in the following, the details of the surface reconstruction \n(i.e. buckling and periodicity of the surface dimers) are\ncrucial ingredients in determining the adsorption structure\nof benzene.\nMoreover, the convergence\nof different properties, such as the binding energies of adsorbed\nmolecules, is rather slow as a function of\nthe cluster size. \n\nIn order to overcome these limitations and to clarify the open issues\ndiscussed above we have performed a full \n{\\em ab initio} study of benzene adsorption on Si(100).\nTotal-energy calculations and molecular dynamics \n(MD) simulations have\nbeen carried out within the Car-Parrinello approach\\cite{CP,CPMD}\nin the framework of the density\nfunctional theory, in the local spin density approximation. \nTests have been also performed using gradient\ncorrections in the BLYP implementation\\cite{BLYP}.\nThe calculations have been carried out considering the $\\Gamma$-point \nonly of the Brillouin zone, and using norm-conserving \npseudopotentials\\cite{Troullier}, with $s$ and $p$ nonlocality for\nC and Si. Wavefunctions were expanded in plane waves with an\nenergy cutoff of 35 Ry.\n\nThe Si(100) surface is modeled with a periodically repeated slab\nof 5 Si layers and a vacuum region of 7 \\AA~ (tests\nhave been also carried out with a vacuum region of 10 \\AA~, without\nany significant change in the results).\nA monolayer of hydrogen atoms is used to saturate the dangling bonds on the\nlower surface of the slab. \nWe have used a supercell with \n$p(\\sqrt8\\times\\sqrt8)R45^{\\circ}$ surface periodicity, corresponding to \n8 Si atoms/layer; however, in order to check\nfinite-size effects, the geometry optimizations have been repeated\nusing a larger $p(4\\times4)$ supercell with 16 atoms/layer.\n\nStructural relaxations of the ionic coordinates are performed \nusing the method of direct inversion in the iterative subspace \\cite{DIIS}. \nDuring ionic relaxations and MD simulations\nthe lowest Si layer and the saturation hydrogens are kept fixed.\nWe verified that, by starting with the unreconstructed,\nclean Si(100) surface, the structural optimization procedure\ncorrectly produces \nasymmetric surface dimers, with a dimer bond length and \nbuckling angle in good agreement with previous, highly\nconverged {\\em ab initio} calculations \\cite{Bertoni}.\nWe have considered different surface periodicities for the dimer reconstruction\nwhich may occur on the Si(100)surface, i.e. \n$(2\\times 1)$, $p(2\\times 2)$ and $c(4\\times 2)$.\nA single benzene molecule is added on top of the slab and\nthe system is then fully relaxed towards the minimum energy configuration.\nTo better explore the complex potential energy surface of this system,\nin most of the cases the optimization procedure\nwas repeated using a simulated-annealing strategy and also starting\nfrom different initial configurations.\n\nWe find that the lowest-energy configurations are given by two\ntetra-$\\sigma$-bonded structures, characterized by the presence\nof one C-C double bond, which we refer to as \n``tight bridge'' (TiB) and\n``twisted bridge'' (TwB) (see Fig. \\ref{struc7}).\nTwB is similar to TiB but the benzene molecule is rotated\nby 90$^{\\circ}$ with respect to the Si surface and is slightly\nhigher in energy (see Table I).\nThis result is in agreement with the findings of Ref. \\onlinecite{Lopinski}\nand turns out to be\nindependent on the size of the supercell used in the\nsimulation and on the different reconstructions of the\nSi(100) surface. It remains true also using BLYP gradient corrections, \nas can be seen in Table I.\n\nWe also find, at somewhat higher energies, three different, \n{\\em metastable} ``butterfly'' structures, characterized by two C-Si\nbonds, which are shown in Fig. \\ref{struc7}.\nOne of them (``standard butterfly'', SB) is the well-known \nconfiguration with the\nbenzene molecule adsorbed on top of a single Si dimer.\nThe others (``tilted-bridge butterfly'', TB, and ``diagonal-bridge \nbutterfly'', DB), which bridge two adjacent surface dimers,\nhave not been reported in any previous study.\n\nThe Si(100) reconstruction crucially affects the occurrence and\nenergetic ordering of the three ``butterfly'' structures.\nIn fact, in the $(2\\times 1)$ reconstruction (with parallel buckled \ndimers), SB and TB are\nthe most stable (almost isoenergetic) ``butterfly'' configurations, while\nDB is considerably less favored; in contrast, with \nreconstructions involving alternating buckled Si dimers,\nsuch as the $p(2\\times 2)$ and the $c(4\\times 2)$, SB and DB\nare the lowest-energy configurations, while the binding energy of TB is\nsignificantly smaller.\nThis clearly happens because the two C-Si bonds\nof the TB structure are more easily created when the benzene molecule\nis adsorbed onto Si(100) $(2\\times 1)$, while the formation \nof the DB structure is favored by the presence of alternating buckled \nSi dimers.\n\nThe other configurations proposed in the literature, that is \nthe ``tilted'' (T) and the ``pedestal'' (P) ones, \nlie higher in energy for all the Si(100) reconstructions considered\n(see Table \\ref{energy}). In particular,\nthe P structure is only found to be stable in the\n$(2\\times 1)$ reconstruction; however, even in this case,\na MD simulation performed at 300 K shows that the \nstructure converts very rapidly (in less than 1 ps) into a DB structure.\nAlthough the P structure has four C-Si bonds,\nit is energetically disfavored because\nit involves the presence of two radical centers.\n\nInspection of the C-C distances for the various stable structures \nreveals the existence of two kind of bonds: a long one (\"single\") \nand a short one (\"double\"), of length $1.49-1.59$ and\n$1.34-1.36$ \\AA, respectively. These values should be compared with\nthe C-C bond length in the isolated benzene molecule, \n1.39 \\AA.\nOne double bond characterizes the TiB and TwB structures,\nwhile two double bonds are found in the ``butterfly'' structures.\nIn contrast, in the P configuration all the C-C bonds are single ones.\nThese conclusions are confirmed by a more quantitative \nanalysis of the electronic orbitals, which we \nperformed by using both the notion of Mayer bond order\\cite{Mayer}\nand the method of the Localized Wannier functions \\cite{Wannier}.\nIn the three ``butterfly'' configurations (SB, TB, DB),\nthe bond angles ($119^{\\circ}$-$122^{\\circ}$) at the C atoms not involved \nin the Si-C bonds are close to that ($120^{\\circ}$)\nof the isolated benzene molecule, while those ($103^{\\circ}$-$113^{\\circ}$) \nat the 4-fold coordinated C atom are closer\nto the ideal tetrahedron ($109.5^{\\circ}$) angle.\nThis clearly indicates $sp^2$ and $sp^3$ hybridization, respectively. \nAfter benzene chemisorption, although the Si-Si dimers are \npreserved, the Si dimer buckling angle is almost reduced to zero,\nwith the exception of the TB and DB structures.\nIn the lowest-energy TiB structure the\nangle between the double bond and the Si(100) surface\nis $45^{\\circ}$ in good agreement with the experimental\nestimate \\cite{Kong}, $\\sim 43^{\\circ}$.\n\nThe structural parameters do not change appreciably when\na larger $p(4\\times4)$ surface supercell is used.\nUse of BLYP gradient corrections makes \nbond lengths about 1-2 \\% longer, while binding energies \nare significantly reduced (see Table \\ref{energy}). \nMoreover, in the $(2\\times 1)$ reconstruction, the P\nconfiguration is no longer stable and, \namong the three ``butterfly'' structures, BLYP favors\nSB, while the binding energy of DB is even smaller than that of the\nT structure.\nNote, however, that TiB and TwB remain the lowest-energy configurations.\n\nAccording to the results of some experiments and theoretical \ncalculations\\cite{Lopinski,Kong}, adsorbed benzene predominantly forms\na ``butterfly'' (SB) configuration, while the TiB one (and perhaps TwB)\nappears in detectable amounts on relatively long timescales only,\nthus indicating the existence of an energy barrier between the two\nstructures.\n\nIn order to identify possible metastable states, occurring in the early\nstages of adsorption, we have tried to find, in the simplest way,\nthe most probable\nstructure of a benzene molecule impinging on the Si(100) surface.\nIf we place the molecule at some distance from the surface\nwe observe that, regardless of the initial position and orientation of\nthe molecule, after full relaxation the final structure \nis almost invariably one of the three \n``butterfly'' configurations.\nThis happens because the dimers are tilted, favoring\nthe formation of the di-bonded ``butterfly'' structures rather than\nthe tetra-bonded ones.\nThe specific ``butterfly'' configuration which is actually \nformed depends critically on the type of\nreconstruction of the Si surface that is considered,\nas already discussed above.\nOn the contrary, there are only very few initial positions which \nlead to the low-energy TiB and TwB configurations.\n\nWe have tried to characterize the energy barrier which must be overcome to\nrelax from the ``butterfly'' configurations\nto the lower-energy TiB and TwB structures. \nTo this aim we started with the benzene molecule \nin the SB configuration.\nLet C$_d$ be one of the C atoms involved in the Si-C bonds.\nMany calculations have been performed in which \nthe ionic coordinates of both the molecule\nand the substrate were optimized under the constraint \nthat the $x,y$ coordinates of the two C$_d$ atoms are held fixed.\nA particular pathway, connecting the SB to the DB structure, \nis shown in Fig. 2, where the reaction coordinate is defined as \nthe distance between the C$_d$-C$_d$ axis of the initial configuration\nand that of the displaced structure.\nThe pronounced energy minimum corresponds to the occurrence, \nduring the transformation, of the lowest-energy Tib structure. Note\nhowever that this is characterized by a very narrow well. \nFrom Fig. 2 a lower bound of $\\sim 0.5$ eV can be inferred for the\nenergy barrier, to be compared with the experimental \nestimates\\cite{Borovsky,Lopinski}, $\\sim$ 0.9-1.0 eV.\nA similar calculation for the TB$\\rightarrow$TwB transition gives\na smaller value of $\\sim 0.4$ eV. As a consequence the conversion\nfrom TB to TwB is expected to be somewhat faster\nthan that from SB to TiB.\n\nA large fraction of experiments on benzene on Si(100)\nis based on STM techniques. However, different interpretations of \nsimilar STM images\nled to contradictory conclusions\\cite{Borovsky,Self,Lopinski}\nabout the adsorption sites and geometry of the adsorbed molecules.\nFor each of the structures reported in Table I we have produced\n``theoretical'' STM images to be compared with the experimental ones,\nfollowing the recipe of Ref. \\onlinecite{Takeuchi}.\nCharge density iso-surfaces have been obtained by including \nelectron states in an\nenergy range down to $\\sim 2$ eV below the highest \noccupied state, which\ncorresponds to typical STM bias voltages. The simulated\nimages are obtained by viewing these iso-surfaces at typical \ntip-surface distances (a few \\AA\\ above the benzene molecule).\n\nOur computed STM image for the TiB structure exhibits a density\nmaximum above one of the two Si dimers involved in bonding with\nbenzene, while the TwB configuration produces a similar\nimage but rotated by 90$^{\\circ}$.\nThese images resemble those obtained by Lopinski \n{\\em et al.}\\cite{Lopinski}.\nThe theoretical STM image for the SB structure is characterized\nby a bright two-lobe protrusion centered symmetrically above \na single Si dimer unit and\noriented orthogonal to the dimer axis, in qualitative agreement with\nthe experimental findings\\cite{Borovsky,Self,Lopinski}.\nInstead, the STM images of the TB and DB structures are quite different\nfrom that of SB. In fact the TB image is qualitatively similar to \nthat of TwB\n(and the experimental STM resolution could be insufficient to \ndistinguish between the two configurations), while\nDB gives rise to a much fainter feature, \nbridging in diagonal two Si dimers,\nwhich is probably hardly visible in experiments.\nThese observations could explain why the DB and TB structures\nhave not been detected in STM experiments.\nThe T configuration produces an asymmetric\n(with respect to Si dimers) image, appearing as \na bright region (placed between two Si dimers) adjacent to a dark region. \nFinally the P structure is characterized \nby two spots corresponding to the \ndangling bonds of benzene; this result supports the conjecture\n\\cite{Self} which rules out the presence of a\nsignificant fraction of benzene molecules adsorbed in the \nP structure because of the absence of such spots in the STM images.\n\nWe have also computed the vibrational spectra for a representative \n``butterfly'' structure, SB, and for the lowest-energy TiB configuration, \nby performing Car-Parrinello MD simulations at room temperature. \nOur results for TiB show a slightly more \nquantitative agreement with the experimental results\n\\cite{Taguchi,Kong} than those for the SB structure, although \nthe main features of the spectra are similar in the two structures.\nLet C$\\,''$ (C$\\,'$) denote a C atom which shares a double\n(single) bond with another C atom.\nThe C$\\,'$-H and C$\\,''$-H frequencies (2880 and 3010 cm$^{-1}$) are \nin agreement with\nthe $sp^3$ and $sp^2$ stretching modes observed\nin recent IR spectroscopy experiments \\cite{Kong}\n(2945 and 3044 cm$^{-1}$), and semiempirical\ncluster calculations \\cite{Lopinski}.\nNote that the C-H vibrations for the isolated benzene molecule are \ncharacterized by a single frequency of 3040 cm$^{-1}$.\nFor the C$\\,'$-C$\\,''$ and C$\\,''$-C$\\,''$ frequencies we find \n1230 and 1520 cm$^{-1}$, respectively, to be compared with \nthe EELS experimental values \\cite{Taguchi}, 1170 and 1625 cm$^{-1}$.\nThe C-H bending modes are found at 900 and 1100 cm$^{-1}$,\nwhereas experimentally \\cite{Taguchi} they are at 910 and 1075 cm$^{-1}$. \n\nIn conclusion, using state-of-the-art {\\em ab initio} simulations,\nwe have shown that a tetra-$\\sigma$ bonded structure \nis the most stable configuration for benzene adsorbed on Si(100).\nHowever, this structure and a very similar one, lying only slightly \nhigher in energy, correspond\nto very narrow wells in the potential energy surface for\na benzene molecule impinging on the surface. Therefore\nit is more likely for the molecule to be adsorbed\ninto one of three different, metastable ``butterfly'' configurations,\nand eventually convert into the lowest-energy structures.\nOur study provides detailed information about structural,\nelectronic, and vibrational properties of the system,\nand allows a critical comparison with results obtained \nfrom different experimental techniques and previous theoretical\ncalculations. \n\nWe thank M. Boero and A. Vittadini for useful discussions.\nThis work is partially supported by INFM\nthrough the Parallel Computing Initiative.\n\n\\begin{references}\n\\bibitem{Taguchi} Y. Taguchi, M. Fujisawa, T. Takaoka, T. Okada, \n M. Nishijima,\n J. Chem. Phys. {\\bf 95}, 6870 (1991).\n\\bibitem{Craig} B. I. Craig,\n Surf. Sci. {\\bf 280}, L279 (1993).\n\\bibitem{Jeong} H. D. Jeong, S. Ryu, Y. S. Lee, S. Kim,\n Surf. Sci. {\\bf 344}, L1226 (1995).\n\\bibitem{Konecny} R. Kone\\v{c}n\\'{y}, D. J. Doren,\n Surf. Sci. {\\bf 417}, 169 (1998).\n\\bibitem{Gokhale} S. Gokhale, P. Trischberger, D. Menzel, W. Widdra, \n H. Dr\\\"oge, H.-P. Steinr\\\"uck, U. Birkenheuer,\n U. Gutdeutsch, N. R\\\"osch,\n J. Chem. Phys. {\\bf 108}, 5554 (1998);\n U. Birkenheuer, U. Gutdeutsch, N. R\\\"osch,\n Surf. Sci. {\\bf 409}, 213 (1998).\n\\bibitem{Borovsky} B. Borovsky, M. Krueger, E. Ganz,\n Phys. Rev. B {\\bf 57}, R4269 (1998). \n\\bibitem{Self} K. W. Self, R. I. Pelzel, J. H. G. Owen, C. Yan,\n W. Widdra, W. H. Weinberg,\n J. Vac. Sci. Technol. A {\\bf 16}, 1031 (1998). \n\\bibitem{Lopinski} G. P. Lopinski, D. J. Moffatt, R. A. Wolkow,\n Chem. Phys. Lett. {\\bf 282}, 305 (1998);\n G. P. Lopinski, T. M. Fortier, D. J. Moffatt, \n R. A. Wolkow,\n J. Vac. Sci. Technol. A {\\bf 16}, 1037 (1998);\n R. A. Wolkow, G. P. Lopinski, D. J. Moffatt,\n Surf. Sci. {\\bf 416}, L1107 (1998). \n\\bibitem{Kong} M. J. Kong, A. V. Teplyakov, J. G. Lyubovitsky,\n S. F. Bent, \n Surf. Sci. {\\bf 411}, 286 (1998).\n\\bibitem{CP} R. Car and M. Parrinello,\n Phys. Rev. Lett. {\\bf 55}, 2471 (1985).\n\\bibitem{CPMD} We have used the code CPMD, versions 3.0 and 3.3, developed\n by J. Hutter {\\em et al.}, at MPI f\\\"ur\n Festk\\\"orperforschung and IBM Research Laboratory\n (1990-1999).\n\\bibitem{BLYP} A. D. Becke,\n Phys. Rev. A {\\bf 38}, 3098 (1988);\n C. Lee, W. Yang, and R. C. Parr,\n Phys. Rev. B {\\bf 37}, 785 (1988).\n\\bibitem{Troullier} N. Troullier and J. Martins,\n Phys. Rev. B {\\bf 43}, 1993 (1991).\n\\bibitem{DIIS} J. Hutter, H. P. L\\\"uthi, and M. Parrinello,\n Comput. Mat. Sci. {\\bf 2}, 244 (1994).\n\\bibitem{Bertoni} A. I. Shkrebtii, R. Di Felice, C. M. Bertoni, R. Del Sole,\n Phys. Rev. B {\\bf 51}, 11201 (1995);\n S. C. A. Gay and G. P. Srivastava,\n Phys. Rev. B {\\bf 60}, 1488 (1999), and further references\n quoted therein.\n\\bibitem{Mayer} I. Mayer, \n Chem. Phys. Lett. {\\bf 97}, 270 (1983).\n\\bibitem{Wannier} N. Marzari, D. Vanderbilt,\n Phys. Rev. B {\\bf 56}, 12847 (1997);\n P. L. Silvestrelli, N. Marzari, D. Vanderbilt,\n M. Parrinello,\n Solid St. Comm. {\\bf 107}, 7 (1998);\n P. L. Silvestrelli,\n Phys. Rev. B {\\bf 59}, 9703 (1999).\n\\bibitem{Takeuchi}N. Takeuchi,\n Phys. Rev. B {\\bf 58}, R7504 (1998).\n\n\\end{references}\n\n\\vfill\n\\eject\n\n\\begin{figure}\n\\caption{The stable structures of benzene adsorbed on Si(100):\nSB=``standard butterfly'', TB=``tilted-bridge butterfly'',\nDB=``diagonal-bridge butterfly'', T=``tilted'', P=``pedestal'',\nTiB=``tight bridge'', TwB=``twisted bridge''.\nFor clarity only the four Si atoms of two dimers and\nfour belonging to the second layer are shown.} \n\\label{struc7}\n\\end{figure}\n\n\\begin{figure}\n\\caption{Total energy along the pathway obtained by shifting the benzene\nmolecule along a dimer row from the SB (at the origin)\nto the DB configuration, going through the lowest-energy TiB configuration\n(on the bottom of the narrow well). \nA $p(\\sqrt8\\times\\sqrt8)R45^{\\circ}$ supercell with a $p(2\\times 2)$\nsurface reconstruction has been used.\nData are represented by symbols, while the line\nis just a guide for the eye.\nThe energies are relative to the SB structure.}\n\\label{path}\n\\end{figure}\n\n\\vfill\n\\eject\n\n\\begin{table}\n\\caption{Binding energies (in eV) of different configurations for\nbenzene adsorbed on Si(100) in the $(2\\times 1)$ and $c(4\\times 2)$\nreconstructions (the nomenclature is the same as in Fig. 1).\nThe $p(\\sqrt8\\times\\sqrt8)R45^{\\circ}$ supercell was used; $L$ denotes\nresults obtained with the larger $p(4\\times4)$ supercell and BLYP\nmeans application of BLYP gradient corrections [12]. \nA missing entry\nindicates that a stable configuration was not obtained by the optimization\nprocess.}\n\\begin{tabular}{lllll}\nConfiguration&$(2\\times 1)$&$(2\\times 1)$ $L$&$(2\\times 1)$ BLYP&$c(4\\times 2)$ $L$\\\\ \\tableline\nSB&2.04&2.06&1.22&2.20 \\\\\nTB&2.10&2.08&1.12&1.99 \\\\\nDB&1.63&1.70&0.41&2.24 \\\\\nT& 1.50&1.55&0.77&1.68 \\\\\nP& 1.51&1.60&---&--- \\\\\nTiB& 2.68&2.77&1.53&2.65 \\\\\nTwB& 2.47&2.53&1.31&2.38 \\\\\n\\end{tabular}\n\\label{energy}\n\\end{table}\n\n\\vfill\n\\eject\n\n\\end{document}\n\n"
}
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[
{
"name": "cond-mat0002007.extracted_bib",
"string": "\\bibitem{Taguchi} Y. Taguchi, M. Fujisawa, T. Takaoka, T. Okada, \n M. Nishijima,\n J. Chem. Phys. {\\bf 95}, 6870 (1991).\n\n\\bibitem{Craig} B. I. Craig,\n Surf. Sci. {\\bf 280}, L279 (1993).\n\n\\bibitem{Jeong} H. D. Jeong, S. Ryu, Y. S. Lee, S. Kim,\n Surf. Sci. {\\bf 344}, L1226 (1995).\n\n\\bibitem{Konecny} R. Kone\\v{c}n\\'{y}, D. J. Doren,\n Surf. Sci. {\\bf 417}, 169 (1998).\n\n\\bibitem{Gokhale} S. Gokhale, P. Trischberger, D. Menzel, W. Widdra, \n H. Dr\\\"oge, H.-P. Steinr\\\"uck, U. Birkenheuer,\n U. Gutdeutsch, N. R\\\"osch,\n J. Chem. Phys. {\\bf 108}, 5554 (1998);\n U. Birkenheuer, U. Gutdeutsch, N. R\\\"osch,\n Surf. Sci. {\\bf 409}, 213 (1998).\n\n\\bibitem{Borovsky} B. Borovsky, M. Krueger, E. Ganz,\n Phys. Rev. B {\\bf 57}, R4269 (1998). \n\n\\bibitem{Self} K. W. Self, R. I. Pelzel, J. H. G. Owen, C. Yan,\n W. Widdra, W. H. Weinberg,\n J. Vac. Sci. Technol. A {\\bf 16}, 1031 (1998). \n\n\\bibitem{Lopinski} G. P. Lopinski, D. J. Moffatt, R. A. Wolkow,\n Chem. Phys. Lett. {\\bf 282}, 305 (1998);\n G. P. Lopinski, T. M. Fortier, D. J. Moffatt, \n R. A. Wolkow,\n J. Vac. Sci. Technol. A {\\bf 16}, 1037 (1998);\n R. A. Wolkow, G. P. Lopinski, D. J. Moffatt,\n Surf. Sci. {\\bf 416}, L1107 (1998). \n\n\\bibitem{Kong} M. J. Kong, A. V. Teplyakov, J. G. Lyubovitsky,\n S. F. Bent, \n Surf. Sci. {\\bf 411}, 286 (1998).\n\n\\bibitem{CP} R. Car and M. Parrinello,\n Phys. Rev. Lett. {\\bf 55}, 2471 (1985).\n\n\\bibitem{CPMD} We have used the code CPMD, versions 3.0 and 3.3, developed\n by J. Hutter {\\em et al.}, at MPI f\\\"ur\n Festk\\\"orperforschung and IBM Research Laboratory\n (1990-1999).\n\n\\bibitem{BLYP} A. D. Becke,\n Phys. Rev. A {\\bf 38}, 3098 (1988);\n C. Lee, W. Yang, and R. C. Parr,\n Phys. Rev. B {\\bf 37}, 785 (1988).\n\n\\bibitem{Troullier} N. Troullier and J. Martins,\n Phys. Rev. B {\\bf 43}, 1993 (1991).\n\n\\bibitem{DIIS} J. Hutter, H. P. L\\\"uthi, and M. Parrinello,\n Comput. Mat. Sci. {\\bf 2}, 244 (1994).\n\n\\bibitem{Bertoni} A. I. Shkrebtii, R. Di Felice, C. M. Bertoni, R. Del Sole,\n Phys. Rev. B {\\bf 51}, 11201 (1995);\n S. C. A. Gay and G. P. Srivastava,\n Phys. Rev. B {\\bf 60}, 1488 (1999), and further references\n quoted therein.\n\n\\bibitem{Mayer} I. Mayer, \n Chem. Phys. Lett. {\\bf 97}, 270 (1983).\n\n\\bibitem{Wannier} N. Marzari, D. Vanderbilt,\n Phys. Rev. B {\\bf 56}, 12847 (1997);\n P. L. Silvestrelli, N. Marzari, D. Vanderbilt,\n M. Parrinello,\n Solid St. Comm. {\\bf 107}, 7 (1998);\n P. L. Silvestrelli,\n Phys. Rev. B {\\bf 59}, 9703 (1999).\n\n\\bibitem{Takeuchi}N. Takeuchi,\n Phys. Rev. B {\\bf 58}, R7504 (1998).\n\n"
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cond-mat0002008
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Hysteresis effect due to the exchange Coulomb interaction in short-period superlattices in tilted magnetic fields
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"author": "Andrei Manolescu$^1$ and Vidar Gudmundsson$^2$"
}
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We calculate the ground-state of a two-dimensional electron gas in a short-period lateral potential in magnetic field, with the Coulomb electron-electron interaction included in the Hartree-Fock approximation. For a sufficiently short period the dominant Coulomb effects are determined by the exchange interaction. We find numerical solutions of the self-consistent equations that have hysteresis properties when the magnetic field is tilted and increased, such that the perpendicular component is always constant. This behavior is a result of the interplay of the exchange interaction with the energy dispersion and the spin splitting. We suggest that hysteresis effects of this type could be observable in magneto-transport and magnetization experiments on quantum-wire and quantum-dot superlattices.
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"name": "lhyst4.tex",
"string": "\\documentstyle[aps,twocolumn,prb,epsf]{revtex}\n%\\documentstyle[aps,preprint,prb,epsfig,rotating]{revtex}\n\n\\draft\n\\preprint{Version \\today}\n\\title{Hysteresis effect due to the exchange Coulomb interaction in \nshort-period superlattices in tilted magnetic fields}\n\n\\author{Andrei Manolescu$^1$ and Vidar Gudmundsson$^2$}\n\n\\address{\n$^1$Institutul Na\\c{t}ional de Fizica Materialelor, C.P. MG-7 \nBucure\\c{s}ti-M\\u{a}gurele, Rom\\^ania,\\\\\n$^2$Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, \nIceland}\n\n\\begin{document}\n%\\tighten\n\\maketitle\n\n\n\\begin{abstract} \n\nWe calculate the ground-state of a two-dimensional electron gas in\na short-period lateral potential in magnetic field, with the Coulomb\nelectron-electron interaction included in the Hartree-Fock approximation.\nFor a sufficiently short period the dominant Coulomb effects are\ndetermined by the exchange interaction. We find numerical solutions\nof the self-consistent equations that have hysteresis properties when\nthe magnetic field is tilted and increased, such that the perpendicular\ncomponent is always constant. This behavior is a result of the interplay\nof the exchange interaction with the energy dispersion and the\nspin splitting. We suggest that hysteresis effects of this type could\nbe observable in magneto-transport and magnetization experiments on \nquantum-wire and quantum-dot superlattices.\n\n\\end{abstract}\n\n\\pacs{71.45.Gm,71.70.Di,73.20.Dx}\n\n\\draft\n\nA well known manifestation of the Coulomb exchange interaction in a\ntwo-dimensional electron gas (2DEG) in a perpendicular magnetic field\nis the enhancement of the Zeeman splitting for odd-integer filling\nfactors,\\cite{Ando74:1044} observable in magetotransport experiments\non GaAs systems. The same mechanism leads to the enhancement of the Landau\ngaps for even-integer filling factors, which can be identified in more\nrecent magnetization measurements.\\cite{Meinel99:819}\n\nIn the presence of a periodic potential the Landau levels become\nperiodic Landau bands, and the calculations based on the Hartree-Fock\napproximation (HFA) show an enhancement of the energy dispersion of the\nbands intersected by the Fermi level.\\cite{Manolescu95:1703} Such an effect\nhas been indirectly observed in the magnetoresistance of short-period\nsuperlattices as an abrupt onset of the spin splitting of the Shubnikov-de\nHaas peaks, occurring only for a sufficiently \nstrong magnetic field. \\cite{Petit97:225} \nIn other words, when the magnetic field increases the\nsystems makes a first-order phase transition from spin-unpolarized to\nspin-polarized states. This effect has also been discussed in other\nforms, for narrow quantum wires, \\cite{Kinaret90:11768} and for edge\nstates.\\cite{Dempsey93:3639,Rijkels94:8629}\n\nIn the spirit of the HFA, the Coulomb interaction can be split into a\ndirect and an exchange component. The direct (classical) interaction\nis repulsive (i.\\ e.\\ positive) and long ranged, while the exchange\n(quantum mechanical) part is attractive (i.\\ e.\\ negative) and short ranged.\nThe direct component is usually much larger than the exchange one. In our\nsystem this is decided by the two lengths involved, the superlattice (or\nmodulation) period $a$, and the magnetic length $\\ell=\\sqrt{\\hbar/eB_0}$\ndetermined by the perpendicular magnetic field $B_0$. For long periods,\n$a\\gg\\ell$, the screening (direct) effects are strong: the width of the\nLandau bands is typically much smaller than the amplitude of the periodic\npotential, except when a gap is eventually present at the Fermi level.\n\\cite{Manolescu97:9707} For periods of the order of $\\ell$ the situation\nbecomes opposite: the screening effect is weak, the exchange interaction\nis the dominant Coulomb manifestation, and the energy dispersion of the\nLandau bands may exceed the amplitude of the periodic potential if the\nlatter is small enough.\\cite{Manolescu99:5426} \n\nIn a recent paper \\cite{Manolescu99:5426} we have studied the numerical\nsolutions of the Hartree-Fock equations in the presence of short-period\npotentials. After preparing the solution for a fixed potential we\nchange the potential amplitude by a small amount and find a new,\nperturbed solution, and then we change again the amplitude, and repeat\nthe scheme. In this way, by increasing and then decreasing the amplitude,\nwe obtain a hysteretic evolution of the ground state due to the combined\neffects of the external potential and of the exchange interaction, on the\nenergy dispersion of the Landau bands. In the present paper we consider\na fixed modulation amplitude, but a tilted magnetic field, such that we\ninclude in the problem, self-consistently, the Zeeman splitting of the\nLandau bands. We hereby intend to suggest further experiments that can \nidentify strong effects of the Coulomb exchange interaction. \nThe material parameters are those for\nGaAs: effective mass $m_{\\mathrm eff}=0.067 m_e$, dielectric constant\n$\\kappa=12.4$, bare g-factor $g=-0.44$, and electron concentration\n$n_s=2.4 \\times 10^{11}$ cm$^{-2}$.\n\nWe fix the component of the magnetic field perpendicular to the 2DEG,\n$B_0$, which determines our filling factors, while the bare Zeeman\nsplitting is given by the total field $B=B_0/\\cos\\phi$, where\n$\\phi$ is the tilt angle. We first consider a periodic potential \nvarying only along one spatial direction, $V\\cos Kx$, where $K=2\\pi/a$,\nand solve for the eigenstates of the Hamiltonian within the thermodynamic\nHFA. We chose the Landau gauge for the vector potential and we diagonalize\nthe Hamiltonian in the Landau basis \n$\\psi_{nX_0}(x,y)=L_y^{-1/2} e^{-iX_0y/\\ell^2} f_n(x-X_0)\\mid\\sigma\\rangle$, \nwhere $X_0$ is the so-called center coordinate, $L_y$ is the linear\ndimension of the 2DEG, $f_n(x-X_0)$ are shifted oscillator \nwave functions, and $\\sigma=\\pm1$ is the spin projection.\n\nWe begin our calculations with $\\phi=0$, and find the numerical\nHFA-eigenstates by an iterative method, starting from the noninteracting\nsolution. Then, we increase $\\phi$ and find a new solution starting\nfrom the previous one. In Fig.\\ 1(a) we show a typical energy spectrum,\ni.\\ e.\\ the Landau bands $E_{n\\sigma X_0}$, $n=0,1,2,...$, within the first\nBrillouin zone, for a small tilt angle, $\\phi < \\phi_1$. Here $B_0=4.1$\nT, and the parameters of the external potential are $a=40$ nm and $V=9$\nmeV.\n\nIn a simplified view, the exchange interaction contributes\nwith a negative amount of energy to the occupied states, which\nenhances the energy dispersion in the vicinity of the Fermi\nlevel.\\cite{Manolescu95:1703,Manolescu99:5426} The classical Hartree\n(positive) energy is small in our case, but it would increase with\nincreasing modulation period and would rapidly flatten the energy\nbands. Also, the spin splitting is almost suppressed for $\\phi < \\phi_1$.\nHowever, for a sufficiently high field, when $\\phi=\\phi_1$, the difference\nin population of the spin-up and spin-down bands exceeds a critical\nvalue, and the spin gap is abruptly amplified by the exchange energy.\nThe spin-up states become self-consistently more populated and lower\nin energy. The energy spectrum becomes like in Fig.\\ 1(b), and keeps\nthis structure when $\\phi$ further increases. Then, we decrease $\\phi$\nstep by step. For low temperatures we find for $\\phi=0$ the solution\nwith large spin gap, similar to Fig.\\ 1(b), while for higher temperatures\nwe may find a transition to the spin-unpolarized state, Fig.\\ 1(a),\nat $\\phi_2 < \\phi_1$.\n\nWe show in Fig.\\ 2 the spin polarization,\n$(n_{\\uparrow}-n_{\\downarrow})/(n_{\\uparrow}+n_{\\downarrow})$, for two\ntemperatures, when $\\phi_2>0$. We consider here the temperature \nonly as an effective parameter, that may also include includes the \neffects of a certain disorder, inherent in any real system. Clearly, \nin the presence of disorder similar results will appear for lower \ntemperatures.\n\nWe have explicitly included disorder in a transport calculation,\nby assuming Gaussian spectral functions, $\\rho_{n\\sigma}(E)=\n(\\Gamma\\sqrt{\\pi/2})^{-1}\\exp[-2(E-E_{n\\sigma})^2/\\Gamma^2]$,\nwhere $\\Gamma$ is the Landau level broadening. We have\ncalculated the conductivity tensor $\\sigma_{\\alpha\\beta}$,\n$\\alpha,\\beta=x,y$, using the standard Kubo formalism for the\nmodulated 2DEG. \\cite{Zhang90:12850,Manolescu97:9707} In Fig.\\\n3 we show the hysteresis loops for the longitudinal resistivities\n$\\rho_{xx,yy}=\\sigma_{yy,xx}/(\\sigma_{xx}\\sigma_{yy}+\\sigma_{xy}^2)$.\nIn our regime $\\sigma_{xy}^2\\gg\\sigma_{xx}\\sigma_{yy}$, such that\n$\\rho_{xx,yy}$ are in fact proportional to $\\sigma_{yy,xx}$. Also,\nthe conductivity in the $y$ direction is dominated by the quasi-free\nnet electron motion along the equipotential lines of the modulation,\nwith group velocity $<v_y>=-(eB_0)^{-1}dE_{nX_0}/dX_0$, known as band\nconductivity,\\cite{Aizin84:1469} \nwhile the conductivity in the $x$ direction is related to\ninter-band scattering processes (scattering conductivity). The band\nand the scattering conductivities are inverse and respectively direct\nproportional to a power of the density of states at the Fermi level\n(DOSF).\\cite{Zhang90:12850,Manolescu97:9707} In the transition between\nspin-unpolarized and spin-polarized states the Fermi level touches\nthe minima of the band $E_{1\\downarrow}$, where DOSF has a van Hove\nsingularity. Therefore, for that situation the band conductivity, and\nthus $\\rho_{xx}$, have a minimum, whereas the scattering conductivity,\nand thus $\\rho_{yy}$, have a maximum, see Fig.\\ 3.\n\nSimilar hysteresis effects can be found in a 2DEG that is modulated\nin two perpendicular spatial directions. \n\\cite{Manolescu99:5426,Gudmundsson95:16744} \nHowever, in this case the picture is further\ncomplicated by the presence of the Hofstadter\\cite{Hofstadter76:2239}\ngaps and their interplay with the spin gaps. Then, since the dispersion\nof the Landau bands is essential for the hysteresis, another complication\nwith the two-dimensional potential is that for an asymmetric unit cell\nthe behavior of the system may not be the same when the magnetic field is\ntilted towards the $x$ or towards the $y$ axis of the plane, reflecting\nthe anisotropy of the Brillouin zones. These details are not addressed in \nthis paper.\n\nIn principle, the effects discussed in the present paper should also\noccur in narrow quantum wires or dots, and not necessarily only in\nperiodic systems, as long as Landau bands with both flat and steep\nregions are generated, as in Fig. \\ 1. In wide wires or dots\nthe electrostatic screening is expected to dominate the exchange\ninteraction, just like in long-period superlattices, and thus the\nLandau bands are smooth, except at the edges. \nRijkels and Bauer \\cite{Rijkels94:8629}\nhave also predicted hysteresis effects in the edge channels of quantum\nwires when a small chemical-potential difference between the spin-up and\nspin-down channels can be controlled. To our knowledge an experimental\nconfirmation has not been reported yet. Instead, several groups have\nbuild short-period superlattices for transport and other experiments,\n\\cite{Petit97:225,Nakamura98:944,Schlosser96:683} which can also be used\nto check our predictions.\n\nIn conclusion, we have found a hysteresis property of the numerical\nsolution of the thermodynamic HFA, with the physical origin in the\nexchange effects of the Coulomb interaction in the quantum Hall regime,\nin the presence of a short-period potential, when the Zeeman splitting\nis changed by tilting the magnetic field with respect to the 2DEG.\nWe suggest that such effects could be observed e.\\ g.\\ in magnetization\nor magnetotransport measurements.\n\n\n\n\n\nA.\\ M.\\ was supported by a NATO fellowship at the Science Institute,\nUniversity of Iceland. The research was partly supported by\nthe Icelandic Natural Science Foundation, and the University of Iceland\nResearch Fund.\n\n\n\\begin{thebibliography}{}\n\n\\bibitem{Ando74:1044}\nT. Ando and Y. Uemura, \nJ. Phys. Soc. Jpn. {\\bf 37}, 1044 (1974).\n\n\\bibitem{Meinel99:819}\nI. Meinel, T. Hengstmann, D. Grundler, and D. Heitmann, \nPhys. Rev. Lett. {\\bf 82}, 819 (1999).\n\n\\bibitem{Manolescu95:1703}\nA. Manolescu and R. R. Gerhardts, \nPhys. Rev. B {\\bf 51}, 1703 (1995).\n\n\\bibitem{Petit97:225}\nF. Petit, L. Sfaxi, F. Lelarge, A. Cavanna, M. Hayne, and B. Etienne, \nEurophys. Lett. {\\bf 38}, 225 (1997).\n\n\\bibitem{Kinaret90:11768}\nJ. M. Kinaret and P. A. Lee,\nPhys. Rev. B {\\bf 42}, 11768 (1990).\n\n\\bibitem{Dempsey93:3639}\nJ. Dempsey, B. Y. Gelfand, and B. I. Halperin,\nPhys. Rev. Lett. {\\bf 70}, 3639 (1993).\n\n\\bibitem{Rijkels94:8629}\nL. Rijkels and G. W. Bauer, Phys. Rev. B {\\bf 50}, 8629 (1994).\n\n\\bibitem{Manolescu97:9707}\nA. Manolescu and R. R. Gerhardts, \nPhys. Rev. B {\\bf 56}, 9707 (1997).\n\n\\bibitem{Manolescu99:5426}\nA. Manolescu and V. Gudmundsson,\nPhys. Rev. B {\\bf 59}, 5426 (1999).\n\n\\bibitem{Zhang90:12850}\nC. Zhang and R. R. Gerhardts\nPhys. Rev. B {\\bf 41}, 12850 (1990).\n\n\\bibitem{Aizin84:1469} \nG. R. Aizin and V. A. Volkov,\nZh. Eksp. Teor. Fiz. {\\bf 87}, 1469 (1984)\n[Sov. Phys. JETP {\\bf 60}, 844 (1984)].\n\n\\bibitem{Gudmundsson95:16744}\nV. Gudmundsson and R. R. Gerhardts,\nPhys. Rev. B {\\bf 52}, 16744 (1995).\n\n\\bibitem{Hofstadter76:2239}\nR. D. Hofstadter,\nPhys. Rev. B {\\bf 14}, 2239 (1976).\n\n\\bibitem{Nakamura98:944}\nY. Nakamura, T. Inoshita, and H. Sakaki, \nPhysica E {\\bf 2}, 944 (1998).\n\n\\bibitem{Schlosser96:683}\nT. Schl\\\"osser, K. Ensslin, J. P. Kotthaus, and M. Holland,\nEurophys. Lett. {\\bf 33}, 683 (1996).\n\n\\end{thebibliography}\n\n%\\newpage\n\n\\vspace{-2.2cm}\n\\begin{figure}\n\\epsfxsize 13.5cm\n\\begin{center}\n \\epsffile{fig1.ps}\n\\end{center}\n\\vspace{-7cm}\n\\caption{Two energy spectra: (a) $\\cos\\phi=0$, (b) $\\cos\\phi=1/6$.\n The dashed lines show the Fermi level.} \n\\end{figure}\n%\n\\vspace{2cm}\n\\begin{figure}\n\\epsfxsize 9cm\n\\begin{center}\n \\epsffile{fig2.ps}\n\\end{center}\n\\caption{Hysteresis loops for the spin polarization for $T=5$~K, \n with solid line, and for $T=7$ K, with dashed line.} \n\\end{figure}\n%\n\\begin{figure}\n\\epsfxsize 10cm\n\\begin{center}\n \\epsffile{fig3.ps}\n\\end{center}\n\\vspace{-3cm}\n\\caption{Hysteresis loops for the resistivities $\\rho_{xx}$ and $\\rho_{yy}$.\n $T=2$ K, $\\Gamma=1$ meV.}\n\\end{figure}\n%\n\\end{document}\n"
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"name": "cond-mat0002008.extracted_bib",
"string": "\\begin{thebibliography}{}\n\n\\bibitem{Ando74:1044}\nT. Ando and Y. Uemura, \nJ. Phys. Soc. Jpn. {\\bf 37}, 1044 (1974).\n\n\\bibitem{Meinel99:819}\nI. Meinel, T. Hengstmann, D. Grundler, and D. Heitmann, \nPhys. Rev. Lett. {\\bf 82}, 819 (1999).\n\n\\bibitem{Manolescu95:1703}\nA. Manolescu and R. R. Gerhardts, \nPhys. Rev. B {\\bf 51}, 1703 (1995).\n\n\\bibitem{Petit97:225}\nF. Petit, L. Sfaxi, F. Lelarge, A. Cavanna, M. Hayne, and B. Etienne, \nEurophys. Lett. {\\bf 38}, 225 (1997).\n\n\\bibitem{Kinaret90:11768}\nJ. M. Kinaret and P. A. Lee,\nPhys. Rev. B {\\bf 42}, 11768 (1990).\n\n\\bibitem{Dempsey93:3639}\nJ. Dempsey, B. Y. Gelfand, and B. I. Halperin,\nPhys. Rev. Lett. {\\bf 70}, 3639 (1993).\n\n\\bibitem{Rijkels94:8629}\nL. Rijkels and G. W. Bauer, Phys. Rev. B {\\bf 50}, 8629 (1994).\n\n\\bibitem{Manolescu97:9707}\nA. Manolescu and R. R. Gerhardts, \nPhys. Rev. B {\\bf 56}, 9707 (1997).\n\n\\bibitem{Manolescu99:5426}\nA. Manolescu and V. Gudmundsson,\nPhys. Rev. B {\\bf 59}, 5426 (1999).\n\n\\bibitem{Zhang90:12850}\nC. Zhang and R. R. Gerhardts\nPhys. Rev. B {\\bf 41}, 12850 (1990).\n\n\\bibitem{Aizin84:1469} \nG. R. Aizin and V. A. Volkov,\nZh. Eksp. Teor. Fiz. {\\bf 87}, 1469 (1984)\n[Sov. Phys. JETP {\\bf 60}, 844 (1984)].\n\n\\bibitem{Gudmundsson95:16744}\nV. Gudmundsson and R. R. Gerhardts,\nPhys. Rev. B {\\bf 52}, 16744 (1995).\n\n\\bibitem{Hofstadter76:2239}\nR. D. Hofstadter,\nPhys. Rev. B {\\bf 14}, 2239 (1976).\n\n\\bibitem{Nakamura98:944}\nY. Nakamura, T. Inoshita, and H. Sakaki, \nPhysica E {\\bf 2}, 944 (1998).\n\n\\bibitem{Schlosser96:683}\nT. Schl\\\"osser, K. Ensslin, J. P. Kotthaus, and M. Holland,\nEurophys. Lett. {\\bf 33}, 683 (1996).\n\n\\end{thebibliography}"
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cond-mat0002009
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Observation of the cluster spin-glass phase in La$_{2-x}$Sr$_{x}$CuO$_{4}$ by anelastic spectroscopy
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"author": "F. Cordero"
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An increase of the acoustic absorption is found in La$_{2-x}$Sr$_{x}$CuO$% _{4} $ ($x=0.019$, 0.03 and 0.06) close to the temperatures at which freezing of the spin fluctuations in antiferromagnetic-correlated clusters is expected to occur. The acoustic absorption is attributed to changes of the sizes of the quasi-frozen clusters induced by the vibration stress through magnetoelastic coupling.
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"string": "\n\\documentstyle[amssymb,prb,aps]{revtex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%TCIDATA{Created=Sun Dec 05 11:50:09 1999}\n%TCIDATA{LastRevised=Fri Jan 28 17:03:03 2000}\n%TCIDATA{Language=American English}\n\n\\begin{document}\n\\title{Observation of the cluster spin-glass phase in La$_{2-x}$Sr$_{x}$CuO$_{4}$\nby anelastic spectroscopy}\n\\author{F. Cordero}\n\\address{CNR, Area di Ricerca di Tor Vergata, Istituto di Acustica ``O.M. Corbino``,\\\\\nVia del Fosso del Cavaliere 100, I-00133 Roma, and INFM, Italy}\n\\author{R. Cantelli, A. Paolone}\n\\address{Universit\\`{a} di Roma ``La Sapienza``, Dipartimento di Fisica, P.le A.\\\\\nMoro 2, I-00185 Roma, and INFM, Italy}\n\\author{M. Ferretti}\n\\address{Universit\\`{a} di Genova, Dipartimento di Chimica e Chimica Fisica,\\\\\nVia Dodecanneso 31, I-16146 Genova, and INFM, Italy}\n\\maketitle\n\n\\begin{abstract}\nAn increase of the acoustic absorption is found in La$_{2-x}$Sr$_{x}$CuO$%\n_{4} $ ($x=0.019$, 0.03 and 0.06) close to the temperatures at which\nfreezing of the spin fluctuations in antiferromagnetic-correlated clusters\nis expected to occur. The acoustic absorption is attributed to changes of\nthe sizes of the quasi-frozen clusters induced by the vibration stress\nthrough magnetoelastic coupling.\n\\end{abstract}\n\n%\\draft\n\\twocolumn\n\n\\section{INTRODUCTION}\n\nThe low doping region of the phase diagram of La$_{2-x}$Sr$_{x}$CuO$_{4}$ is\nattracting considerable interest, due to the appearance of unconventional\ncorrelated spin dynamics and ordering processes (for a review see Ref.~%\n\\onlinecite{RBC98}). In undoped La$_{2}$CuO$_{4}$ the Cu$^{2+}$ spins order\ninto a 3D antiferromagnetic (AF) state with the staggered magnetization in\nthe $ab$ plane.\\cite{VSM87} Doping by Sr rapidly destroys the long range AF\norder, with $T_{N}$ passing from 315$~$K to practically 0$~$K around $%\nx_{c}\\simeq 0.02$. Above this critical value of the Sr content no long range\nAF order is expected at finite temperature. There are also indications that\nthe holes are segregated into domain walls, sometimes identified as charge\nstripes, which separate hole-poor regions where the AF correlations build up.%\n\\cite{CBJ92,BCC95} The holes should be mobile along these ''charge rivers'',\nbut at low $x$ they localize near the Sr atoms below $\\sim 30~$K, causing a\ndistortion of the spin texture of the surrounding Cu$^{2+}$ atoms.\\cite\n{BCC95} For $x<x_{c}$ the spin distortions around the localized holes are\ndecoupled from the AF background, and freeze into a spin-glass (SG) state\nbelow $T_{f}\\left( x\\right) \\simeq \\left( 815~\\text{K}\\right) x$. For $%\nx>x_{c}$ a cluster spin-glass (CSG) state is argued to freeze below $%\nT_{g}\\left( x\\right) \\propto 1/x$ and AF correlations develop within the\ndomains defined by the charge walls, with the easy axes of the staggered\nmagnetization uncorrelated between different clusters. The formation of the\nSG and CSG states are inferred from sharp maxima in the $^{139}$La NQR\\cite\n{CBJ92,RBC98,JBC99} and $\\mu $SR\\cite{NBB98} relaxation rates, which\nindicate the slowing of the AF fluctuations below the measuring frequency ($%\n\\sim 10^{7}-10^{8}$~Hz in those experiments) on cooling, and from the\nobservation of irreversibility, remnant magnetization, and scaling behavior\nin magnetic susceptibility experiments.\\cite{CBK95,WUE99}\n\nHere we report the observation of a step-like increase of the low-frequency\nacoustic absorption close to the temperature at which the spin freezing\nprocess is detected in the NQR measurements. The absorption is ascribed to\nchanges of the sizes of the frozen clusters induced by the vibration stress\nthrough magnetoelastic coupling,\\cite{NB} or equivalently to the motion of\nthe walls between them.\n\n\\section{EXPERIMENTAL AND RESULTS}\n\nThe samples where prepared by standard solid state reaction as described in\nRef.~\\onlinecite{DFF94} and cut in bars approximately $40\\times 4\\times 0.6$%\n\\ mm$^{3}$. The final Sr contents and homogeneities where checked from the\ntemperature position and sharpness of the steps in the Young's modulus and\nacoustic absorption due to the tetragonal (HTT) / orthorhombic (LTO)\ntransition, which occurs at a temperature $T_{t}$ linearly decreasing with\ndoping.\\cite{Joh97} The transitions appear narrower in temperature than the\none of a Sr-free sample, indicating that the width was mostly intrinsic and\nnot due to Sr inhomogeneity, except for the sample at the lowest Sr content.%\n\\cite{76} The Sr concentrations estimated in this way turned out $%\nx=0.0185\\pm 0.0015$, $0.0315\\pm 0.0015$ and $0.0645\\pm 0.002$, in good\nagreement with the nominal compositions. In the following the samples will\nbe referred as $x=0.019$, 0.03 and 0.06.\n\nThe complex Young's modulus $E$ was measured by electrostatically exciting\neither of the lowest three flexural modes and detecting the vibration\namplitude by a frequency modulation technique. The elastic energy loss\ncoefficient (or reciprocal of mechanical $Q$) is related to the imaginary\npart $E^{\\prime \\prime }$ of $E$ by $Q^{-1}\\left( \\omega ,T\\right)\n=E^{\\prime \\prime }\\left( \\omega ,T\\right) /E^{\\prime }\\left( \\omega\n,T\\right) $, and it was measured by the decay of the free oscillations or\nthe width of the resonance peak.\n\nIn Fig. 1 the anelastic spectra of three samples with $x=0.019$, 0.03, 0.06\nbelow $16~$K measured exciting the first flexural mode are reported. A\nstep-like increase of the absorption is observed around or slightly below $%\nT_{g}$ (Ref. \\onlinecite{Joh97}) The gray arrows indicate the values of $%\nT_{g}$ in the magnetic phase diagrams deduced from NQR\\cite{CBJ92} (lower\nvalues) and $\\mu $SR\\cite{NBB98} (higher values) experiments, which are in\nagreement with the data in Ref. \\onlinecite{Joh97} (for the sample with $%\nx=0.019$ the $T_{g}\\left( x=0.02\\right) $ values are indicated). The black\narrows indicate the temperature of the maximum of the $^{139}$La NQR\nrelaxation rate measured on the same samples in a separate study,\\cite{CCC00}\nwhich indicate a freezing in the spin-glass phase, as discussed later. The\ncoincidence of the temperatures of the absorption steps with those of\nfreezing of the spin fluctuations suggest a correlation between the two\nphenomena.\n\nThe sample with $x=0.03$ was outgassed from excess O by heating in vacuum up\nto $790~$K, while the other two samples where in the as-prepared state,\ntherefore containing some interstitial O. The concentration $\\delta $ of\nexcess O is a decreasing function of $x$ (Ref. \\onlinecite{TR}) and should\nbe negligible for $x=0.06$ but not for $x=0.019$. This fact allowed us to\nobserve the absorption step singled out from the high-temperature tail of an\nintense peak that occurs at lower temperature (see the sharp rise of\ndissipation below 3$~$K for $x=0.019$ in Fig. 1). Such a peak has been\nattributed to the tunneling-driven tilt motion of a fraction of the O\noctahedra.\\cite{76,61} The LTO phase is inhomogeneous on a local scale,\\cite\n{61,HSH96,BBK99,68} and a fraction of the octahedra would be unstable\nbetween different tilt orientations, forming tunneling systems which cause\nthe anelastic relaxation process. The interstitial O atoms force the\nsurrounding octahedra into a fixed tilt orientation, resulting in a decrease\nof the fraction of mobile octahedra and therefore in a depression of the\nabsorption peak. In addition, doping shifts the peak to lower temperature at\na very high rate, due to the coupling between the tilted octahedra and the\nhole excitations.\\cite{76} Therefore, it is possible to reduce the weight of\nthe low temperature peak by introducing concentrations of interstitial O\\\natoms that are so small that do not change appreciably the doping level due\nto the Sr substitutionals. Figure 2\\ compares the absorption curves of the $%\nx=0.019$ sample in the as-prepared state with a concentration $\\delta \\simeq\n0.002$ of excess O and after removing it in vacuum at high temperature. The\ninitial concentration $\\delta $ has been estimated from the intensity of the\nanelastic relaxation process due to the hopping of interstitial O,\\cite{63}\nwhose maximum occurs slightly below room temperature at our measuring\nfrequencies (not shown here). The presence of excess O indeed decreases and\nshifts to lower temperature the tail of the peak in Fig.~2, while the effect\non the absorption step is negligible. This justifies the comparison of the\nsample with $x=0.019$ and $\\delta >0$ together with the other samples with $%\n\\delta \\simeq 0$, and demonstrates that the nature of the low temperature\npeak is different from that of the step-like absorption.\n\n\\section{DISCUSSION}\n\nThe present data show the presence of a step in the acoustic absorption at\nthe boundary of the spin-glass quasi-ordered state in the $T,x$ magnetic\nphase diagram. The case of the $x=0.019$ sample is less clear-cut, since the\nstep is rather smooth. Furthermore, the Sr content is within the range $%\n0.018<x<0.02$, at the boundary between the SG and the CSG phases, where the\nphase diagram is largely uncertain.\\cite{CCC00} The $T_{f}\\left( x\\right) $\nline ends at $15~$K for $x\\simeq 0.018$, and the line $T_{g}\\left( x\\right) $\nstarts from 10-12$~$K at $x\\simeq 0.02$ (Refs. \\onlinecite{WUE99,Joh97,CCC00}%\n). A larger spread of experimental data\\cite{Joh97} (from 7.8 to 12.5$~$K)\nis actually observed just at $x=0.02$.\n\nA\\ mechanism which in principle produces acoustic absorption is the slowing\ndown of the magnetic fluctuations toward the spin-glass freezing. When\nmeasuring the spectral density $J_{\\text{spin}}\\left( \\omega ,T\\right) $ of\nthe spin fluctuations (the Fourier transform of the spin-spin correlation\nfunction), e.g. through the $^{139}$La NQR relaxation rate, a peak in $J_{%\n\\text{spin}}$ is found at the temperature at which the fluctuation rate $%\n\\tau ^{-1}\\left( T\\right) $ becomes equal to the measuring angular frequency \n$\\omega $. Near the glass transition the magnetic fluctuation rate was found\nto approximately follow the law\\cite{CBJ92} $\\tau ^{-1}\\propto \\left[ \\left(\nT-T_{g}\\right) /T_{g}\\right] ^{2}$, and the temperature at which the\ncondition $\\omega \\tau =1$ for the maximum of relaxation is satisfied for $%\n\\omega /2\\pi =12-19$~MHz is close to $T_{g}$. A similar peak would be\nobserved in the spectral density of the lattice strain $J_{\\text{latt}%\n}\\left( \\omega ,T\\right) $, if the spin fluctuations cause strain\nfluctuations through magnetoelastic coupling. The acoustic absorption is\nproportional to the spectral density of the strain and hence to $J_{\\text{%\nlatt}}$, $Q^{-1}=\\omega J_{\\text{strain}}/T\\propto \\omega J_{\\text{latt}}/T$%\n, and therefore at our frequencies ($\\omega \\leq 50$~kHz) we should observe\na narrow peak at a temperature slightly lower than the ones detected by NQR\nrelaxation. The absorption steps in Fig.~1 can hardly be identified in a\nstrict way as due to the contribution from the freezing magnetic\nfluctuations because they appear as steps instead of peaks. We propose that\nthe main contribution comes from the stress-induced movement of the domain\nboundaries between the clusters of quasi-frozen antiferromagnetically\ncorrelated spins. The mechanism is well known for ferromagnetic materials,%\n\\cite{NB} but is possible also for an ordered AF state, if an anisotropic\nstrain is coupled with the easy magnetization axis. In this case, the\nelastic energy of domains with different orientations of the easy axis would\nbe differently affected by a shear stress, and the lower energy domains\nwould grow at the expenses of the higher energy ones. The dynamics of the\ndomain boundaries is different from that of the domain fluctuations and\ngenerally produces broad peaks in the susceptibilities. An example is the\nstructural HTT/LTO transformation in the same samples, where the appearance\nof the orthorhombic domains is accompanied by a step-like increase of the\nacoustic absorption.\\cite{LLN} We argue that the features in the anelastic\nspectra just below $T_{g}$ are associated with the stress-induced motion of\nthe walls enclosing the clusters of AF correlated spins. More properly, the\nanelastic relaxation is attributed to the stress-induced changes of the\nsizes of the different domains.\n\nThe $x=0.019$ sample is at the border $x_{c}\\simeq 0.02$ between SG and CSG\nstate. The NQR measurements on the same sample\\cite{CCC00} indicate a\nspin-freezing temperature $\\sim 9~$K, closer to the CSG $T_{g}\\left(\nx_{c}\\right) $ rather then to the SG $T_{f}\\left( x_{c}\\right) $, which is\nconsistent with the presence of moving walls, otherwise absent in the SG\nstate. Nonetheless, following the model proposed by Gooding {\\it et al.}\\cite\n{GSB97} we do not expect a sharp transition between the SG and the CSG\nstates. According to that model, at low temperature the holes localize near\nthe Sr dopants, and in the ground state an isolated hole circulates\nclockwise or anti-clockwise over the four Cu atoms neighbors to Sr. Such a\nstate induces a distortion of the surrounding Cu spins, otherwise aligned\naccording to the prevalent AF order parameter. The spin texture arising from\nthe frustrated combination of the spin distortions from the various\nlocalized holes produces domains with differently oriented AF order\nparameters, which can be identified with the frozen AF spin clusters. The\ndissipative dynamics which we observe in the acoustic response should arise\nfrom the fact that the energy surface of the possible spin textures has many\nclosely spaced minima\\cite{GSB97} and the vibration stress, through\nmagnetoelastic coupling, can favor jumps to different minima. In this\npicture, one could argue that the random distribution of Sr atoms may cause\nthe formation of spin clusters also for $x\\lesssim x_{c}$ and it is possible\nto justify the fact that for $x=0.019$ the absorption step does not start\nbelow the maximum of the $^{139}$La NQR\\ relaxation rate, which signals the\nfreezing of the spin clusters. Rather, the acoustic absorption slowly starts\nincreasing slightly before the $T_{g}$ determined by the NQR maximum is\nreached. This may indicate that the spin dynamics is not only governed by\ncooperative freezing, but is also determined by the local interaction with\nthe holes localized at the surrounding Sr atoms. Then, the regions in which\nthe Sr atoms induce a particularly strong spin-texture could freeze and\ncause anelastic relaxation before the cooperative transition to the glass\nstate is completed. Systematic measurements around the $x=0.02$ doping range\nare necessary to clarify this point.\n\nThe dependence of the intensity of the absorption step on $x$, which is\nsharper and most intense at $x=0.03$, qualitatively supports the above\npicture. In fact, at lower doping one has only few domains embedded in a\nlong range ordered AF background, while above $0.05$ the fraction of walls\nof disordered spins connecting the Sr atoms increases at the expenses of the\nordered domains, with a cross-over to incommensurate spin correlations.\\cite\n{GSB97} The anelasticity due to the stress-induced change of the domain\nsizes is expected to be strongest in correspondence to the greatest fraction\nof ordered spins, namely between $0.03$ and $0.05$, in accordance with the\nspectra in Fig.~1.\n\nFinally we point out the insensitiveness of the absorption step to the\npresence of interstitial O (Fig. 2 and Ref. \\onlinecite{76}), in view of the\nmarked effects that even small quantities of excess O cause to the low\ntemperature peak (Fig. 2) and to the rest of the anelastic spectrum.\\cite\n{61,76} This is consistent with a dissipation mechanism of magnetic rather\nthan of structural origin.\n\n\\section{CONCLUSION}\n\nThe elastic energy loss coefficient of La$_{2-x}$Sr$_{x}$CuO$_{4}$\n(proportional to the imaginary part of the elastic susceptibility) measured\naround $10^{3}$~Hz in samples with $x=0.019$, 0.032 and 0.064 shows a\nstep-like rise below the temperature of the transition to a quasi-frozen\ncluster spin-glass state. The origin of the acoustic absorption is thought\nto be magnetoelastic coupling, namely anisotropic in-plane strain associated\nwith the direction of the local staggered magnetization. The absorption is\nnot peaked at $T_{g}$ and therefore does not directly correspond to the peak\nin the dynamic spin susceptibility due to the spin freezing. Rather, it has\nbeen ascribed to the stress-induced changes of the sizes of the spin\nclusters, or equivalently to the motion of the walls. The phenomenology is\nqualitatively accounted for in the light of the model of Gooding {\\it et al.}%\n\\cite{GSB97} of magnetic correlations of the Cu$^{2+}$ spins induced by the\nholes localized near the Sr dopants.\n\n\\section*{Acknowledgments}\n\nThe authors thank Prof. A. Rigamonti for useful discussions and for a\ncritical review of the manuscript. This work has been done in the framework\nof the Advanced Research Project SPIS of INFM.\n\n%\\bibliographystyle{unsrt}\n%\\bibliography{altro,ccc,htcsc}\n\n\\begin{references}\n\\bibitem{RBC98} A. Rigamonti, F. Borsa and P. Carretta, Rep. Prog. Phys. \n{\\bf 61}, 1367 (1998).\n\n\\bibitem{VSM87} D. Vaknin, S.K. Sinha, D.E. Moncton, D.C. Johnston, J.M.\nNewsam, C.R. Safinya and H.E. King Jr., Phys. Rev. Lett. {\\bf 58}, 2802\n(1987).\n\n\\bibitem{CBJ92} J.H. Cho, F. Borsa, D.C. Johnston and D.R. Torgeson, Phys.\nRev. B {\\bf 46}, 3179 (1992).\n\n\\bibitem{BCC95} F. Borsa, P. Carretta, J.H. Cho, F.C. Chou, Q. Hu, D.C.\nJohnston, A. Lascialfari, D.R. Torgeson, R.J. Gooding, N.M. Salem and K.J.E.\nVos, Phys. Rev. B {\\bf 52}, 7334 (1995).\n\n\\bibitem{JBC99} M.-H. Julien, F. Borsa, P. Carretta, M. Horvati, C.\nBerthier and C.T. Lin, Phys. Rev. Lett. {\\bf 83}, 604 (1999).\n\n\\bibitem{NBB98} Ch. Niedermayer, C. Bernhard, T. Blasius, A. Golnik, A.\nMoodenbaugh and J.I. Budnick, Phys. Rev. Lett. {\\bf 80}, 3843 (1998).\n\n\\bibitem{CBK95} F.C. Chou, N.R. Belk, M.A. Kastner, R.J. Birgeneau and A.\nAharony, Phys. Rev. Lett. {\\bf 75}, 2204 (1995).\n\n\\bibitem{WUE99} S. Wakimoto, S. Ueki, Y. Endoh and K. Yamada,\ncond-mat/9910400.\n\n\\bibitem{NB} A.S. Nowick and B.S. Berry, {\\it Anelastic Relaxation in\nCrystalline Solids}. (Academic Press, New York, 1972).\n\n\\bibitem{DFF94} M. Daturi, M. Ferretti and E.A. Franceschi, Physica C {\\bf %\n235-240}, 347 (1994).\n\n\\bibitem{Joh97} D.C. Johnston, {\\it Handbook of Magnetic Materials}. ed. by\nK.H.J. Buschow, p. 1 (North Holland, 1997).\n\n\\bibitem{76} F. Cordero, R. Cantelli and M. Ferretti, cond-mat/9910402, to\nbe published in Phys. Rev. B.\n\n\\bibitem{CCC00} A. Campana, R. Cantelli, M. Corti, F. Cordero and A.\nRigamonti, unpublished.\n\n\\bibitem{TR} E. Takayama-Muromachi and D.E. Rice, Physica C {\\bf 177}, 195\n(1991).\n\n\\bibitem{61} F. Cordero, C.R. Grandini, G. Cannelli, R. Cantelli, F.\nTrequattrini and M. Ferretti, Phys. Rev. B {\\bf 57}, 8580 (1998).\n\n\\bibitem{HSH96} D. Haskel, E.A. Stern, D.G. Hinks, A.W. Mitchell, J.D.\nJorgensen and J.I. Budnick, Phys. Rev. Lett. {\\bf 76}, 439 (1996).\n\n\\bibitem{BBK99} E.S. Bozin, S.J.L. Billinge, G.H. Kwei and H. Takagi, Phys.\nRev. B {\\bf 59}, 4445 (1999).\n\n\\bibitem{68} F. Cordero, R. Cantelli, M. Corti, M. Campana and A.\nRigamonti, Phys. Rev. B {\\bf 59}, 12078 (1999).\n\n\\bibitem{63} F. Cordero, C.R. Grandini and R. Cantelli, Physica C {\\bf 305}%\n, 251 (1998).\n\n\\bibitem{LLN} W.-K. Lee, M. Lew and A.S. Nowick, Phys. Rev. B {\\bf 41}, 149\n(1990).\n\n\\bibitem{GSB97} R.J. Gooding, N.M. Salem, R.J. Birgeneau and F.C. Chou,\nPhys. Rev. B {\\bf 55}, 6360 (1997); K.S.D. Beach and R.J. Gooding,\ncond-mat/0001095.\n\\end{references}\n\n\\section{Figures captions}\n\nFig. 1 Elastic energy loss coefficient of La$_{2-x}$Sr$_{x}$CuO$_{4}$ with $%\nx=0.019$ ($1.3$~kHz), $x=0.03$ ($1.7$~kHz), $x=0.06$ ($0.43$~kHz). The gray\narrows indicate the temperature $T_{g}$ of freezing into the cluster\nspin-glass state deduced from NQR\\cite{CBJ92} (lower values) and $\\mu $SR%\n\\cite{NBB98} (higher values) experiments The black arrows indicate the\ntemperature of the maximum of the $^{139}$La NQR relaxation rate measured on\nthe same samples.\\cite{CCC00}\n\nFig. 2 Elastic energy loss coefficient of the sample with $x=0.019$ in the\nas prepared state (with interstitial O) and after outgassing the excess O\nmeasured at 1.3~kHz.\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002009.extracted_bib",
"string": "\\bibitem{RBC98} A. Rigamonti, F. Borsa and P. Carretta, Rep. Prog. Phys. \n{\\bf 61}, 1367 (1998).\n\n\n\\bibitem{VSM87} D. Vaknin, S.K. Sinha, D.E. Moncton, D.C. Johnston, J.M.\nNewsam, C.R. Safinya and H.E. King Jr., Phys. Rev. Lett. {\\bf 58}, 2802\n(1987).\n\n\n\\bibitem{CBJ92} J.H. Cho, F. Borsa, D.C. Johnston and D.R. Torgeson, Phys.\nRev. B {\\bf 46}, 3179 (1992).\n\n\n\\bibitem{BCC95} F. Borsa, P. Carretta, J.H. Cho, F.C. Chou, Q. Hu, D.C.\nJohnston, A. Lascialfari, D.R. Torgeson, R.J. Gooding, N.M. Salem and K.J.E.\nVos, Phys. Rev. B {\\bf 52}, 7334 (1995).\n\n\n\\bibitem{JBC99} M.-H. Julien, F. Borsa, P. Carretta, M. Horvati, C.\nBerthier and C.T. Lin, Phys. Rev. Lett. {\\bf 83}, 604 (1999).\n\n\n\\bibitem{NBB98} Ch. Niedermayer, C. Bernhard, T. Blasius, A. Golnik, A.\nMoodenbaugh and J.I. Budnick, Phys. Rev. Lett. {\\bf 80}, 3843 (1998).\n\n\n\\bibitem{CBK95} F.C. Chou, N.R. Belk, M.A. Kastner, R.J. Birgeneau and A.\nAharony, Phys. Rev. Lett. {\\bf 75}, 2204 (1995).\n\n\n\\bibitem{WUE99} S. Wakimoto, S. Ueki, Y. Endoh and K. Yamada,\ncond-mat/9910400.\n\n\n\\bibitem{NB} A.S. Nowick and B.S. Berry, {\\it Anelastic Relaxation in\nCrystalline Solids}. (Academic Press, New York, 1972).\n\n\n\\bibitem{DFF94} M. Daturi, M. Ferretti and E.A. Franceschi, Physica C {\\bf %\n235-240}, 347 (1994).\n\n\n\\bibitem{Joh97} D.C. Johnston, {\\it Handbook of Magnetic Materials}. ed. by\nK.H.J. Buschow, p. 1 (North Holland, 1997).\n\n\n\\bibitem{76} F. Cordero, R. Cantelli and M. Ferretti, cond-mat/9910402, to\nbe published in Phys. Rev. B.\n\n\n\\bibitem{CCC00} A. Campana, R. Cantelli, M. Corti, F. Cordero and A.\nRigamonti, unpublished.\n\n\n\\bibitem{TR} E. Takayama-Muromachi and D.E. Rice, Physica C {\\bf 177}, 195\n(1991).\n\n\n\\bibitem{61} F. Cordero, C.R. Grandini, G. Cannelli, R. Cantelli, F.\nTrequattrini and M. Ferretti, Phys. Rev. B {\\bf 57}, 8580 (1998).\n\n\n\\bibitem{HSH96} D. Haskel, E.A. Stern, D.G. Hinks, A.W. Mitchell, J.D.\nJorgensen and J.I. Budnick, Phys. Rev. Lett. {\\bf 76}, 439 (1996).\n\n\n\\bibitem{BBK99} E.S. Bozin, S.J.L. Billinge, G.H. Kwei and H. Takagi, Phys.\nRev. B {\\bf 59}, 4445 (1999).\n\n\n\\bibitem{68} F. Cordero, R. Cantelli, M. Corti, M. Campana and A.\nRigamonti, Phys. Rev. B {\\bf 59}, 12078 (1999).\n\n\n\\bibitem{63} F. Cordero, C.R. Grandini and R. Cantelli, Physica C {\\bf 305}%\n, 251 (1998).\n\n\n\\bibitem{LLN} W.-K. Lee, M. Lew and A.S. Nowick, Phys. Rev. B {\\bf 41}, 149\n(1990).\n\n\n\\bibitem{GSB97} R.J. Gooding, N.M. Salem, R.J. Birgeneau and F.C. Chou,\nPhys. Rev. B {\\bf 55}, 6360 (1997); K.S.D. Beach and R.J. Gooding,\ncond-mat/0001095.\n"
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cond-mat0002010
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Enhancement of Kondo Effect in Quantum Dots with %\\ an Even Number of Electrons
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"author": "Mikio Eto$^{1,2}$ and Yuli V.\\ Nazarov$^{1}$"
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We investigate the Kondo effect in a quantum dot with almost degenerate spin-singlet and triplet states for an even number of electrons. We show that the Kondo temperature as a function of the energy difference between the states $\Delta$ reaches its maximum around $\Delta=0$ and decreases with increasing $\Delta$. The Kondo effect is thus enhanced by competition between singlet and triplet states. Our results explain recent experimental findings. We evaluate the linear conductance in the perturbative regime.
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"name": "prepri-f.tex",
"string": "% C:\\Applications\\SWP30\\TCITeX\\TeX\\latex209\\contrib\\RevTeX (manaip.tex)\n\\documentstyle[prl,aps,epsfig]{revtex}\n% \\documentstyle[preprint,prb,aps]{revtex}\n% \\documentstyle[preprint,aps,epsfig]{revtex}\n% \\documentstyle[twocolumn,aps,epsfig]{revtex}\n\\begin{document}\n% \\draft command makes pacs numbers print\n\\draft\n% repeat the \\author\\address pair as needed\n\\author{Mikio Eto$^{1,2}$ and Yuli V.\\ Nazarov$^{1}$}\n\\address{$^1$Department of Applied Physics/DIMES,\nDelft University of Technology, \\\\\nLorentzweg 1, 2628 CJ Delft, The Netherlands \\\\\n$^2$Faculty of Science and Technology, Keio\nUniversity, \\\\\n3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan}\n\\title{Enhancement of Kondo Effect in Quantum Dots with %\\\\\nan Even Number of Electrons}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nWe investigate the Kondo effect in a quantum dot with almost\ndegenerate spin-singlet and triplet states for an even number of electrons.\nWe show that the Kondo temperature as a function of\nthe energy difference between\nthe states $\\Delta$ reaches its maximum around $\\Delta=0$\nand decreases with increasing $\\Delta$.\nThe Kondo effect is thus enhanced by competition between singlet and\ntriplet states.\nOur results explain recent experimental findings.\nWe evaluate the linear conductance in the perturbative regime.\n\\end{abstract}\n\\pacs{73.23.Hk, 72.15.Qm, 85.30.Vw}\n\nThe Kondo effect \\cite{classics} takes place when a localized spin $S$\nis brought in contact with electron Fermi sea. The Kondo effect \ngives rise to a new many-body ground\nstate that has a lesser spin. Recently\nthe Kondo effect has been observed in semiconductor quantum dots connected\nto external leads by tunnel junctions \\cite{Kondo1}.\nIn this case the localized spin is formed by electrons in the dot.\nThe number of electrons $N$ is fixed by Coulomb blockade to integer values\nand can be tuned by gate voltage. \nDespite the Coulomb blockade, \nthe ground state of the dot\nis usually similar to what one obtains disregarding the interaction.\nThe discrete spin-degenerate levels in the dot are \nconsecutively occupied, and the total spin is zero or 1/2\nfor an even and odd number of electrons, respectively.\nThen the Kondo effect takes place only in the latter case\n% if there is an odd number of electrons in the dot\n\\cite{Glazman,Ng}.\n% forming a spin-doublet.\n\n\nSignificant deviations from this plain picture \nwere recently observed in so-called\n``vertical'' quantum dots \\cite{Tarucha,Leo}.\nThe strength of the electron-electron Coulomb\ninteraction in such dots is comparable with \nthe spacing of discrete levels, and this may give rise to a complicated\nground state.\nFor example, if two electrons are put into nearly degenerate levels, \nthe exchange interaction favors a spin-triplet state. \nThis state can be changed to a \nspin singlet by applying a magnetic field since the magnetic field\nincreases the level spacing \\cite{Tarucha}.\n\nThis gives a unique possibility to change the spin of the ground state\nduring an experiment and even obtain extra degenerate states\nby tuning the energies of different spin configurations to the same value. \nSuch a possibility hardly exists \nin the traditional solid state context.\nThe Kondo effect in multilevel quantum dots has been investigated\nby several groups \\cite{Inoshita,Pustilnik}.\nIn this letter we examine a novel effect that stems from the competition\nbetween spin-singlet and threefold spin-triplet states\nfor an even number of electrons in a dot.\nSince the energy difference\nbetween the states $\\Delta$ can be\ncontrolled experimentally, \nwe elucidate $\\Delta$ dependence of Kondo\ntemperature as a typical energy scale for the Kondo effect\nand linear conductance through the dot.\nThis enables direct comparison between our calculations and\nrecent experimental results \\cite{exp}.\n\n\nAt large positive $\\Delta$ the dot is in a triplet ground state\nand an extra singlet state can be disregarded. The Kondo effect\nfollows a usual $S=1$ scenario.\nAt large negative $\\Delta$ the ground state of the dot is \na spin singlet and the Kondo effect ceases to exist. \nFrom this one could suggest that the Kondo temperature decreases as \n$\\Delta \\rightarrow 0$ from the positive side. \nOur results\nshow just the opposite. The Kondo temperature $T_K(\\Delta)$ is enhanced\nat small $\\Delta$ and reaches its maximum at $\\Delta \\approx T_K^{max}$.\nAt $\\Delta \\gg T_K^{max}$ the Kondo temperature decreases with increasing\n$\\Delta$ obeying a power law $T_K(\\Delta) \\propto 1/\\Delta^{\\gamma}$. \nThe exponent $\\gamma$ is not universal but depends on model parameters.\nOur results clearly demonstrate the importance of one of the basic\nprinciples of Kondo physics: although the Kondo effect occurs at small\nenergy scale $T_K$, the value of this scale is determined by \nall energies from $T_K$ up to the upper cutoff. In our case, the energies\nfrom $\\Delta$ to the upper cutoff would feel 4-fold degeneracy of the dot\nstates, which enhances the Kondo temperature.\nAt $\\Delta < \\Delta_c$ ($|\\Delta_c| \\sim T_K^{max}$),\nthe Kondo effect is not relevant.\nWe stress the difference between this mechanism and one\nproposed in Ref.\\ \\cite{Pustilnik}, where the Kondo effect arises from extra\ndegeneracy between one component of the spin-triplet state and\na singlet state, which is brought by the Zeeman splitting.\n\nTo model the situation, it is sufficient to consider two extra electrons\nin a quantum dot at the background of a singlet state of all other\n$N-2$ electrons, which we will regard as the vacuum. \nThese two extra electrons occupy two levels of different orbital\nsymmetry %, {\\it e.\\ g.}\\ angular momentum in a circular dot \n\\cite{symmetry}.\nThe energies of the levels are $\\varepsilon_1, \\varepsilon_2$.\nPossible two-electron states are (i) the threefold spin-triplet state, \n(ii) the spin-singlet state of the same orbital symmetry as the triplet state,\n$1/\\sqrt{2}(d_{1 \\uparrow}^{\\dagger} d_{2 \\downarrow}^{\\dagger}\n -d_{1 \\downarrow}^{\\dagger} d_{2 \\uparrow}^{\\dagger}) |0 \\rangle$,\nand (iii) two singlets of different orbital symmetry,\n$d_{1 \\uparrow}^{\\dagger} d_{1 \\downarrow}^{\\dagger} |0 \\rangle$,\n $d_{2 \\uparrow}^{\\dagger} d_{2 \\downarrow}^{\\dagger} |0 \\rangle$.\nAmong the singlet states,\nwe only consider a state of the lowest energy, which belongs \nto the group (iii). Thus we restrict our attention to four \nstates, $|S M \\rangle$: \n\\begin{eqnarray}\n|1 1 \\rangle & = & d_{1 \\uparrow}^{\\dagger} d_{2 \\uparrow}^{\\dagger}\n|0 \\rangle \\\\\n|1 0 \\rangle & = & \\frac{1}{\\sqrt{2}} (d_{1 \\uparrow}^{\\dagger} d_{2\n\\downarrow}^{\\dagger}\n +d_{1 \\downarrow}^{\\dagger} d_{2\n\\uparrow}^{\\dagger}) |0 \\rangle \\\\\n|1 -1 \\rangle & = & d_{1 \\downarrow}^{\\dagger} d_{2 \\downarrow}^{\\dagger}\n|0 \\rangle \\\\\n|0 0 \\rangle & = & \\frac{1}{\\sqrt{2}} (C_1 d_{1 \\uparrow}^{\\dagger}\nd_{1 \\downarrow}^{\\dagger}\n -C_2 d_{2 \\uparrow}^{\\dagger} d_{2 \\downarrow}^{\\dagger}) |0 \\rangle,\n\\label{eq:singlet}\n\\end{eqnarray}\nwhere $d_{i \\sigma}^{\\dagger}$ creates an electron with spin $\\sigma$ in\nlevel $i$.\nTheir energies, $E_{S=1}$, $E_{S=0}$,\nand coefficients in the singlet state,\n$C_1$, $C_2$ ($|C_1|^2+|C_2|^2=2$),\nare determined by the electron-electron interaction\nand one-electron level spacing $\\delta=\\varepsilon_2-\\varepsilon_1$.\nAt the moment, we set $C_1=C_2=1$ \\cite{com1} and show that this is the\ngeneral case afterwards.\nThe energy difference, $\\Delta=E_{S=0}-E_{S=1}$, is changed by applying\na magnetic field $B$ as shown in Fig.\\ 1(a). \nWe disregard the Zeeman splitting of spin states\nsince this is a much smaller energy scale than the orbital effect of the\nmagnetic field in semiconductor heterostructures in use \\cite{Tarucha,Leo}.\nThe exact condition for this is $E_{\\rm{Zeeman}} \\ll T_K$. % \\cite{exp}.\n\nThe dot is connected to two external leads $L$, $R$ with free electrons\nbeing described by\n\\begin{equation}\nH_{\\rm{leads}}=\\sum_{\\alpha=L,R}\\sum_{k \\sigma i} \\varepsilon_{\\alpha k}^{(i)}\nc_{\\alpha, k\\sigma}^{(i) \\dagger}\nc_{\\alpha, k\\sigma}^{(i)}.\n\\end{equation}\nThe tunneling between the dot and the leads is written as\n\\begin{equation}\nH_T=\\sum_{\\alpha=L,R} \\sum_{k \\sigma i} (V_{\\alpha,i}c_{\\alpha,\nk\\sigma}^{(i) \\dagger} d_{i \\sigma} + \\rm{H.c.}).\n\\end{equation}\nHere $c_{\\alpha, k\\sigma}^{(i) \\dagger}$ is the creation operator of an\nelectron in lead $\\alpha$ with\nmomentum $k$, spin $\\sigma$, and orbital symmetry $i$ $(=1,2)$. \nWe assume that the orbital symmetry is conserved\nin the tunneling processes. Therefore we have two electron ``channels''\nin each lead.\n\n\nWe assume that the state of the dot with $N$ electrons is stable, so that\naddition/extraction energies,\n$E^{\\pm} \\equiv E(N \\pm 1) -E(N) \\mp \\mu$\nwhere $\\mu$ is the Fermi energy in the leads,\nare positive. We are interested in the case when\n$E^{\\pm} \\gg |\\Delta|$, $\\delta$ and also exceed the level broadening\n$\\Gamma_{\\alpha}^{i}=\\pi\\nu |V_{\\alpha,i}|^2$\n($\\nu$ being density of states in the leads) and temperature $T$\n(Coulomb blockade region).\nIn this case we can integrate out\nthe states with one or three extra electrons. This is equivalent to\nSchrieffer-Wolf transformation which is used to obtain the conventional\nKondo model \\cite{classics}.\nWe obtain the following effective low-energy Hamiltonian\n\\begin{equation}\nH_{\\rm{eff}}=H^{S=1}+H^{S=1 \\leftrightarrow 0}+H_{\\rm{eff}}^{\\prime}+\nH_{\\rm{dot}}.\n\\end{equation}\nThe first term involves components of the spin-triplet state and resembles\na conventional Kondo Hamiltonian for $S=1$.\n\\begin{eqnarray}\nH^{S=1} & = & \\sum_{k k'} \\sum_{\\alpha \\beta=L,R} \\sum_{i=1,2} J_{\\alpha\n\\beta}^{(i)}\n\\left[ \\hat{S}_{+} c_{\\alpha k' \\downarrow}^{(i) \\dagger} c_{\\beta k\n\\uparrow}^{(i)}\n+\\hat{S}_{-} c_{\\alpha' k \\uparrow}^{(i) \\dagger} c_{\\beta k\n\\downarrow}^{(i)}\n+\\hat{S}_{z} (c_{\\alpha' k \\uparrow}^{(i) \\dagger} c_{\\beta k\n\\uparrow}^{(i)}\n -c_{\\alpha' k \\downarrow}^{(i) \\dagger} c_{\\beta k\n\\downarrow}^{(i)}) \\right] \\nonumber \\\\\n & = & \\sum_{k k'} \\sum_{\\alpha \\beta=L,R} \\sum_{i=1,2} J_{\\alpha\n\\beta}^{(i)}\n\\Bigl[ \\sqrt{2}(f_{11}^{\\dagger}f_{10}+f_{10}^{\\dagger}f_{1 -1})\n c_{\\alpha k' \\downarrow}^{(i) \\dagger} c_{\\beta k\n\\uparrow}^{(i)}\n+\\sqrt{2}(f_{10}^{\\dagger}f_{11}+f_{1 -1}^{\\dagger}f_{10})\n c_{\\alpha k' \\uparrow}^{(i) \\dagger} c_{\\beta k\n\\downarrow}^{(i)} \\nonumber \\\\\n& & +(f_{11}^{\\dagger}f_{11}-f_{1 -1}^{\\dagger}f_{1 -1})\n (c_{\\alpha k' \\uparrow}^{(i) \\dagger} c_{\\beta k\n\\uparrow}^{(i)}\n -c_{\\alpha k' \\downarrow}^{(i) \\dagger} c_{\\beta k\n\\downarrow}^{(i)}) \\Bigr].\n\\end{eqnarray}\nHere we have introduced pseudo-fermion operators $f_{SM}^{\\dagger}$\n($f_{SM}$) which\ncreate (annihilate) the state $|SM \\rangle$. It is required that\n$\\sum_{SM} f_{SM}^{\\dagger} f_{SM} =1$.\nThe second term in $H_{\\rm{eff}}$ describes the conversion between\nthe spin-triplet and singlet states accompanied by interchannel scattering of\nconduction electrons\n\\begin{eqnarray}\nH^{S=1 \\leftrightarrow 0} =\n\\sum_{k k'} \\sum_{\\alpha \\beta=L,R} \\Bigl\\{ \\tilde{J}_{\\alpha \\beta}\n\\Bigl[\\sqrt{2}(f_{11}^{\\dagger}f_{00}-f_{00}^{\\dagger}f_{1 -1})\n c_{\\alpha k' \\downarrow}^{(1) \\dagger} c_{\\beta k\n\\uparrow}^{(2)}\n+\\sqrt{2}(f_{00}^{\\dagger}f_{11}-f_{1 -1}^{\\dagger}f_{00})\n c_{\\alpha k' \\uparrow}^{(1) \\dagger} c_{\\beta k\n\\downarrow}^{(2)} \\nonumber \\\\\n-(f_{10}^{\\dagger}f_{00}+f_{00}^{\\dagger}f_{10})\n (c_{\\alpha k' \\uparrow}^{(1) \\dagger} c_{\\beta k\n\\uparrow}^{(2)}\n -c_{\\alpha k' \\downarrow}^{(1) \\dagger} c_{\\beta k\n\\downarrow}^{(2)}) \\Bigr]\n+\\tilde{J}_{\\alpha \\beta}^{*} [1 \\leftrightarrow 2] \\Bigr\\}.\n\\end{eqnarray}\nThe third term $H_{\\rm{eff}}^{\\prime}$ represents the scattering processes\nwithout change of the\ndot state and is not relevant for the current discussion.\nThe coupling constants are given by\n%\\[\n%J_{\\alpha \\beta}^{(i)} = \\frac{V_{\\alpha,i}V_{\\beta,i}^{*}}{2 E_c}, \\ \\\n%\\tilde{J}_{\\alpha \\beta} = \\frac{V_{\\alpha,1}V_{\\beta,2}^{*}}{2 E_c},\n%\\]\n$\nJ_{\\alpha \\beta}^{(i)} = V_{\\alpha,i}V_{\\beta,i}^{*}/(2 E_c),\n\\tilde{J}_{\\alpha \\beta} = V_{\\alpha,1}V_{\\beta,2}^{*}/(2 E_c),\n$\nwhere $1/E_c=1/E^+ +1/E^-$.\nThe Hamiltonian of the dot reads\n\\begin{equation}\nH_{\\rm{dot}}=\\sum_{S,M} E_S f_{SM}^{\\dagger}f_{SM}.\n\\end{equation}\n\n\nTo avoid the complication due to the fact that there are two leads\n$\\alpha=L,R$, we perform a unitary\ntransformation for electron modes in the leads along the lines of \nRef.\\ \\cite{Glazman};\n$c_{k \\sigma}^{(i)}=(V_{L,i}c_{L,k \\sigma}+V_{R,i}c_{R,k \\sigma})/V_i$,\n$\\bar{c}_{k \\sigma}^{(i)}=(V_{R,i}^*c_{L,k \\sigma}-V_{L,i}^*c_{R,k\n\\sigma})/V_i$,\nwith $V_i=\\sqrt{|V_{L,i}|^2+|V_{R,i}|^2}$. The modes $\\bar{c}_{k \\sigma}^{(i)}$\nare not coupled to the quantum dot and shall be disregarded.\nThe coupling constants for modes $c_{k \\sigma}^{(i)}$\nbecome\n\\begin{eqnarray}\nJ^{(i)} & = & \\frac{|V_{L i}|^2+|V_{R i}|^2}{2E_c}, \\\\\n\\tilde{J} & = & \\frac{1}{2E_c}\\frac{(V_{L 1}^2+V_{R 1}^2)(V_{L 2}^2+V_{R\n2}^2)}\n{\\sqrt{(|V_{L 1}|^2+|V_{R 1}|^2)(|V_{L 2}|^2+|V_{R 2}|^2)}}.\n\\end{eqnarray}\nThe spin-flip processes included in our model are shown in the inset of\nFig.~1(b).\n\nWe calculate the Kondo temperature $T_K$\nwith the poor man's scaling technique \\cite{Anderson,multiK}.\nBy this method, we can properly consider the energies from $T_K$ to\nthe upper cutoff. We concentrate on evaluating the exponential part\nof $T_K$.\nWe assume constant density of states in the leads $\\nu$ in the\nenergy band of\n$[-D, D]$. By changing the energy scale from $D$ to $D-|d D|$, we obtain a\nclosed form of\nthe scaling equations for $J^{(1)}$, $J^{(2)}$, and $\\tilde{J}$ in two limits.\n\nIn the first limit, the energy difference $|\\Delta|$ is negligible \n($|\\Delta| \\ll D$) and $H_{\\rm{dot}}$ can be safely disregarded.\n The scaling equations are best presented in the following matrix form:\n\\begin{equation}\n\\frac{d}{d\\ln D}\n \\left( \\begin{array}{cc}\n J^{(1)} & \\tilde{J} \\\\\n \\tilde{J}^{*} & J^{(2)} \\end{array} \\right)\n=-2 \\nu\n \\left( \\begin{array}{cc}\n J^{(1)} & \\tilde{J} \\\\\n \\tilde{J}^{*} & J^{(2)} \\end{array} \\right)^2.\n\\label{eq:scalA}\n\\end{equation}\nThe equations can be readily rewritten for eigenvalues of the matrix,\n$J_{\\pm}=(J^{(1)}+J^{(2)})/2 \\pm\n\\sqrt{(J^{(1)}-J^{(2)})^2/4+|\\tilde{J}|^2}$.\nThe larger one, $J_{+}$, diverges faster upon decreasing the bandwidth $D$\nand hence\ndetermines $T_K$. %the Kondo temperature.\nIf the equations remain valid till the scaling ends ($|\\Delta| \\ll T_K$),\nthe Kondo temperature is \n$T_K(0)=D_0 \\exp [-1/2\\nu J_+]$. Here \n$D_0$ is the initial bandwidth\ngiven by $\\sqrt{E^+ E^-}$ \\cite{Haldane}.\nIn another limiting case, $\\Delta \\gg D$. In this case\nthe ground state of the dot is spin triplet and the singlet state can be\ndisregarded.\n$J^{(1)}$ and $J^{(2)}$ evolve independently \n\\begin{equation}\n\\frac{d}{d\\ln D} J^{(i)} = -2\\nu J^{(i) 2},\n\\label{eq:scalB}\n\\end{equation}\nand $\\tilde J$ does not change.\nIf these equations \nremain valid in the whole scaling region ($\\Delta > D_0$),\nit yields $T_K(\\infty) =D_0 \\exp [-1/2\\nu J^{(1)}]$.\nHere we assume $J^{(1)} \\ge J^{(2)}$. \nThis is the Kondo temperature for spin-triplet localized spins \\cite{Okada}.\n\nTo determine $T_K$ in the intermediate region,\n$T_K(0) \\ll \\Delta \\ll D_0$, we match\nthe solutions of Eqs.\\ (\\ref{eq:scalA}) and (\\ref{eq:scalB})\nat $D \\simeq \\Delta$. $\\tilde J$ saturates at this point \nwhile $J^{(1)}$ and $J^{(2)}$ continue to grow with decreasing $D$. \nThis yields power law dependence on $\\Delta$\n\\begin{equation}\nT_K(\\Delta)=T_K(0)\\cdot \\left( T_K(0)/\\Delta \\right)^{\\gamma},\n\\label{eq:TK}\n\\end{equation}\nwhere $\n\\sqrt{\\gamma}=|\\tilde{J}|/[\\sqrt{ (J^{(1)}-J^{(2)})^2/4+|\\tilde{J}|^2}+\n|J^{(1)}-J^{(2)}|/2]$. The exponent $\\gamma$ appears to be nonuniversal,\ndepending on a ratio of the initial coupling constants.\nIn general, $0 <\\gamma \\le 1$. \nFor $\\Delta<0$, all the coupling constants saturate \nand no Kondo effect is expected,\nprovided $|\\Delta| \\gg T_K(0)$.\n\nIn a simple case of the identical couplings, $J^{(1)}=J^{(2)}=\\tilde{J}$\n$(\\equiv J)$, $T_K(0)=D_0 \\exp [-1/4\\nu J]$.\nFor $\\Delta>0$, $T_K$ decreases with increasing $\\Delta$ as\n$T_K(\\Delta)=T_K(0)^2/\\Delta$\n($\\gamma=1$ in Eq.\\ (\\ref{eq:TK})) and finally converges to\n$D_0 \\exp [-1/2\\nu J]=T_K(0)^2/D_0$. For $\\Delta<0$, $T_K$ drops to zero\nsuddenly at $|\\Delta| \\sim T_K(0)$.\nThe dependence of the Kondo temperature on $\\Delta$ is schematically shown in\nFig.~1(b).\n\n\n%---------------------------------------------\nWe have discussed so far the case of $C_1=C_2=1$ \nin Eq.\\ (\\ref{eq:singlet}) for the\nspin-singlet state.\nThis is not required by symmetry and $C_1 \\ne C_2$ in general.\nTo justify the assumption we made, let us consider the renormalization\nequations for $C_1 \\ne C_2$.\n% They are more complicated than those we have considered above.\nThe coupling constants\n$\\tilde{J}_1=C_1 \\tilde{J}$ and $\\tilde{J}_2=C_2 \\tilde{J}$ are\nrenormalized now in a different way, involving\nthe scattering processes without\nspin flip in the dot\n\\begin{equation}\nH_{\\rm{eff}}^{\\prime}=\\sum_{k k' \\sigma}\\sum_{i=1,2}\n \\left[ J^{\\prime (i)} c_{k' \\sigma}^{(i) \\dagger} c_{k \\sigma}^{(i)}\n\\sum_M f_{1 M}^{\\dagger}f_{1 M}\n + J^{\\prime\\prime (i)} c_{k' \\sigma}^{(i) \\dagger} c_{k\n\\sigma}^{(i)} f_{00}^{\\dagger}f_{00}\n \\right].\n\\end{equation}\nGeneral scaling equations for $|\\Delta| \\ll D$ are given by\n\\begin{eqnarray}\nd \\tilde{J}_1/ d \\ln D & = & -2\\nu (J^{(1)}+J^{(2)})\\tilde{J}_1 + \\nu J'\n\\tilde{J}_1\n\\label{eq:scalC} \\\\\nd \\tilde{J}_2/ d \\ln D & = & -2\\nu (J^{(1)}+J^{(2)})\\tilde{J}_2 - \\nu J'\n\\tilde{J}_2 \\\\\nd J'/ d \\ln D & = & 8 \\nu ( |\\tilde{J}_1|^2 - |\\tilde{J}_2|^2 ) \\\\\nd J^{(i)} / d \\ln D & = & -2\\nu \\left[ J^{(i) 2} + (|\\tilde{J}_1|^2 +\n|\\tilde{J}_2|^2 )/2 \\right]\n\\label{eq:scalD}\n\\end{eqnarray}\nwhere $J'=J^{\\prime (1)}-J^{\\prime (2)}-J^{\\prime\\prime (1)}+J^{\\prime\\prime\n(2)}$. When $\\Delta \\gg D$,\nthe equations are identical to Eq.\\ (\\ref{eq:scalB}).\nOur point is that\nif we concentrate on the most rapidly divergent\nsolutions of Eqs.\\ (\\ref{eq:scalC})-(\\ref{eq:scalD}), which are proportional\nto $1/\\ln D$, $\\tilde{J}_1$ and $\\tilde{J}_2$\nappear to be the same.\nTo this leading order, the renormalization is the same as given by \nEq.\\ (\\ref{eq:scalA}).\nConsequently the Kondo temperature is the same as that in the case of\n$C_1=C_2=1$, apart from a prefactor.\n%---------------------------------------------\n\n\nWe calculate perturbation corrections to conductance when\n$T_K \\ll T \\ll E_c$.\nThe third order perturbations in $J$'s yield \nthe logarithmic corrections $G_K$\ntypical for the Kondo\neffect \\cite{classics}. At $T \\gg |\\Delta|$,\n\\begin{eqnarray}\nG_K/(2e^2/h)= \\sum_{i=1,2}\n\\frac{4\\Gamma_L^{i}\\Gamma_R^{i}}{(\\Gamma_L^{i}+\\Gamma_R^{i})^2}\n6\\pi^2 J^{(i)}\\nu \\left[\n(J^{(i)}\\nu)^2+|\\tilde{J}\\nu|^2\\right]\n\\ln \\frac{D_0}{T} \\nonumber \\\\\n+\\frac{2(\\Gamma_L^{1}\\Gamma_R^{2}+\\Gamma_L^{2}\\Gamma_R^{1})}\n{(\\Gamma_L^{1}+\\Gamma_R^{1})(\\Gamma_L^{2}+\\Gamma_R^{2})}\n12\\pi^2 \\left[J^{(1)}\\nu+J^{(2)}\\nu \\right]\n|\\tilde{J}\\nu|^2 \\ln \\frac{D_0}{T}.\n\\end{eqnarray}\nThe interplay between\nthe spin-singlet and triplet states largely enhances the conductance.\nIn the opposite case, $T \\ll |\\Delta|$, the interplay becomes \nless effective and the logarithmic corrections become smaller\n\\begin{equation}\nG_K/(2e^2/h)=\\sum_{i=1,2}\n\\frac{4\\Gamma_L^{i}\\Gamma_R^{i}}{(\\Gamma_L^{i}+\\Gamma_R^{i})^2}\n8\\pi^2 J^{(i)}\\nu \\left[\n(J^{(i)}\\nu)^2 \\ln \\frac{D_0}{T}\n+|\\tilde{J}\\nu|^2 \\ln \\frac{D_0}{\\Delta} \\right]\n\\end{equation}\nfor $\\Delta>0$ and\ndisappear\nfor $\\Delta<0$. If one varies $\\Delta$ at fixed $T$, \none sees the enhanced conductance for $|\\Delta| \\ll T$.\n\nWe believe that the conductance approaches the unitary limit $2e^2/h$ at\n$T \\ll T_K$ \\cite{Glazman} in our model. However, not all the terms\nin $G_K$ can be renormalized to a universal function $G(T/T_K)$.\nBecause of the multichannel nature of our model, one should expect\nnonuniversal logarithmic terms along with the universal ones.\nThose, however, will be much smaller than $2e^2/h$.\n\n%Experiment\n\nRecent experiment \\cite{exp} has shown a significant\nenhancement of conductance to the values of the order\nof $e^2/h$, around the crossing point between spin-singlet \nand triplet states, in vertical quantum dots.\nThe conductance remains low in the stability domains\nof singlet or triplet states. Our results provide\na possible theoretical explanation for the results. The conductance\nincrease can be attributed to the Kondo effect enhanced\nby the competition between the two states,\nnear the crossing point. The Kondo temperature\nelsewhere is probably low in comparison with the actual temperature,\nso that no conductance increase is seen. \n\nIn conclusion, we have shown that the competition between spin-singlet\nand triplet states enhances the Kondo effect. This may explain recent\nexperimental observations. We have predicted power law dependence of the Kondo\ntemperature on the energy difference between the states.\n\n%---------------------------------------------\nThe authors are indebted to L.\\ P.\\ Kouwenhoven for suggesting the topic\nof the research presented, S.\\ De Franceschi, J.\\ M.\\\nElzerman, K.\\ Maijala, S.\\ Sasaki,\nW.\\ G.\\ van der Wiel, Y.\\ Tokura, L.\\ I.\\ Glazman, M.\\ Pustilnik,\nand G.\\ E.\\ W.\\ Bauer for\nvaluable discussions.\n% This work is a part of the research program of the ``Stichting voor\n% Fundamenteel Onderzoek der Materie'' (FOM).\nThe authors acknowledge financial support from the\n``Netherlandse Organisatie voor\nWetenschappelijk Onderzoek'' (NWO). M.\\ E.\\ is also grateful for financial\nsupport from the\nJapan Society for the Promotion of Science for his stay at Delft University\nof Technology.\n\n\n\\begin{references}\n\\bibitem{classics}\nJ.\\ Kondo, Prog.\\ Theor.\\ Phys.\\ {\\bf 32}, 37 (1964);\nA.\\ C.\\ Hewson, {\\it The Kondo Problem to Heavy Fermions} (Cambridge, \nCambridge, England, 1993);\nK.\\ Yosida, {\\it Theory of Magnetism} (Springer, New York, 1996);\nD.\\ L.\\ Cox and A.\\ Zawadowski, Adv.\\ Phys.\\ {\\bf 47}, 599 (1998).\n\\bibitem{Kondo1}\nD.\\ Goldhaber-Gordon {\\it et al}., Nature (London) {\\bf 391}, 156 (1998);\nD.\\ Goldhaber-Gordon {\\it et al}., Phys.\\ Rev.\\ Lett.\\ {\\bf 81}, 5225 (1998);\n% \\bibitem{Kondo2}\nS.\\ M.\\ Cronenwett {\\it et al}., %T.\\ H.\\ Oosterkamp, and L.\\ P.\\ Kouwenhoven,\nScience {\\bf 281}, 540 (1998);\n% \\bibitem{Kondo3}\nF.\\ Simmel {\\it et al}.,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 83}, 804 (1999).\n\\bibitem{Glazman}\nL.\\ I.\\ Glazman and M.\\ {\\'E}.\\ Ra{\\u\\i}kh,\nPis'ma Zh.\\ Eksp.\\ Teor.\\ Fiz.\\ {\\bf 47}, 378 (1988)\n[{JETP} Lett.\\ {\\bf 47}, 452 (1988)].\n\\bibitem{Ng} T.\\ K.\\ Ng and P.\\ A.\\ Lee,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 61}, 1768 (1988);\n% \\bibitem{Kawabata}\nA.\\ Kawabata, J.\\ Phys.\\ Soc.\\ Jpn.\\ {\\bf 60}, 3222 (1991);\n% \\bibitem{Hershfield}\nS.\\ Hershfield {\\it et al}., %J.\\ H.\\ Davies, and J.\\ W.\\ Wilkins,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 67}, 3720 (1991);\n% ; Phys.\\ Rev.\\ B {\\bf 46}, 7046 (1992).\n% \\bibitem{Meir}\nY.\\ Meir {\\it et al}., % N.\\ S.\\ Wingreen, and P.\\ A.\\ Lee,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 70}, 2601 (1993).\n\\bibitem{Tarucha}\nS.\\ Tarucha {\\it et al}., Phys.\\ Rev.\\ Lett.\\ {\\bf 77}, 3613 (1996).\n\\bibitem{Leo}\nL.\\ P.\\ Kouwenhoven {\\it et al}.,\nScience {\\bf 278}, 1788 (1997).\n\\bibitem{Inoshita}\nT.\\ Inoshita {\\it et al}.,\nPhys.\\ Rev.\\ B {\\bf 48}, R14725 (1993);\n% \\bibitem{Pohjola}\nT.\\ Pohjola {\\it et al}.,\nEurophys.\\ Lett.\\ {\\bf 40}, 189 (1997);\n% \\bibitem{Izumida}\nW.\\ Izumida {\\it et al}., %, O.\\ Sakai, and Y.\\ Shimizu, \nJ.\\ Phys.\\ Soc.\\ Jpn.\\ {\\bf 67}, 2444 (1998);\n% \\bibitem{Yeyati}\nA.\\ Levy Yeyati {\\it et al}., %F.\\ Flores, and A.\\ Mart\\'in-Rodero, \ncond-mat/9901144.\n\\bibitem{Pustilnik}\nM.\\ Pustilnik {\\it et al}., % Y.\\ Avishai and K.\\ Kikoin,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 84}, 1756 (2000).\n\\bibitem{exp}\nS.\\ Sasaki {\\it et al}., \n% S.\\ De Franceschi, J.\\ M.\\ Elzerman, W.\\ G.\\ van der Wiel,\n% M.\\ Eto, S.\\ Tarucha, and L.\\ P.\\ Kouwenhoven, submitted to Nature.\nNature (London) {\\bf 405}, 764 (2000).\n\\bibitem{symmetry}\nOur main results, enhancement of $T_K$ and its power law dependence\non $\\Delta$, are valid also for a simpler model where the\nlevels belong to the same symmetry representation. In this case, however,\nthe level repulsion makes it difficult \nto organize the degeneracy of spin-singlet and triplet states. This is why we\nprefer the model in use. Besides, it is experimentally relevant \\cite{Tarucha}.\n\\bibitem{com1}\nConsidering the matrix element of the Coulomb interaction between\n$d_{1 \\uparrow}^{\\dagger} d_{1 \\downarrow}^{\\dagger} |0 \\rangle$\nand $d_{2 \\uparrow}^{\\dagger} d_{2 \\downarrow}^{\\dagger} |0 \\rangle$,\n$\\langle 22|e^2/r|11 \\rangle$,\none observes that this is indeed the case for $\\delta =0$.\n\\bibitem{Anderson}\nP.\\ W.\\ Anderson, J.\\ Phys.\\ C {\\bf 3}, 2436 (1970).\n\\bibitem{multiK}\nP.\\ Nozi\\`eres and A.\\ Blandin, J.\\ Phys.\\ (Paris) {\\bf 41}, 193 (1980).\n\\bibitem{Haldane}\nF.\\ D.\\ M.\\ Haldane, J.\\ Phys.\\ C {\\bf 11}, 5015 (1978).\n\\bibitem{Okada}\nI.\\ Okada and K.\\ Yosida, Prog.\\ Theor.\\ Phys.\\ {\\bf 49}, 1483 (1973).\n\\end{references}\n\n\n%---------------------------------------------\n% \\pagebreak\n\n\n \\begin{figure}[hbt]\n \\centering\n \\vspace*{1cm}\n \\epsfig{file=fig1a.eps,width=1.7in} \\hspace{.5cm}\n \\epsfig{file=fig1b.eps,width=3in}\n \\vspace*{1cm}\n% \\epsfig{file=fig1a.eps,width=1in}\n% \\epsfig{file=fig1b.eps,width=1.8in}\n \\caption{(a) The energies of spin-singlet and triplet states in\na quantum dot, as functions of magnetic field $B$.\nThe energy difference, $\\Delta=E_{S=0}-E_{S=1}$, can be controlled by\nchanging $B$.\n$\\Delta=0$ at $B=B_0$, where the transition of the ground state occurs.\n(b) Schematic drawing of the Kondo temperature, $T_K$, as a function of\n$\\Delta$. When $\\Delta>0$,\n$T_K(\\Delta)/T_K(0)=(T_K(0)/\\Delta)^{\\gamma}$ where\n{\\bf a} $\\gamma=1$, {\\bf b} 0.5 and, {\\bf c} 0.25 at \n$T_K(0) \\ll \\Delta \\ll D_0$ (bandwidth), and\n$T_K(\\Delta)$ is a constant at $\\Delta \\gg D_0$. When $\\Delta<0$,\n$T_K(\\Delta)$ drops\nto zero suddenly at $|\\Delta| \\sim T_K(0)$.\nInset: Spin-flip processes in our model. The exchange\ncouplings $J^{(i)}$ involving spin-triplet states only are accompanied \nby scattering of conduction electrons of channel $i$.\nThose involving spin-triplet and singlet states ($\\tilde{J}$, $\\tilde{J}^*$)\nare accompanied by interchannel scattering of conduction electrons.\n}\n \\end{figure}\n\n\\end{document}\n\n\n\n"
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{
"name": "cond-mat0002010.extracted_bib",
"string": "\\bibitem{classics}\nJ.\\ Kondo, Prog.\\ Theor.\\ Phys.\\ {\\bf 32}, 37 (1964);\nA.\\ C.\\ Hewson, {\\it The Kondo Problem to Heavy Fermions} (Cambridge, \nCambridge, England, 1993);\nK.\\ Yosida, {\\it Theory of Magnetism} (Springer, New York, 1996);\nD.\\ L.\\ Cox and A.\\ Zawadowski, Adv.\\ Phys.\\ {\\bf 47}, 599 (1998).\n\n\\bibitem{Kondo1}\nD.\\ Goldhaber-Gordon {\\it et al}., Nature (London) {\\bf 391}, 156 (1998);\nD.\\ Goldhaber-Gordon {\\it et al}., Phys.\\ Rev.\\ Lett.\\ {\\bf 81}, 5225 (1998);\n% \n\\bibitem{Kondo2}\nS.\\ M.\\ Cronenwett {\\it et al}., %T.\\ H.\\ Oosterkamp, and L.\\ P.\\ Kouwenhoven,\nScience {\\bf 281}, 540 (1998);\n% \n\\bibitem{Kondo3}\nF.\\ Simmel {\\it et al}.,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 83}, 804 (1999).\n\n\\bibitem{Glazman}\nL.\\ I.\\ Glazman and M.\\ {\\'E}.\\ Ra{\\u\\i}kh,\nPis'ma Zh.\\ Eksp.\\ Teor.\\ Fiz.\\ {\\bf 47}, 378 (1988)\n[{JETP} Lett.\\ {\\bf 47}, 452 (1988)].\n\n\\bibitem{Ng} T.\\ K.\\ Ng and P.\\ A.\\ Lee,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 61}, 1768 (1988);\n% \n\\bibitem{Kawabata}\nA.\\ Kawabata, J.\\ Phys.\\ Soc.\\ Jpn.\\ {\\bf 60}, 3222 (1991);\n% \n\\bibitem{Hershfield}\nS.\\ Hershfield {\\it et al}., %J.\\ H.\\ Davies, and J.\\ W.\\ Wilkins,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 67}, 3720 (1991);\n% ; Phys.\\ Rev.\\ B {\\bf 46}, 7046 (1992).\n% \n\\bibitem{Meir}\nY.\\ Meir {\\it et al}., % N.\\ S.\\ Wingreen, and P.\\ A.\\ Lee,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 70}, 2601 (1993).\n\n\\bibitem{Tarucha}\nS.\\ Tarucha {\\it et al}., Phys.\\ Rev.\\ Lett.\\ {\\bf 77}, 3613 (1996).\n\n\\bibitem{Leo}\nL.\\ P.\\ Kouwenhoven {\\it et al}.,\nScience {\\bf 278}, 1788 (1997).\n\n\\bibitem{Inoshita}\nT.\\ Inoshita {\\it et al}.,\nPhys.\\ Rev.\\ B {\\bf 48}, R14725 (1993);\n% \n\\bibitem{Pohjola}\nT.\\ Pohjola {\\it et al}.,\nEurophys.\\ Lett.\\ {\\bf 40}, 189 (1997);\n% \n\\bibitem{Izumida}\nW.\\ Izumida {\\it et al}., %, O.\\ Sakai, and Y.\\ Shimizu, \nJ.\\ Phys.\\ Soc.\\ Jpn.\\ {\\bf 67}, 2444 (1998);\n% \n\\bibitem{Yeyati}\nA.\\ Levy Yeyati {\\it et al}., %F.\\ Flores, and A.\\ Mart\\'in-Rodero, \ncond-mat/9901144.\n\n\\bibitem{Pustilnik}\nM.\\ Pustilnik {\\it et al}., % Y.\\ Avishai and K.\\ Kikoin,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 84}, 1756 (2000).\n\n\\bibitem{exp}\nS.\\ Sasaki {\\it et al}., \n% S.\\ De Franceschi, J.\\ M.\\ Elzerman, W.\\ G.\\ van der Wiel,\n% M.\\ Eto, S.\\ Tarucha, and L.\\ P.\\ Kouwenhoven, submitted to Nature.\nNature (London) {\\bf 405}, 764 (2000).\n\n\\bibitem{symmetry}\nOur main results, enhancement of $T_K$ and its power law dependence\non $\\Delta$, are valid also for a simpler model where the\nlevels belong to the same symmetry representation. In this case, however,\nthe level repulsion makes it difficult \nto organize the degeneracy of spin-singlet and triplet states. This is why we\nprefer the model in use. Besides, it is experimentally relevant \\cite{Tarucha}.\n\n\\bibitem{com1}\nConsidering the matrix element of the Coulomb interaction between\n$d_{1 \\uparrow}^{\\dagger} d_{1 \\downarrow}^{\\dagger} |0 \\rangle$\nand $d_{2 \\uparrow}^{\\dagger} d_{2 \\downarrow}^{\\dagger} |0 \\rangle$,\n$\\langle 22|e^2/r|11 \\rangle$,\none observes that this is indeed the case for $\\delta =0$.\n\n\\bibitem{Anderson}\nP.\\ W.\\ Anderson, J.\\ Phys.\\ C {\\bf 3}, 2436 (1970).\n\n\\bibitem{multiK}\nP.\\ Nozi\\`eres and A.\\ Blandin, J.\\ Phys.\\ (Paris) {\\bf 41}, 193 (1980).\n\n\\bibitem{Haldane}\nF.\\ D.\\ M.\\ Haldane, J.\\ Phys.\\ C {\\bf 11}, 5015 (1978).\n\n\\bibitem{Okada}\nI.\\ Okada and K.\\ Yosida, Prog.\\ Theor.\\ Phys.\\ {\\bf 49}, 1483 (1973).\n"
}
] |
cond-mat0002011
|
Evidence for a two component magnetic response in UPt$_3$
|
[
{
"author": "A. Yaouanc"
},
{
"author": "$^1$ P. Dalmas de R\\'eotier"
},
{
"author": "$^1$ F.N. Gygax"
},
{
"author": "$^2$ A. Schenck"
},
{
"author": "$^2$ A. Amato"
},
{
"author": "$^3$ C. Baines"
},
{
"author": "$^3$ P.C.M. Gubbens"
},
{
"author": "$^4$ C.T.~Kaiser"
},
{
"author": "$^4$ A. de Visser"
},
{
"author": "$^5$ R.J. Keizer"
},
{
"author": "$^5$ A. Huxley$^1$ and A.A. Menovsky$^5$"
}
] |
The magnetic response of the heavy fermion superconductor UPt$_3$ has been investigated on a microscopic scale by muon Knight shift studies. Two distinct and isotropic Knight shifts have been found for the field in the basal plane. While the volume fractions associated with the two Knight shifts are approximately equal at low and high temperatures, they show a dramatic and opposite temperature dependence around $T_N$. Our results are independent on the precise muon localization site. We conclude that UPt$_3$ is characterized by a two component magnetic response.
|
[
{
"name": "UPt3_psi_lettre.tex",
"string": "\\documentstyle[prl,aps,epsf]{revtex}\n\n\\begin{document}\n\\draft\n%\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname\n\n\\title{Evidence for a two component magnetic response in UPt$_3$}\n\\author{A. Yaouanc,$^1$ P. Dalmas de R\\'eotier,$^1$ F.N. Gygax,$^2$\nA. Schenck,$^2$ A. Amato,$^3$ C. Baines,$^3$ P.C.M. Gubbens,$^4$ \nC.T.~Kaiser,$^4$ A. de Visser,$^5$ \nR.J. Keizer,$^5$ A. Huxley$^1$ and A.A. Menovsky$^5$}\n\n\\address{$^1$Commissariat \\`a l'Energie Atomique, D\\'epartement de Recherche\nFondamentale sur la Mati\\`ere Condens\\'ee \\\\\nF-38054 Grenoble Cedex 9, France}\n\\address{$^2$Institute for Particle Physics, Eidgn\\\"ossische Technische \nHochschule Zurich, CH-5232 Villigen PSI, Switzerland}\n\\address{$^3$Laboratory for Muon Spectroscopy, Paul Scherrer Institute,\nCH-5232 Villigen PSI, Switzerland} \n\\address{$^4$Interfacultair Reactor Instituut, Delft University of Technology,\n2629 JB Delft, The Netherlands}\n\\address{$^5$Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, \n1018 XE Amsterdam, The Netherlands}\n\\date{\\today} \\maketitle \n%\n%\n\n\\begin{abstract}\nThe magnetic response of the heavy fermion superconductor UPt$_3$ has been \ninvestigated on a microscopic scale by muon Knight shift studies.\nTwo distinct and isotropic Knight shifts have been found for the field in the \nbasal plane. While the volume fractions associated with the two Knight shifts \nare approximately equal at low and high temperatures, they show a \ndramatic and opposite temperature dependence around $T_N$. Our results are \nindependent on the precise muon localization site. We conclude that\nUPt$_3$ is characterized by a two component magnetic response.\n\n\n\\end{abstract}\n\\pacs{PACS numbers : 74.70.Tx, 75.30.Gw, 76.75.+i}\n%\n]\n%\nThe hexagonal heavy fermion superconductor UPt$_3$ is attracting much interest\nbecause it has been established as an unconventional superconductor\nas seen by the existence of three \ndistinct superconducting phases in the magnetic field-temperature plane\n\\cite{Sauls94,Heffner96}. In zero-field the two superconducting phase \ntransitions occur at $\\sim$ 0.475~K and $\\sim$ 0.520~K. \nIt is usually thought that this complex phase \ndiagram arises from the lifting of the degeneracy of a \nmulticomponent superconducting order parameter. \n\nThe most popular candidate for such a symmetry-breaking field is the short \nrange antiferromagnetic order characterized by a N\\'eel temperature of \n$T_N$ $\\simeq$ 6 K and an extremely small ordered magnetic moment \n(0.02 (1) $\\mu_B$/U-atom in the limit $T \\rightarrow 0$ K) oriented along \nthe $a^*$ axis ($\\equiv b$ axis). The magnetic order has only been \nobserved by neutron \\cite{Aeppli89} and magnetic x-ray \\cite{Isaacs95}\ndiffractions. \n\nNuclear magnetic resonance \\cite{Tou} and zero-field muon spin relaxation\n\\cite{Dalmas95} measurements as well as macroscopic studies have failed to \nprove the existence of static antiferromagnetic order on high quality samples \n\\cite{extra}. Here we present transverse high-field muon spin rotation \n($\\mu$SR) data which present anomalies around $T_N$. Moreover, they \nshow that UPt$_3$ is characterized by \na two component magnetic response at least up to 115 K.\n\n\nIn the transverse $\\mu$SR technique \\cite{muon}, polarized muons are \nimplanted into a sample where their spins ${\\bf S}_{\\mu}$ ($S_{\\mu}$ = 1/2)\nprecess in the local magnetic field ${\\bf B}_{\\rm loc}$ until they decay.\nThe sample is polarized by a magnetic field ${\\bf B}_{\\rm ext}$ applied \nperpendicularly to ${\\bf S}_{\\mu}(t=0)$. \n${\\bf S}_{\\mu}(t)$ is monitored through the decay positron.\nBy collecting several million positrons, one can readily obtain an accurate \nvalue for the field at the muon site(s).\n\nWe present results for three samples. Two samples have been grown in \nGrenoble. Each consists of crystals glued on a silver backing plate and put \ntogether to form a disk. They differ by the orientation of the crystal \naxes relative to the normal to the sample plane: either the \n$a^*$ or $c$ axis is parallel to that direction. Measurements have \ntherefore been carried out either with ${\\bf B}_{\\rm ext}$ parallel to $a^*$\nor $c$. The third sample has been prepared in \nAmsterdam. It is a cube of 5 $\\times$ 5 $\\times$ 5 mm$^3$ which has been \nglued to a thin silver rod. The measurements on this sample have \nbeen done only with ${\\bf B}_{\\rm ext}$ $\\parallel a$.\nThe Grenoble samples have already been used for zero-field \\cite{Dalmas95} \nand transverse low-field \\cite{Yaouanc98} $\\mu$SR measurements. Their high \nquality is demonstrated by the splitting of the two zero-field\nsuperconducting transitions as seen by specific heat \\cite{Yaouanc98} \nand the low residual resistivities which are among the lowest ever reported \n($\\rho_c(0)$ = 0.17 $\\mu \\Omega$~cm and $\\rho_{a^*}(0)$ = 0.54 $\\mu \\Omega$~cm\n\\cite{Suderow98}). The Amsterdam sample is of a somewhat lesser\nquality in terms of the residual resistivities which are roughly a\nfactor 3 higher than for the Grenoble sample. \nNevertheless the double superconducting transition is clearly resolved in the \nspecific heat. \n\n\nThe measurements have been performed at the low temperature facility\n(LTF) and at the general purpose spectrometer (GPS) of the $\\mu$SR facility \nlocated at the Paul Scherrer Institute. The \nLTF spectra have been obtained for temperatures between 0.05 K and 10 K and \n$B_{\\rm ext}$ of 2.3 T (only for two measurements), 2 T and 1.5 T. \nThe GPS data have been taken with \n$B_{\\rm ext}$ = 0.6 T for 1.7 K $\\leq$ $T \\leq$ 200 K. A high \nstatistic GPS measurement \nhas been carried out at 50 K with $B_{\\rm ext}$ = 0.45 T. The GPS \nmeasurements have been performed with an electrostatic kicker device \non the beam line which ensures that only one muon at a time is present in \nthe sample \\cite{Abela99}. With such a device, the signal to noise ratio \nis strongly enhanced and the time window is extended to $\\sim$ 18\n$\\mu$s. For both \nspectrometers ${\\bf B}_{\\rm ext}$ has been applied along the muon beam \ndirection and a spin rotator has been used to flip the muon spin away from \nthe muon momentum. \n\nWe expect to observe a sum of oscillating \nsignals, each corresponding to a given type of muon environment. \nAn extra signal originating from muons stopped in the sample\nsurroundings, basically a silver backing plate, is also expected.\n\nIn Fig.~\\ref{fig_fourier} we present two Fourier transforms of spectra. \nTwo lines from the sample are clearly detected for \n${\\bf B}_{\\rm ext}$ $\\parallel a^*$. A symmetric single line is observed \nfor ${\\bf B}_{\\rm ext}$ $\\parallel c$. \nFor the whole temperature range investigated two components are found for \n${\\bf B}_{\\rm ext}$ $\\perp c$ and only \na single component is detected for ${\\bf B}_{\\rm ext}$ $\\parallel c$. \nIn Fig.~\\ref{fig_spectra} we present a time spectrum which clearly shows the \nexistence of the two components far into the paramagnetic regime for \n${\\bf B}_{\\rm ext}$ $\\perp c$.\n \n\\begin{figure}\n\\epsfxsize=82 mm\n\\epsfbox{UPt3_psi_lettre_fourier.ps}\n\\caption{Two Fourier transforms of spectra recorded at 2.6 K with \n$B_{\\rm ext}$ = 2.3 T. ${\\bf B}_{\\rm ext}$ is either parallel to the $a^*$ \nor $c$ axis. The background signal was\nintentionally enlarged in this measurement to evidence the difference\nbetween the applied field and the field at the muon site. This was\nachieved by fixing a 10 mm diameter Ag mask on the sample, \nresulting in a reduced effective sample size.\nThe line at $\\sim$ 303.6 MHz originates from the Ag mask.}\n\\label{fig_fourier}\n\\end{figure}\n\nWe first discuss the spectra recorded for ${\\bf B}_{\\rm ext} \\perp c $\nwhich have been analyzed with the polarization function $P_X(t)$ written as \nthe sum of three components:\n\n\\begin{eqnarray}\naP_X(t) & & = \na_{\\rm F} \\cos \\left( \\omega_{\\rm F} t \\right) \n\\exp \\left (- {\\Delta^2 t^2/ 2} \\right) \n\\cr \n+ a_{\\rm S} \\cos & & \\left (\\omega_{\\rm S} t \\right) \\exp(- \\lambda t) \n+ a_{\\text{bg}}\\cos \\left (\\omega_{\\text{bg}} t \\right) \n\\exp(- \\lambda_{\\text{bg}} t). \n\\label{fit}\n\\end{eqnarray}\n\nThe first two components describe the $\\mu$SR signal from the sample and the \nthird accounts for the muons stopped in the background. The subscripts F and \nS refer to the first and second components, respectively. $a_{\\alpha}$ is the \ninitial asymmetry of component $\\alpha$ oscillating at the pulsation frequency \n$\\omega_{\\alpha}$ = $2 \\pi \\nu_{\\mu,\\alpha}$ = \n$\\gamma_\\mu B_{{\\rm loc}, \\alpha}$ \nwhere $\\nu_{\\mu,\\alpha}$ is the precession frequency of component $\\alpha$\nand $\\gamma_\\mu$ the muon gyromagnetic ratio \n($ \\gamma_ \\mu$ = 851.6 Mrad s$^{ {\\rm -1}}$T$^{-1} $). \n$a_\\alpha$ is proportionnal to the fraction of muons experiencing field\n$B_{{\\rm loc}, \\alpha}$.\nThe envelop of the first component is \nbest fitted by a Gaussian function, while the envelop of the second component\nis better described by an exponential damping. We stress that the measured\ntemperature dependences of the two initial asymmetries and frequencies are not \ninfluenced by the choice of the envelop functions.\n \n\\begin{figure}\n\\epsfxsize=82 mm\n\\epsfbox{UPt3_psi_lettre_spectre.ps}\n\\caption{A spectrum recorded at 50 K in a field of 0.45 T applied \nperpendicular to the $c$ axis of UPt$_3$ and presented in a \nframe rotating at a precession frequency of 60 MHz. This experiment \nwas perfomed\nin a setup designed to get the smallest possible background ($a_{\\rm bg}\n\\simeq$ 0.017). \nThe solid line is a fit \nto a sum of three oscillating components: two from the sample and one from the \nbackground. \nThe reduced asymmetry observed on the plot around 10 $\\mu$s, which is too\nsmall to be explained by a beating of the UPt$_3$ and background\nsignals, reflects the \nbeating of the two signals originating from UPt$_3$. } \n\\label{fig_spectra}\n\\end{figure}\n\n$\\Delta$ is approximately independent of the temperature and amounts to\n$\\simeq$ 0.55 MHz at high field. It roughly scales with $B_{\\rm ext}$.\n$\\lambda$ is independent of $B_{\\rm ext}$ and is equal to \n$\\lambda$ $\\simeq$ 0.14 MHz at the lowest temperature. It decreases when the \ntemperature is increased and becomes so small above 4 K that it can\nbe fixed to zero. The values\nof the damping rates may reflect only partially the intrinsic properties of \nUPt$_3$ because of the field inhomogeneity due to the demagnetization field. \nHowever, $\\Delta$ and $\\lambda$ are remarkably small, indicating that the \nmagnetic inhomogeneity detected for the two components is small.\n\n\nIn Fig.~\\ref{parametre_astar} we display the temperature dependence of the two\ninitial asymmetries and the associated relative frequency shifts, $K_{\\mu}$. \nThese plots \nconcern the spectra taken with ${\\bf B}_{\\rm ext} \\perp c$ and for \n$T \\leq$ 14.7 K. $K_{\\mu}$, which is the local magnetic susceptibility at the \nmuon site, is usually called the Knight shift. It is deduced from the \nmeasured relative frequency shift, $K_{\\rm exp}$, after correcting for the \nLorentz and demagnetization fields. $K_{\\rm exp}$ is \ndefined by $K_{\\rm exp}$ $= {\\bf B}_{\\rm ext} \\cdot ({\\bf B}_{\\rm loc} \n- {\\bf B}_{\\rm ext})/ B_{\\rm ext}^2$. We have determined $B_{\\rm ext}$ with \na gaussmeter or through the pulsation frequency \nof the background: $B_{\\rm ext}$ = $\\omega_{\\text{bg}}/ \\gamma_\\mu$.\nSince the Knight shift of the background is very small\n($K_{\\mu}$ for silver is \n$\\simeq$ 94 ppm \\cite{Schenck82}), this is a very good approximation. \nAlthough the Lorentz and demagnetization correction modifies substantially \nthe absolute value of the Knight shift, qualitatively it does not influence\nits temperature dependence. The conclusions we shall draw from our \ndata are independent of the uncertainty related to the correction. \n\nThe results of Fig.~\\ref{parametre_astar} show that the \nGrenoble and Amsterdam samples yield consistent results. Since, as \nindicated in the figure, the measurements have been done either with\n${\\bf B}_{\\rm ext}$ $\\parallel a^*$ or ${\\bf B}_{\\rm ext}$ $\\parallel a$, \nwe conclude that the $\\mu$SR response is isotropic \nin the basal plane. The data display also two remarkable features.\n\n\n\\begin{figure}\n\\epsfxsize=82 mm\n\\epsfbox{UPt3_psi_lettre_astar.ps}\n\\caption{Temperature dependence of the initial asymmetries and Knight \nshifts $K_{\\mu}$\nwith ${\\bf B}_{\\rm ext}$ perpendicular to the $c$ axis. The data are for \n0.5 K $\\leq$ $T$ $<$ 15 K, three field intensities and two samples denoted as \nGrenoble and Amsterdam. $K_{\\mu}$ is corrected for Lorentz and \ndemagnetization fields. The actual value of $K_\\mu$ is subject to an \nuncertainty due to the demagnetization correction. Nevertheless the shape \nof $K_\\mu(T)$ is independent on this correction.\nThe F and S letters denotes the two components.}\n\\label{parametre_astar}\n\\end{figure}\n\nFirst, the frequency splitting between the two lines is relatively large \nat low $T$, decreases rapidly as $T$ increases up to $\\sim$ 6 K and exhibits \na shallow minimum around 10 K. It increases again for higher temperatures\n(not shown). This explains the possibility of observing the beating between \nthe two oscillating components at 50 K as shown in Fig.~\\ref{fig_spectra}. \nOnly the F shift has a \nstrong temperature dependence below 6 K while the S shift is practically \ntemperature independent down to 0.4 K, below which its absolute value slightly \ndecreases. Since 6 K is the $T_N$ value as determined by neutron \ndiffraction on our samples, the temperature dependence of the F shift provides\na signature of the N\\'eel temperature. \n\nThe second feature is probably the most striking: we observe two muon \nprecession frequencies with approximately equal initial asymmetries in the \nwhole temperature \nrange (0.05 K $\\leq$ $T \\leq$ 200 K, the region $T >$ 15 K is not shown in \nFig.~\\ref{parametre_astar}) except near $T_N$ ($T_N$ $\\pm$ 4 K) \nwhere $a_{\\rm F}$ increases at the expense of $a_{\\rm S}$. \n\nIn this temperature range a\ntrend for a larger difference between these initial asymmetries seems to be\npresent at high field. However this trend \nmight not be meaningful since the signal to noise ratio for \nthe 2.0 T and 1.5 T spectra is not as good as for the 0.6 T spectra. An \neventual field effect on $a_{\\rm F}$ and $a_{\\rm S}$ could only be \nconfirmed by \nmeasurements with the electrostatic kicker device at all fields. Since\nhigh-field neutron diffraction \\cite{Lussier96,Dijk98} did not detect\nany sizeable change in the\nrelative population of the three equivalent antiferromnagnetic domains \nwe do not expect a field effect on the initial asymmetries.\n \n\nThe spectra recorded with ${\\bf B}_{\\rm ext} \\parallel c $ have been analysed \nwith a formula similar to Eq.~\\ref{fit} with $a_{\\rm S}$ = 0. The precession\nfrequency varies smoothly in temperature. \nThe Gaussian damping rate scales again with $B_{\\rm ext}$ ($\\sim$ 0.42\nMHz at 1.5 T) and is essentially temperature independent up to\n$\\sim$ 30 K above which temperature it drops smoothly to very\nsmall values.\n\nIn Fig.~\\ref{jaca} we present the $K_{\\mu}$ data recorded for \n$B_{\\rm ext}$ = 0.6 T with 1.7 K $\\leq T \\leq$ 115 K as a function of the \nbulk susceptibility $\\chi_B$. This is a so called Clogston-Jaccarino plot, the \ntemperature is an implicit parameter. The bulk susceptibilities for the \ndifferent orientations have been measured on the Grenoble samples and are \nsimilar to those of Ref.~\\cite{Frings83}. Classically, we should find \n$K_{\\mu}$ scaling with the susceptibility. This is approximately observed for \n${\\bf B}_{\\rm ext} \\parallel c$ but not for ${\\bf B}_{\\rm ext} \\perp c$. \nIn addition, as already pointed out when discussing $K_{\\mu}(T)$, the \nClogston-Jaccarino plots clearly show that while the F Knight shift provides a \nsignature of $T_N$, such a signature is absent for the S Knight shift. \nThe data of Fig.~\\ref{jaca} suggest that $K_\\mu$ passes smoothly through $T_N$\nfor ${\\bf B}_{\\rm ext} \\parallel c$, although the almost constant value of \n$\\chi_B(T)$ at low temperature does not allow for a definite statement.\n\nWe now discuss the muon diffusion properties and localization site in UPt$_3$. \nThe shape of the zero-field depolarization and the constant value of the \nrelated damping rate show that the muons are static and occupy the same site\nin the muon time scale, at least below 30 K \\cite{Dalmas95}. \nOur transverse field measurements suggest that in fact the muon is diffusing \nonly above 115 K because the frequency splitting collapses above that \ntemperature (not shown). Since we focus on the properties of UPt$_3$ itself, \nwe only consider the data for which the muon is static. Thus the \nanomalous temperature dependence of the two initial asymmetries \naround $T_N$ for \n${\\bf B}_{\\rm ext} \\perp c$ can not be due to muon diffusion. The \neventual existence of two distinct muon sites can not explain our data\nsince their relative occupancy should not change for a static muon. \nInterestingly, the analysis for \nU(Pt$_{0.95}$Pd$_{0.05}$)$_3$ of the angular dependence of $K_{\\mu}$ shows\nthat the muon occupies only one site, the 2a site in Wyckoff notation \n(P6$_3$/mmc space group) in this related compound\n\\cite{Schenckfuture}.\n\n\\begin{figure}\n\\epsfxsize=82 mm\n\\epsfbox{UPt3_psi_lettre_jac.ps}\n\\caption{Clogston-Jaccarino plots obtained for $B_{\\rm ext}$ = 0.6 T. \nFor ${\\bf B}_{\\rm ext} \\perp c$ we use the same symbol convention as in \nFig. \\ref{parametre_astar}. The filled triangles correspond to ${\\bf\nB}_{\\rm ext} \\parallel c$. The temperature is an implicit parameter: \n1.7 K $\\leq$ $T \\leq$ 115 K. $K_\\mu$ is extracted from the reported \n$\\mu$SR measurements and the bulk susceptibility $\\chi_B$ from measurements \non our samples. The lines are guides to the eyes. Note the drastic change of \nslope at $T_N$ for the F set of data. }\n\\label{jaca}\n\\end{figure}\n\nOur results are understood if we suppose that the muon \noccupies only one magnetic site and the sample is intrinsically \ninhomogeneous: it consists of two regions with slightly different magnetic \nresponses and relative volumes which are temperature dependent. While near \n$T_N$ one region dominates, outside that temperature range the two regions \noccupy approximately equal volumes. An alternative explanation for our\ndata could be the existence of a complex magnetic structure leading to\nthe observed $\\mu$SR response. However this would imply a more involved\nmagnetic structure than the one published \n\\cite{Aeppli89,Isaacs95,Lussier96,Dijk98}. In addition it is difficult\nto imagine that a magnetic structure can influence the muon response up \nto at least 20 $T_N$. Therefore we\ndisregard this latter explanation.\n\n\nThe facts that the magnetic phase transition is only detected by transverse \nhigh-field and not by zero-field $\\mu$SR measurements \n\\cite{Dalmas95,Visser97b} \nare not inconsistent. It is not unexpected to observe below $T_N$ a new \nsource of quasi-static magnetic polarization induced by the applied field \nwhich leads to an extra Knight shift. \n\nBulk magnetic susceptibility does not detect the phase transition since the \nrelative sensitivity in these conditions is $\\simeq$ 10$^{-3}$. As shown in \nFig.~\\ref{parametre_astar}, this is not enough.\n\nThe results obtained by the $\\mu$SR and magnetic diffraction techniques are \nnot contradictory. The diffraction results simply mean that the\ndifference in the scattering properties of the two regions may be too subtle\nto be distinguished.\n\nWe now consider the possible origin for the additional Knight shift observed \nbelow $T_N$ for the F component. A change of the magnitude of the \nmoments is excluded since \nhigh-field neutron diffraction measurements do not detect any sizeable \ninfluence of a field up to 12 T \\cite{Lussier96,Dijk98}. Two \nmechanisms producing an additional shift can be imagined. \nThe first mechanism involves the dipolar field produced at the muon site by \nthe ordered uranium moments. A rotation of these moments induced by the \napplied field leads to an additionnal field at the muon site. A small \nrotation is not excluded by neutron diffraction since this technique gives an \nupper bound rotation angle as large as 26$^\\circ$ \\cite{Lussier96}. \nBut it is surprising, for a magnet with moments oriented along the $a^*$ \ndirection to observe the same $K_{\\mu}$ for\n${\\bf B}_{\\rm ext} \\parallel a$ and ${\\bf B}_{\\rm ext} \\parallel a^*$\n(see Fig.~\\ref{parametre_astar}). The second possible origin focuses on the \nitinerant character of the magnetism of UPt$_3$. \nIn this picture the additional shift is a measure of the enhancement of the \nmagnetic susceptibility of the conduction electrons below $T_N$. UPt$_3$ \nbeing a planar magnet with a negligible planar anisotropy, the enhancement \nshould be isotropic in the plane perpendicular to $c$ and no enhancement \nshould be observed for ${\\bf B}_{\\rm ext} \\parallel c$. This is consistent \nwith our data.\n\n\nOur most surprising result is the existence of the two components when \n${\\bf B}_{\\rm ext} \\perp c$. Since the associated damping rates are \nsmall, we infer that the magnetic disorder is small. The near equality in \nmost of the temperature range of the two initial asymmetries suggests that \nthe two regions originate from a periodic modulation. The behaviour\nof the initial asymmetries near $T_N$ implies that \nthe proposed modulation is strongly coupled to the magnetic order parameter. \nThe structural modulation observed by electron microscopy and \ndiffraction some years ago \\cite{Midgley93} might be related to the \nregions discussed here. However it has never been seen \nthereafter including in our samples.\n\n\nIn summary we have discovered by transverse high-field $\\mu$SR measurements\nthe existence of a two component magnetic response. While the volume\nfraction associated with these components is equal below \n$\\sim$ 2 K and above $\\sim$ 10 K, it is strongly temperature dependent \naround $T_N$. \nWe also observe a signature of the \nmagnetic transition for one of the two components. \nOur results are naturally explained if UPt$_3$ is\nintrinsically inhomogeneous at least in a applied field.\n\n\n\\vspace{-.5 cm} % remove this before submission\n\\begin{references}\n\\vspace{-1.2 cm} % remove this before submission\n\n\n\\bibitem{Sauls94} J.A. Sauls, Adv. in Phys. {\\bf 43}, 113 (1994).\n\\bibitem{Heffner96} R.H. Heffner and M.R. Norman, Comm. Condens. Matter Phys.\n{\\bf 17}, 361 (1996).\n\\bibitem{Aeppli89} G. Aeppli {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 60}, 615\n(1989).\n\\bibitem{Isaacs95} E.D. Isaacs {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 75}, 1178\n(1995).\n\\bibitem{Tou} H. Tou {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 77}, 1374 (1996).\n\\bibitem{Dalmas95} P. Dalmas de R\\'eotier {\\it et al.\\/}, Phys. Lett. A \n{\\bf 205}, 239 (1995).\n\\bibitem{extra} The increase of the muon damping rate observed below $T_N$ \nwith a first generation sample \n(e.g. D.W. Cooke {\\it et al.\\/}, Hyperfine Interact. {\\bf 31}, 425 (1986)\nand R.H. Heffner {\\it et al.\\/}, Phys. Rev. B {\\bf 39}, 11345 (1989))\nhas not been reproduced thereafter (see\nRefs. \\protect\\onlinecite{Dalmas95,Visser97b}).\n\\bibitem{muon} A. Schenck, and F.N. Gygax, in {\\sl Handbook of Magnetic \nMaterials\\/}, Vol. 9, edited by K.H.J. Buschow (Elsevier Science B.V., 1995).\n; P. Dalmas de R\\'eotier and A. Yaouanc, J. Phys.: Condens.\nMatter {\\bf 9}, 9113 (1997); A. Amato, Rev. Mod. Phys. {\\bf 69}, 1119 (1997).\n\\bibitem{Yaouanc98} A. Yaouanc {\\it et al.\\/}, J. Phys.: Condens. Matter \n{\\bf 10}, 9791 (1998).\n\\bibitem{Suderow98} H. Suderow {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 80}, 165\n(1998).\n\\bibitem{Abela99} R. Abela {\\it et al.\\/}, Hyperfine Interact. {\\bf 120-121},\n575 (1999).\n\\bibitem{Schenck82} A. Schenck, Helv. Phys. Acta {\\bf 54}, 471 (1982).\n\\bibitem{Lussier96} B. Lussier {\\it et al.\\/}, Phys. Rev. B {\\bf 54}, R6873\n(1996).\n\\bibitem{Dijk98} N.H. van Dijk {\\it et al.\\/}, Phys. Rev. B {\\bf 58}, \n3186 (1998). \n\\bibitem{Frings83} P.H. Frings {\\it et al.\\/}, J. Magn. Magn. Mat. {\\bf 31-34},\n240 (1983).\n\\bibitem{Schenckfuture} A. Schenck {\\it et al.\\/}, cond-mat/9909197. \n\\bibitem{Visser97b} A. de Visser {\\it et al.\\/}, Physica B {\\bf 230-232}, 53\n(1997).\n\\bibitem{Midgley93} P.A. Midgley {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 70},\n678 (1993).\n\n \n\\end{references}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002011.extracted_bib",
"string": "\\bibitem{Sauls94} J.A. Sauls, Adv. in Phys. {\\bf 43}, 113 (1994).\n\n\\bibitem{Heffner96} R.H. Heffner and M.R. Norman, Comm. Condens. Matter Phys.\n{\\bf 17}, 361 (1996).\n\n\\bibitem{Aeppli89} G. Aeppli {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 60}, 615\n(1989).\n\n\\bibitem{Isaacs95} E.D. Isaacs {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 75}, 1178\n(1995).\n\n\\bibitem{Tou} H. Tou {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 77}, 1374 (1996).\n\n\\bibitem{Dalmas95} P. Dalmas de R\\'eotier {\\it et al.\\/}, Phys. Lett. A \n{\\bf 205}, 239 (1995).\n\n\\bibitem{extra} The increase of the muon damping rate observed below $T_N$ \nwith a first generation sample \n(e.g. D.W. Cooke {\\it et al.\\/}, Hyperfine Interact. {\\bf 31}, 425 (1986)\nand R.H. Heffner {\\it et al.\\/}, Phys. Rev. B {\\bf 39}, 11345 (1989))\nhas not been reproduced thereafter (see\nRefs. \\protect\\onlinecite{Dalmas95,Visser97b}).\n\n\\bibitem{muon} A. Schenck, and F.N. Gygax, in {\\sl Handbook of Magnetic \nMaterials\\/}, Vol. 9, edited by K.H.J. Buschow (Elsevier Science B.V., 1995).\n; P. Dalmas de R\\'eotier and A. Yaouanc, J. Phys.: Condens.\nMatter {\\bf 9}, 9113 (1997); A. Amato, Rev. Mod. Phys. {\\bf 69}, 1119 (1997).\n\n\\bibitem{Yaouanc98} A. Yaouanc {\\it et al.\\/}, J. Phys.: Condens. Matter \n{\\bf 10}, 9791 (1998).\n\n\\bibitem{Suderow98} H. Suderow {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 80}, 165\n(1998).\n\n\\bibitem{Abela99} R. Abela {\\it et al.\\/}, Hyperfine Interact. {\\bf 120-121},\n575 (1999).\n\n\\bibitem{Schenck82} A. Schenck, Helv. Phys. Acta {\\bf 54}, 471 (1982).\n\n\\bibitem{Lussier96} B. Lussier {\\it et al.\\/}, Phys. Rev. B {\\bf 54}, R6873\n(1996).\n\n\\bibitem{Dijk98} N.H. van Dijk {\\it et al.\\/}, Phys. Rev. B {\\bf 58}, \n3186 (1998). \n\n\\bibitem{Frings83} P.H. Frings {\\it et al.\\/}, J. Magn. Magn. Mat. {\\bf 31-34},\n240 (1983).\n\n\\bibitem{Schenckfuture} A. Schenck {\\it et al.\\/}, cond-mat/9909197. \n\n\\bibitem{Visser97b} A. de Visser {\\it et al.\\/}, Physica B {\\bf 230-232}, 53\n(1997).\n\n\\bibitem{Midgley93} P.A. Midgley {\\it et al.\\/}, Phys. Rev. Lett. {\\bf 70},\n678 (1993).\n\n \n"
}
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cond-mat0002012
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Experimental Critical Current Patterns in Josephson Junction Ladders
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[
{
"author": "P. Binder"
},
{
"author": "P. Caputo"
},
{
"author": "M. V. Fistul"
},
{
"author": "and A. V. Ustinov"
}
] |
We present an experimental and theoretical study of the magnetic field dependence of the critical current of Josephson junction ladders. At variance with the well-known case of a one-dimensional (1D) parallel array of Josephson junctions the magnetic field patterns display a single minimum even for very low values of the self-inductance parameter $\beta_{L}$. Experiments performed changing both the geometrical value of the inductance and the critical current of the junctions show a good agreement with numerical simulations. We argue that the observed magnetic field patterns are due to a peculiar mapping between the isotropic Josephson ladder and the 1D parallel array with the self-inductance parameter $\beta_{L}^{\/eff}=\beta_{L}+2$.
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[
{
"name": "critic08.tex",
"string": "% LATEX version\n% THIS IS THE FILE critcur.tex, version of 1/2/2000\n% Editing history:\n% First draft from GF, 11/12/1998\n% AU revised version 17/9/1999\n% After new changings from Misha\n% GF revised version 22/12/1999\n% AU revised version 26/12/1999\n\n\\documentstyle[prb,aps,epsfig]{revtex}\n\n\\begin{document}\n\\draft\n\n\\title{Experimental Critical Current Patterns in Josephson Junction\nLadders}\n\n\n\\author{P. Binder, P. Caputo, M. V. Fistul, and A. V. Ustinov}\n\\address{Physikalisches Institut III, Universit\\\"at Erlangen,\n E.-Rommel-Stra\\ss e 1, D-91058 Erlangen, Germany}\n\n\\author{G. Filatrella}\n\\address{INFM Unit Salerno and Science Faculty, University of Sannio,\n Via Port'Arsa 11, I-82100 Benevento, Italy }\n\n\\date{\\today}\n\n\\wideabs{ %REVTeX 3.1 feature\n\n\\maketitle\n\n\\begin{abstract}\n We present an experimental and theoretical study of the magnetic\n field dependence of the critical current of Josephson junction\n ladders. At variance with the well-known case of a one-dimensional\n (1D) parallel array of Josephson junctions the magnetic field\n patterns display a single minimum even for very low values of the\n self-inductance parameter $\\beta_{\\rm L}$. Experiments performed\n changing both the geometrical value of the inductance and the\n critical current of the junctions show a good agreement with\n numerical simulations. We argue that the observed magnetic field\n patterns are due to a peculiar mapping between the isotropic\n Josephson ladder and the 1D parallel array with the self-inductance\n parameter $\\beta_{\\rm L}^{\\rm \\/eff}=\\beta_{\\rm L}+2$.\n\\end{abstract}\n\n\\pacs{74.50.+r, 85.25.Dq}\n\n}%end of WideAbs\n\n\\section{Introduction}\nA Josephson junction ladder is an array of coupled superconducting\nloops containing small Josephson junctions as shown schematically in\nFig.~\\ref{scheme}(a). In the past years, ladders of Josephson junctions\nhave attracted considerable interest for a number of reasons. On one\nhand, more complex systems such as two-dimensional arrays of Josephson\njunctions can be viewed as elementary ladders coupled to each other\n\\cite{filatrella95,basler}. On the other hand, the ladders are an\nideal ground for experimental and theoretical investigations of\ndiscrete nonlinear\nentities~\\cite{bonilla91,ustinov93,watanabe95,caputo98}, such as\nbreathers~\\cite{sfcrw98,fmmfa96,sfms99,binder00&trias00} and vortex\npropagation \\cite{barahona98}. In spite various groups have\nnumerically and theoretically studied the static properties of\nJosephson ladders~\\cite{barahona98,grimaldi96}, so far no systematic\ncomparison with experimental data has been carried out. Grimaldi et\nal.\\cite{grimaldi96} have performed numerical simulations on these\nsystems. Their findings are that the behavior of ladders is quite\ndifferent from that of one-dimensional (1D) parallel arrays. In\ncontrast to a ladder, a 1D parallel array contains only Josephson\njunctions in the direction of the bias current $I_{\\rm B}$ but no one\ntransverse to it. A 1D parallel array of Josephson junctions placed\nin magnetic field shows a pattern of critical current with as many\nminima as the number of loops (for relatively low $\\beta_{\\rm L}$, as\ndefined below) \\cite{miller91,experiments}. In Josephson ladders with\njunctions on the horizontal branches not only the number of minima in\nthe pattern does not correspond to the number of loops even for\nextremely low $\\beta_{\\rm L}$, but also the pattern dependence on the\nparameter $\\beta_{\\rm L}$ is different: the critical current $I_{\\rm\n C}$ never reduces to zero for fully frustrated arrays (i.e.\\ when\nthere is half flux quantum in each cell). Baharona et\nal.~\\cite{barahona98} have shown that one can analytically estimate\nthe depinning current of fluxon trapped into the ladder in the limit\nof zero inductance. They have also computed the onset of instability\nin the case of no fluxons, thus retrieving analytically the numerical\nresult of Ref.~\\onlinecite{grimaldi96} for very low inductance.\nMoreover, the authors of Ref.~\\onlinecite{barahona98} have estimated\nthat the critical current of a ladder with a fluxon trapped in\neach second cell is higher than the depinning from the empty ground\nstate for a moderately high magnetic field.\n\nThe aim of this paper is to present experiments performed on isotropic\nJosephson ladders with various values of the self-inductance parameter\n$\\beta_{\\rm L}$. We call as isotropic a ladder consisting of identical\njunctions. We also make an analysis of the model to explain the\nobserved dependence of the pattern upon $\\beta_{\\rm L}$. The work is\norganized as follows. In Section II we describe a model for the\nJosephson ladders, in Section III we show the experimental findings\nand make the comparison with the numerical predictions. Finally,\nSection IV contains a discussion of our results and Section V the\nconclusion.\n\n\\begin{figure}\n\\vspace{5pt}\n\\centerline{\\epsfig{file=fig1.eps,width=3.4in}}\n\\noindent(a)\\vspace{1.5mm}\n\\centerline{\\epsfig{file=ladder15_small-hell-grey.eps,width=3.4in}}\n\\noindent(b)\n\\vspace{5pt}\n\\caption{(a) The electrical scheme of a Josephson junction ladder;\n crosses ($\\times$) indicate Josephson junctions. (b) Optical image of\n one of studied samples.}\n\\label{scheme}\n\\end{figure}\n\n\n\\section{The model}\n\nTo derive the equations for the ladder, we start from the fluxoid\nquantization over a cell:\n\\begin{equation}\n\\sum_{cell} \\varphi = \\frac{2 \\pi}{\\Phi_0} \\left[ \\Phi^{\\rm ext} +\n\\Phi^{\\rm ind} \\right],\n\\end{equation}\n%\n\\noindent where $\\varphi$ are the phases of all the junctions in the cell,\n$\\Phi_0$ is the flux quantum, and $\\Phi^{\\rm ext}$ and $\\Phi^{\\rm\n ind}$ are the applied and induced flux, respectively. To evaluate\n$\\Phi^{\\rm ind}$ we retain only self-inductance terms, i.e.\\ we assume\nthat $\\Phi^{\\rm ind} = L I^s$, $I^s$ being the screening current\ncirculating in the elementary cell and $L$ is the self inductance of\nthe cell. For the junctions we assume the $RCSJ$ model and suppose\nthat all junctions are identical (isotropic ladder). With these\ningredients it is possible to derive the following set of equations\nfor the gauge invariant phase difference across the vertical\n($\\phi_i$) and horizontal ($\\psi_i$) junctions \\cite{caputo98}:\n\n\\begin{eqnarray}\n\\label{Genequation}\n&\\ddot{\\phi_i} + \\alpha \\dot{\\phi_i} + \\sin{\\phi_i}=&\\nonumber\\\\\n\\lefteqn{=\\frac{1}{\\beta_{\\rm L}} \\left[\\phi_{i-1} - 2\\phi_i + \\phi_{i+1}\n+2 (\\psi_i - \\psi_{i-1}) \\right] + \\gamma, \\,\\,\\,\\,\\, i=2,\\dots,N }\n~~~~~~~~~~&&\\nonumber\\\\\n&\\ddot{\\psi_i} + \\alpha \\dot{\\psi_i} + \\sin{\\psi_i}=&\\nonumber\\\\\n&=\\displaystyle\\frac{1}{\\eta\\beta_{\\rm L}} \\left[\\phi_{i} - \\phi_{i+1}\n-2 \\psi_i +2\\pi f \\right], \\,\\,\\,\\,\\, i=1,\\dots,N.&\\nonumber\\\\\n\\end{eqnarray}\nThe boundary conditions are:\n\\begin{eqnarray}\n\\label{Genequation2}\n&\\ddot{\\phi_1} + \\alpha \\dot{\\phi_1} + \\sin{\\phi_1} =\n\\displaystyle\\frac{1}{\\beta_{\\rm L}} \\left[\\phi_{2} - \\phi_1 +2 \\psi_1\n + 2\\pi f \\right] + \\gamma ,&\\nonumber \\\\\n&\\ddot{\\phi}_{N+1} + \\alpha \\dot{\\phi}_{N+1} + \\sin{\\phi_{N+1}} =&\\nonumber\\\\\n&=\\displaystyle\\frac{1}{\\beta_{\\rm L}} \\left[\\phi_{N} - \\phi_{N+1} \n+2 \\psi_{N} - 2 \\pi f \\right] + \\gamma.&\n\\end{eqnarray}\nHere, $\\gamma = I_{\\rm B}/i_{\\rm c}^{\\rm \\/ver}$ is the bias current\nnormalized to the single vertical junction critical current,\n$\\beta_{\\rm L} = 2\\pi L i_{\\rm c}^{\\rm \\/ver}/\\Phi_0$ is the\nself-inductance parameter (the self-inductance $L$ of a square cell\nwith the side $a$ can be estimated \\cite{jaycox81} as $L=1.25 \\mu_0\na$, where $\\mu_0$ is the magnetic permeability), the ratio\n$\\eta=i_{\\rm c}^{\\rm \\/hor}/i_{\\rm c}^{\\rm \\/ver}$ between the\nhorizontal and the vertical junction critical currents is the\nanisotropy parameter, $\\alpha$ is the normalized dissipation\nparameter, $f=\\Phi^{\\rm ext}/\\Phi_0$ is the normalized external flux\noften noted as frustration, and $N$ is the number of loops. For the\nstatic case the parallel arrays considered in\nRef.~\\onlinecite{ustinov93,watanabe95,miller91} correspond to the\nlimit $\\eta\\rightarrow\\infty$. In deriving\nEqs.~(\\ref{Genequation})--(\\ref{Genequation2}) we take advantage of\nthe fact that, due to the symmetry of the system, the current flowing\nin the top and bottom horizontal junctions of the same cell (see\nFig.~\\ref{scheme}(a)) differs only in direction but not in amplitude,\nand therefore we can write the equation for just one of them\n\\cite{filatrella95}. Finally, we want to stress that in this work we\nare interested only in the transition point from the static to the\ndynamic solutions (the critical current), therefore the dynamics is\nnothing but a computational mean to find the current point at which\nthe static solution becomes unstable. The value of the dissipation\nused in the simulations is fictitious, and it has been chosen equal to\n$1$ for computational convenience. The experimentally studied arrays\nare actually underdamped and therefore have a much smaller damping\ncoefficient, $\\alpha \\simeq 0.005$ -- $0.08$, that can be controlled\nby temperature.\n\n\n\\section{Experimental $I_{\\rm C}$ vs $\\lowercase{f}$ patterns}\n\nWe present an experimental study of 10-cell Josephson junction ladders.\nEach elementary cell of the ladders contains 4 identical small\nNb/Al-AlO$_x$/Nb Josephson tunnel junctions\\cite{Hypres}, which have\nan area of $3\\times 3 \\, \\mu \\rm m^2$. To get different values of\n$\\beta_{\\rm L}$, we used samples with different critical current\ndensity $j_{\\rm c}$ ($100 \\, {\\rm A/cm}^2$ or $1000 \\, {\\rm A/cm}^2$)\nand also varied the loop size $a$ ($2.8\\,\\mu \\rm m$ or $9.9\\,\\mu \\rm\nm$). A SEM image of a typical ladder is shown in Fig.~\\ref{scheme}(b).\n\nWe have measured the ladder critical current, $I_{\\rm C}$, versus\nfrustration $f$ for $4$ isotropic ladders ($\\eta=1$) with different\n$\\beta_{\\rm L}$. The values selected are $\\beta_{\\rm L} = 3$, $0.88$,\n$0.25$ and $0.088$, a range where the peculiar behavior of the ladders\nshould be clearly visible. The measurements were performed in a\ncryoperm shield. The magnetic field $H$ applied perpendicular to the\nsubstrate was provided by a coil placed inside the shield. The uniform\nbias current $I_{\\rm B}$ was injected at every node via on-chip\nresistors. The voltage across the first vertical junction was\nmeasured to define the depinning current of the ladder. Finally, the\n$I_{\\rm C}$ vs $f$ dependencies were measured using GoldExi\nsoftware\\cite{goldexi}.\n\n\\begin{figure}\n\\vspace{5pt}\n\\centerline{\\epsfig{file=fig2v2.eps,width=3.4in}}\n\\vspace{5pt}\n\\caption{Experimental (symbols) and numerical (solid lines) $\\gamma$ vs $f$\n patterns of ladders with different cell sizes. Parameters are: $N =\n 11$; $\\beta_{\\rm L} =3.0$ (squares), $0.88$ (circles), $0.25$ (up\n triangles), $0.088$ (down triangles).}\n\\label{pattern}\n\\end{figure}\n\nIn Fig.~\\ref{pattern} we present two measured features of the ladders.\nIn contrast to the case of a 1D parallel array, there are no\nadditional lobes between $f=0$ and $f=1$. Also, the critical current\n$I_{\\rm C}$ remains relatively large, despite a $\\beta_{\\rm L}$ as low\nas $0.088$. Similarly to Ref.~\\onlinecite{grimaldi96} we numerically\nsolved the Eqs.~(\\ref{Genequation})--(\\ref{Genequation2}) by using the\nsame parameters as in the experiment (except for $\\alpha$, see above).\nIn Fig.~\\ref{pattern}, we compare the numerical simulations (solid\nlines) with the experimental data (symbols), which show good\nagreement. The $\\beta_{\\rm L}$ used in the simulations was calculated\nfrom the critical current of a single junction measured in the\nexperiment. The calculations show some flattening at $f=0.5$ for low\n$\\beta_{\\rm L}$, which is also present in the experimental data. We\nfound in simulations that in this region the ladder gets first filled\nwith flux and only subsequently undergoes the depinning. We have\nobserved this behavior only for low inductance, in good agreement with\nthe analytical prediction of Ref.~\\onlinecite{barahona98} (that\nneglects inductance terms). In experiments with $\\beta_{\\rm L} = 3$\nand $\\beta_{\\rm L} = 0.88$ we note the simultaneous presence of two\ndifferent states at the same frustration value. We suppose that this\nis due to distinctly different initial conditions that can be realized\nin the ladder, while sweeping the bias current $I_{\\rm B}$ through\nzero. This contradicts the prediction of Ref.~\\onlinecite{barahona98}\nwhere it is stated that the depinning of the whole ladder from a state\ndifferent from the empty ground state occurs {\\em only\\/} around\n$f=0.5$. The reason of this disagreement might be the neglect of the\ninductance in their calculations.\n\n\\begin{figure}\n\\vspace{5pt}\n\\centerline{\\epsfig{file=fig3.eps,width=3.4in}}\n\\vspace{5pt}\n\\caption{Experimental (symbols) and numerical (solid lines) $I_{\\rm\n C}$ vs $f$ patterns of one of the ladders measured at different\n temperatures, in order to vary the critical current. The\n temperature has been derived from the gap voltage. Parameters are:\n $N = 11$; $\\beta_{\\rm L} =3.0$ (squares), $2.6$ (circles), $2.1$ (up\n triangles), $1.8$ (down triangles).}\n\\label{temperature}\n\\end{figure}\n\nIn the ladder with the largest $\\beta_{\\rm L}$ parameter ($\\beta_{\\rm\n L} = 3.0$), we have measured the $I_{\\rm C}$ vs $f$ dependence also\nas a function of the temperature $T$. At higher temperatures the\ndecreased critical current causes a decrease of $\\beta_{\\rm L}$. The\nresults are shown in Fig.~\\ref{temperature}, in physical units to\nunderline the actual change of the critical current. Also in this case\nthe agreement between the model and the experiments is rather good.\n\n\n\\section{Discussion}\n\nWe have characterized the static properties of Josephson ladders for\ndifferent values of the self-inductance parameter $\\beta_{\\rm L}$. The\nexperimentally observed dependencies of the critical current on\nfrustration are in good agreement with the numerical simulations and\nshow that the behavior of the ladder is clearly different from that of\nthe 1D parallel array (see Fig.~\\ref{pattern}). As it is well-known,\nin 1D parallel arrays the critical (depinning) current is determined\nby the parameter $\\beta_{\\rm L}$, and in the limit of small\n$\\beta_{\\rm L}$ the minimum frustration-dependent critical current is\nvery small. Instead, the ladder critical current, even in the case of\nsmall $\\beta_{\\rm L}$, never goes to zero. As it was already pointed\nout in Ref.\\onlinecite{grimaldi96}, with respect to 1D parallel\narrays, the presence of the horizontal junctions in the ladder leads\nto an \"effective\" increased $\\beta_{\\rm L}^{\\rm \\/eff}$, which for\nsmall discreteness can be by up to two orders of magnitude larger than\nthe {\\em natural\\/} $\\beta_{\\rm L}$ of the system, calculated (similar\nto 1D arrays) from the junction critical current and the cell\ninductance.\n\nIn order to show the particular mapping between Josephson ladders and\n1D parallel arrays, we carry out a simple quantitative analysis of the\nEqs.~(\\ref{Genequation}). Let us consider the static case, when all\nJosephson junction phases are independent of time and satisfy the\nsystem of nonlinear equations:\n\\begin{equation}\n\\label{PhaseConn}\n\\eta(\\sin \\psi_{i-1}-\\sin \\psi_{i})~=~\\sin \\phi_i-\\gamma,~~~~~~ i=2, N.\n\\end{equation}\nBy making use of the particular assumption that the horizontal phases\n$\\psi_i$ are small, we can eliminate the phases $\\psi_i$ from all\nequations and write the system of equations for phases $\\phi_i$ in the\nform:\n%\n\\begin{equation}\n\\label{VertPhase}\n\\sin \\phi_i ~=~ \\frac{\\eta}{\\eta\\beta_{\\rm L}+2}\n\\left[\\phi_{i-1} - 2\\phi_i + \\phi_{i+1}\n \\right] + \\gamma\n\\end{equation}\n%\nThis system of equations coincides with the one describing the static\nproperties of 1D parallel array (with horizontal\njunctions replaced by superconducting electrodes). The difference\nbetween a ladder and 1D parallel array is that for the\nladder we have now to use an effective parameter\n\\begin{equation}\n\\label{Mapp}\n\\beta_{\\rm L}^{\\rm \\/eff}=\\beta_{\\rm L}+2/\\eta.\n\\end{equation}\nThe deviation of $\\beta_{\\rm L}^{\\rm \\/eff}$ from $\\beta_{\\rm L}$\noriginates from an additional shielding (and vortex pinning) due to\nthe presence of horizontal junctions, i.e.\\ the horizontal junctions\ncan accommodate part of the phase change. Thus, we expect that\nthis deviation disappears in anisotropic ladders when the critical\ncurrent of horizontal junctions $i_{\\rm c}^{\\rm \\/hor}$ is much larger\nthan the critical current of vertical junction $i_{\\rm c}^{\\rm \\/ver}$\n($\\eta\\gg1$).\n\n\\begin{figure}\n\\vspace{5pt}\n\\centerline{\\epsfig{file=fig4v2.eps,width=3.4in}}\n\\vspace{5pt}\n\\caption{$I_{\\rm C}$ vs $f$ dependencies of the 1D parallel \n array with $\\beta_{\\rm L} \\,=\\,2.7$ (solid circles) and the ladder\n with $\\beta_{\\rm L} =0.88$ (open circles).}\n\\label{ladder-array}\n\\end{figure}\n\nTo verify the mapping given by Eq.(\\ref{Mapp}), in\nFig.~\\ref{ladder-array} we compare the patterns of a 1D parallel array\nwith $\\beta_{\\rm L}^{\\rm \\/1D} = 2.7$ and an isotropic ladder with\n$\\beta_{\\rm L} = 0.88$. For the ladder we expect $\\beta_{\\rm L}^{\\rm\n \\/eff}\\approx\\beta_{\\rm L}^{\\rm \\/1D}$. The agreement is\nparticularly good at low frustration, but not in the vicinity of\n$f\\,=\\,0.5$, where the most critical assumption of our theory, i.e.\\ \nis small values of the horizontal junction phases, breaks down.\n\nWe would like to note here that a similar analysis for anisotropic\nladders in the limit of small $\\beta_{\\rm L}$ has been carried out in\nRef.\\onlinecite{Stroud}. Moreover, for the case of ladder with three\njunctions per cell the mapping takes the form: $\\beta_{\\rm L}^{\\rm\n \\/eff}=\\beta_{\\rm L}+1/\\eta$. This mapping is in good accord with\npreviously published data on the $I_{\\rm C}(f)$ dependence for ladder\nwith three junctions per cell \\cite{CaputoApplSuper}. We want to\nstress out here that this mapping is supposed to work only for the\nstatic case. In the dynamic state, when the Josephson vortices\npropagate in the ladder, the phases of the horizontal junctions start\nto oscillate and Eqs.~(\\ref{PhaseConn}) are not valid anymore,\nespecially in the regime of large vortex velocities. Theoretical and\nexperimental investigation of vortex propagation in Josephson ladders\nwill be reported elsewhere \\cite{schuster00}.\n\n\n\\section{Conclusion}\nWe have reported measurements of the critical current in ladders of\nJosephson junctions. The $\\beta_{\\rm L}$ parameter has been varied by\nchanging both the geometrical inductance and the critical current of\nthe junctions. The results are in good agreement with numerical\nsimulations, and show a behavior clearly distinct from the case of the\n1D parallel Josephson junction array without junctions in the\nhorizontal branches. Using a simple quantitative analysis, we have\nshown that the static properties of 1D parallel arrays and ladders can\nbe mapped by properly scaling the self-inductance parameter\n$\\beta_{\\rm L}$. This analysis well agrees with experimental data.\n\n\n\\section*{Acknowledgments}\n\nFinancial support from the DAAD Vigoni cooperation program is\nacknowledged. P.C.\\ and M.V.F.\\ thank, respectively, the European\nOffice of Scientific Research (EOARD) and the Alexander von Humboldt\nStiftung for supporting this work. The samples were fabricated at\nHypres, Elmsford, New York.\n\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{filatrella95} G. Filatrella and K. Wiesenfeld, J. Appl. Phys.\n {\\bf 78}, 1878 (1995).\n\n\\bibitem{basler} M. Basler, W. Krech, and K. Yu. Platov, Phys. Rev. {\\bf\n B58}, 3409 (1998).\n\n\\bibitem{bonilla91}L.L. Bonilla and B.A. Malomed, Phys. Rev . {\\bf\n B43}, 11539 (1991).\n\n\\bibitem{ustinov93} A.V. Ustinov, M. Cirillo, and B.A. Malomed, Phys.\n Rev. {\\bf B47}, 8357 (1993).\n\n\\bibitem{watanabe95}S. Watanabe, S.H. Strogatz, H.S.J. van der Zant,\n and T.P. Orlando, Phys. Rev. Lett. {\\bf 74}, 379 (1995).\n\n\\bibitem{caputo98}P. Caputo, M.V. Fistul, A.V. Ustinov, B.A. Malomed, and\n S. Flach, Phys. Rev. {\\bf B59}, 14050 (1999). \n\n\\bibitem{sfcrw98}\n S.~Flach and C.~R. Willis,\n \\newblock {\\em Phys. Rep.} {\\bf 295}, 181 (1998).\n\n\\bibitem{fmmfa96}\n L.~M. Floria, J.~L. Marin, P.~J. Martinez, F.~Falo, and S.~Aubry,\n \\newblock {\\em Europhys. Lett.} {\\bf 36}, 539 (1996).\n\n\\bibitem{sfms99}\n S.~Flach and M.~Spicci,\n \\newblock {\\em J. Phys.: Condens. Matter} {\\bf 11}, 321 (1999).\n \n\\bibitem{binder00&trias00} E.~Tr\\'\\i as, J.~J.~Mazo, and T.~P.~Orlando,\n Phys.\\ Rev.\\ Lett.\\ {\\bf 84}, 741 (2000).\\\\\n P.~Binder, D.~Abraimov, A.~V.~Ustinov, S.~ Flach, and Y.~Zolotaryuk,\n Phys.\\ Rev.\\ Lett.\\ {\\bf 84}, 745 (2000).\n\n\\bibitem{barahona98} M. Barahona, S.H. Strogatz, and T.P. Orlando,\n Phys. Rev. {\\bf B57}, 1181 (1998).\n\n\\bibitem{grimaldi96} G. Grimaldi, G. Filatrella, S. Pace, and U.\n Gambardella, Phys. Lett. A {\\bf 223}, 463 (1996).\n\n\\bibitem{miller91}J.H. Miller, H.H. Guratne, J. Huang, and T.D.\n Golding, Appl. Phys. Lett. {\\bf 59}, 3330 (1991).\n\n\\bibitem{experiments} U. Gambardella, P. Caputo, V. Boffa,\n G. Costabile, and S. Pace, J. Appl. Phys. {\\bf 79}, 322 (1996).\n \n\\bibitem{jaycox81} J.M. Jaycox and M.B. Ketchen, IEEE Trans. Magn.\n {\\bf 17}, 400 (1981).\n \n\\bibitem{Hypres} HYPRES Inc., Elmsford, NY 10523.\n\n\\bibitem{goldexi}\n \"http://www.geocities.com/SiliconValley/Hights/7318\".\n\n\\bibitem{Stroud} S. Ryu, W. Yu, and D. Stroud, Phys. Rev. E {\\bf 53},\n 2190 (1996).\n\n\\bibitem{CaputoApplSuper} P. Caputo, A. E. Duwel, T. P. Orlando,\n A. V. Ustinov, N. C. H. Lin, S. P. Yukon, Proc. of ISEC'97, Berlin (1997).\n\n\\bibitem{schuster00} M. Schuster, P. Binder, M. Fistul, and\n A. V. Ustinov, unpublished.\n\n\n\\end{thebibliography}\n\n\n\n% \\section*{Figure Captions}\n\n% \\begin{enumerate}\n% \\item\n% \\label{scheme}\n% (a) The electrical scheme of a Josephson junction ladder; crosses\n% ($\\times$) indicate Josephson junctions. (b) SEM image of one of\n% studied samples.\n% \\item\n% \\label{pattern}\n% Experimental (symbols) and numerical (solid lines) $\\gamma$ vs $f$\n% patterns of ladders with different cell sizes. Parameters are: $N =\n% 11$; $\\beta_{\\rm L} =3.0$ (squares), $0.88$ (circles), $0.25$ (up\n% triangles), $0.088$ (down triangles).\n% \\item\n% \\label{temperature}\n% Experimental (symbols) and numerical (solid lines) $I_{\\rm C}$ vs $f$\n% patterns of one of the ladders measured at different temperatures, in\n% order to vary the critical current. The temperature has been derived\n% from the gap voltage. Parameters are: $N = 11$; $\\beta_{\\rm L} =3.0$\n% (squares), $2.6$ (circles), $2.1$ (up triangles), $1.8$ (down\n% triangles).\n% \\item\n% \\label{ladder-array}\n% $I_{\\rm C}$ vs $f$ dependencies of the 1D parallel array with\n% $\\beta_{\\rm L} \\,=\\,2.7$ (solid circles) and the ladder with\n% $\\beta_{\\rm L} =0.88$ (open circles).\n% \\end{enumerate}\n%(squares), (triangles), (circles), (crosses), (diamonds)\n\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002012.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{filatrella95} G. Filatrella and K. Wiesenfeld, J. Appl. Phys.\n {\\bf 78}, 1878 (1995).\n\n\\bibitem{basler} M. Basler, W. Krech, and K. Yu. Platov, Phys. Rev. {\\bf\n B58}, 3409 (1998).\n\n\\bibitem{bonilla91}L.L. Bonilla and B.A. Malomed, Phys. Rev . {\\bf\n B43}, 11539 (1991).\n\n\\bibitem{ustinov93} A.V. Ustinov, M. Cirillo, and B.A. Malomed, Phys.\n Rev. {\\bf B47}, 8357 (1993).\n\n\\bibitem{watanabe95}S. Watanabe, S.H. Strogatz, H.S.J. van der Zant,\n and T.P. Orlando, Phys. Rev. Lett. {\\bf 74}, 379 (1995).\n\n\\bibitem{caputo98}P. Caputo, M.V. Fistul, A.V. Ustinov, B.A. Malomed, and\n S. Flach, Phys. Rev. {\\bf B59}, 14050 (1999). \n\n\\bibitem{sfcrw98}\n S.~Flach and C.~R. Willis,\n \\newblock {\\em Phys. Rep.} {\\bf 295}, 181 (1998).\n\n\\bibitem{fmmfa96}\n L.~M. Floria, J.~L. Marin, P.~J. Martinez, F.~Falo, and S.~Aubry,\n \\newblock {\\em Europhys. Lett.} {\\bf 36}, 539 (1996).\n\n\\bibitem{sfms99}\n S.~Flach and M.~Spicci,\n \\newblock {\\em J. Phys.: Condens. Matter} {\\bf 11}, 321 (1999).\n \n\\bibitem{binder00&trias00} E.~Tr\\'\\i as, J.~J.~Mazo, and T.~P.~Orlando,\n Phys.\\ Rev.\\ Lett.\\ {\\bf 84}, 741 (2000).\\\\\n P.~Binder, D.~Abraimov, A.~V.~Ustinov, S.~ Flach, and Y.~Zolotaryuk,\n Phys.\\ Rev.\\ Lett.\\ {\\bf 84}, 745 (2000).\n\n\\bibitem{barahona98} M. Barahona, S.H. Strogatz, and T.P. Orlando,\n Phys. Rev. {\\bf B57}, 1181 (1998).\n\n\\bibitem{grimaldi96} G. Grimaldi, G. Filatrella, S. Pace, and U.\n Gambardella, Phys. Lett. A {\\bf 223}, 463 (1996).\n\n\\bibitem{miller91}J.H. Miller, H.H. Guratne, J. Huang, and T.D.\n Golding, Appl. Phys. Lett. {\\bf 59}, 3330 (1991).\n\n\\bibitem{experiments} U. Gambardella, P. Caputo, V. Boffa,\n G. Costabile, and S. Pace, J. Appl. Phys. {\\bf 79}, 322 (1996).\n \n\\bibitem{jaycox81} J.M. Jaycox and M.B. Ketchen, IEEE Trans. Magn.\n {\\bf 17}, 400 (1981).\n \n\\bibitem{Hypres} HYPRES Inc., Elmsford, NY 10523.\n\n\\bibitem{goldexi}\n \"http://www.geocities.com/SiliconValley/Hights/7318\".\n\n\\bibitem{Stroud} S. Ryu, W. Yu, and D. Stroud, Phys. Rev. E {\\bf 53},\n 2190 (1996).\n\n\\bibitem{CaputoApplSuper} P. Caputo, A. E. Duwel, T. P. Orlando,\n A. V. Ustinov, N. C. H. Lin, S. P. Yukon, Proc. of ISEC'97, Berlin (1997).\n\n\\bibitem{schuster00} M. Schuster, P. Binder, M. Fistul, and\n A. V. Ustinov, unpublished.\n\n\n\\end{thebibliography}"
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cond-mat0002013
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Resonant Spin Excitation in an Overdoped High Temperature Superconductor
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[
{
"author": "H. He$^{1}$"
},
{
"author": "Y. ~Sidis$^{2}$"
},
{
"author": "P.~Bourges$^{2}$"
},
{
"author": "G.D. Gu$^{3}$"
},
{
"author": "A.Ivanov$^{4}$%"
},
{
"author": "N. Koshizuka$^{5}$"
},
{
"author": "B. Liang$^{1}$"
},
{
"author": "C.T. Lin$^{1}$"
},
{
"author": "L. P. Regnault$^{6}$"
},
{
"author": "E. Schoenherr$^{1}$"
},
{
"author": "and B. Keimer$% ^{1,7}$"
}
] |
An inelastic neutron scattering study of overdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta} $ (T$_{{c}}=83$ K) has revealed a resonant spin excitation in the superconducting state. The mode energy is E$_{{res}}=38.0$ meV, significantly lower than in optimally doped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$ (T$_{{c}}=91$ K, E$_{{res}}=42.4$ meV). This observation, which indicates a constant ratio E$_{res}$/$k_B {T}_{{c}}$ $\sim$ 5.4, helps resolve a long-standing controversy about the origin of the resonant spin excitation in high-temperature superconductors.
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[
{
"name": "paper.tex",
"string": "%\\documentstyle[preprint,aps]{revtex}\n\n\n\\documentstyle[amssymb,prl,aps,epsfig]{revtex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%TCIDATA{OutputFilter=LATEX.DLL}\n%TCIDATA{LastRevised=Sun Nov 19 07:21:29 2000}\n%TCIDATA{<META NAME=\"GraphicsSave\" CONTENT=\"32\">}\n%TCIDATA{Language=American English}\n%TCIDATA{CSTFile=revtex.cst}\n\n\\begin{document}\n\\title{Resonant Spin Excitation in an Overdoped High Temperature Superconductor}\n\\author{H. He$^{1}$, Y. ~Sidis$^{2}$, P.~Bourges$^{2}$, G.D. Gu$^{3}$, A.Ivanov$^{4}$%\n, N. Koshizuka$^{5}$, B. Liang$^{1}$,\\\\\nC.T. Lin$^{1}$, L. P. Regnault$^{6}$, E. Schoenherr$^{1}$, and B. Keimer$%\n^{1,7}$}\n\\address{$^{1}$Max-Planck-Institut f\\\"{u}r Festk\\\"{o}rperforschung, 70569\nStuttgart,Germany.\\\\\n$^{2}$Laboratoire L\\'{e}on Brillouin, CEA-CNRS, CE-Saclay, 91191 Gif sur\nYvette, France.\\\\\n$^{3}$Department of Advanced Electronic Materials, School of Physics,\nUniversity of New South Wales, Sydney 2052, Australia.\\\\\n$^{4}$Institut Laue Langevin, 156X, 38042 Grenoble cedex 9, France.\\\\\n$^{5}$SRL/ISTEC, 10-13, Shinonome 1-chome, Koto-ku, Tokyo 135, Japan.\\\\\n$^{6}$CEA Grenoble, D\\'{e}partement de Recherche Fondamentale sur la\nMati\\`{e}re Conden\\'{e}e, 38054 Grenoble cedex 9, France.\\\\\n$^{7}$Department of Physics, Princeton University, Princeton, NJ 08544}\n\\date{\\today}\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname\n\n\\maketitle\n\n\\begin{abstract}\nAn inelastic neutron scattering study of overdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\delta} $ \n(T$_{{\\rm c}}=83$ K) has revealed a resonant spin excitation in the\nsuperconducting state. The mode energy is E$_{{\\rm res}}=38.0$ meV,\nsignificantly lower than in optimally doped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\delta }$ \n(T$_{{\\rm c}}=91$ K, E$_{{\\rm res}}=42.4$ meV). This observation, \nwhich indicates a constant ratio E$_{\\rm res}$/$k_B {\\rm T}_{{\\rm c}}$ $\\sim$ 5.4, \nhelps resolve a long-standing controversy about the origin of\nthe resonant spin excitation in high-temperature superconductors.\n\\end{abstract}\n\n\\pacs{PACS numbers: 74.25.Ha, 74.25.Hs, 75.40.Gb, 74.20.Mn} \n]\n\nA resonant spin excitation \\cite\n{Rossat91,Mook93,Fong95,Bourges96,revue-cargese,Fong99_1,dai99,Fong99_2}\nwith wave vector ($\\pi ,\\pi )$ has recently emerged as a key factor in the\nphenomenology of the copper oxide superconductors. In particular, prominent\nfeatures in angle-resolved photoemission \\cite{Shen97,Norman98,Chubukov99}\nand optical conductivity \\cite{Munzar99,Carbotte99} spectra have been\nattributed to interactions of this bosonic mode with fermionic\nquasiparticles. The implications of these observations for the mechanism of\nhigh temperature superconductivity are under intense scrutiny, especially\nfollowing suggestions that the spectral weight of the mode (which, at least in\noptimally doped YBa$_{2}$Cu$_{3}$O$_{6+x}$, is present only below the\nsuperconducting transition temperature, {\\rm T}$_{{\\rm c}}$ \\cite\n{Fong95,Bourges96}) provides a measure of the condensation energy \\cite\n{Scalapino98,Demler98} or condensate fraction \\cite{chakravarty99} of the\nsuperconducting state. Several fundamentally different microscopic\ndescriptions of the neutron data have been proposed. Some of these \\cite\n{Fong95,ph} attribute the resonance peak to the threshold of the\nparticle-hole (ph) spin-flip continuum at $\\lesssim 2\\Delta _{{\\rm SC}}$\nwhere $\\Delta _{{\\rm SC}}$ is the energy gap in the superconducting state,\nothers \\cite{Chubukov99,coll1,coll2} to a magnon-like collective mode whose\nenergy is bounded by the gap. Although the starting points of these\ncalculations are disparate (itinerant band electrons in \\cite\n{Chubukov99,ph,coll1}, localized electrons in \\cite{coll2}), the excitations\ncorresponding to the neutron peak are described by the same quantum numbers\n(spin 1 and charge 0). In a completely different approach \\cite{Demler95},\nthe neutron data are interpreted in terms of a collective mode in the\nparticle-particle (pp) channel whose quantum numbers are spin 1 and charge\n2. The pp continuum that provides the upper bound for the pp resonance in\nthe latter model is unaffected by superconductivity. Despite the central\nsignificance of this issue, there is still no ``smoking gun'' experiment\nselecting the correct theoretical approach.\n\nA careful measurement of the doping dependence of the mode energy, ${\\rm E}_{%\n{\\rm res}}$, can help resolve this issue. In Refs. \\cite\n{coll1,coll2,Demler95}, the mode is interpreted as a soft mode whose energy\nis expected to decrease as a magnetic instability is approached with\ndecreasing hole content. This is made explicit in an expression derived from\nthe pp model which predicts that ${\\rm E}_{{\\rm res}}$ is proportional to\nthe doping level \\cite{Demler95}. The behavior observed in underdoped YBa$%\n_{2}$Cu$_{3}$O$_{6+x}$ \\cite{revue-cargese,Fong99_1,dai99} is consistent\nwith that prediction. This alone, however, does not amount to a ``smoking\ngun'' because it can also be understood in the framework of the simple ph\npair production model where ${\\rm E}_{{\\rm res}}\\propto {\\rm T}_{{\\rm c}}$.\nIn underdoped samples, ${\\rm T}_{{\\rm c}}$ in turn is monotonically related\nto the hole content. Further difficulties derive from ambiguities associated\nwith the distinction between the normal-state ``pseudo-gap'' in the charge\nsector and the true superconducting gap in the underdoped state. These are\nmirrored in the spin sector by uncertainties regarding the relationship\nbetween a broad peak observed by neutron scattering in the normal state \\cite\n{revue-cargese,Fong99_1,dai99} and the sharp resonant peak in the\nsuperconducting state.\n\nThe neutron data on underdoped samples therefore do not discriminate clearly\nbetween the very different theories of the resonance peak. Here we report a\nneutron scattering study in the overdoped state of Bi$_{2}$Sr$_{2}$CaCu$_{2}$%\nO$_{8+\\delta }$ where none of these complicating factors are present. In\nparticular, T$_{{\\rm c}}$ is reduced while the hole content keeps\nincreasing, and the normal-state pseudogap disappears.\n\nTo this end, we used an array comprising eight small (individual volumes $%\n\\sim 0.03$ ${\\rm cm}^{3}$), high quality overdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$%\nO$_{8+\\delta }$ single crystals grown by the floating-zone method \\cite{Gu98}%\n. In their as-grown state, the crystals were optimally doped, with T$%\n_{c}\\sim 91$ K, as was the sample used for our previous neutron study \\cite\n{Fong99_2}. Using established procedures \\cite{Han95}, they were\nsubsequently annealed at 650$^{{\\rm o}}$C under oxygen flow for 200 hours.\n(The long annealing time was used as a precaution. Previous studies \\cite\n{Han95} have shown that 20 hours are sufficient to achieve a homogeneous\noxygen content for identically prepared samples.) Following this, the\nindividual samples exhibited sharp (width 5-7 K) superconducting transitions\nat 83 K, in excellent agreement with prior results \\cite{Han95}.\nRepresentative data measured by SQUID magnetometry are shown in the inset to\nFig. \\ref{fig4}. The crystals were co-aligned by x-ray Laue diffraction and\nmounted in an aluminum holder. The overall mosaicity of the array, $\\sim\n5^\\circ$, was comparable to the angular dependence of the magnetic signal of\nthe previous study \\cite{Fong99_2} and therefore of little consequence for\nthe signal intensity. However, compared to our previous experiment on a\nmonolithic, optimally doped single crystal, the imperfect alignment of the\ncrystal array would introduce additional uncertainties into an absolute\nintensity unit calibration which will therefore not be given here. In order\nto establish an optimal basis for a comparison of the results on optimally\ndoped and overdoped samples, the experimental setup precisely duplicated the\none used for the previous study \\cite{Fong99_2}. The experiments were\nconducted on the triple axis spectrometer IN8 (at the Institut Laue-Langevin\nin Grenoble, France) in a focusing configuration with Cu(111) monochromator,\npyrolytic graphite (002) analyzer, and 35 meV fixed final energy. The wave\nvector $Q=(H,K,L)$ is given in reciprocal lattice units (r.l.u.), that is,\nin units of the reciprocal lattice vectors $a^{\\ast }\\sim b^{\\ast }\\sim 1.64$\n\\AA $^{-1}$ and $c^{\\ast }\\sim 0.20$ \\AA $^{-1}$. In these units, the\nin-plane wave vector $(\\pi ,\\pi )$ is equivalent to ($\\frac{h}{2},\\frac{k}{2}\n$)\\ with $h,k$ odd integers. The data were taken with $L$ set close to the\nmaximum of the intensity modulation due to magnetic coupling of the bilayers\n($L=-13.2$ or $L=-14$ for Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\delta }$ \\cite\n{Fong99_2}).\n\nBi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\delta }$ is a highly complex material with\na multitude of densely spaced phonon branches, not to mention the additional\nlattice dynamical complexity due to the incommensurate modulation of the\nBi-O layer. Raw neutron data therefore show a large, featureless background\npredominantly due to unresolved single-phonon events. An example is given in\nFig. \\ref{fig1}. Building on lessons drawn from work on YBa$_{2}$Cu$_{3}$O$%\n_{6+x}$, we have previously established \\cite{Fong99_2} how the magnetic\nsignal can be separated from this background by virtue of its characteristic\nenergy, momentum, and temperature dependences. Specifically, the magnetic\nresonance peak that is the primary focus of the present study gives rise to\na magnetic signal at wave vector $Q=(\\pi ,\\pi )$ that shows a sharp upturn\nbelow {\\rm T}$_{{\\rm c}}$ (Refs. \\cite\n{Fong95,Bourges96,revue-cargese,Fong99_1,Fong99_2}). The first step\ntherefore involves taking the difference between the measured spectra in the\nsuperconducting and normal states and studying the energy and wave vector\ndependence of the enhanced signal. Figs. \\ref{fig2} and \\ref{fig3} show that\nthis is confined to a narrow region in energy and wave vector centered at $%\nE=38$ meV and $Q=(\\pi ,\\pi )$, while the background away from this region is\nreduced upon cooling. (The temperature dependence of the background becomes\nmore pronounced at low energies, because the phonon scattering follows the\nBose population factor $(1-\\exp (-E/k_{B}T))^{-1}$.) This is precisely the\nsignature of the magnetic resonance peak observed in YBa$_{2}$Cu$_{3}$O$%\n_{6+x}$ and optimally doped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\delta }$.\n\nThe data on overdoped and optimally doped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$%\n_{8+\\delta }$ (also shown for comparison in Fig. \\ref{fig2}) were fitted to\na Gaussian magnetic resonant mode on top of a phonon background whose energy\ndependence is determined independently from scans at high temperatures and\nfrom constant-q scans away from $(\\pi,\\pi)$. The phonon background is\nmultiplied by the Bose population factor, and the difference between the\nBose factors at low and high temperatures gives rise to the negative signal\nat low energies in the difference plots. (The fact that the high temperature\nscan was taken at 100 K for the material with ${\\rm T_c =91}$ K, and at 90 K\nfor the one with ${\\rm T_c =83}$ K, was taken into account in this analysis\nand did not influence the result.) Apart from an overall scale factor, there\nare two free parameters in the fit: The intensity of the resonant mode with\nrespect to the phonon background, and its position. The results of these\nfits are shown in Fig. \\ref{fig2}. The resonance energies extracted in this\nmanner for optimally doped and overdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$%\n_{8+\\delta }$ are $42.4 \\pm 0.8$ meV and $38.0 \\pm 0.6$ meV, respectively\n(95\\% confidence limits). If the assumption of equal widths of the resonant\nmode in both samples is relaxed, the relative position extracted from the\nfits is hardly affected. Likewise, a Lorentzian profile for the resonant\nmode also gave the same result within the error. The null hypothesis (no\nenergy shift) can therefore be ruled out with a statistical confidence well\nexceeding 95\\%.\n\nBefore discussing the implications of these data, we proceed to the second\nstep in the identification of the resonance peak, namely, the determination\nof the onset temperature of the magnetic signal. The temperature dependence\nof the peak magnetic intensity, shown in Fig. \\ref{fig4}, indeed exhibits\nthe strong upturn around {\\rm T}$_{{\\rm c}}=83$ K that characterizes the\nresonance peak. Interestingly, this upturn is even sharper here than in the\noptimally doped sample. As in optimally doped YBa$_{2}$Cu$_{3}$O$_{7}$\n(Refs. \\cite{Fong95,Bourges96}) and Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\delta }$\n(Ref. \\cite{Fong99_2}), there is no evidence of magnetic scattering above \n{\\rm T}$_{{\\rm c}}$ although this determination is limited by the high\nnuclear background.\n\nThe data shown in Figs. \\ref{fig2}-\\ref{fig4} are an essential complement to\nan extensive data set on the resonance peak in underdoped YBa$_{2}$Cu$_{3}$O$%\n_{6+x}$. (Note that some \\cite{revue-cargese}, but not all \\cite{Fong95},\ndata on the resonant mode in slightly overdoped YBa$_{2}$Cu$_{3}$O$_{7}$\nalso exhibit a subtle trend towards lower energies. However, the data of\nRef. \\cite{revue-cargese} were taken under conditions different from those\non underdoped and optimally doped YBa$_{2}$Cu$_{3}$O$_{6+x}$, which prevents\nan accurate comparison of the resonance energies.) A representative subset \n\\cite{revue-cargese,Fong99_1,dai99} is shown in Fig. \\ref{fig5} along with\nthe presently available data on Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\delta }$.\nWhile we did not confirm the linear relationship between ${\\rm E}_{{\\rm res}}\n$ and the doping level predicted by the pp model of the resonance peak \\cite\n{Demler95}, Fig. \\ref{fig5} suggests that the parameter controlling ${\\rm E}%\n_{{\\rm res}}$ is actually the transition temperature {\\rm T}$_{{\\rm c}}$,\nwith E$_{{\\rm res}}$/$k_{B}{\\rm T}_{{\\rm c}}$ $\\sim $ 5.4. Since at least \nin the underdoped regime the superconducting gap does not\nscale with ${\\rm T}_{{\\rm c}}$, this observation is not naturally understood\nwithin the ph scenario either. Along with some aspects of the sharp \n``quasiparticle peak'' observed in photoemission data in the \nsuperconducting state \\cite{Feng00}, the neutron resonance thus appears\nto be one of very few spectral features of the superconducting cuprates \nthat scale with ${\\rm T}_{{\\rm c}}$. While this may indicate a smooth \ncrossover between a magnon-like collective mode below the ph continuum in \nthe underdoped regime and a simple ph pair production scenario in the \noverdoped regime, a quantitative theory of such a crossover has thus far \nnot been reported. \n\nOur result supports the conclusion of a neutron scattering study \\cite\n{ybco-ni} of 3\\%-Ni substituted YBa$_2$Cu$_3$O$_7$ (T$_{{\\rm c}}$=80 K). Ni\nsubstitution is known to reduce T$_{{\\rm c}}$ but does not affect the hole\ncontent. In YBa$_2$(Cu$_{0.97}$Ni$_{0.03}$)$_3$O$_7$, E$_{{\\rm res}}$ is\nshifted from 40 meV to $\\sim$ 35 meV so that the ratio E$_{{\\rm res}}$/T$_{%\n{\\rm c}}$ is preserved, as observed here for overdoped Bi$_{2}$Sr$_{2}$CaCu$%\n_{2}$O$_{8+\\delta }$.\n\nIn conclusion, we have shown that the energy of the magnetic resonant mode\nscales with the superconducting transition temperature in both the\nunderdoped and the overdoped regimes. This result is important for\ncurrent theoretical efforts \\cite{Shen97,Norman98,Chubukov99,Munzar99,Carbotte99} \nto develop a unified phenomenology of magnetic and charge spectroscopies of the cuprates.\n\nWe gratefully acknowledge discussions with P.W. Anderson, S. Chakravarty, A.\nChubukov, E. Demler, W. Hanke, D. Morr, F. Onufrieva, P. Pfeuty, D. Pines,\nS. Sachdev, and S.C. Zhang. The work at Princeton University was supported by the\nNational Science Foundation under DMR-9809483.\n\n\\begin{references}\n\\bibitem{Rossat91} J. Rossat-Mignod {\\it et al.}, Physica C {\\bf 185-189},\n86 (1991).\n\n\\bibitem{Mook93} H.A. Mook {\\it et al.}, Phys. Rev. Lett. {\\bf 70}, 3490\n(1993).\n\n\\bibitem{Fong95} H.F. Fong {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 316\n(1995); Phys. Rev. B {\\bf 54}, 6708 (1996).\n\n\\bibitem{Bourges96} P. Bourges {\\it et al.}, Phys. Rev. B {\\bf 53}, 876\n(1996).\n\n\\bibitem{revue-cargese} P. Bourges, in {\\it The Gap Symmetry and\nFluctuations in High Temperature Superconductors}, edited by J. Bok, G,\nDeutscher, D. Pavuna and S.A. Wolf. (Plenum Press, 1998) 349.\n\n\\bibitem{Fong99_1} H.F. Fong {\\it et al}., Phys. Rev. B {\\bf 61}, 14773\n(2000), and references therein.\n\n\\bibitem{dai99} P. Dai {\\it et al.}, Science {\\bf 284}, 1344 (1999).\n\n\\bibitem{Fong99_2} H.F. Fong {\\it et al.}, Nature {\\bf 398}, 588 (1999).\n\n\\bibitem{Shen97} Z.X. Shen and J.R. Schrieffer, Phys. Rev. Lett. {\\bf 78},\n1771 (1997).\n\n\\bibitem{Norman98} J.C. Campuzano {\\it et al}., Phys. Rev. Lett. {\\bf 83},\n3709 (1999); M.R. Norman and H. Ding, Phys. Rev. B {\\bf 57}, R11089 (1998).\n\n\\bibitem{Chubukov99} A. Abanov and A.V. Chubukov, Phys. Rev. Lett. {\\bf 83}%\n, 1652 (1999).\n\n\\bibitem{Munzar99} D. Munzar, C. Bernhard, and M. Cardona, Physica C {\\bf %\n318}, 547 (1999).\n\n\\bibitem{Carbotte99} J.P. Carbotte, E. Schachinger, and D.N. Basov, Nature \n{\\bf 401}, 354 (1999).\n\n\\bibitem{Scalapino98} D.J. Scalapino and S.R. White, Phys. Rev. B {\\bf 58},\n8222 (1998).\n\n\\bibitem{Demler98} E. Demler and S.C. Zhang, Nature {\\bf 396}, 733 (1998).\n\n\\bibitem{chakravarty99} S. Chakravarty and H.K. Kee, Phys. Rev. B {\\bf 61},\n14821 (2000).\n\n\\bibitem{ph} See, {\\it e.g.}, L. Yin, S. Chakravarty and P.W. Anderson,\nPhys. Rev. Lett. {\\bf 78}, 3559 (1997); A.A. Abrikosov, Phys. Rev. B {\\bf 57}%\n, 8656 (1998).\n\n\\bibitem{coll1} See, {\\it e.g.}, I.I. Mazin and V.M. Yakovenko, Phys. Rev.\nLett. {\\bf 75}, 4134 (1995); F. Onufrieva, Physica C {\\bf 251}, 348 (1995);\nD.L. Liu, Y. Zha and K. Levin, Phys. Rev. Lett. {\\bf 75}, 4130 (1995); N.\nBulut and D.J. Scalapino, Phys. Rev. B {\\bf 53}, 5149 (1996); A.J. Millis\nand H. Monien, Phys. Rev. B {\\bf 54}, 16172 (1996); J. Brinckmann and P.A.\nLee, Phys. Rev. Lett. {\\bf 82}, 2915 (1999); F. Onufrieva and P. Pfeuty,\ncond-mat/9903097.\n\n\\bibitem{coll2} A.V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B {\\bf 49}%\n, 11919 (1994); D.K. Morr and D. Pines, Phys. Rev. Lett. {\\bf 81}, 1086\n(1998); S. Sachdev, C. Buragohain, and M. Vojta, Science {\\bf 286}, 2479\n(1999).\n\n\\bibitem{Demler95} E. Demler and S.C. Zhang, Phys. Rev. Lett. {\\bf 75},\n4126 (1995); E. Demler, H. Kohno, and S.C. Zhang, Phys. Rev. B {\\bf 58},\n5719 (1998).\n\n\\bibitem{Gu98} G.D. Gu, K. Takamuku, N. Koshizuka, and S. Tanaka, J.\nCrystal Growth {\\bf 130}, 325 (1998).\n\n\\bibitem{Han95} S.H. Han {\\it et al}., Physica C {\\bf 246}, 22 (1995).\n\n\\bibitem{ybco-ni} Y. Sidis {\\it et al}., Phys. Rev. Lett. {\\bf 84}, 5900\n(2000).\n\n\\bibitem{Feng00} D.L. Feng {\\it et al.}, Science {\\bf 289}, 277 (2000).\n\\end{references}\n\n\\begin{figure}[tbp]\n\\caption{ Raw (unsubtracted) spectra at wave vector $Q=(0.5,0.5,-13.2)$ and\ntemperatures 10K and 100K. The large background is mostly due to a multitude\nof unresolved phonons (see text). The error bars are smaller than the symbol\nsize.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Closed symbols: Difference spectrum of the neutron intensities at T\n= 5 K (${\\rm < T_c}$) and T = 90 K (${\\rm > T_c}$), at wave vector ${\\bf Q}\n= (0.5,0.5,-13.2)$ for overdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\protect%\n\\delta}$. Open symbols: Data taken under identical conditions in optimally\ndoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\protect\\delta}$ (Ref. \\protect\\cite\n{Fong99_2}). The bar respresents the instrumental energy resolution, The\nsolid lines are the results of fits as described in the text. The dashed\nline is the resonance energy extracted from fitting the data on optimally\ndoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\protect\\delta}$.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{Difference spectrum of the neutron intensities at T = 5 K (${\\rm <\nT_c}$) and T = 90 K (${\\rm > T_c}$), at energy 38 meV. The bar respresents\nthe instrumental momentum resolution, the line is a guide-to-the-eye.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{ Temperature dependence of the neutron intensity at energy 38 meV\nand wave vector ${\\bf Q} = (0.5,0.5,-13.2)$. The intensity falls to\nbackground level above ${\\rm T_c = 83}$ K. The line is a guide-to-the-eye.\nThe insert shows a measurement of the uniform magnetization of a sample from\nthe same batch in an applied field of 10G.}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{ A synopsis of the resonance peak energy ${\\rm E_{res}}$ in\nunderdoped and optimally doped YBa$_{2}$Cu$_{3}$O$_{6+x}$ (open symbols,\nwith squares from Ref. \\protect\\cite{revue-cargese}, circles from Ref. \n\\protect\\cite{Fong99_1}, and diamonds from Ref. \\protect\\cite{dai99}) and\noptimally doped and overdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\\protect\\delta\n}$ (closed symbols, from Ref. \\protect\\cite{Fong99_2} and present work),\nplotted as a function of the superconducting transition temperature ${\\rm %\nT_{c}}$. The shaded areas indicate measures of (or upper bounds on)\nthe intrinsic width of the peak. The error bar on the peak\nposition is of the order of the symbol size. }\n\\label{fig5}\n\\end{figure}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002013.extracted_bib",
"string": "\\bibitem{Rossat91} J. Rossat-Mignod {\\it et al.}, Physica C {\\bf 185-189},\n86 (1991).\n\n\n\\bibitem{Mook93} H.A. Mook {\\it et al.}, Phys. Rev. Lett. {\\bf 70}, 3490\n(1993).\n\n\n\\bibitem{Fong95} H.F. Fong {\\it et al.}, Phys. Rev. Lett. {\\bf 75}, 316\n(1995); Phys. Rev. B {\\bf 54}, 6708 (1996).\n\n\n\\bibitem{Bourges96} P. Bourges {\\it et al.}, Phys. Rev. B {\\bf 53}, 876\n(1996).\n\n\n\\bibitem{revue-cargese} P. Bourges, in {\\it The Gap Symmetry and\nFluctuations in High Temperature Superconductors}, edited by J. Bok, G,\nDeutscher, D. Pavuna and S.A. Wolf. (Plenum Press, 1998) 349.\n\n\n\\bibitem{Fong99_1} H.F. Fong {\\it et al}., Phys. Rev. B {\\bf 61}, 14773\n(2000), and references therein.\n\n\n\\bibitem{dai99} P. Dai {\\it et al.}, Science {\\bf 284}, 1344 (1999).\n\n\n\\bibitem{Fong99_2} H.F. Fong {\\it et al.}, Nature {\\bf 398}, 588 (1999).\n\n\n\\bibitem{Shen97} Z.X. Shen and J.R. Schrieffer, Phys. Rev. Lett. {\\bf 78},\n1771 (1997).\n\n\n\\bibitem{Norman98} J.C. Campuzano {\\it et al}., Phys. Rev. Lett. {\\bf 83},\n3709 (1999); M.R. Norman and H. Ding, Phys. Rev. B {\\bf 57}, R11089 (1998).\n\n\n\\bibitem{Chubukov99} A. Abanov and A.V. Chubukov, Phys. Rev. Lett. {\\bf 83}%\n, 1652 (1999).\n\n\n\\bibitem{Munzar99} D. Munzar, C. Bernhard, and M. Cardona, Physica C {\\bf %\n318}, 547 (1999).\n\n\n\\bibitem{Carbotte99} J.P. Carbotte, E. Schachinger, and D.N. Basov, Nature \n{\\bf 401}, 354 (1999).\n\n\n\\bibitem{Scalapino98} D.J. Scalapino and S.R. White, Phys. Rev. B {\\bf 58},\n8222 (1998).\n\n\n\\bibitem{Demler98} E. Demler and S.C. Zhang, Nature {\\bf 396}, 733 (1998).\n\n\n\\bibitem{chakravarty99} S. Chakravarty and H.K. Kee, Phys. Rev. B {\\bf 61},\n14821 (2000).\n\n\n\\bibitem{ph} See, {\\it e.g.}, L. Yin, S. Chakravarty and P.W. Anderson,\nPhys. Rev. Lett. {\\bf 78}, 3559 (1997); A.A. Abrikosov, Phys. Rev. B {\\bf 57}%\n, 8656 (1998).\n\n\n\\bibitem{coll1} See, {\\it e.g.}, I.I. Mazin and V.M. Yakovenko, Phys. Rev.\nLett. {\\bf 75}, 4134 (1995); F. Onufrieva, Physica C {\\bf 251}, 348 (1995);\nD.L. Liu, Y. Zha and K. Levin, Phys. Rev. Lett. {\\bf 75}, 4130 (1995); N.\nBulut and D.J. Scalapino, Phys. Rev. B {\\bf 53}, 5149 (1996); A.J. Millis\nand H. Monien, Phys. Rev. B {\\bf 54}, 16172 (1996); J. Brinckmann and P.A.\nLee, Phys. Rev. Lett. {\\bf 82}, 2915 (1999); F. Onufrieva and P. Pfeuty,\ncond-mat/9903097.\n\n\n\\bibitem{coll2} A.V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B {\\bf 49}%\n, 11919 (1994); D.K. Morr and D. Pines, Phys. Rev. Lett. {\\bf 81}, 1086\n(1998); S. Sachdev, C. Buragohain, and M. Vojta, Science {\\bf 286}, 2479\n(1999).\n\n\n\\bibitem{Demler95} E. Demler and S.C. Zhang, Phys. Rev. Lett. {\\bf 75},\n4126 (1995); E. Demler, H. Kohno, and S.C. Zhang, Phys. Rev. B {\\bf 58},\n5719 (1998).\n\n\n\\bibitem{Gu98} G.D. Gu, K. Takamuku, N. Koshizuka, and S. Tanaka, J.\nCrystal Growth {\\bf 130}, 325 (1998).\n\n\n\\bibitem{Han95} S.H. Han {\\it et al}., Physica C {\\bf 246}, 22 (1995).\n\n\n\\bibitem{ybco-ni} Y. Sidis {\\it et al}., Phys. Rev. Lett. {\\bf 84}, 5900\n(2000).\n\n\n\\bibitem{Feng00} D.L. Feng {\\it et al.}, Science {\\bf 289}, 277 (2000).\n"
}
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cond-mat0002014
|
Spin Dynamics and Orbital State in LaTiO$_{3}$
|
[
{
"author": "B. Keimer$^{1,2}$"
},
{
"author": "D. Casa$^{2}$"
},
{
"author": "A. Ivanov$^{3}$"
},
{
"author": "J.W. Lynn$^{4}$"
},
{
"author": "M. v. Zimmermann$^{5}$"
},
{
"author": "J.P. Hill$^{5}$"
},
{
"author": "D. Gibbs$^{5}$"
},
{
"author": "Y. Taguchi$^{6}$"
},
{
"author": "and Y. Tokura$^{6}$"
}
] |
A neutron scattering study of the Mott-Hubbard insulator LaTiO$_{3}$ (T$_{{N}}=132$ K) reveals a spin wave spectrum that is well described by a nearest-neighbor superexchange constant $J=15.5$ meV and a small Dzyaloshinskii-Moriya interaction ($D=1.1$ meV). The nearly isotropic spin wave spectrum is surprising in view of the absence of a static Jahn-Teller distortion that could quench the orbital angular momentum, and it may indicate strong orbital fluctuations. A resonant x-ray scattering study has uncovered no evidence of orbital order in LaTiO$_{3}$.
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[
{
"name": "paper.tex",
"string": "\\documentstyle[prl,aps,epsfig]{revtex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%TCIDATA{OutputFilter=Latex.dll}\n%TCIDATA{LastRevised=Mon Aug 07 14:50:35 2000}\n%TCIDATA{<META NAME=\"GraphicsSave\" CONTENT=\"32\">}\n%TCIDATA{CSTFile=revtex.cst}\n\n\\input epsf\n\n\\begin{document}\n\\title{Spin Dynamics and Orbital State in LaTiO$_{3}$}\n\\author{B. Keimer$^{1,2}$, D. Casa$^{2}$, A. Ivanov$^{3}$, J.W. Lynn$^{4}$, M. v.\nZimmermann$^{5}$,\\\\\nJ.P. Hill$^{5}$, D. Gibbs$^{5}$, Y. Taguchi$^{6}$, and Y. Tokura$^{6}$}\n\\address{$^1$ Max-Planck-Institut f\\\"{u}r Festk\\\"{o}rperforschung,\n70569 Stuttgart, Germany\\\\\n$^2$ Department of Physics, Princeton University, Princeton, NJ 08544\\\\\n$^3$ Institut Laue-Langevin, 156X, 38042 Grenoble Cedex 9, France\\\\\n$^4$ NIST Center for Neutron Research, National Institute of Standards and\nTechnology, Gaithersburg, MD 20899\\\\\n$^5$ Department of Physics, Brookhaven National Laboratory, Upton, NY 11973\\\\\n$^6$ Department of Applied Physics, University of Tokyo, Tokyo 113, Japan}\n\\date{\\today}\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname\n\n\\maketitle\n \n\\begin{abstract}\nA neutron scattering study of the Mott-Hubbard insulator LaTiO$_{3}$ (T$_{{\\rm N}}=132$ K) \nreveals a spin wave spectrum that is well described by a\nnearest-neighbor superexchange constant $J=15.5$ meV and a small\nDzyaloshinskii-Moriya interaction ($D=1.1$ meV). The nearly isotropic spin\nwave spectrum is surprising in view of the absence of a static Jahn-Teller\ndistortion that could quench the orbital angular momentum, \nand it may indicate strong orbital fluctuations. A resonant x-ray scattering study\nhas uncovered no evidence of orbital order in LaTiO$_{3}$.\n\\end{abstract}\n\n\\pacs{PACS numbers: 75.30.Ds, 75.50.Ee, 75.30.Et, 75.30.Gw}\n\n]\n\nIn the layered cuprates exemplified by the series La$_{2-x}$Sr$_{x}$ CuO$%\n_{4+\\delta }$, the transition from a $3d^{9}$ antiferromagnetic (AF)\ninsulator at $x=\\delta =0$ into an unconventional metallic and\nsuperconducting state with increasing hole concentration ($x,$ $\\delta >0)$\nhas received an enormous amount of attention. The magnetic spectra of these\nmaterials, revealed by inelastic neutron scattering, have played a key role\nin efforts to arrive at a theoretical explanation of this transition. The\npseudocubic perovskite La$_{1-x}$Sr$_{x}$TiO$_{3+\\delta}$ undergoes an\nanalogous transition from a $3d^{1}$ AF insulator at $x=\\delta =0$ to a\nmetallic state with increasing hole concentration \\cite{tokura93}. In the\ntitanates, however, the metallic state shows conventional Fermi liquid\nbehavior, and no superconductivity is found \\cite{tokura93}.\nMomentum-resolved probes such as angle-resolved photoemission spectroscopy\nand inelastic neutron scattering have thus far not been applied to the\ntitanates, and the origin of the very different behavior of the metallic\ncuprates and titanates is still largely unexplored. Here we report an\ninelastic neutron scattering and anomalous x-ray scattering study of the\nparent compound of the titanate series, LaTiO$_{3}$, that provides insight\ninto the microscopic interactions underlying this behavior.\n\nOrbital degrees of freedom, quenched in the layered cuprates by a large\nJahn-Teller (JT) distortion of the CuO$_{6}$ octahedra, are likely to be a\nkey factor in the phenomenology of the titanates. While the TiO$_{6}$\noctahedra are {\\it tilted} in a GdFeO$_{3}$-type structure, their {\\it %\ndistortion} is small and essentially undetectable in neutron powder\ndiffraction experiments on LaTiO$_{3}$ (Ref. \\cite{greedan79}). The crystal\nfield acting on the Ti$^{3+}$ ion is therefore nearly cubic, and\nheuristically one expects a quadruply degenerate single-ion ground state\nwith unquenched orbital angular momentum opposite to the spin angular\nmomentum due to the spin-orbit interaction. In other perovskites such as\nLaMnO$_3$, such spin-orbital degeneracies are broken by successive orbital\nand magnetic ordering transitions \\cite{murakami98}. In the orbitally and\nmagnetically ordered state of LaMnO$_{3}$, the spin wave spectrum is highly\nanisotropic reflecting the different relative orientations of the orbitals\non nearest-neighbor Mn atoms in different crystallographic directions \\cite\n{moussa96}.\n\nThe reduced ordered moment ($\\mu _{0}\\sim 0.45\\mu _{B}$, Ref. \\cite\n{greedan83}) in the G-Type AF structure of LaTiO$_{3}$ (inset in Fig.~\\ref\n{fig1}) at first sight appears consistent with a conventional scenario in\nwhich the orbital occupancies at every site are established at some high\ntemperature, and the magnetic degrees of freedom (coupled spin and orbital\nangular momenta) order at a lower temperature. Full theoretical\ncalculations, however, generally predict a ferromagnetic spin structure for\nLaTiO$_{3}$ \\cite{fujimori96}. As in LaMnO$_{3}$, the spin dynamics of LaTiO$%\n_{3}$ are highly sensitive to the orbital occupancies and can provide\nimportant information in this regard. We find that the exchange anisotropy\nis small and hence inconsistent with the presence of an appreciable\nunquenched orbital moment. At the same time, synchrotron x-ray scattering\nexperiments have not revealed any evidence of reflections showing a resonant\nenhancement at the Ti K-edge, unlike other perovskites in which orbital\norder (OO) is present. These observations, along with previously puzzling\nRaman scattering data \\cite{reedyk97}, indicate strong fluctuations in the\norbital sector of LaTiO$_{3}$.\n\nThe neutron scattering experiments were conducted on the BT2 and BT4 triple\naxis spectrometers at the NIST research reactor and at the IN8 spectrometer\nat the Institut Laue-Langevin. For excitation energies up to 25 meV, we used\nhigh resolution configurations with vertically focusing pyrolytic graphite\n(PG) (002) monochromator and PG (002) analyser crystals set for final\nneutron energies of 14.7 meV or 30.5 meV and horizontally collimated beams\nat both NIST and ILL. For excitation energies of 20 meV and higher, we used\na double-focusing analyser on IN8 with open collimations, and with a Cu\n(111) monochromator and a PG (002) analyser set for a final neutron energy\nof 35 meV. PG filters were used in the scattered beam to reduce higher order\ncontamination. Data obtained on the different spectrometers and with\ndifferent configurations were in good agreement.\n\nThe sample was a single crystal of volume 0.2 cm$^{3}$ and mosaicity 0.5$%\n^{\\circ}$ grown by the floating zone technique. It is semiconducting, and\nneutron diffraction (Fig.~\\ref{fig1}) shows a sharp, second-order N\\'{e}el\ntransition at T$_{{\\rm N}}=132$ K, implying a highly homogeneous oxygen\ncontent ($\\delta \\sim 0.01$ in LaTiO$_{3+\\delta}$) \\cite{tokura93}. The\ndiffraction pattern is consistent with the G-type structure found previously \n\\cite{greedan83}, and a small uncompensated moment ($\\sim 10^{-2}\\mu _{B}$\nper Ti spin at low temperatures) appears below T$_{{\\rm N}}$ due to spin\ncanting, as observed by magnetization measurements. The\nnuclear structure of LaTiO$_{3}$ is orthorhombic (space group Pnma \\cite\n{greedan79}), but the crystal is fully twinned. Because of the isotropy of the\nspin wave dispersions (see below), twinning did not influence the\nneutron measurements. For simplicity we express the\nwave vectors in the pseudocubic notation with lattice constant $a\\sim 3.95$%\n\\AA. In this notation, AF Bragg reflections are located at ($h/2,k/2,l/2$)\nwith $h,k,l$ odd. Data were taken with the crystal in two different\norientations in which wave vectors of the form ($h,h,l$) or ($h,k,(h+k)/2$),\nrespectively, were accessible.\n\nFig.~\\ref{fig2} shows inelastic neutron scattering data obtained in constant-%\n{\\bf q} mode with high resolution near the AF zone center. The dispersing\npeak shown disappears above the N\\'{e}el temperature, thus clearly\nidentifying itself as a spin wave excitation. In Fig.~\\ref{fig3},\nconstant-energy scans obtained in the high-intensity configuration with\nrelaxed resolution are presented. The profile shapes are strongly influenced\nby the spectrometer resolution, and a deconvolution is required to\naccurately extract the positions of the spin wave peaks. For the\nhigh-resolution configuration with only vertical focusing we used the\nstandard Cooper-Nathans procedure while a Monte-Carlo ray-tracing routine\nwas used for the doubly focused geometry \\cite{kulda}. An analytical\napproximation to the Ti$^{3+}$ form factor and the standard intensity\nfactors for AF magnons were incorporated in the programs \\cite{wilson}. The\npeak positions thus obtained are shown in Fig.~\\ref{fig4} in the (111)\ndirection.\n\nA very good global fit to all data was obtained by convoluting the\nresolution function with a single spin wave branch of the generic form \n%TCIMACRO{\\UNICODE{0x127}}%\n%BeginExpansion\nh\\hskip-.2em\\llap{\\protect\\rule[1.1ex]{.325em}{.1ex}}\\hskip.2em%\n%EndExpansion\n$\\omega =zSJ\\sqrt{(1+\\epsilon )^{2}-\\gamma ^{2}\\text{ }}$where \n%TCIMACRO{\\UNICODE{0x127}}%\n%BeginExpansion\nh\\hskip-.2em\\llap{\\protect\\rule[1.1ex]{.325em}{.1ex}}\\hskip.2em%\n%EndExpansion\n$\\omega $ is the spin wave energy, $z=6$ is the coordination number, $S=1/2$\nis the Ti spin, $J=15.5\\pm 1$ meV is the isotropic (Heisenberg) part of the\nnearest-neighbor superexchange \\cite{note}, $\\gamma =\\frac{1}{3}[\\cos\n(q_{x}a)+\\cos (q_{y}a)+\\cos (q_{z}a)]$ with the magnon wave vector {\\bf q}\nmeasured from the magnetic zone center, and the zone center gap is $\\Delta\n\\sim zSJ\\sqrt{2\\epsilon }=3.3\\pm 0.3$ meV. The solid lines in Figs.~\\ref\n{fig2}-\\ref{fig4} result from this global fit and obviously provide a good\ndescription of all data. Inclusion of further-neighbor interactions, damping \nparameters above the instrumental resolution, or other\n(nondegenerate) spin wave branches did not improve the fit.\n\nThe Heisenberg exchange constant $J$ is in fair agreement with predictions\nbased on a comparison of the N\\'{e}el temperature with numerical simulations\n(T$_{{\\rm N}}=0.946J/k_{B}\\sim 170$ K for spins-1/2 on a simple cubic\nlattice \\cite{sandvik99}). In general, the spin wave gap is determined by\nsymmetric and antisymmetric (Dzyaloshinskii-Moriya) anisotropy terms in the\nsuperexchange matrix, by terms originating from direct exchange, by dipolar\ninteractions, and by the single ion anisotropy. The latter two effects are\nnegligible and nonexistent, respectively, in spin-1/2 systems. In the GdFeO$%\n_{3}$ structure, antisymmetric exchange is allowed by symmetry, and its\nmagnitude is expected to scale with the tilt angle of the ${\\rm TiO_{6}}$\noctahedra. Because of the large tilt angle ($11.5^{\\circ }$), we expect this\neffect to dominate over the more subtle direct exchange terms. Theories of\nsuperexchange anisotropies \\cite{moriya60} were recently reexamined \\cite\n{aharony95} in the light of neutron scattering data on the layered cuprates.\nIt was shown that the symmetric and antisymmetric terms are related by a\nhidden symmetry so that the two zone-center spin wave gaps depend on the\nrelationship between Dzyaloshinkii-Moriya (DM) vectors centered on each\nmagnetic bond. Specifically, for two-dimensional (2D) spin structures (such\nas the one of La$_{2}$CuO$_{4}$) the gaps are degenerate if all DM vectors\nhave the same magnitude, and the degenerate gaps are nonzero if, in\naddition, not all vectors have the same orientation. The bond-dependent DM\nvectors for LaMnO$_{3}$ (isostructural to LaTiO$_{3}$) were given in Ref. \n\\cite{solovyev96}. Although a detailed analysis of the spin dynamics for\nthese 3D systems has not been reported, we note that both of the above\ncriteria are fulfilled for LaTiO$_{3}$. In particular, the DM vectors\ncentered on the six Ti-O-Ti bonds have different orientations but are\nexpected to have the same magnitude because of the practically equal bond\nlengths and bond angles \\cite{greedan79}. Our observation of degenerate but\nnonzero spin wave gaps therefore provides support for the predictions of\nRef. \\cite{aharony95} in a 3D, non-cuprate system.\n\nCarrying the analogy to the 2D cuprates one step further, we expect that the\nanisotropy gap is $\\Delta \\sim zSD$, and hence $D\\sim 1.1$ meV is the net DM\ninteraction per Ti spin. Due to spin canting, the net ferromagnetic moment\nper spin should therefore be $\\sim \\mu _{0}D/2J=1.5\\times 10^{-2}\\mu _{B}$,\nin good agreement with the observed value. This supports our assumption that\nthe DM interaction provides the dominant contribution to the spin wave gap.\nA microscopic calculation of $D$ would be a further interesting test of the\nformalism developed in Ref. \\cite{aharony95}.\n\nThe small easy-axis anisotropy in the exchange Hamiltonian of LaTiO$_{3}$ is\ndifficult to understand based on simple crystal-field considerations for the\nTi$^{3+}$ ion. The antisymmetric exchange is generally of order $(\\Delta\ng/g)J$ where $g$ is the free-electron Land\\'{e} factor and $\\Delta g$ is its\nshift in the crystalline environment \\cite{moriya60}. Based on our data, we\ntherefore estimate $\\Delta g/g\\sim 0.05.$ On the other hand, in the absence\nof any appreciable static JT distortion as observed experimentally \\cite\n{greedan79}, one expects that the spin-orbit interaction ( $\\Lambda \\sim +20$\nmeV \\cite{abragam}) splits the $t_{2g}$ multiplet of the cubic crystal field\nHamiltonian into a quadruply degenerate ground state and a higher-lying\nKramers doublet. In a simple crystal field model, this ground state is\ncharacterized by an unquenched orbital moment equal and antiparallel to the\nspin moment ($\\Delta g/g=1$). More elaborate Hartree-Fock calculations \\cite\n{fujimori96} do not change this picture qualitatively: Even if a static JT\ndistortion at the limits of the experimental error bars \\cite{greedan79} is\nincluded, the orbital contribution to the moment remains comparable to the\nspin moment ($\\Delta g/g\\sim 0.5$). There is thus an order-of-magnitude\ndiscrepancy between the predictions of conventional models and the neutron\nscattering obervations. The smallness of the spin anisotropy in LaTiO$_{3}$\nis underscored by a different comparison: The $D/J$ ratio of LaTiO$%\n_{3}$ differs only by a factor of 3 from that of La$_{2}$CuO$_{4}$, whose\nlow-temperature ordered moment is in near-perfect agreement with the\nspin-only prediction. Since $D/J$ scales with the tilt angle of the\noctahedra which is a factor of $\\sim 3$ larger in LaTiO$_{3}$, the relative\nmagnitude of this quantity in the two materials can be accounted for without\ninvoking a large orbital moment in LaTiO$_{3}$.\n\nInterestingly, the large discrepancy between the predictions of conventional\nmodels and the neutron scattering observations has a close analogy in the\nelectron spin resonance (ESR) literature. The description of ESR data on Ti$%\n^{3+}$ impurities embedded into perovskite lattices in fact commonly\nrequires $g$-factors that are much more isotropic than predicted by simple\ncrystal field calculations \\cite{abragam}. According to a widely used model \n\\cite{ham65}, this is attributed to the dynamical JT effect where the\norbital degeneracy is lifted by coupling to zero-point lattice vibrations.\n\nWhile the dynamical JT effect is well established in impurity systems, it\nhas thus far not been reported in lattice systems which commonly exhibit\nstatic, cooperative JT distortions associated with OO. Anomalous x-ray\ndiffraction with photon energies near an absorption edge of the transition\nmetal ion has recently been established as a direct probe of OO in\nperovskites \\cite{murakami98} whose sensitivity far exceeds conventional\ndiffraction techniques that probe OO indirectly through associated lattice\ndistortions. We have therefore carried out an extensive search for\nreflections characteristic of orbital ordering near the Ti K-edge (4.966\nkeV) at beamline X22C at the National Synchrotron Light Source, with energy\nresolution $\\sim 5$ eV. The experiments were performed at low temperature\n(T=10K) on a polished (1,1,0) surface of the same crystal that was also used\nfor the neutron measurements. No evidence for resonant reflections at\nseveral high symmetry positions (such as ($\\frac12$,$\\frac12$,0), the\nordering wave vector expected for $t_{2g}$ orbitals with a G-type spin\nstructure \\cite{ren98}) was found under the same conditions that enabled\ntheir positive identification in LaMnO$_{3}$ \\cite{murakami98}, YTiO$_{3}$ \n\\cite{ytio3} and related materials. If OO is present in LaTiO$_{3}$, \nwe can hence conclude that its\norder parameter is much smaller than in comparable perovskites. On general\ngrounds, the reduction of the order parameter should be accompanied by\nenhanced orbital zero-point fluctuations. These may have already been\ndetected (though not identified as such): A large electronic background and\npronounced Fano-type phonon anomalies were observed by Raman scattering in\nnominally stoichiometric, {\\it insulating} titanates and are most pronounced in\nLaTiO$_{3}$ \\cite{reedyk97}. In the light of our observations, it is of\ncourse important to revisit these experiments and rule out any possible role\nof residual oxygen defects, inhomogeneity, etc. The presently available data\nare, however, naturally interpreted as arising from orbital fluctuations\ncoupled to lattice vibrations.\n\nAn observation not explained by these qualitative considerations is the \nsmall ordered moment in the AF state. If the orbital moment is indeed \nlargely quenched, one would naively expect a spin-only moment of $0.85 \\mu _{B}$ \n\\cite{anderson52}, in contrast to the experimental observation of $0.45 \\mu_{B}$. \nA recently proposed full theory of the interplay between the orbital and spin \ndynamics in LaTiO$_3$ has yielded a prediction of the ordered moment that is \nin quantitative agreement with experiment \\cite{khaliullin}.\n\nIn conclusion, several lines of evidence from neutron, x-ray and Raman\nscattering can be self-consistently interpreted in terms of an unusual many\nbody state with AF long range order but strong orbital fluctuations. This\nshould be an interesting subject of theoretical research. The orbital\nfluctuations are expected to be enhanced in the presence of itinerant charge\ncarriers and therefore to strongly influence the character of the\ninsulator-metal transition. The present study provides a starting point for\nfurther investigations in doped titanates.\n\nWe thank A. Aharony, M. Cardona, P. Horsch, D. Khomskii, E.\nM\\\"{u}ller-Hartmann, A. Oles, G. Sawatzky, and especially G. Khaliullin for\ndiscussions, and J. Kulda for gracious assistance with his resolution\nprogram. The work was supported by the US-NSF under grant No. DMR-9701991,\nby the US-DOE under contrat No. DE-AC02-98CH10886, and by NEDO and\nGrants-In-Aid from the Ministry of Education, Japan.\n\n\\begin{references}\n\\bibitem{tokura93} Y. Taguchi {\\it et al}., Phys. Rev. B {\\bf 59}, 7917\n(1999); Y. Tokura {\\it et al}., Phys. Rev. Lett. {\\bf 70}, 2126 (1993).\n\n\\bibitem{greedan79} D.A. MacLean, H.N. Ng, and J.E. Greedan, J. Solid State\nChem. {\\bf 30}, 35 (1979); M. Eitel and J.E. Greedan, J. Less Common Met. \n{\\bf 116}, 95 (1986).\n\n\\bibitem{murakami98} Y. Murakami {\\it et al.}, Phys. Rev. Lett. {\\bf 81},\n582 (1998).\n\n\\bibitem{moussa96} F. Moussa {\\it et al.}, Phys. Rev. B {\\bf 54}, 15149\n(1996).\n\n\\bibitem{greedan83} J.P. Goral and J.E. Greedan, J. Mag. Mag. Mat. {\\bf 37}%\n, 315 (1983).\n\n\\bibitem{fujimori96} T. Mizokawa and A. Fujimori, Phys. Rev. B {\\bf 54},\n5368 (1996); T. Mizokawa, D.I. Khomskii, and G. Sawatzky, {\\it ibid.} {\\bf 60%\n}, 7309 (1999).\n\n\\bibitem{reedyk97} M. Reedyk {\\it et al.}, Phys. Rev. B {\\bf 55}, 1442\n(1997).\n\n\\bibitem{kulda} J. Kulda, private communication.\n\n\\bibitem{wilson} P.J. Brown in {\\it International Tables for\nCrystallography,} edited by A.J.C. Wilson (Reidel, Dordrecht, 1995), Vol. C,\npp. 391-99.\n\n\\bibitem{note} $J$ is the superexchange energy per nearest-neighbor pair.\nIn this notation $J\\sim 100$ meV in La$_{2}$CuO$_{4}$.\n\n\\bibitem{sandvik99} A.W. Sandvik, Phys. Rev. Lett. {\\bf 80}, 5196 (1999).\n\n\\bibitem{moriya60} T. Moriya, Phys.\\ Rev. {\\bf 120}, 91 (1960).\n\n\\bibitem{aharony95} T. Yildirim {\\it et al}., Phys. Rev. B {\\bf 52}, 10239\n(1995).\n\n\\bibitem{solovyev96} I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev.\nLett. {\\bf 76}, 4825 (1996).\n\n\\bibitem{abragam} A. Abragam and B. Bleaney, {\\it Electron Paramagnetic\nResonance of Transition Ions }(Oxford University Press, New York, 1970).\n\n\\bibitem{ham65} F.S. Ham, Phys. Rev. {\\bf 138}, A1727 (1965); R.M.\nMacfarlane, J.Y. Wong, and M.D. Sturge, {\\it ibid.} {\\bf 166}, 250 (1968).\n\n\\bibitem{ren98} See, {\\it e.g.}, Y. Ren {\\it et al.}, Nature {\\bf 396}, 441\n(1998).\n\n\\bibitem{ytio3} H. Nakao {\\it et al.}, to be published.\n\n\\bibitem{anderson52} P.W. Anderson, Phys. Rev. {\\bf 86}, 694 (1952).\n\n\\bibitem{khaliullin} G. Khaliullin and S. Maekawa, Phys. Rev. Lett. {\\bf 85}, 3950 \n(2000).\n\\end{references}\n\n\n\\begin{figure}\n\\caption{\nIntegrated intensity of the (0.5,0.5,0.5) AF Bragg reflection as a\nfunction of temperature. The inset shows the G-type spin structure.\n}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\nTypical constant-{\\bf q} scans near the AF zone center. The upper\nprofile is offset by 200 units. The lines are the result of a convolution of\na spin wave cross section with the instrumental resolution function, as\ndescribed in the text.\n}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\nTypical constant-energy scans in two different directions of\nreciprocal space. The profiles are labeled by the excitation energy in meV.\nThe lines are the result of a convolution of a spin wave cross section with\nthe instrumental resolution function, as described in the text.\n}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\nFitted spin wave peak positions in the (1,1,1) direction of\nreciprocal space. The line is the magnon dispersion curve described in the\ntext.\n}\n\\label{fig4}\n\\end{figure}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002014.extracted_bib",
"string": "\\bibitem{tokura93} Y. Taguchi {\\it et al}., Phys. Rev. B {\\bf 59}, 7917\n(1999); Y. Tokura {\\it et al}., Phys. Rev. Lett. {\\bf 70}, 2126 (1993).\n\n\n\\bibitem{greedan79} D.A. MacLean, H.N. Ng, and J.E. Greedan, J. Solid State\nChem. {\\bf 30}, 35 (1979); M. Eitel and J.E. Greedan, J. Less Common Met. \n{\\bf 116}, 95 (1986).\n\n\n\\bibitem{murakami98} Y. Murakami {\\it et al.}, Phys. Rev. Lett. {\\bf 81},\n582 (1998).\n\n\n\\bibitem{moussa96} F. Moussa {\\it et al.}, Phys. Rev. B {\\bf 54}, 15149\n(1996).\n\n\n\\bibitem{greedan83} J.P. Goral and J.E. Greedan, J. Mag. Mag. Mat. {\\bf 37}%\n, 315 (1983).\n\n\n\\bibitem{fujimori96} T. Mizokawa and A. Fujimori, Phys. Rev. B {\\bf 54},\n5368 (1996); T. Mizokawa, D.I. Khomskii, and G. Sawatzky, {\\it ibid.} {\\bf 60%\n}, 7309 (1999).\n\n\n\\bibitem{reedyk97} M. Reedyk {\\it et al.}, Phys. Rev. B {\\bf 55}, 1442\n(1997).\n\n\n\\bibitem{kulda} J. Kulda, private communication.\n\n\n\\bibitem{wilson} P.J. Brown in {\\it International Tables for\nCrystallography,} edited by A.J.C. Wilson (Reidel, Dordrecht, 1995), Vol. C,\npp. 391-99.\n\n\n\\bibitem{note} $J$ is the superexchange energy per nearest-neighbor pair.\nIn this notation $J\\sim 100$ meV in La$_{2}$CuO$_{4}$.\n\n\n\\bibitem{sandvik99} A.W. Sandvik, Phys. Rev. Lett. {\\bf 80}, 5196 (1999).\n\n\n\\bibitem{moriya60} T. Moriya, Phys.\\ Rev. {\\bf 120}, 91 (1960).\n\n\n\\bibitem{aharony95} T. Yildirim {\\it et al}., Phys. Rev. B {\\bf 52}, 10239\n(1995).\n\n\n\\bibitem{solovyev96} I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev.\nLett. {\\bf 76}, 4825 (1996).\n\n\n\\bibitem{abragam} A. Abragam and B. Bleaney, {\\it Electron Paramagnetic\nResonance of Transition Ions }(Oxford University Press, New York, 1970).\n\n\n\\bibitem{ham65} F.S. Ham, Phys. Rev. {\\bf 138}, A1727 (1965); R.M.\nMacfarlane, J.Y. Wong, and M.D. Sturge, {\\it ibid.} {\\bf 166}, 250 (1968).\n\n\n\\bibitem{ren98} See, {\\it e.g.}, Y. Ren {\\it et al.}, Nature {\\bf 396}, 441\n(1998).\n\n\n\\bibitem{ytio3} H. Nakao {\\it et al.}, to be published.\n\n\n\\bibitem{anderson52} P.W. Anderson, Phys. Rev. {\\bf 86}, 694 (1952).\n\n\n\\bibitem{khaliullin} G. Khaliullin and S. Maekawa, Phys. Rev. Lett. {\\bf 85}, 3950 \n(2000).\n"
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cond-mat0002015
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Multiscale Finite-Difference-Diffusion-Monte-Carlo Method for Simulating Dendritic Solidification
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[
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"author": "Mathis Plapp and Alain Karma"
}
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We present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field. The first key feature of this method is an efficient multiscale diffusion Monte-Carlo (DMC) algorithm which allows off-lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid-liquid interface. As a result, the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution. The second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface. Inside this layer and in the solid, the diffusion equation is solved using a standard finite-difference algorithm that is interfaced with the DMC algorithm using the local conservation law for the diffusing quantity. Here we combine this algorithm with a previously developed phase-field formulation of the interface dynamics and demonstrate that it can accurately simulate three-dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance.
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"name": "jcp.tex",
"string": "\\documentstyle[twocolumn,aps,psfig,floats]{revtex} % <---| galley\n\n\\begin{document} \t % |\n\n\\def\\be{\\begin{equation}}\n\\def\\ee{\\end{equation}}\n\\def\\vnab{{\\vec\\nabla}}\n\\draft \t % |\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname %\t | \n@twocolumnfalse\\endcsname\n\n\\title{Multiscale Finite-Difference-Diffusion-Monte-Carlo Method for Simulating Dendritic Solidification}\n\n\\author{Mathis Plapp and Alain Karma}\n\n\\address{\nPhysics Department and Center for Interdisciplinary Research\non Complex Systems, \\\\\nNortheastern University, Boston, Massachusetts 02115 \n}\n\n\\date{January 11, 2000}\n\n\\maketitle\n\n\\begin{abstract}\nWe present a novel hybrid computational method\nto simulate accurately dendritic solidification in the\nlow undercooling limit where the dendrite tip radius \nis one or more orders of magnitude smaller than the \ncharacteristic spatial scale of variation of the \nsurrounding thermal or solutal diffusion field. \nThe first key feature of this method is an efficient multiscale \ndiffusion Monte-Carlo (DMC) algorithm which allows off-lattice random \nwalkers to take longer and concomitantly rarer steps with increasing\ndistance away from the solid-liquid interface. \nAs a result, the computational cost of evolving the \nlarge scale diffusion field becomes insignificant when\ncompared to that of calculating the interface evolution. \nThe second key feature is that random walks are only permitted\noutside of a thin liquid layer surrounding the interface. Inside\nthis layer and in the solid, the diffusion equation is solved \nusing a standard finite-difference algorithm that is interfaced \nwith the DMC algorithm using the local conservation law for \nthe diffusing quantity.\nHere we combine this algorithm with a previously developed \nphase-field formulation of the interface dynamics and demonstrate\nthat it can accurately simulate three-dimensional dendritic growth\nin a previously unreachable range of low undercoolings\nthat is of direct experimental relevance.\n\\end{abstract}\n\\pacs{}\n]\n\n\\section{Introduction}\n\\nobreak\n\nDiffusion-limited pattern formation, which leads to the\nspontaneous emergence of complex branched structures, \noccurs in numerous contexts. A few examples include \ndendritic solidification \\cite{Kurz}, electrochemical deposition\n\\cite{eldep} and corrosion, \nand the growth of bacterial colonies \\cite{BenJacob}.\nTwo distinct length scales are typically involved in this\nclass of problems: one that characterizes \nthe pattern itself, such as the thickness of a \nbranch, and one that characterizes the diffusion \nfield associated with the transport of heat or matter. \nIn many cases, these two scales are vastly different.\nFor example, in solidification, the decay length of\nthe thermal or solutal field ahead of a growing\ndendrite (in a pure or alloy melt) can be one to three orders\nof magnitude larger than the tip radius of one of its primary \nbranches. Non-trivial pattern formation dynamics \ncan be expected to occur on all intermediate scales.\nThis poses a serious challenge for numerical\nsimulations since a precise integration of\nthe equations of motion on the pattern scale requires a good resolution \nof the interfacial region, and such a resolution is completely inefficient\n(i.e. much too fine) to treat the large scale \ndiffusion field. Therefore, in order to retain this precision\non the small scale and, at the same time, simulate the pattern evolution\non sufficiently large length and time scales, it is \nnecessary to use some form of multiscale algorithm. \n\nMulti-grid and finite element methods with non-uniform meshing\nrepresent one possible solution for this type of problems.\nTheir application, however, in the context of growth\nsimulations faces the additional difficulty of a\nmoving interface, which implies that the structure \nof the simulation grid has to be dynamically adapted.\nFor the classic problem of dendritic \ncrystal growth, several multi-grid \\cite{Braun97}\nor adaptive meshing algorithms \\cite{Schmidt96} have been\nproposed in recent years. The most precise to date is the method\nof Provatas {\\it et al.} which uses the phase-field model\non a regular grid on the scale of the dendrite,\nwhereas the diffusion field is integrated on an adaptive\nmesh using finite element techniques \\cite{Provatas99}. \nWhile this method appears to be promising, it has yet \nto be implemented in three dimensions where the difficulty\nof adaptive meshing becomes significantly enhanced.\n\nWe present in this paper an alternative solution to\nsolve this computational challenge and we illustrate its\napplication in the context of\nthe dendritic crystallization of a pure substance\nfrom its undercooled melt, even though this algorithm can \nbe applied to any diffusion-limited growth problem \nfor which an explicit solver of the interface \ndynamics is available. The idea is to use a hybrid\napproach. The interface dynamics is treated using\ndeterministic equations of motion, in particular \nthose of the phase-field model for the dendritic\ngrowth problem considered here.\nOn the other hand, the large-scale diffusion field\nis represented by an ensemble of off-lattice random \nwalkers and is evolved using a Diffusion Monte Carlo \n(DMC) algorithm. The two solutions are connected at \nsome distance from the moving interface.\nThe key point for rendering our method efficient is that\nwe use random walkers which dynamically adapt the average \nlength of their random steps. Far from the interface,\nthe walkers can make large jumps and hence be updated\nonly rarely without affecting the quality of the solution\nnear the growing interface. In some sense, our\nmethod can be seen as an ``adaptive grid algorithm\nwithout grid''. The DMC algorithm and the connection\nbetween deterministic and stochastic parts are rather\nsimple and straightforward to implement in both two \nand three dimensions, both on single-processor and\nparallel architectures. We demonstrate in this paper\nthat our method is precise, robust, and reliable,\nand hence constitutes a powerful alternative to\nstate of the art adaptive meshing techniques.\nTechnically, the algorithm bears \nmany similarities to quantum Monte Carlo methods \n\\cite{Koonin}. It is therefore remarkable \nthat the gap between mesoscopic and macroscopic length scales \ncan be bridged using a method borrowed from \nmicroscopic physics in an interfacial pattern \nformation context, which was not {\\it a priori} obvious to \nus at the start of this investigation.\n\nOur algorithm builds on ideas of earlier \nrandom walk algorithms for simulating pattern \nformation during viscous fingering \\cite{Kadanoff85,Liang86} \nand solidification \\cite{Vicsek84,Nittmann86,Karma87,Saito89},\nbut introduces two essential new features. Firstly,\nrandom walks with variable step size have been used\npreviously in simulations of large-scale diffusion-limited\naggregation \\cite{Meakin83}, but only one walker at a\ntime was simulated, and hence the time variable did not\nexplicitly appear in the treatment of the walkers.\nIn the present diffusive case, the memory of the past \nhistory, stored in the diffusion field, is essential to \nthe problem. Our DMC algorithm works with a whole ensemble\nof walkers in ``physical'' time and hence constitutes a \ntrue multiscale solver for the full diffusion problem.\nSecondly, the algorithms mentioned above use a lattice\nboth to evolve the walkers and to represent the position\nof the interface by the bonds between occupied (solid)\nand empty (liquid) sites. Walkers are created or absorbed \ndirectly at this interface. The discretization of space and \nthe stochastic creation and absorption of walkers make it \ndifficult to control accurately the interfacial anisotropy \nand the noise that both play a crucial role in\ndendritic evolution \\cite{Nittmann86,Brener96}.\nConsequently, the algorithms aimed at describing dendritic \ngrowth \\cite{Vicsek84,Nittmann86,Saito89},\nwhile correctly reproducing all the qualitative features\nof the growth process, are unable to yield quantitative\nresults that can be tested against experiments. We solve \nboth problems by creating and absorbing walkers not at the\nsolid-liquid interface, but at a ``conversion boundary''\nat some fixed distance from the interface. This means\nthat the stochastic representation of the diffusion\nand the motion of the interface can be treated separately,\nwhich allows us to evolve the interface accurately\nby the phase-field method using a finite difference\nrepresentation of controlled precision.\nAt the same time, the stochastic noise created by the\nDMC algorithm is rapidly damped by the deterministic diffusion\nin the ``buffer layer'' between the conversion boundary\nand the solid-liquid interface, and hence the amplitude of the \nfluctuations {\\em at the solid-liquid interface} can be reduced \nto a prescribed level without much cost in computation\ntime by increasing the thickness of the buffer layer.\nThis is an important issue for simulations of dendritic growth, \nbecause the amplification of microscopic fluctuations\nof the interface is believed to be the main cause for the\nformation of secondary dendrite branches \\cite{Langer87},\nand it is well known that numerical noise can lead to the\nformation of spurious sidebranches in simulations. \nConsequently, we have to demonstrate that the walker \nnoise of our algorithm can be reduced to a level that does \nnot affect the pattern evolution.\n\nAnother benefit of the buffer layer is that it makes the \nalgorithm very versatile. Away from the interface, only the \nstandard diffusion equation has to be solved. Therefore, the \nDMC part of the algorithm and the conversion process between\ndeterministic and stochastic solutions are completely\nindependent from the method used for simulating the\ninterface dynamics, and can easily be carried over to\nother free boundary problems.\n\nThe purpose of the present paper is to describe the \nalgorithm in detail and to demonstrate its reliability\nand precision by benchmark simulations. Some results\nconcerning three-dimensional crystal growth at low\nundercoolings have already been presented elsewhere\n\\cite{Plapp99,Tip99}, and hence we will focus here\non the computational aspects of the problem. \nSection II contains a brief introduction to dendritic \nsolidification and the basic equations of motion,\nand describes the phase-field method. In section III, \nthe DMC algorithm and its interfacing with the \nphase-field equations are described in detail.\nIn section IV, we present results of benchmark simulations, \nassess the efficiency of the code and the magnitude\nof numerical noise, and present\nsimulations of three-dimensional dendritic growth.\nSection V contains a conclusion and the outline of future work.\n\n\\section{Dendritic growth and the phase-field method}\n\\nobreak\n\nWhen a crystal grows from an undercooled melt, it develops\ninto an intricate branched structure, called a dendrite.\nThis phenomenon has been of central importance to the\nunderstanding of spontaneous pattern formation during\nphase transformations and the emergence of branched\nstructures \\cite{Langer80,Kessler88,Brener91}.\nIn addition, it is of considerable practical \ninterest, because dendrites form during the\nsolidification of many commercially important alloys\nand influence the mechanical properties of the \nfinished material.\n\nWe will focus on the dendritic solidification \nof a pure substance from its homogeneously undercooled \nmelt, starting from a single supercritical nucleus \n\\cite{Huang81,Rubinstein91,Muschol92,Glicksman94,Bisang95}. \nThis situation is well described by the\nsymmetric model of solidification, which assumes\nthat the diffusivity and thermophysical quantities \nsuch as the specific heat and the density are equal for \nthe solid and the liquid phases.\nDuring the growth of the crystal, the latent\nheat of melting is released, and in the absence\nof convection, the growth becomes limited by the\ndiffusion of heat away from the growing dendrite.\nThe state of the system at any time is described\nby the temperature field $T(\\vec x,t)$ and the\nshape $\\Gamma(t)$ of the boundary between solid \nand liquid. It is customary to define a dimensionless \ntemperature field\n\\be\nu(\\vec x,t) = {T(\\vec x,t)-T_m\\over L/c_p},\n\\ee\nwhere $L$ and $c_p$ are the latent heat of melting\nand the specific heat, respectively, and $T_m$ is the\nmelting temperature. In terms of this field, the \nequations of motion of the symmetric model are\n\\be\n\\partial_t u = D{\\vnab}^2 u,\n\\label{diffuu}\n\\ee\n\\be\nv_n = D \\hat n\\cdot\\left({\\vec\\nabla u |}_S - \n {\\vec\\nabla u |}_L\\right),\n\\label{stefanu}\n\\ee\n\\be\nu_\\Gamma = - d_0 \\,\\sum_{i=1}^{d-1}\\,\n\\left[a(\\hat n)+\\frac{\\partial^2a(\\hat n)}\n{\\partial \\theta_i^2}\\right]{1\\over R_i}-\\beta(\\hat n) v_n,\n\\label{githou}\n\\ee\nwhere $D$ is the thermal diffusivity, $v_n$ is the\nnormal velocity of the interface, and $\\hat n$ \nis the unit normal vector of the surface $\\Gamma$ \npointing towards the liquid. The diffusion equation,\nEq. (\\ref{diffuu}) is valid everywhere (in the liquid\nand in the solid) except on the surface $\\Gamma$.\nThe Stefan condition, Eq. (\\ref{stefanu}), valid on\n$\\Gamma(t)$, expresses the conservation of enthalpy\nat the moving phase boundary. Here, $\\vnab u|_S$\nand $\\vnab u|_L$ denote the limits of the\ntemperature gradient when $\\Gamma$ is approached from the\nsolid and the liquid side, respectively, and the\nequation states that the local heat flux at the\ninterface must be equal to the latent heat generated\nor consumed during the phase transformation; $v_n$\nis positive if the solid grows (i.e. freezing).\nThe dimensionless temperature at the interface $u_\\Gamma$ is\ngiven by the generalized Gibbs-Thomson condition Eq. (\\ref{githou}).\nThe first term on the right hand side (RHS) is the anisotropic \nform of the local equilibrium condition (Gibbs-Thomson\ncondition) which relates the temperature to the curvature\nof the interface and the anisotropic surface tension\n$\\gamma(\\hat n)=\\gamma_0 a(\\hat n)$. For a crystal with\ncubic symmetry in three dimensions, the anisotropy\nfunction $a(\\hat n)$ is usually written as\n\\be\na(\\hat n) = (1-3\\epsilon_4)\\,\n\\left[1+\\frac{4\\epsilon_4}\n{1-3\\epsilon_4}\\left(n_x^4+n_y^4+n_z^4\\right)\\right],\n\\label{ani}\n\\ee\nwhere $\\epsilon_4$ is the anisotropy parameter.\nNote that in two dimensions ($d=2$), this expression \nreduces to\n\\be\na(\\theta) = 1 + \\epsilon_4 \\cos(4\\theta),\n\\label{ani2d}\n\\ee\nwhere $\\theta$ is the angle between the normal and one \nof the axes of symmetry. On the RHS of Eq. (\\ref{githou}),\n\\be\nd_0 = {\\gamma_0 T_m c_p\\over L^2}\n\\ee\nis the capillary length, $d$ is the spatial dimension,\n$\\theta_i$ are the angles between the normal $\\hat n$ \nand the two local principal directions on $\\Gamma$, \nand $R_i$ are the principal radii of curvature. \nFinally, the second term on the RHS of Eq. (\\ref{githou})\ndescribes the shift of the interface temperature \ndue to molecular attachment kinetics, and $\\beta(\\hat n)$\nis the orientation-dependent linear kinetic coefficient.\nKinetic effects are believed to be small for the range of\nsolidification speeds of interest here. We will\ntherefore focus on the case where the interface \nkinetics vanish ($\\beta(\\hat n) \\equiv 0$), which \ncorresponds to local equilibrium at the interface.\nIn this case, the physical length and time scales are \nset by the capillary length and the diffusivity,\nand the control parameters of the problem are\nthe anisotropy $\\epsilon_4$ and the dimensionless \nundercooling\n\\be\n\\Delta = {T_m - T_0\\over L/c_p},\n\\ee\nwhere $T_0$ is the initial temperature, $T(\\vec x,0)=T_0$,\nwhich provides the thermodynamic driving force for \nsolidification. We assume that the dendrite grows into an \ninfinite volume of liquid, and hence $u(\\vec x,t)\\to -\\Delta$\nas $|\\vec x|\\to \\infty \\, \\forall \\,t$. Typical experimental\nvalues for $\\Delta$ range from $0.001$ to $0.1$. \nThe length scales involved in the problem \nare (i) the capillary length $d_0$,\n(ii) a typical scale of the pattern such as the radius\nof curvature at a tip $\\rho$, and (iii) the length scale of the\ndiffusion field $l_D$. To fix the ideas, let us consider\nthe measurements of Rubinstein and Glicksman on pivalic\nacid (PVA) \\cite{Rubinstein91}. For a dimensionless undercooling \nof $\\Delta=0.075$, $\\rho = 8.5 \\,\\mu m$, and the speed of the\ntips is $v=390 \\,\\mu m/s$, which gives a diffusion length\n$l_D = 2D/v = 0.38 \\,mm$, whereas $d_0 = 3.8 \\,nm$. The\nmultiscale character of this situation is obvious: \n$l_D$ and $d_0$ differ by five\norders of magnitude, and $l_D$ is forty times larger \nthan $\\rho$. These ratios become even larger for lower\nundercoolings.\n\nThe above equations constitute a notoriously difficult\nfree boundary problem. To simplify the task, theoretical and \nnumerical efforts first concentrated on the treatment of a\nsingle needle crystal growing at constant velocity.\nThis situation can be treated by boundary integral\nmethods \\cite{Kessler88}, which are exact in two\ndimensions (2-d) but have remained approximate in three\ndimensions (3-d). More recently, time-dependent methods \nhave been developed to describe the full growth dynamics\n\\cite{Sethian92,Ihle94,Karma98}. Of those,\nthe phase-field method \\cite{pf} seems presently\nthe most compact and precise approach. We use\na recent efficient formulation of this method,\nwhich has been benchmarked against boundary\nintegral calculations \\cite{Karma98}.\nAn ``order parameter'', or phase-field \n$\\psi(\\vec x,t)$ is introduced, which is an indicator \nfield distinguishing the solid ($\\psi=1$) and\nthe liquid ($\\psi=-1$) phase.\nThe two-phase system is described by a free energy\nfunctional of Ginzburg-Landau type,\n\\be\n{\\cal F}=\\int dV\\,[\\,W^2({\\hat n})|\\vnab\\psi|^2+f(\\psi,u)],\n\\label{freeen}\n\\ee\nwhere $W({\\hat n})$ is the orientation-dependent interface\nthickness, i.e. the spatial scale on which the phase-field\nvaries smoothly between its equilibrium values $\\psi=\\pm 1$,\nand $f(\\psi,u)$ is the free energy density. The\nequations of motion are\n\\be\n\\tau(\\hat n)\\partial_t \\psi = -{\\delta {\\cal F} \\over \\delta \\psi(\\vec x,t)},\n\\label{pfpsi}\n\\ee\nwhere $\\delta{\\cal F}/\\delta\\psi$ denotes the functional derivative, and\n\\be\n\\partial_t u = D\\,\\vnab^2 u+\\frac{1}{2} \\partial_t \\psi.\n\\label{pfu}\n\\ee\nThe phase-field relaxes to its local minimum free energy \nconfiguration, which depends on the local temperature\nfield, with an orientation-dependent relaxation time \n$\\tau(\\hat n)$. The diffusion equation contains a source\nterm to account for the latent heat released or consumed\nduring the phase transformation.\nFor a suitable choice of the functions $f(\\psi,u)$, \n$W(\\hat n)$ and $\\tau(\\hat n)$, these equations \nreduce precisely to the free boundary\nproblem given by Eqs. (\\ref{diffuu}) to (\\ref{githou})\nin the limit where the interface thickness is small\ncompared to the radii of curvature \\cite{Karma98}. A brief\ndescription of the model used for our simulations\nand its relation to the macroscopic free boundary problem\nis given in the appendix. The key point is that the\nphase-field equations of motion are partial\ndifferential equations which can be integrated on\na regular grid on the scale of the dendrite, without\nknowing explicitly where the solid-liquid interface\nis located. The phase field rapidly decays to its\nequilibrium values $\\psi=\\pm 1$ away from the interface.\nTherefore, well within the bulk phases, Eq. (\\ref{pfpsi}) \nbecomes trivial and Eq. (\\ref{pfu}) reduces to the \nordinary diffusion equation.\n\n\\section{Diffusion Monte Carlo algorithm}\n\\nobreak\n\\subsection{Outline}\n\\nobreak\n\\noindent\nOur goal is to combine the precision of the \nphase-field method and the efficiency\nof a DMC treatment for the diffusion field. \n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=fig1b.eps,width=.4\\textwidth}}\n\\caption{Simulation of two-dimensional dendritic \ngrowth for a dimensionless undercooling $\\Delta=0.1$\nand a surface tension anisotropy \n$\\epsilon_4=0.025$. The solid line is the solid-liquid \ninterface $\\Gamma$, the dashed line is the conversion boundary $\\Gamma'$\nbetween the inner (deterministic) and outer (stochastic) domains, \nand the dots show the positions of random walkers (only one walker\nout of 50 is shown for clarity).}\n\\label{figillu}\n\\end{figure}\n%-------------------------------------------\nThis is achieved by dividing the \nsimulation domain into an ``inner'' and \nan ``outer'' region as shown in Fig. \\ref{figillu}. \nIn the inner region, consisting of the growing structure and \na thin ``buffer layer'' of liquid, we integrate the \nphase-field equations described above. In the outer \nregion, the diffusion field is represented by an ensemble \nof random walkers. Walkers are created and absorbed at the \nboundary between inner and outer domains at a rate which\nis proportional to the local diffusion flux. \nThe value of the diffusion field in the outer \ndomain is related to the local density of walkers, \nand the boundary conditions for the integration in \nthe inner region are obtained by averaging this density \nover coarse-grained boxes close to the boundary.\nWe will now describe in detail the DMC algorithm\nfor the evolution of the random walkers and the\nconnection of the two solutions.\n\nLet us start by recalling some well-known facts about random\nwalkers. Consider first a single point particle performing\na Brownian motion in continuous space and time. The conditional\nprobability $P({\\vec x}',t'|\\vec x,t)$ of finding \nthe particle at position ${\\vec x}'$ at time $t'$, \ngiven that it started from position $\\vec x$\nat time $t$, is identical to the diffusion kernel,\n\\be\nP({\\vec x}',t'|\\vec x,t) = \n {1\\over \\left[4\\pi D(t'-t)\\right]^{d/2}}\n \\exp{\\left[-{|{\\vec x}'-{\\vec x}|}^2\\over 4D(t'-t)\\right]},\n\\label{diffker}\n\\ee\nwhere $D$ is the diffusion coefficient and $d$ is the\nspatial dimension. This kernel satisfies the well-known\nconvolution relation\n\\begin{eqnarray}\nP({\\vec x}'',t''|\\vec x,t) &=& \n \\int\\,P({\\vec x}'',t''|{\\vec x}',t')P({\\vec x}',t'|\\vec x,t)\\,d{\\vec x}'\n \\nonumber\\\\\n && \\quad\\forall\\, t<t'<t''.\n\\label{convo}\n\\end{eqnarray}\nTherefore, a realization of a random walk, i.e. the position\nof a walker as a function of time, represented by a time-dependent\nvector of real numbers $\\vec x(t)$, can be obtained on a \ncomputer by successive steps. The position of the walker \nis updated following the scheme\n\\be\n\\vec x(t+\\tau) = \\vec x(t) + \\ell \\vec \\xi,\n\\label{scheme}\n\\ee\nwhere the components of the random vector $\\vec\\xi$ are\nindependent Gaussian random variables of unit variance.\nThe time increment $\\tau$ (not to be confused with \nthe phase-field relaxation time $\\tau(\\hat n)$ \ndefined in the preceding section) and the\nstep size $\\ell$ must satisfy the relation\n\\be\n{\\ell^2\\over\\tau} = 2 D.\n\\label{taucond}\n\\ee\nSince time is continuous and Eq. (\\ref{convo}) is not\nrestricted to $t''-t'=t'-t$, successive \nsteps may have different time increments (and concomitantly\nuse different step lengths) if Eq. (\\ref{taucond})\nis satisfied for each update.\n\nThe basic idea of Diffusion Monte Carlo simulations is to\nsample many realizations of diffusion paths. The density\nof random walkers then satisfies a stochastic differential\nequation which converges to the deterministic diffusion\nequation in the limit of an infinite number of walkers.\nA density of walkers can be defined by a suitable \ncoarse-graining procedure on a scale $L_{cg}$, i.e.\nby dividing space into cells of volume $L_{cg}^d$ and\ncounting the number of walkers within each cell. \nIf the coarse-graining length is chosen larger than\nthe average step length $\\ell$, this density evolves\nsmoothly on the scale of $L_{cg}$ over times of order\n$L_{cg}^2/D$, the time for one walker to diffuse\nthrough a coarse cell.\n\nFrom the above considerations, it is clear that the\ncharacteristic length and time scales that can be\nresolved by a stochastic DMC algorithm are set by\nthe step size $\\ell$ and the time increment $\\tau$,\nrespectively. The key point is that for the present\napplication a high spatial and temporal resolution\nis needed {\\em only close to the interface}, whereas\nfar from the dendrite, the coarse-graining length\nand hence the step size can become much larger than\nthe fine features of the growing crystal. In practice,\nwe choose the step size to be approximately proportional\nto the distance $d_{cb}$ of the walker from the conversion\nboundary between the inner (deterministic) and outer \n(stochastic) regions, i.e.\n\\be\n\\ell \\approx c\\, d_{cb}\n\\label{ldist}\n\\ee\nwith a constant $c\\ll 1$. According to Eq. (\\ref{taucond}), \nthe time increment between updates grows as the square \nof the step size, and hence the walkers\nfar from the dendrite have to be updated only rarely.\nWe use dynamical lists to efficiently handle the updating\nprocess, as will be described in more detail in Sec. \\ref{secwalk}.\nFor low undercoolings, where the scale of the diffusion field\nis much larger than the dendrite itself and most of\nthe walkers need only be updated sporadically,\nwe obtain enormous savings of computational \ntime over a straightforward integration of\nthe diffusion equation.\n\nLet us now discuss how the inner and outer regions\nare interfaced. Two essential goals have to be accomplished.\nFirstly, we have to supply a boundary condition at the\nconversion boundary for the integration of the deterministic\nequations in the inner region, and secondly we need to\ncreate and absorb walkers at a rate which is proportional\nto the local heat flux across this boundary. \n\n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=grids2.eps,width=.45\\textwidth}}\n\\medskip\n\\caption{Sketch of a small part of the conversion boundary in\ntwo dimensions for $n=4$. Each cell of the coarse grid (thick\nlines) contains $16$ points of the fine grid (thin lines). \nThe fine grid is shown only in the inner region for clarity.\nThe shaded cells are conversion cells, and walkers are \nrepresented by black dots. The boundary $\\Gamma'$ between\ninner and outer regions is indicated by a dashed line.}\n\\label{figgrids}\n\\end{figure}\n%-------------------------------------------\nThe phase-field equations are\nintegrated in the inner region on a regular cubic grid,\nhenceforth called ``fine grid'', with spacing $\\Delta x$.\nEach node on this grid contains the local values of the\nphase field $\\psi$ and the temperature field $u$.\nWe superimpose on this grid another, coarser grid,\nof mesh size $L_{cg}=n\\Delta x$, such that the links \nof the coarse grid intersect the links of the fine grid\nas shown in Fig. \\ref{figgrids}. The first purpose of\nthis grid is to define the geometries of the two\nsimulation regions and of the conversion boundary.\nWe describe the ``state'' of each coarse cell by an\ninteger status variable $S_{\\alpha\\beta\\gamma}^t$. \nHere and in the following, greek indices ($\\alpha$,\n$\\beta$, $\\gamma$) label the {\\em cells} of the coarse grid \nalong the $x$-, $y$-, and $z$-directions, whereas\nlatin indices ($i$, $j$, $k$) label the {\\em nodes}\nof the fine grid. All cells which contain at\nleast one node of the fine grid where $\\psi>0$ are\nassigned the status ``solid'' ($S=-2$). \nAll cells with a center-to-center distance to the \nnearest solid cell smaller than a prescribed length \n$L_b$ are ``buffer cells'' ($S=-1$),\nwhereas all other cells belong to the outer region.\nCells of the outer region which have at least one nearest\nneighbor with buffer status are called {\\em conversion\ncells} ($S=0$) and play the central role in interfacing the\ntwo solutions. The dividing surface $\\Gamma'$ between\ninner and outer regions is the union of all the links \n(or plaquettes in three dimensions) of the coarse grid \nwhich separate conversion from buffer cells \n(see Fig. \\ref{figgrids}). Evidently, as the crystal\ngrows, the geometry of the two regions changes, which\nmeans that the status variables must be periodically updated.\nDetails on this procedure are given in Sec. \\ref{secupdate}.\n\nWe always choose $L_b$ sufficiently large to ensure that\nthe phase field is already close to its liquid equilibrium \nvalue, $\\psi \\approx -1$, at the conversion boundary. \nHence we can set $\\psi=-1$ in the entire outer region\nand treat only the standard diffusion equation there. \nIn the initial state, the entire system is undercooled to\n$u=-\\Delta$, and no walkers are present. When the\ncrystal grows, it releases latent heat which diffuses\naway from the interface, and hence the inner region \nbecomes a heat source for the outer region. This heat \nflux is converted into walkers, each walker representing\na certain discrete amount of heat. We define\nin each coarse cell an integer variable \n$m_{\\alpha\\beta\\gamma}^t$ which contains the number\nof walkers being within this cell at time $t$. For a specific\nheat which is independent of temperature, the density\nof walkers is proportional to the difference between\nthe actual and the initial temperatures, i.e. the\ntemperature in the outer region is related to the\nnumber of walkers by\n\\be\nu_{\\alpha\\beta\\gamma}^t = \n -\\Delta\\left(1-{m_{\\alpha\\beta\\gamma}^t\\over M}\\right),\n\\label{coarsetemp}\n\\ee\nwhere the constant $M$ fixes the number of walkers in a\ncell that corresponds to the melting temperature $u=0$.\n\nThe inner region is completely delimited by conversion\ncells. To fix the boundary condition for the integration \non the fine grid, it is therefore sufficient to set the \nfield $u$ on all nodes of the fine grid in each conversion \ncell to the value specified by Eq. (\\ref{coarsetemp}).\nThe diffusion equation is then timestepped in the inner\nregion using the standard explicit scheme\n\\begin{eqnarray}\nu_{ijk}^{t+\\Delta t} & = & u_{ijk}^t + {D\\Delta t\\over (\\Delta x)^2}\n \\nonumber \\\\\n & & \\mbox{}\\times\n \\left(u_{i+1jk}^t+u_{i-1jk}^t+u_{ij+1k}^t+u_{ij-1k}^t\\right. \\nonumber \\\\\n & & \\quad\\mbox{} +\n \\left.u_{ijk+1}^t+u_{ijk-1}^t-6u_{ijk}^t\\right).\n\\label{diffudis}\n\\end{eqnarray}\nNote that we have omitted for simplicity the source terms due\nto the phase field, which are zero at the conversion boundary. \nSeen on a discrete level, this equation can be interpreted\nas a ``pipe flow'' equation: the local change of $u$ is given\nby the sum of the ``flow'' through all the discrete links \n(``pipes''), where, for example, the ``flow'' through a link \nalong $x$ during a timestep is given by \n$D\\Delta t(u_{i+1jk}-u_{ijk})/(\\Delta x)^2$.\nFor nodes at the boundary of the inner region, some links\ncross the conversion boundary $\\Gamma'$, which means that\nthere is exchange of heat with the neighboring conversion\ncell. This heat flux is collected by the conversion\ncell and stored in a heat reservoir variable \n$H_{\\alpha\\beta\\gamma}^t$. A symbolic manner to describe\nthe updating of $H_{\\alpha\\beta\\gamma}^t$ is\n\\be\nH_{\\alpha\\beta\\gamma}^{t+\\Delta t} = H_{\\alpha\\beta\\gamma}^t\n + {D\\Delta t\\over (\\Delta x)^2} \n \\left(\\sum_{\\rm bonds} u_{\\rm grid}^t-u_{cc}^t\\right),\n\\ee\nwhere the sum runs over all the bonds of the fine grid \nthat cross $\\Gamma'$, $u_{\\rm grid}$ is the temperature \non a node of the fine grid and $u_{cc}$ is the temperature \nin the conversion cell given by Eq. (ref{coarsetemp}). \nFor example, for a conversion cell $(\\alpha,\\beta,\\gamma)$ \nin contact with a buffer cell $(\\alpha-1,\\beta,\\gamma)$, \nwe have (we recall that the linear dimension of a coarse\ncell is $L_{cg}=n\\Delta x$):\n\\begin{eqnarray}\n & & H_{\\alpha\\beta\\gamma}^{t+\\Delta t} = \n H_{\\alpha\\beta\\gamma}^t + \\nonumber \\\\\n & & \\quad\\sum_{j=(\\beta-1)n+1}^{\\beta n}\\,\\, \\sum_{k=(\\gamma-1)n+1}^{\\gamma n}\n {D\\Delta t\\over (\\Delta x)^2} (u_{i-1jk}^t-u_{ijk}^t) \\nonumber \\\\\n & & \\quad {\\rm with}\\quad i=(\\alpha-1)n +1.\n\\label{resupdate}\n\\end{eqnarray}\nIf the stored quantity of heat exceeds a critical value $H_c$\ngiven by\n\\be\nH_c = {n^d \\Delta \\over M},\n\\ee\na walker is created at the center of the conversion cell\nand $H_c$ is subtracted from $H_{\\alpha\\beta\\gamma}$.\nConversely, if the local heat flux is negative (heat\nis locally flowing {\\em towards} the dendrite) and\n$H_{\\alpha\\beta\\gamma}$ falls below $-H_c$,\na walker is removed from the cell and $H_c$ is added to\nthe reservoir. This algorithm exactly conserves the\ntotal heat if the contributions of the fine grid, the\nreservoir variables and the walkers are added.\nIn dimensional quantities, each walker\nis equivalent to an amount of heat $\\Delta Q$ equal to\n\\be\n\\Delta Q = {L (n\\Delta x)^d \\Delta \\over M}. \n\\ee\n\nThe walkers are restricted to the outer region. If a\nwalker attempts to jump across the conversion boundary,\nthe move is discarded and the walker stays at its old \nposition until the next update. If $c$ in \nEq. (\\ref{ldist}) is small enough, such jumps are \nattempted almost only by walkers close to\nthe conversion boundary. Accordingly, this procedure is\na convenient way of implementing the re-absorption\nof walkers: if a walker stays in a conversion cell,\nthe heat flux is more likely to be directed towards\nthe inner region, which increases the chances for\nthe walker to be absorbed. An alternative method,\nnamely to deposit all the heat contained in a walker\nin the fine grid and remove the walker upon its crossing\nof the boundary, would create stronger temperature\nfluctuations on the fine grid close to the conversion\nboundary.\n\nIn summary, the conversion process is handled using three\nauxiliary fields on the coarse grid: the status field\n$S_{\\alpha\\beta\\gamma}$ which encodes the geometry of \nthe buffer layer and the conversion boundary, \nthe field $m_{\\alpha\\beta\\gamma}$ that contains the number\nof walkers in each cell and is zero in the inner region, \nand the heat reservoir field $H_{\\alpha\\beta\\gamma}$, which\nis different from zero only in conversion cells. \nLet us comment on the size of the grids and the resulting\nmemory usage. The fine grid needs to be large enough \nto accommodate the dendrite and the liquid buffer layer \nduring the whole time of the simulation. Especially in \nthree dimensions, the restrictions on storage space make \nit necessary to fully use the fine grid.\nThe coarse grid needs to cover at least the same space\nregion as the fine grid. As will be detailed below,\nfor an efficient handling of the walkers close to the \nconversion boundary, it is desirable to always have some\nportion of coarse grid in front of the conversion boundary,\nand hence the coarse grid should actually cover a \nslightly larger region of space than the fine grid. \nSince the coarse grid has far less nodes than the fine \ngrid ($1$ node of coarse grid for $n^d$ nodes of fine grid), \nthis does not significantly increase the storage requirement.\nIn addition, we need an array to store the positions of\nthe walkers. The latter are represented by ``continuous'' \npositions and need no grid for their evolution. The\nwalkers can therefore leave the region of space where the \ngrids are defined and diffuse arbitrarily far away from the\ndendrite, allowing us to simulate growth into an infinite medium.\nThe most storage-intensive part is the fine grid. In fact,\nthe limiting factor for most of our three-dimensional\nsimulations is not so much computation time, but rather the\nstorage space needed to accommodate large dendrites.\n\nFinally, let us describe how the different parts of the \nalgorithm are connected. The program runs through the\nfollowing steps:\n\\begin{enumerate}\n\\item Setup (or update) the status field $S_{\\alpha\\beta\\gamma}$\non the coarse grid to fix the geometry of the conversion boundary \n\\item Calculate the temperature in each conversion cell and set\nthe boundary condition for the inner region on the fine grid\n\\item Timestep the phase-field equations on the fine grid and\ncalculate the heat flux between the inner region and the conversion\ncells\n\\item Update the heat reservoir variables $H_{\\alpha\\beta\\gamma}$\nand create or absorb walkers in the conversion cells\n\\item Advance the walkers\n\\item Repeat steps 2 through 5. From time to time, extract the\nshape of the dendrite and store it for future processing. If\nthe phase boundary has moved by more than a coarse cell size,\ngo back to step 1.\n\\end{enumerate}\n\nIn the following subsections, we will give more details on\nsome features of our implementation, such as the updating \nof the walkers, the updating of the geometry, the choice\nof parameters, and parallelization.\n\n\\subsection{Updating random walkers}\n\\label{secwalk}\n\\nobreak\n\\noindent\nBefore going into details, let us briefly point out\nsimilarities and differences between our method and\nother DMC algorithms. Such methods are widespread in\nQuantum Monte Carlo (QMC) calculations where they are used\nto solve the Schroedinger equation in imaginary time \\cite{Koonin}.\nEach walker represents a configuration in a usually\nhigh-dimensional Hilbert space, and the density of\nwalkers is proportional to the square amplitude of\nthe wave function. In contrast, in\nour method the walkers evolve in real space, and their\ndensity represents the temperature field.\nThe most important difference, however, is that in QMC\nall walkers are usually updated at the same time, whereas\nin our method some walkers are updated much more rarely\nthan others. Therefore, it would be very inefficient \nto visit every walker in each timestep. Instead, we \nwork with dynamical lists.\n\nTo simplify the bookkeeping of the different update times,\nwe enforce that updating takes place only at the discrete\ntimes when the fine grid is updated, i.e. for $t=i\\Delta t$,\n$i=1,2,\\ldots$. Then, we can make a list for every timestep\ncontaining all the walkers that have to be updated at that\nmoment. However, these lists greatly vary in length and\ncan therefore not easily be accommodated in standard arrays \nof variables. Therefore, we define a data structure that \ncontains the coordinates of one walker plus a pointer \nvariable. Within a given list, the pointer associated with one\nwalker indicates the next element of the list, or contains\nan end of list tag if the corresponding walker is the last\none of the list. An array of pointer variables indicates\nfor each timestep the first element of the corresponding\nlist. This array is the ``backbone'' of the list structure.\nIt is easy to add new walkers to a list: the pointer\nof the new walker is set to the former first element of\nthe list, and the pointer of the backbone is set to the\nnew walker (see Fig. \\ref{figwalk}). Lists of arbitrary \nlength can be constructed, and every walker is visited \nonly when it actually has to be updated.\n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=walkers.eps,width=.4\\textwidth}}\n\\caption{Sketch illustrating the configuration of the dynamical \nwalker lists. Each box stands for a walker, and the full arrows\nindicate pointer variables; the ``backbone'' array of pointers\nis represented by the downward arrow on the left. At time $t$, \nwalkers are updated and prepended to the lists corresponding to\ntheir next update time, as indicated by broken arrows.}\n\\label{figwalk}\n\\end{figure}\n%-------------------------------------------\n\nAt a given time $t$, the program works through the\ncorresponding list of walkers. The treatment of each \nwalker starts by looking up the status of the coarse \ngrid cell corresponding to its position. If the walker \nis inside a buffer cell because the conversion boundary \nhas moved since its last update, it is removed.\nThis removal does not violate heat conservation because\nthe heat associated with the walker \nis accounted for in the initialization of the temperature\nfield inside newly created buffer cells\n(see Sec. \\ref{secupdate}, Eq. (\\ref{initialnewbuff}) below).\nIf it is inside a conversion cell, and the corresponding \nreservoir variable $H_{\\alpha\\beta\\gamma}^t<-H_c$,\nthe walker is removed and $H_c$ is added to the \nreservoir. In all other cases, the jump distance\n$\\ell$ and corresponding time increment are \ndetermined and a new position is selected according\nto Eq. (\\ref{scheme}). To apply Eq. (\\ref{ldist}) \nfor the jump distance $\\ell$, we need to determine\nthe distance of a walker to the conversion boundary.\nIt would be very inefficient to calculate this distance\nfor each walker separately, especially when the shape\nof the boundary becomes complex. Therefore, we use the\nstatus field $S^t_{\\alpha\\beta\\gamma}$ on the coarse grid in \nthe outer region to store an approximate value for this \ndistance, which can then be easily looked up by each walker\nbefore a jump. Some more details are given in Sec. \\ref{secupdate}.\n\nAs mentioned above, we restrict the walker updating to\na discrete set of times. Therefore, the time increment \n$\\tau$ in Eq. (\\ref{scheme}) has to be an integer\nmultiple of the time step $\\Delta t$, which would not \nbe the case if we directly applied Eqs. (\\ref{ldist}) \nand (\\ref{taucond}). We solve this problem by defining \na lower cutoff for the jump distances,\n\\be\n\\ell_{\\rm min} = \\sqrt{2Dn_t\\Delta t}\n\\label{lmin}\n\\ee\nwhere $n_t$ is a fixed integer, and replace the jump \ndistances $\\ell$ found from Eq. (\\ref{ldist}) by\nthe closest integer multiple of $\\ell_{\\rm min}$.\nWe also define a maximum jump length $\\ell_{\\rm max}$,\nmainly to limit the size of the backbone pointer array:\nwith a maximum jump distance $\\ell_{\\rm max}$, each\nwalker is at least updated every $\\ell^2_{\\rm max}/( 2 D\\Delta t)$\ntimesteps. Consequently, the discrete time modulo this number\ncan be used to index the pointer variables in the backbone array.\n\nIt should be mentioned that in our list structure, it is\ndifficult to find a walker which is close to a given\nposition, because all sublists must be searched. This is \nimportant because the number of walkers in the conversion\ncells has to be known for the interfacing with the inner\nsolution. To avoid time-consuming sweeps through the walker\nlists, we update the walker number field $m^t_{\\alpha\\beta\\gamma}$\non the coarse grid whenever a walker jumps.\n\n\\subsection{Updating the geometry}\n\\label{secupdate}\n\\nobreak\n\\noindent\nWe now describe more in detail how the status field on the\ncoarse grid is setup and adapted to the changing geometry.\nWhen the dendrite grows, the configuration of the buffer layer \nand the conversion boundary has to change in order to maintain a \nconstant thickness $L_b$ of the buffer layer. Cells which are \npart of the outer region at the beginning of the simulation \nmay become conversion cells, then part of the buffer layer,\nand finally part of the dendrite. Under the conditions\nwe want to simulate, the crystal may locally melt\nback, but no large regions of space\nwill undergo the transition from solid to liquid, and\nhence we do not consider the inverse status change \n(from buffer to conversion cell, for example). \nTypically, at low undercoolings a readjustment of the \ngeometry becomes necessary only after $1000$ to $10000$\ntimesteps. Therefore, the efficiency requirements are\nnot as stringent as in the other parts of the program.\n\nThe procedure starts with a sweep through the fine grid. \nEvery cell of the coarse grid which contains at least one \nnode of the fine grid where $\\psi>0$ is assigned \nthe status ``solid'' ($S_{\\alpha\\beta\\gamma}^t=-2$).\nNext, the solid cells at the boundary of the dendrite\n(i.e. each solid cell which has at least one neighboring cell\nwhich is not solid) are used to define the buffer region:\nall cells with a center-to-center distance less than $L_b$ \nof a boundary cell which are not solid are assigned the \nstatus ``buffer'' ($S_{\\alpha\\beta\\gamma}^t=-1$). \nWhen a conversion cell or a cell of the outer \nregion becomes a buffer cell, we need to define the initial\nvalues of the two fields on the fine grid. The phase\nfield is set to its liquid value, $\\psi=-1$. The \ntemperature is calculated from the total heat \ncontained in the cell, taking into account both the \nwalkers and the heat reservoir variables in the conversion \ncells in order to ensure that the total amount of heat \nremains conserved, i.e.\n\\be\nu_{\\rm init}={\\Delta\\over M}\\left(m^t_{\\alpha\\beta\\gamma} + \n H^t_{\\alpha\\beta\\gamma}/H_c\\right).\n\\label{initialnewbuff}\n\\ee\nAll nodes of the fine grid within the new buffer cell \nare initially assigned this value. The walkers contained in \nthe cell are removed.\n\nAll cells of the outer region which are adjacent to the buffer, \ni.e. which have at least one neighbor with buffer status, are\nconversion cells ($S_{\\alpha\\beta\\gamma}^t=0$). When a cell \nof the outer region becomes a conversion cell, its heat \nreservoir variable is initialized at zero.\n\nFinally, in the outer region, which is comprised of all\nthe other cells, the status field is used to store an\napproximate value for the distance from the conversion\nboundary. A precise determination of this\ndistance is rather costly in computation time,\nbecause for each cell in the outer region, we \nmust calculate the distance to all conversion cells \nand retain the minimum value. A much cheaper, albeit \napproximate method is the following. As mentioned,\nin a conversion cell we have $S_{\\alpha\\beta\\gamma}^t=0$.\nWe assign to all cells adjacent to a conversion cell \nthe value $S_{\\alpha\\beta\\gamma}^t=1$. Neighbors\nof the latter receive the value $S_{\\alpha\\beta\\gamma}^t=2$,\nand we continue this process outward by assigning the value\n$S_{\\alpha\\beta\\gamma}^t=i+1$ to all cells adjacent\nto a cell with $S_{\\alpha\\beta\\gamma}^t=i$. For a relatively \nsimple geometry such as a single growing dendrite, the\nstatus field can be correctly set up on the whole lattice\nduring a single outward sweep, starting from the center\nof the dendrite. The number assigned to a given cell\ncan be used as a measure for the distance. Note that\nthe exact relationship of the number to the distance\ndepends on the direction with respect to the axes of\nthe coarse grid; our numerical tests below show, however, \nthat this anisotropy in the distance function does not\nsignificantly influence the dendrite shapes.\n\nIf we follow this procedure, the coarse grid needs to\ncover the entire region of space where the jump distance\nvaries. Even though we introduce a large-scale cutoff\n$\\ell_{\\rm max}$, this would become prohibitive in terms\nof memory usage for truly multi-scale problems. Fortunately,\nsuch a sophisticated scheme for the determination of the\ndistance is mainly needed close to the dendrite (for example,\na walker that enters in the space between two dendrite\narms needs to make small steps). Once a walker has left\nthe vicinity of the dendrite, this rather complicated\nestimate for the distance to the conversion boundary can \nbe replaced by a simpler one, for example the distance \nto the closest dendrite tip. In consequence, the coarse\ngrid needs to cover only a slightly larger region of\nspace than the fine grid.\n\nFinally, let us comment on the integration of the\nphase-field equations in the inner region. We need to\nknow which part of the fine grid must be timestepped.\nThis information is encoded in the status field\n$S_{\\alpha\\beta\\gamma}^t$ on the coarse grid.\nIt would, however, be rather inefficient in terms of \nmemory access time to integrate the inner region \n``coarse cell by coarse cell''. Instead, integration \nproceeds along the spatial direction\ncorresponding to successive memory locations, which is\nthe x-direction in our implementation. During the updating\nof the status field, the program determines for each $y$ and $z$\ncoordinate the range(s) to be integrated along $x$ and keeps\nthis information in a lookup table. This table is updated\nevery time the status field changes.\n\n\\subsection{Choice of computational parameters}\n\\nobreak\n\\noindent\nThere are number of parameters in our algorithm which\ncan be adjusted to maximize the computational efficiency.\nHowever, certain restrictions apply. Firstly, there are\nvarious length scales. In order of increasing magnitude,\nthose are:\n\\begin{enumerate}\n\\item the lattice spacing of the fine grid, $\\Delta x$,\n\\item the minimum jump length of the walkers, $\\ell_{\\rm min}$,\n\\item the size of a coarse-grained cell, $L_{cg}=n\\Delta x$, and\n\\item the buffer thickness $L_b$.\n\\end{enumerate}\nThe minimum jump length should be of the order of the\ninner grid spacing to assure a precise interfacing between\ninner and outer solutions. On the other hand, a larger\n$\\ell_{\\rm min}$ means less frequent walker updating.\nWe usually worked with $\\ell_{\\rm min}\\approx 2\\Delta x$,\nor $n_t\\approx 10$ in Eq. (\\ref{lmin}). On the other hand,\n$\\ell_{\\rm min}$ has to be smaller than $L_{cg}$ in order\nto achieve a well-defined coarse-graining. The \ncoarse-graining length, in turn, is limited by geometrical\nconstraints. The conversion boundary appears ``jagged'' on \nthe scale of $L_{cg}$ (see Fig. \\ref{figgrids}). In order\nto render the effects of this coarse geometry irrelevant\nfor the interface evolution, the buffer thickness must\nbe much larger than this scale, $L_b\\gg L_{cg}$. We\nfound that $L_{cg}\\approx 0.1\\,L_b$ is sufficient to\nachieve this goal. In our simulations, we mostly worked\nwith $n=4$ ($L_{cg}\\approx 2\\ell_{\\rm min}$) and $n=8$\nfor larger buffer sizes. \n\nNext, consider the constant of proportionality $c$ between\nthe walker jump length and the distance to the conversion\nboundary, $d_{cb}$. Since the Gaussian random vector $\\vec \\xi$\nin Eq. (\\ref{scheme}) has no cutoff, steps of arbitrary length\nare possible, and hence even a walker which is far away\ncan jump directly to the conversion boundary. The number\nof such events has to be kept small, because otherwise\nthe conversion process is influenced by the far field\nwith its coarse length and time scales. This goal can be\nnaturally achieved by choosing $c$ small enough. For \nexample, for $c=0.1$, only jumps with a length of more\nthan $10$ standard deviations can reach the conversion\nboundary, which represents a negligible fraction. On\nthe other hand, the increase of $\\ell$ with distance\ndetermines the efficiency of the algorithm, and hence\n$c$ should be chosen as large as possible. We usually\nworked with $c=0.1$, which seems to provide a good \ncompromise.\n\nFinally, the parameter $M$ determines the number of\nwalkers per coarse cell and hence the precision of the\nstochastic representation for the temperature field and\nthe diffusion equation. Considering Eq. (\\ref{coarsetemp}),\nwe see that the temperature at the boundary of the inner\nregion takes discrete values spaced by $\\Delta/M$. In addition,\nfor a homogeneous distribution of walkers in a system\nat $u=0$, the temperature fluctuations are of order\n$\\Delta/\\sqrt{M}$. On the other hand, increasing $M$ means\nlonger computation time because more random walks have\nto be performed. In addition, the total number of walkers $N$\nnecessary to simulate a dendrite of final volume $V$ is\n\\be\nN = {M V\\over (n\\Delta x)^d\\Delta},\n\\ee\nwhich means that high values of $M$ become prohibitive,\nespecially at low undercooling. Fortunately, a good precision\nof the solution can be obtained also by increasing $L_b$,\nas will be described in Sec. \\ref{sectests}. In practice,\nwe worked with values of $M$ ranging between $25$ and $100$.\n\n\\subsection{Boundary conditions and symmetries}\n\\nobreak\n\\noindent\nFor a two-dimensional dendrite seeded at the origin and\nwith arms growing along the $x$- and $y$-directions, the\nsimulations can be accelerated by taking advantage of the\ncubic symmetry. There are several symmetry axes, and\nconsequently it is sufficient to integrate the equations\nin a part of the plane while imposing reflective boundary\nconditions at the proper axes to enforce the symmetry.\nThese boundary conditions have to be imposed both on the\nfine grid and for the walkers. For the symmetry axes at \n$x=0$ and $y=0$, this can be easily achieved by choosing\none of the nodes of the coarse grid to coincide with the\norigin. Then, the two symmetry axes coincide with bonds\nin the coarse grid. On the fine grid, the nodes\noutside the simulation domain but adjacent to the boundary\nare set to the values of their mirror images inside the\nsimulation domain after each timestep. Walkers that attempt \nto cross the boundaries are reflected, i.e. instead of their\n``true'' final position outside the simulation domain, its\nmirror image with respect to the symmetry axis is chosen. \nAnother interpretation of this boundary condition for the \nwalkers is to imagine that there exists an ensemble of \n``mirror walkers'' which are the images of the walkers \ninside the simulation domain. When a walker jumps outside\nof the simulation domain, its mirror image jumps inside, \nand interchanging the walker and its mirror, we just \nobtain a reflection of the walker at the boundary as above.\n\n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=mirror.eps,width=.3\\textwidth}}\n\\smallskip\n\\caption{Sketch illustrating the implementation of reflecting\nboundary conditions at the symmetry axis $x=y$. Shown is a\ncell of the coarse grid (solid lines) on the diagonal $x=y$\n(dashed line). A walker inside the simulation domain ($x>0$, \n$0<y<x$) enters the cell. An accompanying ``mirror walker'' \n(open circles), the image of the walker with respect to the \nsymmetry axis, enters the same cell.}\n\\label{figmirror}\n\\end{figure}\n%-------------------------------------------\nThe latter view is useful when considering the last symmetry\naxis, the diagonal $x=y$. While the boundary conditions on\nthe fine grid and for the walkers can be implemented as before,\nthe conversion process requires special attention, because\nthe symmetry axis does not coincide with the boundaries\nof a coarse cell. When a walker enters a coarse cell situated\non the diagonal, there is an additional ``mirror walker''\nentering {\\em the same} coarse cell (see Fig. \\ref{figmirror}), \nand hence the number of walkers $m^t_{\\alpha\\beta\\gamma}$ \nhas to be increased by two \n(or, equivalently, decreased by two if a walker leaves the cell). \nSimilarly, walkers are created and absorbed in\npairs, which means that walker creation in such a cell can\noccur only when the heat reservoir exceeds twice the equivalent\nof one walker. In addition, when calculating the heat flux\nreceived by conversion cells on the diagonal, both the ``real''\nand the ``mirror'' flux has to be taken into account.\nIt is clear that this procedure induces an\nanisotropy in the conversion process; our tests showed,\nhowever, that its effect is undetectable for reasonable\nbuffer thickness.\n\nIn three dimensions, the reduction in computational resources\nis even more dramatic. For example, using the symmetry planes\n$y=0$, $x=y$, and $x=z$, i.e. integrating only the domain\n$x>0$, $0<y<x$, $z>x$, we need only integrate $1/48$ of the\nfull space, i.e. one eighth of one dendrite arm. The planes\n$x=y$ and $x=z$ can be handled as described above, with the\nexception of cells on the diagonal $x=y=z$. Such cells\nactually have only $1/6$ of their volume within the simulation\ndomain, and for each walker entering a cell, there are $5$\nmirror walkers to be considered.\n\n\\subsection{Parallelization}\n\\nobreak\n\\noindent\nEven though our algorithm is very efficient as will be shown\nbelow, the demands on computation time and RAM storage space\nrapidly increase when the undercooling is lowered. Therefore,\nwe have developed a parallel version of our code for the \nCray T3E at the National Energy Research Scientific Computing\nCenter (NERSC), using the shared memory library SHMEM.\n\nWe are mainly interested in the development of a single\nprimary dendrite branch. Hence, an efficient method\nof parallelization is to divide the simulation domain in\n``slices'' normal to the growth direction, and to distribute\nthe slices among the processors. In the inner region, the\nintegration of the partial differential equations makes it\nnecessary to exchange the boundary values between neighboring\nprocessors after each timestep. This is a standard procedure.\nThe more delicate points are the handling of the walkers\nand the updating of the geometry.\n\nEach processor stores only the parts of the fine grid \nit has to integrate, along with the values of the status\nfield in the whole simulation domain. The latter is\nnecessary to correctly handle the walkers. For the\nwalkers which are far from the dendrite, the average\njump distance may become much larger than the thickness\nof a computational slice. But if a walker approaches \nthe conversion boundary, the conversion process has to\nbe handled by the ``local'' processor which contains the\nappropriate part of the fine grid. Therefore, the walkers\nneed to be redistributed after their jumps. We have found\nit sufficient to implement ``exchange lists'' between\nneighboring processors, i.e. processors which contain\nadjacent parts of fine grid. If a walker jumps to a\nposition outside of the local slice, it is stored\nin one of two lists, corresponding to ``upward'' and \n``downward'' motion. After each timestep, these lists\nare exchanged between neighboring processors. As most\nof the walkers make several small steps before reaching\nthe conversion boundary, this procedure assures the\ncorrect redistribution of walkers with insignificantly \nfew errors, which arise in the rare case that a walker\narrives at the conversion boundary after several large \njumps.\n\nThe only step of the algorithm which needs massive exchange\nof data between the processors is the updating of the \ngeometry: each processor has to determine locally the\n``solid'' part of its computation domain, and this\ninformation has to be exchanged in order to correctly\nsetup the whole status field on each processor. However,\nas mentioned earlier, the geometry is updated only rarely,\nand therefore this part of the algorithm does not\nrepresent a significant computational burden. We have\nfound that the parallel version of our code showed\nsatisfactory execution time scaling when the number of \nprocessors is increased.\n\n\\section{Numerical tests}\n\\label{sectests}\n\\nobreak\n\\noindent\nThe accuracy of the standard phase-field method has been\nassessed in detail by comparison to boundary integral \nresults \\cite{Karma98}. Therefore, to test the stochastic\nalgorithm it is sufficient to check its results against\ndirect simulations of the standard deterministic \nphase-field equations. The most critical questions\nare whether the use of the rather coarse lattice for the \nconversion introduces spurious anisotropy, and what is\nthe magnitude of the temperature fluctuations generated\nby the stochastic treatment of the far field. The main\nparameters which control both of these effects are the\nthickness of the buffer layer and the number $M$ of walkers\ngenerated per coarse cell. The boundary condition\nfor the inner region is imposed on a coarse geometry\nwith a cutoff scale of $n\\Delta x$, and the temperature\nat the boundary is a stochastic variable which changes\nas walkers are created, absorbed, enter, or leave a \nconversion box, and which assumes discrete values spaced \nby $\\Delta/M$. When the buffer layer is much larger than \nthe size of a coarse cell, $L_b \\gg n\\Delta x$,\nthe field is ``smoothed out'' in space and time by the\ndiffusive dynamics. We expect high spatial and temporal\nfrequencies to decay rapidly through the buffer layer,\nand hence the evolution of the interface to become\nsmoother as $L_b$ is increased.\n\n%----------------table-------------------\n\\begin{table} \n\\caption{\\label{benchtable} Computational parameters for the\nbenchmark simulations in two dimensions.}\n\n\\begin{center}\n\\begin{tabular}{l|c|c}\nQuantity & Symbol & Value\\\\\n\\hline\nInterface thickness & $W_0$ & $1$ \\\\\nAnisotropy & $\\epsilon_4$ & $0.025666$, $0.000666$ \\\\\nEffective Anisotropy & $\\epsilon_4^e$ & $0.025$, $0.0$ \\\\\nRelaxation time & $\\tau_0$ & $1$ \\\\\nKinetic anisotropy & $\\delta_4$ & $0$ \\\\\nGrid spacing & $\\Delta x$ & $0.4$ \\\\\nTimestep & $\\Delta t$ & $0.003$ \\\\\nDiffusion coefficient & $D$ & $10$ \\\\\nCoupling constant & $\\lambda$ & $15.957$ \\\\\nCapillary length & $d_0$ & $0.0554$ \\\\\nKinetic coefficient & $\\beta_0$ & $0$ \\\\\nUndercooling & $\\Delta$ & $0.3$ \\\\\nCoarse cell size & $n$ & $4$ \\\\\nNumber of walkers per coarse cell & $M$ & $50$ \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n%----------------------------------------\n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=bench1.eps,width=.35\\textwidth}}\n\\smallskip\n\\centerline{\n\\psfig{file=bench2.eps,width=.35\\textwidth}}\n\\caption{Comparison of standard (deterministic) phase-field\nand random walker method in two dimensions for $\\Delta = 0.3$ \nand $\\epsilon_4=0.025$. (a) Dendrite shapes, represented\nby the contour line $\\phi=0$, after 200000\niterations, (b) tip velocity versus time.}\n\\label{figtest2d}\n\\end{figure}\n%-------------------------------------------\nWe conducted two-dimensional simulations at an intermediate\nundercooling, $\\Delta=0.3$. At this value of $\\Delta$, the\nstandard phase-field method can still be used to simulate\nnon-trivial length and time scales of dendritic evolution,\nbut the length scale of the diffusion field is large enough\nto provide a serious test for the random walker method,\ni.e. the diffusion length is much larger than the thickness\nof the buffer layer. Table \\ref{benchtable} shows the \ncomputational parameters that were used for these tests.\nOnly the first quadrant was simulated,\nwith reflecting boundary conditions at $x=0$ and $y=0$.\nFig. \\ref{figtest2d}(a) shows a comparison of dendrite\nshapes obtained from the standard phase-field and from\nour algorithm with different buffer sizes. While the\nshapes slightly differ for $L_b/\\Delta x = 20$, the\ncurve for $L_b/\\Delta x = 40$ is almost undistinguishable \nfrom the deterministic shape. Fig. \\ref{figtest2d}(b)\nshows the velocity of the dendrite tip along the \n$x$-direction, measured over periods of 500 iterations,\nversus time. The fluctuations around the deterministic\nvalue are much larger for $L_b/\\Delta x = 20$ than for\n$L_b/\\Delta x = 40$, and for $L_b/\\Delta x = 80$ (not\nshown) the curve obtained from the stochastic method\nis very close to the deterministic data. For comparison,\nthe diffusion length $2D/v$ at the end of the run is\nabout $400\\,\\Delta x$.\n\n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=bench3.eps,width=.3\\textwidth}}\n\\caption{Comparison of ``dendrite'' shapes without fourfold\nanisotropy after 500000 iterations.}\n\\label{figcirc}\n\\end{figure}\n%-------------------------------------------\nA particularly sensitive test for the anisotropy of the\nconversion process is the growth of a circular germ \nwithout anisotropy, because such a germ\nis unstable against even smallest perturbations. This\ncan be clearly seen from Fig. \\ref{figcirc}: even\nthough we completely screen the fourfold anisotropy\ncreated by the lattice ($\\epsilon_4^e = 0$), the weak\nnext harmonic of the lattice anisotropy, proportional\nto $\\cos 8\\theta$, destabilizes the circle and leads\nto the formation of bulges in the $(21)$- and \n$(12)$-directions. For $L_b/\\Delta x = 80$, the stochastic\nalgorithm perfectly reproduces this trend, and we can\nhence conclude that the anisotropy created by the\ncoarse structure of the conversion boundary is \nnegligibly small. Note that the diffusion field\nextends to a distance of more than 1000 lattice units at \nthe end of this run, which means that the larger \npart of the simulation domain is integrated by the\nstochastic method.\n\n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=fluct.eps,width=.35\\textwidth}}\n\\caption{Variance of temperature fluctuations, \n$\\left<u^2\\right>$, as a function of the distance \nfrom the conversion boundary for two values of\nthe walker parameter $M$.}\n\\label{figfluct}\n\\end{figure}\n%-------------------------------------------\nTo quantify the numerical noise, we performed 2-d\nsimulations of the simple diffusion equation in a \nsystem of $N\\times N$ lattice sites with $N=160$. \nOne half of the system ($x<0$) was integrated by the\nstochastic algorithm, whereas in the other half ($x>0$)\nwe used a standard Euler algorithm. The conversion\nboundary $\\Gamma'$ hence coincides with the $y$-axis,\nand there is a single column of conversion cells\nalong this axis. We used $\\Delta x=1$,\n$\\Delta t=0.02$, $D=1$, $\\Delta=1$, and applied no-flux \nboundary conditions at $x=\\pm N/2$ and periodic boundary\nconditions along $y$. The system was initialized \nat $u=0$ everywhere, i.e. in the walker region we \nrandomly placed $M$ walkers in each coarse cell. \nWhen the walkers evolve, fluctuations are created\nin the deterministic region, which plays the role of\nthe buffer layer. We recorded $u^2$ as a function\nof $x$ and averaged over a time which is long compared\nto the diffusive relaxation time of the system, $N^2/D$.\nThe results for two different choices of the walker\nparameter $M$ are shown in Fig. \\ref{figfluct}. \nIn an infinite homogeneous system filled \nwith walkers, the distribution of the number\nof walkers in a given coarse cell is Poissonian,\nwhich means that the fluctuations in the walker\nnumbers are of order $\\sqrt M$. If this scaling\nremains valid for the conversion cells in our\nhybrid system, we expect $\\left<u^2\\right> \\sim 1/M$ \nclose to the conversion boundary, which is indeed \nwell satisfied. As shown in Fig \\ref{figfluct},\nthe variance of the temperature fluctuations\nrapidly decreases with the distance from the\nconversion layer -- by four orders of magnitude\nover the distance of $80$ lattice sites. No\nsimple functional dependence of $\\left<u^2\\right>$\non $x$ is observed. We expect high spatial and\ntemporal frequencies to be rapidly damped. A theoretical\ncalculation of $\\left<u(x)^2\\right>$ seems possible\nbut non-trivial because the random variables which\nare the sources of the fluctuations in the deterministic\nregion are correlated in space and time by the exchange\nof walkers through the stochastic region and the\ndiffusion of heat through the deterministic region.\nFor our present purpose, we can draw two important\nconclusions. Firstly, for a reasonable thickness of\nthe buffer layer, fluctuations are damped by several\norders of magnitude. The residual fluctuations are\nmuch smaller than the thermal fluctuations represented\nby Langevin forces that have to be introduced in\nthe equations of motion to observe a noticeable\nsidebranching activity \\cite{Karma99}. Indeed,\nfor sufficiently large buffer layers we always\nobserve needle crystals without sidebranches. \nSecondly, the fluctuations at the solid-liquid\ninterface can be reduced both by increasing the\nnumber of walkers and by increasing the thickness\nof the buffer layer, which allows to accurately\nsimulate dendritic evolution with a reasonable\nnumber of walkers.\n\n%----------------table-------------------\n\\begin{table} \n\\caption{\\label{timetab} Execution times \nof the benchmark simulations for various sets of\ncomputational parameters.}\n\n\\begin{center}\n\\begin{tabular}{l|c|c|c}\n & $M$ & $L_b/\\Delta x$ & CPU time (min) \\\\\n\\hline\nDeterministic & -- & -- & $1950$ \\\\\n & $50$ & $20$ & $89$ \\\\\n & $50$ & $40$ & $110$ \\\\\n & $100$ & $40$ & $119$ \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n%----------------------------------------\nIn Table \\ref{timetab}, we compare the run\ntimes of our code on a DEC Alpha 533 MHz workstation along\nwith the run time of the deterministic phase\nfield reference simulation. The gain in computational\nefficiency is obvious. Increasing the buffer layer\nfrom $L_b=20\\,\\Delta x$ to $L_b=40\\,\\Delta x$\nreduces the amplitude of the temperature fluctuations\nat the solid-liquid interface by more than an order of\nmagnitude, whereas the computation time increases by\nonly $25$\\%. Comparing the runs with different\nvalues of $M$, we see that the walker part of the\nprogram accounts only for a small part of the total\nrun time.\n\nFrom these results, we can conclude that the computational\neffort that has to be invested to simulate a given time\nincrement scales approximately as the size of the fine\ngrid region, i.e. as the size of the dendrite. This is a\nmajor advance with respect to the standard phase-field implementation\non a uniform grid,\nwhere the computation time scales with the volume enclosing the\ndiffusion field. The spatial and temporal scales of\ndendritic evolution that can be simulated with our\nmethod are hence limited by the integration of the \nphase-field equations on the scale of the dendrite.\n\nAll the data shown so far are for two-dimensional\nsimulations. We repeated similar tests in three \ndimensions and obtained comparable results for the \nquality of the solution and the efficiency of the \ncode. We will not display the details of these \ncomparisons here, but rather show an example of\na three-dimensional simulation under realistic\nconditions to demonstrate that our method is capable \nof yielding quantitative results in a regime of\nparameters that was inaccessible up to now.\nIn Fig. \\ref{figsnap}, we show snapshot pictures\nof a three-dimensional dendrite growing at an\nundercooling of $\\Delta=0.1$ and for a surface\ntension anisotropy $\\epsilon_4^e=0.025$, which is the\nvalue measured for PVA \\cite{Muschol92}. The other\ncomputational parameters are $W_0=1$, $\\Delta x=0.8$, \n$\\epsilon_4=0.0284$, $\\tau_0=0.965$, $\\delta=0.0364$, \n$n=4$, $M=50$, $L_b/\\Delta x = 48$, $D=24$, $\\Delta t=0.004$, \nand $\\lambda=39.6$ (giving $d_0 = 0.0223$, $\\beta_0=0$),\nand the simulation was started from a homogeneously\nundercooled melt with an initial solid germ of radius\n$r=2\\Delta x$ centered at the origin.\n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=snaprv.eps,width=.4\\textwidth}}\n\\smallskip\n\\caption{Snapshots of a three-dimensional dendrite at\n$\\Delta=0.1$ after 60000, 120000, 200000, 300000,\nand 650000 timesteps (from top left).}\n\\label{figsnap}\n\\end{figure}\n%-------------------------------------------\nDuring the run, we recorded the velocity $v(t)$ and\nthe radius of curvature $\\rho(t)$ of the dendrite\ntip. The latter was calculated using the method \ndescribed in Ref. \\cite{Karma98}. With these two\nquantities, we can calculate the time-dependent\ntip selection parameter\n\\be\n\\sigma^*(t) = {2Dd_0\\over \\left[\\rho(t)\\right]^2 v(t)}.\n\\ee\nThe results are shown in Fig. \\ref{figplot}.\nIn the initial stage during which the arms emerge\nfrom the initial sphere, growth is very rapid. \nSubsequently, the tips slow down while the \ndiffusion field builds up around the crystal.\n%-------------------------------------------\n\\begin{figure}\n\\centerline{\n\\psfig{file=sigplot.eps,width=.4\\textwidth}}\n\\caption{Tip velocity, tip radius and selection parameter\nversus time for the run of Fig. \\protect\\ref{figsnap}.\nArrows mark the times of the snapshot pictures.\nLength and time are rescaled by $d_0$ and $d_0^2/D$,\nrespectively. The steady-state velocity $v_ss$ was\ncalculated using a boundary integral \nmethod \\protect\\cite{Lee99}.}\n\\label{figplot}\n\\end{figure}\n%-------------------------------------------\nAt the end of the run, the velocity has almost converged\nto a constant value that is in excellent quantitative\nagreement with the velocity predicted by the boundary integral\nsolution of the sharp-interface steady-state growth equations \nassuming an axisymmetric surface energy and tip shape (i.e. the \nmost accurate numerical implementation of solvability theory \nto date \\cite{Lee99}). This velocity is also \nin reasonably good agreement with the velocity predicted \nby the linearized solvability theory of Barbieri and Langer\n\\cite{Barbieri89}, even though the actual tip radius in both \nthe phase-field simulation and the boundary integral calculation \ndiffer from the tip radius of the paraboloidal shape assumed \nin this theory. A more detailed discussion of \nthis point and the entire steady-state tip morphology \ncan be found in Ref. \\cite{Tip99}.\n\nRemarkably, the selection constant $\\sigma^*$ becomes\nalmost constant long before the velocity and the tip\nradius have reached their steady-state values. This\nis in good agreement with the concepts of solvability\ntheory, which stipulates that the selection of the\ntip parameters is governed by the balance between\nthe anisotropic surface tension and the local diffusion\nfield at the tip. To establish the correct local balance,\ndiffusion is necessary only over a distance of a few\ntip radii, whereas the buildup of the complete\ndiffusion field around an arm requires heat transport\nover the scale of the diffusion length, $D/v$. Our\nsimulation shows that $\\sigma^*$ indeed becomes essentially\nconstant soon after the formation of the primary arms.\nThis fact can be used to derive scaling laws for the\nevolution of the dendrite arms at low undercooling\nduring the transient that leads to steady-state\ngrowth \\cite{Plapp99}. Finally, even at the end of\nthe simulations, where the dendrite arms are well\ndeveloped, no sidebranches are visible. We repeated\nthe same simulation for different thickness of the\nbuffer layer, and observed no changes in the morphology.\nTiny ripples can in fact be seen close to the base of\nthe dendrite shaft, but the amplitude of these\nperturbations does not depend on the noise strength.\nWe therefore believe that this is rather a deterministic\ninstability due to the complicated shape of the\ndendrite base that a beginning of noise-induced \nsidebranching. In summary, there are at present no\nindications that the noise created by the walkers\nhas a noticeable effect on the morphological evolution.\n\n\\section{Conclusions}\n\\nobreak\n\\noindent\nWe have presented a new computational approach for\nmulti-scale pattern formation in solidification.\nThe method is efficient, robust, precise, easy to implement in \nboth two and three dimensions, and parallelizable. Hence, it \nconstitutes a powerful alternative to state of the art\nadaptive meshing and finite element techniques. We have\nillustrated its usefulness by simulating dendritic growth \nof a pure substance from its undercooled melt\nin an infinite geometry. Due to the fact that only\na very limited amount of ``geometry bookkeeping''\nis required, our method can be easily adapted to other\nexperimental settings, such as directional solidification.\nIn addition, the DMC algorithm is not limited to the\npresent combination with the phase-field method, but \ncan be used in conjunction with any method to solve \nthe interface dynamics, as long as the diffusion equation \nis explicitly solved. The adaptation of our method to other\ndiffusion-limited free boundary problems is straightforward;\nproblems with several diffusion fields can be handled by\nintroducing multiple species of walkers.\n\nIn view of the results presented here,\nthere is a realistic prospect for direct simulations\nof solidification microstructures for experimentally\nrelevant control parameters. An especially interesting prospect \nis to combine our method with a recently developed approach to\n{\\it quantitatively} incorporate thermal fluctuations \\cite{Karma99}\nin the phase-field model. Such an extension should make it possible\nto test noise-induced sidebranching theories \\cite{Langer87,Brener95} \nin three dimensions and for an undercooling range where detailed \nmeasurements of sidebranching characteristics are available \n\\cite{Huang81,Bisang95,Dougherty87,Li98}. \nIf thermal noise in the liquid region outside the\nbuffer layer turns out to be unimportant for sidebranching, \nthe straightforward addition of Langevin forces as in \nRef. \\cite{Karma99} in the finite-difference region \n(i.e. the buffer region plus the solid) should suffice for\nthis extension. In contrast, if the noise from this region is\nimportant, a method to produce the correct level of noise in \nthe walker region will need to be developed. Work concerning \nthis issue is currently in progress. \n\nTo conclude, let us comment on some possible extensions and\nimprovements of our method which will be necessary to address \ncertain questions. Firstly, we have described the method\nhere using an explicit integration scheme on the fine grid in \nthe inner region, which enforces rather small time steps. We \nalso tested an alternating direction Crank-Nicholson scheme\nin 2-d, which speeds up the calculations but makes it\nnecessary to introduce corrective terms at the conversion\nboundary to guarantee the local heat conservation. \nSecondly, for the moment we use the stochastic algorithm \nonly at the {\\it exterior} of the dendrite; for other\ngeometries, such as directional solidification where\nthe volumes of solid and liquid are comparable, it might \nbe useful to introduce a second stochastic region in the \nsolid. It would also be desirable to combine our algorithm \nwith more efficient memory managing techniques to overcome\nthe limitations due to storage space. Finally, a\ncompletely open question is whether it is possible to\ncombine our stochastic algorithm with a suitable method\nfor simulating hydrodynamic equations. This would open\nthe way for studies of the influence of convection on dendritic\nevolution at low undercooling, thereby extending in a non-trivial\nway recent studies that have been restricted \nto a relatively high undercooling regime \\cite{Crisandme}.\n\n\\acknowledgments\n\nThis research was supported by U.S. DOE Grant \nNo. DE-FG02-92ER45471 and benefited from \ncomputer time at the National Energy Research \nScientific Computing Center (NERSC) at\nLawrence Berkeley National Laboratory and\nthe Northeastern University Advanced Scientific\nComputation Center (NU-ASCC). We thank Youngyih\nLee for providing the boundary integral results\nand Flavio Fenton for his help with the \nvisualization. Fig. \\ref{figsnap} was created\nusing Advanced Visual Systems' AVS.\n\n\\appendix\n\n\\section{Phase-field method}\n\\nobreak\n\\noindent\nWe will briefly outline the main features of the\nphase-field method used for our simulations. More\ndetails can be found in Ref. \\cite{Karma98}.\n\nThe starting point is the free energy functional,\nEq. (\\ref{freeen}), together with the equations\nof motion for the phase field and the temperature\nfield, Eqs. (\\ref{pfpsi}) and (\\ref{pfu}). The\nfree energy density in Eq. (\\ref{freeen}) is\nchosen to be of the form\n\\be\nf(\\psi,u)=-{\\psi^2\\over 2}+{\\psi^4\\over 4}+\\lambda\\,u\\,\\psi\\,\n\\left(1-2{\\psi^2\\over 3}+{\\psi^4\\over 5}\\right).\n\\label{freeden}\n\\ee\nThis function has the shape of a double well, with minima \nat $\\psi = \\pm 1$ corresponding to the solid and the liquid\nphases, respectively. Here, $u$ is the dimensionless \ntemperature field, $\\lambda$ is a dimensionless coupling\nconstant, and the term proportional to $u$ on the RHS of \nEq. (\\ref{freeden}) ``tilts'' the double well in order to favor \nthe solid (liquid) minimum when the temperature is below (above)\nthe melting temperature. The coefficient $W(\\hat n)$ of\nthe gradient term in the free energy (\\ref{freeen}) determines \nthe thickness of the diffuse interface, i.e. the scale on\nwhich the phase field varies rapidly to connect the\ntwo equilibrium values. In addition, $W$ is related to\nthe surface tension, and exploiting its dependence on the\norientation of the interface allows to recover the \nanisotropic surface tension of Eq. (\\ref{ani}) by choosing\n\\be\nW(\\hat n) = W_0 {\\gamma(\\hat n)\\over \\gamma_0}.\n\\label{wthet}\n\\ee\nThe orientation $\\hat n$ is defined in terms of the phase field by\n\\be\n\\hat n = {\\vnab \\psi\\over | \\vnab\\psi|}\\,.\n\\ee\nNote that this dependence of $W$ on $\\psi$ has to be taken\ninto account in performing the functional derivative, such\nthat the explicit form of Eq. (\\ref{pfpsi}) becomes\n\\begin{eqnarray}\n\\tau(\\hat n)\\partial_t\\psi & = & \n [\\psi - \\lambda u(1-\\psi^2)](1-\\psi^2) + \n \\vnab\\cdot[W(\\hat n)^2\\vnab\\psi] \\nonumber \\\\\n & & \\mbox{} + \\partial_x \\left({|\\vnab\\psi|}^2 W(\\hat n)\n {\\partial W(\\hat n) \\over \\partial(\\partial_x \\psi)}\\right) \\nonumber \\\\\n & & \\mbox{} + \\partial_y \\left({|\\vnab\\psi|}^2 W(\\hat n)\n {\\partial W(\\hat n) \\over \\partial(\\partial_y \\psi)}\\right) \\nonumber \\\\\n & & \\mbox{} + \\partial_z \\left({|\\vnab\\psi|}^2 W(\\hat n)\n {\\partial W(\\hat n) \\over \\partial(\\partial_z \\psi)}\\right).\n\\label{pfnum}\n\\end{eqnarray}\n\nNext, we need to specify the orientation-dependent \nrelaxation time $\\tau(\\hat n)$ of the phase-field.\nIn analogy with Eqs. (\\ref{wthet}) and (\\ref{ani})\nwe choose\n\\be\n\\tau(\\hat n) = \\tau_0\\,(1-3\\delta_4)\\,\n\\left[1+\\frac{4\\delta_4}\n{1-3\\delta_4}\\left(n_x^4+n_y^4+n_z^4\\right)\\right],\n\\label{anikin}\n\\ee\nwhere $\\delta_4$ is the kinetic anisotropy.\n\nThe phase-field equations can be related to the original\nfree boundary problem by the technique of matched asymptotic\nexpansions. Details on this procedure can be found in\nRef. \\cite{Karma98}. As a result, we obtain expressions\nfor the capillary length and the kinetic coefficient\nin terms of the phase-field parameters $W_0$ and \n$\\tau(\\hat n)$:\n\\be\nd_0 = {a_1W_0\\over \\lambda}\n\\label{d0}\n\\ee\n\\be\n\\beta(\\hat n) = {a_1\\over\\lambda}{\\tau(\\hat n)\\over W_0}\n \\left(1-a_2 \\lambda{W(\\hat n)^2\\over D\\tau(\\hat n)}\\right),\n\\label{beta}\n\\ee\nwhere $a_1 = 0.8839$ and $a_2 = 0.6267$ are numerical \nconstants fixed by a solvability condition.\nThere is an important difference between this result and\nearlier matched asymptotic expansions of the phase-field\nequations, due to a different choice of the expansion\nparameter. If the coupling constant $\\lambda$ is used\nas the expansion parameter, the first order\nin $\\lambda$ gives only the first term in \nEq. (\\ref{beta}), while the complete expression\nis the result of an expansion to first order\nin the interface P{\\'e}clet number, which is defined\nas the ratio of the interface thickness and a relevant\nmacroscopic scale of the pattern (local radius of curvature\nor diffusion length). An important consequence of\nEq. (\\ref{beta}) is that the kinetic coefficient and\nits anisotropy can be set to arbitrary values by a \nsuitable choice of $\\lambda$ and $\\tau(\\hat n)$,\nand in particular we can achieve vanishing kinetics\n($\\beta(\\hat n) = 0$). Note that for a $\\tau(\\hat n)$\nas given by Eq. (\\ref{anikin}), the kinetic coefficient\ncannot be made to vanish simultaneously in all directions,\nbut for small anisotropies choosing $\\delta_4=2\\epsilon_4$\nis a sufficiently accurate approximation.\nFurthermore, the ratio $d_0/W_0$\ncan be decreased without changing the kinetics by \nsimultaneously increasing $\\lambda$ and the diffusivity $D$. \nThis method dramatically increases the computational\nefficiency of the phase-field approach, because the\ninterface width $W_0$ determines the grid spacing which\nmust be used for an accurate numerical solution. For\na physical system with fixed capillary length $d_0$,\nthe number of floating point operations necessary to\nsimulate dendritic evolution for some fixed time interval\nand system size scales $\\sim(d_0/W_0)^{d+3}$ for the choice\nof phase-field parameters where the interface kinetics vanish\n(i.e. $D\\tau/W_0^2 \\sim \\lambda \\sim W_0/d_0$), where $d$ is the \nspatial dimension \\cite{Karma98}.\n\nWe integrate the phase-field equations on a cubic grid \nwith spacing $\\Delta x$, All spatial derivatives\nare discretized using $(\\Delta x)^2$-accurate finite \ndifference formulas, and timestepping is performed by a \nstandard Euler algorithm. The use of a regular grid\ninduces small anisotropies in the surface tension and\nthe kinetic coefficient. These effects have been precisely\nquantified in Ref. \\cite{Karma98}. Since the grid has the same \nsymmetry as the crystal we want to simulate, the presence of\nthe lattice simply leads to small shifts in the surface\ntension anisotropy and in the kinetic parameters. \nFor example, we obtain an effective \nsurface tension anisotropy $\\epsilon_4^e$ which is slightly \nsmaller than the ``bare'' value $\\epsilon_4$. Evidently,\nthe use of this method restricts the simulation to crystals\nwith symmetry axes aligned to the lattice, but this is\nnot a severe limitation in the present study which focuses \non the growth of single crystals.\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Kurz} W. Kurz and D.~J. Fisher, \n ``Fundamentals of solidification'', \n Trans Tech, Aedermannsdorf, Switzerland (1992).\n\n\\bibitem{eldep} M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo,\n and Y. Sawada, \n ``Fractal structures of zinc metal leaves grown by electrodeposition'',\n Phys. Rev. Lett. {\\bf 53}, 286 (1984).\n\n\\bibitem{BenJacob} E. Ben-Jacob, H. Shmueli, O. Shochet, and\n A. Tenenbaum, \n ``Adaptive self-organization during growth of bacterial colonies'',\n Physica A {\\bf 187}, 378 (1992).\n\n\\bibitem{Braun97} R.~J.~Braun and M.~T.~Murray,\n J.~Cryst. Growth {\\bf 174}, 41 (1997).\n\n\\bibitem{Schmidt96} A.~Schmidt:\n ``Computations of Three-Dimensional Dendrites with Finite Elements'',\n J.~Comp. Phys. {\\bf 125}, 293 (1996).\n\n\\bibitem{Provatas99} N.~Provatas, N.~Goldenfeld, and J.~Dantzig,\n ``Efficient computation of dendritic microstructures using adaptive\n mesh refinement'',\n Phys. Rev. Lett. {\\bf 80}, 3308 (1998),\n ``Adaptive Mesh Refinement Computation of Solidification\n Microstructures Using Dynamic Data Structures'',\n J.~Comp. Phys. {\\bf 148}, 265 (1999).\n\n\\bibitem{Koonin} See for example S.~E. Koonin and D. C. Meredith,\n ``Computational Physics'', \n Addison-Wesley, Redwood City (1992), chapter 8.\n\n\\bibitem{Kadanoff85} L.~P.~Kadanoff, \n ``Simulating Hydrodynamics: A Pedestrian Model'',\n J.~Stat.~Phys. {\\bf 39}, 267 (1985).\n\n\\bibitem{Liang86} S.~Liang,\n ``Random-walk simulations of flow in Hele Shaw cells'',\n Phys.~Rev.~A {\\bf 33}, 2663 (1986).\n\n\\bibitem{Vicsek84} T.~Vicsek,\n ``Pattern Formation in Diffusion-Limited Aggregation'',\n Phys. Rev. Lett. {\\bf 53}, 2281 (1984).\n\n\\bibitem{Nittmann86} J.~Nittmann and H.~E.~Stanley,\n ``Tip splitting without interfacial tension and dendritic growth \n patterns arising from molecular anisotropy'',\n Nature {\\bf 321}, 663 (1986).\n\n\\bibitem{Karma87} A. Karma,\n ``Beyond Steady-State Lamellar Eutectic Growth'',\n Phys. Rev. Lett. {\\bf 59}, 71 (1987).\n\n\\bibitem{Saito89} Y. Saito and T. Ueta,\n ``Monte Carlo studies of equilibrium and growth shapes of a crystal'',\n Phys. Rev. A {\\bf 40}, 3408 (1989)\n\n\\bibitem{Meakin83} P. Meakin,\n ``Diffusion-controlled cluster formation in 2--6-dimensional space'',\n Phys. Rev. A {\\bf 27}, 1495 (1983).\n\n\\bibitem{Brener96} E. Brener, H. M{\\\"u}ller-Krumbhaar, D. Temkin,\n ``Structure formation and the morphology diagram of possible \n structures in two-dimensional diffusional growth'',\n Phys. Rev. E {\\bf 54}, 2714 (1996).\n\n\\bibitem{Langer87} J.~S. Langer,\n ``Dendritic sidebranching in the three-dimensional symmetric model \n in the presence of noise''\n Phys. Rev. A {\\bf 36}, 3350 (1987).\n\n\\bibitem{Plapp99} M. Plapp and A. Karma,\n ``Multiscale random-walk algorithm for simulating interfacial \n pattern formation'',\n cond-mat/9906370\n% Phys. Rev. Lett., in press.\n\n\\bibitem{Tip99} A. Karma, Y. Lee, and M. Plapp,\n ``Three-dimensional dendrite tip morphology at low undercooling'',\n cond-mat/9909021\n% Phys. Rev. E, in press.\n\n\\bibitem{Langer80} J.~S.~Langer,\n ``Instabilities and pattern formation in crystal growth'',\n Rev. Mod. Phys. {\\bf 52}, 1 (1980).\n\n\\bibitem{Kessler88} D.~A.~Kessler, J.~Koplik, and H.~Levine,\n ``Pattern selection in fingered growth phenomena'',\n Adv. Phys. {\\bf 37}, 255 (1988).\n\n\\bibitem{Brener91} E.~A.~Brener and V.~I.~Mel'nikov,\n ``Pattern selection in two-dimensional dendritic growth'',\n Adv. Phys. {\\bf 40}, 53 (1991).\n\n\\bibitem{Huang81} S.-C. Huang and M.~E. Glicksman,\n ``Fundamentals of Dendritic Solidification I. Steady-State Tip Growth'',\n Acta Metall. {\\bf 29}, 701 (1981).\n\n\\bibitem{Rubinstein91} E.~R.~Rubinstein and M.~E.Glicksman,\n ``Dendritic growth kinetics and structure I. Pivalic acid'',\n J. Cryst. Growth {\\bf 112}, 84 (1991).\n \n\\bibitem{Muschol92} M.~Muschol, D.~Liu, and H.~Z.~Cummins,\n ``Surface-tension-anisotropy measurements of succinonitrile and \n pivalic acid: Comparison with microscopic solvability theory'',\n Phys. Rev. A {\\bf 46}, 1038 (1992).\n\n\\bibitem{Glicksman94} M.~E.~Glicksman, M.~B.~Koss, and E.~A.~Winsa,\n ``Dendritic Growth Velocities in Microgravity'',\n Phys. Rev. Lett. {\\bf 73}, 573 (1994).\n\n\\bibitem{Bisang95} U.~Bisang and J.~Bilgram,\n ``Shape of the tip and the formation of sidebranches of xenon dendrites'',\n Phys. Rev. E {\\bf 54}, 5309 (1996).\n\n\\bibitem{Sethian92} J.~A.~Sethian and J.~Strain,\n ``Crystal Growth and Dendritic Solidification'',\n J. Comp. Phys. {\\bf 98}, 231 (1992).\n\n\\bibitem{Ihle94} T.~Ihle and H.~M{\\\"u}ller-Krumbhaar,\n ``Fractal and compact growth morphologies in phase transitions \n with diffusion transport'',\n Phys. Rev. E {\\bf 49}, 2972 (1994).\n\n\\bibitem{Karma98} A.~Karma and W.-J. Rappel,\n ``Quantitative phase-field modeling of dendritic growth in\n two and three dimensions'',\n Phys. Rev. E {\\bf 57}, 4323 (1998).\n\n\\bibitem{pf} For a brief historic overview and a comprehensive list\n of references concerning the phase-field method, \n see Ref. \\cite{Karma98}.\n\n\\bibitem{Karma99} A.~Karma and W.-J.~Rappel,\n ``Phase-Field Model of Dendritic Sidebranching with Thermal Noise'',\n cond-mat/9902017 (1999).\n\n\\bibitem{Lee99} Youngyih Lee, private communication.\n\n\\bibitem{Barbieri89} A. Barbieri and J.~S.~Langer, \n ``Prediction of dendritic growth rates in the linearized \n solvability theory'',\n Phys. Rev. A {\\bf 39}, 5314 (1989).\n\n\\bibitem{Brener95} E. Brener and D. Temkin,\n ``Noise-induced sidebranching in the three-dimensional \n nonaxisymmetric dendritic growth'',\n Phys. Rev. E {\\bf 51}, 351 (1995).\n\n\\bibitem{Dougherty87} A. Dougherty, P.~D. Kaplan, and J.~P. Gollub,\n ``Development of Side Branching in Dendrite Crystal Growth'',\n Phys. Rev. Lett. {\\bf 58}, 1652 (1987)\n\n\\bibitem{Li98} Q. Li and C. Beckermann,\n ``Scaling behavior of three-dimensional dendrites'',\n Phys. Rev. E {\\bf 57}, 3176 (1998)\n\n\\bibitem{Crisandme} C. Beckermann, H.-J. Diepers, I. Steinbach, \n A. Karma, and X. Tong, \n ``Modeling Melt Convection in Phase-Field Simulations of Solidification'',\n J. Comp. Phys. {\\bf 154}, 468-496 (1999).\n\n\\end{thebibliography}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002015.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Kurz} W. Kurz and D.~J. Fisher, \n ``Fundamentals of solidification'', \n Trans Tech, Aedermannsdorf, Switzerland (1992).\n\n\\bibitem{eldep} M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo,\n and Y. Sawada, \n ``Fractal structures of zinc metal leaves grown by electrodeposition'',\n Phys. Rev. Lett. {\\bf 53}, 286 (1984).\n\n\\bibitem{BenJacob} E. Ben-Jacob, H. Shmueli, O. Shochet, and\n A. Tenenbaum, \n ``Adaptive self-organization during growth of bacterial colonies'',\n Physica A {\\bf 187}, 378 (1992).\n\n\\bibitem{Braun97} R.~J.~Braun and M.~T.~Murray,\n J.~Cryst. Growth {\\bf 174}, 41 (1997).\n\n\\bibitem{Schmidt96} A.~Schmidt:\n ``Computations of Three-Dimensional Dendrites with Finite Elements'',\n J.~Comp. Phys. {\\bf 125}, 293 (1996).\n\n\\bibitem{Provatas99} N.~Provatas, N.~Goldenfeld, and J.~Dantzig,\n ``Efficient computation of dendritic microstructures using adaptive\n mesh refinement'',\n Phys. Rev. Lett. {\\bf 80}, 3308 (1998),\n ``Adaptive Mesh Refinement Computation of Solidification\n Microstructures Using Dynamic Data Structures'',\n J.~Comp. Phys. {\\bf 148}, 265 (1999).\n\n\\bibitem{Koonin} See for example S.~E. Koonin and D. C. Meredith,\n ``Computational Physics'', \n Addison-Wesley, Redwood City (1992), chapter 8.\n\n\\bibitem{Kadanoff85} L.~P.~Kadanoff, \n ``Simulating Hydrodynamics: A Pedestrian Model'',\n J.~Stat.~Phys. {\\bf 39}, 267 (1985).\n\n\\bibitem{Liang86} S.~Liang,\n ``Random-walk simulations of flow in Hele Shaw cells'',\n Phys.~Rev.~A {\\bf 33}, 2663 (1986).\n\n\\bibitem{Vicsek84} T.~Vicsek,\n ``Pattern Formation in Diffusion-Limited Aggregation'',\n Phys. Rev. Lett. {\\bf 53}, 2281 (1984).\n\n\\bibitem{Nittmann86} J.~Nittmann and H.~E.~Stanley,\n ``Tip splitting without interfacial tension and dendritic growth \n patterns arising from molecular anisotropy'',\n Nature {\\bf 321}, 663 (1986).\n\n\\bibitem{Karma87} A. Karma,\n ``Beyond Steady-State Lamellar Eutectic Growth'',\n Phys. Rev. Lett. {\\bf 59}, 71 (1987).\n\n\\bibitem{Saito89} Y. Saito and T. Ueta,\n ``Monte Carlo studies of equilibrium and growth shapes of a crystal'',\n Phys. Rev. A {\\bf 40}, 3408 (1989)\n\n\\bibitem{Meakin83} P. Meakin,\n ``Diffusion-controlled cluster formation in 2--6-dimensional space'',\n Phys. Rev. A {\\bf 27}, 1495 (1983).\n\n\\bibitem{Brener96} E. Brener, H. M{\\\"u}ller-Krumbhaar, D. Temkin,\n ``Structure formation and the morphology diagram of possible \n structures in two-dimensional diffusional growth'',\n Phys. Rev. E {\\bf 54}, 2714 (1996).\n\n\\bibitem{Langer87} J.~S. Langer,\n ``Dendritic sidebranching in the three-dimensional symmetric model \n in the presence of noise''\n Phys. Rev. A {\\bf 36}, 3350 (1987).\n\n\\bibitem{Plapp99} M. Plapp and A. Karma,\n ``Multiscale random-walk algorithm for simulating interfacial \n pattern formation'',\n cond-mat/9906370\n% Phys. Rev. Lett., in press.\n\n\\bibitem{Tip99} A. Karma, Y. Lee, and M. Plapp,\n ``Three-dimensional dendrite tip morphology at low undercooling'',\n cond-mat/9909021\n% Phys. Rev. E, in press.\n\n\\bibitem{Langer80} J.~S.~Langer,\n ``Instabilities and pattern formation in crystal growth'',\n Rev. Mod. Phys. {\\bf 52}, 1 (1980).\n\n\\bibitem{Kessler88} D.~A.~Kessler, J.~Koplik, and H.~Levine,\n ``Pattern selection in fingered growth phenomena'',\n Adv. Phys. {\\bf 37}, 255 (1988).\n\n\\bibitem{Brener91} E.~A.~Brener and V.~I.~Mel'nikov,\n ``Pattern selection in two-dimensional dendritic growth'',\n Adv. Phys. {\\bf 40}, 53 (1991).\n\n\\bibitem{Huang81} S.-C. Huang and M.~E. Glicksman,\n ``Fundamentals of Dendritic Solidification I. Steady-State Tip Growth'',\n Acta Metall. {\\bf 29}, 701 (1981).\n\n\\bibitem{Rubinstein91} E.~R.~Rubinstein and M.~E.Glicksman,\n ``Dendritic growth kinetics and structure I. Pivalic acid'',\n J. Cryst. Growth {\\bf 112}, 84 (1991).\n \n\\bibitem{Muschol92} M.~Muschol, D.~Liu, and H.~Z.~Cummins,\n ``Surface-tension-anisotropy measurements of succinonitrile and \n pivalic acid: Comparison with microscopic solvability theory'',\n Phys. Rev. A {\\bf 46}, 1038 (1992).\n\n\\bibitem{Glicksman94} M.~E.~Glicksman, M.~B.~Koss, and E.~A.~Winsa,\n ``Dendritic Growth Velocities in Microgravity'',\n Phys. Rev. Lett. {\\bf 73}, 573 (1994).\n\n\\bibitem{Bisang95} U.~Bisang and J.~Bilgram,\n ``Shape of the tip and the formation of sidebranches of xenon dendrites'',\n Phys. Rev. E {\\bf 54}, 5309 (1996).\n\n\\bibitem{Sethian92} J.~A.~Sethian and J.~Strain,\n ``Crystal Growth and Dendritic Solidification'',\n J. Comp. Phys. {\\bf 98}, 231 (1992).\n\n\\bibitem{Ihle94} T.~Ihle and H.~M{\\\"u}ller-Krumbhaar,\n ``Fractal and compact growth morphologies in phase transitions \n with diffusion transport'',\n Phys. Rev. E {\\bf 49}, 2972 (1994).\n\n\\bibitem{Karma98} A.~Karma and W.-J. Rappel,\n ``Quantitative phase-field modeling of dendritic growth in\n two and three dimensions'',\n Phys. Rev. E {\\bf 57}, 4323 (1998).\n\n\\bibitem{pf} For a brief historic overview and a comprehensive list\n of references concerning the phase-field method, \n see Ref. \\cite{Karma98}.\n\n\\bibitem{Karma99} A.~Karma and W.-J.~Rappel,\n ``Phase-Field Model of Dendritic Sidebranching with Thermal Noise'',\n cond-mat/9902017 (1999).\n\n\\bibitem{Lee99} Youngyih Lee, private communication.\n\n\\bibitem{Barbieri89} A. Barbieri and J.~S.~Langer, \n ``Prediction of dendritic growth rates in the linearized \n solvability theory'',\n Phys. Rev. A {\\bf 39}, 5314 (1989).\n\n\\bibitem{Brener95} E. Brener and D. Temkin,\n ``Noise-induced sidebranching in the three-dimensional \n nonaxisymmetric dendritic growth'',\n Phys. Rev. E {\\bf 51}, 351 (1995).\n\n\\bibitem{Dougherty87} A. Dougherty, P.~D. Kaplan, and J.~P. Gollub,\n ``Development of Side Branching in Dendrite Crystal Growth'',\n Phys. Rev. Lett. {\\bf 58}, 1652 (1987)\n\n\\bibitem{Li98} Q. Li and C. Beckermann,\n ``Scaling behavior of three-dimensional dendrites'',\n Phys. Rev. E {\\bf 57}, 3176 (1998)\n\n\\bibitem{Crisandme} C. Beckermann, H.-J. Diepers, I. Steinbach, \n A. Karma, and X. Tong, \n ``Modeling Melt Convection in Phase-Field Simulations of Solidification'',\n J. Comp. Phys. {\\bf 154}, 468-496 (1999).\n\n\\end{thebibliography}"
}
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cond-mat0002016
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Localization and delocalization in dirty superconducting wires
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[
{
"author": "P.\\ W.\\ Brouwer$^{a}$"
},
{
"author": "A.\\ Furusaki$^{b}$"
},
{
"author": "I.\\ A.\\ Gruzberg$^{c}$"
},
{
"author": "and C.\\ Mudry$^{d}$"
}
] |
[
{
"name": "cond-mat0002016.tex",
"string": "\\documentstyle[aps,prl,epsf,twocolumn,floats,amssymb,amsfonts]{revtex}\n\n\\begin{document}\n\\draft\n\\title{\nLocalization and delocalization in dirty superconducting \nwires}\n\n\\author{P.\\ W.\\ Brouwer$^{a}$, A.\\ Furusaki$^{b}$, I.\\ A.\\ Gruzberg$^{c}$, \nand C.\\ Mudry$^{d}$}\n\\address{$^{a}$Laboratory of Atomic and Solid State Physics,\nCornell University, Ithaca, NY 14853-2501\\\\\n$^{b}$Yukawa Institute for Theoretical Physics, Kyoto University,\nKyoto 606-8502, Japan\\\\\n$^{c}$Institute for Theoretical Physics, University of California, \nSanta Barbara, CA 93106-4030\\\\\n$^{d}$Paul Scherrer Institut, CH-5232, Villigen PSI, \nSwitzerland\\\\\n{\\rm February 1, 2000}\n\\medskip \\\\ \\parbox{14cm}{\\rm \nWe present Fokker-Planck equations that describe transport of heat\nand spin in dirty unconventional superconducting quantum wires.\nFour symmetry classes are distinguished, depending on the presence\nor absence of time-reversal and spin rotation invariance. In the absence \nof spin-rotation symmetry, heat transport is anomalous in that the \nmean conductance decays like $1/\\sqrt{L}$ instead of exponentially \nfast for large enough length $L$ of the wire. \nThe Fokker-Planck equations in the presence of time-reversal symmetry\nare solved exactly and the mean conductance for quasiparticle\ntransport is calculated for the crossover from the diffusive to the\nlocalized regime.\n\\smallskip \\\\\nPACS numbers: 72.15.Rn, 73.20.Fz, 73.23.-b, 74.25.Fy.}}\n\n\\maketitle \n\n% 70. CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND\n% OPTICAL PROPERTIES\n% 71. Electronic structure\n% (see also 73.20 Surface and interface electron states)\n% 71.10.Fd Lattice fermion models (Hubbard model, etc.)\n% 71.23.-k Electronic structure of disordered solids\n% 71.55.-i Impurity and defect levels\n% 71.55.Jv Disordered structures; amorphous and glassy solids\n% 72. Electronic transport in condensed matter (for electronic transport \n% in surfaces, interfaces, and thin films, see 73; for thermal \n% conduction in nonmetallic liquids, see 66.60; for electrical \n% properties related to treatment conditions, see 81.40.R)\n% 72.15.Rn Quantum localization\n% 00. GENERAL\n% 02. Mathematical methods in physics\n% 02.50.Ey Stochastic processes\n% 10. THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS (for cosmic rays, see\n% 96.40; for experimental methods and instrumentation, see 29)\n% 11. General theory of fields and particles (see also 03.65 Quantum \n% mechanics, 03.70 Theory of quantized fields, 03.80 General theory \n% of scattering)\n% 11.30.Rd Chiral symmetries\n\n\\narrowtext\n\nThe discovery of the $d$-wave nature of the order parameter in high\n$T_c$ materials has renewed interest in unconventional superconductors\nwith low energy quasiparticles near the Fermi energy $\\varepsilon_F$.\nAn important question is how disorder affects the quasiparticle\ndynamics and the corresponding low-temperature properties of the\nsuperconductor. In Ref.\\ \\onlinecite{Gorkov} it is predicted that, on\nenergy scales $|\\varepsilon- \\varepsilon_F|$ less than the inverse mean\nfree time and on length scales beyond the mean free path, weak\nimpurity scattering leads to a finite density of states (DoS)\nand to a diffusive dynamics of quasiparticles. In normal metals, it\nhas been known for a long time that quantum interference imposes\ncorrections to this picture, in the form of weak localization, and\neventually, for dimensions $\\le 2$, exponential (Anderson)\nlocalization. The analogous question for low energy quasiparticles in\nunconventional superconductors has been considered only recently\n\\cite{HigherD,Bundschuh,Senthil,spinQH,classD,Bocquet}.\n\nThe crucial distinction between quasiparticles in a superconductor and\nin a normal metal, is that the former are described by a Hamiltonian\nof Bogoliubov-de Gennes (BdG) type. Such a Hamiltonian has an\nadditional particle-hole grading, accompanied by a discrete\nparticle-hole symmetry, which is absent in the Hamiltonian for\n(electron-like) quasiparticles in a normal metal. Symmetry plays a\ncrucial role in the problem of Anderson localization. A classification\nof the symmetry classes for BdG Hamiltonians, depending on the\npresence or absence of time-reversal (TR) and spin-rotation (SR) symmetry,\nhas been given by Altland and Zirnbauer \\cite{AltlandZirnbauer}. The\nfour possibilities are denoted C, CI, D, and DIII, see table\n\\ref{tab:1}. Ref.\\ \\onlinecite{AltlandZirnbauer} addressed the\n``zero-dimensional'' \n(0D)\ncase of chaotic quantum dots with\nsuperconducting leads. The higher dimensional realizations of the BdG\nsymmetry classes, relevant for the question of localization, were\nstudied in Refs.\\ \n\\onlinecite{HigherD,Bundschuh,Senthil,spinQH,classD,Bocquet}, mainly\nby field-theoretical methods involving construction and analysis of\nnon-linear sigma models with appropriate symmetries.\n\nIn this Letter we study localization in the BdG symmetry classes for\nthe geometry of a quantum wire, i.e., in quasi-one-dimension (quasi-1D). \nFor this\npurpose, we use the Fokker-Planck (FP) approach\n\\cite{DorokhovMPK,Transport}, which is complementary to the non-linear\nsigma model of Refs.\\\n\\onlinecite{HigherD,Bundschuh,Senthil,classD,Bocquet}. Using\nthe\nclassification scheme of Ref.\\ \\onlinecite{AltlandZirnbauer}, we\nobtain four FP equations that control quasiparticle\ntransport at the Fermi level in a dirty superconducting wire. Our\nfindings are remarkable: While for classes C and CI the mean and\ntypical values of the quasiparticle conductance $g$ decay\nexponentially with the length $L$ of the wire for large $L$, the\nsituation in classes D and DIII is quite\ndifferent. There the mean $\\langle g \\rangle$ decays only\nalgebraically to zero for large $L$ and $\\ln g$ is not self-averaging,\nindicating a very broad distribution of the conductance and the\nabsence of the exponential localization of the quasiparticle states\nat $\\varepsilon_F$. (The absence of exponential localization for \nclass D has been announced independently in Ref.\\ \\onlinecite{Bocquet}.)\n\nIt should be stressed that the BdG Hamiltonians do not conserve\ncharge. Instead, the conserved densities are those of the energy (in\nall four classes) and spin (when the SR symmetry is present). Thus,\nthe transport properties (the conductance $g$) studied in this\nLetter refer to transport of heat and spin.\n\nWe now proceed with a detailed statement of our results and their\nderivation. The model that we consider is that of a disordered quantum\nwire, with a Hamiltonian of the BdG form. We distinguish gradings\ncorresponding to spin up/down, particle/hole, left/right\nmovers. Denoting these with Pauli matrices $\\sigma$, $\\gamma$, and\n$\\tau$, respectively, we write our model Hamiltonian as\n\\label{eq:our model}\n\\begin{equation}\n{\\cal H} = {\\cal K} + {\\cal V}, \\qquad \n{\\cal K} = i v_F \\partial_x\n \\sigma_0 \\otimes \\gamma_0 \\otimes \\tau_3 \\otimes \\openone_{N},\n \\label{eq:cal H}\n\\end{equation}\nwhere $\\sigma_0$ is the $2 \\times 2$ unit matrix in the spin grading\netc. The kinetic energy ${\\cal K}$ describes the propagation of right\nand left moving quasiparticles in $N$ channels at the Fermi level. The\n``potential'' ${\\cal V}(x)$ is an $8N \\times 8N$ matrix that accounts\nboth for the presence of disorder and of superconducting\ncorrelations. In particle/hole ($\\gamma$) grading it reads\n\\begin{equation}\n {\\cal V} =\n\\pmatrix{\nv&\\Delta\\cr\n-\\Delta^*&-v^{\\rm T}\\cr\n},\n\\label{eq:def V}\n\\end{equation}\nwhere $v$ ($\\Delta$) is a hermitian (antisymmetric) $4N \\times 4N$\nmatrix, representing the impurity potential (superconducting order\nparameter). The form (\\ref{eq:def V}) of the potential ${\\cal V}$\nensures that the Hamiltonian ${\\cal H}$ obeys particle-hole symmetry,\n${\\cal H} = - \\gamma_1 {\\cal H}^{\\rm T} \\gamma_1$ \\cite{AltlandZirnbauer}.\nIn addition, ${\\cal H}$ (and hence ${\\cal V})$ may obey TR\ninvariance ${\\cal H} = {\\cal T}{\\cal H}^* {\\cal T}^{-1}$, with ${\\cal\nT}=i\\tau_1\\otimes\\sigma_2$, and/or SR invariance\n${\\cal H} = -\\gamma_2 {\\cal H}^{\\rm T} \\gamma_2$. \n\n\\begin{table}\n\\begin{tabular}{l|c|c|c|c|c|c|c}\nClass & {SR} & {TR} & ${\\cal L}$ & \n ${\\cal G}$ & $m_0$ & $m_l$ & $d$ \\\\ \\hline\n{ C} & Yes & No & ${\\rm Sp}(N,N)$ &\n ${\\rm Sp}(N)\\times {\\rm Sp}(N)$ & 4 & 3 & 4 \\\\\n{ CI} & Yes & Yes & ${\\rm Sp}(N,{\\Bbb C})$ &\n ${\\rm Sp}(N) $ & 2 & 2 & 4 \\\\\n{ D} & No & No & ${\\rm O}(4N,4N)$ & \n ${\\rm O}(4N)\\times {\\rm O}(4N)$ & 1 & 0 & 1 \\\\\n{ DIII} & No & Yes & ${\\rm O}(4N,{\\Bbb C})$ &\n ${\\rm O}(4N)$ & 2 & 0 & 2 \\\\\n\\end{tabular}\\hfill\\\\\n\\caption{\\label{tab:1}\nDefinition of the symmetry classes for a dirty superconducting quantum\nwire, with respect to the presence of spin-rotation (SR) and\ntime-reversal (TR) symmetry. The table lists the Lie group ${\\cal L}$\nof the transfer matrix ${\\cal M}$, the factor group ${\\cal G}$ of\nangular degrees of freedom of ${\\cal M}$, the multiplicities of the\nroots of the symmetric space ${\\cal L}/{\\cal G}$, and the degeneracies\nof the radial coordinates $x_j$ of the transfer matrix ${\\cal M}$.}\n\\end{table}\n\nSpatial fluctuations of the order parameter $\\Delta$ and the potential\n$v$ are taken into account by assuming that ${\\cal V}$ is a Gaussian\nrandom variable with vanishing mean, i.e.\\ that its probability\nfunctional $P[{\\cal V}]$ is of the form\n\\begin{equation}\n P[{\\cal V}] \\propto \\exp\\left[-{\\gamma \\ell \\over 4c}\\int_0^L dx\\,\n {\\rm tr}\\,{\\cal V}^2 (x)\\right],\n \\label{eq:prob dist}\n\\end{equation}\nwhere $\\ell$ is the mean free path, $\\gamma$ is a numerical\nconstant to be defined below, and $c=1$ ($2$) for class C/D (CI/DIII). \nThe transport properties of the\nHamiltonian ${\\cal H}$ describe the transport of spin and heat by\nquasiparticles in a disordered superconducting quantum wire.\n\nBefore we continue with the analysis of our model (\\ref{eq:cal\nH}--\\ref{eq:prob dist}), some remarks about its validity and relevance\nare in place. One key property of the model is that, apart from \ncorrections at very low energies due to quasiparticle localization\n\\cite{Senthil} or the appearance of a critical state, the DoS of the\nHamiltonian ${\\cal H}$ near the Fermi level $\\varepsilon_F$ is\nnonzero and finite.\nThis is related to the fact that the statistical average of\nthe order parameter $\\Delta$ is zero in our model, cf.\\ Eq.\\\n(\\ref{eq:prob dist}). For a dirty superconductor, such behavior is\nplausible if the order parameter is unconventional, as in\n$d$-wave superconductors, or when it breaks TR\nsymmetry, as is believed to be the case for, e.g., the\nruthenates \\cite{Ruthenates}, vortex lines in a (conventional)\nsuperconductor \\cite{Bundschuh}, \nor a normal metal wire with magnetic impurities that is\nweakly connected to a superconducting substrate.\nIn all these cases the disorder leads to the existence of low-energy \nquasiparticle states \\cite{Gorkov,Senthil}. \nThe Hamiltonian (\\ref{eq:cal H}) \nthen describes diffusion and localization of these\n``disorder-facilitated'' quasiparticles. \nAn altogether different scenario is that of a wire made out of an\nunconventional superconductor with very weak disorder. If boundary\nconditions are suitably chosen, one or several propagating \nmodes can exist at $\\varepsilon_F$, whose localization properties\nare described by Eq.\\ (\\ref{eq:cal H}). \nIn any case, one should view ${\\cal H}$ as an effective or coarse\ngrained Hamiltonian, whose validity is restricted to length scales\nbeyond the microscopic mean free path $\\ell$. It is universal in the\nsense that its form is determined solely by the symmetry, and the\ndistribution (\\ref{eq:prob dist}) provides for the existence of the\ndiffusive regime with a finite DoS at the proper energy scale. (Note\nthat the restrictions to the validity of our model are not different\nfrom those of related field theoretic descriptions appearing in the\nliterature \\cite{Bundschuh,Senthil,classD,Bocquet}.)\n\nWe describe transport properties of the model (\\ref{eq:cal\nH}--\\ref{eq:prob dist}) through its $8N \\times 8N$ transfer matrix\n${\\cal M}$ that encodes the $x$-dependence of an $8N$-component\nquasiparticle wavefunction $\\psi$ satisfying the Schr\\\"odinger\nequation ${\\cal H} \\psi = \\varepsilon \\psi$ at $\\varepsilon=0$,\n%\\begin{equation}\n$\n \\psi(x+L) = {\\cal M}(x+L,x) \\psi(x).\n$\n%\\end{equation}\nFormally, ${\\cal M}$ is related to the Hamiltonian (\\ref{eq:cal H}) as\n\\begin{equation}\n {\\cal M}(x+L;x)=\n {\\rm T}_{y}\n \\exp\n \\left[\n i\\int_x^{x+L} dy\\, \\tau_3\\, {\\cal V}(y) \\right],\n\\label{eq:def cal M}\n\\end{equation}\nwhere ${\\rm T}_{y}$ denotes the path ordering operator for the\n$y$-integration along the wire. {}From Eq.\\ (\\ref{eq:def cal M}) one\nfinds that flux conservation (i.e., Hermiticity of ${\\cal H}$) and \nparticle-hole symmetry imply that ${\\cal M}^{\\dagger} \\tau_3 {\\cal M} =\n\\tau_3$ and $\\gamma_1 {\\cal M} \\gamma_1 = {\\cal M}^*$,\nrespectively. Further, TR invariance requires $ {\\cal T} {\\cal M}\n{\\cal T}^{-1} = {\\cal M}^*$, while SR invariance is obeyed if\n$\\gamma_2 {\\cal M} \\gamma_2 = {\\cal M}^*$. The transfer matrix ${\\cal\nM}$ obeys the multiplicative rule ${\\cal M}(z,x) = {\\cal M}(z,y) {\\cal\nM}(y,x)$ for $x < y < z$ and hence is an element of a certain Lie\ngroup ${\\cal L}$. The appropriate Lie groups for the four symmetry\nclasses are listed in Table \\ref{tab:1}. We note that the actual\ntransfer matrix group is an $8N$-dimensional representation of the Lie\ngroup ${\\cal L}$, where ${\\cal L}$ also allows a lower dimensional\n(irreducible) representation for the classes C, CI, and DIII\n\\cite{LieGroupExample}. Elements of ${\\cal L}$ are conveniently\nparameterized in terms of their polar decomposition, which, in an\nirreducible representation, takes the form\n\\begin{eqnarray*}\n&\n\\pmatrix{V_1&0\\cr0&\\!\\! V_2\\cr}\n\\pmatrix{\\cosh X&\\sinh X\\cr\\sinh X&\\cosh X\\cr}\n\\pmatrix{V_3&0\\cr0&\\!\\! V_4\\cr} \\ \\ & (\\mbox{C, D}),\n\\nonumber \\\\\n&\nV_1\n\\pmatrix{\\cosh X&i\\sinh X\\cr-i\\sinh X&\\cosh X\\cr}\nV_2 \\ \\ & (\\mbox{CI, DIII}).\n\\end{eqnarray*} \nHere $V_i \\in {\\rm O}(4N)$ [Sp$(N)$] for classes D/DIII [C/CI], for\nall $i=1,2,3,4$, and $X$ is a diagonal matrix with positive\nentries $x_j$. (By Kramers' degeneracy, the elements\nof $X$ occur in pairs in class C.) The $x_j$ serve as radial\ncoordinates on the Lie Group ${\\cal L}$. One verifies that the\neigenvalues of the true $8N \\times 8N$ transfer matrix ${\\cal M} {\\cal\nM}^{\\dagger}$ occur in $d$-fold degenerate inverse pairs $\\exp(\\pm 2\nx_j)$, where the degeneracy $d$ is listed in Table \\ref{tab:1}. Hence\nthe number of independent $x_j$'s is $4N/d$. Finally, we note that the\n$x_j$ are related to the conductance $g$ through \\cite{Transport}\n%\\vspace{-0.5cm}\n\\begin{equation}\n g = d \\sum_{j=1}^{4N/d} \\cosh^{-2} x_j.\n\\end{equation}\n\nOur aim is to find the probability distribution of the $x_j$ for a\ntransfer matrix corresponding to the model \n(\\ref{eq:cal H}--\\ref{eq:prob dist}).\nIncreasing the length $L$ of the wire by a small increment $\\delta L$\namounts to multiplication of its transfer matrix ${\\cal M}(L) = {\\cal\nM}(x+L,x)$ by a transfer matrix ${\\cal M}' = {\\cal\nM}(x+L+\\delta L,x+L)$. Since ${{\\cal M}}'$ is close to the unit\nmatrix, random, and statistically independent from ${\\cal M}(L)$, we\nfind that as a function of $L$, ${\\cal M}(L)$ performs a random\ntrajectory on its Lie group ${\\cal L}$. Actually, we do not need to\nknow the full trajectory on ${\\cal L}$ if we are only interested in\nthe conductance $g$. It is sufficient to know the\ntrajectory of the radial coordinates $x_j$ of ${\\cal M}(L)$ after\ndividing out a maximal compact subgroup ${\\cal G}$ of ${\\cal L}$\ncorresponding to the angular degrees freedom of ${\\cal M}$ that leave\nthe product ${\\cal M} {\\cal M}^{\\dagger}$ invariant, or, in other\nwords, to know the trajectory of the $x_j$ in the symmetric space\n${\\cal L}/{\\cal G}$ \\cite{Caselle}. The subgroups ${\\cal G}$ are\nlisted in Table \\ref{tab:1}.\nStarting from the microscopic model (\\ref{eq:cal H}-\\ref{eq:prob\ndist}), one can show that the trajectory obeyed by the $x_j$ is a\nBrownian motion on the coset space ${\\cal L}/{\\cal G}$ described by\nthe joint probability distribution $P(x_1,\\ldots,x_{4N/d};L)$. The\n$L$-evolution of $P$ is described by a \nFP equation, which\nfollows either from a direct calculation starting from Eq.\\\n(\\ref{eq:cal H}), or from the general theory of symmetric spaces\n\\cite{Caselle}. In both cases we find\n\\begin{eqnarray}\n{\\partial P\\over\\partial L} &=&\n{1\\over2 \\gamma \\ell}\n\\sum_{j=1}^{4N/d}\n{\\partial\\over\\partial x_j}\n\\left[\nJ\n\\left(\n{\\partial\\over\\partial x_j} J^{-1}P\n\\right)\n\\right],\n\\label{eq:FPbdg}\\\\\nJ &= &\n\\prod_{j=1}^{4N/d}\n|\\sinh 2x_j|^{m_l}\n\\prod_{k>j}^{4N/d} \\prod_{\\pm}\n|\n\\sinh(x_j\\pm x_k)\n|^{m_o}, \\nonumber\n\\end{eqnarray}\nwhere the numbers $m_{l}$ and $m_{o}$ are the long and ordinary\nroot multiplicities in the symmetric spaces ${\\cal L}/{\\cal G}$,\nsee Table \\ref{tab:1}, and $\\gamma = (4N m_o/d) + 1 - m_o + m_l$.\nThe FP equation (\\ref{eq:FPbdg}) is supplemented with\nthe boundary condition ${\\partial P/\\partial x_j} = \n(P/J) {\\partial J / \\partial x_j}$ at $x_j = 0$.\nThe initial condition ${\\cal M} = 1$ for $L=0$ corresponds to\n$P(x_1,\\ldots,x_{4N/d};0) = \\prod_{j} \\delta(x_j)$.\n\nThe FP equation (\\ref{eq:FPbdg}) is the fundamental\nequation that governs quasiparticle transport and localization in\nquantum wires of the symmetry classes C, CI, D, and DIII. \n\nIn the localized regime $L \\gg N\\ell$, typically all $x_j$ and their\nspacings are much bigger than unity, and the conductance is governed\nby the smallest coordinate $x_1$. That coordinate has a Gaussian\ndistribution, with mean $m_l L/\\gamma \\ell$ and variance $L/\\gamma\n\\ell$. For classes C and CI this implies that \n$g$ is exponentially small, with\n\\begin{equation}\n \\langle \\ln g \\rangle = - {2 m_l L \\over \\gamma \\ell}, \\qquad\n \\mbox{var}\\, \\ln g = {4 L \\over \\gamma \\ell},\\ \\ \n\\end{equation}\nwith $m_l = 2$ for class CI and $m_l = 3$ for class C. Exponential\nlocalization for class C in quasi-1D was previously obtained by \nBundschuh {\\em et al.} \\cite{Bundschuh}, using the non-linear sigma\nmodel.\nFor class D and DIII however, $m_l = 0$, so that there is no exponential \nlocalization. Instead, $g$ has a very broad distribution (broader than \nlog-normal), with an algebraic decay of the mean and the variance\nand an $L^{1/2}$-dependence of $\\ln g$,\n\\begin{eqnarray}\n && \\langle g \\rangle = d \\sqrt{2 \\gamma \\ell \\over \\pi L}, \\qquad\n \\mbox{var}\\, g = {2 d \\over 3} \\langle g \\rangle,\n \\nonumber \\\\\n\\label{eq:localizedD}\n && \\langle \\ln g \\rangle = -4 \\sqrt{L \\over 2 \\pi \\gamma \\ell}, \\qquad\n \\mbox{var}\\, \\ln g = {4 (\\pi - 2) L \\over \\pi \\gamma \\ell}.\n\\end{eqnarray}\nHence in classes D and DIII, quasiparticle states are not localized at\nthe Fermi level. Since they are neither truly extended (typically $g\n\\ll 1$ in class D and DIII), we label them critical, following\nterminology from the case of quantum wires with off-diagonal disorder,\nwhere similar behavior is found at the center of\nthe band \\cite{Dyson53,BMSA}.\n\n\nThe effect of disorder is much less pronounced in the \ndiffusive regime $\\ell\\ll L\\ll N\\ell$. Here, the conductance has only small \nfluctuations around its mean. Following the method of \nmoments \\cite{Transport}, we find $\\langle g \\rangle$ from the FP \nequation by construction of\nevolution equations for the moments of $g_a=d \\sum_j\\cosh^{-2a} x_j$,\n$a=1,2,\\ldots$,\n\\label{eq:dif D DIII}\n\\begin{eqnarray}\n{\\gamma \\ell \\over a}{\\partial\n\\left\\langle\ng_a\n\\right\\rangle\n\\over\n\\partial L\n}\n&=&\n{m_o \\over d}\\sum_{n=1}^{a-1} \\langle g_{a-n} g_{n} \\rangle\n - {m_o \\over d} \\sum_{n=1}^{a} \\langle g_{a-n+1} g_{n} \\rangle\n \\nonumber \\\\ && \\mbox{}\n + (a m_o - 2a-1 + m_l) \\langle g_{a+1} \\rangle\n \\nonumber \\\\ && \\mbox{}\n + (2a- a m_o + m_o - 2 m_l) \\langle g_{a } \\rangle.\n\\end{eqnarray}\nIn the diffusive regime one may replace the average of a product by\nthe product of the averages, and hence one finds for $\\ell \\ll L \\ll\nN \\ell$\n\\begin{eqnarray} \\label{eq:WL}\n \\langle g \\rangle = {4N\\ell \\over L + \\ell} + \n {d (m_o - 2 m_l) \\over 3 m_o} + {\\cal O}(\\ell/L,L/N \\ell).\n\\end{eqnarray} \nThe first term in Eq.\\ (\\ref{eq:WL}) is the Drude conductance, while\nthe second term is the first quantum interference correction to the\naverage conductance. For classes C, CI, D, and DIII it takes the\nvalues $-2/3$, $-4/3$, $1/3$, and $2/3$, respectively. The weak\nlocalization correction for class C was obtained earlier in Ref.\\\n\\onlinecite{Bundschuh}. For the classes D and DIII, the correction is\npositive, i.e.,\\ quantum interference enhances the conductance\nrelative to the classical Drude-like leading behavior (see also \nRef.\\ \\onlinecite{Bundschuh}). This\nis similar to the phenomenon of anti-localization in the\nstandard symplectic symmetry class, though, as pointed out by Bocquet\net al.\\cite{Bocquet}, here it is a precursor of the breakdown of\nexponential localization, while in the standard symplectic class\nlocalization takes over in higher order quantum corrections.\n\nIn the presence of TR symmetry, Eq.~(\\ref{eq:FPbdg}) is soluble. This\nis in contrast to the case of the FP equations for the standard and\nchiral symmetry classes, where only the case of broken TR symmetry was\nexactly solvable \\cite{Transport,BMSA}. The first step\n\\cite{Transport} is a map of Eq.~(\\ref{eq:FPbdg}) onto a Schr\\\"\nodinger equation in imaginary time for the wave function\n$\\Psi(\\{x_j\\};s) = \\exp\\left[-{1\\over2}\\ln\nJ(\\{x_j\\})\\right]P(\\{x_j\\};s)$ for $4N/d$ fermions in one dimension\nwith coordinates on the half-line $x>0$. They interact through a\ntwo-body potential proportional to $m_o-2$ in the presence of a\none-body potential proportional to $(m_l-2)m_l$. Hence, for classes\nDIII and CI these fermions are free and they only differ by the\nboundary condition obeyed by their wave functions $\\Psi$ at the\norigin. We thus find the solutions\n\\begin{eqnarray}\nP & \\propto &\n \\prod_{j} (x_j \\sinh 2 x_j)^{m_l/2} e^{- \\gamma x^2_j \\ell/2L}\n \\nonumber \\\\ && \\mbox{} \\times\n\\prod_{j < k} \n\\left(x_k^2 - x_j^2 \\right) \\left(\\sinh^2 x_k - \\sinh^2 x_j \n\\right).\n\\end{eqnarray}\nUsing the method of bi-orthogonal functions\\cite{MuttalibFrahm} \nit is then possible to calculate the average conductance $\\langle g \\rangle$\nfor all $N$ and $L$. Here we report the result for the limit of large\n$N$, leaving the results for finite $N$ for a future \npublication,\n\\begin{eqnarray}\n\\langle g\\rangle &=&\n{1 \\over s}\n-\n{4\\over3}\n+ 4\n\\sum_{n=1}^{\\infty} e^{-\\pi^2 n^2/4s} \\left( {1 \\over s} + {2 \\over \\pi^2 \nn^2}\\right) \\ \\ \\mbox{CI}, \\nonumber\\\\\n\\langle g\\rangle &=&\n{1 \\over s}\n+\n{2\\over3}\n-\n4 \\sum_{n=1}^{\\infty} e^{-\\pi^2n^2/2s} {1 \\over \\pi^2 n^2} \\ \\ \\ \\ \\ \\\n \\ \\ \\ \\ \\ \\ \\ \\mbox{DIII},\n\\label{eq:exact}\n\\end{eqnarray}\nwhere $s = L/(4 N \\ell)$. Note the agreement with Eq.\\ (\\ref{eq:WL})\nin the diffusive regime $s \\ll 1$. In the\nlocalized regime $s \\gg 1$, Eq.\\ (\\ref{eq:exact}) may be resummed, and\nthe asymptotic result (\\ref{eq:localizedD}) is reproduced for class DIII,\nwhile for class CI one finds $\\langle g \\rangle = 8(\\pi s)^{-1/2} \\exp(-4 s)$.\nQuite remarkably, the exact results (\\ref{eq:exact}) for $\\langle g \\rangle$ \nin classes CI and DIII are related to the average conductance $\\langle g \n\\rangle_{\\rm ch}$ in the chiral unitary ensemble for odd channel \nnumber\\cite{BMSA} as $\\langle g (s/2) \\rangle_{\\rm CI} + 2 \\langle g (s) \n\\rangle_{\\rm DIII} = 4 \\langle g (s) \\rangle_{\\rm ch}$. \n\nThe absence of localization in wires of classes D and DIII may have\nimportant implications for higher dimensions, provided our results can\nbe extended beyond 1D, and provided they are not restricted to the\nregime of weak disorder. With respect to the latter restriction, we\ncan point to the close formal similarity of the delocalization for the\nFP equations of class D/DIII and the \ncorresponding \nFP equation for the\nchiral symmetry classes with odd $N$, where it is understood that the\nabsence of localization holds both for weak and strong disorder\n\\cite{Dyson53,Fisher}. Thus arguing that quasiparticle states at the Fermi\nlevel remain delocalized for arbitrary disorder strength and\ndimensionality in the D-classes, our result suggests a possible\nresolution of a controversy in the literature surrounding \n2D disordered superconductors of class\nD \\cite{classD,Bocquet}. While all Refs.\\ \\onlinecite{classD,Bocquet}\nassumed existence of two localized phases, distinguished by the\nquantized value of the Hall conductivity $\\sigma_{xy}$, and a metallic\nphase,\nthe proposed\nglobal phase diagrams and \ntransitions between the phases differ considerably. We suggest that\nthe solution might simply lie in the absence of\nlocalized phases for classes D and DIII in any dimension $\\ge 1$.\n\nIn conclusion, we considered quasiparticle transport and localization\nin disordered quasi-1D superconducting wires at the Fermi level for the four\nBogoliubov-de Gennes symmetry classes C, CI, D, and DIII. We obtained\nand solved the Fokker-Planck equations for the probability of the radial\ncoordinates of the transfer matrix. While quasiparticle states are\nlocalized in classes C/CI, localization is absent if\nspin-rotation symmetry is broken (classes D/DIII).\n\nWe thank A.\\ Altland, L. Balents, M.\\ P.\\ A.\\ Fisher, N.\\\nRead, T.\\ Senthil, and M.\\ Sigrist for valuable discussions. PWB\ngratefully acknowledges that this problem was suggested to him in an\nearlier stage by A.\\ Altland. Close to completion of this work, we\nlearned that J.\\ T.\\ Chalker and coworkers obtained independently\nsimilar results for class D, see also Ref.\\ \\onlinecite{Bocquet}.\nThis work was supported by a Grant-in-Aid for Scientific Research from\nJapan Society for the Promotion of Science No.\\ 11740199 (AF), and by\nthe NSF under grant No.\\ DMR-9528578 (IAG).\n\\vspace{-0.6cm}\n\n\\begin{references}\\vspace{-1.6cm}\n\n\\bibitem{Gorkov} L. P. Gorkov and P. A. Kalugin, \n%\tPis'ma Zh. \\'Eksp. Teor. Fiz. {\\bf 41}, 208 (1985) [\n\tJETP Lett. {\\bf 41}, 253 (1985);\n S. Schmitt-Rink {\\em et al.}, \n% K. Miyake, and C. M. Varma, \n Phys. Rev. Lett. {\\bf 57}, 2575 (1986);\n\tP. A. Lee, {\\it ibid.} {\\bf 71}, 1887 (1993).\n\n\n\\bibitem{HigherD}\n\tA. Altland {\\em et al.}, \n% B. D. Simons, and D. Taras-Semchuk, \n JETP Lett. {\\bf 67}, 22 (1998).\n\n\\bibitem{Bundschuh}\n\tR. Bundschuh {\\it et al.}, \n% C. Casanello, D. Serban, and M. R. Zirnbauer, \n Nucl.\\ Phys.\\ B {\\bf 532}, 689 (1998); \n\tPhys. Rev. B {\\bf 59}, 4382 (1999).\n\n\\bibitem{Senthil} \n\tT. Senthil {\\it et al.}, Phys. Rev. Lett. {\\bf 81}, 4704 (1998);\n\tT. Senthil and M. P.\\ A. Fisher, Phys. Rev. B {\\bf 60}, 6893 (1999). \n\n\\bibitem{spinQH} \n\tV. Kagalovsky {\\it et al.}, Phys. Rev. Lett. {\\bf 82}, 3516 (1999); \n\tT. Senthil {\\em et al.}, \n% J. B. Marston, and M. P. A. Fisher, \n Phys. Rev. B {\\bf 60}, 4245 (1999);\n\tI. A. Gruzberg {\\em et al.}, \n% A. W. W. Ludwig, and N. Read, \n Phys. Rev. Lett. {\\bf 82}, 4524 (1999).\n Y.\\ Morita and Y.\\ Hatsugai, cond-mat/9907001. \n\\bibitem{classD}\n\tT. Senthil and M. P. A. Fisher, cond-mat/9906290;\n\tN. Read and D. Green, cond-mat/9906453.\n\n\\bibitem{Bocquet}\n M.\\ Bocquet {\\em et al.}, \n% D.\\ Serban, and M. R. Zirnbauer, \n cond-mat/9910480.\n\n\\bibitem{AltlandZirnbauer}\n A. Altland and M. R. Zirnbauer, Phys. Rev. Lett. {\\bf 76}, 3420 \n (1996); Phys. Rev. B {\\bf 55}, 1142 (1997).\n\n\\bibitem{DorokhovMPK} \n O.\\ N.\\ Dorokhov, \n% Pis'ma Zh. Eksp. Teor. Fiz. {\\bf 36}, 259 (1982) [\n JETP Letters {\\bf 36}, 318 (1982);\n P.\\ A.\\ Mello {\\em et al.}, \n% P.\\ Pereyra, and N.\\ Kumar, \n Ann.\\ Phys.\\ (NY) {\\bf 181}, 290 (1988).\n\n\\bibitem{Transport} \n C.\\ W.\\ J.\\ Beenakker, Rev.\\ Mod.\\ Phys.\\ {\\bf 69}, 731 (1997).\n\n\n\\bibitem{Ruthenates}\n G.\\ M.\\ Luke {\\em et al.}, Nature (London) {\\bf 394}, 558 (1998);\n M.\\ Sigrist {\\em et al.}, Physica C {\\bf 317-318}, 134 (1999).\n\n\n\\bibitem{LieGroupExample} \n For instance, in the presence of\n SR invariance, ${\\cal M}$ separates into two identical\n blocks of size $4N \\times 4N$. \n\n\\bibitem{Caselle} \n A. H\\\"uffmann, J. Phys. A {\\bf 23}, 5733 (1990);\n M. Caselle, cond-mat/9610017.\n\n\\bibitem{Dyson53} \n F.\\ J.\\ Dyson, Phys.\\ Rev.\\ {\\bf 92}, 1331 (1953);\n A.\\ D.\\ Stone and J.\\ D.\\ Joannopoulos,\n Phys. Rev. B {\\bf 24}, 3592 (1981).\n\n\\bibitem{BMSA} \n P.\\ W.\\ Brouwer {\\it et al.}, \n% C.\\ Mudry, B.\\ D.\\ Simons, and A.\\ Altland, \n Phys.\\ Rev.\\ Lett.\\ {\\bf 81}, 862 (1998);\n C.\\ Mudry {\\em et al.},\n% P.\\ W.\\ Brouwer, and A.\\ Furusaki,\n Phys. Rev. B {\\bf 59}, 13\\,221 (1999).\n\n\\bibitem{MuttalibFrahm}\n K.\\ A.\\ Muttalib, J.\\ Phys.\\ A {\\bf 28}, L159 (1995);\n K.\\ Frahm, Phys.\\ Rev.\\ Lett.\\ {\\bf 74}, 4706 (1995).\n\n\\bibitem{Fisher} \n D. S. Fisher, Phys. Rev. B {\\bf 50}, 3799 (1994); \n {\\bf 51}, 6411 (1995). \n\n\\end{references}\n\n\\narrowtext\n\n\n\\end{document}\n\n"
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[
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"name": "cond-mat0002016.extracted_bib",
"string": "\\bibitem{Gorkov} L. P. Gorkov and P. A. Kalugin, \n%\tPis'ma Zh. \\'Eksp. Teor. Fiz. {\\bf 41}, 208 (1985) [\n\tJETP Lett. {\\bf 41}, 253 (1985);\n S. Schmitt-Rink {\\em et al.}, \n% K. Miyake, and C. M. Varma, \n Phys. Rev. Lett. {\\bf 57}, 2575 (1986);\n\tP. A. Lee, {\\it ibid.} {\\bf 71}, 1887 (1993).\n\n\n\n\\bibitem{HigherD}\n\tA. Altland {\\em et al.}, \n% B. D. Simons, and D. Taras-Semchuk, \n JETP Lett. {\\bf 67}, 22 (1998).\n\n\n\\bibitem{Bundschuh}\n\tR. Bundschuh {\\it et al.}, \n% C. Casanello, D. Serban, and M. R. Zirnbauer, \n Nucl.\\ Phys.\\ B {\\bf 532}, 689 (1998); \n\tPhys. Rev. B {\\bf 59}, 4382 (1999).\n\n\n\\bibitem{Senthil} \n\tT. Senthil {\\it et al.}, Phys. Rev. Lett. {\\bf 81}, 4704 (1998);\n\tT. Senthil and M. P.\\ A. Fisher, Phys. Rev. B {\\bf 60}, 6893 (1999). \n\n\n\\bibitem{spinQH} \n\tV. Kagalovsky {\\it et al.}, Phys. Rev. Lett. {\\bf 82}, 3516 (1999); \n\tT. Senthil {\\em et al.}, \n% J. B. Marston, and M. P. A. Fisher, \n Phys. Rev. B {\\bf 60}, 4245 (1999);\n\tI. A. Gruzberg {\\em et al.}, \n% A. W. W. Ludwig, and N. Read, \n Phys. Rev. Lett. {\\bf 82}, 4524 (1999).\n Y.\\ Morita and Y.\\ Hatsugai, cond-mat/9907001. \n\n\\bibitem{classD}\n\tT. Senthil and M. P. A. Fisher, cond-mat/9906290;\n\tN. Read and D. Green, cond-mat/9906453.\n\n\n\\bibitem{Bocquet}\n M.\\ Bocquet {\\em et al.}, \n% D.\\ Serban, and M. R. Zirnbauer, \n cond-mat/9910480.\n\n\n\\bibitem{AltlandZirnbauer}\n A. Altland and M. R. Zirnbauer, Phys. Rev. Lett. {\\bf 76}, 3420 \n (1996); Phys. Rev. B {\\bf 55}, 1142 (1997).\n\n\n\\bibitem{DorokhovMPK} \n O.\\ N.\\ Dorokhov, \n% Pis'ma Zh. Eksp. Teor. Fiz. {\\bf 36}, 259 (1982) [\n JETP Letters {\\bf 36}, 318 (1982);\n P.\\ A.\\ Mello {\\em et al.}, \n% P.\\ Pereyra, and N.\\ Kumar, \n Ann.\\ Phys.\\ (NY) {\\bf 181}, 290 (1988).\n\n\n\\bibitem{Transport} \n C.\\ W.\\ J.\\ Beenakker, Rev.\\ Mod.\\ Phys.\\ {\\bf 69}, 731 (1997).\n\n\n\n\\bibitem{Ruthenates}\n G.\\ M.\\ Luke {\\em et al.}, Nature (London) {\\bf 394}, 558 (1998);\n M.\\ Sigrist {\\em et al.}, Physica C {\\bf 317-318}, 134 (1999).\n\n\n\n\\bibitem{LieGroupExample} \n For instance, in the presence of\n SR invariance, ${\\cal M}$ separates into two identical\n blocks of size $4N \\times 4N$. \n\n\n\\bibitem{Caselle} \n A. H\\\"uffmann, J. Phys. A {\\bf 23}, 5733 (1990);\n M. Caselle, cond-mat/9610017.\n\n\n\\bibitem{Dyson53} \n F.\\ J.\\ Dyson, Phys.\\ Rev.\\ {\\bf 92}, 1331 (1953);\n A.\\ D.\\ Stone and J.\\ D.\\ Joannopoulos,\n Phys. Rev. B {\\bf 24}, 3592 (1981).\n\n\n\\bibitem{BMSA} \n P.\\ W.\\ Brouwer {\\it et al.}, \n% C.\\ Mudry, B.\\ D.\\ Simons, and A.\\ Altland, \n Phys.\\ Rev.\\ Lett.\\ {\\bf 81}, 862 (1998);\n C.\\ Mudry {\\em et al.},\n% P.\\ W.\\ Brouwer, and A.\\ Furusaki,\n Phys. Rev. B {\\bf 59}, 13\\,221 (1999).\n\n\n\\bibitem{MuttalibFrahm}\n K.\\ A.\\ Muttalib, J.\\ Phys.\\ A {\\bf 28}, L159 (1995);\n K.\\ Frahm, Phys.\\ Rev.\\ Lett.\\ {\\bf 74}, 4706 (1995).\n\n\n\\bibitem{Fisher} \n D. S. Fisher, Phys. Rev. B {\\bf 50}, 3799 (1994); \n {\\bf 51}, 6411 (1995). \n\n"
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cond-mat0002017
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Dynamical ordering in the c-axis in 3D driven vortex lattices
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"author": "Alejandro B. Kolton"
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"author": "Daniel Dom\\'{\\i}nguez \\address{Centro At\\'{o}mico Bariloche,"
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"author": "8400 S. C. de Bariloche, Rio Negro, Argentina} % %\\thanks{Footnotes should appear on the first page only to % indicate your present address (if different from your % normal address)"
},
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"author": "research grant"
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"author": "sponsoring agency"
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"author": "Davis, California 95616, USA} \\address{NERSC, Lawrence Berkeley National Laboratory,"
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"author": "Berkeley, California 94720, USA}"
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We present molecular dynamics simulations of driven vortices in layered superconductors in the presence of an external homogeneous force and point disorder. We use a model introduced by J.R.Clem for describing 3D vortex lines as stacks of 2D pancake vortices where only magnetic interactions are considered and the Josephson interlayer coupling is neglected. We numerically evaluate the long-range magnetic interaction between pancake vortices exactly. We analyze the vortex correlation along the field direction on (c-axis). We find that above the critical current, in the ``plastic flow'' regime, pancakes are completely uncorrelated in the c-direction. When increasing the current, there is an onset of correlation along the c-axis at the transition from plastic flow to a moving smectic phase. This transition coincides with the peak in the differential resistance. \vspace{1pc}
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"name": "hou.tex",
"string": "%%%%%%%%%% espcrc2.tex %%%%%%%%%%\n%\n% $Id: espcrc2.tex 1.1 1999/07/26 10:28:22 Simon Exp spepping $\n%\n%\\documentclass[twoside]{article}\n%\\usepackage{fleqn,espcrc2}\n\n% change this to the following line for use with LaTeX2.09\n \\documentstyle[twoside,fleqn,espcrc2,epsf]{article}\n\n% if you want to include PostScript figures\n%\\usepackage{graphicx}\n%\\usepackage{epsfig}\n\n% if you have landscape tables\n%\\usepackage[figuresright]{rotating}\n\n% put your own definitions here:\n% \\newcommand{\\cZ}{\\cal{Z}}\n% \\newtheorem{def}{Definition}[section]\n% ...\n\\newcommand{\\ttbs}{\\char'134}\n\\newcommand{\\AmS}{{\\protect\\the\\textfont2\n A\\kern-.1667em\\lower.5ex\\hbox{M}\\kern-.125emS}}\n\n% add words to TeX's hyphenation exception list\n%\\hyphenation{author another created financial paper re-commend-ed Post-Script}\n\n% declarations for front matter\n\\title{Dynamical ordering in the c-axis in 3D driven vortex\nlattices}\n\n\\author{\n\tAlejandro B. Kolton, Daniel Dom\\'{\\i}nguez \n\t\\address{Centro At\\'{o}mico Bariloche,\\\\ 8400 S. C. de Bariloche,\n\tRio Negro, Argentina}\t\t\n\t%\n %\\thanks{Footnotes should appear on the first page only to\n % indicate your present address (if different from your\n % normal address), research grant, sponsoring agency, etc.\n % These are obtained with the {\\tt\\ttbs thanks} command.}\n and \n Niels Gr{\\o}nbech-Jensen\t\n\t\\address{Department of Applied Science, University of California, \\\\\n\tDavis, California 95616, USA}\n\t\\address{NERSC, Lawrence Berkeley National Laboratory,\\\\\n Berkeley, California 94720, USA}\n\t}\n \n\\begin{document}\n\n\\begin{abstract}\nWe present molecular dynamics simulations of driven vortices in layered\nsuperconductors in the presence of an external homogeneous force and point\ndisorder. We use a model introduced by J.R.Clem for describing 3D vortex\nlines as stacks of 2D pancake vortices where only magnetic interactions\nare considered and the Josephson interlayer coupling is neglected. We\nnumerically evaluate the long-range magnetic interaction between pancake\nvortices exactly. We analyze the vortex correlation along the field\ndirection on (c-axis). We find that above the critical current, in the\n``plastic flow'' regime, pancakes are completely uncorrelated in the\nc-direction. When increasing the current, there is an onset of correlation\nalong the c-axis at the transition from plastic flow\n to a moving smectic phase. This transition coincides with the peak in the\ndifferential resistance.\n\\vspace{1pc}\n\\end{abstract}\n\n% typeset front matter (including abstract)\n\\maketitle\n\n\\section{INTRODUCTION}\nThe prediction \\cite{KV} of a {\\it dynamical phase transition} upon increasing \ndrive, from a fluidlike plastic flow regime \\cite{plastico} to a \ncoherently moving solid \\cite{KV} in a moving vortex lattice has motivated many \nrecent theoretical \\cite{theo}, experimental \n\\cite{bhatta,hellerq,pardo}, and \nsimulation work \\cite{2d,ryu,olson2,kolton,mingo,tdgl,3dxy}. \nIn a previous work we have studied the dynamical regimes in the velocity-force\ncurve (voltage-current) in 2D thin films \\cite{kolton} and found two\ndynamical phase transitions above the critical force. The first transition, from a \nplastic flow regime to a smectic flow regime, is characterized by the \nsimultaneous occurrence of a peak in differential resistance, isotropic low \nfrequency voltage noise and maximum transverse diffusion. \nThe second transition, from a smectic flow regime to a frozen transverse\nsolid, is a freezing transition in the transverse direction where transverse \ndiffusion vanishes abruptly and the Hall noise drops many orders of magnitude.\nIn other 2D simulations the peak in differential\nresistance was found to coincide with the onset of orientational order \\cite{ryu} \nand a maximum number of defects \\cite{olson2}. Experimentally, the position of \nthis peak was taken by Hellerqvist {\\it et al.} \\cite{hellerq} as an \nindication of a dynamical phase transition. \nIn this paper we show that in driven 3D layered superconductors, the peak in \ndifferential resistance also \ncoincides with the onset of correlation along the c-axis. \n\\section{MODEL}\nSimulations of vortices in 3D layered superconductors have been done previously using \ntime--dependent Ginzburg--Landau--Lawrence--Doniach equations (two layers), \n\\cite{tdgl}, the 3D XY model \\cite{3dxy} and Langevin dynamics of interacting \nparticles \\cite{reefman,srld}. \nHere we study the motion of pancakes vortices in a layered superconductor with\ndisorder, with an applied magnetic field in the c-direction and with an external \nhomogeneous current in the layers (ab-planes). We use a model introduced by \nJ.R. Clem for a layered superconductor with vortices in the limit of zero \nJosephson-coupling between \nlayers \\cite{clem}. \nWe simulate a stack of equally spaced superconducting layers with interlayer periodicity $s$, \neach layer containing the same number of pancake vortices. The equation of motion for a \npancake located in position ${\\bf R_{i}}=({\\bf r_{i}},z_i)=(x_{i},y_{i},n_i s)$ \n(with z-axis in c-direction) is:\n\\begin{equation}\n\\eta \\frac{ d{\\bf r_i}}{dt} = \\sum_{j\\not= i}{\\bf F_v}(\\rho_{ij},z_{ij})\n+\\sum_p{\\bf F_p}(\\rho_{ip}) + \\bf{F} \\; ,\n\\end{equation}\nwhere $\\rho_{ij}=|{\\bf r}_i-{\\bf r}_j|$ and $z_{ij}=|z_i-z_j|$ are \nthe in-plane and inter-plane distance between pancakes $i,j$, $r_{ip}=\n|{\\bf r}_i-{\\bf r}_p|$ is the in-plane distance between the vortex $i$ and a pinning \nsite at ${\\bf R_p}=({\\bf r}_p,z_i)$ (pancakes interact only with pinning centers within the same \nlayer),\n $\\eta$ is the Bardeen-Stephen friction,\n and ${\\bf F}=\\frac{\\Phi_0}{c}{\\bf J}\\times{\\bf z}$\nis the driving force due to an applied in-plane current density ${\\bf J}$. \nWe consider a random uniform distribution of attractive pinning centers in each layer with \n${\\bf F_p}=-A_p e^{-(r/r_p)^2} {\\bf r}/r_p^2$, where $r_p$\nis the pinning range.\nThe magnetic interaction between pancakes ${\\bf F}_v(\\rho,z)=F_{\\rho}(\\rho,z) \\hat{r}$ is\ngiven by:\n\\begin{equation}\nF_{\\rho}(\\rho,0)=(\\phi_0^2 / 4 \\pi^2 \\Lambda \\rho)[1-(\\lambda_{\\parallel}\n/\\Lambda)(1-e^{-\\rho / \\lambda_{\\parallel}})]\n\\end{equation}\n\\begin{equation}\nF_{\\rho}(\\rho,z)=(\\phi_0^2 \\lambda_{\\parallel}/ 4 \\pi^2 \\Lambda^2 \\rho)\n[e^{-z/\\lambda_{\\parallel}}-e^{-R/\\lambda_{\\parallel}}] \\; .\n\\end{equation}\nHere, $R=\\sqrt{z^2+\\rho^2}$, $\\lambda_{\\parallel}$ is the penetration length parallel to\nthe layers, and $\\Lambda=2 \\lambda_{\\parallel}^2/s$ is the 2D thin-film screening\nlength. A analogous model to Eqs.\\ (2-3) was used in \\cite{reefman}. \nWe normalize length scales by $\\lambda_{\\parallel}$, energy scales by \n$A_v=\\phi_0^2 / 4 \\pi^2 \\Lambda$, \nand time is normalized by \n$\\tau=\\eta \\lambda_{\\parallel}^2/A_v$. We consider $N_v$ pancake vortices and $N_p$ pinning\ncenters per layer in $N_l$ rectangular layers of size $L_x\\times L_y$, \nand the normalized vortex density is $n_v=N_v\\ \\lambda_{\\parallel}^2 /\n L_xL_y=B\\ \\lambda_{\\parallel}^2 /\\Phi_0$.\nMoving pancake vortices induce a total electric field ${\\bf\nE}=\\frac{B}{c}{\\bf v}\\times{\\bf z}$, with ${\\bf v}=\\frac{1}{N_v N_l}\\sum_i \n{\\bf v}_i$.\nWe study the dynamical regimes in the velocity-force \ncurve at $T=0$, solving Eq.\\ (1) for increasing values of ${\\bf F}=F{\\bf y}$. \nWe consider a constant vortex density $n_v=0.1$ in\n$N_l=5$ layers with $L_x/L_y=\\sqrt{3}/2$, $s=0.01$, and \n$N_v=36$ pancake vortices per layer. \nWe take a pinning range of $r_p=0.2$, pinning strengh of $A_p/A_v=0.2$,\nwith a density of pinning centers $n_p=0.65$ on each layer. We use periodic boundary\nconditions in all directions and the periodic long-range in-plane interaction is dealt\nwith exactly using an exact and fast converging sum \\cite{log}. \nThe equations are integrated with a time step of $\\Delta t=0.01\\tau$ and averages are\nevaluated in $16384$ integration steps after $2000$ iterations for \nequilibration (when the total energy reaches a stationary value). \nEach simulation is started at $F=0$ with an ordered\ntriangular vortex lattice (perfectly correlated in c-direction) and slowly increasing \nthe force in steps of $\\Delta F= 0.1$ up to values as high as $F=8$.\n\n\\section{RESULTS}\n\nWe start by looking at the vortex trajectories in the steady state \nphases.\nIn Figure 1(a-b) we show a top view snapshot of the instantaneous pancake \nconfiguration for two typical values of $F$. In Figure 2(a-b) we show the vortex \ntrajectories $\\{ {\\bf R}_i(t)\\}$ for the same two typical values of $F$ by \nplotting all the positions of the pancakes in all the layers for all the time \niteration steps. In Fig.\\ 3(a) we plot the average vortex velocity \n$V=\\langle V_y(t)\\rangle=\\langle\\frac{1}{N_v}\\sum_i \\frac{dy_i}{dt}\\rangle$,\nin the direction of the force as a function of $F$ and its corresponding\nderivative $dV/dF$. We also study the pair distribution function:\n\\begin{equation} \ng(\\rho,n)=\\frac{1}{N_p N_l}\\langle \\sum_{i<j}\n\\delta(\\rho-\\rho_{ij}) \\delta(ns-z_{ij}) \\rangle.\n\\end{equation}\nIn Fig.\\ 3(b) we plot a correlation parameter along c-axis defined as:\n\\begin{equation} \nC^n_z=\\lim_{\\rho \\rightarrow 0} g(\\rho,n) \\; ,\n\\end{equation}\nas a function of $F$ for $n=1,2$. \nBelow a critical force, $F_c \\approx 0.4$, all the pancakes are\npinned and there is no motion. At the characteristic force,\n$F_p \\approx 0.8$, we observe a peak in the differential resistance. \nAt $F_c$ pancake vortices start to move in a few channels \n, as was also seen in 2D vortex simulations \\cite{plastico}. A typical \nsituation is shown in Fig.\\ 2(a). In this plastic flow regime \nwe observe that the motion is completely uncorrelated along c-direction, \nwith $C^n_z \\approx 0$ for $F_c< F <F_p$ \nas shown in Fig.\\ 3(b). In Fig.\\ 1(a) and 2(a) we see that this situation corresponds to \na disordered configuration of pancakes and to an uncorrelated structure of plastic \nchannels along c-axis. At $F_p$ there is an onset of correlation along the \nc-axis and pancakes vortices start to align forming well defined stacks or \nvortex lines. This onset of c-axis correlation corresponds to the transition \nfrom plastic flow to a moving smectic phase\n(a complete discussion of the translational \nas well as temporal order will be discussed elsewhere \\cite{veremos}). For $F > F_p$ we \nobserve that the structure of smectic channels is very correlated in the c-direction. \n\\begin{figure}[htb]\n%\\centerline{\\epsfxsize=7cm \\epsfbox{fig1hou.ps}}\n\\centerline{\\epsfxsize=8cm \\epsfbox{fig1hou.ps}}\n\\caption{Pancake configuration (top view of the layers) for two typical values \nof $F$, (a) $F_c< F=0.7 <F_p$, (b) $F_p < F=2.5$. Diamonds ($\\Diamond$) represent pancake \nvortices and asterisks ($*$) represent pinning centers.}\n\\label{fig:largenenough}\n\\end{figure}\n\\begin{figure}[htb]\n\\centerline{\\epsfxsize=8cm \\epsfbox{fig2hou.ps}}\n%\\centerline{\\epsfxsize=8cm \\epsfbox{fig1hou.ps}}\n\\caption{Pancake trajectories for two typical values of\n$F$, (a) $F_c< F=0.7 < F_p$, (b) $F_p<F=2.5$, obtained plotting all the\npositions of the pancakes for all time iteration steps.}\n\\label{fig:toosmall}\n\\end{figure}\n\\begin{figure}[htb]\n\\centerline{\\epsfxsize=7cm \\epsfbox{fig3hou.ps}}\n\\caption{(a) Velocity-force curve (voltage-current characteristics), left scale,\nblack points, $dV/dF$ (differential-resistance), right scale, white points. (b)\nCorrelation parameter along c-axis $C^n_z$ for $n=1,2$.}\n\\label{fig:toosmall2}\n\\end{figure}\n\\section{DISCUSSION}\n%In \\cite{tdgl} was studied the phase diagram for a driven vortex lattice in a bilayered\n%system as a prototype of the multilayered superconductor using time-dependent Ginzburg-Landau \n%equations. They examined both a dynamic decoupling transition and a dynamic melting transition. They \n%found that c-axis correlations increase with increasing drive but do not observe a sharp \n%alignment transition. At higher currents they found a sharp dynamic melting transition. \n%Neglecting Josephson coupling between layers \nIn a system with $N_l=5$ layers we have found that there is a clear onset of c-axis correlation \nwith increasing driving force, where the pancake vortices start to align forming well defined stacks \nmoving in smectic flow channels. Below this transition, in the plastic flow regime, these stacks of pancakes \nare unstable. Also, in \\cite{tdgl} an enhancement of c-axis correlations with increasing drive was\nobserved in a bilayered system.\n\nWe have further found \\cite{veremos} that the in-plane properties are in well correspondence\n with the ones obtained in 2D thin films simulations \\cite{kolton}. A better understanding \n of the effects of c-axis correlations in pancake motion on each layer can be obtained \n by studying translational and temporal \norder in larger systems, through the analysis of the structure factor, voltage noise, and \nin-plane \\cite{mingo} as well as {\\it inter-plane} velocity-velocity \ncorrelation functions \\cite{veremos}.\n\n\nIn conclusion we have analyzed the vortex correlation along the field\ndirection (c-axis) in the velocity-force characteristics at $T=0$ and found that \nabove the critical current there is an onset of c-axis correlation in the \ntransition between a plastic flow regime to a smectic flow regime. This transition \ncoincides with the peak in the differential resistance. Experimentally, this effect \ncould be studied with measurements of c-axis resistivity as a function of an applied \ncurrent parallel to the layers \\cite{lamenghi}. \\newline \n\nWe acknowledge discussions with L.N.\\ Bulaevskii, P.S.\\ Cornaglia, F.\\ de la Cruz, Y.\\ Fasano, \nM.\\ Menghini and C.J.\\ Olson. \nThis work has been supported by a grant from ANPCYT (Argentina), \nProy. 03-00000-01034. \nD.D.\\ and A.B.K.\\ acknowledge support from Fundaci\\'on Antorchas\n(Proy. A-13532/1-96), Conicet, CNEA and FOMEC \n(Argentina).\nThis work was also supported by the Director, Office of Advanced Scientific\nComputing Research, Division of Mathematical, Information, and \nComputational Sciences of the U.S.\\ Department of Energy under contract\nnumber DE-AC03-76SF00098.\n\n\\begin{thebibliography}{9}\n\n\\bibitem{KV} A. E. Koshelev and V. M. Vinokur, Phys. Rev. Lett.\n{\\bf 73}, 3580 (1994).\n\n\\bibitem{plastico}\nH.\\ J.\\ Jensen {\\it et al.}, Phys.\\ Rev.\\ Lett.\\ {\\bf 60},\n1676 (1988); \nA.-C.\\ Shi and A.\\ J.\\ Berlinsky,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 67}, 1926 (1991).\n\n\\bibitem{theo} T. Giamarchi and P. Le Doussal, \nPhys. Rev. Lett. {\\bf 76}, 3408 (1996);\nP. Le Doussal and T. Giamarchi, Phys. Rev. B {\\bf 57}, 11356\n(1998); L. Balents, M. C. Marchetti and L. Radzihovsky, Phys. Rev. B\n{\\bf 57}, 7705 (1998); S. Scheidl and V. M. Vinokur, \nPhys. Rev. E {\\bf 57}, 2574 (1998); Phys. Rev. B {\\bf 57}, 13800\n(1998).\n\n\\bibitem{bhatta} \nM. J. Higgins and S. Bhattacharya, Physica C {\\bf 257}, 232 (1996).\n\n\\bibitem{hellerq} M.\\ C.\\ Hellerqvist {\\it et al.},\nPhys. Rev. Lett. {\\bf 76}, 4022 (1996).\n\n\\bibitem{pardo} F. Pardo {\\it et al}, Nature (London) {\\bf 396}, 348 (1998).\n\n\\bibitem{2d} K. Moon, R. T. Scalettar and G. Zim\\'{a}nyi, \nPhys. Rev. Lett. {\\bf 77}, 2778 (1996); \nS. Spencer and H. J. Jensen, Phys. Rev. B {\\bf 55}, 8473 (1997).\n\n\\bibitem{ryu} S. Ryu {\\it et al.}, Phys. Rev. Lett. {\\bf 77}, 5114 (1996).\n\n\\bibitem{olson2} C. J. Olson, C. Reichhardt and F. Nori,\nPhys. Rev. Lett. {\\bf 81}, 3757 (1998).\n\n\\bibitem{kolton} A. B. Kolton, D. Dom\\'{\\i}nguez, N. Gr\\o nbech-Jensen, \nPhys. Rev. Lett. {\\bf 83}, 3061 (1999).\n\n\\bibitem{mingo} D. Dom\\'{\\i}nguez, Phys. Rev. Lett. {\\bf 82}, 181 (1999).\n\n\\bibitem{tdgl}\nI. Aranson, A. Koshelev and V. Vinokur, \nPhys. Rev. B {\\bf 56}, 5136 (1997).\n\n\\bibitem{3dxy} D. Dom\\'{\\i}nguez, N. Gr\\o nbech-Jensen and A.R. Bishop,\nPhys. Rev. Lett. {\\bf 78}, 2644 (1997).\n\n\\bibitem{reefman}\nD. Reefman, H. B. Brom, Physica C {\\bf 213}, 229 (1993).\n\n\n\\bibitem{srld} \nS. Ryu, D. Stroud, Phys. Rev. B, {\\bf 54}, 1320 (1996);\nN. K. Wilkin, H. J. Jensen, Phys. Rev. Lett. {\\bf 21}, 4254 (1997); \nA. van Otterlo, R. T. Scalettar, G. T. Zim\\'amyi, Phys. Rev. Lett. {\\bf 81},\n1497 (1998);\nC. J. Olson, R. T. Scalettar, G. T. Zim\\'anyi, cond-mat/9909454.\n\n\n\\bibitem{clem} J. R. Clem, Phys. Rev. B.\n{\\bf 43}, 7837 (1990).\n\n\\bibitem{log} N. Gr\\o nbech-Jensen, Int. J. Mod. Phys. C {\\bf 7}, 873 (1996);\nComp. Phys. Comm. {\\bf 119}, 115 (1999).\n\n\\bibitem{veremos} A. B. Kolton, D. Dom\\'{\\i}nguez, N. Gr\\o nbech-Jensen, \nto be published.\n\n\\bibitem{lamenghi} M. Menghini (private communication).\n\n\n\n\\end{thebibliography}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002017.extracted_bib",
"string": "\\begin{thebibliography}{9}\n\n\\bibitem{KV} A. E. Koshelev and V. M. Vinokur, Phys. Rev. Lett.\n{\\bf 73}, 3580 (1994).\n\n\\bibitem{plastico}\nH.\\ J.\\ Jensen {\\it et al.}, Phys.\\ Rev.\\ Lett.\\ {\\bf 60},\n1676 (1988); \nA.-C.\\ Shi and A.\\ J.\\ Berlinsky,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 67}, 1926 (1991).\n\n\\bibitem{theo} T. Giamarchi and P. Le Doussal, \nPhys. Rev. Lett. {\\bf 76}, 3408 (1996);\nP. Le Doussal and T. Giamarchi, Phys. Rev. B {\\bf 57}, 11356\n(1998); L. Balents, M. C. Marchetti and L. Radzihovsky, Phys. Rev. B\n{\\bf 57}, 7705 (1998); S. Scheidl and V. M. Vinokur, \nPhys. Rev. E {\\bf 57}, 2574 (1998); Phys. Rev. B {\\bf 57}, 13800\n(1998).\n\n\\bibitem{bhatta} \nM. J. Higgins and S. Bhattacharya, Physica C {\\bf 257}, 232 (1996).\n\n\\bibitem{hellerq} M.\\ C.\\ Hellerqvist {\\it et al.},\nPhys. Rev. Lett. {\\bf 76}, 4022 (1996).\n\n\\bibitem{pardo} F. Pardo {\\it et al}, Nature (London) {\\bf 396}, 348 (1998).\n\n\\bibitem{2d} K. Moon, R. T. Scalettar and G. Zim\\'{a}nyi, \nPhys. Rev. Lett. {\\bf 77}, 2778 (1996); \nS. Spencer and H. J. Jensen, Phys. Rev. B {\\bf 55}, 8473 (1997).\n\n\\bibitem{ryu} S. Ryu {\\it et al.}, Phys. Rev. Lett. {\\bf 77}, 5114 (1996).\n\n\\bibitem{olson2} C. J. Olson, C. Reichhardt and F. Nori,\nPhys. Rev. Lett. {\\bf 81}, 3757 (1998).\n\n\\bibitem{kolton} A. B. Kolton, D. Dom\\'{\\i}nguez, N. Gr\\o nbech-Jensen, \nPhys. Rev. Lett. {\\bf 83}, 3061 (1999).\n\n\\bibitem{mingo} D. Dom\\'{\\i}nguez, Phys. Rev. Lett. {\\bf 82}, 181 (1999).\n\n\\bibitem{tdgl}\nI. Aranson, A. Koshelev and V. Vinokur, \nPhys. Rev. B {\\bf 56}, 5136 (1997).\n\n\\bibitem{3dxy} D. Dom\\'{\\i}nguez, N. Gr\\o nbech-Jensen and A.R. Bishop,\nPhys. Rev. Lett. {\\bf 78}, 2644 (1997).\n\n\\bibitem{reefman}\nD. Reefman, H. B. Brom, Physica C {\\bf 213}, 229 (1993).\n\n\n\\bibitem{srld} \nS. Ryu, D. Stroud, Phys. Rev. B, {\\bf 54}, 1320 (1996);\nN. K. Wilkin, H. J. Jensen, Phys. Rev. Lett. {\\bf 21}, 4254 (1997); \nA. van Otterlo, R. T. Scalettar, G. T. Zim\\'amyi, Phys. Rev. Lett. {\\bf 81},\n1497 (1998);\nC. J. Olson, R. T. Scalettar, G. T. Zim\\'anyi, cond-mat/9909454.\n\n\n\\bibitem{clem} J. R. Clem, Phys. Rev. B.\n{\\bf 43}, 7837 (1990).\n\n\\bibitem{log} N. Gr\\o nbech-Jensen, Int. J. Mod. Phys. C {\\bf 7}, 873 (1996);\nComp. Phys. Comm. {\\bf 119}, 115 (1999).\n\n\\bibitem{veremos} A. B. Kolton, D. Dom\\'{\\i}nguez, N. Gr\\o nbech-Jensen, \nto be published.\n\n\\bibitem{lamenghi} M. Menghini (private communication).\n\n\n\n\\end{thebibliography}"
}
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cond-mat0002018
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The effect of the annealing temperature on the local distortion of La$_{0.67}$Ca$_{0.33}$MnO$_3$ thin films
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[
{
"author": "D. Cao$^1$"
},
{
"author": "F. Bridges$^1$"
},
{
"author": "D.~C. Worledge$^2$"
},
{
"author": "C.~H. Booth$^3$"
},
{
"author": "T. Geballe$^2$"
}
] |
Mn $K$-edge fluorescence data are presented for thin film samples (3000~\AA) of Colossal Magnetoresistive (CMR) La$_{0.67}$Ca$_{0.33}$MnO$_3$: as-deposited, and post-annealed at 1000~K and 1200~K. The local distortion is analyzed in terms of three contributions: static, phonon, and an extra, temperature-dependent, polaron term. The polaron distortion is very small for the as-deposited sample and increases with the annealing temperature. In contrast, the static distortion in the samples decreases with the annealing temperature. Although the local structure of the as-deposited sample shows very little temperature dependence, the change in resistivity with temperature is the largest of these three thin film samples. The as-deposited sample also has the highest magnetoresistance (MR), which indicates some other mechanism may also contribute to the transport properties of CMR samples. We also discuss the relationship between local distortion and the magnetization of the sample.
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[
{
"name": "dcao1_paper.tex",
"string": "% Physical Review B:\n%\n% Dear editor:\n% PLease find below the paper \"The effect of the annealing temperature\n% on the local distortion of La$_{0.67}$Ca$_{0.33}$MnO$_3$thin films\"\n% by D Cao et al, (BG7525) which we are resubmitted for publication in PRB.\n%\tOur response to the referee's comments are given below.\n%\n% Sincerely\n% Daliang Cao\n%\n% Corresponding author: Daliang Cao\n% Email: dcao@wu.ucsc.edu\n% Address: Physics Department, Kerr Hall\n% Santa Cruz, CA 95064\n% Phone number: (831) 459 3646\n% FAX number: (831) 459 3043.\n%\n% Article type: Letter\n% Experiment or theory: Experiment\n%\n% Suggested section: L-6 Condensed matter: structure, etc.\n%\n% Suggested principal PACS No.: 75.70.Pa\n% Additional PACS No(s).: 61.10.Ht 71.30.+h 71.38.+i\n%\n%\tRESPONSE to REFEREE REPORT.\n%\t\n%\tWe have addressed all the points that the referee brought to our \n%\tattention. We have expanded it a little to make it more understandable\n%\tto people who are not XAFS experts, and thank the referee for bringing\n%\tthese changes to our attention. \n%\t\n%\tSummary of the changes we have made:\n%\t\n%\tFirst, we have added one more paragraph in the INTRODUCTION section\n%\tto give readers some more information about the comparison between\n%\tthe structure as determined by diffraction techniques and the local\n%\tstructure as determined by XAFS. Much of this is already published but\n%\twe agree a short summary is important for the non-expert.\n%\n%\tSecond, In the fourth section (XAFS DATA ANALYSIS AND DISCUSSION), we\n%\tadd more background about the XAFS technique in the first paragraph,\n%\tto provide a better understanding of how we analyze our data.\n%\t\n%\tThird, in the same section (fourth section) we have added another \n%\tparagraph, to explain what the pair distribution function width (sigma)\n%\tmeans and how it changes with various types of disorder - - local \n%\tdistortions, thermal vibrations etc.\n%\n%\tFourth, In order to make it clearer that the local structure of \n%\tMn-O bond in the fully annealed thin film sample can be almost as \n%\tordered as CaMnO3 from the perspective of the width of the pair\n%\tdistribution, we provide some discussion as to why the presence\n%\tof slow moving polarons results in distortions (see below for additional\n%\tdetails.)\n%\n%\tSome specific answers to the referee's questions:\n%\n%\tWe have added more information about how we analyze our data and extract\n%\tthe correlated width, sigma, of the pair distribution function. \n%\tSome discussion of how different broadening mechanisms contribute to the \n%\ttotal value of sigma^2 have been include to make our model easier \n%\tto understand.\n%\n%\tRegarding the definitions of \"defects, distortion and disorder\": in our\n%\tpaper, defects represent interstitials, vacancies, dislocations and \n%\tsubstitutional atoms (here Ca); distortion normally means the change\n%\tin bond length or angle of the local structure in a shell of atoms, - for \n%\texample, the Jahn-Teller distortion has 2 long Mn-O bonds and 4 shorter \n%\tbonds. It can usually be observed as short-range disorder, with a \n%\tdistribution of Mn-O bond lengths, but such distortions may have no \n%\tlong-range order. Consequently, this correlated distortion of the Mn-O \n%\tbond lengths may not be easily observed using diffraction \n%\ttechniques. For example, If the distance between the Mn does not change\n%\tmuch (say less tha 0.01A) and the O atom between Mn atoms randomly moves\n%\ttowards one of the Mn, there will be a distribution of long and short \n%\tMn-O bonds. For such a case, the change in sigma would be significantly \n%\tlarger than the corresponding change in the thermal parameters, U, for \n%\tdiffraction.\n%\n%\tDisorder is used as a more general term which includes both local\n%\tdistortions as well as the presence of defects which can also induce\n%\tdistortions. In most of the paper we have restricted the discussion to\n%\tonly distortions of the bond length.\n%\n%\tA comparison of the XAFS data with diffraction results has been\n%\tdone in several previous papers, it shows that the XAFS results are \n%\tvery similar to the structure as determined by diffraction. The main \n%\tdifference is the larger change in sigma^2 compared to U.\n%\n%\tThe referee states that \"the CaMnO3 and 33% doped LCMO CMR sample can`t \n%\tcontain the same amount of static distortion\". First we reemphasize that\n%\tin this part of the discussion, we are only talking about the \n%\tMn-O local distortion at very low temperatures, and specifically the \n%\tcorrelated width of the pair distribution function. The surprising result\n%\tobserved previously is that sigma for the substituted samples is only \n%\tslightly larger than for CaMnO3. This means that to first order, the Mn-O\n%\tbond lengths at low T are all the same (within a value of order 0.01A) and \n%\tvariations must be small compared to the value of sigma (0.04A) needed to \n%\tmodel zero-point motions and some random strains that are always present. \n%\tFor the CMR samples at low T, this result is attributed to the rapid motion\n%\tof the electrons in the metallic regime. Electrons can hop very rapid from \n%\tone Mn site to another, on a time scale comparable or faster than phonon\n%\ttime scales. Consequently, a given electron is not localized long enough for\n%\ta Jahn-Teller (JT) distortion to form at low temperatures. However, JT \n%\tdistortions are present at high T in the semi-conducting (insulating) phase,\n%\tand the value of sigma increases significantly above Tc.\n\n%\tFor the thin film samples, the high temperature annealing process appears \n%\tto remove nearly all of the static defects/distortions introduced during\n%\tfilm growth, and at low T the annealed film samples have similar properties\n%\t(as far as structure is concerned) to powder samples.\n%\n%\tWe hope that we have answered all the reviewers comments and that the paper\n%\tis now acceptible for publication.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\tolerance 10000\n%\\documentstyle[prl,aps,preprint,tighten,twocolumn]{revtex}\n%\\documentstyle[aps,twocolumn,epsf]{revtex}\n%\\documentstyle[aps,prb,twocolumn]{revtex}\n%\\documentstyle[prb,aps,epsf]{revtex}\n%\\documentstyle[twoside,fleqn,espcrc2]{article}\n%\\documentstyle[aps,twocolumn]{revtex}\n%\\documentstyle[spie]{article}\n\\documentstyle[aps,prb,preprint,tighten]{revtex}\n\n\n%\\input{psfig}\n\n\\begin{document}\n\n\\draft\n\\title{\n\\large\\bf The effect of the annealing temperature on the local distortion of\nLa$_{0.67}$Ca$_{0.33}$MnO$_3$ thin films}\n\\author{\\rm D. Cao$^1$, F. Bridges$^1$, D.~C. Worledge$^2$, C.~H. Booth$^3$, \nT. Geballe$^2$} \n\n\\address{\\rm\n$^{(1)}$Department of Physics, University of California Santa\nCruz, Santa Cruz, CA 95064}\n\n\\address{\\rm\n$^{(2)}$Department of Applied Physics, Stanford University, Stanford, CA\n94305-4090}\n\n\\address{\\rm\n$^{(3)}$Los Alamos National Laboratory, Los Alamos, NM 87545}\n\n\\date{draft: \\today}\n\n\\maketitle\n\n\\begin{abstract}\n\nMn $K$-edge fluorescence data are presented for thin film\nsamples (3000~\\AA) of Colossal Magnetoresistive (CMR) \nLa$_{0.67}$Ca$_{0.33}$MnO$_3$: as-deposited, and post-annealed\nat 1000~K and 1200~K. The local distortion is analyzed in terms of three \ncontributions: static, phonon, and an extra, temperature-dependent, polaron term.\nThe polaron distortion is very small for the as-deposited sample and increases\nwith the annealing temperature. In contrast, the static distortion in the \nsamples \ndecreases with the annealing temperature. Although the local structure of the \nas-deposited sample shows very little temperature dependence, the \nchange in resistivity with temperature is the largest of these three\nthin film samples. The as-deposited sample also has the highest \nmagnetoresistance (MR), which indicates some other mechanism may also \ncontribute to the transport properties of CMR samples. We also discuss the \nrelationship between local distortion and the magnetization of the sample.\n\n\\end{abstract}\n\n\\pacs{Keywords: CMR, XAFS, thin film, annealing}\n\n\\narrowtext\n\n\\section{Introduction}\n\nThe Double Exchange (DE) mechanism\\cite{Zener51,Anderson55,deGennes60}\nwas originally considered to be the main\ninteraction contributing to the colossal magnetoresistance\n(CMR). In the DE model, if the spins of \ntwo neighboring Mn ions are aligned, then an electron will require less \nenergy to hop from one Mn site to another. Consequently, at low temperature, \nthe lattice will have ferromagnetic (FM) order such that the total system \n(both local and itinerant sub-systems) has the lowest energy.\nAlthough the DE model can explain many properties of CMR materials,\nthe magnitude of\nthe MR calculated from the DE model is much smaller than the actual measured \nMR\\cite{Millis95}.\nMillis {\\it et al.} suggested that local $\\it{Jahn-Teller}$\ndistortions also play an important role in CMR materials, and are needed\nto explain the large magnitude of the MR\\cite{Millis95} in these materials.\n\nBoth X-ray Absorption Fine Structure \n(XAFS)\\cite{Booth98a,Booth98b}\nand pair-distribution function (PDF) analysis of neutron diffraction \ndata\\cite{Billinge96,Louca97}, \nhave been done to study the local structure of the CMR materials and \nan important relationship between the local distortions\nand magnetism in these materials has been found\\cite{Booth98a,Booth98b}.\nThese new experiments investigate the local structure of thin films of\nLa$_{0.67}$Ca$_{0.33}$MnO$_3$ (LCMO) to \nunderstand more about this relationship.\n\nRecent experiments show that the \nnon-fully annealed thin film samples are oxygen\ndeficient\\cite{Worledge96,Worledge98}; the Curie temperature, T$_c$, the \nsaturated magnetization, M$_0$, and the resistivity \npeak temperature, T$_{MI}$, of the \nsamples increase with increasing the oxygen stoichiometry, \nwhile the resistivity of the \nsamples decreases\\cite{Worledge96,Worledge98,Ju95,Malde98}. \nFor the fully\nannealed thin film sample, T$_c$ and T$_{MI}$ are almost the same\nas those of the bulk material. It is not yet clear what\nmechanisms are responsible for suppressing T$_c$, M$_0$ and T$_{MI}$\nin the other samples. Our experiments\non transport and magnetization measurements show similar annealing effects\nwith significant changes in the resistivity and the magnetization.\nThey also show that a huge MR occurs for the films,\nespecially for the as-deposited sample at low temperature.\n\n\nXAFS experiments on these thin film\nsamples allow us to observe the local structure around the Mn sites, primarily\nthe local\ndistortion of the Mn-O bonds. This paper focuses on the $\\it{changes}$ in the \nlocal\nstructure of CMR thin films that are induced by annealing at different\ntemperatures. These results may also help us to better understand how annealing\nmodifies other sample properties.\n\nDiffraction studies of LaMnO$_3$ and CaMnO$_3$ have been carried out\nby several groups\\cite{Norby95,Mitchell96,Poeppelmeier82}. \nThey show that there are three groups of \nMn-O bonds in LaMnO$_3$ with different lengths: 1.91, 1.97 and \n2.17 \\AA{}\\cite{Mitchell96}, while in CaMnO$_3$ all bond lengths nearly the \nsame at \n1.90 \\AA{} \\cite{Poeppelmeier82}(variation is within 0.01 \\AA{}).\nOur previous work has compared our XAFS data with the diffraction results,\nand found that the local structure around the Mn site in LCMO CMR samples is \nvery similar to the average structure determined by the diffraction data.\nMore comparison details are shown in reference \\onlinecite{Booth98b}.\n\n\nIn Sec. II, we provide a brief description of the samples and some \nexperimental details. We present the magnetization and transport property data\nin Sec. III and our XAFS results in Sec. IV.\nThe conclusions are given in Sec. V. \n\n\\section{Samples and Experiments}\n\nThe La$_{0.67}$Ca$_{0.33}$MnO$_3$ thin-film samples were deposited on\nSrTiO$_3$ substrates using PLD; each film is 3000~\\AA{} thick.\nSee Ref. \\onlinecite{Worledge98} \nfor additional details. The samples we chose for the XAFS studies were: \nas-deposited at\n750~K, annealed at 1000~K and annealed at 1200~K. The annealed samples were\nheld at their respective temperatures for 10 hours in flowing oxygen, and\nwere heated and cooled at 2~K per minute.\n\nThe XAFS experiments were done on beamline 10-2 at SSRL using Si $<$220$>$\nmonochromator crystals and a 13-element Ge detector to collect\nMn $K_\\alpha$ fluorescence data. The thin film samples were aligned at \n$\\sim$55.0$^\\circ$\nwith respect to the X-ray beam to make x, y and z axes equivalent, and thus\ncorrespond to a powder. This angle comes from\nthe polarization dependence of the photoelectric effect\\cite{Pettifer90}.\nFor each sample, we made\nfour sweeps at each temperature; for two of these sweeps, we rotated the\nsample by 1.5$^\\circ$ in order to determine the position of glitches, which\nmust be removed.\n\n\\section{Magnetization and Resistivity}\n\nThe magnetization $vs$ temperature data for these thin film samples are \nshown in Fig. \\ref{magnetization}. All samples have broad \nferromagnetic-to-paramagnetic phase \ntransitions and T$_c$ increases with the annealing temperature. From these \nmeasurements, we extract T$_c$ for these samples (see Table I). \n\nThe saturated magnetization of the samples increases with annealing temperature.\nThis indicates that at higher annealing temperature, a larger fraction of\nthe sample becomes ferromagnetic in the FM phase at low temperature. Our \nprevious experiments show that the 30\\% Ca doped LCMO powder sample has a Curie\ntemperature of $\\sim$ 260~K\\cite{Schiffer95}, which is very close \nto The T$_c$ of the fully annealed thin film sample. \n\n\nFig. \\ref{rho} is a plot of log$_e$ of the resistivity $vs.$ temperature\nfor all three samples. We show both the data in zero field and\nat 5.5 Tesla. \nThe resistivity decreases with annealing temperature. At zero field,\na large peak is present for the 1000~K annealed sample; the resistivity\ndecreases and the peak moves to higher temperature for\nthe 1200~K annealed sample. There is no metal-to-insulator phase transition for\nthe as-deposited sample; this sample also shows a very large resistivity at low\ntemperature. When the external magnetic field is raised to 5.5 Tesla,\nthe resistivity drops\ndramatically with magnetic field for the 1000~K and 1200~K annealed samples\nat temperatures near the resistivity peak, and the\nresistivity peaks move to higher temperature. For the as-deposited\nsample, the biggest change in MR occurs at the lowest measuring temperature \n(70~K). It should also be noted that the peak in resistivity is very close to\nT$_c$ for the 1200~K annealed sample, but well below T$_c$ (over 10~K\ndifference) for the 1000~K annealed sample.\n\nFig. \\ref{mr} shows CMR\\% (expressed as a percentage) $vs.$ temperature data \nfor these samples on a log$_e$\nscale. Here, we define CMR\\% to be:\n\n\\vspace*{0.2in}\nCMR\\% = 100$\\cdot$( R$_0$ - R$_H$ )/R$_H$\n\\vspace*{0.2in}\n\n\\noindent{where R$_0$ is the resistivity without magnetic field and R$_H$ is\nthe resistivity with a 5.5~T field.}\n\nThe magnetoresistance for these thin films is very large compared to that of\nthe corresponding powder samples\\cite{Booth98b}. This is especially \ntrue for the as-deposited sample\nat low temperature---the change of magnetoresistance at 70~K is about 3200\\%.\nThe maximum CMR\\% is about 1000\\% for\nthe 1000~K sample and about 250\\% for the 1200~K sample.\n\nThere is a double peak structure for the 1200~K annealed \nsample clearly visible in Fig.~\\ref{mr}. As mentioned earlier,\nour thin film samples are deposited on SrTiO$_3$ (STO) substrates. When the \nsample is deposited on STO, some Sr diffuses into the first few hundred \\AA~ \nof the film and changes the stoichiometry\nof that layer. This Sr diffusion could produce a resistance peak at a lower\ntemperature and may be responsible for this extra peak. We chose to use STO \nsubstrates instead of LaAlO$_3$ (LAO) \nsubstrates, which have no diffusion problem, because the large La \n$\\it{L_I}$-edge XAFS\nfrom the substrate would interfere with our Mn K edge XAFS data. Our XAFS \ndata are sensitive to this Sr diffusion layer only if we focus on the \nfurther neighbors such as Mn-La, Mn-Ca and Mn-Sr. The Mn-O bond distance does\nnot change much between Ca and Sr substitution. Consequently for the Mn-O\npair distribution function which is the focus of this paper, we are not \nvery sensitive to Sr in the lower 10\\% of the film.\n\n\n\\section{XAFS Data Analysis and Discussion}\n\nWe collected all Mn-$K$ edge data in fluorescence mode. First the pre-edge\nabsorption (absorptions from other edges) was removed using the\nVictoreen formula. Next we extract k$\\chi$(k) where the photoelectron \nwave vector k is obtained from\nk=$\\protect\\sqrt{2m_e(E-E_0)/h^2}$ and the XAFS function $\\chi$(k)\nis defined as $\\chi$(k)=$\\mu$(k)/$\\mu_0$(k)-1. We fit a 7-knot spline to\n$\\mu$(k) (the $K$-edge absorption data above the edge) to obtain the \nbackground function $\\mu_0$(k) (embedded atom function).\nAn example of such data\nis shown in Fig. \\ref{kspace}. Next we obtain the r-space data from the\nFourier Transform (FT) of k$\\chi$(k); fits to the data were carried out \nin r-space\\cite{Hayes82}. \nSome details of these fits are shown later in this paper.\nSee references \\onlinecite{Booth98a,Booth98b,Rehr94}\nfor additional details. \n\n\nIn Fig. \\ref{rspace}, we show the Mn $K$-edge Fourier transformed (FT) \n$r$-space data for\nall three thin-film samples. In this figure, the position of each peak \ncorresponds to an atom pair shifted by a well understood phase shift, \n$\\Delta$r; for example, \nthe first peak corresponds to the\nMn-O pairs and the second peak, which is near 3 \\AA{}, \ncorresponds to the Mn-La \n(Mn-Ca) pairs. The width, $\\sigma$, of the pair distribution function is a \nmeasure of the local distortions in a shell of neighboring atoms. In XAFS,\na large $\\sigma$ leads to a decrease in the amplitude of the $r$-space peak.\nWe obtain some results from these data without having to resort to curve\nfitting. First, the amplitude of the Mn-O peak (the first peak) decreases\nwith increasing temperature. That means there are increasing distortions in \neach sample with increasing temperature.\nSecond, for the higher annealing temperature, the \namplitude at low temperature increases and the\nchange of the amplitude with temperature for the Mn-O peak is larger than \nthat for the other samples. The amount of distortion removed at low\ntemperature is smaller for the 1000~K annealed sample, \nwhile most of the distortion in the\nas-deposited sample is still present at $T$ = 20~K. The amplitude of the r-space\ndata at high temperature (310~K) is almost the same for all three samples;\nthis suggests that the amount of distortion at high temperatures\ndoesn't change very much with the annealing temperature; however it is still\nclearly less than the local distortion observed previously in LaMnO$_3$ at\n300~K\\cite{Booth98a,Booth98b}. \n\n\nOur r-space data were fit using a Gaussian pair distribution function for the\nMn-O, Mn-Ca/La and Mn-Mn shells. The pair-distribution width,\n$\\sigma$(T), (for Mn-O) was determined from these detailed fits \nto the data, which were carried out using FEFF6 theoretical\nfunctions\\cite{Zabinsky95} (See Fig. \\ref{s2}). In the fit, S$_0^2$N was \nfixed at 4.3, where N is the\nnumber of nearest neighbors (6 O neighbors), and S$_0^2$ \nis an amplitude reduction factor. k$\\chi$(k) arrives from single-electron\nprocess and is normalized to the step height; S$_0^2$ corrects for the fact\nthat the measured step height also includes multi-electron process. \nThere is an absolute uncertainty in S$_0^2$ of roughly 10\\%, however small \nchanges in S$_0^2$ move all the curves up or down in Fig. \\ref{s2} and do not \nchange the shape or relative position.\n\nThe values of $\\sigma^2$ obtained from the fits provide a measure of the\ndistortion of the Mn-O bond. Different contributions to the broadening add in\nquadrature and hence $\\sigma^2$ has the general form:\n\n\\[\n\\sigma^2 = \\sigma^2_{static} + \\sigma^2_{phonons} + \\sigma^2_{other-mechanisms}\n\\]\n\nAt low temperature, $\\sigma^2$ is dominated by zero-point motion and some \nstatic distortions. For all the manganites, the smallest value for $\\sigma$ is \nabout 0.04 \\AA{}; consequently small variations in bond lengths, such as \noccurring for CaMnO$_3$ are not directly observed in $\\sigma$. It is surprising\nthat the net distortion of the substituted CMR samples is about as small as\nthat observed for the more ordered CaMnO$_3$ structure at 20~K.\n\nFor each temperature 4 traces are analyzed\nand averaged. The relative errors shown in Fig. \\ref{s2} are the \nroot mean square (rms) variation\nof the fit result at each temperature. For the as-deposited sample,\n$\\sigma^2$ has a very small change with decreasing temperature; there is a \nlarger change for the 1000~K annealed sample, and an even larger \ntemperature dependence for the 1200~K annealed sample.\nWe also find in Fig. \\ref{s2} that, at 320~K, $\\sigma^2$\nincreases as the annealing temperature is lowered. This indicates that some of \nthe static defects in the as-deposited sample can be removed during the \nannealing process.\n\nThe solid line in Fig. \\ref{s2} corresponds to the data for the Mn-O bond \nin CaMnO$_3$, which has a high\nDebye temperature (950~K); $\\sigma^2$ for this sample will be denoted\n$\\sigma_T^2$ (The CaMnO$_3$ data we use in this \nfigure are obtained from \na powder sample\\cite{Booth98b}. There might be a difference up to \n10\\% between powder samples\nand thin film samples since they are in different form and this difference \nmay change the effective value of S$_0^2$ for the fit process.)\n\nThe difference between $\\sigma^2_{data}$ and $\\sigma^2_T$ at low temperature\nis due to a static distortion $\\sigma^2_{static}$, which is defined as:\n\n\\[\n\\sigma^2_{static} = \\sigma^2_{data}(20K) - \\sigma^2_T(20K).\n\\]\n\nTo estimate this quantity, we\nshift the solid line vertically until it fits the low temperature data for\nthe 1200~K annealed sample This yields the dashed line in Fig. \\ref{s2}, which\nis defined to be $\\sigma^2_T$ + $\\sigma^2_{static}$.\nAlthough $\\sigma^2_{static}$ is almost zero for the 1200~K annealed sample,\nit is large for the other samples. We include it here for the 1200~K annealed\nsample to clarify its definition.\nThe contribution to $\\sigma^2_{data}$ above the dashed line is attributed to a \npolaron distortion, where the full (maximum) polaron distortion, \n$\\sigma^2_{FP}$, is defined\nby:\n\n\\[\n\\sigma^2_{FP} = \\sigma^2_{data}(300K) - \\sigma^2_T(300K) - \\sigma^2_{static}\n\\]\n\nWe have found in previous work\\cite{Booth98a,Booth98b} that a useful \nparameter is the distortion removed as T drops below T$_c$, $\\Delta\\sigma^2$,\nwhich we define below. First,\nthe $\\sigma^2_T$ curve is shifted vertically (by an amount\n$\\sigma^2_{FP}$ + $\\sigma^2_{static}$) such that it\nfits the high temperature data. This yields the dotted line shown in \nFig. \\ref{s2} which is $\\sigma^2_T$ + $\\sigma^2_{static}$ + $\\sigma^2_{FP}$.\nThis dotted line represents the expected Debye behavior plus static \ndistortion if no polaron distortion were removed upon cooling.\nWe define $\\Delta\\sigma^2$ as the difference \nbetween the dotted line and the data:\n\n\n\\[\n\\Delta\\sigma^2 = \\sigma^2_T + \\sigma^2_{FP} + \\sigma^2_{static} - \\sigma^2_{data}\n\\]\n%\n\nA similar analysis is carried out for the as deposited\nsample and the 1000~K annealed sample (corresponding curves for $\\sigma_T^2$\n+ $\\sigma_{static}^2$ and\n$\\sigma_T^2$ + $\\sigma_{static}^2$ + $\\sigma_{FP}^2$ are not shown in \nFig. \\ref{s2}). It is also important to point out that the\ndifference between $\\sigma^2$ for the 1200~K annealed sample and that of\nCaMnO$_3$ at 20~K is very small, which suggests that the Mn-O local structure \nof the fully annealed sample can be as ordered as that of CaMnO$_3$ even though\nit is a doped sample. The same\nresult was obtained for La$_{1-x}$Ca$_x$MnO$_3$ CMR powder samples \n(x = 0.2$\\sim$0.5) from our previous experiments\\cite{Booth98a,Booth98b}.\nThe reason for this phenomena for the thin films can be explained as follows:\nfirst, the high temperature annealing process appears to remove most of the\nstatic defects such as dislocations and vacancies; second, at very low \ntemperatures, there is almost no difference\nbetween the two type of Mn sites in the DE model, the electron moves \nrapidly from one site to another on a time scale fast compared to the \nappropriate phonon\nfrequency. Consequently, Jahn-Teller distortions don't have time to form.\n\nFor the as-deposited sample, the large CMR\\% occurs when \nthere is a large value for $\\sigma^2$ at low temperature; this suggests that\nthe distortion in the sample may, in part, be the origin of the unusually large \nmagnetoresistance in thin-film samples. We have recently observed similar\nresults in our study of Ti and Ga doped LCMO powder samples\\cite{Dcao99}.\n\n\nIn Fig. \\ref{lns2}, we plot $\\ln \\Delta\\sigma^2$ {\\it vs} $M/M_0$.\nOur previous studies\nof La$_{1-x}$Ca$_x$MnO$_3$ powder samples showed that there is a linear\nrelationship between $\\ln \\Delta\\sigma^2$ and the\nmagnetization\\cite{Booth98a,Booth98b}, which\nprovides evidence that there is a strong connection between the local distortion\nand magnetism in these materials. The solid squares in Fig. \\ref{lns2} show\nthe linear\nrelationship for a La$_{0.70}$Ca$_{0.30}$MnO$_3$ powder sample from our previous\nwork\\cite{Booth98a,Booth98b}. For the thin film samples, there is a similar \nconnection\nbetween local distortion and magnetization.\nFor the 1000~K and 1200~K annealed samples, we find a small deviation from\na straight line below M/M$_0$ $\\sim$0.3. Since the error of the data in this\nrange (M/M$_0$ $<$ 0.3) is large, it is not clear\nif this is a real effect or not. The data for the as-deposited sample\nappear to follow a straight line, but the errors in the difference are too\nlarge to draw conclusions. Also, we find that the data for the 1200~K annealed\nthin-film and the powder sample (both La$_{0.70}$Ca$_{0.30}$MnO$_3$) almost \noverlap each other. This suggests that the local structure of the fully annealed\nthin-film behaves similarly to that of the corresponding bulk sample.\n\n\nIn order to see the effect of annealing temperature on the local distortion more\nclearly, we plot the distortion contributions, $\\sigma_{static}^2$ and \n$\\sigma_{FP}^2$, as a function of annealing temperature in\nFig. \\ref{disorder}. This figure shows that the static\ndistortion decreases with annealing temperature, while the polaron contribution \nincreases with annealing temperature. This suggests that part of the static \ndistortions observed in the as-deposited sample become dynamic, polaron-like,\ndistortions after annealing, for T $>$ T$_c$. \n\nIt should be noted that although the magnetization\nonly drops by roughly 50\\% for the as-deposited sample, the $\\sigma_{FP}^2$ \ncontribution becomes much smaller ( of order 5\\%). Consequently, there must be \nstatically\ndistorted regions in the as-deposited sample that are also ferromagnetic.\nThe reduction of the saturated magnetization can arise in several ways.\nFirst, because of the inhomogeneous material, small regions may be \nantiferromagnetic\n(AF). Second, it has been suggested that the crystal field, particularly \nin regimes where\nthe inhomogeneity causes a local reduction of the tolerance factor, can result\nin a low spin Mn ($^4$t$_{2g}$) configuration with a local 50\\% decrease in \nMn moment\\cite{Geballe98}. Third, the spins in a small domain may not be \nexactly parallel.\nHowever, to explain the entire decrease in saturated magnetization would require\nvery large canting angles. Fourth, the magnetization vectors of each domain\nmay not be aligned. The number of domain-walls may be important for calculating\nthe resistivity. However, for domains large compared to a unit cell, slightly\ncanted spins within a domain or a lack of alignment of the magnetization of \nvarious domains would not lead to a significant decrease in the polaron \ncontribution to the broadening in XAFS. Consequently, the presence of static \ndistortions but essentially no polaron-like contributions in the as-deposited\nmaterial indicate both a significant fraction of AF material, and the \npresence of some low spin Mn sites, possibly induced by the disorder \n(interstitials, vacancies, inhomogeneous Ca concentrations etc.).\nThis disorder must pin the local distortions which would also suppress the \nelectron\nhopping frequency, thereby reducing the effectiveness of the DE interaction. \n\n\nAll three thin film samples were prepared in the same way, except for the\nannealing temperature. During the annealing process, part of the static\ndefects in the sample such as vacancies and interstitials can be removed.\nIn addition, the annealing process can also change the amount of oxygen in\nthe sample\\cite{Worledge96,Worledge98,Ju95,Malde98}. The sample is slightly\noxygen deficient before the annealing process, and oxygen is\nincorporated during annealing. The fully annealed sample\n(1200~K) is expected to be fully\nstoichiometric ( similar to the corresponding powder sample).\nIt is well documented\nthat samples without sufficient oxygen can have higher resistivities, a lower\nresistance peak temperature, a lower T$_c$ and a lower saturated\nmagnetization\\cite{Worledge96,Worledge98,Ju95,Malde98}.\nWe have the same trends in our experiments.\n\n\nCompared to the 1200~K annealed sample,\nthe samples annealed at lower temperatures have a large decrease in \n$\\sigma^2_{FP}$\nand a large increase in resistivity, while the saturated magnetization changes\nby less than a factor of two. This suggests that much of the distorted \nmagnetic regions \nprobably do not contribute to the conductivity and that the fraction of \nconducting material is very low for the as-deposited sample. However, the six\norder of magnitude increase in resistivity is much larger than the volume \nreduction of the regions that still have a polaron-like distortion. \nConsequently, it is likely that percolation also plays a role for transport in \nthe as-deposited sample. Then the magnetic field may play two roles for this \nsample; it will decrease the resistivity for the conducting pathways that \nexist at B=0 and may also make some \"marginal\" pathways become conducting.\n\n\\section{Conclusion}\n\nFrom our analysis, we find that the annealing temperature of the thin films\naffects the local distortion of the materials appreciably.\nThe large change in resistivity and the small change in local structure with\ntemperature for the\nas-deposited sample suggest that only small regions are contributing to the\nresistivity and percolation may play a role. We also find that\nthere is still a strong connection between local distortions, resistivity and\nmagnetism in the fully annealed thin-film materials, which behaves much \nlike a powder sample. For the 1000~K annealed sample, the resistivity and MR\npeaks are well below T$_c$. In this case (including the as-deposited sample),\nthe local distortions correlate well with the magnetization, but there is no \nfeature in $\\sigma$ that occurs at the temperature at which the resistivity \nhas a peak.\n\n\\acknowledgements{\nThis work was supported in part by NSF grant DMR9705117.\nThe experiments were performed at SSRL, which is operated by the DOE,\nDivision of Chemical Sciences, and by the NIH, Biomedical Resource Technology\nProgram, Division of Research Resources.} \n\n\\begin{thebibliography}{10}\n\n\\bibitem{Zener51}\nC. Zener, Phys. Rev. {\\bf 82}, 403 (1951).\n\n\\bibitem{Anderson55}\nP.~W. Anderson, H. Hasegawa, Phys. Rev. {\\bf 100}, 675 (1955).\n\n\\bibitem{deGennes60}\nP.~G. de~Gennes, Phys. Rev. {\\bf 118}, 141 (1960).\n\n\\bibitem{Millis95}\nA.~J. Millis, P.~B. Littlewood, and B.~I. Shraiman, Phys. Rev. Lett. {\\bf 74},\n 5144 (1995).\n\n\\bibitem{Booth98a}\nC.~H. Booth, F. Bridges, G.~H. Kwei, J.~M. Lawrence, A.~L. Cornelius, and J.~J.\n Neumeier, Phys. Rev. Lett. {\\bf 80}, 853 (1998).\n\n\\bibitem{Booth98b}\nC.~H. Booth, F. Bridges, G.~H. Kwei, J.~M. Lawrence, A.~L. Cornelius, and J.~J.\n Neumeier, Phys. Rev. B {\\bf 57}, 10440 (1998).\n\n\\bibitem{Billinge96}\nS.~J.~L. Billinge, R~.G. DiFrancesco, G.~H. Kwei, J.~J. Neumeier, and J.~D. \nThompson, Phys. Rev. Lett. {\\bf 77}, 715 (1996).\n\n\\bibitem{Louca97}\nD. Louca, T. Egami, E.~L. Brosha, H. R{\\\"{o}}der, and A.~R. Bishop,\nPhys. Rev. B {\\bf 56}, R8475 (1997).\n\n\\bibitem{Worledge96}\nD.~C. Worledge, G.~Jeffrey Snyder, M.~R. Beasley, and T.~H. Geballe,\nJ. Appl. Phys. {\\bf 80}, 5158 (1996). \n\n\\bibitem{Worledge98}\nD.~C. Worledge, L. Mieville, T.~H. Geballe,\nJ. Appl. Phys. {\\bf 83}, 5913 (1998).\n\n\\bibitem{Ju95}\nH.~L. Ju, J.~Gopalakrishnan, J.~L. Peng, Qi~Li, G.~C. Xiong, T.~Venkatesan \nand R.~L. Greene, Phys. Rev. B {\\bf 51}, 6143 (1995). \n\n\\bibitem{Malde98}\nN. Malde, P. S. I. P. N. De Silva, A. K. M. Akther Hossain, L. F. Cohen,\nK. A. Thomas, J. L. MacManus-Driscoll, N. D. Mathur and M. G. Blamire,\nSolid State Comm. {\\bf 105}, 643 (1998).\n\n\\bibitem{Norby95}\nP. Norby, I.~G. {Krogh Andersen}, E. {Krogh Andersen}, and N.~H. Andersen, J.\n Solid State Chem. {\\bf 119}, 191 (1995).\n\n\\bibitem{Mitchell96}\nJ.~F. Mitchell, D.~N. Argyriou, C.~D. Potter, D.~G. Hinks, J.~D. Jorgensen, and\n S.~D. Bader, Phys. Rev. B {\\bf 54}, 6172 (1996).\n\n\\bibitem{Poeppelmeier82}\nK.~R. Poeppelmeier, M.~E. Leonowicz, J.~C. Scanlon, and J.~M. Longo, J. Solid\n State Chem. {\\bf 45}, 71 (1982).\n\n\\bibitem{Pettifer90}\nR.F. Pettifer, C. Brouder, M. Benfatto, C.~R. Natoli, C. Hermes,\nM.~F. Ruiz Lopez, Phys. Rev. B {\\bf 42}, 37 (1990).\n\n\\bibitem{Schiffer95}\nP. Schiffer, A. Ramirez, W. Bao, and S-W. Choeng, Phys. Rev. lett. {\\bf 75},\n3336 (1995).\n\n\\bibitem{Hayes82}\nT. M. Hayes and J. B. Boyce, Solid State Physics {\\bf 37}, 173 (1982).\n\n\\bibitem{Rehr94}\nJ.~J. Rehr, C.~H. Booth, F. Bridges, S.~I. Zabinsky,\nPhys. Rev. B {\\bf 49}, 12347 (1994).\n\n\\bibitem{Zabinsky95}\nS.~I. Zabinsky, A. Ankudinov, J.~J. Rehr, R.~C. Albers,\nPhys. Rev. B {\\bf 52}, 2995 (1995).\n\n\\bibitem{Dcao99}\nD. Cao, F. Bridges, A.~P. Ramirez, M. Olapinski, M.~A. Subramanian,\nC.~H. Booth, G. Kwei, Unpublished.\n\n\\bibitem{Geballe98}\nT.~H. Geballe, B.~Y. Moyzches, 5th International Workshop in Oxide\nElectronics, Dec. 7 1998, U. of Maryland, Unpublished.\n\n\\end{thebibliography}\n\n\\begin{table}\n\\label{Tc}\n\\caption{The Curie temperature for each thin\nfilm sample.}\n\\vspace*{0.1in}\n\\begin{tabular}{|l|c|c|c|}\n sample & As-deposited & Annealed 1000K & Annealed 1200K \\\\\n\\hline\n T$_c$ (K)& 164(5) & 202(5) & 257(5) \\\\\n\\end{tabular}\n\\end{table}\n\n%\\vspace*{-1.2in}\n\\begin{figure}\n%\\hspace*{-2.4in}\n%\\vspace*{-7.15in}\n%\\psfig{file=fig1.ps,width=3.3in}\n\\vspace*{0.15in}\n\\caption{A plot of magnetization $vs$ temperature for\nLa$_{0.67}$Ca$_{0.33}$MnO$_3$ thin film samples, with an applied field of\n H=5000~Oe.}\n\\label{magnetization}\n\\end{figure}\n\n%\\vspace*{-1.2in}\n\\begin{figure}\n%\\hspace*{-2.4in}\n%\\vspace*{-7.15in}\n%\\psfig{file=fig2.ps,width=3.3in}\n\\vspace*{0.15in}\n\\caption{This plot shows the resistivity data (ln scale) for three thin film\nsamples, with and without magnetic field. The open symbol corresponds\nto data without field and the solid symbol corresponds to the data with a\nfield of 5.5~T.\nThere is a double peak structure\nfor the 1200~K annealed sample around 150~K, which is just visible in this\nfigure.\n}\n\\label{rho}\n\\end{figure}\n\n%\\vspace*{-1.2in}\n\\begin{figure}\n%\\hspace*{-2.4in}\n%\\vspace*{-7.15in}\n%\\psfig{file=fig3.ps,width=3.3in}\n\\vspace*{0.15in}\n\\caption{A plot of CMR\\% (ln scale) $vs$ temperature for\nLa$_{0.67}$Ca$_{0.33}$MnO$_3$\nthin film samples in a field of 5.5~T. The double peak structure\nfor the 1200~K annealed sample is obvious in this plot.\n}\n\\label{mr}\n\\end{figure}\n\n%\\vspace*{-1.2in}\n\\begin{figure}\n%\\hspace*{-2.4in}\n%\\vspace*{-7.15in}\n%\\psfig{file=fig4.ps,width=3.3in}\n\\vspace*{0.15in}\n\\caption{A plot of the $k$-space data for the 1200~K annealed sample at 20~K\nto show the quality of the data. Although there is some noise in the data, the\nquality is good up to 11 \\AA$^{-1}$\n}\n\\label{kspace}\n\\end{figure}\n\n%\\vspace*{-0.3in}\n\\begin{figure}\n%\\hspace*{-2.4in}\n%\\vspace*{-6.5in}\n%\\psfig{file=fig5.ps,width=3.3in}\n\\vspace*{0.15in}\n\\caption{A comparison of the change in the FT, $r$-space, data with temperature\nfor the La$_{0.67}$Ca$_{0.33}$MnO$_3$ thin-film samples with different annealing\ntemperatures. Top: 750~K anneal (as deposited), middle: 1000~K anneal, bottom:\n1200~K anneal. FT range is 3.3--10.5~\\AA$^{-1}$, with 0.3~\\AA$^{-1}$ Gaussian\nbroadening. The curve inside the envelope, with a higher frequency, is the\nreal part of the FT (FT$_R$).\nThe envelope is defined as:\n$\\pm \\protect\\sqrt{FT_R^2 + FT_I^2}$\nwhere FT$_I$\nis the imaginary part of the FT.\n}\n\\label{rspace}\n\\end{figure}\n\n%\\vspace*{-1.0in}\n\\begin{figure}\n%\\hspace*{-2.4in}\n%\\vspace*{-6.750in}\n%\\psfig{file=fig6.ps,width=3.3in}\n\\vspace*{0.15in}\n\\caption{A plot of $\\sigma^2$ $vs$ temperature for the as-deposited,\n1000~K and 1200~K annealed La$_{0.67}$Ca$_{0.33}$MnO$_3$ thin-film samples.\n(Here $\\sigma$ is the pair-distribution width of the Mn-O peak)\nThe solid line is the thermal contribution $\\sigma_T^2$ from \nCaMnO$_3$ \\protect\\cite{Booth98a,Booth98b}\nthe dashed line corresponds to $\\sigma_T^2$ + $\\sigma_{static}^2$ and the\ndotted line is $\\sigma_T^2$ + $\\sigma_{static}^2$ + $\\sigma_{FP}^2$ for the\nannealed 1200~K sample (see text).}\n\\label{s2}\n\\end{figure}\n\n\\begin{figure}\n%\\hspace*{-2.4in}\n%\\vspace*{-7.00in}\n%\\psfig{file=fig7.ps,width=3.3in}\n\\vspace*{0.15in}\n\\caption{A plot of ln$\\Delta\\sigma^2$ $vs$ $M/M_0$ is presented\nfor the as-deposited, 1000~K,\nand 1200~K annealed La$_{0.67}$Ca$_{0.33}$MnO$_3$ thin-film samples, as well as\na La$_{0.70}$Ca$_{0.30}$MnO$_3$ powder sample. M is the\nmeasured magnetization and M$_0$ is the saturated magnetization;\nM/M$_0$ is the relative magnetization.}\n\\label{lns2}\n\\end{figure}\n\n%\\vspace*{-1.0in}\n\\begin{figure}\n%\\hspace*{-2.4in}\n%\\vspace*{-6.750in}\n%\\psfig{file=fig8.ps,width=3.3in}\n\\vspace*{0.15in}\n\\caption{Static and polaron distortion $vs$ annealing temperature for all\nthree thin film samples. The solid triangle symbol represents the static\ndistortion, $\\sigma^2_{static}$; the open square symbol represents the\nfull-polaron distortion, $\\sigma^2_{FP}$.}\n\\label{disorder}\n\\end{figure}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002018.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{Zener51}\nC. Zener, Phys. Rev. {\\bf 82}, 403 (1951).\n\n\\bibitem{Anderson55}\nP.~W. Anderson, H. Hasegawa, Phys. Rev. {\\bf 100}, 675 (1955).\n\n\\bibitem{deGennes60}\nP.~G. de~Gennes, Phys. Rev. {\\bf 118}, 141 (1960).\n\n\\bibitem{Millis95}\nA.~J. Millis, P.~B. Littlewood, and B.~I. Shraiman, Phys. Rev. Lett. {\\bf 74},\n 5144 (1995).\n\n\\bibitem{Booth98a}\nC.~H. Booth, F. Bridges, G.~H. Kwei, J.~M. Lawrence, A.~L. Cornelius, and J.~J.\n Neumeier, Phys. Rev. Lett. {\\bf 80}, 853 (1998).\n\n\\bibitem{Booth98b}\nC.~H. Booth, F. Bridges, G.~H. Kwei, J.~M. Lawrence, A.~L. Cornelius, and J.~J.\n Neumeier, Phys. Rev. B {\\bf 57}, 10440 (1998).\n\n\\bibitem{Billinge96}\nS.~J.~L. Billinge, R~.G. DiFrancesco, G.~H. Kwei, J.~J. Neumeier, and J.~D. \nThompson, Phys. Rev. Lett. {\\bf 77}, 715 (1996).\n\n\\bibitem{Louca97}\nD. Louca, T. Egami, E.~L. Brosha, H. R{\\\"{o}}der, and A.~R. Bishop,\nPhys. Rev. B {\\bf 56}, R8475 (1997).\n\n\\bibitem{Worledge96}\nD.~C. Worledge, G.~Jeffrey Snyder, M.~R. Beasley, and T.~H. Geballe,\nJ. Appl. Phys. {\\bf 80}, 5158 (1996). \n\n\\bibitem{Worledge98}\nD.~C. Worledge, L. Mieville, T.~H. Geballe,\nJ. Appl. Phys. {\\bf 83}, 5913 (1998).\n\n\\bibitem{Ju95}\nH.~L. Ju, J.~Gopalakrishnan, J.~L. Peng, Qi~Li, G.~C. Xiong, T.~Venkatesan \nand R.~L. Greene, Phys. Rev. B {\\bf 51}, 6143 (1995). \n\n\\bibitem{Malde98}\nN. Malde, P. S. I. P. N. De Silva, A. K. M. Akther Hossain, L. F. Cohen,\nK. A. Thomas, J. L. MacManus-Driscoll, N. D. Mathur and M. G. Blamire,\nSolid State Comm. {\\bf 105}, 643 (1998).\n\n\\bibitem{Norby95}\nP. Norby, I.~G. {Krogh Andersen}, E. {Krogh Andersen}, and N.~H. Andersen, J.\n Solid State Chem. {\\bf 119}, 191 (1995).\n\n\\bibitem{Mitchell96}\nJ.~F. Mitchell, D.~N. Argyriou, C.~D. Potter, D.~G. Hinks, J.~D. Jorgensen, and\n S.~D. Bader, Phys. Rev. B {\\bf 54}, 6172 (1996).\n\n\\bibitem{Poeppelmeier82}\nK.~R. Poeppelmeier, M.~E. Leonowicz, J.~C. Scanlon, and J.~M. Longo, J. Solid\n State Chem. {\\bf 45}, 71 (1982).\n\n\\bibitem{Pettifer90}\nR.F. Pettifer, C. Brouder, M. Benfatto, C.~R. Natoli, C. Hermes,\nM.~F. Ruiz Lopez, Phys. Rev. B {\\bf 42}, 37 (1990).\n\n\\bibitem{Schiffer95}\nP. Schiffer, A. Ramirez, W. Bao, and S-W. Choeng, Phys. Rev. lett. {\\bf 75},\n3336 (1995).\n\n\\bibitem{Hayes82}\nT. M. Hayes and J. B. Boyce, Solid State Physics {\\bf 37}, 173 (1982).\n\n\\bibitem{Rehr94}\nJ.~J. Rehr, C.~H. Booth, F. Bridges, S.~I. Zabinsky,\nPhys. Rev. B {\\bf 49}, 12347 (1994).\n\n\\bibitem{Zabinsky95}\nS.~I. Zabinsky, A. Ankudinov, J.~J. Rehr, R.~C. Albers,\nPhys. Rev. B {\\bf 52}, 2995 (1995).\n\n\\bibitem{Dcao99}\nD. Cao, F. Bridges, A.~P. Ramirez, M. Olapinski, M.~A. Subramanian,\nC.~H. Booth, G. Kwei, Unpublished.\n\n\\bibitem{Geballe98}\nT.~H. Geballe, B.~Y. Moyzches, 5th International Workshop in Oxide\nElectronics, Dec. 7 1998, U. of Maryland, Unpublished.\n\n\\end{thebibliography}"
}
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cond-mat0002019
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Vortex Lattice Depinning vs. Vortex Lattice Melting: a pinning-based explanation of the equilibrium magnetization jump
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[
{
"author": "Y. Kopelevich\\thanksref{kope}"
}
] |
In this communication we argue that the Vortex Lattice Melting scenario fails to explain several key experimental results published in the literature. From a careful analysis of these results we conclude that the Flux Line Lattice (FLL) does not melt along a material- and sample-dependent boundary $H_j(T)$ but the opposite, it de-couples from the superconducting matrix becoming more ordered. When the FLL depinning is sharp, the difference between the equilibrium magnetization $M_{eq}(T,H)$ of the pinned and unpinned FLL leads to the observed step-like change $\Delta M_{eq}(T,H)$. We demonstrate that the experimentally obtained $\Delta M_{eq}(T,H)$ can be well accounted for by a variation of the pinning efficiency of vortices along the $H_j(T)$ boundary.
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[
{
"name": "deltam.tex",
"string": "\\documentstyle{elsart}\n\\input psfig\n\n\\begin{document}\n\\begin{frontmatter}\n\\title{Vortex Lattice Depinning vs. Vortex Lattice Melting: a pinning-based explanation of the equilibrium \nmagnetization jump}\n\n\\author[leipzig]{Y. Kopelevich\\thanksref{kope}} and\n\\author[leipzig]{P. Esquinazi}\n\\address[leipzig]{Department of Superconductivity and Magnetism, \nInstitut f\\\"ur Experimentelle Physik II, \nUniversit\\\"at Leipzig, Linn\\'estr. 5, D-04103 \nLeipzig, Germany}\n\\thanks[kope]{On leave from Instituto de Fisica, Unicamp, \n13083-970 Campinas, Sao Paulo, Brasil. Supported by the \nDeutsche \nForschungsgemeinschaft under DFG IK 24/B1-1,\nProject H, and by CAPES proc. No. 077/99.} \n\n\n\\begin{abstract}\nIn this communication we argue that the Vortex Lattice Melting scenario fails to explain several \nkey experimental\nresults published \nin the literature. From a careful analysis of these results \nwe conclude that the Flux Line Lattice (FLL) does not melt along a material- and sample-dependent \nboundary $H_j(T)$ but the opposite, it de-couples from the superconducting matrix becoming \nmore ordered. When the FLL depinning is sharp, the difference between the \nequilibrium magnetization $M_{eq}(T,H)$ of the pinned and unpinned FLL leads to the observed\nstep-like change $\\Delta M_{eq}(T,H)$. We demonstrate that the experimentally obtained \n$\\Delta M_{eq}(T,H)$ can be well accounted for by a variation of the pinning efficiency \nof vortices along the $H_j(T)$ boundary.\n\\end{abstract}\n\\begin{keyword}\nA. superconductors, D. flux pinning, D. phase transitions \n\\end{keyword}\n\\end{frontmatter}\n\n%\\pacs{74.60.Ge, 74.60.Jg., 74.76.Db}\n\nThe phenomenological description of superconductors is based on the \nknowledge of their magnetic field-temperature $(H-T)$ phase diagram. The equilibrium behavior \nof conventional type-II superconductors in an applied magnetic field is well known: At fields \nbelow the lower critical field $H_{c1}(T)$, the superconductor is in the Meissner-Ochsenfeld phase in \nwhich surface currents screen the magnetic field from the interior of the sample. Above $H_{c1}(T)$ the \nfield penetrates the superconductor in the form of a lattice of vortices, the so-called \nAbrikosov vortex lattice. This flux-line-lattice (FLL) persists up to the upper critical field $H_{c2}(T)$ \nwhere \nsuperconductivity vanishes in the bulk of the sample. \n\nIn high-temperature superconductors (HTS), however, due to strong \nthermal fluctuations, a first-order phase transition \nof the FLL to a liquid-like state, the ``melting'' of the FLL, has been predicted to occur \\cite{nelson,brandt}\n well below $H_{c2}(T)$. Since then, an enormous amount of experimental and theoretical work \nhas been done trying to find this, or other more sophisticated transitions and to extend the original \ntheoretical treatment\\cite{blatter,brandt2}. At the beginning of these research activities it was claimed \nthat the melting transition of the FLL in HTS manifests itself at the damping peak of vibrating HTS in \na magnetic field\\cite{gammel}. However, this interpretation was controversial \\cite{brandt3}.\nWe know nowadays that the damping peaks in vibrating superconductors \nattributed to the melting transition \\cite{gammel} can be explained quantitatively\nassuming thermally activated depinning and the diffusive motion of the FLL under a \nsmall perturbation generated by the vibration of the sample \\cite{ziese}. To find the true \nexperimental evidence for the melting transition of the FLL is by no means simple: Because \nthe FLL interacts with the superconducting matrix through pinning centers (atomic lattice defects, surface \nbarriers, etc.) every property of the FLL one measures will be influenced by the pinning and, therefore, \nno direct and straightforward proof of the melting phase transition can be achieved. \n\nSeveral years after the above cited first experimental attempt, a striking jump in the equilibrium magnetization \nin Bi$_2$Sr$_2$CaCu$_2$O$_8$ (Bi2212) single crystals has been measured using a SQUID\\cite{pastoriza} \nas well as sensitive micro-Hall sensors\\cite{zeldov}. This magnetization jump, which was interpreted as \na first-order transition of the FLL, lies at more than one order of magnitude lower fields than the \nthermally activated depinning line measured with vibrating superconductors\\cite{gammel,ziese}. \nThese interesting results\\cite{pastoriza,zeldov}are important because if the melting transition would \nbe of the first order, a discontinuous change in the equilibrium magnetization $M_{eq}(T,H)$ at the \ntransition is expected. The step-like increase of $M_{eq}(T,H)$ (a decrease in absolute value) with \nincreasing magnetic field and temperature has been also observed in YBa$_2$Cu$_3$O$_7$ (Y123) \nsingle crystals\\cite{welp}.\n \nApart from the clearly defined magnetization jump, another important fact was revealed by \nthe experiments. It has been found that the line $H_j(T)$ along which $M_{eq}(T,H)$ jumps, \nand the temperature dependence of the field $H_{\\rm SMP}(T)$ where an anomalous \nmaximum in the width of the magnetization hysteresis loop takes place (the so-called ``second \nmagnetization peak'' \n(SMP)), define a unique boundary on the $H-T$ plane for a given sample,\\cite{khaykovich,nishi}\n demonstrating their intimate relationship. In order to account for this behavior, a second-order phase \ntransition associated with the increase of the critical current density $j_c(T,H)$ was suggested as the \norigin of SMP based on a pinning-induced disordering of the FLL\\cite{ertas,giller,vinokur}.\n \nRecently, a similar SMP occurring at a temperature dependent field $H_{\\rm SMP}(T) \\ll H_{c2}(T)$ \nwas also measured in conventional Nb superconducting films\\cite{kope,esqui}. The studies of the \nSMP performed on Nb films,\\cite{kope,esqui} Bi2212 single crystals,\\cite{kope2} as well as on \na non-cuprate isotropic single crystalline Ba$_{0.63}$K$_{0.37}$BiO$_3$ thick film\\cite{galkin}\n provide a clear evidence that the SMP is not related to a critical current enhancement, but originates \nfrom a thermomagnetic instability effect and/or a non-uniform current distribution,\\cite{gurevich} \nleading to the ``hollow'' in $M(H)$ at $H < H_{\\rm SMP}(T)$. In agreement with the models, the \nSMP vanishes in all these superconductors when the lateral sample size becomes less \nthan $\\sim 100~\\mu$m \\cite{kope,esqui,kope2,galkin}. \nBecause $100~\\mu$m is much larger than all relevant vortex-pinning-related characteristic lengths, \nthe strong influence of the sample geometry on the SMP cannot be explained by a change of \nthe pinning efficiency of vortices. Besides, the results\\cite{esqui,kope2} suggest that the interaction \nbetween vortices starts to dominate that between vortices and the matrix at $H > H_{\\rm SMP}(T)$. \nWe note that the results obtained in Bi2212 crystals\\cite{kope2} are actually in good agreement \nwith the second-order diffraction small-angle neutron scattering (SANS) experiments which revealed a \nwell-ordered FLL at $H > H_{\\rm SMP}(T)$ \\cite{forgan}. Moreover, the formation of a more \nordered FLL with increasing temperature due to thermal depinning has been found in the above mentioned \nSANS experiments\\cite{forgan} at $H > H_{\\rm SMP}(T)$ and for intermediate temperatures. \nA similar result was obtained by means of Lorentz microscopy in Bi2212 thin films near the low \nfield - high temperature portion of the ``irreversibility line''\\cite{harada}. \nThe high resolution SANS measurements, recently reported for Y123 crystals\\cite{johnson} also \nrevealed a well defined FLL up to a field of 4 T and at low temperatures, i.e. above the \n$H_{\\rm SMP}(T)$-line measured in similar crystals\\cite{nishi}. \n\nBased on this experimental evidence and the intimate relationship between SMP- and the \nmagnetization-jump-lines,\\cite{khaykovich,nishi} we propose that the jump in $M_{eq}(T,H)$ \nresults from a magnetic-field- and temperature-driven FLL depinning transition to a more {\\em ordered}\n state of the FLL, effectively de-coupled from the atomic lattice. \n\nThere exist already experimental\\cite{puzniak,van,li} as well as theoretical\\cite{bula} works \nthat show that the interaction of vortices with pinning centers {\\em increases} $|M_{eq}(T,H)|$, indicating \nclearly that pinning influences the thermodynamic, equilibrium properties of superconductors. If the \nFLL depinning is sharp,\\cite{kos,wag} then one expects a step-like change of the equilibrium \nmagnetization $|\\Delta M_{eq}(T,H)| = |M_{eq}^{dis}(T, H) - M_{eq}(T, H)|$ along the \n$H_j(T)$ boundary. Here $|M_{eq}^{dis}(T, H)| > |M_{eq}(T, H)|$ is the absolute \nequilibrium magnetization in the presence of the quenched disorder which is measured below \n$H_j(T)$. We note further that the sharpness of the vortex depinning onset, irrespectively of \nthe underlying mechanism, manifests itself as a sudden increase in the electrical resistivity at $H_j(T)$, \nbelow which the vortex behavior is irreversible\\cite{fuchs}.\n\nIn what follows we present a phenomenological approach to describe the magnetization jump observed \nat the depinning transition. The equilibrium magnetization of an ordered, unpinned FLL in the \nLondon regime and neglecting fluctuations\\cite{bula2} is given by the equation \n\\begin{equation}\n M_{eq} = - \\frac{\\phi_0}{2(4\\pi\\lambda)^2} \\ln(\\eta H_{c2}/H)\\,,\t \n\\label{eq1}\n\\end{equation}\nwhere $\\lambda(T) \\equiv \\lambda_{ab}(T)$ is the in-plane London penetration depth, \n$\\phi_0$ is the flux quantum, and $\\eta$ is a parameter analogous to the \nAbrikosov ratio $\\beta_A = <|\\Psi|^4>/<|\\Psi^2|>^2 (\\Psi$ being the superconducting \norder parameter) that depends on the FLL structure\\cite{abri}. \nWe assume now that the $\\Delta M_{eq}$ results from a change of the \nparameter $\\eta$ due to a change in the vortex arrangement triggered by the \ninteraction of the FLL with pinning centers. Hence, the magnetization jump can be written as \n\\begin{equation}\n \\Delta M_{eq} = \\frac{\\phi_0}{2(4\\pi \\lambda)^2} \\ln(\\eta^{dis}/\\eta_0)\\,,\n\t\\label{eq2}\n\\end{equation}\nwhere the parameter $\\eta^{dis} \\ge \\eta_0$ is related to the strength of the quenched \ndisorder, and $\\eta_0$ applies for the FLL above the $H_j(T)$. Therefore, it is reasonable \nto assume that $\\eta^{dis}$ is proportional to the critical current density $j_c(T,H)$ which \nmeasures the vortex pinning strength. In fact, the correlation between $\\Delta M_{eq}$ and \nthe pinning of the FLL has been observed experimentally\\cite{farell}. We emphasise that \nalthough the relationship between $\\eta$ and $\\beta_A$ is unknown, one expects an \nincrease of the parameter $\\eta$ in the vortex liquid state compared to that of \nthe vortex solid, similar to the results obtained for $\\beta_A$ \\cite{hikami}.\n\\begin{figure}\n\\centerline{\\psfig{file=fig1.eps,height=5.2in}}\n\\caption{The jump $\\Delta B$ in the induction (a) and equilibrium magnetization \n$4\\pi \\Delta M$ (b) as a function of temperature measured for Bi$_2$Sr$_2$CaCu$_2$O$_8$ \n(Bi2212) \\protect\\cite{zeldov} and YBa$_2$Cu$_3$O$_{7-\\delta}$ (Y123)\\protect\\cite{welp}\n single crystals, respectively. The solid lines are obtained from Eq.~(2) with \n$\\eta^{dis} = \\eta_0[1 + aj_{c0}(1 - T/T_0)^n/(H^{\\alpha} + H_0)]$, with the fitting \nparameters $n = 1, \\alpha = 1.8, H_0 = 200~$Oe$^{1.8}, T_0 = 90$~K, $aj_{c0} = \n9.3 \\times 10^4~$Oe$^{1.8}$ for Bi2212 and $n = 1, \\alpha = 1, H_0 = 3.5 \\times 10^3~$Oe, \n$T_0 = 90.1~$K, $aj_{c0} = 2.6 \\times 10^5$~Oe for \nY123. Dotted line (b) corresponds to the equation $4 \\pi \\Delta M \n= 1.32 (1 - T/T_c)^{2/3} (Tc = 92.9~$K), according to the theoretical result from\nmelting theory \\cite{dogson}. \n}\n\\end{figure}\n \nWe show in Fig.~1 the jumps of the induction $\\Delta B(T,H)$ and of the magnetization \n$4\\pi \\Delta M(T, H)$ measured along the $H_j(T)$ boundaries in Bi2212 (a)\\cite{zeldov}\n and Y123 (b)\\cite{welp} single crystals, respectively. The difference in the \nbehavior of $\\Delta M_{eq}(T,H)$ in Bi2212 and Y123 crystals (see Fig.~1) can be easily \nunderstood noting that $H_j(T) < H^{\\ast} = \\phi_0/\\lambda^2$ for Bi2212, \nwhereas $H_j(T) \\gg H^{\\ast}$ in the case of Y123. At fields $H < H^{\\ast}$ the FLL \nshear modulus exponentially decreases with field as $c_{66} \\simeq \n(\\epsilon_0/\\lambda^2)(H\\lambda^2/\\phi_0)^{1/4}\n\\exp(-\\phi_0/H\\lambda^2)$, whereas at $H > H^{\\ast}$ it is proportional to the \nfield $c_{66} \\simeq (\\epsilon_0/4\\phi_0)H$ \\cite{wag}.\n \nThe exponential decrease of $c_{66}$ with decreasing $H_j$ (increasing $T_j$) in the case \nof Bi2212, implies an enhancement of the interaction between vortices and the \nquenched disorder (pinning centers) which leads to an increase of $\\Delta M_{eq}$ increasing $T_j$ \n(or decreasing $H_j$), see Fig.~1(a). As temperature tends to the critical one $T_c$, the \npinning of vortices vanishes. Therefore, above a certain temperature, $\\Delta M_{eq}(T,H)$ \ndecreases with temperature and tends to zero. \n\nIn the case of Y123, however, and due to a relatively weak field dependence of $c_{66}$ \nalong the $H_j(T)$ line, the reduction of the vortex pinning efficiency with temperature \nis the dominant effect which explains the monotonous decrease of $\\Delta M_{eq}(T,H)$ with \ntemperature, see Fig.~1(b). We stress that the vanishing of the magnetization jump \n$(\\Delta M_{eq}(T, H))$ at $T_0 \\simeq 90~$K, i.e. approximately 3 K below the \nsuperconducting transition temperature $T_c = 92.9~$K\\cite{welp} (similar result was obtained in \nanother untwinned Y123 crystal\\cite{schil}), can be also explained naturally by the \neffect of thermal fluctuations which smear out the pinning potential. On the other hand, the \ntheory\\cite{dogson} based on the FLL melting hypothesis predicts $\\Delta\n M_{eq} \\sim \\phi_0/\\lambda^2(T)$\n (dotted line in Fig.~1(b)) which implies the vanishing of $\\Delta M_{eq}$ at $T_c$. While the \nFLL-melting theory\\cite{dogson} requires different approaches in order to explain $\\Delta M_{eq}(T, H)$ \nin Y123 and Bi2212, our analysis can equally well be applied to both weakly (Y123) and \nstrongly (Bi2212) \nanisotropic superconductors.\n\nIn order to use Eq.~(2) to calculate the magnetization jump we need the relationship between $\\eta^{dis}$ \nand the critical current density $j_c(T, H)$ at the boundary $H_j(T)$, which is unknown at present. \nNevertheless and as an illustration of our ideas we present here a simple fitting approach. The solid lines \nin Fig.~1(a,b) were obtained from Eq.~(2) with $\\eta^{dis} = \\eta_0[1 + aj_c(T, H)] = \\eta_0[1 + \naj_{c0}(1 - T/T_0)^n/(H^\\alpha + H_0)]$ calculated at the boundary $H_j(T)$, where $T_0$ \ncorresponds to the temperature at which $j_c = 0$, and $a, j_{c0}, H_0$ are \nmodel-dependent constants, $n$ and $\\alpha$ are pinning related exponents. Note, that the \nhere used $j_c(T,H)$ is a rather general expression which reflects the well-known experimental fact that \nthe critical current density generally decreases with temperature and increasing field. \nIn our fits (solid lines in Fig.~1) we have set the exponents $n = 1, \\alpha = 1.8 (1)$, and \nused $\\lambda(T) = 250 (1- T/T_c)^{-1/3}~$nm ($140 (1-T/T_c)^{-1/3}$~nm) for\n Bi2212 (Y123)\\cite{kamal}; other fitting parameters close to those used give also satisfactory fits. \n\nFurthermore, within the here proposed physical picture we expect a reduction and ultimately the \nvanishing of $\\Delta M_{eq}(T, H)$ if by applying external driving forces one de-couples the \nFLL from the matrix. Such effect was observed in Bi2212 crystals, indeed\\cite{farell}. \nAlso, within our picture we expect that the FLL remains in a more ordered, metastable state if the sample \nis field cooled as compared to the zero-field-cooled state. Several published results have indicated such \na behavior, both directly (see, e.g. Ref.~\\cite{oral}) and indirectly. Among them, the \npioneer work\\cite{pastoriza} which demonstrates that the magnetization jump at the irreversibility \nline can be much larger in the zero-field-cooled sample compared to the field-cooled one, providing \na clear evidence that $\\Delta M_{eq}(T, H)$ is essentially related to the competition between \nvortex-vortex and vortex-pinning centers interactions. Supporting the above ideas, the magnetic-field-driven \ntransition from a disorder-dominated vortex state to a moving well-ordered FLL was observed in NbSe$_2$ \nlow-$T_c$ layered superconductor \\cite{pardo}.\n \nFinally, we would like to point out that \nthe equilibrium magnetization jump at a \nfirst-order depinning transition would imply the use of the Clausius-Clapeyron relation. \nHowever, we are not aware of any prediction on the \nentropy change at the depinning transition which we could\nuse for comparison. Therefore, we believe that it has little sense to comment here \non the use of the Clausius-Clapeyron equation at this stage of our study. \n\nTo summarise, based essentially on experimental facts we propose the magnetic-field and \ntemperature-driven vortex-lattice-ordering transition as an alternative to the FLL melting\nscenario in high-$T_c$ \nsuperconductors. Simple arguments allowed us to account for the equilibrium \nmagnetization jump associated \nwith the FLL depinning transition. This transition awaits for a rigorous theoretical treatment.\n\n\n\n\\ack\nWe acknowledge illuminating discussions with G. Blatter, E. H. Brandt, A.M. Campbell, G. Carneiro, \nF. de la Cruz, M. D\\\"aumling, R. Doyle, E.M. Forgan, D. Fuchs, A. Gurevich, B. Horovitz, \nM. Konczykowski, \nA. H. MacDonald, A. Schilling, T. Tamegai, V. Vinokur, E. Zeldov, and M. Ziese. This work \nwas supported by \nthe German-Israeli-Foundation for Scientific Research and Development and the Deutsche \nForschungsgemeinschaft.\n\n\n\n\n\n\n\n\\begin{thebibliography}{9}\n\\bibitem{nelson} D. R. Nelson, Phys. Rev. Lett. {\\bf 60}, 1973 (1988). \n\\bibitem{brandt}E. H. Brandt, Phys. Rev. Lett. {\\bf 63}, 1106 (1989). \n\\bibitem{blatter}G. Blatter, M. V. Feigel'man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, \nRev. Mod. Phys. {\\bf 66}, 1125 (1994).\n\\bibitem{brandt2}E. H. Brandt, Rep. Prog. Phys. 58, 1465-1594 (1995). \n\\bibitem{gammel}P. L. Gammel, L. F. Schneemeyer, J. V. Waszczak, and D. Bishop, \nPhys. Rev. Lett. {\\bf 61}, 1666-1669 (1988).\n\\bibitem{brandt3}E. H. Brandt, P. Esquinazi, and G. Weiss, Comment on Ref. 5. Phys. \nRev. Lett {\\bf 62}, 2330 (1989).\n\\bibitem{ziese}M. Ziese, P. Esquinazi, and H. F. Braun, Supercond. Sci. Technol. {\\bf 7}, \n869 (1994).\n\\bibitem{pastoriza}H. Pastoriza, M. F. Goffman, A. Arrib�re and F. de la Cruz, Phys. Rev. \nLett. {\\bf 72}, 2951 (1994). \n\\bibitem{zeldov}E. Zeldov, et al., Nature {\\bf 375}, 373 (1995). \n\\bibitem{welp}U. Welp et al., Phys. Rev. Lett. {\\bf 76}, 4809 (1996). \n\\bibitem{khaykovich}B. Khaykovich et al., Phys. Rev. Lett. {\\bf 76}, 2555 (1996). \n\\bibitem{nishi}T. Nishizaki, T. Naito, and N. Kobayashi, Phys. Rev. B {\\bf 58}, 11169 (1998).\n\\bibitem{ertas}D. Ertas, and D. R. Nelson, Physica C {\\bf 272}, 79, (1996).\n\\bibitem{giller}D. Giller et al., Phys. Rev. Lett. {\\bf 79}, 2542 (1997). \n\\bibitem{vinokur}V. Vinokur et al., Physica C {\\bf 295}, 209 (1998). \n\\bibitem{kope}Y. Kopelevich and P. Esquinazi, J. Low Temp. Phys. {\\bf 113}, 1 (1998).\n\\bibitem{esqui}P. Esquinazi et al., Phys. Rev. B {\\bf 60}, 12454 (1999). \n\\bibitem{kope2}Y. Kopelevich, S. Moehlecke, J. H. S. Torres, R. Ricardo da Silva, and P. \nEsquinazi, J. Low Temp. Phys. {\\bf 116}, 261 (1999). \n\\bibitem{galkin}A. Yu. Galkin et al., Solid State Commun. (2000, in press). \nWe note that the occurrence of similar $H_{\\rm SMP}(T)$-lines as well as the vanishing of the \nSMP for small enough samples in both layered and isotropic HTS indicate that the layer-decoupling \nscenario [see B. Horovitz, Phys. Rev. B {\\bf 60}, 9939 (1999)] does not apply. \n\\bibitem{gurevich}A. Gurevich, V. M. Vinokur, Phys. Rev. Lett. {\\bf 83}, 3037 (1999). \n\\bibitem{forgan}E. M. Forgan, M. T. Wylie, S. Lloyd, S. L. Lee, and R. Cubitt, Proc. LT-21: \nCzechoslovak Journal of Physics {\\bf 46}, 1571 (1996). \n\\bibitem{harada}K. Harada et al., Phys. Rev. Lett. {\\bf 71}, 3371 (1993). We note \nthat the arguments used by these authors regarding \nthe decrease in contrast of the signal for visualising the FLL by increasing field or\n temperature may, \nin principle, be applied to other techniques like neutron diffraction, muon spin \nrotation, and scanning Hall \nprobe microscope (see Ref.~\\protect\\cite{oral}) studies.\n\\bibitem{johnson}S. T. Johnson et al., Phys. Rev. Lett. {\\bf 82}, 2792 (1999).\n\\bibitem{puzniak}R. Pu$\\acute{\\rm z}$niak, J. Ricketts, J. Sch\\\"utzmann, G. D. Gu, and N. Koshizuka, \nPhys. Rev. B {\\bf 52}, 7042 (1995). \n\\bibitem{van}C. J. van der Beek, M. Konczykowski, T. W. Li, P. H. Kes, and W. \nBenoit, Phys. Rev. B {\\bf 54}, 792 (1996). \n\\bibitem{li}Q. Li et al., Phys. Rev. B {\\bf 54}, 788 (1996). \n\\bibitem{bula}L. N. Bulaevskii, V. M. Vinokur, and M. P. Maley, Phys. Rev. Lett. \n{\\bf 77}, 936 (1996). \n\\bibitem{kos}A. E. Koshelev, and P. H. Kes, Phys. Rev. B {\\bf 48}, 6539 (1993).\n\\bibitem{wag}O. S. Wagner, G. Burkard, V. B. Geshkenbein, and G. Blatter, Phys. Rev. \nLett. {\\bf 81}, 906 (1998). \n\\bibitem{fuchs}D. T. Fuchs et al., Phys. Rev. B {\\bf 54}, 796 (1996). \n\\bibitem{bula2}L. N. Bulaevskii, M. Ledvij, and V. G. Kogan, Phys. Rev. Lett. {\\bf 68}, \n3773 (1992). \n\\bibitem{abri}A. A. Abrikosov, in {\\it Fundamentals of the Theory of Metals}, North \nHolland (1998), pp. 414, 425. \n\\bibitem{farell}D. E. Farell et al., Phys. Rev. B {\\bf 53}, 11807 (1996). \n\\bibitem{hikami}S. Hikami, A. Fujita, and A. I. Larkin, Phys. Rev. B {\\bf 44}, 10400 (1991). \n\\bibitem{schil}A. Schilling et al., Nature {\\bf 382}, 791 (1996).\n\\bibitem{dogson}M. J. W. Dodgson, V. B. Geshkenbein, H. Nordborg, and G. Blatter, \nPhys. Rev. Lett. {\\bf 80}, 837 (1998). \n\\bibitem{kamal}S. Kamal et al., Phys. Rev. Lett. {\\bf 73}, 1845 (1994). \n\\bibitem{oral}A. Oral et al., Phys. Rev. Lett. {\\bf 80}, 3610 (1998).\n\\bibitem{pardo}F. Pardo, F. de la Cruz, P. L. Gammel, E. Bucher, and D. J. Bishop, \nNature {\\bf 396}, 348 (1998).\n\n\n\\end{thebibliography}\n\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002019.extracted_bib",
"string": "\\begin{thebibliography}{9}\n\\bibitem{nelson} D. R. Nelson, Phys. Rev. Lett. {\\bf 60}, 1973 (1988). \n\\bibitem{brandt}E. H. Brandt, Phys. Rev. Lett. {\\bf 63}, 1106 (1989). \n\\bibitem{blatter}G. Blatter, M. V. Feigel'man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, \nRev. Mod. Phys. {\\bf 66}, 1125 (1994).\n\\bibitem{brandt2}E. H. Brandt, Rep. Prog. Phys. 58, 1465-1594 (1995). \n\\bibitem{gammel}P. L. Gammel, L. F. Schneemeyer, J. V. Waszczak, and D. Bishop, \nPhys. Rev. Lett. {\\bf 61}, 1666-1669 (1988).\n\\bibitem{brandt3}E. H. Brandt, P. Esquinazi, and G. Weiss, Comment on Ref. 5. Phys. \nRev. Lett {\\bf 62}, 2330 (1989).\n\\bibitem{ziese}M. Ziese, P. Esquinazi, and H. F. Braun, Supercond. Sci. Technol. {\\bf 7}, \n869 (1994).\n\\bibitem{pastoriza}H. Pastoriza, M. F. Goffman, A. Arrib�re and F. de la Cruz, Phys. Rev. \nLett. {\\bf 72}, 2951 (1994). \n\\bibitem{zeldov}E. Zeldov, et al., Nature {\\bf 375}, 373 (1995). \n\\bibitem{welp}U. Welp et al., Phys. Rev. Lett. {\\bf 76}, 4809 (1996). \n\\bibitem{khaykovich}B. Khaykovich et al., Phys. Rev. Lett. {\\bf 76}, 2555 (1996). \n\\bibitem{nishi}T. Nishizaki, T. Naito, and N. Kobayashi, Phys. Rev. B {\\bf 58}, 11169 (1998).\n\\bibitem{ertas}D. Ertas, and D. R. Nelson, Physica C {\\bf 272}, 79, (1996).\n\\bibitem{giller}D. Giller et al., Phys. Rev. Lett. {\\bf 79}, 2542 (1997). \n\\bibitem{vinokur}V. Vinokur et al., Physica C {\\bf 295}, 209 (1998). \n\\bibitem{kope}Y. Kopelevich and P. Esquinazi, J. Low Temp. Phys. {\\bf 113}, 1 (1998).\n\\bibitem{esqui}P. Esquinazi et al., Phys. Rev. B {\\bf 60}, 12454 (1999). \n\\bibitem{kope2}Y. Kopelevich, S. Moehlecke, J. H. S. Torres, R. Ricardo da Silva, and P. \nEsquinazi, J. Low Temp. Phys. {\\bf 116}, 261 (1999). \n\\bibitem{galkin}A. Yu. Galkin et al., Solid State Commun. (2000, in press). \nWe note that the occurrence of similar $H_{\\rm SMP}(T)$-lines as well as the vanishing of the \nSMP for small enough samples in both layered and isotropic HTS indicate that the layer-decoupling \nscenario [see B. Horovitz, Phys. Rev. B {\\bf 60}, 9939 (1999)] does not apply. \n\\bibitem{gurevich}A. Gurevich, V. M. Vinokur, Phys. Rev. Lett. {\\bf 83}, 3037 (1999). \n\\bibitem{forgan}E. M. Forgan, M. T. Wylie, S. Lloyd, S. L. Lee, and R. Cubitt, Proc. LT-21: \nCzechoslovak Journal of Physics {\\bf 46}, 1571 (1996). \n\\bibitem{harada}K. Harada et al., Phys. Rev. Lett. {\\bf 71}, 3371 (1993). We note \nthat the arguments used by these authors regarding \nthe decrease in contrast of the signal for visualising the FLL by increasing field or\n temperature may, \nin principle, be applied to other techniques like neutron diffraction, muon spin \nrotation, and scanning Hall \nprobe microscope (see Ref.~\\protect\\cite{oral}) studies.\n\\bibitem{johnson}S. T. Johnson et al., Phys. Rev. Lett. {\\bf 82}, 2792 (1999).\n\\bibitem{puzniak}R. Pu$\\acute{\\rm z}$niak, J. Ricketts, J. Sch\\\"utzmann, G. D. Gu, and N. Koshizuka, \nPhys. Rev. B {\\bf 52}, 7042 (1995). \n\\bibitem{van}C. J. van der Beek, M. Konczykowski, T. W. Li, P. H. Kes, and W. \nBenoit, Phys. Rev. B {\\bf 54}, 792 (1996). \n\\bibitem{li}Q. Li et al., Phys. Rev. B {\\bf 54}, 788 (1996). \n\\bibitem{bula}L. N. Bulaevskii, V. M. Vinokur, and M. P. Maley, Phys. Rev. Lett. \n{\\bf 77}, 936 (1996). \n\\bibitem{kos}A. E. Koshelev, and P. H. Kes, Phys. Rev. B {\\bf 48}, 6539 (1993).\n\\bibitem{wag}O. S. Wagner, G. Burkard, V. B. Geshkenbein, and G. Blatter, Phys. Rev. \nLett. {\\bf 81}, 906 (1998). \n\\bibitem{fuchs}D. T. Fuchs et al., Phys. Rev. B {\\bf 54}, 796 (1996). \n\\bibitem{bula2}L. N. Bulaevskii, M. Ledvij, and V. G. Kogan, Phys. Rev. Lett. {\\bf 68}, \n3773 (1992). \n\\bibitem{abri}A. A. Abrikosov, in {\\it Fundamentals of the Theory of Metals}, North \nHolland (1998), pp. 414, 425. \n\\bibitem{farell}D. E. Farell et al., Phys. Rev. B {\\bf 53}, 11807 (1996). \n\\bibitem{hikami}S. Hikami, A. Fujita, and A. I. Larkin, Phys. Rev. B {\\bf 44}, 10400 (1991). \n\\bibitem{schil}A. Schilling et al., Nature {\\bf 382}, 791 (1996).\n\\bibitem{dogson}M. J. W. Dodgson, V. B. Geshkenbein, H. Nordborg, and G. Blatter, \nPhys. Rev. Lett. {\\bf 80}, 837 (1998). \n\\bibitem{kamal}S. Kamal et al., Phys. Rev. Lett. {\\bf 73}, 1845 (1994). \n\\bibitem{oral}A. Oral et al., Phys. Rev. Lett. {\\bf 80}, 3610 (1998).\n\\bibitem{pardo}F. Pardo, F. de la Cruz, P. L. Gammel, E. Bucher, and D. J. Bishop, \nNature {\\bf 396}, 348 (1998).\n\n\n\\end{thebibliography}"
}
] |
cond-mat0002020
|
Fractional Kramers Equation
|
[
{
"author": "Department of Chemistry"
},
{
"author": "and"
},
{
"author": "Center for Materials Science and Engineering"
},
{
"author": "M.I.T. Cambridge"
},
{
"author": "MA"
}
] |
We introduce a fractional Kramers equation for a particle interacting with a thermal heat bath and external non--linear force field. For the force free case the velocity damping follows the Mittag--Leffler relaxation and the diffusion is enhanced. The equation obeys the generalized Einstein relation, and its stationary solution is the Boltzmann distribution. Our results are compared to previous results on enhanced L\'evy type of diffusion derived from stochastic collision models.
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[
{
"name": "1kramers.tex",
"string": "%\n% \n%\\documentstyle[twocolumn,aps,prl]{revtex}\n%\\documentstyle[prl,aps,epsf,twocolumn]{revtex}\n%\\documentstyle[12pt]{article}\n\\documentstyle[preprint,eqsecnum,aps]{revtex}\n\\tightenlines\n\\begin{document}\n%\\draft\n%\n%\\def\\d{{\\rm d}}\n%\\def\\LFP{L_{\\rm FP}}\n%\n\\title{\nFractional Kramers Equation\n}\n\\author{\nE. Barkai\n\\footnote{corresponding author. Fax: +1-617-253-7030; e-mail: barkai@mit.edu}\n and R.J. Silbey\n\\\\Department of Chemistry\\\\and\\\\\nCenter for Materials Science and Engineering\\\\\nM.I.T. Cambridge, MA}\n\n%\\address{\n%$\n%Department of Chemistry\n%$\n%and \n%$\n%Center for Materials Science and Engineering.\n%$\n%Massachusetts Institute of Technology}\n%$\n\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\n\n We introduce a fractional Kramers equation for a \nparticle interacting with a thermal heat bath\nand external non--linear force field. For the\nforce free case the velocity damping follows\nthe Mittag--Leffler relaxation and the\ndiffusion is enhanced.\nThe equation obeys the generalized Einstein relation,\nand its stationary solution is the Boltzmann distribution.\nOur results are compared to previous results\non enhanced L\\'evy type of diffusion \nderived from stochastic collision models.\n\n\\end{abstract}\n\n\\pacs{02.50.-a,r05.40.Fb,05.30.Pr}\n\n%05.45.Df Fractals\n%05.40.Fb Random walks and Levy flights\n%05.20.Dd Kinetic theory\n%\n%05.30.Pr Fractional statistics systems (anyons, etc.)\n%05.60.-k Transport processes\n%02.50.-r Probability theory, stochastic processes, and \n% statistics (see also 05 Statistical physics, \n% thermodynamics, and nonlinear dynamical systems)\n\n\n\\section{Introduction}\n\n\n The pioneering work of Scher and Montroll \\cite{SM}\n and Scher and Lax \\cite{SL} on the \ncontinuous time random walk \\cite{Weiss1} applied to diffusion problems led to a revolution \nin our understanding of anomalous diffusion processes. \nAnomalous diffusion is now a well established phenomenon, \nfound in a broad range\nof fields \\cite{Weiss1,Bouch,Klafter1,Balescu,barkai8}. It is characterized by\na mean square displacement\n%\n\\begin{equation}\n\\langle x^2 \\rangle \\sim t^{\\delta}\n\\label{eqInt1}\n\\end{equation}\n%\nwith $\\delta \\ne 1$. \nVarious mechanisms are known to lead to\nenhanced diffusion $\\delta>1$ or sub-diffusion\n$\\delta<1$. Usually such processes are non-Gaussian\nmeaning that the standard central limit theorem cannot be used\nto analyze the long time behavior of these phenomena.\nIn order to describe such anomalous processes, fractional kinetic equations \nwere recently introduced by several authors\n\\cite{schneider,Gl,fogedby,zaslavsky,saichev,MBK,Hui,Grig,kuz}.\nWithin this approach,\nfractional space and/or time derivatives replace\nthe ordinary time and/or space derivatives in the\nstandard kinetic equation (e.g., Fokker--Planck equation).\nExamples include kinetics of viscoelastic media \\cite{Gl},\nL\\'evy flights in random\nenvironments \\cite{fogedby}, chaotic Hamiltonian dynamics \\cite{zaslavsky},\nand Quantum L\\'evy processes \\cite{kuz}. \nFor a discussion on L\\'evy statistics and continuous time random walk\nin the context of single molecule spectroscopy see \\cite{Zum,SB,MUK}.\n\n In this paper, we introduce a fractional Kramers equation\ndescribing both the velocity $v$ and coordinate $x$ of\na particle exhibiting anomalous diffusion in an external\nforce field $F(x)$.\nIn the absence of the external force field the equation describes\nenhanced diffusion.\nThe new equation we propose is\n%\n$$ {\\partial P(x,v,t) \\over \\partial t} + v {\\partial P \\over \\partial x} + {F(x)\\over M}\n{\\partial P(x,v,t) \\over \\partial v} = $$\n\\begin{equation}\n\\gamma_\\alpha\\ _0 D_t^{1-\\alpha} \\hat{L}_{fp} P(x,v,t),\n\\label{eqKr01a}\n\\end{equation}\n%\nwith $0< \\alpha < 1$ and\n%\n\\begin{equation}\n\\hat{L}_{fp}=\n{\\partial \\over \\partial v} v + { k_b T \\over M} { \\partial^2 \\over \\partial v^2} \n\\label{eqR01a}\n\\end{equation}\n%\n the dimensionless Fokker--Planck operator.\nHere we employ the fractional \nLiouville--Riemann operator \\cite{oldham} $\\ _0 D_t^{1-\\alpha} $\nin Eq. (\\ref{eqKr01a}) which we define later in\nEq. (\\ref{eqL02}). \n$\\gamma_{\\alpha}$ is a damping coefficient whose units\nare $[1/\\mbox{sec}^{\\alpha}]$.\nIn the presence of a bounding force\n$F(x)=-V'(x)$ the stationary solution of the fractional \nKramers equation (\\ref{eqKr01a})\nis the Maxwell--Boltzmann distribution. \nWhen $\\alpha=1$ we recover the standard Kramers equation.\n\n Fractional kinetic equations are related to Montroll-Weiss continuous time\nrandom walk (CTRW) \\cite{Hilfer,compte,West,barkai9}. \nHere we show that the fractional Kramers equation is related\nto the coupled CTRW in the enhanced diffusion regime $\\delta > 1$.\nThis case corresponds to the so called L\\'evy walks that are\nobserved in a number of systems[4,5].\nThe different limits of the CTRW were used to model\ndiverse physical phenomenon (when $F(x)=0$)\nand therefore fractional kinetic equations in general\nand the fractional Kramers equation in particular are believed\nto be of physical significance. \n\n \n The basis of the fractional Kramers equation is the \nfractional Ornstein--Uhlenbeck process described by\nthe fractional Fokker--Planck equation\n%\n\\begin{equation}\n{\\partial Q(v,t) \\over \\partial t} = \n\\gamma_\\alpha \\ _0 D_t^{1-\\alpha}\\hat{L}_{fp}Q(v,t),\n\\label{eqR04aa}\n\\end{equation}\n%\n$Q(v,t)$ the probability density of finding the particle at\ntime $t$ with velocity $v$ when $F(x)=0$.\nWe see that the fractional Kramers equation is a natural\nextension of Eq. (\\ref{eqR04aa}) in which the streaming terms\ndescribing Newtonian evolution are added in the standard way\n(i.e., as in the Boltzmann or Liouville equations).\nWhen $\\alpha=1$ we get the standard Ornstein--Uhlenbeck process\nwhich has a fundamental role in non equilibrium statistical mechanics.\n\n The Rayleigh model was used to derive Eq. (\\ref{eqR04aa})\nfor the normal case of $\\alpha=1$ \n\\cite{Ral,Kampen}.\nThe Rayleigh model for Brownian motion, also called the Rayleigh piston,\nconsiders a one-dimensional heavy particle \nof mass $M$ colliding with light non-interacting bath particles of mass $m$\nwhich are always thermalized. \nAccording to this model the moments of time intervals between collision events\nare finite.\nWhat happens when the variance of time intervals between collision events \ndiverges? In this case we anticipate a non Gaussian behavior.\nThis case has been investigated by Barkai and Fleurov \\cite{barkai3,barkai2}\nand as we shall show in subsection \n\\ref{secCM}, for a non-stationary model,\nsuch an anomalous case corresponds to the\nfractional Fokker--Planck equation (\\ref{eqR04aa}). \nHowever, as we shall show, the usual Rayleigh limit $m/M \\to 0$ \nin the non-Gaussian case is not as straightforward as for the\nordinary Gaussian case.\n\n The more fundamental question of the necessary conditions for transport to\nbe described by diverging variance of time intervals between collision events,\nor how to derive $\\alpha$ from first principles is not addressed in this\npaper.\nIn this context we note that\nseveral mechanisms in which the variance of time between collisions\n(or turning events in random walk schemes) \ndiverge and which lead to anomalous type of diffusion are known in the \nliterature \\cite{Bouch,Klafter1,barkai8}.\n\n Previous approaches \n\\cite{schneider,Gl,fogedby,zaslavsky,saichev,MBK,Hui,Grig}\nhave considered fractional kinetic equations\nin which the coordinate and/or time acquire the fractional character while in\nour approach the velocities\nare the variables that acquire fractional character. \nFor $F(x)=0$ we find $1\\le \\delta=2-\\alpha\\le 2$.\nThe lower bound $\\delta=1$ corresponds to normal diffusion, \nthe upper bound $\\delta=2$ can also be easily understood.\nFor a system close to thermal equilibrium we expect\n%\n\\begin{equation}\n\\langle x^2 \\rangle\\le (k_b T/M)t^2,\n\\label{bound}\n\\end{equation}\nand hence $\\delta\\le 2$.\n\n Previously \nKusnezov, Bulgac and Do Dang\n\\cite{kuz}\nhave suggested a fractional Kramers equation which was derived\nfor the classical limit of a Quantum L\\'evy process. Also in this work do the \nvelocities acquire fractional character; however this approach\nis very different\nthan ours, since it is based on Reisz fractional operators and\n gives $\\langle x^2(t) \\rangle=\\infty$ (i.e., $\\delta=\\infty$). \nRecently Metzler and Klafter \\cite{MKp} considered a fractional kinetic equation\nthat they also called a fractional Kramers equation. Their equation\ndescribes sub-diffusion ($\\delta < 1$) and is very different from our equation\nwhich described enhanced diffusion ($\\delta>1$).\n\n This paper is organized as follows.\nIn section \\ref{sectwo} we introduce the fractional\nOrnstein--Uhlenbeck process described by Eq. (\\ref{eqR04aa}),\na brief introduction to fractional calculus is given.\nIn subsection\n\\ref{secsol} the solution of the fractional Fokker--Planck \nEq. (\\ref{eqR04aa}) is presented and in subsection\n\\ref{secCM} we derive this equation based on stochastic\ncollision model. In section \n\\ref{Kra} we consider the fractional Kramers equation\n(\\ref{eqKr01a}), general properties of this equation are given\nand the force free case is investigated in some detail.\nWe end the paper with a short summary in section\n\\ref{secSum}.\n\n\n\\section{Fractional Ornstein--Uhlenbeck process}\n\\label{sectwo}\n\n\nLet $Q(v,t)$ be the probability density describing the velocity\n$v$ of a macroscopic Brownian particle, with mass $M$,\ninteracting with a thermal heat bath.\nThe Fokker--Planck equation describing the time evolution\nof $Q(v,t)$ with initial conditions $Q(v,t=0)$\nis given by \\cite{Kampen,Risken}\n%\n\\begin{equation}\n{\\partial Q\\left( v, t \\right) \\over \\partial t} = \\gamma_1 \\hat{L}_{fp} \nQ\\left(v,t\\right).\n\\label{eqR01}\n\\end{equation}\n%\nWe shall call such a Fokker--Planck equation standard or\nordinary.\nAccording to Eq. (\\ref{eqR01}) the damping law \nfor the averaged velocity is linear $\\langle \\dot{v}(t) \\rangle=-\n\\gamma_1 \\langle v(t) \\rangle$, and the velocity fluctuations\nare thermal. The stationary solution \nof Eq. (\\ref{eqR01}) is the Maxwell's density\ndefined with the thermal energy\n$k_b T$. \nThe corresponding Langevin equation is \n%\n\\begin{equation}\n\\dot{v}=-\\gamma_1 v + \\xi(t)\n\\label{eqLang}\n\\end{equation}\n%\nand $\\xi(t)$ is Gaussian white noise \\cite{Kampen,Risken}. \nThe stochastic process described by the Langevin equation is\nthe well known Ornstein--Uhlenbeck process.\nOriginally Eq. (\\ref{eqR01})\nwas derived by Rayleigh \\cite{Ral}\nfor a particle interacting with a gas consisting of\nlight particles, however the scope\nof Eq. (\\ref{eqR01}) is much wider. It is used to \nmodel Brownian motion in dense environments when memory effects\nare negligible.\n\n% Rayleigh \\cite{Ral} derived his equation base upon a one dimensional\n%collision model called the Rayleigh piston. A particle\n%with mass $M$ collides ellasticaly with non interacting\n%gas particles whose mass is $m<<M$. The velocity $\\tilde{v}$ of the\n%uncorrelated gas particles are randomly distributed described by\n%Maxwell's probability density. The gas particles are uniformally\n%distributed in space and events for which a light particle\n%collides more than once with the massive Brownian particle\n%are neglected. Other more sophisticated models \\cite{Oppenheim}\n%of Brownian motion\n%are known\n%to produce the dynamics described in\n%Eq. (\\ref{eqR01}) at least in certain limits. \n\n In this section we generalize the Fokker--Planck equation (\\ref{eqR01}) using fractional\ncalculus.\nFirst we give some mathematical definitions and tools.\n\n\n The Liouville--Riemann fractional integral operator \\cite{oldham}\nof order $\\alpha>0$ is defined by\n%\n\\begin{equation}\n_0 D_t^{-\\alpha} f(t) \\equiv \\int_0^t{\\left( t - t'\\right)^{\\alpha-1} \\over \\Gamma(\\alpha)} f(t') d t'.\n\\label{eqL01}\n\\end{equation}\n%\nFor integer values $\\alpha=n$; $_0 D_t^{-n}$ is the Riemann\nintegral operator of order $n$. Fractional differentiation\nof order $\\alpha>0$ is defined by\n%\n\\begin{equation}\n_0 D_t^{\\alpha} f(t) \\equiv {d^n \\over dt^n} \\left[ _0 D^{\\alpha-n}_t f\\left(t\\right)\\right] ,\n\\label{eqL02}\n\\end{equation}\n%\nwhere $n-1\\le \\alpha < n$ and $d^n/dt^n$ is ordinary differentiation\nof order $n$. Within this fractional calculus\n%\n\\begin{equation}\n_0 D^{\\pm \\alpha}_t t^{\\mu} = { \\Gamma\\left( \\mu+1 \\right) \\over \\Gamma\\left( \\mu \\mp \\alpha +1 \\right) } t^{\\mu \\mp \\alpha}\n\\label{eqL03}\n\\end{equation}\n%\nwhen $\\mu>-1$. Notice that $_0 D^{\\pm \\alpha}_t 1 \\sim t^{\\mp \\alpha}$\nwhen $0<\\alpha<1$. The Laplace transform\n%\n\\begin{equation}\nf(u) = \\int_0^{\\infty} e^{ - u t } f(t) dt= {\\cal L}[f(t)]\n\\label{eqL04}\n\\end{equation}\n%\nof a fractional Liouville--Riemann operator is\n%\n\\begin{equation}\n{\\cal L} \\left[ _0 D^{\\alpha}_t f(t)\\right] = u^{\\alpha} f(u) - \\sum_{k=0}^{n-1} u^{k}\\ _0 D_t^{\\alpha- 1 - k } f(t)|_{ t = 0}.\n\\label{eqL05}\n\\end{equation}\n%\n$n$ is an integer satisfying $n-1< \\alpha \\le n$.\nFor fractional integrals $\\alpha\\le 0$\nthe sum on the RHS of Eq. (\\ref{eqL05})\nvanishes. From Eq. (\\ref{eqL05}) \nwe see that the Laplace transform of a fractional\nderivative of $f(t)$ depends on fractional derivatives\nof that function at time $t=0$.\nIn this work we use the convention that the arguments\nof a function indicate the space in which the function\nis defined, e.g. the Laplace transform of $Q(v,t)$ is\n$Q(v,u)$.\n\n The Fokker--Planck equation (\\ref{eqR01})\nis rewritten in the integral form\n%\n\\begin{equation}\nQ(v,t) - Q(v,t=0)= \\gamma_1\\ _0 D_t^{-1}\\hat{L}_{fp} Q(v,t)\n\\label{eqR02}\n\\end{equation}\n%\nWe now replace the integer integral operator $_0 D_t^{-1}$ \nin Eq. (\\ref{eqR02}) with a fractional integral operator\n$_0 D_t^{-\\alpha}$ and $0<\\alpha\\le 1$. The result\nis\n%\n%\n\\begin{equation}\nQ(v,t) - Q(v,t=0)= \\gamma_\\alpha \\ _0 D_t^{-\\alpha}\\hat{L}_{fp} Q(v,t)\n\\label{eqR03}\n\\end{equation}\n%\nwhere $\\gamma_{\\alpha}$ is a generalized damping coefficient\nwith units $[\\mbox{sec}]^{-\\alpha}$.\nOrdinary differentiation of Eq. (\\ref{eqR03})\ngives the fractional Fokker--Planck equation\n%\n\\begin{equation}\n{\\partial Q(v,t) \\over \\partial t} = \\gamma_\\alpha \\ _0 D_t^{1-\\alpha}\\left[{\\partial \\over \\partial v} v + { k_b T \\over M} { \\partial^2 \\over \\partial v^2} \\right] Q(v,t).\n\\label{eqR04}\n\\end{equation}\n%\nWhen $\\alpha=1$ the ordinary Fokker--Planck Eq.\n(\\ref{eqR01})\nis obtained.\nIn Eq. (\\ref{eqR04}) we use the natural boundary conditions\n%\n\\begin{equation}\n\\lim_{v\\to \\pm \\infty} Q(v,t)=\\lim_{v\\to \\pm \\infty} \\partial Q(v,t)/\\partial v=0.\n\\label{eqR04a}\n\\end{equation}\n%\nLater we shall show that the solution of Eq. (\\ref{eqR04})\nis non-negative and normalized. Eq. (\\ref{eqR04}) describes a fractional\nOrnstein--Uhlenbeck process. When the velocity $v$ is replaced \nwith a coordinate\n$x$, the equation describes an anomalously over damped\n harmonic oscillator as investigated in \\cite{MBK}.\n\n\n Eqs. (\\ref{eqR03}) and (\\ref{eqR04}) are initial\nvalue problems.\nWhile Eq. (\\ref{eqR03}) depends on a single initial condition\n[$Q(v,t=0)$ on the LHS of the equation],\nin solving Eq. (\\ref{eqR04}) two initial conditions\nhave to be specified \\cite{oldham},\nthese being $Q(v,t=0)$ and $_0D_t^{-\\alpha}Q(v,t)|_{t=0}$.\nWhen setting $_0D_t^{-\\alpha}Q(v,t)|_{t=0}$ equal to zero, the two\nequations are equivalent\n\\cite{remark3}.\n\n Multiplying Eq. (\\ref{eqR04}) by $v$ and integrating over \n$v$ we find\nthat the mean velocity is described by\n%\n\\begin{equation}\n\\langle \\dot{v}\\left( t \\right) \\rangle= \n- \\gamma_{\\alpha}\\ _0 D^{ 1 - \\alpha}_t \\langle v\\left( t \\right) \\rangle.\\label{eqR07}\n\\end{equation}\n%\nIn Laplace space Eq. (\\ref{eqR07}) reads\n%\n\\begin{equation}\n\\langle v(u) \\rangle ={ v_0 \\over u + \\gamma_{\\alpha} u^{ 1 - \\alpha}}.\n\\label{eqR08}\n\\end{equation}\n%\n$v_0$ is the initial velocity.\nThe inverse Laplace transform of Eq. (\\ref{eqR08}) is\n%\n\\begin{equation}\n\\langle v(t) \\rangle = v_0 E_{\\alpha}\\left( - \\gamma_{\\alpha} t^{\\alpha} \\right),\n\\label{eqR09}\n\\end{equation}\n%\nand \n\\begin{equation}\nE_{\\alpha}\\left( z \\right) =\\sum_{n = 0 }^{\\infty} { z^n \\over \\Gamma\\left( 1 + \\alpha n \\right)}\n\\label{eqR10}\n\\end{equation}\n%\nis the Mittag--Leffler function.\nWhen $\\alpha=1$ the Mittag--Leffler reduces to the\nexponential. For large $t$, \nEq. (\\ref{eqR09}) exhibits a power law decay\n%\n\\begin{equation}\n\\langle v(t) \\rangle \\sim {v_0 \\left( \\gamma_{\\alpha} t\\right)^{- \\alpha} \\over \\Gamma\\left( 1 - \\alpha\\right) } \n\\label{eqvvv}\n\\end{equation}\n%\nand for short times the relaxation is a stretched exponential\n%\n\\begin{equation}\n\\langle v(t ) \\rangle \\simeq v_0 \\exp\\left[ - { \\gamma_{\\alpha} t^{\\alpha} \\over \\Gamma\\left( 1 + \\alpha\\right) } \\right].\n\\label{eqsstt}\n\\end{equation}\nIn a similar way we find the second moment\n%\n\\begin{equation}\n\\langle v^2(t) \\rangle = v_0^2 E_{\\alpha}\\left( - 2 \\gamma_{\\alpha} t^{\\alpha}\\right)+ { k_b T \\over M} \\left[ 1 - E_{\\alpha} \\left( - 2 \\gamma_{\\alpha} t^{\\alpha} \\right)\\right],\n\\label{eqv22}\n\\end{equation}\n%\nexhibiting power law decay towards the thermal equilibrium value\n$\\langle v^2 \\left(t=\\infty\\right)\\rangle =k_b T /M$.\nFrom Eqs. (\\ref{eqR09}) and\n(\\ref{eqv22}) we see that the Mittag--Leffler relaxation\nreplaces the ordinary exponential relaxation found for ordinary Brownian motion.\nThese equations were derived in \\cite{barkai3} based upon a stochastic collision model\nwhich we will discuss in subsection \\ref{secCM}.\n\n\n\\subsection{Solution}\n\\label{secsol}\n\n Our aims are {\\bf (a)} to find a solution of the fractional Fokker--Planck\nEq. (\\ref{eqR04}) and {\\bf (b)} show that $Q(v,t)$ in Eq.\n(\\ref{eqR04}) is a probability density. \nThe solution we find is an integral of a product of two\nwell known functions.\nWe use the initial conditions\n$Q(v,t=0)= \\delta(v - v_0)$ the generalization for other\ninitial conditions is carried out in the usual way.\n\n We first show that the solution is normalized.\nThe Laplace transform of Eq. \n(\\ref{eqR04}) is\n%\n\\begin{equation}\nuQ(v,u)-\\delta(v-v_0)=\\gamma_{\\alpha}u^{1 - \\alpha}\\hat{L}_{fp} Q(v,u).\n\\label{eqA01}\n\\end{equation}\n%\nIntegrating Eq. (\\ref{eqA01}) with respect to $v$,\nusing the boundary conditions\nin Eq. (\\ref{eqR04a}) and the normalized initial condition\nwe find\n%\n\\begin{equation}\n\\int_{-\\infty}^{\\infty} Q(v,u) dv= {1 \\over u}.\n\\label{eqANOR}\n\\end{equation}\n%\nSince ${\\cal L}(1,u) = 1/u$ we see that $Q(v,t)$ in \nEq. (\\ref{eqR04}) is normalized.\n\n Let us now find the solution in Laplace $u$ space.\nWe write $Q(v,u)$ as\n%\n\\begin{equation}\nQ(v,u)=\\int_0^{\\infty} R_{s}(u) G_{s}( v ) d s\n\\label{eqA04}\n\\end{equation}\n%\nwhere\n%\n\\begin{equation}\n\\hat{L}_{fp} G_{s} \\left( v \\right) = { \\partial \\over \\partial s} G_{s} \\left( v \\right)\n\\label{eqA05}\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n G_0\\left( v \\right) = \\delta( v - v_0).\n\\label{eqA05ic}\n\\end{equation}\n%\n%\nEq. (\\ref{eqA05}) is the dimensionless ordinary \nFokker--Planck Eq. (\\ref{eqR01}),\nwith solution \\cite{Kampen,Risken}\n%\n$$ G_{s} \\left( v \\right) = $$\n\\begin{equation}\n{ \\sqrt{M} \\over \\sqrt{ 2 \\pi k_b T \\left( 1 - e^{ - 2 s} \\right)}}\n\\exp\\left[ - { M\\left( v - v_0e^{ - s} \\right)^2 \\over\n2 k_b T \\left( 1 - e^{ - 2 s} \\right) } \\right].\n\\label{eqA06m}\n\\end{equation}\n%\nWe see that $G_s(v)$ is a non-negative probability density function,\ndescribing the standard \n Ornstein--Uhlenbeck process, and normalized according to\n%\n\\begin{equation}\n\\int_{-\\infty}^{\\infty} G_s(v) dv = 1.\n\\label{eqadd}\n\\end{equation} \n\n\n $R_{s}\\left( u \\right)$ in Eq.\n(\\ref{eqA04}) must satisfy a normalization condition. Using Eqs. \n(\\ref{eqANOR}) and (\\ref{eqadd}), we have\n%\n\\begin{equation}\n\\int_0^{\\infty} R_{s} \\left( u \\right)d s= {1 \\over u}.\n\\label{eqA07}\n\\end{equation}\n%\nInserting Eq. (\\ref{eqA04}) in Eq. (\\ref{eqA01}), using\nEq. (\\ref{eqA05}), and integrating by parts, we find \n%\n$$ u\\int_0^{\\infty} R_{s} \\left( u \\right) G_{s} ( v ) d s- \\delta(v-v_0) = $$\n%\n$$ \\gamma_{\\alpha} u^{1 - \\alpha} \\int_0^{\\infty} R_{s} \\left( u \\right) \\hat{L}_{fp} G_{s}\\left( v \\right) d s =$$\n%\n%\n$$ \\gamma_{\\alpha} u^{1 - \\alpha} \\int_0^{\\infty} R_{s} \\left( u \\right) \n{ \\partial \\over \\partial s} G_{s}\\left( v \\right) d s \n= $$\n%\n$$ \\gamma_{\\alpha} u^{1 - \\alpha} \\left[ R_{\\infty}\\left( u \\right) \nG_{\\infty}\\left( v \\right) - R_0\\left( u \\right) G_0\\left( v \\right) \\right]-$$\n\\begin{equation}\n\\gamma_{\\alpha} u^{ 1 - \\alpha} \\int_0^{\\infty}\n \\left[ {\\partial \\over \\partial s} R_{s}\\left( u \\right) \\right] G_{s}\\left( v \\right) d s.\n\\label{eqA08}\n\\end{equation}\n%\nAccording to Eq. (\\ref{eqA07}) the boundary term \n$R_{\\infty}(u)$\nin\nEq. (\\ref{eqA08}) is zero.\nUsing the initial condition Eq.\n(\\ref{eqA05ic})\n in Eq. \n(\\ref{eqA08})\nwe find\n%\n$$ \\int_0^{\\infty}\\left\\{ u R_{s}\\left(u\\right)+\\gamma_{\\alpha} u^{1-\\alpha}\\left[\n{\\partial \\over \\partial\ns} R_{s}\\left( u \\right) \\right] \\right\\} G_{s}\\left( v \\right) d s=$$\n\\begin{equation}\n\\left[1 - \\gamma_{\\alpha}u^{1 - \\alpha} R_0(s)\\right]\\delta(v - v_0).\n\\label{eqexp}\n\\end{equation}\n%\nEq. (\\ref{eqexp}) is solved once both sides of it are equal\nto zero; therefore, two conditions must be satisfied, the first being\n%\n\\begin{equation}\n\\gamma_{\\alpha} u^{1 - \\alpha} R_0 \\left( u \\right) = 1\n\\label{eqA09}\n\\end{equation}\n%\nand the second being\n%\n\\begin{equation}\n- \\gamma_{\\alpha} u^{ 1 - \\alpha} { \\partial \\over \\partial s} R_{s} \\left( u \\right) = u R_{s} \\left( u \\right).\n\\label{eqA10}\n\\end{equation}\n%\nThe solution of Eq. (\\ref{eqA10}) with the condition Eq. (\\ref{eqA09})\nis\n%\n\\begin{equation}\nR_{s}\\left( u \\right) = { 1 \\over \\gamma_{\\alpha} u^{ 1 - \\alpha} } \\exp\\left( \n- { s u^{\\alpha} \\over \\gamma_{\\alpha} } \\right).\n\\label{eqA11m}\n\\end{equation}\n%\nIt is easy to check that $R_{s}\\left( u \\right)$ is normalized according\nto Eq. \n(\\ref{eqA07}). \n\n\n The solution of the problem in $t$ space is the inverse Laplace of Eq.\n(\\ref{eqA04})\n%\n\\begin{equation}\nQ(v,t)=\\int_0^{\\infty} R_{s}(t) G_{s}( v ) d s,\n\\label{eqA04mt}\n\\end{equation}\n%\nwhere $R_{s}(t)$ is the inverse Laplace transform of $R_{s}(u)$ given by\n%\n\\begin{equation}\nR_{s}\\left( t \\right) = { 1 \\over \\alpha \\gamma_{\\alpha} t^{\\alpha}}\nz^{\\alpha + 1} l_{\\alpha}\\left( z \\right),\n\\label{eqlevy}\n\\end{equation}\n%\nand \n%\n\\begin{equation}\nz={ \\left( \\gamma_{\\alpha}\\right)^{1/\\alpha} t \\over s^{1/\\alpha} }.\n\\label{eqlevyz}\n\\end{equation}\n%\nProperties of $R_s(t)$ are discussed by Saichev and Zaslavsky \\cite{saichev}.\n$l_{\\alpha}\\left( z \\right)$ in Eq. (\\ref{eqlevy}) is one sided L\\'evy stable\ndensity \n\\cite{Feller}, \nwhose Laplace transform is \n%\n\\begin{equation}\nl_{\\alpha}(u)=\\int_0^{\\infty} \\exp\\left( - u z\\right) l_{\\alpha}\\left( z \\right) dz =\\exp( -u^{\\alpha}).\n\\label{eqadd1}\n\\end{equation}\n%\nThe proof that $R_{s}( t )$ Eq. (\\ref{eqlevy}) and \n$R_{s}( u )$ Eq.(\\ref{eqA11m}) are a Laplace pair is given in \nAppendix A (and see also \\cite{saichev}).\n\n A few features of the solution Eq. (\\ref{eqA04mt}) can now be discussed.\nAn interpretation of Eq.(\\ref{eqA04mt}) in terms of a stochastic \ncollision model will be given in the next subsection.\n\n When $0< \\alpha \\le 1$; $R_{s}(t)$ is a probability density normalized\naccording to $\\int_0^{\\infty} d s R_{s}(t)=1$.\nSince $G_{s}(v)$ is also a probability density\nthe solution Eq. (\\ref{eqA04mt}) is normalized and non negative.\nThis justifies our interpretation of $Q(v,t)$ \nas a probability density. \n\n When $\\alpha=1$ the solution reduces to the well known\nsolution of the ordinary Fokker--Planck equation. \nTo see this, note that the inverse Laplace of Eq.\n(\\ref{eqA11m}) for $\\alpha=1$ is\n$R_{s}(t) = {1/\\gamma_1} \\delta\\left( t -s/\\gamma_1 \\right)$,\nand then use definition Eq.\n(\\ref{eqA04mt}).\n When $\\alpha=1/2$ we have \n%\n\\begin{equation}\nl_{1/2}(z)={1 \\over 2 \\sqrt{\\pi}} z^{-3/2} \\exp\\left( - {1 \\over 4 z}\\right)\n\\label{eqlevyha}\n\\end{equation}\n%\nwith $z>0$, then $R_{s}(t)$ \nEq. (\\ref{eqlevy})\nis a one sided\nGaussian\n%\n\\begin{equation}\nR_{s}(t)=\\sqrt{{ 1 \\over \\pi \\gamma_{1/2}^2 t}} \\exp\\left( - { s^2 \\over 4 \\gamma_{1/2}^2 t } \\right).\n\\label{eqpthalf}\n\\end{equation}\n%\nTwo other closed forms of one sided L\\'evy probability\ndensities $l_{2/3}(z)$ and $l_{1/3}(z)$ can be found\nin \\cite{Zol,EWM}. Series expansions of L\\'evy\nstable density are in Feller's book \\cite{Feller} chapter\n{\\bf XVII.6}. \n\n In Fig. 1, we show the solution \nfor the case $\\alpha = 1/2$ and for different\ntimes. The solution is found with numerical integration\nof \n(\\ref{eqA04mt}) using Eqs. \n(\\ref{eqA06m}) and\n(\\ref{eqpthalf}). We choose the initial condition\n$Q(v,t=0)=\\delta(v-1)$, (i.e., $v_0 = 1$) and \n$k_b T / M = 1$. The solution exhibits a slow power law decay towards \nthermal equilibrium. We observe a cusp at $v=v_0=1$; thus, initial conditions\nhave a strong signature on the shape of $Q(v,t)$. \nA close look at the figure shows that for short times (i.e., $t\\le 2$) the peak of\n$Q(v,t)$ is at $v=v_0$. This is very different from Gaussian evolution\nfor which the peak is always on $\\langle v(t) \\rangle$.\n\n\n% this is four figures combined.\n% see more details in Kr*/Gra*\n% the solutions were found using a numerical integration\n% (for details see Kr*/math1\n\n\n% this is four figures combined.\n% see more details in Kr*/Gra*\n% the solutions were found using a numerical integration\n% (for details see Kr*/math1\n\n\n% We note that when $0<\\alpha\\le 1/2$,\n%$R_{s}(t)$ Eq. (\\ref{eqlevy}) is the Green function solution\n%of the one-dimensional \n%fractional diffusion equation investigated by Schneider and Wyss\n%\\cite{schneider}, the Laplace inversion\n%of $R_{s}(u)$ was also investigated\n%by Saichev and Zaslavsky \\cite{saichev} in the context of fractional diffusion\n%equations. In these investigation the meaning of $R_{\\tau}(t)$ is\n%different then what we find here.\n\n\n%Using theorem {\\bf 3.1} and Eq. $3.13$\n%of Schneider and Wyss \\cite{schneider} \n%the solution Eq. (\\ref{eqlevy}) can be reformulated in terms of a\n%Fox function\n%\\cite{mathai,srivastava} \n%%\n%\\begin{equation}\n%R_{s} \\left( t \\right) = { 1 \\over \\alpha \\sqrt{\\pi}} { 1 \\over s}\n%H^{20}_{12} \\left( 2^{-1/\\alpha} z^{-1} \\left| \\right.\n%\\begin{array}{l}\n%\\left( 1, 1 \\right)\\\\\n%\\left( {1 \\over 2}, { 1 \\over 2 \\alpha} \\right), \\left( 1, { 1 \\over 2 \\alpha}\\right)\n%\\end{array}\n%\\right)\n%\\label{eqFox}\n%\\end{equation}\n%%\n%with $z$ defined in \n%Eq. (\\ref{eqlevyz}).\n\n Metzler, Barkai and Klafter \\cite{MBK}\nhave shown that a fractional Fokker--Planck equation,\nwhich describes sub-diffusion $\\delta<1$,\ncan be solved using an eigenfunction expansion which is identical \nto the ordinary expansion of the Fokker--Planck solution \\cite{Risken};\nbut in which the exponential relaxation of eigenmodes is replaced with a\nMittag--Leffler relaxation.\nWe can use the eigenfunction expansion in \\cite{MBK}\nto find a second representation of $Q(v,t)$ \nin terms of a sum of Hermite polynomials. Expanding $G_{s}\\left( v \\right)$,\nusing the standard eigenfunction technique of Fokker--Planck\nsolutions, we write\n%\n$$G_s\\left( v \\right) = \\sqrt{{ M \\over 2 \\pi k_b T}} \\exp\\left( - { M v^2 \\over 2 k_b T } \\right) \\times $$\n\\begin{equation}\n\\sum_{n = 0}^{\\infty} { 1 \\over 2^n n! } H_n\\left(\\sqrt{{M \\over 2 k_b T}} v \\right)\nH_n\\left(\\sqrt{{M \\over 2 k_b T}} v_0 \\right) \\exp\\left(-n s\\right)\n\\label{eqexp1}\n\\end{equation}\n%\nwhere $H_n$ are Hermite polynomials. We insert the expansion\nEq. (\\ref{eqexp1}) in Eq.\n(\\ref{eqA04mt}) and use\n%\n\\begin{equation}\n\\int_0^{\\infty} R_{s}\\left( t \\right) \\exp\\left( - n s\\right) d s=\nE_{\\alpha}\\left( - n \\gamma_{\\alpha} t^{\\alpha} \\right),\n\\label{eqidentity}\n\\end{equation}\n%\nto find the eigen function expansion \n%\n$$Q \\left(v, t \\right) = \\sqrt{{ M \\over 2 \\pi k_b T}} \\exp\\left( - { M v^2 \\over 2 k_b T } \\right) \\times $$\n\\begin{equation}\n\\sum_{n = 0}^{\\infty} { 1 \\over 2^n n! } H_n\\left(\\sqrt{{M \\over 2 k_b T}} v \\right)\nH_n\\left(\\sqrt{{M \\over 2 k_b T}} v_0 \\right) E_{\\alpha} \\left(-n \\gamma_{\\alpha} t^{\\alpha} \\right).\n\\label{eqexp2}\n\\end{equation}\n%\nThe stationary solution, determined by the smallest eigen value $n=0$,\nis the Maxwell distribution which is independent of $\\gamma_{\\alpha}$\nand $\\alpha$.\n\n\n An extension of the fractional Fokker--Planck equation\n(\\ref{eqR04}) to higher dimensions is carried out by replacing the\none-dimensional Fokker--Planck operator, Eq. \n(\\ref{eqKr01a})\n with the appropriate\n$d$ dimensional Fokker--Planck operator (e.g., replace ${\\partial / \\partial v}$\nwith $\\nabla$). The solution for such a $d$ dimension equation\nis then found to be Eq. \n(\\ref{eqA04mt}) in which Eq.\n(\\ref{eqA06m}) must be replaced with the appropriate solution\nof the $d$ dimensional ordinary Fokker--Planck equation.\n\n \n\n\\subsection{Collision Model}\n\\label{secCM}\n\n As mentioned above, the ordinary Fokker--Planck equation\n(\\ref{eqR01}) was derived by\nRayleigh over a century ago. Briefly, the Rayleigh model\nfor Brownian motion considers a one dimensional test particle with mass $M$\ncolliding with bath particles of mass $m$ and $\\epsilon\\equiv m/M<<1$.\nWhen collisions are frequent but weak, the ordinary Fokker--Planck\nequation is valid \\cite{Kampen}. \nHere we consider the case when the concept of\ncollision rate does not hold and \nthe mean time\nintervals between collision events diverges.\nAs mentioned in the introduction this case was investigated in \\cite{barkai3,barkai2}\nand as we shall show now such a case corresponds to the\nfractional Fokker--Planck equation \n(\\ref{eqR04}).\n\n We consider a particle of mass $M$ which moves freely in\none dimension and at random times it collides elastically\nwith bath particles of mass $m$.\nBath particles are assumed to be much faster than the test particle. \nCollisions are elastic and one-dimensional and therefore the velocity\n$V_M^+$ of the test particle immediately after a\ncollision event can be related to the velocity $V_M^-$ of the\ntest particle just before the collision event \naccording to\n%\n\\begin{equation}\nV_M^+ = \\left({ 1- \\epsilon \\over 1 + \\epsilon}\\right) V_M^-\n + {2 \\epsilon \\over 1 + \\epsilon} \\tilde{v}_m.\n\\label{eqCol01}\n\\end{equation}\n%\nwhere ${\\epsilon} = m/M$ and $\\tilde{v}_m$ is the velocity of bath particle\ndistributed according to Maxwell's distribution.\n\n The times between collision events are assumed to be independent\nidentically distributed random variables implying\nthat the number of collisions in a time interval\n$(0,t)$ is a renewal process. This is reasonable\nwhen the bath particles thermalize very quickly and when\nthe test particle is slow. According to these assumptions the\ntimes between collision events $\\{ \\tau_i \\}$ are described by a probability\ndensity $\\psi\\left( \\tau \\right)$ which is independent \nof the mechanical state of the test particle. Therefore the process\nis characterized by free motion with constant velocity for\ntime $\\tau_1$ then a collision event described by Eq. (\\ref{eqCol01}) \nand then a free evolution for a period $\\tau_2$, then again a collision etc.\nThe most important ingredient of the model is the assumption\nthat $\\psi(\\tau)$ decays like a power law for long time\n%\n\\begin{equation}\n\\psi(\\tau) \\sim \\tau^{ - 1 - \\alpha}\n\\label{eqCol02}\n\\end{equation}\n% \nwith $0<\\alpha <1$. From Eq. (\\ref{eqCol02}) we learn that the mean time\nbetween collision diverges, $\\int^{\\infty}_0 \\tau \\psi(\\tau) d\\tau=\\infty$.\nSince in such a problem there is no characteristic time scale, the number\nof collisions in an interval $(0,t)$ is not proportional to $t$ for large\ntimes. In other words the law of large numbers is not valid for the \nchoice Eq. (\\ref{eqCol02}), leading to non normal behavior.\nSimilar waiting times distributions were used within\nthe CTRW to model anomalous diffusion for the past three decades.\nWhen $\\psi(\\tau)$ is exponential and in the presence of an external force field $F(x)$,\nthis model was investigated extensively in the context of\nreaction rate theory \\cite{Hangi,Mont,Knessel,Bork,barkaif,Ber1,Ber2}.\n\n Such a model is non stationary and the probability of a collision\nevent in a small time interval $(t,t+dt)$ is time dependent\neven in the limit of large times. This is a consequence of the diverging\nfirst moment.\n\n Let $Q_{col}\\left( v, t \\right)$ be the probability density for\nfinding the test particle with velocity $v$ at time $t$ and initially\n$v=v_0$. Using the model assumptions\n%\n\\begin{equation}\nQ_{col}\\left( v , t \\right) = \\sum_{s=0}^{\\infty} \\tilde{R}_s\\left( t \\right) \\tilde{G}_s \\left( v \\right)\n\\label{eqCol03}\n\\end{equation}\n%\nwith $\\tilde{R}_s(t)$ is the probability that $s$ collision\nevents have occurred in the interval $(0,t)$\nand $\\tilde{G}_s\\left( v \\right)$ is the conditional probability \ndensity of finding the particle with velocity $v$ after $s$ collision\nevents. We note that Eq. (\\ref{eqCol03}) is the discrete version\nof Eq. \n(\\ref{eqA04mt}).\n \nUsing the map Eq. (\\ref{eqCol01}), it can be shown that\n%\n\\begin{equation}\n\\tilde{G}_s\\left( v \\right) = { \\sqrt{M} \\over \\sqrt{ 2 \\pi k_b T \\left( 1 - \\mu_1^{2s} \\right)}}\n\\exp\\left[ - { M \\left( v - v_0\\mu_1^s\\right)^2 \\over 2 k_b T \\left( 1 - \\mu_1^{2 s} \\right)} \\right]\n\\label{eqCol04}\n\\end{equation}\n%\nwith $\\mu=(1-\\epsilon)/ ( 1 + \\epsilon)$.\nNot surprisingly $\\tilde{G}_s(v)$ is Gaussian since velocities of colliding\nparticles are Gaussian random variables. We note that\n%\n\\begin{equation}\n{\\partial \\tilde{G}_s\\left( v \\right) \\over \\partial s} =\n \\ln\\left(\\mu_1^{-1} \\right)\\hat{L}_{fp}\\tilde{G}_s\\left( v \\right) \n\\label{eqCol05}\n\\end{equation}\n%\nwith the initial condition\n%\n\\begin{equation}\n\\tilde{G}_0\\left( v \\right) = \\delta\\left( v - v_0\\right)\n\\label{eqCol05a}\n\\end{equation}\n%\nand $\\ln\\left( \\mu_1^{-1} \\right) \\sim 2 \\epsilon$. Eq. (\\ref{eqCol05}) is \na Fokker--Planck equation in which $s$, the collision number, plays the role of dimensionless\ntime.\n\n The Laplace $t \\to u$ transform of $\\tilde{R}_s\\left( t \\right)$, \n$\\tilde{R}_s\\left( u \\right)$\ncan be calculated using renewal theory\n%\n\\begin{equation}\n\\tilde{R}_s\\left( u \\right)= { 1 - \\psi(u) \\over u} \\psi^s(u)\n\\label{eqCol06}\n\\end{equation}\n%\nand $\\psi(u)= \\cal{L}[\\psi(\\tau)]$. From Eq.\n(\\ref{eqCol03}) we have in Laplace $t \\to u$ space\n%\n\\begin{equation}\nQ_{col}\\left( v , u \\right) = \\sum_{s=0}^{\\infty} \\tilde{R}_s\\left( u \\right) \\tilde{G}_s\\left( v \\right),\n\\label{eqCol07}\n\\end{equation}\n%\nmultiplying this equation from the left with\n$\\hat{L}_{fp}$, using Eq.(\\ref{eqCol05}), and integrating by parts, we have\n%\n$$\\hat{L}_{fp} Q_{col}\\left( v , u \\right) ={1 \\over 2 \\epsilon} \\sum_{s=0}^{\\infty} \\tilde{R}_s\\left( u \\right) \n{\\partial \\tilde{G}_s\\left( v \\right) \\over \\partial s}$$\n%\n\\begin{equation}\n= -{ 1 \\over 2 \\epsilon} \\tilde{R}_0\\left( u \\right) \\delta \\left( v- v_0 \\right) \n- { 1 \\over 2 \\epsilon} \\sum_{s = 0 } ^\\infty \n\\left[ {\\partial \\over \\partial s} \\tilde{R}_s \n\\left( u \\right) \\right] \\tilde{G}_s \\left( v \\right),\n\\label{eqCol09}\n\\end{equation}\n%\nWe have used Eq.\n(\\ref{eqCol05a}) and the the boundary condition \n$\\tilde{R}_{\\infty}\\left( u \\right) = 0$ for $u \\ne 0$.\n\n According to Eq. \n(\\ref{eqCol06}) \n%\n\\begin{equation}\n{\\partial \\over \\partial s} \\tilde{R}_s\\left( u \\right)= \n\\ln \\left[ \\psi( u ) \\right] \\tilde{R}_s\\left( u \\right)\n\\label{eqCol10}\n\\end{equation}\n%\nand since according to\n(\\ref{eqCol02})\n$\\psi(\\tau) \\sim \\tau^{ - 1 - \\alpha}$ we have\n%\n\\begin{equation}\n\\psi\\left( u \\right) \\sim 1 - A u^{\\alpha}\n\\label{eqCol11}\n\\end{equation}\n%\n(here A is a parameter with units $t^{\\alpha}$)\nvalid for small $u$, inserting Eq. (\\ref{eqCol11}) in Eq. (\\ref{eqCol10})\nwe have\n%\n\\begin{equation}\n{\\partial \\over \\partial s} \\tilde{R}_s\\left( u \\right)\\sim - Au^{\\alpha } R_s\\left( u \\right)\n\\label{eqCol12}\n\\end{equation}\n%\nand from Eq.\n(\\ref{eqCol06}) \n%\n\\begin{equation}\n\\tilde{R}_0(u)\\sim Au^{\\alpha - 1}.\n\\label{eqadd2}\n\\end{equation}\n\nWe are now ready to derive the fractional Fokker--Planck equation.\nInserting Eqs. (\\ref{eqCol12}) and\n(\\ref{eqadd2})\n in \nEq. (\\ref{eqCol09}) \nwe find in the limit of small $A u^{\\alpha} $ and $\\epsilon$\n%\n\\begin{equation}\nuQ_{col}\\left( v, u \\right) - \\delta\\left( v - v_0\\right)= \n\\gamma_{\\alpha}u^{1 - \\alpha} \\hat{L}_{fp} Q_{col}\\left( v , u \\right)\n\\label{eqColl13}\n\\end{equation}\nwith\n%\n\\begin{equation}\n\\gamma_{\\alpha} = \\lim_{A\\to 0, \\epsilon \\to 0} { 2 \\epsilon \\over A}.\n\\label{eqColl14}\n\\end{equation}\nEq. (\\ref{eqColl13}) is the fractional Fokker--Planck equation \n(\\ref{eqR01})\nin Laplace\nspace.\n We note that the moments of the collision model\nin the limit $\\epsilon \\to 0$ found in \\cite{barkai3} \nare identical to those found here based upon the fractional Fokker--Planck equation\nEqs. \n(\\ref{eqR09}) and\n(\\ref{eqv22}) as they should be.\n\n\\section{Fractional Kramers Equation}\n\\label{Kra}\n\n\n Let $P(x,v,t)$ be the joint probability density function\ndescribing both the position $x$ and the velocity $v$ of a Brownian\nparticle subjected to an external force field $F(x)$.\nThe one dimensional Kramers\nequation models such stochastic motion according to\n%\n$$ {\\partial P(x,v,t) \\over \\partial t} + v {\\partial P \\over \\partial x} + {F(x)\\over M}\n{\\partial P(x,v,t) \\over \\partial v} = $$\n\\begin{equation}\n\\gamma_1 \\hat{L}_{fp} P(x,v,t),\n\\label{eqKr01}\n\\end{equation}\n%\nwhere the Fokker--Planck operator $\\hat{L}_{fp}$ \nis given in Eq. \n(\\ref{eqR01a}). \nThe Kramers Eq. (\\ref{eqKr01}) implies that noise is white\nand Gaussian and it describes under--damped motion close\nto thermal equilibrium. \nEq. (\\ref{eqKr01}) \nis an extension of Eq. (\\ref{eqR01}) which includes the coordinate\n$x$ as well as the effects\nof $F(x)$. We generalize Kramers equation in the same way as above,\n and consider\n%\n$$ {\\partial P(x,v,t) \\over \\partial t} + v {\\partial P(x,v,t) \\over \\partial x} + {F(x)\\over M} \n{\\partial P(x,v,t) \\over \\partial v} = $$\n\\begin{equation}\n\\gamma_\\alpha\\ _0 D_t^{1-\\alpha} \\hat{L}_{fp} P(x,v,t),\n\\label{eqKr02}\n\\end{equation}\n%\nwith $0<\\alpha<1$.\nThe terms on the LHS of the equation\nare the standard streaming terms describing\nreversible dynamics according to Newton's second law of motion.\nThe term\non the RHS of the equation describes an interaction\nwith a bath, it can be considered as a generalized collision\noperator replacing the ordinary collision operator found\nin the standard Fokker--Planck equation. \nAs mentioned in the introduction the stationary \nsolution of Eq. (\\ref{eqKr02}) is the \nMaxwell--Boltzmann distribution and\nwhen $\\alpha=1$ we recover the ordinary Kramers equation.\n \n A formal solution of the fractional Kramers equation can be found in\nterms of the solution of the ordinary Kramers equation.\nWe denote the solution of the fractional Kramers equation\nwith $P_{\\alpha} \\left( x,v,t,\\gamma_{\\alpha}\\right)$ instead of\n$P(x,v,t)$ we have used so far. \nThe Laplace transform \nof Eq. (\\ref{eqKr02}) is \n%\n%\n$$u P_{\\alpha}(x,v,u,\\gamma_{\\alpha}) -P_{\\alpha}\\left( x,v,t=0,\\gamma_{\\alpha}\\right)$$\n$$ + v {\\partial P_{\\alpha}\\left(x,v,u,\\gamma_{\\alpha}\\right) \\over \\partial x} + {F(x)\\over M} \n{\\partial P_{\\alpha}(x,v,u,\\gamma_{\\alpha}) \\over \\partial v} = $$\n\\begin{equation}\n\\gamma_\\alpha\\ u^{1-\\alpha} \\hat{L}_{fp} P_{\\alpha}(x,v,u,\\gamma_{\\alpha}),\n\\label{eqKr02a}\n\\end{equation}\n%\nand $P_{\\alpha}\\left(x,v,u,\\gamma_{\\alpha}\\right)$ \nis the Laplace transform of $P_{\\alpha}\\left(x,v,t,\\gamma_{\\alpha}\\right)$. \nFrom Eq. (\\ref{eqKr02a}) we learn that \n$P_{\\alpha}(x,v,u,\\gamma_{\\alpha})$ solves an ordinary Kramers equation \nin which the damping coefficient $\\gamma_1$ was transformed according to\n%\n\\begin{equation}\n\\gamma_1 \\to \\gamma_{\\alpha} u^{1 - \\alpha}.\n\\label{eqwww}\n\\end{equation}\nWe therefore find\n%\n\\begin{equation}\nP_{\\alpha} \\left( x,v , u , \\gamma_{\\alpha}\\right) = \nP_1\\left( x, v, u, \\gamma_{\\alpha} u^{1 - \\alpha}\\right),\n\\label{eqwww11}\n\\end{equation}\n%\nassuming the initial conditions are identical\nfor both solutions. Transforming to the time domain we find the formal solution\n%\n\\begin{equation}\nP_{\\alpha}\\left( x, v, t, \\gamma_{\\alpha} \\right) =\n{\\cal L}^{-1} \\left[ P_1\\left( x, v, u, \\gamma_{\\alpha}u^{1 - \\alpha} \\right) \\right]\n\\label{eqwww2}\n\\end{equation}\n%\nwith ${\\cal L}^{-1}$ being the inverse Laplace transform.\nClosed form solutions of ordinary Kramers equation, $P_1\\left( x, v, t, \\gamma_1\\right)$,\nare known only for a handful of cases, approximate solutions can be found\nusing methods specified in \\cite{Risken}.\nIn some cases\nEq. (\\ref{eqwww2}) can be used to find moments of the solution of fractional\nKramers equation, $\\langle x^n v^m \\rangle$, in a straight forward way.\nIn what follows we shall revert to the \nnotation $P(x,v,t)$ instead of $P_{\\alpha}(x,v,t,\\gamma_{\\alpha})$.\n\n\n\n\n\n\n\\subsection{Force free case 1}\n\n We consider the force free case $F(x)=0$.\nAs usual $\\langle \\dot{x}(t) \\rangle = \\langle v(t) \\rangle$,\nwith the mean velocity $\\langle v(t) \\rangle$ given in Eq.\n(\\ref{eqR09}), hence\n%\n\\begin{equation}\n\\langle x(t) \\rangle = v_0 t E_{\\alpha,2} \\left( - \\gamma_{\\alpha} t^{\\alpha} \\right)\\label{eqKr03}\n\\end{equation}\n%\nwhere\n%\n\\begin{equation}\nE_{\\alpha,\\beta}(z)=\\sum_{k=0}^{\\infty} { z^k \\over \\Gamma\\left( \\alpha + \\beta k \\right)}\n\\label{eqKr04}\n\\end{equation}\n%\nis a generalized \nMittag--Leffler function \\cite{Erde} satisfying\n%\n\\begin{equation}\nE_{\\alpha,\\beta}\\left( z \\right) = - \\sum_{n=1}^{N-1} { z^{-n} \\over \\Gamma\\left( \\beta - \\alpha n\\right) } + {\\cal O} \\left( z^{-n}\\right)\n\\label{eqKr05}\n\\end{equation}\n%\nwith $z \\to \\infty$. For short times\n%\n\\begin{equation}\n\\langle x(t) \\rangle \\sim v_0 t\n\\label{eqKr05a}\n\\end{equation}\n%\nas expected for a pure ballistic\npropagation, and for long times\n%\n\\begin{equation}\n\\langle x(t) \\rangle \\sim {v_0 t^{1-\\alpha}\\over \\gamma_{\\alpha} \\Gamma\\left( 2 - \\alpha\\right)}.\n\\label{eqKr0r65}\n\\end{equation}\n%\nThe particle exhibits a net drift in a direction determined\nby the initial velocity $v_0$. Of course,\nwhen averaging over initial conditions, using\nthermal equilibrium condition, no net drift is\nobserved as expected from symmetry. The mean\nsquare displacement is determined by $\\langle \\dot{x}^2(t)\\rangle=\n\\langle 2 x(t) v(t) \\rangle$. A short calculation using the Laplace transform\n of Eq. \n(\\ref{eqKr02}) shows\n%\n\\begin{equation}\n\\langle x^2(t) \\rangle_{eq} = 2 {k_b T \\over M} t^2 E_{\\alpha, 3}\\left( - \\gamma_{\\alpha}t^{\\alpha} \\right), \n\\label{eqKr07}\n\\end{equation}\n%\nwhere the subscript $_{eq}$ means that thermal initial\nconditions are considered (i.e., $\\langle v_0^2 \\rangle_{eq}=k_b T / M$).\nFor short times\n%\n\\begin{equation}\n\\langle x^2(t) \\rangle_{eq} \\sim {k_b T \\over M} t^2, \n\\label{eqKr08}\n\\end{equation}\n%\nwhile for long times\n%\n\\begin{equation}\n\\langle x^2(t) \\rangle_{eq} \\sim 2 D_{\\alpha} t^{ 2 - \\alpha} \n\\label{eqKr09}\n\\end{equation}\n%\nwhere\n%\n\\begin{equation}\n D_{\\alpha} = { k_b T \\over \\gamma_{\\alpha} M \\Gamma\\left( 3 - \\alpha\\right) }.\n\\label{eqKr10}\n\\end{equation}\n%\nEq. (\\ref{eqKr09}) exhibits an enhanced diffusion when $0<\\alpha<1$.\nEq. (\\ref{eqKr10}) is the (first) generalized Einstein relation and\nwhen $\\alpha=1$ we recover the well known Einstein relation\n$D_1=(k_b T ) /(M \\gamma_1)$.\n\nIt is straightforward to prove the more general Einstein relation \n%\n\\begin{equation}\n\\langle x^2 \\left( t \\right) \\rangle_{eq} = 2 \\int_0^t dt' \\int_0^{t'} \\langle \nv(\\tau) v(0) \\rangle_{eq} d\\tau,\n\\label{eqER03}\n\\end{equation}\n%\nwhere according to\nEq. (\\ref{eqR09})\n%\n\\begin{equation}\n\\langle v(\\tau) v(0)\\rangle_{eq}={k_b T\\over M} E_{\\alpha}\\left( -\\gamma \\tau^{\\alpha} \\right)\n\\label{eqVACF}\n\\end{equation}\n%\nis the velocity autocorrelation function. \nWe note that relation Eq. (\\ref{eqER03}) is valid for stationary\nprocesses \\cite{remark4}, while the collision model we investigated\nin previous section is non-stationary.\n\n Now consider the constant force field $F(x)=F$, \nusing the fractional Kramers Eq. (\\ref{eqKr02}). We can show\nthat a second generalized Einstein relation \n between\nthe drift in the presence of the driving field, $\\langle x(t) \\rangle_F$,\nand the mean square displacement, Eq. \n(\\ref{eqKr07}), in the absence of the field is valid\n%\n\\begin{equation}\n\\langle x(t) \\rangle_F= F {\\langle x^2(t) \\rangle_{eq}\\over 2 k_b T}.\n\\label{eqEin}\n\\end{equation}\n%\nThis relation suggests that the fractional Kramers equation\nis compatible with linear response theory \\cite{barkaiE,Berlin}. \n\n\n\\subsection{Force free case 2}\n\n We shall now use the formal solution Eq.\n(\\ref{eqwww2}) to find moments $\\langle x^{2n}(t)\\rangle_{eq}$\nfor the case $F(x)=0$. Odd moments are equal zero.\nConsider the reduced probability density $W_{eq}(x,t)$\nof finding the particle on $x$ at time $t$, defined according to\n%\n\\begin{equation}\nW_{eq}(x,t)=\\int_{-\\infty}^{\\infty} dv \\int_{-\\infty}^{\\infty} dv_0 P(x,v,t) M(v_0)\n\\label{eqFFC01}\n\\end{equation}\n%\n$P(x,v,t)$ is the solution of the fractional Kramers equation\nwith initial conditions concentrated on $x_0$ and $v_0$.\n$M(v_0)$ is Maxwell's probability density implying an equilibrium\ninitial condition for the initial velocity $v_0$. \n\nFor the standard \ncase $\\alpha = 1$, $W_{eq}(x,t)$\nis a Gaussian, therefore \n%\n$$ \\langle x^{2 n } \\left( t \\right) \\rangle_{eq} =\n\\int_{-\\infty}^{\\infty} x^{2n} W_{eq}\\left( x , t \\right) dx=$$\n\\begin{equation}\n {\\left( 2 n \\right)! \\over 2^n n!} \\langle x^2(t) \\rangle^n_{eq}\n\\label{eqfff01}\n\\end{equation}\n%\nand according to \\cite{Risken}\n%\n\\begin{equation}\n\\langle x^2\\left( t \\right) \\rangle_{eq}=2 {k_b T \\over M} \n{ \\left[ \\gamma_1 t - 1 + \\exp\\left( - \\gamma_1 t \\right) \\right]\n\\over\n\\gamma_1^2}\n\\label{eqfff02}\n\\end{equation}\n%\nwhich in the long time limit gives (only for $\\alpha = 1$)\n%\n\\begin{equation}\n\\langle x^2\\left( t \\right) \\rangle_{eq}\\sim 2 {k_b T \\over M \\gamma_1} t.\n\\label{eqfff03}\n\\end{equation}\n%\n\n\n According to Eqs. (\\ref{eqwww}-\\ref{eqwww2}) the calculation of\n$\\langle x^{2 n } \\left( t \\right) \\rangle_{eq}$, for $0< \\alpha< 1$,\nfollows three steps. First find the Laplace transform of the Gaussian\nmoments,\n$\\langle x^{2 n } \\left( t \\right) \\rangle_{eq}$\nEq.\n(\\ref{eqfff01}).\nSince we shall be interested in the long time behavior of\n$\\langle x^{2 n } \\left( t \\right) \\rangle_{eq}$\nit is sufficient to consider only the long time\nbehavior of the standard $\\alpha=1$ case, namely\nwe use the asymptotic Eq. (\\ref{eqfff03}) instead of\nEq. (\\ref{eqfff02}). We note that the inclusion of the short\ntime behavior is also straightforward, but of less interest to us here.\nIt is easy to show that \n%\n\\begin{equation}\n\\langle x^{2 n } \\left( u \\right) \\rangle_{eq}\\sim \\left( 2 n \\right)! \n\\left({k_b T \\over M \\gamma_1}\\right)^n { 1 \\over u^{n + 1 } }\n\\label{eqfff04}\n\\end{equation}\n%\nvalid for small $u$ and $\\alpha=1$. The second step is to \n transform\nEq. (\\ref{eqfff04}) using\nEq. (\\ref{eqwww}) and \nthe last step is to invert Laplace transform the result from the second step,\nas in\nEq. (\\ref{eqwww2}). We find\n%\n\\begin{equation}\n\\langle x^{2n} \\left( t \\right) \\rangle_{eq} \\sim \\left( 2 n \\right)! \n\\left({ k_b T \\over M \\gamma_{\\alpha}}\\right)^n \n{ t^{n(2 - \\alpha)} \\over \\Gamma\\left( 2 n - n \\alpha +1\\right)}.\n\\label{eqfff05}\n\\end{equation}\n%\nIt is easy to check that the moments in (\\ref{eqfff05}) can\nalso be calculated based on\n%\n\\begin{equation}\n\\langle x^{2n} \\left( t \\right) \\rangle \\sim {\\cal L}^{-1} \\left[ \\left( { d \\over i dk}\\right)^{2n}\n\\left({ u^{1 - \\alpha} \\over u^{2 - \\alpha} + {k_b T \\over M \\gamma_{\\alpha}} k^2}\\right)\n\\left|_{k=0} \\right. \\right] .\n\\label{eqfff06}\n\\end{equation}\n%\nThis result is also to be expected. \nIn Fourier--Laplace space the well known $\\alpha=1$ solution\nis\n%\n\\begin{equation}\nW_{eq}\\left( k, u \\right) \\sim {1 \\over u + {k_b T \\over M \\gamma_1} k^2},\n\\label{eqFFC02}\n\\end{equation}\n%\nif we\nuse the transformation \n(\\ref{eqwww}) in Eq. (\\ref{eqFFC02}) we find\n%\n\\begin{equation}\nW_{eq}\\left( k, u \\right) \\sim {u^{1-\\alpha} \\over u^{2 - \\alpha} + {k_b T \\over M \\gamma_\\alpha} k^2}.\n\\label{eqFFC03}\n\\end{equation}\n%\nAnd this is the moment generating function in Eq. (\\ref{eqfff06}).\nWe note that in general there is no guarantee that \nthe transformation Eq. (\\ref{eqwww}) can be made after\nthe small $u$ limit of the $\\alpha=1$ solution is taken,\ninstead the small $u$ limit must be taken only after the transformation\nEq. (\\ref{eqwww}) takes place.\n The small $(k,u)$ expansion a particular\n coupled CTRW also known as L\\'evy walks\nhas the\nsame form as Eq. (\\ref{eqFFC03}) \\{see Eq. $38$ in \\cite{KBS}\\}\n.\n\n\n According to Eq. (\\ref{eqFFC03}) $W_{eq}(x,t)$ satisfies the following\nfractional diffusion equation\n%\n\\begin{equation}\n{\\partial W_{eq}\\left(x,t\\right) \\over \\partial t} = \n{k_b T \\over M \\gamma_\\alpha}\\ _0 D_t^{-(1 - \\alpha)}\n{\\partial^2 \\over \\partial x^2} W_{eq}(x,t)\n\\label{eqFFC04a}\n\\end{equation}\n%\nwhich is expected to work well only at large times.\nEq. (\\ref{eqFFC04a}) was investigated by Schneider and Wyss \\cite{schneider}.\n\nFor one dimension the inverse Laplace--Fourier transform of Eq.\n(\\ref{eqFFC03}) is\n%\n\\begin{equation}\nW_{eq}\\left(x,t\\right)\\sim \n{ \\sqrt{\\gamma_{\\alpha} M} \\over \\sqrt{2 k_b T}\\left(1 - \\alpha/2\\right)} \n\\left( {z^{2 - \\alpha/2} \\over t^{ 1 - \\alpha/2} } \\right) \nl_{1 - \\alpha/2}\\left( z \\right)\n\\label{eqFFC04}\n\\end{equation}\n%\nwith \n%\n\\begin{equation}\nz=t\\left[{ \\sqrt{k_b T } \\over \n\\left( \\sqrt{2 M \\gamma_{\\alpha}} |x|\\right) } \n\\right]^{{1 \\over 1 - \\alpha/2}}\n\\label{eqFFC05}\n\\end{equation}\n%\nThe inversion of the two and three dimensional solutions\nis not as straightforward as for the one dimensional case, and \nwe leave the details for a future publication.\n\n\n\n\\subsection{Collision Model}\n\n We have derived the fractional Fokker--Planck equation \n(\\ref{eqR04})\nfrom a non-stationary stochastic collision model in the limit of $\\epsilon = m/M\\to 0$.\nEq. (\\ref{eqR04}) describing a fractional Ornstein--Uhlenbeck process,\nis the basis of the fractional Kramers equation which is reached\nafter adding the standard streaming terms (i.e. Newton's law\nof motion).\nIn \\cite{barkai3,barkai2} \nthe mean square displacement, $\\langle x^2(t) \\rangle_{col}$ of the test particle was calculated\nalso\nfor mass ratios $\\epsilon$ not tending to zero. \nIt was shown the collision process gives $\\langle x^2 \\rangle_{col} \\sim t^2$\nfor $0<\\alpha< 1$ and finite $\\epsilon$, a behavior different from\nthe asymptotic behavior we have found in Eq.\n(\\ref{eqKr09})\n based on the fractional Kramers equation,\n$\\langle x^2 \\rangle \\sim t^{ 2 - \\alpha}$. The $\\langle x^2 \\rangle_{col} \\sim t^2$\nbehavior of the collision model can be understood by the fact\nthat within the collision process, one typically finds long \n(collision-free) time intervals,\nof the order of the observation time $t$, in which the motion\nis ballistic so that $x^2 \\sim t^2$. A similar ballistic behavior\nis known from the L\\'evy walk model with diverging time intervals \nbetween turning points.\nAs $\\epsilon$ becomes smaller,\nthe correction to the ballistic term becomes important\nfor longer times and these correction terms behave \nlike $\\langle x^2 \\rangle \\sim t^{ 2 - \\alpha}$.\nAlso it is clear that the two limits of $t \\to \\infty$ and $\\epsilon \\to 0$ do not commute.\nIf we first take $t\\to \\infty$ and only then $\\epsilon \\to 0$ we find\n%\n\\begin{equation}\n\\langle x^2 (t) \\rangle_{col} \\sim t^2. \n\\label{eqcol2}\n\\end{equation}\n%\nHence the derivation of the fractional Kramers equation\nfrom the stochastic collision model is a delicate matter, with a result\ndepending on the order in which limits are taken. Another difficulty in our\nderivation is that our starting point is a non-stationary process. It is\nstill unclear if stationary models can lead to dynamics\ndescribed by the fractional Kramers equation.\n\n\n%\\section{Harmonic Oscillator}\n\n% I have a few result for the averaged displacement $\\langle x(t) \\rangle$\n%and averaged velocity $\\langle v(t) \\rangle$ for the case when\n%the potential is $V(x)=(1/2)M \\omega_0^2 x^2$. Then for normal\n%relaxation in harmonic oscillator we get two well known\n%behaviors, under and over damped motions,\n%depending on the value of friction $\\gamma_1$ and\n%oscillator frequency $\\omega_0$.\n%When $\\alpha\\ne 1$ we get a generalizations of\n%these two types of motion. However things are\n%not as simple as in the standard case $\\alpha =1$.\n%\n\\section{Summary and Discussion}\n\\label{secSum}\n\n We have investigated a fractional Kramers equation\nwhich has the following properties:\\\\\n{\\bf (a)} the velocity of the particle evolves according\nto a\nfractional Ornstein--Uhlenbeck process described by Eq. \n(\\ref{eqR04}),\nthe velocity moments decay according to a Mittag--Leffler relaxation, \nnamely as a stretched exponential (Kohlrausch form) for short times and as\na power law for long times,\\\\\n{\\bf (b)} in the absence of a force field diffusion is enhanced, $1< \\delta < 2$,\\\\\n{\\bf (c)} the stationary solution of the fractional Kramers equation\n is the Maxwell-Boltzmann distribution,\\\\\n{\\bf (d)} Einstein relations are obeyed in consistency\nwith fluctuation dissipation theorem and linear response theory,\\\\\n{\\bf (e)} in Laplace space a simple transformation of solutions\nof ordinary Kramers equation gives the solution of\nfractional Kramers equation.\\\\\n\n As mentioned in the\nintroduction fractional kinetic equations in the literature\nare related to the CTRW.\nWe showed here that in the small $(k,u)$ limit\n$W_{eq}(k,u)$ has the same form as a particular coupled Levy walk CTRW.\nOther limits of the CTRW are shown to correspond to\nother fractional kinetic equations. \nAs pointed out in \\cite{compte}\nthe fractional diffusion equation \\cite{schneider}, describing the sub-diffusion, corresponds\nto the uncoupled CTRW in the limit of small $(k,u)$.\nA comparison between the uncoupled CTRW and solution of fractional\ndiffusion equation in $(x, t )$ space was carried out in \\cite{barkai9}.\nAnother fractional\nequation in \n\\cite{fogedby,zaslavsky}\ndescribes L\\'evy flights for which $\\langle x^2 \\rangle = \\infty$,\nsuch a fractional equation is related to the decoupled limit of \nthe CTRW with diverging jump\nlengths.\n\n One may ask if\nit is worthwhile to introduce fractional\nderivatives, given that the older CTRW approach is so successful.\nBesides the fact that fractional equations are beautiful and \nsimple (i.e., in some cases\nthey are solvable),\nthese equations can incorporate the\neffect of an external potential field. To us, this extension\nseems important although not explored in depth in the present paper.\nLittle is known on anomalous diffusion in an external force field.\n\n Metzler, Barkai and Klafter \\cite{MBK} \nhave investigated a fractional Fokker--Planck \nequation defined with the fractional Liouville\n--Riemann operator. In the absence of an\nexternal force the fractional Fokker--Planck equation\ninvestigated in \\cite{MBK} describes a sub--diffusive behavior\n($\\delta < 1$). The equation considers a type of over damped dynamics\nin which only the coordinate $x$ is considered not the velocity $v$.\nThe fractional Fokker--Planck equation in \\cite{MBK} together with\nthe fractional Kramers equation investigated here give a stochastic \nframework for both sub and enhanced\ndiffusion in an external field. We believe that both approaches\nwill find their application.\n\n% It is well known that when $F(x)=0$\n%the long time solution of the Kramers equation, after integrating over velocity,\n%exhibits a Gaussian evolution which is described also by the\n%simpler diffusion equation (the diffusion equation considers only\n%the coordinate and in in this sense it is simpler than the Kramers\n%equation). Similarly we show that under certain conditions\n%the non-Gaussian solution of the fractional Kramers equation\n%can be described by a simpler fractional diffusion equation\n%solved already by Schneider and Wyss\n%\\cite{schneider}.\n\n{\\em Note added in proof.} Recently related \nwork on fractional diffusion was published \\cite{add}.\n\n\n\\section{Acknowledgments}\n\nEB thanks J. Klafter and R. Metzler for helpful discussions. \n This research was \nsupported in part by a grant from the NSF.\n\n\\section{Appendix A}\n\n We find the Laplace transform of \nEq. (\\ref{eqlevy}) \n%\n\\begin{equation}\n R_{s}( u ) = \\int_0^{\\infty} e^{ - u t} {1 \\over \\alpha \\gamma_{\\alpha} t^{\\alpha} } z^{ \\alpha + 1} l_{\\alpha} \\left( z \\right) dt,\n\\label{eqAn01}\n\\end{equation}\n%\nwith $z$ defined in Eq. (\\ref{eqlevyz}).\nUsing the change of variables\n$t=y (s/\\gamma_{\\alpha})^{1 /\\alpha}$ it is easy to show\n$(s \\ne 0)$\n%\n$$ R_{s}\\left( u \\right) = -{ 1 \\over \\alpha s} {d \\over du}\\int_0^{\\infty}\n\\exp\\left[ - y \\left({ s\\over \\gamma_{\\alpha} } \\right)^{ 1 / \\alpha} u \\right]\nl_{\\alpha}\\left( y \\right) dy =$$\n%\n$$ -\\left({ 1 \\over \\alpha s} \\right) {d \\over du} \\exp\\left( - { s \\over \\gamma_{\\alpha}} u^{\\alpha} \\right) = $$\n%\n\\begin{equation}\n{u^{\\alpha - 1} \\over \\gamma_{\\alpha} } \\exp\\left( - { s u^{\\alpha} \\over \\gamma_{\\alpha} } \\right),\n\\label{eqAn02}\n\\end{equation}\n%\nwhich is $R_{s}( u )$, Eq. \n(\\ref{eqA11m}). \nFrom Eq. (\\ref{eqA09}) we learn that Eq. (\\ref{eqA11m}) is valid also for $s=0$.\n \n \n%\\section{Appendix B}\n\n% We shall use the following mathematical tools and definitions.\\\\\n%{\\bf (a)} The Fox function is defined for \n%$z \\ne 0$\n%%\n%\\begin{equation}\n%H^{mn}_{pq}\\left( z \\left| \\right. \n%\\begin{array}{l}\n%\\left( a_1,\\alpha_1\\right)\\cdots\\left(a_p,\\alpha_p\\right) \\\\\n%\\left( b_1,\\beta_1\\right)\\cdots\\left(b_q,\\beta_q\\right) \n%\\end{array}\n%\\right)=\n%{1 \\over 2 \\pi i} \\int_c h(s) z^s ds,\n%\\label{eqB01}\n%\\end{equation}\n%%\n%with\n%%\n%\\begin{equation}\n%h\\left( s \\right) = { \\prod_{j=1}^n \\Gamma\\left( 1 - a_j + \\alpha_j s \\right) \\prod_{j=1}^m \\Gamma\\left( b_j - \\beta_j s\\right) \\over\n%\\prod_{j=m+1}^q \\Gamma\\left( 1 - b_j + \\beta_js\\right) \\prod_{j =n + 1}^p \\Gamma\\left( a_j -\\alpha_j s \\right) }.\n%\\label{eqB02}\n%\\end{equation}\n%%\n%$p,q,m$ and $n$ are integers satisfying $0\\le n \\le p$ and \n%$1 \\le m \\le q$. The $\\alpha_j$'s and $\\beta_j$'s \n%are positive numbers, while the\n%$a_j$'s and $b_j$'s are complex numbers.\n%These parameters satisfy\n%%\n%\\begin{equation}\n%\\alpha_j\\left( b_h + \\nu \\right) \\ne \\beta_h \\left( \\alpha_j - 1 - \\Lambda\\right)\n%\\label{eqB03}\n%\\end{equation}\n%%\n%for $\\nu,\\Lambda=0,1,\\cdots$; $h=1,\\cdots,m$ and $j=1,\\cdots,n$.\n%The contour $C$ in the complex plane\n%separates the poles in such a way that the poles\n%of $\\Gamma\\left( b_j - \\beta_j s\\right)$\n%lie to the right and the poles of\n%$\\Gamma\\left( 1 - a_j + \\alpha_j s \\right)$ lie to the left of \n%the contour.\n%The Fox function is an analytical\n%function of $z\\ne 0$ if \n%%\n%\\begin{equation}\n%\\mu = \\sum_{j = 1}^q \\beta_j - \\sum_{j = 1}^p \\alpha_j > 0..\n%\\label{eqB04}\n%\\end{equation}\n%%\n%The case $\\mu=0$ is discussed in $\\cdots$. \n%\\\\\n%{\\bf (b)} The Mellin transform is defined by\n%%\n%\\begin{equation}\n%f(s)={\\cal M}\\left(f(t),s\\right)=\\int_0^{\\infty}t^{s - 1} f(t) dt,\n%\\label{eqB05}\n%\\end{equation}\n%%\n%the inverse Mellin transform\n%%\n%\\begin{equation}\n%f(t)={ 1 \\over 2 \\pi i} \\int_{c - i \\infty}^{c+i \\infty}f(s) t^{-s} ds.\n%\\label{eqB06}\n%\\end{equation}\n%%\n%The relation between the Laplace and Mellin transform is \n%%\n%\\begin{equation}\n%{\\cal M}(f(t),s) = {1 / \\Gamma(1 - s)}{\\cal M}\\left( {\\cal L}\\left( f(t),u\\right),1-s\\right).\n%\\label{eqB07}\n%\\end{equation}\n%%\n%\n% Using Eqs. \n%(\\ref{eqA11m}),\n%(\\ref{eqB05}) and\n%(\\ref{eqB07})\n%we find\n%%\n%\\begin{equation}\n%\\gamma_{\\alpha} P_{s}\\left( s \\right) = { 1 \\over \\Gamma\\left( 1 - s \\right)}\n%\\int_0^{\\infty}u^{\\alpha-1-s} \\exp\\left( - B u^\\alpha\\right) du.\n%\\label{eqB08}\n%\\end{equation}\n%%\n%Using the inverse Mellin transform it can be\n%shown that\n%%\n%\\begin{equation}\n%\\gamma_{\\alpha} P_{s}\\left( t \\right) = { 1 \\over B} { 1 \\over 2 \\pi i} \n%\\int_C{ \\Gamma\\left( 1 - s \\right) \\over \\Gamma\\left( 1 - \\alpha s \\right)}\n%\\left({ B \\over t^\\alpha}\\right)^s ds.\n%\\label{eqB09}\n%\\end{equation}\n%%\n%which according to definition of the Fox function gives the result\n%\\label{eqB10m}.\n%%\n%%\\begin{equation}\n%%P_{s}\\left( t \\right) = { 1 \\over \\alpha s} H_{11}^{10}\\left( {B^{1/\\alpha}\\over t} \\left|\\right.\\begin{array}{l}\n%%\\left( 1, 1 \\right)\\\\\n%%\\left( 1 , 1/\\alpha\\right)\n%%\\end{array}\n%%\\right).\n%%\\label{eqB10}\n%%\\end{equation}\n%%\n\n%\n\\begin{thebibliography}{99}\n\n\\bibitem{SM} H. 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Lett., {\\bf 310} 287 (1999).\n\n\\bibitem{MUK} V. Chernyak, M. Schultz and S. Mukamel, J. Chemical Physics,\n{\\bf 111} 7416 (1999)\n\n\\bibitem{oldham} K. B. Oldham and J. Spanier, {\\em The\nFractional Calculus} Academic Press, (New York) 1974.\n\n\\bibitem{Hilfer} R. Hilfer and L. Anton, Phys. Rev. E. {\\bf 51},\nR848 (1995)\n \n\n\\bibitem{compte} A. Compte, Phys. Rev. E {\\bf 53}, 4191 (1996)\n\n\\bibitem{West} B. J. West, P. Grigolini, R. Metzler and T. F. Nonnenmacher,\nPhys. Rev. E {\\bf 55}, 99 (1997)\n\n\n\\bibitem{barkai9} E. Barkai, R. Metzler and J. Klafter, Phys. Rev. E.\nPhys. Rev. E {\\bf 61} 132 (2000)\n\n\n%\n\n\\bibitem{Ral} Lord Rayleigh {\\em Phil. Mag.} {\\bf 32} 424 (1891). [{\\em Scientific\nPapers} (Cambridge 1902) {\\bf 3}, 473]\n\n%\n\\bibitem{Kampen} N.G. van Kampen {\\em Stochastic Processes in Physics and\n Chemistry} North Holland (Amsterdam -- New York -- Oxford, 1981)\n\n\n\n\\bibitem{barkai3} E. Barkai and V. N. Fleurov, {\\em J. Chem. Phys.} {\\bf 212},\n 69, (1996) \n\n%Levy Wals\n\\bibitem{barkai2} E. Barkai and V. N. Fleurov, {\\em Phys. Rev. E.} {\\bf 56},\n 6355, (1997) \n\n\\bibitem{MKp} R. Metzler and J. Klafter (preprint)\n\n\\bibitem{Risken} H. Risken, {\\it The Fokker--Planck\nequation\\/} (Springer, Berlin, 1989)\n\n\n\\bibitem{remark3} To see this Laplace transform Eq. (\\ref{eqR04}) using Eq.\n(\\ref{eqL05})\n%\n$$ u Q(u,v) - Q(v,t=0) = $$\n$$\\gamma_{\\alpha} \\hat{L}_{fp}\n\\left[ u^{1 - \\alpha} Q(v,u) - _0D_t^{ - \\alpha} Q(v,t) |_{t = 0} \\right], $$\nsetting $_0D_t^{ - \\alpha} Q(v,t) |_{t = 0}=0$ \nwe get an expression equivalent to\nto the Laplace transform of\nEq. (\\ref{eqR03}).\n\n\\bibitem{Feller} W. Feller,{\\em An introduction to probability Theory and\nIts Applications} Vol. 2 (John Wiley and Sons, New York 1970).\n\n\n%\n%\\bibitem{Silbey} J. Klafter and R. Silbey,\n% {\\em Phys. Rev. Lett.} {\\bf 44}, 55 (1980)\n\n\n%\\bibitem{Balakrishnan} V. Balakrishnan, Physica A {\\bf 132} 569 (1985)\n\n%\\bibitem{Wyss} W. Wyss, J. Math. Phys.\n%{\\bf 27}, 2782 (1986)\n\n%\\bibitem{mathai} A. M. Mathai and R. K. Saxena, {\\it The\n%$H$--function with Applications in Statistics and Other\n%Disciplines\\/} (Wiley Eastern Ltd, New Delhi, 1978)\n\n%\\bibitem{srivastava} H. M. Srivastava, K. C. Gupta and S. P. Goyal,\n%{\\it The $H$--functions of one and two variables\n%with applications\\/} (South Asian Publishers, New Delhi, 1982)\n\n% Comment\n%\\bibitem{barkai7} E. Barkai and J. Klafter, Phys. Rev. Lett. {\\bf 81},\n%1134 (1998)\n\n\n\\bibitem{Zol} V. M. Zoltarev, Dokl. Acad. Nauk. USSR {\\bf 98}, 715 (1954)\n\n\\bibitem{EWM} E. W. Montroll and B. J. West in {\\it Fluctuation Phenomena} Eds. E. W. Montroll\nand J. L. Lebowitz (North Holland, Amsterdam, 1987)\n\n\\bibitem{Hangi} P. H$\\ddot{a}$nggi, P. Talkner and M. Borkovec, Rev. Mod. Phys.\n{\\bf 62}, 251 (1990)\n\n\\bibitem{Mont} J. A. Montgomery, D. Chandler, and B. J. Berne,\nJ. Chem. Phys. {\\bf 70}, 4056 (1979)\n\n\\bibitem{Knessel} C. Knessel, M. Mangel, B. J. Matkowsky, A. Schuss,\nand C. Tier, J. Chem. Phys. {\\bf 81} 1285 (1984)\n\n\\bibitem{Bork} M. Borkovek, J. E. Straub, and B. J. Berne,\nJ. Chem. Phys. {\\bf 85}, 146 (1986)\n\n\n\\bibitem{barkaif} E. Barkai and V. Fleurov Phys. Rev. E {\\bf 52} 1558\n(1995)\n\n\\bibitem{Ber1} D. J. Bicout, A. M. Berezhkovskii, A. Szabo and G. H. Weiss, \nPhys. Rev. E. {\\bf 59} 3702 (1999) \n\n\\bibitem{Ber2} A. M. Berezhkovskii, D. J. Bicout and G. H. Weiss,\nJ. Chem. Phys. {\\bf 111} 11050 (1999) \n\n\\bibitem{Erde} A. Erdelyi Ed. {\\it Higher Transcendental Functions} McGraw-Hill, New York (1955) \n\n\n\\bibitem{remark4}\nTo see this write\n%\\begin{equation}\n$$\\langle x^2 (t) \\rangle_{eq}= \\langle \\left[\\int_0^t v\\left(t_1\\right) d t_1\\right]\n\\left[\\int_0^t v\\left(t_2\\right) d t_2\\right] \\rangle_{eq},$$\n%\nif the process is stationary \n$\\langle v(t_1) v(t_2)\\rangle_{eq} = \\langle v(\\tau) v(0)\\rangle_{eq}$\n%\nwith $\\tau=|t_1 - t_2|$, and only then Eq. (\\ref{eqER03}) valid.\n\n% Einstein Relation\n\\bibitem{barkaiE} \nE. Barkai and V. N. Fleurov, Phys. Rev. E {\\bf 58},\n1296 (1998)\n\n\\bibitem{Berlin} \nY. A. Berlin, L. D. A. Siebbeles and A. A. Zharikov,\nChem. Phys. Lett. {\\bf 305} 123 (1999)\n\n\\bibitem{KBS} \nJ. Klafter, A. Blumen and M. F. Shlesinger, \nPhys. Rev. A {\\bf 35}, 3081 (1987)\n\n\n%\n\n%\\bibitem{tsallis1} C. Tsallis, S. V. F. Levy, A. M. C. Souza\n%and R. Maynard, Phys. Rev. Lett. {\\bf 75}, 3589 (1995); D. H.\n%Zanette and P. A. Alemany, Phys. Rev. Lett. {\\bf 75} 366 (1995);\n%M. Buiatti, P. Grigolini and A. Monagnini, Phys. Rev. Lett. {\\bf 26}\n%3383 (1999); D. H. Zanette cond-mat/9905064 May 1999 and invited review in Braz. J. Phys.\n%\n\\bibitem{add}\nM. Bologna, P. Grigolini P and J. Riccardi,\nPhys. Rev. E {\\bf 60} 6435 (1999).\nR. Kutner, K. Wysocki\n{\\em Physica A} {\\bf 274} 67 (1999).\nV. V. Yanovsky, A. V. Chechkin, D. Schertzer\nand A. V. Tour nlin/001035 (to appear in Physica A)\n\n\n\n\n\\end{thebibliography}\n\n\\newpage\n\n{\\bf Figure Caption}\n\nFigure 1: The dynamics of $Q(v,t)$ for the fractional Ornstein-Uhlenbeck\nprocess with $\\alpha = 1/2$ and for times, t = 0.02,0.2,2, 20 \n(solid, dashed, dotted, and dot-dash lines, respectively). Also shown (fine\ndotted curve) is the stationary solution which is Maxwell's distribution. \nNotice the cusp on $v = v_0 = 1$ as well as the non-symmetrical shape \nof $Q(v,t)$. \n\n\\end{document}\n\n"
}
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[
{
"name": "cond-mat0002020.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{SM} H. Scher and E. Montroll, Phys. Rev. {\\bf B12},2455 (1975) \n\n\\bibitem{SL} H. Scher and M. Lax, Phys. Rev. {\\bf B7}, 4491 (1973); Phys. Rev\n{\\bf B7}, 4502 (1973)\n\n\\bibitem{Weiss1} G. H. Weiss {\\em Aspects and Applications of the Random Walk}\n North Holland (Amsterdam -- New York -- Oxford, 1994)\n\n\\bibitem{Bouch} J.--P. Bouchaud and A. Georges, Phys. Rep.\n{\\bf 195}, 127 (1990)\n\n%\n\\bibitem{Klafter1} J. Klafter, M. F. Shlesinger and G. Zumofen, {\\em Phys. Today\n} {\\bf 49}, 33 (1996).\n\n\\bibitem{Balescu} R. Balescu, {\\it Statistical Dynamics; Matter Out of Equilibrium} Imperial College Press, World scientific, Singapore (1997).\n\n\\bibitem{barkai8} E. Barkai and J. Klafter, \nLecture Notes in Physics, S. Benkadda and G. M. Zaslavsky\nEd. Chaos, Kinetics and Non-linear Dynamics in Fluids and Plasmas\n(Springer-Verlag, Berlin 1998).\n\n\\bibitem{schneider} W. R. Schneider and W. Wyss, J. Math. Phys.\n{\\bf 30}, 134 (1989)\n\n\\bibitem{Gl} W. G. Gl$\\ddot{o}$ckle and T. F. Nonnenmacher, Macromolecules, {\\bf 24}\n6426 (1991)\n\n\\bibitem{fogedby} H. C. Fogedby, Phys. Rev. Lett. {\\bf 73}, 2517 (1994);\nPhys. Rev. E {\\bf 58}, 1690 (1998);S. Jespersen, R. Metzler and H. C. Fogedby, Phys. Rev. E {\\bf 59},\n2736 (1999)\n\n\n\\bibitem{zaslavsky}\nG. M. Zaslavsky, M. Edelman and B. A. Niyazov,\nChaos {\\bf 7}, 159 (1997)\n\n\\bibitem{saichev} A. I. Saichev and M. Zaslavsky, Chaos {\\bf 7} 4 (1997)\n\n\n\\bibitem{MBK} R. Metzler, E. Barkai and J. Klafter, Phys. Rev. Lett. {\\bf 82},\n 3563 (1999) \n\n\n\\bibitem{Hui} T. Huillet, J. Phys. A {\\bf 32} 7225 (1999)\n\n\\bibitem{Grig} Grigolini P, Rocco A, West B. J., Phys. Rev. E. \n{\\bf 59} 2603 (1999)\n\n\\bibitem{kuz} D. Kusnezov. A. Bulgac and G. Do Dang,\nPhys. Rev. Lett. {\\bf 82}, 1136 (1999)\n\n\\bibitem{Zum} G. Zumofen and J. Klafter\n{\\em Chem. Phys. Lett.} {\\bf 219} 303 (1994) \n\n\\bibitem{SB} E. Barkai and R. Silbey, Chem. Phys. Lett., {\\bf 310} 287 (1999).\n\n\\bibitem{MUK} V. Chernyak, M. Schultz and S. Mukamel, J. Chemical Physics,\n{\\bf 111} 7416 (1999)\n\n\\bibitem{oldham} K. B. Oldham and J. Spanier, {\\em The\nFractional Calculus} Academic Press, (New York) 1974.\n\n\\bibitem{Hilfer} R. Hilfer and L. Anton, Phys. Rev. E. {\\bf 51},\nR848 (1995)\n \n\n\\bibitem{compte} A. Compte, Phys. Rev. E {\\bf 53}, 4191 (1996)\n\n\\bibitem{West} B. J. West, P. Grigolini, R. Metzler and T. F. Nonnenmacher,\nPhys. Rev. E {\\bf 55}, 99 (1997)\n\n\n\\bibitem{barkai9} E. Barkai, R. Metzler and J. Klafter, Phys. Rev. E.\nPhys. Rev. E {\\bf 61} 132 (2000)\n\n\n%\n\n\\bibitem{Ral} Lord Rayleigh {\\em Phil. Mag.} {\\bf 32} 424 (1891). [{\\em Scientific\nPapers} (Cambridge 1902) {\\bf 3}, 473]\n\n%\n\\bibitem{Kampen} N.G. van Kampen {\\em Stochastic Processes in Physics and\n Chemistry} North Holland (Amsterdam -- New York -- Oxford, 1981)\n\n\n\n\\bibitem{barkai3} E. Barkai and V. N. Fleurov, {\\em J. Chem. Phys.} {\\bf 212},\n 69, (1996) \n\n%Levy Wals\n\\bibitem{barkai2} E. Barkai and V. N. Fleurov, {\\em Phys. Rev. E.} {\\bf 56},\n 6355, (1997) \n\n\\bibitem{MKp} R. Metzler and J. Klafter (preprint)\n\n\\bibitem{Risken} H. Risken, {\\it The Fokker--Planck\nequation\\/} (Springer, Berlin, 1989)\n\n\n\\bibitem{remark3} To see this Laplace transform Eq. (\\ref{eqR04}) using Eq.\n(\\ref{eqL05})\n%\n$$ u Q(u,v) - Q(v,t=0) = $$\n$$\\gamma_{\\alpha} \\hat{L}_{fp}\n\\left[ u^{1 - \\alpha} Q(v,u) - _0D_t^{ - \\alpha} Q(v,t) |_{t = 0} \\right], $$\nsetting $_0D_t^{ - \\alpha} Q(v,t) |_{t = 0}=0$ \nwe get an expression equivalent to\nto the Laplace transform of\nEq. (\\ref{eqR03}).\n\n\\bibitem{Feller} W. Feller,{\\em An introduction to probability Theory and\nIts Applications} Vol. 2 (John Wiley and Sons, New York 1970).\n\n\n%\n%\\bibitem{Silbey} J. Klafter and R. Silbey,\n% {\\em Phys. Rev. Lett.} {\\bf 44}, 55 (1980)\n\n\n%\\bibitem{Balakrishnan} V. Balakrishnan, Physica A {\\bf 132} 569 (1985)\n\n%\\bibitem{Wyss} W. Wyss, J. Math. Phys.\n%{\\bf 27}, 2782 (1986)\n\n%\\bibitem{mathai} A. M. Mathai and R. K. Saxena, {\\it The\n%$H$--function with Applications in Statistics and Other\n%Disciplines\\/} (Wiley Eastern Ltd, New Delhi, 1978)\n\n%\\bibitem{srivastava} H. M. Srivastava, K. C. Gupta and S. P. Goyal,\n%{\\it The $H$--functions of one and two variables\n%with applications\\/} (South Asian Publishers, New Delhi, 1982)\n\n% Comment\n%\\bibitem{barkai7} E. Barkai and J. Klafter, Phys. Rev. Lett. {\\bf 81},\n%1134 (1998)\n\n\n\\bibitem{Zol} V. M. Zoltarev, Dokl. Acad. Nauk. USSR {\\bf 98}, 715 (1954)\n\n\\bibitem{EWM} E. W. Montroll and B. J. West in {\\it Fluctuation Phenomena} Eds. E. W. Montroll\nand J. L. Lebowitz (North Holland, Amsterdam, 1987)\n\n\\bibitem{Hangi} P. H$\\ddot{a}$nggi, P. Talkner and M. Borkovec, Rev. Mod. Phys.\n{\\bf 62}, 251 (1990)\n\n\\bibitem{Mont} J. A. Montgomery, D. Chandler, and B. J. Berne,\nJ. Chem. Phys. {\\bf 70}, 4056 (1979)\n\n\\bibitem{Knessel} C. Knessel, M. Mangel, B. J. Matkowsky, A. Schuss,\nand C. Tier, J. Chem. Phys. {\\bf 81} 1285 (1984)\n\n\\bibitem{Bork} M. Borkovek, J. E. Straub, and B. J. Berne,\nJ. Chem. Phys. {\\bf 85}, 146 (1986)\n\n\n\\bibitem{barkaif} E. Barkai and V. Fleurov Phys. Rev. E {\\bf 52} 1558\n(1995)\n\n\\bibitem{Ber1} D. J. Bicout, A. M. Berezhkovskii, A. Szabo and G. H. Weiss, \nPhys. Rev. E. {\\bf 59} 3702 (1999) \n\n\\bibitem{Ber2} A. M. Berezhkovskii, D. J. Bicout and G. H. Weiss,\nJ. Chem. Phys. {\\bf 111} 11050 (1999) \n\n\\bibitem{Erde} A. Erdelyi Ed. {\\it Higher Transcendental Functions} McGraw-Hill, New York (1955) \n\n\n\\bibitem{remark4}\nTo see this write\n%\\begin{equation}\n$$\\langle x^2 (t) \\rangle_{eq}= \\langle \\left[\\int_0^t v\\left(t_1\\right) d t_1\\right]\n\\left[\\int_0^t v\\left(t_2\\right) d t_2\\right] \\rangle_{eq},$$\n%\nif the process is stationary \n$\\langle v(t_1) v(t_2)\\rangle_{eq} = \\langle v(\\tau) v(0)\\rangle_{eq}$\n%\nwith $\\tau=|t_1 - t_2|$, and only then Eq. (\\ref{eqER03}) valid.\n\n% Einstein Relation\n\\bibitem{barkaiE} \nE. Barkai and V. N. Fleurov, Phys. Rev. E {\\bf 58},\n1296 (1998)\n\n\\bibitem{Berlin} \nY. A. Berlin, L. D. A. Siebbeles and A. A. Zharikov,\nChem. Phys. Lett. {\\bf 305} 123 (1999)\n\n\\bibitem{KBS} \nJ. Klafter, A. Blumen and M. F. Shlesinger, \nPhys. Rev. A {\\bf 35}, 3081 (1987)\n\n\n%\n\n%\\bibitem{tsallis1} C. Tsallis, S. V. F. Levy, A. M. C. Souza\n%and R. Maynard, Phys. Rev. Lett. {\\bf 75}, 3589 (1995); D. H.\n%Zanette and P. A. Alemany, Phys. Rev. Lett. {\\bf 75} 366 (1995);\n%M. Buiatti, P. Grigolini and A. Monagnini, Phys. Rev. Lett. {\\bf 26}\n%3383 (1999); D. H. Zanette cond-mat/9905064 May 1999 and invited review in Braz. J. Phys.\n%\n\\bibitem{add}\nM. Bologna, P. Grigolini P and J. Riccardi,\nPhys. Rev. E {\\bf 60} 6435 (1999).\nR. Kutner, K. Wysocki\n{\\em Physica A} {\\bf 274} 67 (1999).\nV. V. Yanovsky, A. V. Chechkin, D. Schertzer\nand A. V. Tour nlin/001035 (to appear in Physica A)\n\n\n\n\n\\end{thebibliography}"
}
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cond-mat0002021
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Scaling Law and Aging Phenomena in the Random Energy Model
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[
{
"author": "Munetaka {\\sc Sasaki} and Koji {\\sc Nemoto}"
}
] |
[
{
"name": "text.tex",
"string": "\\documentstyle[seceq,epsbox,preprint]{jpsj}\n\\renewcommand{\\baselinestretch}{1}\n\\newcommand{\\df}{%\n{\\rm d}}\n\\def\\runtitle\n{Scaling Law and Aging Phenomena in the Random Energy Model}\n\\def\\runauthor{Munetaka {\\sc Sasaki} and Koji {\\sc Nemoto}}\n\\title{Scaling Law and Aging Phenomena in the Random Energy Model}\n\\author{Munetaka {\\sc Sasaki} and Koji {\\sc Nemoto}}\n\n\\inst{Division of Physics, Hokkaido University, Sapporo 060-0810}\n\\recdate{\\phantom{October 4, 1999}}\n\\abst{We study the effect of temperature shift on aging phenomena in the \nRandom Energy Model (REM). From calculation on the correlation function and \nsimulation on the Zero-Field-Cooled magnetization, we find that the REM \nsatisfies a scaling relation even if temperature is shifted. Furthermore, this \nscaling property naturally leads to results obtained in experiment \nand the droplet theory. \n}\n\\kword{aging,scaling,random energy model,zero-field-cooled magnetization,\ndroplet theory\n}\n\\begin{document}\n\\sloppy\n\\maketitle\n\\section{Introduction}\\label{sec:introduction}\nThe aging, which are dynamical behaviors largely depending on the \nhistory of system, is one of the most striking phenomena in the complex \nsystems such as spin glasses, glasses, polymers and proteins, and \nhas been studied vigorously from both theoretical and experimental aspects. \nIn experiments on spin glasses, one of the most familiar methods for the \nstudy of the aging phenomena is the measurement of the Zero-Field-Cooled \n(ZFC) magnetization\\cite{ZFC1,ZFC2}. \nIn this measurement, the sample is quenched to a temperature \\( T \\) \nbelow \\( T_{\\rm c} \\) in zero field. After a waiting time \\( t_{\\rm w} \\), \na weak magnetic field is applied and the magnetization \\( M(t)\\) \nis recorded as a \nfunction of observation time \\( t \\). In this experiment, \nthe length of \\( t_{\\rm w} \\) determines the degree of equilibration \nand aging appears as the \\( t_{\\rm w} \\) dependence, especially, \nthe peak of the relaxation rate \\( S(t) \\equiv \\frac{\\partial M(t)}\n{\\partial t} \\) emerges around \\( t_{\\rm w} \\) as if the system remembers \nhow the equilibration has proceeded before the observation. \n\nGranberg {\\it et al.}\\cite{ZFCTS} modified this experiment by shifting the \nwaiting temperature from \\( T \\) to \\( T-\\Delta T \\) and observed that, \nalthough \\( t_{\\rm w}\\) was unchanged, the maximum of \\( S(t) \\) \nshifted to left with increasing \\( \\Delta T \\) as if \\( t_{\\rm w} \\)\ndecreased. We refer to the position of this \nmaximum as apparent wait time \\( t_{\\rm w}^{\\rm app} \\). This suggests \nthat the equilibration at \\( T-\\Delta T \\) during \\( t_{\\rm w} \\) \nonly corresponds to the one at \\( T \\) during \\( t_{\\rm w}^{\\rm app} \\). \nFor \\( \\Delta T < 0 \\), the position shifts to right. \nThey also investigated the relation between \\( t_{\\rm w} \\) and \n\\( \\Delta T \\) and found\n\\begin{equation}\n\\log(t_{\\rm w}^{\\rm app}/t_{\\rm w}) \\approx \n-\\alpha(t_{\\rm w}) \\Delta T,\n\\label{eqn:relation1}\n\\end{equation}\nwith the result that the coefficient \\( \\alpha(t_{\\rm w}) \\) is a monotonically increasing function. \n\nFrom the theoretical aspects, this phenomenon is demonstrated by the droplet \ntheory very well\\cite{droplet1}, and is reproduced in the study of the two and \nthree dimensional EA ising spin glass model\\cite{MCS} and a hierarchical \ndiffusion model\\cite{hierarchical}. In this manuscript, we study this \nphenomenon with the Random Energy Model (REM)\\cite{Bouchaud,REM2,REM3}. \nAt first, we calculate a correlation function with the temperature shift \nand confirm that this correlation function satisfies a scaling law. \nNext, we carry out the simulation to observe the temperature shift ZFC \nmagnetization. As a result, we confirm that this magnetization also satisfies \nthe same scaling law as the correlation function, which naturally leads to \nthe relation eq.~(\\ref{eqn:relation1}). \n\nThe organization of this manuscript is following. In section 2, we introduce \nthe REM. In section 3, we define the temperature shift correlation function \nand calculate it. In section 4, we observe the temperature shift ZFC \nmagnetization by simulation and analyze the mechanism of this phenomenon. \nIn section 6, we summarize this paper with some discussions. \n\n\\section{The Random Energy Model}\nThe REM is schematically shown in Fig.~1\\cite{Bouchaud}. The bottom points represent the \naccessible states of this system. The length \nof each branch represents the barrier energy \\( E \\), over which the system \ngoes from one state to another. This energy is an independent random variable \ndistributed as\n\\begin{equation}\n\\rho (E) {\\rm d}E = \\frac{{\\rm d} E}{T_{\\rm c}}\\exp[-E/T_{\\rm c} ],\n\\label{eqn:simplerho}\n\\end{equation}\nwhere \\( T_{\\rm c} \\) is the transition temperature. \n\nFrom the Arrhenius law, \nthe relaxation time \\( \\tau(\\alpha:T) \\), the average time\n for the system to escape from the state \\( \\alpha \\) at \\( T \\), \nis related to \\( E(\\alpha) \\) as\n\\[ \\tau(\\alpha:T) = \\tau_0 \\exp[ E(\\alpha)/T], \\]\nwhere \\( \\tau_0 \\) is a microscopic time scale. Hereafter \\( \\tau_0 \\) \nis used as the unit of time and set to \\( 1 \\). \nFrom \neq.~(\\ref{eqn:simplerho}), the distribution of \\( \\tau \\) can be written as\n\\begin{equation}\np_x(\\tau) {\\rm d} \\tau = \\frac {x }{\\tau^{x+1}}{\\rm d} \\tau \n\\hspace{1cm}(\\tau \\geq 1),\n\\label{eqn:sinpleptau}\n\\end{equation}\nwhere \\( x \\equiv T/T_{\\rm c} \\).\nFrom eq.~(\\ref{eqn:sinpleptau}), \nit is easily shown that the averaged relaxation time \\(\\langle \\tau \\rangle \\) \nis \\( x/(x-1) \\) for \\( x>1 \\) and infinite for \\( x \\le 1 \\). \nThis means that the transition from the ergodic phase to the non-ergodic phase \noccurs at \\( T_{\\rm c} \\). \n\nFor dynamics, we employ the following simple Markoff process. \nFirst, the system is thermally activated \nfrom \\( \\alpha \\) in unit time with the probability\n\\begin{equation}\nW(\\alpha:T)=\\exp[-E(\\alpha)/T].\n\\label{eqn:defofW}\n\\end{equation}\nSince the time evolution is Markoffian, we can easily evaluate the probability \n\\( q(\\alpha,t:T) {\\rm d}t \\) that \nfor the event to occur during \\( t+t_0 \\) and \\( t+t_0+{\\rm d}t \\) with \nknowing the system is in the state \\( \\alpha \\) at \\( t_0 \\) as\n\\begin{equation}\nq(\\alpha,t:T) {\\rm d}t=W(\\alpha:T)\\exp[-W(\\alpha:T)t]{\\rm d}t.\n\\label{eqn:defofq}\n\\end{equation}\nWe generate an event time \\( t \\) according to \\( q(\\alpha,t:T) \\) to perform \nevent-driven Monte Carlo simulation. When the magnetic field \\( H \\) \nis applied, \nwe replace \\( E(\\alpha) \\) in eq.~(\\ref{eqn:defofW}) with \\( E+HM_{\\alpha} \\), \nwhere \\( M_{\\alpha} \\) is the magnetization of the state \\( \\alpha \\). \nThe magnetization is chosen randomly and independently from a distribution \n\\( D(M) \\) with zero mean. In this simulation, we choose the uniform \ndistribution between \\( -1 \\) and \\( +1 \\) for \\( D(M) \\). \n\nAfter the activation, the system falls to one of all states with equal \nprobability. In the limit \n\\(N \\rightarrow \\infty\\), we can neglect the possibility that \nthe system has stayed one state more than twice in our finite time simulation. \nTherefore, in this simulation, we create a new state by using the distribution \nfunctions \\( \\rho(E) \\) and \\( D(M) \\) whenever activation occurs. \n\\section{The Temperature Shift Correlation Function}\nAt first, we define the correlation function mentioned in \n\\S\\ref{sec:introduction}. \nThis correlation function is observed in the following procedure. \nWe consider the infinitely high temperature limit for the initial \ncondition. In other words, the initial state is chosen randomly at \\( t=0 \\). \nThen, the system is kept at \\( T-\\Delta T \\) during \\( t_{\\rm w} \\) and \nheated (or cooled) to \\( T \\) after that. Here, the magnetic field \\( H \\) is always zero. \nIn this procedure, we observe the correlation of magnetization \nbetween the time \\( t_{\\rm w} \\) and \\( t_{\\rm w}+t \\), which we \nhereafter refer to as \\( C_{\\Delta T}(t,t_{\\rm w}) \\). It is explicitly \nwritten as\n\\begin{equation}\nC_{\\Delta T}(t,t_{\\rm w}) = \\sum_{\\alpha,\\beta} M_{\\beta}\nG_{\\beta\\alpha}(t)M_{\\alpha} P_{\\alpha}(t_{\\rm w}),\n\\label{eqn:CF1}\n\\end{equation} \nwhere \\( P_{\\alpha}(t_{\\rm w}) \\) is the probability that the system \nis found at state \\( \\alpha \\) at \\( t_{\\rm w} \\) and \n\\( G_{\\beta\\alpha}(t) \\) is the probability that the system which \ninitially stays at \\( \\alpha \\) reaches \\( \\beta \\) at \\(t\\). \nBecause \\( M_{\\alpha} \\) is \nindependent random variable with zero mean, eq.~(\\ref{eqn:CF1}) is reduced \nto the autocorrelation function by taking the average over the distribution of \nthe magnetization: \n\\begin{equation}\nC_{\\Delta T}(t,t_{\\rm w}) = \\overline{M^2}\\sum_{\\alpha} \nG_{\\alpha\\alpha}(t)P_{\\alpha}(t_{\\rm w}),\n\\label{eqn:CF2}\n\\end{equation} \nwhere \\( \\overline{M^2} \\) is the variance of the distribution \\( D(M) \\). \nFinally, we ignore the possibility that the system is activated \nfrom \\( \\alpha \\) \nbetween \\( t_{\\rm w} \\) and \\( t+t_{\\rm w} \\) and we still find the system \nat \\( \\alpha \\) at time \\( t+t_{\\rm w} \\) in the limit \n\\( N\\rightarrow \\infty \\). As the result, eq.~(\\ref{eqn:CF2}) is further \nreduced to \n\\begin{equation}\nC_{\\Delta T}(t,t_{\\rm w}) =\\overline{M^2}\\sum_{\\alpha} \n\\exp[-W(\\alpha:T)t]P_{\\alpha}(t_{\\rm w}).\n\\label{eqn:CF3}\n\\end{equation} \n\nBouchaud and Dean\\cite{Bouchaud} calculated the Laplace transformation of \n\\( P_{\\alpha}(t_{\\rm w}) \\) and obtain the result, \n\\begin{equation}\n{\\hat P}_{\\alpha}(E)=\\frac{E^{x_1}}{Nx_1c(x_1)}\\frac{\\tau(\\alpha:T-\\Delta T)}\n{\\{E\\tau(\\alpha:T-\\Delta T)+1\\}},\n\\label{eqn:hatPalpha}\n\\end{equation}\nwhere \\( x_1 \\equiv \\frac{T-\\Delta T}{T_{\\rm c}} \\) and\n\\begin{equation}\nc(x)\\equiv\\Gamma(x)\\Gamma(1-x).\n\\end{equation}\nThe inverse Laplace transformation of eq.~(\\ref{eqn:hatPalpha}) leads us to \n\\begin{equation}\nP_{\\alpha}(t_{\\rm w})=\\frac{1}{Nx_1c(x_1)\\Gamma(x_1)}\n\\int_0^{t_{\\rm w}} {\\rm d}s s^{x_1-1}\\exp\\left[-\\frac{t_{\\rm w}-s}\n{\\tau(\\alpha:T-\\Delta T)}\\right].\n\\label{eqn:solP}\n\\end{equation}\n%There are \n%two possibilities for the system to stay at the state \\( \\alpha \\) \n%at \\( t_{\\rm w} \\). \n%The first one is that the initial state is \\( \\alpha \\) by chance \n%and the system is never activated during \\( t_{\\rm w} \\). \n%The second one is that the system starts from \\( \\beta \\) and is activated \n%from there at \\( t' \\) \\( (<t_{\\rm w}) \\). \n%In this case, the probability that the system stays at \\( \\alpha \\) \n%at \\( t_{\\rm w} \\) is \\( P_{\\alpha}(t_{\\rm w}-t') \\) because the next \n%state after activation is chosen randomly as the same for the initial \n%condition. Therefore, we obtain the following integral equation:\n%\\begin{equation}\n%P_{\\alpha}(t_{\\rm w})=\\frac{\\exp[-W(\\alpha:T-\\Delta T)]}{N}\n%+\\frac1N \\sum_{\\beta}\\int_{0}^{t_{\\rm w}} {\\rm d}t' q(\\beta,t':T-\\Delta T)\n%P_{\\alpha}(t_{\\rm w}-t').\n%\\end{equation}\n%The Laplace transformation of this equation will lead us to\n%\\begin{eqnarray}\n%{\\hat P}(E)&\\equiv&\\int_0^{\\infty}{\\rm d}t'\\exp[-Et']P_{\\alpha}(t')\n%\\nonumber\\\\\n%&=& \n%\\frac{\\displaystyle{\\frac1N \\frac{\\tau(\\alpha:T-\\Delta T)}\n%{E\\tau(\\alpha:T-\\Delta T)+1}}}\n%{\\displaystyle{\\frac1N \\sum_{\\beta}\\frac{E\\tau(\\beta:T-\\Delta T)}\n%{E\\tau(\\beta:T-\\Delta T)+1}}},\n%\\label{eqn:hatP1}\n%\\end{eqnarray}\n%where we used eq.~(\\ref{eqn:defofq}) and the fact \\( W(\\beta:T)=1/\\tau(\\beta:T) \\). In the limit \\( N\\rightarrow \\infty \\), we can approximate that \n%\\begin{equation}\n%\\frac1N \\sum_{\\beta}\\frac{E\\tau(\\beta:T-\\Delta T)}{E\\tau(\\beta:T-\\Delta T)+1} \n%= \\int_1^{\\infty}{\\rm d}\\tau \n%p_{x_1}(\\tau) \\frac{E\\tau}{E\\tau+1},\n%\\label{eqn:LTofP1}\n%\\end{equation}\n%where \\( x_1 \\equiv \\frac{T-\\Delta T}{T_{\\rm c}} \\). In the regime \n%\\( E \\ll 1 \\) ( equivalent to \\( t_{\\rm w} \\gg 1 \\)) and \\( x_1 < 1 \\), \n%we can estimate eq.~(\\ref{eqn:LTofP1}) as\n%\\begin{equation}\n%\\int_1^{\\infty}{\\rm d}\\tau p_{x_1}(\\tau) \\frac{E\\tau}{E\\tau+1}\n%= x_1 E^{x_1}c(x_1),\n%\\label{eqn:LTofP2}\n%\\end{equation}\n%where\n%\\begin{equation}\n%c(x)\\equiv\\Gamma(x)\\Gamma(1-x).\n%\\end{equation}\n%Using this result to eq.~(\\ref{eqn:hatP1}) \n%and taking the inverse Laplace transformation, we find that\n\nNow, let us rewrite \\( W(\\alpha:T) \\) as\n\\begin{equation}\nW(\\alpha:T)=\\tau(\\alpha:T-\\Delta T)^{-\\gamma},\n\\end{equation}\nwhere \\( \\gamma=\\frac{T-\\Delta T}{T} \\). Substituting this equation and \neq.~(\\ref{eqn:solP}) into eq.~(\\ref{eqn:CF3}), we obtain\n\\begin{equation}\nC_{\\Delta T}(t,t_{\\rm w})=\\frac{\\overline{M^2}}{c(x_1)\\Gamma(x_1)}\n\\int_{1}^{\\infty}{\\rm d}\\tau \\tau^{-x_1-1}\n\\exp[-\\tau^{-\\gamma}t]\n\\int_0^{t_{\\rm w}}{\\rm d}s s^{x_1-1}\\exp\\left[-\\frac{t_{\\rm w}-s}{\\tau}\\right].\n\\end{equation}\nBy changing the variables as \\( u\\equiv t_{\\rm w}/\\tau, v\\equiv s/\\tau \\) \nand replacing the upper limit of $u$-integral with \\( \\infty \\) in the \nassumption \\( t_{\\rm w} \\gg 1 \\), we find the following scaling relation, \n\\begin{equation}\nC_{\\Delta T}(t,t_{\\rm w})={\\tilde C}_{\\Delta T}(t/t_{\\rm S}),\n\\end{equation}\nwhere \n\\begin{equation}\nt_{\\rm S}=t_{\\rm w}^{\\gamma},\n\\end{equation}\nand\n\\begin{equation}\n{\\tilde C}_{\\Delta T}(X)=\\frac{\\overline{M^2}}{c(x_1)\\Gamma(x_1)}\n\\int_0^{\\infty} \\frac{{\\rm d} u}{u}\\exp[-Xu^{\\gamma}-u]\\int_0^{u}\n\\frac{{\\rm d} v}{v} v^{x_1} {\\rm e}^v.\n\\end{equation}\nWe can easily check that for the case \\( \\Delta T = 0 \\), \nthis relation is reduced to the \\( t/t_{\\rm w} \\) scaling as shown in \nref.~\\citen{Bouchaud}. \n\nNow, we are interested in the asymptotic behaviors of \n\\( {\\tilde C}_{\\Delta T}(X) \\) in the limit \\( X\\ll 1\\) and \\( X \\gg 1 \\). \nFor \\( X \\gg 1 \\), we obtain\n\\begin{subequations}\n\\begin{eqnarray}\n\\frac{C_{\\Delta T}(t,t_{\\rm w})}{\\overline{M^2}}\n&=&\\frac{1}{\\gamma c(x_1)\\Gamma(x_1)}\n\\int_0^{\\infty}\\frac{{\\rm d}s}{s}\\exp[-s-(s/X)^{\\frac{1}{\\gamma}}]\n\\int_0^{(s/X)^{\\frac{1}{\\gamma}}}\\frac{{\\rm d}v}{v} v^{x_1}{\\rm e}^v\\nonumber\\\\\n&\\approx &\\frac{1}{x_1 \\gamma c(x_1)\\Gamma(x_1)}\n\\int_0^{\\infty}\\frac{{\\rm d}s}{s}{\\rm e}^{-s}(s/X)^{x_1}\\nonumber \\\\\n&=& \\frac{\\Gamma(x_2)}{x_1 \\gamma c(x_1)\\Gamma(x_1)}X^{-x_2},\n\\label{eqn:asymptA}\n\\end{eqnarray}\nwhere \\( x_2 \\equiv \\frac{T}{T_{\\rm c}} \\). In the case \\( X\\ll 1 \\) and \\( x_1+\\gamma > 1 \\), we find\n\\begin{eqnarray}\n\\frac{C_{\\Delta T}(t,t_{\\rm w})}{\\overline{M^2}} &=& 1-\\frac{1}{\\gamma c(x_1)\\Gamma(x_1)}\n\\int_0^{\\infty}\\frac{{\\rm d}s}{s}\\exp[-(s/X)^{\\frac{1}{\\gamma}}]\n\\{ 1-{\\rm e^{-s}} \\}\n\\int_0^{(s/X)^{\\frac{1}{\\gamma}}}\\frac{{\\rm d}v}{v} v^{x_1}{\\rm e}^v\\nonumber\\\\\n&\\approx& 1-\\frac{1}{\\gamma c(x_1)\\Gamma(x_1)}\\int_0^{\\infty}\n\\frac{{\\rm d}s}{s}\\{ 1-{\\rm e^{-s}} \\}(s/X)^{\\frac{x_1-1}{\\gamma}}\\nonumber \\\\\n&=& 1-\\frac{\\Gamma(\\frac{x_1+\\gamma-1}{\\gamma})}{(1-x_1) c(x_1)\\Gamma(x_1)}\nX^{\\frac{1-x_1}{\\gamma}}.\n\\label{eqn:asymptB}\n\\end{eqnarray}\nFor \\( X\\ll 1 \\) and \\( x_1+\\gamma < 1 \\),\n\\begin{eqnarray}\n\\frac{C_{\\Delta T}(t,t_{\\rm w})}{\\overline{M^2}} \n&=& 1-\\frac{1}{c(x_1)\\Gamma(x_1)}\n\\int_0^1\\frac{{\\rm d} s}{s}s^{x_1}\\int_0^{\\infty}\\frac{{\\rm d} u}{u}\nu^{x_1}{\\rm e}^{-u(1-s)} \\{1-\\exp[-Xu^{\\gamma}]\\} \\nonumber\\\\\n&\\approx&1-\\frac{X}{c(x_1)\\Gamma(x_1)}\\int_0^1\\frac{{\\rm d} s}{s}s^{x_1}\n\\int_0^{\\infty}\\frac{{\\rm d} u}{u}u^{x_1+\\gamma}{\\rm e}^{-u(1-s)}\\nonumber\\\\\n&=&1-\\frac{c(x_1+\\gamma)}{c(x_1)} X.\n\\label{eqn:asymptC}\n\\end{eqnarray}\n\\end{subequations}\n\nNext, we evaluate eq.~(\\ref{eqn:CF3}) by simulation to check the\nvalidity of these results. \nRandom average is taken over \\( 10^6 \\) samples. In this simulation, \nwe fix \\( x_1 \\) to \\( 0.4 \\) \nand change \\( x_2 \\) and \\( t_{\\rm w} \\) as \\( 0.2,0.4,0.6,0.8,1.2 \\) \nand \\( 10,10^2,10^3,10^4,10^5 \\), respectively. \n%The initial condition \n%in the high temperature limit is realized by deciding the energy of the first \n%state according to the density of state \\( \\rho(E) \\). \nIn Fig.~2, the scaling \nplot for all \\( x_2 \\) is shown for (a) \\(t/t_{\\rm S} \\gg 1 \\) and \n(b) \\( t/t_{\\rm S}\\ll 1 \\). We can see that the scaling holds \nvery well and the exponent in eqs.~(\\ref{eqn:asymptA}),\n ~(\\ref{eqn:asymptB}) and~(\\ref{eqn:asymptC}) are rather correct, \nalthough a little deviation is observed on the coefficients \nfor \\( t/t_{\\rm S}\\ll 1 \\). \n\n\\section{The Result of the Temperature Shift ZFC Simulation}\nIn this section, we show the result of simulations on the temperature shift \nZFC magnetization with \\( T_{\\rm c}=1.0 \\) and \\( T=0.5 \\). \nThe number of samples for random average is \\( 5\\times10^7 \\). \nThe amplitude of field is \\( 0.1 \\). \nWe prepare the initial condition in the infinitely high temperature limit, \nthe same as the temperature shift correlation function. \nDuring \\( t_{\\rm w} \\), the temperature of the system is kept at \n\\( T-\\Delta T \\). After that, the temperature is changed to \\( T \\) and \nthe field \\( H \\) is applied. We performed the simulations for the 36 \ndifferent cases with \\( t_{\\rm w}=10^3,3\\times10^3,10^4,3\\times 10^4 \\) and \n\\( \\Delta T = 0.08,0.06,\\ldots,0,-0.02,\\ldots,-0.08 \\). \n\nIn Fig.~3, we plot the relaxation rate \\( S(t) \\) for \n\\( t_{\\rm w}=3\\times 10^4 \\) and all \\( \\Delta T \\). \nWe can see the position of the peak shifts to left with increasing \n\\( T-\\Delta T \\), although its shape becomes broader. \nNext, we checked whether the same scaling relation as \n\\( C_{\\Delta T}(t,t_{\\rm w}) \\) holds or not. In Fig.~4, we plot \n\\( S(t) \\) as a function of \\( t/t_{\\rm S} \\) for all \\( t_{\\rm w} \\) and \n\\( \\Delta T \\). We can confirm the scaling law. \nAlthough the position of peak depends on \\( \\Delta T \\) to some extent \nin our simulation, we consider that this dependence vanishes if \nwe take \\( \\Delta T/T \\) as small as that used in\nexperiment\\cite{ZFCTS}. \nBy neglecting this \\( \\Delta T \\) \ndependence, we can roughly estimate the position of peak as \n\\( t_{\\rm w}^{\\rm app}\\approx t_{\\rm S}=t_{\\rm w}^{\\frac{T-\\Delta T}{T}} \\) \nand obtain the relation eq.~(\\ref{eqn:relation1}) with \n\\( \\alpha (t_{\\rm w})= \\frac{\\log (t_{\\rm w})}{T} \\). This result is the same \nas the one from the droplet theory including the coefficient \n\\( \\alpha(t_{\\rm w}) \\)\\cite{MCS}. \n\nIn order to investigate how the equilibration proceeds during \n\\( t_{\\rm w} \\), we examine the energy distribution \\(P(E,t)(t\\le t_{\\rm w})\\) \nwhich is defined as the probability density that the system is found \nat time \\( t \\) in one of the states whose energy is \\( E \\). In Fig.~5, \nwe show how \\( P(E,t) \\) changes with time. At first sight, we can see \nthat \\( P(E,t) \\) has \na peak at some point \\( E^* \\), which shifts to right with increasing \\( t \\). \nThe value of \\( E^* \\) is roughly determined how far the system \ncan be activated during time interval \\( t \\), and estimated as\n\\begin{equation}\nE^* \\approx T \\log t.\n\\label{eqn:shiftspeed}\n\\end{equation}\nFor \\( E \\le E^* \\) the distribution is well equilibrated, so that \nthe exponent \\( \\alpha_1 \\) is given as\n\\begin{equation}\n\\alpha_1 = \\frac1T -\\frac{1}{T_{\\rm c}}. \n\\end{equation}\nNote that this exponent changes its sign at \\( T = T_{\\rm c} \\). \nWhile, the other part \\( E > E^* \\) is not equilibrated and \nthe exponent \\( \\alpha_2 \\) is equal to that of \\( \\rho(E) \\), {\\rm i.e.},\n\\begin{equation}\n\\alpha_2 = -\\frac{1}{T_{\\rm c}}.\n\\end{equation}\n\nEquation~(\\ref{eqn:shiftspeed}) suggests that the speed of the peak shift \nvaries with the temperature. In Fig.~6, we show \\( P(E,t) \\) for \n\\( t=10^4 \\) and all \\( \\Delta T \\). We can see that the position of \nthe peak shifts to right just as Fig.~3. From these nature of \n\\( P(E,t) \\), we understand the relation eq.~(\\ref{eqn:relation1}) \nas follows: During \\( t_{\\rm w} \\) at \\( T-\\Delta T \\), the peak shifts to \n\\( E^* \\approx (T-\\Delta T)\\log(t_{\\rm w}) \\). We can expect that, in this \nexperiment, the magnetization most strongly changes at the time scale \nin which the system can be activated to \\( E^* \\) at \\(T\\). Therefore, \nwe can estimate \\( t_{\\rm w}^{\\rm app} \\) from the condition\n\\begin{equation}\n(T-\\Delta T)\\log(t_{\\rm w}) \\approx T \\log(t_{\\rm w}^{\\rm app}),\n\\label{eqn:CDfortwapp}\n\\end{equation}\nwhich leads to eq.~(\\ref{eqn:relation1}) \nwith \\( \\alpha (t_{\\rm w})= \\frac{\\log (t_{\\rm w})}{T} \\), \nas mentioned above. \n\\section{Summary and Discussions}\nWe have shown that the REM satisfies a scaling law even if the temperature is \nshifted and time \\( t \\) is scaled by \n\\( t_{\\rm S}=t_{\\rm w}^{1-\\frac{\\Delta T}{T}} \\). For the case \n\\( \\Delta T =0 \\), \\( t/t_{\\rm w} \\)-type scaling is reported in the \nexperiment\\cite{scaling1} and simulation on the three-dimensional \n\\( \\pm J \\) Ising spin-glass model\\cite{scaling2}, although some \nsystematic deviations from the scaling remain on the experiment. It is interesting \nto check whether these systems satisfy the scaling like the REM or not. \n\nNext, we consider why the droplet theory leads to the same result \nas the REM. In the droplet theory, the crossover from equilibrium relaxation \nto non-equilibrium one occurs when the droplet created after the \nfield application grows to the size of the droplet created before that, \nand we can expect that peak of \\( S(t) \\) emerges there. Therefore, \n\\( t_{\\rm w}^{\\rm app} \\) is determined by the condition\n\\label{eqn:DLcondition}\n\\begin{equation}\nL(t_{\\rm w},T-\\Delta T)\\approx L(t_{\\rm w}^{\\rm app},T),\n\\end{equation}\nwhere \\( L(t,T) \\) denotes the characteristic size of the droplet \nafter the age \\( t \\) at \\( T \\), and is given from the condition\n\\begin{equation}\nB(T,L(t,T))=T\\log(t),\n\\end{equation}\nand \\( B(T,L) \\) denotes the barrier energy that the system has to \novercome to create the droplet of size \\( L \\), and is given as \n\\( B(T,L)=V(T) L^{\\psi} \\) explicitly. If \\( T \\) dependence of \n\\( V(T) \\) is week enough, the condition eq.~(\\ref{eqn:DLcondition}) \nis reduced to eq.~(\\ref{eqn:CDfortwapp}). \n\nRecently it was reported that \\( t_{\\rm w}^{\\rm app} \\) converges \nto \\( 0 \\) for somewhat large \\( |\\Delta T| \\) (\\( |\\Delta T/T|>0.07 \\) in \nthis experiment) regardless of its sign\\cite{ZFCTS2}. \nThis suggests that the equilibration \nat \\( T-\\Delta T \\) does affect the one at \\( T \\). In the droplet \ntheory\\cite{chaoticDP1,chaoticDP2}, \nit is considered that the {\\it chaotic} change of the equilibrium spin \nconfiguration against the temperature variation causes this phenomenon. But \nthe REM has no reason why this phenomenon is observed. Actually, \nthis phenomenon is not observed in our simulation even in the region \n\\( |\\Delta T/T| \\le 0.16 \\). But we can show that \nthis situation is improved by treating \nthe Multi-layer Random Energy Model (MREM), \nthe details of which will be presented elsewhere\\cite{tobesubmitted}.\n\\section*{Acknowledgements}\nWe would like to thank H. Yoshino for fruitful discussions and\nsuggestions on the manuscript. The numerical calculations were performed \non an Origin 2000 at Division of Physics, Graduate school of Science, \nHokkaido University. \n\\begin{thebibliography}{99}\n\\bibitem{ZFC1}P. Svedlindh, P. Granberg, P. Nordblad, L. Lundgren \nand H. S. Chen: Phys. Rev B {\\bf 35}(1987)268.\n%\n\\bibitem{ZFC2}L. Lundgren, P. Svedlindh and O. Beckman: \nPhys. Rev B {\\bf 26}(1982)3990.\n%\n\\bibitem{ZFCTS}P. Granberg, L. Sandlund, P. Nordblad, P. Svedlindh and \nL. Lundgren: Phys. Rev. B {\\bf 38}(1988)7097.\n%\n\\bibitem{droplet1}D. S. Fisher and D. A. Huse: Phys. Rev. B {\\bf 38}(1988)373.\n%\n%\\bibitem{droplet2}G. J. M. Koper and H. J. Hilhorst: J. Phys. (Paris) \n%{\\bf 49}(1988)429.\n%\n\\bibitem{MCS}J. O. Andersson, J. Mattsson and P. Svedlindh: Phys. Rev. B \n{\\bf 46}(1992)8297.\n%\n\\bibitem{hierarchical}C. Shulze, K. H. Hoffmann and P. Sibani: \nEurophys. Lett {\\bf 15}(1991)361.\n%\n\\bibitem{Bouchaud}J. P. Bouchaud and D. S. Dean: \nJ. Phys. I France {\\bf 5}(1995)265.\n%\n\\bibitem{REM2}B. Derrida: Phys. Rev. B {\\bf 24}(1981)2613.\n%\n\\bibitem{REM3}G. J. M. Koper and H. J. Hilhorst: Europhys. Lett \n{\\bf 3}(1987)1213.\n%\n\\bibitem{scaling1}E. Vincent, J. Hammann, M. Ocio, J. P. Bouchaud and \nL. F. Cugliandolo: {\\it Slow dynamics and aging} (Springer-Verlag, 1996) \ned. M. Rubi Sitges Conference on Glassy Systems; {\\it cond-mat/9607224}.\n%\n\\bibitem{scaling2}Heiko Rieger: J. Phys. A {\\bf 26}(1993)L615.\n%\n\\bibitem{ZFCTS2}P. Nordblad and P. Svendlidh: in {\\it Spin-glasses and \nrandom fields}, edited by A. P. Young, (World Scientific, Singapore, 1997); \n{\\it cond-mat/9810314}. \n%\n\\bibitem{chaoticDP1}A. J. Bray and M. A. Moore: Phys. Rev. Lett \n{\\bf 58}(1987)57.\n%\n\\bibitem{chaoticDP2}D. S. Fisher and D. A. Huse: Phys. Rev. Lett \n{\\bf 56}(1986)1601.\n%\n\\bibitem{tobesubmitted}M. Sasaki and K. Nemoto: to be submitted to \nJ. Phys. Soc. Jpn.\n%\n\\end{thebibliography}\n\\newpage\n\\noindent\n{\\bf \\large FIGURE CAPTIONS}\n\n\\vspace*{3mm}\\noindent\n\n\\vspace*{3mm}\\noindent\nFig.~1 Structure of the Random Energy Model. \n\n\\vspace*{3mm}\\noindent\nFig.~2 (a) \\( C_{\\Delta T}(t,t_{\\rm w}) \\) and (b) \n \\( 1-C_{\\Delta T}(t,t_{\\rm w}) \\) is plotted as a function of \n \\( t/t_{\\rm S} \\) for all \\( t_{\\rm w} \\) and \\( x_2 \\). \n Each line is the asymptotic behavior estimated in \n eqs.~(\\ref{eqn:asymptA}),~(\\ref{eqn:asymptB}) and~(\\ref{eqn:asymptC}).\n\n\\vspace*{3mm}\\noindent\nFig.~3 \\( S(t) \\) vs. \\( t \\) for \\( t_{\\rm w}=3\\times 10^4 \\) and \n \\( \\Delta T =0.08,0.06,\\ldots,0,-0.02,\\ldots,-0.08 \\) \n (from left to right). \n\n\\vspace*{3mm}\\noindent\nFig.~4 The scaling plot of \\( S(t) \\) for all \\( t_{\\rm w} \\) and \n \\( \\Delta T \\). \n\n\\vspace*{3mm}\\noindent\nFig.~5 \\( P(E,t) \\) for \\( T=0.5 \\) at \n \\( t=10^{0.5},10,\\ldots,10^{3.5} \\) (from left to right).\n \n\n\\vspace*{3mm}\\noindent\nFig.~6 \\( P(E,t) \\) for \\( t=10^4 \\) and \\( \\Delta T= 0.08,0.06,\\ldots,0,\n -0.02,\\ldots,-0.08 \\) (from left to right). \n\n\\newpage\n\\begin{center}\n\\epsfile{file=simpleREM.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.~1}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=CFfig3.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.~2(a)}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=CFfig4.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.~2(b)}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=ZFCfig1.ps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.~3}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=ZFCfig2.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.~4}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=Pfig1.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.~5}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=Pfig2.ps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.~6}\n\\end{center}\n\\end{document}"
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{
"name": "cond-mat0002021.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{ZFC1}P. Svedlindh, P. Granberg, P. Nordblad, L. Lundgren \nand H. S. Chen: Phys. Rev B {\\bf 35}(1987)268.\n%\n\\bibitem{ZFC2}L. Lundgren, P. Svedlindh and O. Beckman: \nPhys. Rev B {\\bf 26}(1982)3990.\n%\n\\bibitem{ZFCTS}P. Granberg, L. Sandlund, P. Nordblad, P. Svedlindh and \nL. Lundgren: Phys. Rev. B {\\bf 38}(1988)7097.\n%\n\\bibitem{droplet1}D. S. Fisher and D. A. Huse: Phys. Rev. B {\\bf 38}(1988)373.\n%\n%\\bibitem{droplet2}G. J. M. Koper and H. J. Hilhorst: J. Phys. (Paris) \n%{\\bf 49}(1988)429.\n%\n\\bibitem{MCS}J. O. Andersson, J. Mattsson and P. Svedlindh: Phys. Rev. B \n{\\bf 46}(1992)8297.\n%\n\\bibitem{hierarchical}C. Shulze, K. H. Hoffmann and P. Sibani: \nEurophys. Lett {\\bf 15}(1991)361.\n%\n\\bibitem{Bouchaud}J. P. Bouchaud and D. S. Dean: \nJ. Phys. I France {\\bf 5}(1995)265.\n%\n\\bibitem{REM2}B. Derrida: Phys. Rev. B {\\bf 24}(1981)2613.\n%\n\\bibitem{REM3}G. J. M. Koper and H. J. Hilhorst: Europhys. Lett \n{\\bf 3}(1987)1213.\n%\n\\bibitem{scaling1}E. Vincent, J. Hammann, M. Ocio, J. P. Bouchaud and \nL. F. Cugliandolo: {\\it Slow dynamics and aging} (Springer-Verlag, 1996) \ned. M. Rubi Sitges Conference on Glassy Systems; {\\it cond-mat/9607224}.\n%\n\\bibitem{scaling2}Heiko Rieger: J. Phys. A {\\bf 26}(1993)L615.\n%\n\\bibitem{ZFCTS2}P. Nordblad and P. Svendlidh: in {\\it Spin-glasses and \nrandom fields}, edited by A. P. Young, (World Scientific, Singapore, 1997); \n{\\it cond-mat/9810314}. \n%\n\\bibitem{chaoticDP1}A. J. Bray and M. A. Moore: Phys. Rev. Lett \n{\\bf 58}(1987)57.\n%\n\\bibitem{chaoticDP2}D. S. Fisher and D. A. Huse: Phys. Rev. Lett \n{\\bf 56}(1986)1601.\n%\n\\bibitem{tobesubmitted}M. Sasaki and K. Nemoto: to be submitted to \nJ. Phys. Soc. Jpn.\n%\n\\end{thebibliography}"
}
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cond-mat0002022
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[
{
"name": "ac.tex",
"string": "\\documentstyle [12pt]{article}\n\\textheight 22.5cm\n\\textwidth 16.5cm\n\\topmargin=-2cm\n\\oddsidemargin -.5cm\n\\begin{document}\n\\baselineskip .3in\n\\begin{titlepage}\n\\begin{center}{\\large {\\bf The travelling salesman problem on randomly diluted \nlattices: Results for small-size systems}}\n\\vskip .2in\n{\\bf Anirban Chakraborti}$~^{(1)}$ and\n{\\bf Bikas K. Chakrabarti}$~^{(2)}$\\\\\n{\\it Saha Institute of Nuclear Physics},\\\\\n{\\it 1/AF Bidhan Nagar, Calcutta 700 064, India.}\\\\\n\\end{center}\n\\vskip .3in\n{\\bf Abstract}\\\\\n\\noindent\nIf one places $N$ cities randomly on a lattice of size $L$,\nwe find that $\\bar l_E\\sqrt p$ and $\\bar l_M\\sqrt p$ vary with \nthe city concentration $p=N/L^2$, where $\\bar l_E$ is the average optimal travel \ndistance per city in the Euclidean metric \nand $\\bar l_M$ is the same in the Manhattan metric. We have studied such optimum\ntours for visiting all the cities using a branch and bound algorithm, giving the\nexact optimized tours for small system sizes ($N\\leq 100$) and near-optimal\ntours for bigger system sizes ($100<N\\leq 256$). \nExtrapolating the results for $N\\rightarrow \\infty$, we find that $\\bar l_E\\sqrt\np = \\bar l_M\\sqrt p = 1$ for $p=1$, and $\\bar l_E\\sqrt p=0.73\\pm 0.01$ and \n$\\bar l_M\\sqrt p=0.93\\pm 0.02$ with $\\bar l_M/\\bar l_E \\simeq 4/\\pi$ as \n$p\\rightarrow 0$. Although the problem is \ntrivial for $p=1$, for $p\\rightarrow 0$ it certainly reduces to the standard \ntravelling salesman problem on continuum which is NP- hard.\nWe did not observe any irregular behaviour at any intermediate point.\nThe crossover from the triviality to the NP- hard \nproblem presumably occurs at $p=1$.\n\\vskip 2.5in\n\\noindent\n{\\bf PACS No. :} 05.50+q\n\\end{titlepage}\n\\newpage\n\\noindent\n{\\bf 1 Introduction}\\\\\n\\noindent\nThe travelling salesman problem (TSP) is a simple example of a multivariable combinatorial \noptmization problem and perhaps the most famous one. Given a certain set of \ncities and the intercity distance metric, a travelling salesman must find the \nshortest tour in which he visits all the cities and comes back to his starting \npoint. It is a non-deterministic polynomial complete (NP- complete) problem \n[1-3]. \nIn the standard formulation of TSP, we have $N$ number of cities distributed \nrandomly on a continuum plane and we determine the average optimal\n travel distance per city $\\bar l_E$ in the Euclidean metric (with \n$\\Delta r_E= \\sqrt {\\Delta x^2+\\Delta y^2}$), or $\\bar l_M$ in the Manhattan \nmetric (with $\\Delta r_M= |\\Delta x|+|\\Delta y|$).\nSince the average distance per city scales (for fixed area) with the number of \ncities $N$ as $1/ \\sqrt N$, we find that the normalized travel distance per \ncity $\\Omega_E=\\bar l_E \\sqrt N$ or\n$\\Omega_M=\\bar l_M \\sqrt N$ become the optimized constants and their values\ndepend on the method used to optimize the travel distance. Extending the \nanalytic estimates of the average nearest neighbour distances, in particular\nwithin a strip and varying the width of the strip to extremize (single parameter\noptimization approximation), one gets $\\frac {5}{8}<\\Omega_E<0.92$ [4] and\n$\\frac {5}{2\\pi}<\\Omega_M<1.17$ [5]. Careful (scaling, etc.) analysis of the \nnumerical results obtained so far indicates that $\\Omega_E\\simeq 0.72$ [6].\n\n\nSimilar to many of the \nstatistical physics problems redefined on the lattices, e.g., the statistics of \nself-avoiding walks on lattices (for investigating the linear polymer \nconformational statistics), the TSP can also be \ndefined on randomly dilute lattices. The (percolation) cluster statistics of \nsuch dilute lattices is now extensively studied [7].\nThe salesman's optimized path on a dilute lattice is necessarily a\n self-avoiding one; for optimized tour the salesman cannot afford to visit any\ncity more than once and obviously it is one where the path is non-intersecting.\nThe statistics of self-avoiding walks on dilute lattices has also been \nstudied quite a bit (see e.g., [8]). However, this knowledge \n is not sufficient to understand the TSP on similar lattices. The TSP on dilute\nlattices is a very intriguing one, but has not been studied intensively so far.\n\n\nThe lattice version of the TSP was first studied by Chakrabarti [9].\nIn the lattice version of the TSP, the $N$ cities are represented by randomly \noccupied lattice sites of a two- dimensional square lattice ($L \\times L$), \nthe fraction of sites occupied being $p$ ($=N/L^2$, the lattice occupation \nconcentration). One must then find the shortest tour in which the salesman \nvisits each city only once and comes back to its starting point. The average \noptimal travel distance in the Euclidean metric \n$\\bar l_E$, and in the Manhattan metric $\\bar l_M$, are functions of the lattice\noccupation concentration $p$ [10]. We intend to study here the variation of the \nnormalised travel distance per city, \n$~\\Omega_E=\\bar l_E \\sqrt p$ and $\\Omega_M=\\bar l_M \\sqrt p$, \nwith the lattice concentration $p$ for different system sizes.\nIt is obvious that at $p=1$, all the self-avoiding walks passing through all the\noccupied sites will satisfy the requirements of TSP and $\\Omega_E=1=\\Omega_M$ \n(the distance between the neighbouring cities is equal to the unit lattice \nconstant and the path between neighbouring sites makes discrete angles of \nof $\\pi /2 $ or its multiples with the Cartesian axes). The problem becomes \nnontrivial as $p$ decreases from unity: isolated occupied cities and branching \nconfigurations of occupied cities are found here with finite probabilities and\nself-avoiding walks through all the occupied cities, and only through the \noccupied cities, become impossible.\nAs $p$ decreases from unity, the discreteness of the distance of the path\nconnecting the two cities and of the angle which the path makes with the Cartesian\naxes, tend to disappear. The problem reduces to the standard TSP on the \ncontinuum in the $p\\rightarrow 0$ limit when all the continuous sets of \ndistances and angles become possible. \nWe study here the TSP on dilute lattice employing a\ncomputer algorithm which gives the exact optimized tours for small system sizes\n($N\\leq 100$) and near-optimal tours for bigger system sizes ($100<N\\leq 256$).\n Our study indeed indicates that $\\Omega_E$ and $\\Omega_M$ vary \nwith $p$ and $\\Omega_E\\simeq 0.73$ and $\\Omega_M\\simeq 0.93$ as \n$p\\rightarrow 0$. \n\n\\noindent\n{\\bf 2 Computer Simulation and Results}\\\\\n\\noindent\nWe generate the randomly diluted lattice configurations following the standard\nMonte Carlo procedure for different system sizes. For each system size $N$, we \nvary the lattice size $L$ so that the lattice concentration $p$ varies. \nFor each such lattice configuration, the optimum tour with open boundary\nconditions, is obtained with the help of the {\\it GNU tsp\\_ solve} [11] \ndeveloped using a branch and bound algorithm (see Fig. 1). \nIt claims to give exact results for $N\\leq 100$ and near-optimal solutions for \n$100<N\\leq 256$. It may be noted that the program works \nessentially with the Euclidean distance. However there exists a geometric \nrelationship between the Euclidean distance and the Manhattan distance. We may\nwrite $l_E=\\sum_{i=1}^{N} r_i$, and $l_M=\\sum_{i=1}^N r_i \\alpha_i$, where $r_i$\nis the magnitude of the Euclidean path vector between two neighbouring cities\nand $r_i\\alpha_i=r_i(|\\sin \\theta_i |+|\\cos \\theta_i |)$ is the sum of the components \nof the Euclidean path projected along the Cartesian axes. \nNaturally, $1\\leq \\alpha_i \\leq \\sqrt 2$. If $l_E$ corresponds to the shortest\nEuclidean path, then $\\sum_{i=1}^N r_i^{\\prime} > \\sum_{i=1}^N r_i$ , \nfor any other path denoted by the primed set. If the optimized Euclidean\npath does not correspond to the optimized Manhattan path, then one will have \n$\\sum_{i=1}^N r_i^{\\prime } \\alpha_i^{\\prime } < \\sum_{i=1}^N r_i\\alpha_i$,\nwhere all the $\\alpha_i$ and $\\alpha_i^{\\prime}$ satisfy the previous bounds.\nAdditionally, for random orientation of the Euclidean distance with respect to \nthe Cartesian axes, $\\langle \\alpha_i \\rangle =\\langle \\alpha_i^{\\prime}\\rangle=\n(2/\\pi) \\int_0^{\\pi /2} (\\sin \\theta + \\cos \\theta ) d\\theta =4/\\pi $. It seems,\nwith all these constraints on $\\alpha$'s and $\\alpha^{\\prime}$'s, it would be \nimpossible to satisfy the above inequalities on \n$\\sum r_i$, and $\\sum r_i \\alpha_i$. In fact, we checked for a set of $50$ \nrandom optimized Euclidean tours for small $N$ ($<10$), obtained using the algorithm, whether the \noptimized Manhattan tours correspond to different sequence (of visiting the \ncities), and did not find any. We believe that the optimized Euclidean tour\nnecessarily corresponds to the optimized Manhattan tour.\nWe then calculate $l_E$ and $l_M$ for each such optimized tour. \n\n\nAt each lattice concentration $p$, we take about $100$\nlattice configurations (about 150 configurations at some special points near \n$p\\rightarrow 0$) and then obtain the averages $\\bar l_E$ and \n$\\bar l_M$. We then determine $~\\Omega_E=\\bar l_E ~\\sqrt p~$ and\n$~\\Omega_M=\\bar l_M ~\\sqrt p~$ and study the variations of $\\Omega_E$ and\n$\\Omega_M$, and of the ratio $\\Omega_M /\\Omega_E$ with $p$. \nWe find that $\\Omega_E$ and $\\Omega_M$ both have variations starting \nfrom the exact result of unity for $p=1$ to the respective constants \nin the $p\\rightarrow 0$ limit. In fact we noted that although $\\Omega_M$ \ncontinuously decreases as $p\\rightarrow 0$, it\nremains close to unity for all values of $p$. \nWe studied the numerical results for $N=~64,~81,~100,~121,~144,~169,\n~196,~225~\\rm {and}~256$. \nThe results for $N=64~\\rm {and}~100$ have been shown in Figs. 2 and 3\nrespectively. \nWe have studied the variations in the values of $\\Omega_E$ and $\\Omega_M$ \nagainst $1/N$ for $p\\rightarrow 0$, to extrapolate its value in the \n$N\\rightarrow \\infty$ limit. It appears that for the large $N$ limit (see \nFig. 4), $\\Omega_E(p\\rightarrow 0)$ and $\\Omega_M(p\\rightarrow 0)$ eventually \nextrapolate to $0.73\\pm 0.01$ (as in \ncontinuum TSP) and to $0.93\\pm 0.02$, respectively. \nThis result for $\\Omega_E$ (at $p\\rightarrow 0$) compares very well with the \nprevious estimates [6]. As $p$ changes from $1$ to $0$, the ratio $\\Omega_M /\n \\Omega_E $ changes continuously from $1$ to about $1.27~(\\simeq 4/\\pi )$ (see \nFig. 4), which is the average ratio of the Manhattan distance between two random\n points in a plane and the Euclidean distance between them [10, 5]. \n\n\n\\noindent\n{\\bf 3 Conclusions}\\\\\n\\noindent\nWe note that the TSP \non randomly diluted lattice is certainly a trivial problem when $p=1$ (lattice \nlimit) as it reduces to the one-dimensional TSP (the connections in the optimal\n tour are between the nearest neighbours along the lattice).\nHere $\\Omega_E(p)=\\Omega_M(p)=1$. \n However, it is certainly NP- hard at the $p\\rightarrow 0$ (continuum) limit,\nwhere $\\Omega_E \\simeq 0.73$ and $\\Omega_M \\simeq 0.93$ (extrapolated for large \nsystem sizes $N$). We note that\n$\\Omega_M$ remains practically close to unity for all values of \n$p<1$. Our numerical results also suggest that $\\Omega_M/\\Omega_E\\simeq 4/\\pi$\nas $p\\rightarrow 0$.\nIt is clear that the problem crosses from triviality (for $p=1$) to the NP- hard\nproblem (for $p\\rightarrow 0$) at a certain value of $p$. \nWe did not find any irregularity in the variation of $\\Omega$ at any $p$.\nA naive expectation might be that around \nthe percolation point, beyond which the marginally connected lattice spanning \npath is snapped off [7], the $\\Omega_E$ or $\\Omega_M$ suffers some irregularity.\nThe absence of any such irregularity can also be justified easily: the \ntravelling salesman\nhas to visit all the occupied lattice sites (cities), not necessarily those on \nthe spanning cluster. Also, the TSP on dilute lattices has got to accomodate \nthe same kind of frustration as the (compact) self-avoiding chains on dilute \n(percolating) lattices, although there the (collapsed) polymer is confined only\n to the spanning cluster. This indicates that the \ntransition occurs either at $p=1_-$ or at $p=0_+$. From the consideration of \nfrustration for the TSP even at $p=1_-$, it is almost certain that the \ntransition occurs at $p=1$. \nHowever, this point requires further investigations.\n\n\\vskip 0.3in\n\\noindent\n{\\bf Acknowledgement} : We are grateful to O. C. Martin and A. Percus\nfor very useful comments and suggestions.\n\n\\newpage\n\\noindent\n{\\bf References}\\\\\n\\vskip .1in\n\\noindent\n{\\it e-mail addresses} :\n\n\\noindent\n$^{(1)}$anirban@cmp.saha.ernet.in\n\n\\noindent\n$^{(2)}$bikas@cmp.saha.ernet.in\n\\vskip .2 in\n\n\\noindent\n1. M. R. Garey and D. S. Johnson, {\\it Computers and Intractability: A Guide\nto the Theory of NP- Completeness} (Freeman; San Franscisco) (1979).\\\\\n\\noindent\n2. S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, {\\it Science},\n{\\bf 220}, 671 (1983).\\\\\n\\noindent\n3. M. Mezard, G. Parisi and M. A. Virasoro, {\\it Spin Glass Theory and Beyond}\n(World Scientific; Singapore) (1987).\\\\\n\\noindent\n4. J. Beardwood, J. H. Halton and J. M. Hammersley, {\\it Proc. Camb. Phil. \nSoc.} {\\bf 55}, 299 (1959); R. S. Armour and J. A. Wheeler, {\\it Am. J. Phys.}\n{\\bf 51}, 405 (1983).\\\\ \n\\noindent\n5. A. Chakraborti and B. K. Chakrabarti, {\\it cond-mat/0001069} (2000).\\\\ \n\\noindent\n6. A. Percus and O. C. Martin, {\\it Phys. Rev. Lett.}, {\\bf 76}, 1188 (1996).\\\\\n\\noindent\n7. D. Stauffer and A. Aharony, {\\it Introduction to Percolation Theory} (Taylor\nand Francis; London) (1985).\\\\\n\\noindent\n8. K. Barat and B. K. Chakrabarti, {\\it Phys. Rep.}, {\\bf 258}, 377 (1995).\\\\\n\\noindent\n9. B. K. Chakrabarti, {\\it J. Phys. A: Math. Gen.}, {\\bf 19}, 1273 (1986).\\\\\n\\noindent\n10. D. Dhar, M. Barma, B. K. Chakrabarti and A. Tarapder, \n{\\it J. Phys. A: Math. Gen.}, {\\bf 20}, 5289 (1987);\nM. Ghosh, S. S. Manna and B. K. Chakrabarti,\n{\\it J. Phys. A: Math. Gen.}, {\\bf 21}, 1483 (1988);\nP. Sen and B. K. Chakrabarti,\n{\\it J. Phys. (Paris)}, {\\bf 50}, 255, 1581 (1989).\\\\\n\\noindent\n11. C. Hurtwitz, {\\it GNU tsp\\_ solve}, available at: http://www.cs.sunysb.edu/\\~\\\\\n\\noindent\nalgorith/implement/tsp/implement.shtml\\\\\n\n\\newpage\n\\noindent\n{\\bf Figure captions}\\\\\n\\vskip .1 in\n\\noindent\n{\\bf Fig. 1} : A typical TSP for ($N=)~64$ cities on a \ndilute lattice of size $L=30$. The cities are represented by black dots which \nare randomly occupied sites of the lattice with concentration $p=N/L^2\\simeq \n0.07$. The optimized Euclidean path is indicated.\\\\\n\\noindent\n{\\bf Fig. 2} : Plot of $\\Omega_E$, $\\Omega_M$ and $\\Omega_M/\\Omega_E$ against \n$p$ for $N=64$ cities, obtained using the optimization programs (exact).\n The error bars are due to configurational fluctuations.\nThe extrapolated values of $\\Omega_E$, $\\Omega_M$ and $\\Omega_M/\\Omega_E$ are\nindicated by horizontal arrows on the y-axis.\\\\ \n\\noindent\n{\\bf Fig. 3} : Plot of $\\Omega_E$, $\\Omega_M$ and $\\Omega_M/\\Omega_E$ against \n$p$ for $N=100$ cities, obtained using the optimization programs (exact).\n The error bars are due to configurational fluctuations.\nThe extrapolated values of $\\Omega_E$, $\\Omega_M$ and $\\Omega_M/\\Omega_E$ are\nindicated by horizontal arrows on the y-axis.\\\\ \n\\noindent\n{\\bf Fig. 4 } : Plots of $\\Omega_E$ , $\\Omega_M $ and of $\\Omega_M/\\Omega_E$ in\nthe $p\\rightarrow 0$ limit, against $1/N$.\n The error bars are due to configurational fluctuations. \nThe extrapolated value of $\\Omega_E$ , $\\Omega_M$ and $\\Omega_M/\\Omega_E$ in \nthis $p\\rightarrow 0$ limit for $N\\rightarrow \\infty$ are indicated by \nhorizontal arrows on the y-axis.\\\\ \n \n\\end{document}\n"
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cond-mat0002023
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Charge and spin ordering in Nd$_{1/3}$Sr$_{2/3}$FeO$_{3}$
|
[
{
"author": "Y. Oohara$^{a}$"
},
{
"author": "M. Kubota$^{a}$"
},
{
"author": "H. Yoshizawa$^{a}$"
},
{
"author": "S. K. Park$^{b}$"
},
{
"author": "Y. Taguchi$^{b}$"
},
{
"author": "Y. Tokura$^{b}$"
}
] |
We have investigated the charge and spin ordering in Nd$_{1/3}$Sr$_{2/3}$FeO$_{3}$ with neutron diffraction technique. This sample undergoes a charge ordering transition accompanying charge disproportionation of $\mbox{2Fe}^{4+} \to \mbox{Fe}^{3+} + \mbox{Fe}^{5+}$. We measured the superlattice reflections due to the charge and spin ordering, and confirmed that charges and spins order simultaneously at $T_{CO} = 185$ K. The ordering pattern of charges and spins in this sample can be viewed as three dimensional stripe order, and is compared with two dimensional stripe order observed in other transition metal oxides.
|
[
{
"name": "crest.tex",
"string": "%&latex209\n\\documentstyle[aps,twocolumn,psfig]{revtex}\n%\\documentstyle[aps,twocolumn,eclepsf]{revtex}\n\n\\begin{document}\n\n\\draft\n\n\\wideabs{\n\n\\title{Charge and spin ordering in Nd$_{1/3}$Sr$_{2/3}$FeO$_{3}$}\n\n\\author{R. Kajimoto$^{\\rm a}$\\thanks{Corresponding\nauthor. Fax: +81-29-283-3922. {\\it E-mail address:}\nkaji\\@red.issp.u-tokyo.ac.jp (R. Kajimoto)}, Y. Oohara$^{\\rm a}$,\nM. Kubota$^{\\rm a}$, H. Yoshizawa$^{\\rm a}$, \nS. K. Park$^{\\rm b}$, Y. Taguchi$^{\\rm b}$, Y. Tokura$^{\\rm b}$}\n\n\\address{$^{\\rm a}$Neutron Scattering Laboratory, I. S. S. P.,\nUniversity of Tokyo, Tokai, Ibaraki, 319-1106, Japan \\\\\n$^{\\rm b}$Department of Applied Physics, University of\nTokyo, Bunkyo-ku, Tokyo 113-8656, Japan}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\n We have investigated the charge and spin ordering in\n Nd$_{1/3}$Sr$_{2/3}$FeO$_{3}$ with neutron diffraction technique. This\n sample undergoes a charge ordering transition accompanying charge\n disproportionation of $\\mbox{2Fe}^{4+} \\to \\mbox{Fe}^{3+} +\n \\mbox{Fe}^{5+}$. We measured the superlattice reflections due to the\n charge and spin ordering, and confirmed that charges and spins order\n simultaneously at $T_{\\rm CO} = 185$ K. The ordering pattern of charges\n and spins in this sample can be viewed as three dimensional stripe\n order, and is compared with two dimensional stripe order observed in\n other transition metal oxides.\n\\end{abstract}\n\n\\pacs{{\\it keywords:} A. oxides, B. crystal growth, C. neutron\nscattering, D. charge-density wave, magnetic structure}\n\n}\n\n%\\section{Introduction}\n\nCharge ordering is widely seen in hole doped transition metal oxides,\nsuch as cuprates (e.g. La$_{2-x}$Sr$_{x}$CuO$_{4}$ \\cite{tranquada_cu}),\nnickelates (e.g. La$_{2-x}$Sr$_{x}$NiO$_{4+\\delta}$ \\cite{tranquada_ni})\nand manganites (e.g. Pr$_{1/2}$Ca$_{1/2}$MnO$_{3}$ \\cite{tomioka95} and\nNd$_{1/2}$Sr$_{1/2}$MnO$_{3}$ \\cite{kuwahara95}). Because the charge\nordering transition occurs at the region near the insulator-metal\ntransition or superconducting transition, a number of works concentrated\non the study of this phenomenon in order to clarify a key to understand\nthe origin of the insulator-metal transition or superconducting\ntransition. From these studies, it is widely recognized that the charge\nordering phenomenon is a consequence of the coupling or the competition\namong the degrees of freedom of charge, spin, lattice, or orbitals\n\n\nHole doped perovskite-type $R_{1/3}$Sr$_{2/3}$FeO$_{3}$ is one of the\nsystem which undergoes a charge ordering transition. In this system, the\ncharge ordering accompanies charge disproportionation of\n$\\mbox{2Fe}^{4+} \\to \\mbox{Fe}^{3+} + \\mbox{Fe}^{5+}$ and simultaneous\nantiferromagnetic spin ordering. A pioneering work with M\\\"{o}ssbauer\nspectroscopy on La$_{1/3}$Sr$_{2/3}$Fe$_{3}$ by Takano {\\it et al.}\nrevealed that there are two kinds of Fe ions with the ratio of $2:1$,\nand they were attributed to Fe$^{3+}$ and Fe$^{5+}$\n\\cite{takano83}. This charge disproportionation state was confirmed by\nBattle {\\it et al.} with the neutron powder diffraction measurements on\nthe same compound. They observed the antiferromagnetic spin ordering\nwith sixfold periodicity along the cubic [111] direction and showed that\nthis magnetic structure was generated from the ordering of the layers of\nthe Fe$^{3+}$ ions and the Fe$^{5+}$ ions along the cubic [111]\ndirection in a sequence of\n$\\cdots\\mbox{Fe}^{3+}\\mbox{Fe}^{3+}\\mbox{Fe}^{5+}\\cdots$. However, they\ncould not observe the superlattice reflections due to the charge\nordering presumably because the weak intensity of the reflections in the\npowder sample, although it is well known that in many charge ordered\nsystems the ordering of charges strongly couples with lattice and\nproduces periodic modulation in the crystal structure\n\\cite{tranquada_cu,tranquada_ni,yoshi00,tomioka95,kuwahara95}. Recently\nLi {\\it et al.} observed the superlattice reflections of the charge\nordering in La$_{1-x}$Sr$_{x}$FeO$_{3}$ single crystals for the first\ntime by electron diffraction measurements \\cite{li97}. Park {\\it et al.}\nshowed that similar charge and spin ordering also exists in\n$R_{1/3}$Sr$_{2/3}$FeO$_{3}$ single crystals where $R$ is rare-earth\natoms other than La \\cite{park99}.\n\nNeutron diffraction is very useful to investigated the spin and charge\ncoupled physics such as charge ordering, because it can detect the\ndirect evidence of the magnetic ordering and the structural modulations\nrelated to the ordering of charges which is formed in a bulk\nsample. Therefore we performed the neutron diffraction experiments on\none of the 2/3-hole-doped Fe oxides, Nd$_{1/3}$Sr$_{2/3}$FeO$_{3}$. By\nutilizing a single crystal sample, we could find the superlattice\nreflections due to the structural modulations by the charge ordering\ntogether with magnetic reflections. We could also detect a subtle change\nof the nature of the charge ordering as a function of temperature.\n\n\n%\\section{Experimental procedure}\n\nA single crystal sample studied in the present study was grown by the\nfloating zone method in oxygen atmosphere with a traveling speed of 1.0\nmm/h. The detailed procedures of the sample preparation have already\nbeen described elsewhere \\cite{park99}. The quality of the sample was\nchecked by x-ray powder diffraction measurements and by electron probe\nmicroanalysis.\n\nNeutron diffraction experiments were performed using triple axis\nspectrometer GPTAS installed at the JRR-3M reactor in JAERI, Tokai,\nJapan with fixed incident neutron momentums $k_{\\rm i} = 2.66$\n\\AA$^{-1}$ and 3.83 \\AA$^{-1}$. The combination of collimators were\n20$^{\\prime}$-40$^{\\prime}$-20$^{\\prime}$-open and\n40$^{\\prime}$-80$^{\\prime}$-80$^{\\prime}$-80$^{\\prime}$ (from\nmonochromator to detector). Although the crystal structure of the\nsample has a slight rhombohedral distortion along the cubic [111]\ndirection ($\\alpha=60.1^{\\circ}$ at 300 K), we employed the cubic\nlattice ($a \\sim 3.85$ \\AA) notation of the scattering plane. The\nsample was mounted in an Al can filled with He gas, and was attached to\nthe cold head of a closed-cycle helium gas refrigerator. The temperature\nwas controlled within an accuracy of 0.2 K.\n\n%\\section{Results and discussion}\n\nFirst we show the results of resistivity and magnetization measurements\nwhich were performed on the same crystal used in the neutron diffraction\nstudy\\cite{taguchi_un}. Figure \\ref{transport} shows the temperature\ndependence of the resistivity and the spontaneous magnetization. The\nresistivity at room temperature is relatively low and gradually\nincreases as temperature decreases. However, it shows a steep increase\nbelow $T_{\\rm CO} = 185$ K because of the charge ordering transition. As\nshown in Fig. \\ref{transport}, small spontaneous magnetization appears\nbelow $T_{\\rm CO}$, signaling the onset of the antiferromagnetic\nordering with minute spin canting.\n\n\\begin{figure}\n \\centering \\leavevmode\n \\psfig{file=transport.eps,width=0.8\\hsize}\n \\vspace{2mm}\n \\caption{Temperature dependence of resistivity and spontaneous\n magnetization for Nd$_{1/3}$Sr$_{2/3}$FeO$_{3}$.}\n \\label{transport}\n\\end{figure}\n\nIn order to characterize the charge and spin ordering, we surveyed the\n$(hhl)$ scattering plane and found some superlattice reflections below\n$T_{\\rm CO}$. For an example of such survey scans, we show in\nFig. \\ref{profile} profiles of line scans along $[11\\bar{1}]$ direction\nmeasured at 200 K and 10 K. One can see that at 10 K superlattice\nreflections appear at $(\\frac{1}{6},\\frac{1}{6},\\frac{5}{6})$,\n$(\\frac{1}{3},\\frac{1}{3},\\frac{2}{3})$,\n$(\\frac{1}{2},\\frac{1}{2},\\frac{1}{2})$,\n$(\\frac{2}{3},\\frac{2}{3},\\frac{1}{3})$, and\n$(\\frac{5}{6},\\frac{5}{6},\\frac{1}{6})$. The observed superlattice\nreflections can be classified into three groups according to their\nmodulation vectors {\\boldmath $q$}: $\\mbox{\\boldmath $q$}_{\\frac{1}{6}}\n= a^{*}(\\frac{1}{6},\\,\\frac{1}{6},\\,\\frac{1}{6})$, $\\mbox{\\boldmath\n$q$}_{\\frac{1}{3}} = a^{*}(\\frac{1}{3},\\,\\frac{1}{3},\\,\\frac{1}{3})$,\nand $\\mbox{\\boldmath $q$}_{\\frac{1}{2}} =\na^{*}(\\frac{1}{2},\\,\\frac{1}{2},\\,\\frac{1}{2})$. Due to the twin\ndomains of the cubic structure, we also observed reflections with\n$\\mbox{\\boldmath $q$} = a^{*}(\\frac{1}{6},\\,\\frac{1}{6},\\,-\\frac{1}{6})$\netc. in the $(hhl)$ zone.\n\n\\begin{figure}\n \\centering \\leavevmode\n \\psfig{file=profile.eps,width=\\hsize}\n \\vspace{-1mm}\n \\caption{A profile of the line scan along the $[11\\bar{1}]$ direction\n at 200 K and 10 K. Shaded peaks represent the superlattice reflections\n due to the magnetic and charge ordering.}\n \\label{profile}\n\\end{figure}\n\n$\\mbox{\\boldmath $q$}_{\\frac{1}{3}}$ reflections were observed also by\nelectron diffraction measurements \\cite{li97,park99} and they indicate\nlattice modulation along the cubic [111] direction whose period was\nthree times larger than the lattice spacing of the (111) plane. This\nmodulation should originate from the charge ordering of\n$\\cdots\\mbox{Fe}^{3+}\\mbox{Fe}^{3+}\\mbox{Fe}^{5+}\\cdots$ and\naccompanying local lattice distortion. The intensity of $\\mbox{\\boldmath\n$q$}_{\\frac{1}{6}}$ and $\\mbox{\\boldmath $q$}_{\\frac{1}{2}}$ reflections\nobey the $Q$ dependence of the magnetic form factor of the Fe\nion. Therefore they can be attributed to the antiferromagnetic ordering,\nwhich is consistent with the previous powder neutron diffraction\nmeasurements on La$_{1/3}$Sr$_{2/3}$FeO$_{3}$ \\cite{battle90}. Note that\nnuclear reflections were observed at\n$(\\frac{h}{2},\\frac{h}{2},\\frac{l}{2})$ even at $T > T_{\\rm CO}$\n(Fig. \\ref{profile}). These reflections are forbidden in the previously\nproposed rhombohedral symmetry $R\\bar{3}c$, and indicate that the true\ncrystal symmetry of the present sample is lower than $R\\bar{3}c$\n\\cite{battle90,li97,park99}.\n\nThe existence of $\\mbox{\\boldmath $q$}_{\\frac{1}{6}}$ and\n$\\mbox{\\boldmath $q$}_{\\frac{1}{2}}$ magnetic modulation vectors means\nthat the distribution of magnetic moments can be represented by the sum\nof two Fourier components, each has a wave vector of $\\mbox{\\boldmath\n$q$}_{\\frac{1}{6}}$ and $\\mbox{\\boldmath $q$}_{\\frac{1}{2}}$. As a\nconsequence, if one assume the magnetic moments are localized on the Fe\nsites, there are two Fe sites in the ratio of $2:1$ in the sequence of\n$\\cdots \\Uparrow\\, \\Downarrow\\, \\downarrow\\, \\Downarrow\\, \\Uparrow\\,\n\\uparrow\\, \\Uparrow \\cdots$. The sites denoted by $\\Uparrow$ and those\nby $\\uparrow$ may be attributed to Fe$^{3+}$ sites and Fe$^{5+}$ sites,\nrespectively.\n\nIn order to analyze the magnetic moments for two Fe sites, we have\nperformed neutron powder diffraction measurements to avoid difficulty in\nanalyzing the single crystal caused by the domain distribution. The\nmeasurement was performed at 50 K, because below that temperature, the\nreflections by the ordering of the magnetic moments of Nd$^{3+}$ ions\nsuperpose the $\\mbox{\\boldmath $q$}_{\\frac{1}{6}}$ magnetic reflections\n(see below). The obtained value of the magnetic moments are 3.7\n$\\mu_{\\rm B}$ for Fe$^{3+}$ site and 2.3 $\\mu_{\\rm B}$ for Fe$^{5+}$\nsite assuming the spins lie in the (111) plane. The direction of the\nspins in the (111) plane could not be determined due to the high\nsymmetry. These values are similar to those obtained for\nLa$_{1/3}$Sr$_{2/3}$FeO$_{3}$ (3.61 $\\mu_{\\rm B}$ for Fe$^{3+}$ site and\n2.72 $\\mu_{\\rm B}$ for Fe$^{5+}$ site) \\cite{battle90}. The smaller\nobserved magnetic moments than their nominal values of 5 $\\mu_{\\rm B}$\nand 3 $\\mu_{\\rm B}$ indicate the strong hybridization of the Fe 3d\norbitals and the O 2d orbitals because of the small charge-transfer\nenergy of Fe oxides \\cite{bocquet92,matsuno99}.\n\nFigure \\ref{Tdep} shows temperature dependence of the intensity of (a)\nthe $\\mbox{\\boldmath $q$}_{\\frac{1}{3}}$ charge superlattice reflection,\n(b) $\\mbox{\\boldmath $q$}_{\\frac{1}{6}}$ (open circles) and\n$\\mbox{\\boldmath $q$}_{\\frac{1}{2}}$ (closed circles) spin superlattice\nreflections. As decreasing temperature, all the reflection start to\ndevelop at the same temperature $T_{\\rm CO} = 185$ K, indicating that\ncharges and spins order simultaneously, which is consistent with the\nresistivity and magnetization data shown in Fig. \\ref{transport}. The\ntransition at $T_{\\rm CO}$ is first order with small hysteresis of $\\sim\n4$ K. The increase of the intensity of the\n$(\\frac{5}{6}\\frac{5}{6}\\frac{5}{6})$ reflection below $T \\sim 50$ K\ncomes from the ordering of the spins of Nd$^{3+}$ ions.\n\n\\begin{figure}\n \\centering \\leavevmode\n \\psfig{file=Tdep.eps,width=0.85\\hsize}\n \\vspace{2mm}\n \\caption{Temperature dependence of the intensity of the spin and charge\n super lattice reflections. (a) Charge superlattice reflection\n $(\\frac{4}{3}\\frac{4}{3}\\frac{4}{3})$. (b) Spin superlattice reflection\n $(\\frac{5}{6}\\frac{5}{6}\\frac{5}{6})$ (open circles) and\n $(\\frac{1}{2}\\frac{1}{2}\\frac{1}{2})$ (closed circles).}\n \\label{Tdep}\n\\end{figure}\n\nThe development of the charge ordering almost saturates around $T_{\\rm\ns} = 150$ K. The character of the spin ordering also changes at this\ntemperature. The slope of the curve for the $\\mbox{\\boldmath\n$q$}_{\\frac{1}{2}}$ reflection becomes steeper below $T_{\\rm s}$, while\nthe growth of the $\\mbox{\\boldmath $q$}_{\\frac{1}{6}}$ reflection is\nslightly suppressed around $T_{\\rm s}$.\n\nThe anomaly in the spin superlattice reflections at $T_{\\rm s}$ can not\nbe explained by the rotation of the spin orientations. Because the\nscattering vectors of the two kinds of the spin reflections shown in\nFig. \\ref{Tdep} (b) are parallel, the rotation of the spins should\nproduce the same effect on the intensity of both spin\nreflections. Therefore, the anomaly at $T_{\\rm s}$ should be attributed\nto the change of the ratio of the two Fourier component of the spin\ndensity wave at this temperature. If one assume the moments are\nlocalized at Fe sites, this means that the moments on Fe$^{3+}$ sites\nincrease while those on Fe$^{5+}$ sites decrease, suggesting the\nenhancement of the charge disproportionation. Of course this\ninterpretation is too na\\\"{\\i}ve because the holes may locate on oxygen\nsites and the magnetic moments distribute continuously from site to site\ndue to the hybridization of the Fe 3d orbitals and the O 2p\norbitals. Nevertheless, from the fact that the charge ordering almost\nsaturates around $T_{\\rm s}$, one can say that the change of the\ndistribution of the magnetic moments are correlated with the charge\nordering, and the nature of the charge and the spin ordering changes\nwhen the charge ordering sufficiently develops. We should note that we\nalso observed the similar anomaly at $T < T_{\\rm CO}$ in another\n2/3-hole-doped charge disproportionated Fe oxides\nPr$_{1/3}$Sr$_{2/3}$FeO$_{3}$ \\cite{oohara_un}.\n\nOne of the most interesting phenomena in the charge ordering transition\nin the hole-doped transition metal oxides is stripe order, where the\ndoped holes align to form domain walls and spins order\nantiferromagnetically. Most of the stripe ordering observed so far, e.g.\nin La$_{2-x}$Sr$_{x}$CuO$_{4}$ \\cite{tranquada_cu} or\nLa$_{2-x}$Sr$_{x}$NiO$_{4+\\delta}$ \\cite{tranquada_ni,yoshi00}, is two\ndimensional (2 D) which has one dimensional domain walls. On the other\nhand, the charge and spin ordering pattern of the 2/3-hole-doped cubic\nperovskite Fe oxides including Nd$_{1/3}$Sr$_{2/3}$FeO$_{3}$ can be\nviewed as a three dimensional (3 D) stripe order propagating the [111]\ndirection with 2 D domain walls parallel to (111) planes,\n\nOne of the clear differences between the 2 D and the 3 D stripe order is\nthe relation between the charge ordering temperature $T_{\\rm CO}$ and\nthe spin ordering temperature $T_{\\rm N}$. In the 2 D stripe order,\nspins always order {\\em after} the charges order, while in the 3 D\nstripe order, spins and charges order {\\em simultaneously}. For the\nformation of the stripe order, the importance of the superexchange\ninteraction between the spins as well as the Coulomb interaction between\ncharges is widely recognized, although there is still a dispute about\nthe driving force for the stripe order. Zachar {\\it et al.} proposed a\nphase diagram for the stripe order based on Landau\ntheory\\cite{zachar98}. They claimed that there are several ways in the\ntransition to the stripe ordered state: a charge driven transition, a\ncharge-spin coupling driven transition, and a spin driven transition. In\nthe charge driven transition, $T_{\\rm N}$ should be lower than $T_{\\rm\nCO}$, while in the charge-spin coupled transition and in the spin driven\ntransition $T_{\\rm N}$ should be same as $T_{\\rm CO}$. Moreover, the\nspin-charge coupled transition should be the first order while the spin\ndriven transition should be the second order. In their framework, the 2\nD stripe order observed in cuprates or nickelates is classified into the\ncharge driven transition, and the 3 D stripe order observed in the\npresent study can be classified into the spin-charge coupling driven\ntransition. In any event, the fact that spins and charges order\nsimultaneously in the 3 D stripe order means that the role of the spin\nordering in the stripe order becomes relatively important compared to\nthe 2 D stripe.\n\nWe think the increase of the relative importance of the spin ordering in\nthe 3 D stripe can be interpreted as a consequence of the difference in\nthe dimensionality between the 2 D and the 3 D stripe. In the 2 D stripe\norder, the number of the nearest holes around a hole in a domain wall\nbecomes larger because the domain walls become 2 D. Therefore, the\nenergy loss by the Coulomb repulsion between the holes may become larger\nas compared to the 3 D stripe order. On the other hand, the number of\nthe nearest sites for the undoped region also becomes larger because of\nthe three dimensionality, which may increase the energy gain by the\nsuperexchange interaction between spins.\n\nIn order to verify the above scenario, the energy scale of the Coulomb\ninteraction and the superexchange interaction should be examined. We\nwould like to note that by recent Hartree-Fock calculations, it has been\nshown that the 3 D stripe ordering observed in Fe oxides can be well\nexplained by the superexchange interaction \\cite{mizokawa98}, which\nindicates the importance of the spin interaction for the formation of\nthe stripe order.\n\n\n%\\section{Conclusions}\n\nIn summary, we have investigated the charge and spin ordering in a\nNd$_{1/3}$Sr$_{2/3}$MnO$_{3}$ crystal using neutron diffraction\ntechnique. We measured the superlattice reflections due to the charge\nand spin ordering, and confirmed that charges and spins order\nsimultaneously at $T_{\\rm CO} = 185$ K. The character of the charge and\nspin ordering changes at 150 K when the charge ordering almost\nsaturates. The pattern of the charge and spin ordering in this sample\ncan be viewed as the 3 D stripe order, and the transition may be driven\nby the spin-charge coupling.\n\n%\\section*{Acknowledgements}\n\nThis work was supported by a Grant-In-Aid for Scientific Research from\nthe Ministry of Education, Science, Sports and Culture, Japan and by the\nNew Energy and Industrial Technology Development Organization (NEDO) of\nJapan.\n\n\\begin{references}\n\n \\bibitem{tranquada_cu} J. M. Tranquada, B. J. Sternlieb, J. D. Axe,\n Y. Nakamura, and S. Uchida, Nature (London) {\\bf 375} (1995) 561.\n\n \\bibitem{tranquada_ni} J. M. Tranquada, D. J. Buttrey, and V. Sachan,\n Phys. Rev. B {\\bf 54} (1996) 12318; P. Wochner, J. M. Tranquada,\n D. J. Buttrey, and V. Sachan, Phys. Rev. B {\\bf 57} (1998) 1066.\n\n \\bibitem{yoshi00} H. Yoshizawa, T. Kakeshita, R. Kajimoto, T. Tanabe,\n T. Katsufuji, and Y. Tokura, Phys. Rev. B {\\bf 61} (2000) R854.\n\n \\bibitem{tomioka95} Y. Tomioka, A. Asamitsu, Y. Moritomo, H. Kuwahara,\n and Y. Tokura, Phys. Rev. Lett. {\\bf 74} (1995) 5108.\n\n \\bibitem{kuwahara95} H. Kuwahara, Y. Tomioka, A. Asamitsu, Y. Moritomo,\n and Y. Tokura, Science {\\bf 270} (1995) 961.\n\n \\bibitem{takano83} M. Takano and Y. Takeda, Bull. Inst. Chem. Res.,\n Kyoto Univ. {\\bf 61} (1983) 406.\n\n \\bibitem{battle90} P. D. Battle, T. C. Gibb, and P. Lightfoot, J. Solid\n State Chem. {\\bf 84} (1990) 271.\n\n \\bibitem{li97} J. Q. Li, Y. Matsui, S. K. Park, and Y. Tokura,\n Phys. Rev. Lett. {\\bf 79} (1997) 297.\n\n \\bibitem{park99} S. K. Park, T. Ishikawa, Y. Tokura, J. Q. Li, and\n Y. Matsui, Phys. Rev. B {\\bf 60} (1999) 10788.\n\n \\bibitem{taguchi_un} Y. Taguchi, S. K. Park, and Y. Tokura, unpublished.\n\n \\bibitem{bocquet92} A. E. Bocquet, A. Fujimori, T. Mizokawa, T. Saitoh,\n H. Namatame, S. Suga, N. Kimizuka, Y. Takeda, and M. Takano,\n Phys. Rev. B {\\bf 45} (1992) 1561.\n\n \\bibitem{matsuno99} J. Matsuno, T. Mizokawa, and A. Fujimori,\n K. Mamiya, Y. Takeda, S. Kawasaki, and M. Takano, Phys. Rev. B {\\bf\n 60} (1999) 4605.\n\n \\bibitem{oohara_un} Y. Oohara, M. Kubota, H. Yoshizawa, S. K. Park, and\n Y. Tokura, unpublished.\n\n \\bibitem{mizokawa98} T. Mizokawa and A. Fujimori, Phys. Rev. Lett. {\\bf\n 80} (1998) 1320.\n\n \\bibitem{zachar98} O. Zachar, S. A. Kivelson, and V. J. Emery,\n Phys. Rev. B {\\bf 57} (1998) 1422.\n\n\\end{references}\n\n\\end{document}\n"
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"name": "cond-mat0002023.extracted_bib",
"string": "\\bibitem{tranquada_cu} J. M. Tranquada, B. J. Sternlieb, J. D. Axe,\n Y. Nakamura, and S. Uchida, Nature (London) {\\bf 375} (1995) 561.\n\n \n\\bibitem{tranquada_ni} J. M. Tranquada, D. J. Buttrey, and V. Sachan,\n Phys. Rev. B {\\bf 54} (1996) 12318; P. Wochner, J. M. Tranquada,\n D. J. Buttrey, and V. Sachan, Phys. Rev. B {\\bf 57} (1998) 1066.\n\n \n\\bibitem{yoshi00} H. Yoshizawa, T. Kakeshita, R. Kajimoto, T. Tanabe,\n T. Katsufuji, and Y. Tokura, Phys. Rev. B {\\bf 61} (2000) R854.\n\n \n\\bibitem{tomioka95} Y. Tomioka, A. Asamitsu, Y. Moritomo, H. Kuwahara,\n and Y. Tokura, Phys. Rev. Lett. {\\bf 74} (1995) 5108.\n\n \n\\bibitem{kuwahara95} H. Kuwahara, Y. Tomioka, A. Asamitsu, Y. Moritomo,\n and Y. Tokura, Science {\\bf 270} (1995) 961.\n\n \n\\bibitem{takano83} M. Takano and Y. Takeda, Bull. Inst. Chem. Res.,\n Kyoto Univ. {\\bf 61} (1983) 406.\n\n \n\\bibitem{battle90} P. D. Battle, T. C. Gibb, and P. Lightfoot, J. Solid\n State Chem. {\\bf 84} (1990) 271.\n\n \n\\bibitem{li97} J. Q. Li, Y. Matsui, S. K. Park, and Y. Tokura,\n Phys. Rev. Lett. {\\bf 79} (1997) 297.\n\n \n\\bibitem{park99} S. K. Park, T. Ishikawa, Y. Tokura, J. Q. Li, and\n Y. Matsui, Phys. Rev. B {\\bf 60} (1999) 10788.\n\n \n\\bibitem{taguchi_un} Y. Taguchi, S. K. Park, and Y. Tokura, unpublished.\n\n \n\\bibitem{bocquet92} A. E. Bocquet, A. Fujimori, T. Mizokawa, T. Saitoh,\n H. Namatame, S. Suga, N. Kimizuka, Y. Takeda, and M. Takano,\n Phys. Rev. B {\\bf 45} (1992) 1561.\n\n \n\\bibitem{matsuno99} J. Matsuno, T. Mizokawa, and A. Fujimori,\n K. Mamiya, Y. Takeda, S. Kawasaki, and M. Takano, Phys. Rev. B {\\bf\n 60} (1999) 4605.\n\n \n\\bibitem{oohara_un} Y. Oohara, M. Kubota, H. Yoshizawa, S. K. Park, and\n Y. Tokura, unpublished.\n\n \n\\bibitem{mizokawa98} T. Mizokawa and A. Fujimori, Phys. Rev. Lett. {\\bf\n 80} (1998) 1320.\n\n \n\\bibitem{zachar98} O. Zachar, S. A. Kivelson, and V. J. Emery,\n Phys. Rev. B {\\bf 57} (1998) 1422.\n\n"
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cond-mat0002024
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Memory Effect, Rejuvenation and Chaos Effect in the Multi-layer Random Energy Model
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"author": "Munetaka {\\sc Sasaki} and Koji {\\sc Nemoto}"
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"string": "\\documentstyle[seceq,epsbox,preprint]{jpsj}\n\\renewcommand{\\baselinestretch}{1}\n\\newcommand{\\df}{%\n{\\rm d}}\n\\def\\runtitle\n{Memory Effect, Rejuvenation and Chaos Effect in the MREM}\n\\def\\runauthor{Munetaka {\\sc Sasaki} and Koji {\\sc Nemoto}}\n\\title{Memory Effect, Rejuvenation and Chaos Effect in the Multi-layer Random Energy Model}\n\\author{Munetaka {\\sc Sasaki} and Koji {\\sc Nemoto}}\n\n\\inst{Division of Physics, Hokkaido University, Sapporo 060-0810}\n\\recdate{\\phantom{October 4, 1999}}\n\\abst{\nWe introduce magnetization to the Multi-layer Random Energy Model\nwhich has a hierarchical structure, and perform Monte Carlo \nsimulation to observe the behavior of ac-susceptibility.\nWe find that this model is able to reproduce three prominent features\nof spin glasses, i.e., memory effect, rejuvenation and chaos effect,\nwhich were found recently\nby various experiments on aging phenomena with temperature variations.\n}\n\\kword{aging, multi-layer random energy model, hierarchical picture, memory effect, rejuvenation, chaos effect }\n\\begin{document}\n\\sloppy\n\\maketitle\n\\section{Introduction}\\label{sec:introduction}\nThe free energy of complex systems such as spin glasses,\n polymers and proteins, is considered\n to have a very complex structure with numerous local minima. \nDue to this complexity,\n it costs very long time for equilibration of these systems and\nvarious non-equilibrium relaxation phenomena are observed. \nFrom experiments focused on these non-equilibrium relaxations, \nthe aging phenomena, which are dynamical behaviors\n largely depending on the history of system, were found and \nhave been studied vigorously from both theoretical \nand experimental aspects, in particular, in spin glass systems\n\\cite{ZFCexperiment1,ZFCexperiment2,TRMexperiment,ZFCTS,ZFCTC,MCS}. \n\n\nRecently the following interesting experiment has been reported \nin spin glasses\\cite{acTC1,acTC2,acTC3}. \nIn this experiment, the relaxation of the out-of-phase \nac-susceptibility \\( \\chi'' \\) is observed in the following \nthree stage. In the first stage, the sample is quenched from above \n\\( T_{\\rm c} \\) down to a temperature \\( T \\) below \\( T_{\\rm c} \\) \nand is kept at this temperature during \\( t_1\\). \nThen the sample is perturbed by changing temperature from \\( T \\) \nto \\( T\\pm \\Delta T\\) during \\( t_2\\) in the following second\nstage and the temperature is returned to \\( T \\) (the third stage). \nThe effect of the perturbation is examined by comparing the \nperturbed data and unperturbed (\\( t_2 =0 \\)) data \nin the third stage. In the case \\( -\\Delta T \\), both data coincide \nexcept at the very beginning. This means that the relaxation \nat \\( T -\\Delta T \\) never affects the relaxation at \\( T \\) and \nthe system remembers the relaxation during \\( t_1 \\) after \nperturbation. We call this as memory effect hereafter. \nIn the case \\( +\\Delta T \\), on the other hand, \nwe can see that \\( \\chi'' \\) becomes larger than the unperturbed \ncorrespondence and the rejuvenation occurred during \\( t_2 \\). \nThese 'memory effect' and 'rejuvenation' are observed in other \nexperiments\\cite{TRMexperiment}. \n\nFrom the theoretical point of view,\n the aging phenomena have been studied along \ntwo different pictures. One is the droplet \npicture\\cite{droplet1,droplet2}, where the behavior in the \nspin glass phase is governed by the low laying and large length scale \nexcitation. The other is the so-called hierarchical picture\n\\cite{hierarchical1,hierarchical2,hierarchical3,hierarchical4,Bouchaud}, \nfrom which we will try to understand the phenomena in this manuscript.\n\nAccording to the Parisi mean-field solution of the SK model, \nthe free energy in the spin-glass phase has a very complex structure \nwith numerous local minima. These valleys are considered to \nbe divided into smaller valleys as the system is cooled \nand micro phase transitions occur with this continuous multifurcation. \nFrom this hierarchical structure,\n the memory effect and rejuvenation mentioned above can be\nexplained as follows\\cite{acTC2}. \n%Assume that the structure of the free energy in some region changes \n%as Fig.~1 as the system is cooled.\n%When the temperature is cooled from temperature above \\( T_{\\rm c} \\)\n% down to \\( T \\), the system falls into one of the numerous local minima.\n%Let this local minimum be \\( \\alpha \\). Then, as the age increases, \n%the system relaxes toward the equilibrium in which each of states \n%\\( \\alpha, \\beta \\) and \\( \\gamma \\) is occupied with a weight proportional \n%to the corresponding Boltzmann factor.\n%Now we consider to change the temperature from \\( T \\) to \n%\\( T\\pm\\Delta T \\). \n%In the case \\( -\\Delta T \\), \n%each state at \\( T \\) is divided into a set of the states \n%\\( \\{\\alpha_i\\}, \\{\\beta_i\\} \\) and \\( \\{\\gamma_i\\} \\). \n%If \\( \\Delta T\\) is large enough and the time \\( t_2 \\) \n%is not so long, we can ignore the possibility that the system overcomes \n%the barrier which separates the sets \\( \\{\\alpha_i\\}, \\{\\beta_i\\} \\) \n%and \\( \\{\\gamma_i\\} \\). Therefore, when the temperature is raised back to \n%\\( T \\), we find that the ratio of the weights among the states \n%\\( \\alpha, \\beta \\) and \\( \\gamma \\) are unchanged during \\( t_2 \\) and \n%the memory effect appears. In the case \\( +\\Delta T \\), on the other hand,\n%the states \\( \\alpha, \\beta \\) and \\( \\gamma \\) merge into one state\n% and the system forgets how long the relaxation proceeded at \\( T \\).\n%Thus the system is rejuvenated. \nWhen the temperature is cooled from temperature above \\( T_{\\rm c} \\) \ndown to \\( T \\), the system falls into one of the numerous local minima. \nThen, the system is equilibrated in some region of the phase space \nincluding several valleys, \\( {\\cal R} \\), during \\( t_1 \\). \nNow we consider to change the temperature from \\( T \\) to \n\\( T\\pm\\Delta T \\). In the case \\( -\\Delta T \\), \neach valley at \\( T \\) is divided into smaller valleys. \nIf \\( \\Delta T\\) is large enough and the time \\( t_2 \\) \nis not so long, the equilibration proceeds only in the smaller valleys \nand we can ignore the possibility that the system overcomes \nthe barrier which separates the bigger valleys at \\( T \\). \nTherefore, when the temperature is raised back to \n\\( T \\) and smaller valleys merge, we find that \nthe system are unchanged during \\( t_2 \\) and \nthe memory effect appears. In the case \\( +\\Delta T \\), on the other hand, \nseveral valleys in \\( {\\cal R} \\) merge into one valley \nand the system forges the equilibration at \\( T \\). \nThus the system is rejuvenated. \n\nThis explanation is, however, quite qualitative\n and there exists no theoretical model which reproduces\n these features along this hierarchical picture.\n In this manuscript, we study the \nMulti-layer Random Energy Model (MREM)\\cite{Bouchaud,MREM2,MREM3} \nwhich has a hierarchical structure causing a continuous phase transition. \nWe introduce magnetization to this model so that we can carry out \nthe simulation to observe the ac-susceptibility, and\nfind that the memory effect and rejuvenation are\n indeed reproducible. \nFurthermore, we analyze the mechanism of these phenomena and \nshow that the above-mentioned scenario realizes these features, \nat least, in this model. \n%\\subsection{Chaos effects in spin-glasses}\\label{subsec:chaos}\n\nAnother feature we are interested in is the {\\em chaos} effect found\nin experiments on the ac-susceptibility\\cite{Jonason}.\nThe fact found is that\nthe equilibration at the temperature \\( T \\) does not affect \nthe relaxation at a lower temperature \\( T-\\Delta T \\).\nThis can be interpreted as follows.\nIf the structure of the free energy at \\( T-\\Delta T \\)\n is quite different from that at \\( T \\), the equilibration \n at \\( T \\) never helps the equilibration at \\( T-\\Delta T \\). \nBut this reorganization of structure makes us unable to explain\n the memory effect, which is also observed in the same experiment.\nThere seem to exist few theoretical \nexplanations\nfor these apparently conflicting \nfeatures, i.e., memory and {\\em chaos} effects.\nWe find that these conflicting features hold \nsimultaneously in the MREM in a situation slightly different \nfrom the experiment in ref.~\\citen{Jonason}. \nWe also analyze the mechanism of these phenomena in this model. \n\n\\section{The Multi-layer Random Energy Model}\nLet us first explain the Simple-layer Random Energy Model (SREM)\nillustrated in Fig.~1.\n The bottom points represent the accessible states of this system. \nWe hereafter consider the case that the number of states is very large. \nThe length of each branch represents the barrier energy $E$, over which\nthe system goes from one state to another.\nThis energy is an independent random variable distributed as\n\\begin{equation}\n\\rho (E) {\\rm d}E = \\frac{{\\rm d} E}{T_{\\rm c}}\\exp[-E/T_{\\rm c} ],\n\\label{eqn:simplerho}\n\\end{equation}\nwhere \\( T_{\\rm c} \\) is the transition temperature. \n\nFrom the Arrhenius law, \nthe relaxation time \\( \\tau(\\alpha) \\), i.e., the average time\n for the system to escape from the state \\( \\alpha \\), \nis related to \\( E(\\alpha) \\) as\n\\[ \\tau(\\alpha) = \\tau_0 \\exp[ E(\\alpha)/T], \\]\nwhere \\( \\tau_0 \\) is a microscopic time scale. From \neq.~(\\ref{eqn:simplerho}), the distribution of \\( \\tau \\) can be written as\n\\begin{equation}\np(\\tau) {\\rm d} \\tau = \\frac {x \\tau_0^x}{\\tau^{x+1}}{\\rm d} \\tau \n\\hspace{1cm}(\\tau \\geq \\tau_0),\n\\label{eqn:sinpleptau}\n\\end{equation}\nwhere \\( x \\equiv T/T_{\\rm c} \\).\nFrom eq.~(\\ref{eqn:sinpleptau}), \nit is easily shown that the averaged relaxation time \\( \\langle\\tau\\rangle \\) \nis \\( \\tau_0 x/(x-1) \\) for \\( x>1 \\) and infinite for \\( x \\le 1 \\). \nThis means that the transition from the ergodic phase to the non-ergodic phase \noccurs at \\( T_{\\rm c} \\). \n\nNow we construct the Multi-layer Random Energy Model (MREM). As shown in \nFig.~2,\n this model is constituted by piling up the SREM $L$ times hierarchically. \nThe energy of the \\( n \\)-th layer counted from the bottom, \\( E_n \\),\n is given according to the \ndistribution \\( \\rho_n(E_n)=\\exp[-E_n/ T_{\\rm c}(n)]/T_{\\rm c}(n) \\) \nand each layer has a different transition temperature chosen so as to satisfy\n\\( T_{\\rm c}(1)<T_{\\rm c}(2)<\\cdots <T_{\\rm c}(L) \\). \nTherefore, in this model the transition occurs continuously \nfrom the uppermost (the \\( L \\)-th) layer to the lowest one.\n%We will see that this nature plays a very important rule. \n\n\\subsection{The dynamics}\nIn this simulation, we use the following Markoff process for dynamics. \nFirst the system is thermally activated from \\( \\alpha \\) to \n\\( \\alpha_1 \\), where \\( \\alpha_n \\) is the $n$-th {\\em ancestor}\n of \\( \\alpha \\) (see Fig.~2).\nThe probability that \nthis event occurs at time $t$ in unit time is defined as\n\\begin{equation}\n\\omega(\\alpha;t) = \\tau_0^{-1} \\exp[-E_1(\\alpha)/T].\n\\label{eqn:omega}\n\\end{equation}\nThe $t$ dependence comes implicitly from time variation of\ntemperature and magnetic field. \nWe will introduce the effect of magnetic field in the following subsection. \nSince the time evolution is Markoffian, we can easily evaluate\nthe probability $q(\\alpha;t_0,t){\\rm d}t$ for\nthe event to occur during\n\\( t \\) and \\( t+{\\rm d} t \\)\n with knowing that the system is in the state $\\alpha$ at time $t_0$ as\n\\begin{equation}\nq(\\alpha;t_0,t){\\rm d}t=\\omega(\\alpha;t){\\rm d}t\\exp[-\\int_{t_0}^{t}{\\rm d}t'\n\\omega(\\alpha;t')].\n\\label{eqn:Etime}\n\\end{equation}\nWe generate an event time $t$ according to \\( q(\\alpha;t_0,t) \\) to\nperform event-driven Monte Carlo simulation. \n\nOnce the event occurs and the system is activated to $\\alpha_1$,\nit may be activated further to \n\\( \\alpha_2 \\), $\\alpha_3$ or higher branching point. \nWe accept the excitation to \\( \\alpha_2 \\) with the probability \n\\( \\exp[-E_2(\\alpha)/T] \\).\nIf it is accepted, the next trial to $\\alpha_3$ follows.\nBy repeating these procedures, we obtain\nthe probability \\( H(n) \\) that the system is activated from $\\alpha_1$\nup to $\\alpha_n$ as\n%\\begin{eqnarray}\n%H(n)&=& \\prod_{k=2}^n \\exp[-E_k(\\alpha)/T]\\nonumber\\\\\n%&&\\times\\left(1-\\exp[-E_{n+1}(\\alpha)/T]\\right),\\quad(n=1,2,\\ldots,L),\n%\\end{eqnarray}\n\\begin{equation}\nH(n) =\n \\left(1-\\exp[-E_{n+1}(\\alpha)/T]\\right) \\prod_{k=2}^n \\exp[-E_k(\\alpha)/T],\n\\label{eqn:eqforHL}\n\\end{equation}\nwhere \\( E_{L+1}(\\alpha) \\equiv \\infty \\). \n\nFinally, we adopt the simplest falling process from \\( \\alpha_n \\). \nThe system falls to one of all states under \\( \\alpha_n \\) with \nequal probability. In the limit that the number of branches in \neach layer is infinite, we can neglect the possibility that the system has \nvisited one state more than twice in our finite time simulation. \nTherefore, in this simulation, we create a new \nstate \\( \\beta \\) after each activation. Note that if the system \nis activated to $n$-th layer, \nwe set \\( E_k(\\beta)=E_k(\\alpha) \\) for \\( k \\ge n+1 \\), although \nfor \\( k\\le n \\), \\( E_k \\) is updated according to the density of state \n\\( \\rho_k(E) \\). \n\n\nWe note that the whole process drives the system to the equilibrium\ndistribution proportional to $\\exp[\\sum_n E_n/T]$.\n\n\\subsection{Assignment of magnetization}\nIn order to observe the magnetic response, we introduce magnetization to the MREM.\nIt is natural to suppose that\nthe nearer two states are located in the phase space\n the stronger the correlation of the two magnetizations is, and that \n the distance in the MREM can be measured in terms of barrier height.\nIn fact the barrier height is related with the overlap in the SK model\\cite{Nemoto}.\nTo incorporate this aspect, we assign the value of magnetization to state $\\alpha$, $M_\\alpha$, as\n%Now we consider to give states \\( \\alpha \\) and \\( \\beta \\) the \n%magnetization \\( M_{\\alpha} \\) and \\( M_{\\beta} \\). \n%We denote the barrier energy between \\( \\alpha \\) and \\( \\beta \\) as \n%\\(B_{\\alpha \\beta}\\). If \\(B_{\\alpha \\beta}\\) is small, it is natural to \n%consider that these two states are similar. Actually the relation between \n%\\( B_{\\alpha \\beta} \\) and the overlap \\( q_{\\alpha \\beta} \\) are \n%examined numerically for the SK model and confirmed that \n%the relation mentioned above holds\\cite{Nemoto}. \n%Furthermore, it is probable that the correlation between \\( M_{\\alpha} \\) and \n%\\( M_{\\beta} \\) is strong if these two state is similar. \n%To adopt this effect, we consider that the magnetization \\( M_{\\alpha} \\) \n%consists of \\( L \\) different contributions as \n\\begin{equation}\nM_{\\alpha} ={\\cal M}_0(\\alpha)+{\\cal M}_1(\\alpha_1)+\\cdots\n+{\\cal M}_{L-1}(\\alpha_{L-1}),\n\\end{equation}\nwhere \\( {\\cal M}_k(\\alpha_k) \\) is a contribution \nfrom the branch point $\\alpha_k$ given as a independent random variable with zero mean.\nThe correlation between $M_\\alpha$ and $M_\\beta$ with the lowest common branch point $\\alpha_k$\ncomes from the common contributions of ${\\cal M}_n, (n=k,k+1,\\cdots)$ to these magnetization.\nIt decreases monotonically as $k$ increases and the barrier becomes higher.\n\nIn this simulation, we choose a uniform distribution for ${\\cal M}$ of the $n$-th layer as\n\\begin{equation} \nD_n({\\cal M}) = \\left\\{ \n \\begin{array}{cl}\n\\frac{\\sqrt{L}}{2} &\\mbox{($|{\\cal M}| \\leq \\frac{1}{\\sqrt{L}} $)}, \\\\\n0 &\\mbox{($ |{\\cal M}| > \\frac{1}{\\sqrt{L}} $)}.\n\\end{array}\\right.\n\\end{equation}\nThe range of distribution is determined so that \nthe variance of $M_\\alpha$ is independent of $L$.\n\nThe Zeeman energy due to applying magnetic field $H(t)$ is attributed to the lowest layer and\n\\( E_1(\\alpha) \\) in eq.~(\\ref{eqn:omega}) is replaced\nwith \\( E_1(\\alpha)+H(t)M_{\\alpha} \\). \n\nTo summarize, we introduce the concrete procedure of our simulation.\n\\begin{itemize}\n\\item[(i)] Choose the initial state \\( \\alpha \\). Because we consider the \nsituation that the system is quenched from infinitely high temperature, we \ndetermine \\( E_k(\\alpha) \\) \\((k=1,\\ldots,L)\\) according to \\( \\rho_k(E) \\). \nWe also determine \\( {\\cal M}_k(\\alpha) \\) from \\( D_k({\\cal M}) \\). \n\\item[(ii)]Determine the time \\( t_{\\rm stay} \\) that the system stay at \n\\( \\alpha \\) from eq.~(\\ref{eqn:Etime}).\n\\item[(iii)]Determine how far the system is activated by eq.~(\\ref{eqn:eqforHL}). \n\\item[(iv)]Give the energy and magnetization of new state \\( \\beta \\). \nWhen the system is activated to \\( \\alpha_n \\), we set \n\\( E_k(\\beta)=E_k(\\alpha) \\) and \\( {\\cal M}(\\beta)={\\cal M}(\\alpha) \\) \nfor \\( k \\ge n+1 \\) and give \\( E_k(\\beta) \\) and \\( {\\cal M}(\\beta) \\)\nfrom \\( \\rho_k(E) \\) and \\( D({\\cal M}) \\) for \\( k \\le n \\). \n\\item[(v)] Set the time ahead by \\( t_{\\rm stay} \\) and return to (ii).\n\\end{itemize}\n\nWe show an example of simulation for \\( L=2 \\) with a weak ac-field. \nWe plot \\( {\\cal M}_k(t) \\) \\( (k=1,2) \\) in one trial in Fig.3(a), and \naveraged magnetization over \\( 2\\times 10^7 \\) trials in Fig.3(b). \nWe can see that in one trial, \\( {\\cal M}_k \\) keeps a constant during \nthe system stays at a state and changes discontinuously when activation \noccurs. We can see that the activation to the \\( 1 \\)-st layer and the one \nto the \\( 2 \\)-nd layer occurs at \\( t_2 \\) and \\( t_1 \\) respectively. \nFrom this figure, we can not find the effect of ac-field. But this effect \nemerges by taking average over numerous trials, as shown in Fig.~3(b). \nWe estimate \\( \\chi''(t) \\) directly from data of this kind. \n\n\\section{The Result of Simulations}\nIn the following we show the result of simulations performed on the MREM with\n$L=2$, $T_{\\rm c}(1)=0.6$ and $T_{\\rm c}(2)=1.0$.\nThe number of samples used for random average is typically $10^7$.\nThe amplitude and the period of applying ac-field are fixed to $0.1$ and $100\\tau_0$, respectively.\nHereafter this period is used as the unit of time. \n\n\\subsection{The case $-\\Delta T$}\nIn the first stage the system evolves in temperature $T=0.85$ for time interval $t_1$.\nTemperature is then reduced to $T-\\Delta T=0.5$ for $t_2$ (the second stage), \nand brought back to $T$ in the following third stage.\nNote that these temperatures are set so as to satisfy\n\\[ T-\\Delta T < T_{\\rm c}(1) < T < T_{\\rm c}(2). \\]\nIn Fig.~4, we plot imaginary part of ac-susceptibility, $\\chi''$,\n for two different intervals of the first stage, $t_1=20$ and $100$.\nIn inset we plot \\( t_1\\) and \\( t_3\\) part of data. \nWe can see that the memory effect takes place. \nWe also find in the data for $t_1=100$ the slight jump\nat the beginning of the second stage as seen in experiment\\cite{acTC1}.\n\nIn Fig.~5, we plot \\( \\chi_0'' \\) and \\( \\chi_1'' \\) evaluated \nfrom \\( {\\cal M}_0 \\) and \\( {\\cal M}_1 \\), respectively ($\\chi''=\\chi_0''+\\chi_1''$). \nWe can see that \\( \\chi_0'' \\) remains almost constant in the \nfirst and third stage because \\( T>T_{\\rm c}(1) \\) \nso that the first layer quickly relaxes to the equilibrium.\nThe long-time dependence of relaxation is \ndominated by the second layer there. In contrast the first layer is\ndominant in the second stage because $T-\\Delta T$ is too low for the second layer\nto be activated and \\( \\chi_1'' \\) is much smaller than \n\\( \\chi_0'' \\).\n From these results, we notice that the dominant layer changes as\ntemperature varies. % as proposed by Bouchaud and Dean\\cite{Bouchaud}.\n\n\nwe can also see that the slight jump of \\( \\chi'' \\) mentioned above \nis brought from \\( \\chi_0''\\).\nThis sudden increase of \n\\( \\chi_0'' \\) is due to the quenching of the first layer across its transition temperature \\( T_{\\rm c}(1) \\). \n\nIn Fig.~6, \\( \\chi_s'' \\) in the second stage is magnified for $t_1=20$ and $t_1=100$. \nThere are few differences on both \\( \\chi_0''\\) and \\( \\chi_1'' \\).\nThis means that the length of the first stage, \\( t_1 \\),\ndoes not affect the relaxation in the second stage.\nIn this sense, we may say that the system has\nthe {\\em chaos} effect.\nOn the other hand, \\( t_1 \\) reflects on relaxation \nin the third stage and the memory effect appears.\nThus the MREM has these two apparently conflicting \nfeatures simultaneously. % as mentioned in \\( \\S \\)1. \n\nIn order to investigate which states contribute to $\\chi''$ in more detail,\nwe examine the energy distribution\n \\( P_n(E_n,t) \\) which is defined as the probability density\nthat the system is found at time $t$ in one of the states whose energy of the $n$-th layer is $E_n$.\nBy the definition, we have $P_n(E_n,t=0)=\\rho_n(E_n)$ since the initial state is chosen randomly.\n\nIn Fig.~7, we show the time dependence of \\( P_n(E_n,t) \\) in the first stage.\nWe can see that the distribution consists of two exponential functions which are connected to each other \nat some point, $E^*$.\nThe value of $E^*$ roughly represents the energy level up to which the system can be activated during time\ninterval $t$, and is estimated as\n\\begin{equation}\nE^*\\approx T\\log[t/\\tau_0].\n\\end{equation}\nFor \\( E_n \\le E^* \\) the distribution is aged or equilibrated, so that\nthe exponent $\\alpha_1(n)$ is given as\n\\begin{equation}\n\\alpha_1(n)=\\frac1T-\\frac{1}{T_{\\rm c}(n)},\n\\end{equation}\nwhile the other part $E_n \\ge E^*$ leaves untouched and\nthe exponent $\\alpha_2(n)$ is equal to that of $\\rho_n(E)$, i.e.,\n\\begin{equation}\n\\alpha_2(n)=-\\frac{1}{T_{\\rm c}(n)}.\n\\end{equation}\n\nFor the first layer we have $\\alpha_1(1) < 0$ since \\( T > T_{\\rm c}(1) \\), and\nthe distribution quickly converges to the equilibrium one (Fig.7, left).\nNote that the discrepancy for $E_1>E^*$ is negligible in magnitude.\nFor the second layer where $\\alpha_1(2)>0$, on the other hand, \\( P_2(E_2,t) \\) \nhas a peak at $E^*$ moving right with time (Fig.~7, right).\nThis is why\n\\( \\chi_1'' \\) \ncontinues to change with time\nwhile\n \\( \\chi_0'' \\) remains almost constant\n in the first stage.\n\nExperiments on the Zero-Field-Cooled (ZFC) magnetization in \nspin glasses\\cite{ZFCexperiment1,ZFCexperiment2} showed that\nthe distribution of the relaxation time \n\\( \\tau \\), which has a peak at $\\tau_{\\rm max}$,\n depends on the waiting time \\( t_{\\rm w} \\) \nand \\( \\tau_{\\rm max} \\) appears near \\( t_{\\rm w} \\), which \njust corresponds to the shift of peak shown in the right of Fig.~7.\n\nIn Fig.~8, we show the time dependence of \\(P_n(E_n,t)\\) in the second stage. \nWe can see that a peak appears and moves to right for the first layer \n(Fig.~8, left).\nAs for \\( P_2(E_2,t)\\), the global aspects such as the position of\npeak formed in the first stage do not change\nalthough the distribution of the lower energy\ndecreases gradually. This brings the memory effect to the system. \n\nTo see how $t_1$ affects the energy distribution, \nwe plot in Fig.~9 $P_n(E_n,t)$ for \n\\( t_1=20 \\) and \\( t_1=10^2 \\) with the same $t$ in the second stage.\nAs for the first layer, the both distributions are \nalmost the same since the layer is well equilibrated in the first stage.\nIn the second stage, this layer is dominant and \nthe behaviors of \\( \\chi'' \\) are almost the same in the two cases.\nOn the other hand, the difference of the peak position quenched since the end of first stage,\nmakes the shapes of\n$P_2(E_2,t)$ very different from each other, and this causes, in turn, \nthe difference in the behavior of $\\chi''$ in the third stage.\n\n\\subsection{The case $+\\Delta T$}\nIn Fig.~10, the $\\chi''$ for the case \\( +\\Delta T \\) is shown. The heating\nis done twice in this case.\nWe set the temperatures as \\( T=0.5\\), \\( T+\\Delta T=0.85 \\), which satisfy\nthe relation $T<T_{\\rm c}(1)<T+\\Delta T<T_{\\rm c}(2)$.\nWe can see from this figure that the rejuvenation occurs.\n\nWe show the behavior of \\( \\chi_0'' \\) and \\( \\chi_1'' \\) in Fig.~11.\nIn this case \nwe can see that the $\\chi_1''$ dominates the slow relaxation in the second and fourth stages\nas expected.\nThe rejuvenation occurs only in $\\chi_0''$ but not in $\\chi_1''$.\n\nIn Fig.~12, we compare the behavior of \\( \\chi'' \\) in the first, third and \nfifth stages. Although the value in the first stage is slightly larger than that in the third,\nthose in the third and fifth stages are collapsed to a single curve,\nwhich means the perfect re-initialization.\n\nWe again examine \\( P_n(E_n,t) \\) in this case.\n In Fig.~13, the time evolution of \\( P_1(E_1,t) \\) in the second stage is shown.\nThe peak created in the first stage is rapidly \ndestroyed and the information on the relaxation \n in the first stage is completely forgotten.\nIn fact we find that the peak starts to disappear at \\( t\\simeq t_1+1 \\)\n, i.e., just after the second stage begins. \n\n\\section{Conclusions and Discussions}\nWe have shown that the present MREM can reproduce the prominent behaviors \nfound in spin glasses such as the memory effect and the rejuvenation although\nthe number of layers is only two.\nThis indicates that the alternation of the activated layer is important to\nexplain these features in this model.\nWe expect that the phenomena can be observed at any $T<T_{\\rm c}$\nin the MREM with large $L$.\n\nNow let us discuss what happens when $L\\gg 1$ along the scenario proposed \nby Bouchaud and Dean\\cite{Bouchaud}. \nFor a given temperature $T<T_{\\rm c}(L)=T_{\\rm c}$, there exists the $n$-th layer\nsuch that $T_{\\rm c}(n-1)<T<T_{\\rm c}(n)$.\nThe essential point is the fact that layers below $n$ are quickly equilibrated and\nthey forget what happened before the temperature is set, while those above $n$ are\nalmost quenched and they behave as if the time evolution stops.\nIn this sense the $n$-th layer is the {\\em activated} one and dominates the relaxation\nof the system.\nThis mechanism certainly explains the memory effect and the rejuvenation as shown in\nthe present work.\nIn this context, the {\\em chaos} effect means that the layer which is to be activated when\ntemperature decreases is not capable to remember the previous situation and it relaxes from\n{\\em tabulae rasae}.\nTherefore it is not necessary to introduce a {\\em chaotic} reorganization of hierarchy\nfor description of the phenomena.\n\n%It is challenging to find some hierarchical structure, if really exists in spin glasses, to which\n%the real spin space is mapped, although this mapping seems to be far from complete even for the SK model.\n\nFinally, let us compare the droplet model with the MREM and try to \nfuse these models. In the droplet model, \nsystem {\\em ages} by growth of droplets. On the other hand, aging means \nthe shift of the peak of \\( P(E) \\) in the {\\em frozen} layers \n(the layers which satisfy \\( T < T_{\\rm c}(n) \\)) in the MREM. \nIn both model, aging means to seek more stable states with time and \nthese processes make the system stiffer and the response to \nan external field weaker. This is the reason why \nthe ac-susceptibility decreases monotonously with age. \nMoreover, as discussed Jonason {\\it et al}\\cite{Jonason}, it may be possible \nto map the MREM into the droplet model as following. We now consider \nwhat will happen when we cool real spin glass systems across the \n\\( T_{\\rm c} \\) (which corresponds to \\(T_{\\rm c}(L)\\) in the MREM). Above \n\\( T_{\\rm c} \\), all the spins flip rather free. When the system is cooled \njust below \\( T_{\\rm c} \\), we can consider that larger droplets are blocked \nat first and the spins in these droplets begin to flip collectively. In this \nstage, the spins in smaller droplet are still free relatively. \nIn the MREM, the larger droplets correspond to the upper {\\em frozen} layers \nand the smaller droplets correspond to the lower {\\em unfrozen} layers. \nAs the temperature is lowered, smaller droplets begin to be blocked and flip \ncollectively. But the larger droplets which blocked earlier are almost frozen \nand they can hardly flip at this temperature. \nWe expect that these successive \nprocesses bring the memory effect, rejuvenation and {\\em chaos} effect \nas discussed in this manuscript. Furthermore, the idea of \n{\\em droplets within droplets}\\cite{DPinDP} will naturally lead to the hierarchical \norganization of droplets. 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O. Andersson, J. Mattsson and P. Svedlindh: Phys. Rev. B \n{\\bf 46}(1992)8297.\n\n\\bibitem{acTC1}E. Vincent, J. P. Bouchaud, J. Hammann and F. Lefloch: \nPhil. Mag. B {\\bf 71}(1995)489.\n\n\\bibitem{acTC2}F. Lefloch, J. Hammann, M. Ocio and E. Vincent: \nEurophys. Lett {\\bf 18}(1992)647.\n\n\\bibitem{acTC3}J. O. Andersson, J. Mattsson and P. Nordblad: \nPhys. Rev. B {\\bf 48}(1993)13977.\n\n\\bibitem{droplet1}D. S. Fisher and D. A. Huse: Phys. Rev. B {\\bf 38}(1988)373.\n\n\\bibitem{droplet2}G. J. M. Koper and H. J. Hilhorst: J. Phys. (Paris) \n{\\bf 49}(1988)429.\n\n\\bibitem{hierarchical1}M. Sasaki and K. Nemoto: J. Phys. Soc. Jpn \n{\\bf 68}(1999)1148. \n\n\\bibitem{hierarchical2}H. Yoshino: J. Phys. A {\\bf 30}(1997)1143.\n\n\\bibitem{hierarchical3}P. Sibani and K. H. Hoffmann: Phys. Rev. Lett \n{\\bf 63}(1989)2853.\n\n\\bibitem{hierarchical4}C. Shulze, K. H. Hoffmann and P. Sibani: \nEurophys. Lett {\\bf 15}(1991)361.\n\n\\bibitem{Bouchaud}J. P. Bouchaud and D. S. Dean: \nJ. Phys. I France {\\bf 5}(1995)265.\n\n\\bibitem{MREM2}B. Derrida: J. Phys. Lett. France {\\bf 46}(1985)401.\n\n\\bibitem{MREM3}B. Derrida and E. Gardner: J. Phys. C {\\bf 19}(1986)2253.\n\n\\bibitem{Jonason}K. Jonason, E. Vincent, J. Hammann, J. P. Bouchaud and \nP. Nordblad: Phys. Rev. Lett. {\\bf 81}(1998)3243.\n\n\\bibitem{Nemoto}K. Nemoto: J. Phys. A {\\bf 21}(1988)L287.\n\n\\bibitem{DPinDP}J. Villain: J. Phys. France {\\bf 46}(1985)1843.\n\n\\end{thebibliography}\n\n\\newpage\n\\noindent\n{\\bf \\large FIGURE CAPTIONS}\n\n%\\vspace*{3mm}\\noindent\n%Fig.~1 Schematic structure of free energy.\n%\n\\vspace*{3mm}\\noindent\nFig.~1 Structure of the Simple-layer Random Energy Model.\n\n\\vspace*{3mm}\\noindent\nFig.~2 Structure of the Multi-layer Random Energy Model with $L=2$.\n\n\\vspace*{3mm}\\noindent\nFig.~3 (a) ${\\cal M}_1(t)$ (dashed line) and ${\\cal M}_2(t)$ (solid line) \n observed in one trial and (b) averaged \n ${\\cal M}_k(t)$ over $2\\times 10^7$ trials. \n\n\\vspace*{3mm}\\noindent\nFig.~4 $\\chi''$ for the case \\( -\\Delta T \\) with\n \\( t_1=20 \\) (dashed line) and \\( t_1=100 \\) (solid line). \n In the inset, the data in the first and third stage \n are plotted. \n\n\\vspace*{3mm}\\noindent\nFig.~5 \\( \\chi_0'' \\) (solid line) and \\( \\chi_1'' \\) (dashed line) for the \n case $-\\Delta T$.\n\n\\vspace*{3mm}\\noindent\nFig.~6 $\\chi''$ in the second stage. \\( \\chi_0'' \\) with \\( t_1=20 \\), \n \\( \\chi_0'' \\) with \\( t_1=100 \\), \\( \\chi_1'' \\) with \\( t_1=20 \\)\n and \\( \\chi_1'' \\) with \\( t_1=100 \\) (from top to bottom).\n\n\\vspace*{3mm}\\noindent\nFig.~7 \\( P_n(E_n,t) \\) in the first stage at\n \\( t=1,10^{0.5},10,\\ldots,10^2 \\) (from left to right).\n\n\\vspace*{3mm}\\noindent\nFig.~8 \\( P_n(E_n,t) \\) in the second stage at\n \\( t=0,10^{-1.5},10^{-1},\\ldots,10^2 \\) (from left to right).\n\n\\vspace*{3mm}\\noindent\nFig.~9 \\( P_n(E_n,t) \\) for \\( t_1=20 \\) (left) and \n \\( t_1=100 \\) (right) in the second stage at \\( t=t_1+10^{1.5} \\).\n\n\\vspace*{3mm}\\noindent\nFig.~10 $\\chi''$ for the case \\( +\\Delta T \\). In the inset,\n the tree parts of perturbed data and unperturbed data are shown.\n\n\\vspace*{3mm}\\noindent\nFig.~11 \\( \\chi_0'' \\) (upper) and \\( \\chi_1'' \\) (lower) for the case \\( +\n \\Delta T \\).\n\n\\vspace*{3mm}\\noindent\nFig.~12 \\( \\chi'' \\) in the first, third and fifth stages.\n\n\\vspace*{3mm}\\noindent\nFig.~13 \\( P_1(E_1,t) \\) in the second stage for the \n case \\( +\\Delta T \\) at time \\( t=t_1+\\Delta t \\) with $\\Delta t = \n 10^{-1.5},10^{-1},\\ldots,10^{2}$.\n\\newpage\n\\begin{center}\n\\epsfile{file=simpleREM.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.1}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=multiREM.eps,width=16cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.2}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=OSmag1.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.3(a)}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=OSmag2.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.3(b)}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=hchfig1.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.4}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=hchfig2.ps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.5}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=hchfig3.ps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.6}\n\\end{center}\n\\newpage\n\\begin{center}\n\\hspace*{-1cm}\n\\epsfile{file=test1.ps,width=17.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.7}\n\\end{center}\n\\newpage\n\\begin{center}\n\\hspace*{-1cm}\n\\epsfile{file=test2.ps,width=17.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.8}\n\\end{center}\n\\newpage\n\\begin{center}\n\\hspace*{-1cm}\n\\epsfile{file=test3.ps,width=17.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.9}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=chcfig1.eps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.10}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=chcfig2.ps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.11}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=chcfig3.ps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.12}\n\\end{center}\n\\newpage\n\\begin{center}\n\\epsfile{file=chcfig4.ps,width=16.5cm}\n\\end{center}\n\\vspace{2cm}\n\\begin{center}\n{\\LARGE Fig.13}\n\\end{center}\n\n\\end{document}\n"
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[
{
"name": "cond-mat0002024.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{ZFCexperiment1}P. Svedlindh, P. Granberg, P. Nordblad, L. Lundgren \nand H. S. Chen: Phys. Rev B {\\bf 35}(1987)268.\n\n\\bibitem{ZFCexperiment2}L. Lundgren, P. Svedlindh and O. Beckman: \nPhys. Rev B {\\bf 26}(1982)3990.\n\n\\bibitem{TRMexperiment}Ph. Refreggier, E. Vincent, J. Hammann and M. Ocio: \nJ. Phys. (Paris) {\\bf 48}(1987)1533.\n\n\\bibitem{ZFCTS}P. Granberg, L. Sandlund, P. Nordblad, P. Svedlindh and \nL. Lundgren: Phys. Rev. B {\\bf 38}(1988)7097.\n\n\\bibitem{ZFCTC}P. Granberg, L. Lundgren and P. Nordblad: J. Magn. Magn. Mater \n{\\bf 92}(1990)228.\n\n\\bibitem{MCS}J. O. Andersson, J. Mattsson and P. Svedlindh: Phys. Rev. B \n{\\bf 46}(1992)8297.\n\n\\bibitem{acTC1}E. Vincent, J. P. Bouchaud, J. Hammann and F. Lefloch: \nPhil. Mag. B {\\bf 71}(1995)489.\n\n\\bibitem{acTC2}F. Lefloch, J. Hammann, M. Ocio and E. Vincent: \nEurophys. Lett {\\bf 18}(1992)647.\n\n\\bibitem{acTC3}J. O. Andersson, J. Mattsson and P. Nordblad: \nPhys. Rev. B {\\bf 48}(1993)13977.\n\n\\bibitem{droplet1}D. S. Fisher and D. A. Huse: Phys. Rev. B {\\bf 38}(1988)373.\n\n\\bibitem{droplet2}G. J. M. Koper and H. J. Hilhorst: J. Phys. (Paris) \n{\\bf 49}(1988)429.\n\n\\bibitem{hierarchical1}M. Sasaki and K. Nemoto: J. Phys. Soc. Jpn \n{\\bf 68}(1999)1148. \n\n\\bibitem{hierarchical2}H. Yoshino: J. Phys. A {\\bf 30}(1997)1143.\n\n\\bibitem{hierarchical3}P. Sibani and K. H. Hoffmann: Phys. Rev. Lett \n{\\bf 63}(1989)2853.\n\n\\bibitem{hierarchical4}C. Shulze, K. H. Hoffmann and P. Sibani: \nEurophys. Lett {\\bf 15}(1991)361.\n\n\\bibitem{Bouchaud}J. P. Bouchaud and D. S. Dean: \nJ. Phys. I France {\\bf 5}(1995)265.\n\n\\bibitem{MREM2}B. Derrida: J. Phys. Lett. France {\\bf 46}(1985)401.\n\n\\bibitem{MREM3}B. Derrida and E. Gardner: J. Phys. C {\\bf 19}(1986)2253.\n\n\\bibitem{Jonason}K. Jonason, E. Vincent, J. Hammann, J. P. Bouchaud and \nP. Nordblad: Phys. Rev. Lett. {\\bf 81}(1998)3243.\n\n\\bibitem{Nemoto}K. Nemoto: J. Phys. A {\\bf 21}(1988)L287.\n\n\\bibitem{DPinDP}J. Villain: J. Phys. France {\\bf 46}(1985)1843.\n\n\\end{thebibliography}"
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cond-mat0002025
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Self-Duality in Superconductor-Insulator Quantum Phase Transitions
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"author": "Adriaan M. J. Schakel"
}
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It is argued that close to a Coulomb interacting quantum critical point, the interaction between two vortices in a disordered superconducting thin film separated by a distance $r$ changes from logarithmic in the mean-field region to $1/r$ in the region dominated by quantum critical fluctuations. This gives support to the charge-vortex duality picture of the observed reflection symmetry in the current-voltage characteristics on both sides of the transition.
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"name": "cond-mat0002025.tex",
"string": "\\documentstyle[aps,times,epsf,multicol]{revtex}\n\n\\renewcommand{\\narrowtext}\n{\\begin{multicols}{2} \\global\\columnwidth20.5pc}\n\n\\begin{document}\n\n\\draft\n\n\\title{Self-Duality in Superconductor-Insulator Quantum Phase\nTransitions}\n%\n\\author{Adriaan M. J. Schakel} \n%\n\\address{Low Temperature Laboratory, Helsinki University of Technology,\nP.O. Box 2200, FIN-02015 HUT, Finland \\\\ and \\\\ Institut f\\\"ur\nTheoretische Physik, Freie Universit\\\"at Berlin, Arnimallee 14, 14195\nBerlin, Germany }\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\nIt is argued that close to a Coulomb interacting quantum critical point,\nthe interaction between two vortices in a disordered superconducting\nthin film separated by a distance $r$ changes from logarithmic in the\nmean-field region to $1/r$ in the region dominated by quantum critical\nfluctuations. This gives support to the charge-vortex duality picture\nof the observed reflection symmetry in the current-voltage\ncharacteristics on both sides of the transition.\n\\end{abstract}\n\n\\pacs{74.40.+k, 71.30.+h, 64.60.Fr }\n\n\\narrowtext\n\nOne of the most intriguing results found in experiments on quantum phase\ntransitions in superconducting films, 2-dimensional Josephson-junction\narrays \\cite{Delft}, quantum Hall systems \\cite{IVQH}, and 2-dimensional\nelectron systems \\cite{KSSMF} is the striking similarity in the\ncurrent-voltage ($I$-$V$) characteristics on both sides of the\ntransition. By interchanging the $I$ and $V$ axes in one phase, an\n$I$-$V$ characteristic of that phase at a given value of the applied\nmagnetic field (in superconducting films, 2-dimensional\nJosephson-junction arrays, and quantum Hall systems) or charge-carrier\ndensity (in 2-dimensional electron systems) can be mapped onto an\n$I$-$V$ characteristic of the other phase at a different value of the\nmagnetic field or charge-carrier density. This reflection symmetry\nhints at a deep connection between the conduction mechanisms in the two\nphases that can be understood by invoking a duality transformation\n\\cite{MPAFisher,WeZe}. Whereas the conducting phase is most succinctly\ndescribed in terms of charge carriers of the system, the insulating\nphase is best formulated in terms of vortices, which behave as quantum\npoint particles in these systems. The duality transformation links the\ntwo surprisingly similar looking descriptions.\n\nThere appears to be, however, one disturbing difference. Whereas \ncharges interact via the usual 3-dimensional $1/r$ Coulomb potential,\nvortices are believed to interact via a logarithmic potential--at least\nfor distances smaller than the transverse magnetic penetration depth\n$\\lambda_\\perp$, which is typically larger than the sample size\n\\cite{BMO}. This is disturbing because the difference should spoil the\nexperimentally observed reflection symmetry.\n\nIt is this fundamental problem we wish to address in this Letter. It\nwill be shown that close to a Coulomb interacting quantum critical point\n(CQCP), the interaction between vortices in disordered superconducting\nfilms changes from logarithmic in the mean-field region to $1/r$ in the\nregion dominated by quantum critical fluctuations. This conclusion is\nan exact result, depending only on the presence of a CQCP.\n\nA common characteristic of the systems mentioned is, apart from\nimpurities, the presence of charge carriers confined to move in a\n2-dimensional plane. As the $1/r$ Coulomb repulsion between \ncharges is genuine 3-dimensional, we assume this interaction not to be\naffected by what happens in the film, which constitutes a mere slice of\n3-dimensional space. In contrast to this, the interaction between\nvortices is susceptible to the presence of a CQCP. This is because the\nvortex interaction is a result of currents around the vortex cores which\nare confined to the plane.\n\nAs starting point, we take the observation (for a review, see Ref.\\\n\\cite{SGCS}) that close to a CQCP, the electric field $E$ scales with\nthe correlation length $\\xi$ as $E \\sim \\xi_t^{-1} \\xi^{-1} \\sim\n\\xi^{-(z+1)}$. Here, $\\xi_t$ denotes the correlation time, indicating\nthe time period over which the system fluctuates coherently, and $z$ is\nthe dynamic exponent. Thus conductivity measurements \\cite{YaKa,KSSMF}\nclose to a CQCP collapse onto a single curve when plotted as function of\nthe dimensionless combination $\\delta^{\\nu (z+1)}/E$, where\n$\\delta=(K-K_{\\rm c})/K_{\\rm c}$ measures the distance from the critical\npoint $K_{\\rm c}$, and $\\nu$ is the correlation length exponent, $\\xi\n\\sim \\delta^{- \\nu}$. (For a field-controlled transition, $K$ stands\nfor the applied magnetic field, while for a density-controlled\ntransition it stands for the charge-carrier density.) The scaling of\nthe electric field with the correlation length expresses the more\nfundamental result that the anomalous scaling dimension $d_{\\bf A}$ of\nthe magnetic vector potential ${\\bf A}$ is unity, $d_{\\bf A} = 1$.\n\nIn addition, because the magnetic vector potential always appears in the\ngauge-invariant combination $\\nabla - q {\\bf A}$, the anomalous scaling\ndimension of the electric charge $q$ of the charge carriers times the\nvector potential is unity too, $d_{q {\\bf A}} = 1$. Writing the\nanomalous scaling dimension of the vector potential as a sum $d_{\\bf A}\n= d^0_{\\bf A} + \\case{1}{2} \\eta_{\\bf A}$ of its canonical scaling\ndimension $d^0_{\\bf A} = \\case{1}{2} (d + z -2)$, obtained by simple\npower counting, and (half) the critical exponent $\\eta_{\\bf A}$,\ndescribing how the correlation function decays at the critical point, we\nconclude that $d_q = d^0_q - \\frac{1}{2}\\eta_{\\bf A}$. Here, $d^0_q = 1\n- d^0_{\\bf A}$ stands for the canonical scaling dimension of the\nelectric charge. Now, for a $1/r$ Coulomb potential, the charge scales\nas $q^2 \\sim \\xi^{1-z}$ independent of the number $d$ of space\ndimensions \\cite{FGG}. Combined with the previous result, this fixes\nthe value of the critical point decay exponent $\\eta_{\\bf A}$ in terms\nof the number of space dimensions and the dynamic exponent:\n\\begin{equation} \\label{eta}\n\\eta_{\\bf A}=5 -d - 2z.\n\\end{equation} \n\nIn Ref.\\ \\cite{FGG} it was further argued that in the presence of\ndisorder, the electric charge is finite at a CQCP, so that $z=1$. This\nprediction was first confirmed for disordered superconducting films\n\\cite{HPsu1}, and subsequently also for 2-dimensional Josephson-junction\narrays \\cite{Delft}, quantum Hall systems \\cite{WET}, and 2-dimensional\nelectron systems \\cite{KSSMF}. With $z=1$, the value of the\ncritical-point decay exponent becomes $\\eta_{\\bf A}=1$ in $d=2$. As we\nwill now demonstrate, this leads to a qualitative change in the\ninteraction potential between two vortices from logarithmic in the\nmean-field region, where $\\eta_{\\bf A} = 0$, to $1/r$ in the vicinity of\nthe CQCP, where $\\eta_{\\bf A}=1$.\n\nTo set the stage, let us first consider a bulk superconductor with two\nstatic vortices directed along the $x_3$-axis and separated a distance\n$r$. For our purposes, the effective phase-only \\cite{NgSu} Hamiltonian\n${\\cal H}_{\\rm eff} = (\\rho_{\\rm s}/2 m^2) (\\nabla \\varphi - q {\\bf\nA})^2$ in terms of the phase $\\varphi$ of the superconducting order\nparameter---the so-called Anderson-Bogoliubov mode---suffices (for\nreviews, see Ref.\\ \\cite{reviews}). Here, $\\rho_{\\rm s}$ is the\nsuperconducting mass density, which scales as $\\rho_{\\rm s} \\sim\n\\xi^{2-(d+z)}$ \\cite{FF}, and $m$ is the mass of the charge carriers.\nThe interaction potential can be extracted from the magnetic part of the\neffective action $S_{\\rm mag}$. Written as a functional integral over\nthe the magnetic vector potential, it is given in the Coulomb gauge\n$\\nabla \\cdot {\\bf A}=0$ by\n\\begin{mathletters}\n\\begin{equation} \\label{mag}\n{\\rm e}^{i S_{\\rm mag}} = \\int \\mbox{D} {\\bf A} \\, {\\rm e}^{i \\int\n\\mbox{d} t \\, \\mbox{d}^3 x [-\\frac{1}{2}(\\nabla \\times {\\bf A} - {\\bf\nB}^{\\rm P})^2 - \\frac{1}{2} \\lambda^{-2}{\\bf A}^2 ]},\n\\end{equation} \nwith $\\lambda$ the magnetic penetration depth, which is related to\n$\\rho_{\\rm s}$ via $\\lambda^{-2} = q^2\\rho_{\\rm s}/m^2$. The mass\nterm is generated through the Anderson-Higgs mechanism by integrating\nout the phase mode $\\varphi$. The so-called plastic field ${\\bf B}^{\\rm\nP}$ \\cite{GFCM}\n\\begin{equation} \\label{BP}\nB_i^{\\rm P} = - \\Phi_0 \\sum_{\\alpha} \\int_{C_\\alpha} \\mbox{d} x_i^\\alpha\n\\, \\delta({\\bf x} - {\\bf x}^\\alpha),\n\\end{equation} \n\\end{mathletters}\nwith $\\Phi_0 = 2 \\pi/q$ the magnetic flux quantum in units where the\nspeed of light and Planck's constant $\\hbar$ is set to unity, describes\nthe two vortices located along the lines $C_\\alpha$ ($\\alpha=1,2$).\n\nNote that since the anomalous scaling dimension of the magnetic vector\npotential is unity, the dimension of the Maxwell term is 4, implying\nthat in $d=2$ it is an irrelevant operator in the renormalization-group\nsense. This term is, however, important when considering the\ninteraction between vortices.\n\nTo facilitate the calculation in the case of a superconducting film\nbelow, we linearize the first term in Eq.\\ (\\ref{mag}) by introducing an\nauxiliary field $\\tilde{\\bf h}$ via a Hubbard-Stratonovich\ntransformation to obtain the combination $i (\\nabla \\times {\\bf A} -\n{\\bf B}^{\\rm P}) \\cdot \\tilde{\\bf h} - \\case{1}{2} \\tilde{\\bf h}^2$.\nAfter integrating out the magnetic vector potential, we arrive at a form\nappropriate for a dual description in terms of magnetic vortices rather\nthan electric charges \\cite{KKS}\n\\begin{equation} \\label{3d}\n{\\rm e}^{i S_{\\rm mag}} = \\int \\mbox{D} \\tilde{\\bf h} \\, {\\rm e}^{i \\int\n\\mbox{d} t \\, \\mbox{d}^3 x [-\\case{1}{2} \\lambda^2 (\\nabla \\times\n\\tilde{\\bf h})^2 - \\case{1}{2}\\tilde{\\bf h}^2 - i \\tilde{\\bf h} \\cdot\n{\\bf B}^{\\rm P} ] }.\n\\end{equation} \nPhysically, $\\tilde{\\bf h}$ represents ($i$ times) the fluctuating local\ninduction; it satisfies the condition $\\nabla \\cdot \\tilde{\\bf h}=0$.\nThe vortices couple with a coupling constant $g=\\Phi_0/\\lambda$\nindependent of the electric charge to $\\tilde{\\bf h}$. Observe the\nclose similarity between the original (\\ref{mag}) and the dual form\n(\\ref{3d}). This becomes even more so when an external electric current\n${\\bf j}^{\\rm P}$ is coupled to the ${\\bf A}$ field by including a term\n$- {\\bf A} \\cdot {\\bf j}^{\\rm P}$ in Eq.\\ (\\ref{mag}), and ${\\bf\nB}^{\\rm P}$ describing the vortices is set to zero there. \n\nIntegrating out the local induction, one obtains the well-known\nBiot-Savart law for the interaction potential $S_{\\rm mag} = - \\int\n\\mbox{d} t V$ between two static vortices in a bulk superconductor\n\\cite{deGennes},\n\\begin{eqnarray} \\label{V3d}\nV(r) &=& \\frac{1}{2\\lambda^2} \\int \\mbox{d}^3 x \\, \\mbox{d}^3 y B_i^{\\rm\nP}({\\bf x}) G({\\bf x} - {\\bf y}) B_i^{\\rm P}({\\bf y}) \\nonumber \\\\ &=&\n\\frac{g^2}{4 \\pi} \\int_{C_1} \\int_{C_2} \\mbox{d} {\\bf l}^1 \\cdot\n\\mbox{d} {\\bf l}^2 \\; \\frac{{\\rm e}^{-R/\\lambda}}{R} \\nonumber \\\\ &=& -\n\\frac{g^2}{2 \\pi} L \\left[\\ln(r/2\\lambda) + \\gamma \\right] + {\\cal\nO}(r/\\lambda)^2\n\\end{eqnarray} \nwhere we ignored the self-interaction. In Eq.\\ (\\ref{V3d}), $G({\\bf\nx})$ is the correlation function whose Fourier transform reads $G({\\bf\nk})=1/({\\bf k}^2 + \\lambda^{-2} )$, $R$ denotes the distance between the\ndifferential lengths $\\mbox{d} {\\bf l}^1$ and $\\mbox{d} {\\bf l}^2$, $L$\nis the length of each of the two vortices, and $\\gamma$ is Euler's\nconstant. For distances smaller than the magnetic penetration depth,\nwhich is the length scale for variations in the current and the magnetic\nfield, the interaction is logarithmic as in a superfluid. If the system\nsize is smaller than $\\lambda$, it will replace $\\lambda$ as infra-red\ncutoff in the logarithm, and there will be no reference to the electric\ncharge anymore.\n\nTo describe magnetic vortices in a film of thickness $w$ \\cite{Pearl},\nthe bulk result (\\ref{3d}) has to be adjusted in two ways\nto account for the fact that both the vortices and the screening\ncurrents, which produce the second term in (\\ref{3d}), are confined to\nthe plane. This is achieved by including a Dirac delta function $w\n\\delta(x_3)$ in the second and third term. Instead of Eq.\\ (\\ref{3d}),\none then arrives at the interaction potential \\cite{Pearl,deGennes}\n\\begin{mathletters}\n\\begin{eqnarray} \\label{V2d}\nV_\\perp(r) &=& \\frac{1}{2 \\lambda_\\perp} \\int \\mbox{d}^2 x_\\perp\n\\mbox{d}^2 y_\\perp B_\\perp^{\\rm P}({\\bf x}_\\perp) G_\\perp({\\bf x}_\\perp\n- {\\bf y}_\\perp) B_\\perp^{\\rm P}({\\bf y}_\\perp) \\nonumber \\\\ &=&\n-\\frac{g_\\perp^2}{2 \\pi} \\left[ \\ln(r/4 \\lambda_\\perp) + \\gamma\\right] +\n{\\cal O}(r/\\lambda_\\perp)^2,\n\\end{eqnarray} \nwhere $B_\\perp^{\\rm P} = - \\Phi_0 \\sum_{\\alpha} \\delta({\\bf x}_\\perp -\n{\\bf x}_\\perp^\\alpha)$ describes the vortices in the film with\ncoordinates ${\\bf x}_\\perp$, $\\lambda_\\perp = \\lambda^2/w$ is the\ntransverse magnetic penetration depth, $g_\\perp^2 =\n\\Phi_0^2/\\lambda_\\perp$ the coupling constant squared, and\n\\begin{eqnarray} \nG_\\perp({\\bf x}_\\perp) &=& \\int \\mbox{d} x_3 \\, G_\\perp({\\bf x}_\\perp,\nx_3) \\nonumber \\\\ &=& \\int \\frac{\\mbox{d}^2 k_\\perp}{(2 \\pi)^2} \\, {\\rm\ne}^{- i {\\bf k}_\\perp \\cdot {\\bf x}_\\perp} G_\\perp({\\bf k}_\\perp,0),\n\\end{eqnarray} \n\\end{mathletters}\nwith $G_\\perp({\\bf k}_\\perp,0) = 2/ k_\\perp (2 k_\\perp\n+\\lambda_\\perp^{-1})$. For small distances, the interaction is seen to\nbe identical to that in a bulk superconductor \\cite{Pearl}, and also to\nthat in a superfluid film. As in the bulk, the vortex coupling constant\n$g_\\perp$ in the film is independent of the electric charge, $g_\\perp^2\n= \\Phi_0^2/\\lambda_\\perp = (2 \\pi)^2 \\rho_{\\rm s} w /m^2$, with\n$\\rho_{\\rm s}$ the bulk superconducting mass density.\n\nThe above results are valid in the mean-field region, where $\\eta_{\\bf\nA}=0$. In the critical region governed by a CQCP, the value of this\nexponent is unity, and the correlation function becomes\n\\begin{equation} \\label{Gr} \nG_\\perp({\\bf k}_\\perp,0) = \\frac{2}{k_\\perp } \\frac{Z_{\\bf\nA}}{ 2 k_\\perp +\\lambda_\\perp^{-1}},\n\\end{equation} \nwith $Z_{\\bf A} \\sim k_\\perp^{\\eta_{\\bf A}}$ the field renormalization\nfactor. Because the magnetic vector potential and the local induction\nrenormalize in the same way, their renormalization factor is identical.\nDue to this extra factor, the interaction between two vortices in the\nfilm takes the form of a $1/r$ Coulomb potential\n\\begin{equation} \\label{renV}\nV_\\perp(r) = \\frac{g_\\perp^2}{2 \\pi} \\frac{a}{r},\n\\end{equation} \nwhere $a$ is some microscopic length scale which accompanies the\nrenormalization factor $Z_{\\bf A}$ for dimensional reasons\n\\cite{Goldenfeld}. \n\nSince the electric charge is finite at the CQCP, the penetration depth\n$\\lambda_\\perp \\propto 1/\\rho_{\\rm s}$ scales with the correlation\nlength as $\\lambda_\\perp \\sim \\xi$. In the correlation function\n(\\ref{Gr}) we thus have the combination $1/(2 k_\\perp +\\xi^{-1})$ which\nshould be compared with $1/({\\bf k}^2 + \\xi^{-2})$ for a bulk\nsuperconductor.\n\nThe absence of any reference to the electric charge in the renormalized\nand bare interaction (at least for small enough systems) implies that\nthe same results should be derivable from our starting Hamiltonian\nrestricted to two dimensions and with $q$ set to zero: ${\\cal H}_\\perp =\n(\\rho_{\\rm s} w/2 m^2) (\\nabla_\\perp \\varphi - \\bbox{\\varphi}_\\perp^{\\rm\nP})^2$. The plastic field $\\bbox{\\varphi}_\\perp^{\\rm P}$, with\n$\\nabla_\\perp \\times \\bbox{\\varphi}_\\perp^{\\rm P} = -2 \\pi \\sum_\\alpha\n\\delta({\\bf x}_\\perp - {\\bf x}_\\perp^\\alpha)$ describes vortices in a\nsuperfluid \\cite{GFCM}. It is obtained from the description involving\nthe plastic field $B_\\perp^{\\rm P}$ by a canonical transformation of the\nvector potential. By directly integrating out the Anderson-Bogoliubov\nmode, and ignoring the $k_\\perp$ dependence of $\\rho_{\\rm s}$, which is\nvalid outside of the critical region, one easily reproduces the bare\ninteraction potential (\\ref{V2d}). The renormalized interaction\n(\\ref{renV}) is obtained by realizing that the anomalous scaling\ndimension of the superconducting mass density is $d_{\\rho_{\\rm s}} =\n(d+z)-2$ \\cite{FF}, so that in our case $\\rho_{\\rm s} \\sim k_\\perp$. In\nother words, the extra factor of $k_\\perp$ that came in via the\nrenormalization factor $Z_{\\bf A}$ in our first calculation to produce\nthe $1/r$ potential, comes in via $\\rho_{\\rm s}$ here \\cite{note}.\n\nA similar change in the $r$-dependence of the interaction between two\nvortices upon entering a critical region has been observed numerically\nin the 3-dimensional Ginzburg-Landau model \\cite{OlTe}. Near the\ncharged fixed point of that theory, $\\eta_{\\bf a}=1$ \\cite{HeTe}, as in\nour case. \n\nThis is a very pleasing coincidence as the (2+1)-dimensional\nGinzburg-Landau model constitutes the dual formulation of the system.\nTo appreciate the basic elements of the dual theory, note that the {\\it\ndynamics} of the charged degrees of freedom is described by the\neffective Lagrangian\n\\begin{equation} \\label{effprime}\n{\\cal L}_{\\perp,{\\rm eff}} = \\frac{\\rho_{\\rm s}w}{2m^2}\\left[\n\\frac{1}{c^2} (\\partial_t\\varphi + \\varphi_t^{\\rm P})^2 -(\\nabla_\\perp\n\\varphi - \\bbox{\\varphi}_\\perp^{\\rm P})^2 \\right],\n\\end{equation} \nwith $c$ the speed of sound. In accord with the above findings, we have\nignored the coupling to the magnetic vector potential, so that Eq.\\\n(\\ref{effprime}) essentially describes a superfluid. Although the\ncomplete effective theory is Galilei invariant \\cite{GWW,nr}, the\nlinearized form (\\ref{effprime}) is invariant under Lorentz\ntransformations, with $c$ replacing the speed of light.\n\nIn the dual formulation, where the roles of charges and vortices are\ninterchanged, the Anderson-Bogoliubov mode mediating the interaction\nbetween two vortices is represented as a photon associated with a\nfictitious gauge field $a_\\mu$, i.e., (in relativistic notation)\n$\\partial_\\mu \\varphi \\sim \\epsilon_{\\mu \\nu \\lambda} \\partial^\\nu\na^\\lambda$. In 2+1 dimensions, a photon has only one transverse\ndirection and thus only one degree of freedom---as has the\nAnderson-Bogoliubov mode. The elementary excitations of the dual theory\nare the vortices, described by a complex scalar field $\\psi$.\nSpecifically, the (well-known) dual theory of Eq.\\ (\\ref{effprime}) is\nthe Ginzburg-Landau model \\cite{dualGL,GFCM,WeZe,KKS}\n\\begin{equation} \n{\\cal L}_{\\rm dual} = -\\case{1}{4} f_{\\mu \\nu}^2 + |(\\partial_\\mu -i g\na_\\mu) \\psi|^2 - m_\\psi^2 |\\psi|^2 - \\case{1}{4} u|\\psi|^4,\n\\end{equation} \nwith $f_{\\mu \\nu} = \\partial_\\mu a_\\nu - \\partial_\\nu a_\\mu$, $m_\\psi$ a\nmass parameter, and $u$ the strength of the self-coupling. Both the\ngauge part as well as the matter part of the dual theory are of a\nrelativistic form. The gauge part is because the effective theory\n(\\ref{effprime}) is Lorentz invariant, while the matter part is because\nvortices of positive and negative circulation can annihilate, and can\nalso be created. In this sense they behave as relativistic particles.\nAs was pointed out in Ref.\\ \\cite{WeZe}, the speed of ``light'' in the\ngauge and matter part in general differ.\n\nThe interaction potential (\\ref{V2d}) between two external vortices is\nnow being interpreted as the 2-dimensional Coulomb potential between\ncharges. The observation concerning the critical behavior of the\nGinzburg-Landau model implies that the qualitative change in $V(r)$\nupon entering the critical region is properly represented in the dual\nformulation.\n\nWhereas in the conducting phase, the charges are condensed, in the\ninsulating phase, the vortices are condensed \\cite{MPAFisher}. In the\ndual theory, the vortex condensate is represented by a nonzero\nexpectation value of the $\\psi$ field, which in turn leads via the\nAnderson-Higgs mechanism to a mass term for the gauge field $a_\\mu$.\nBecause $(\\epsilon_{\\mu \\nu \\lambda} \\partial^\\nu a^\\lambda)^2 \\sim\n(\\partial_\\mu \\varphi)^2$, the mass term $a_\\mu^2$ with two derivatives\nless implies that the Anderson-Bogoliubov mode has acquired an energy\ngap. That is to say, the phase where the vortices are condensed is\nindeed an insulator. Since electric charges are seen by the dual theory\nas flux quanta, they are expelled from the system as long as the dual\ntheory is in the Meissner state. Above the critical field $h =\n\\nabla_\\perp \\times {\\bf a} = h_{c_1}$ they start penetrating the system\nand form an Abrikosov lattice. In the original formulation, this\ncorresponds to a Wigner crystal of the charges. Finally, when more\ncharges are added and the dual field reaches the critical value\n$h_{c_2}$, the lattice melts and the charges condense in the superfluid\nphase described by the effective theory (\\ref{effprime}).\n\n\n\\acknowledgements I'm grateful to M. Krusius for the kind hospitality at\nthe Low Temperature Laboratory in Helsinki. I wish to acknowledge\nuseful conversations with J.\\ Hove, N.\\ Kopnin, M.\\ Krusius, K.\\ Nguyen,\nM.\\ Paalanen, A.\\ Sudb\\o, and especially G. Volovik.\n\nThis work was funded in part by the EU sponsored programme Transfer and\nMobility of Researchers under contract No.\\ ERBFMGECT980122.\n\n%\n\\begin{thebibliography}{99}\n\\bibitem{Delft} H. S. J. van der Zant, F. C. Fritschy, W. J. Elion,\nL. J. Geerligs, and J. E. Mooij, Phys. Rev. Lett. {\\bf 69}, 2971\n(1992).\n\\bibitem{IVQH} D. Shahar, D. C. Tsui, M. Shayegan, E. Shimshoni, and\nS. L. Sondhi, Science {\\bf 274}, 589 (1996).\n\\bibitem{KSSMF} S. V. Kravchenko, D. Simonian, M. P. Sarachik,\nW. E. Mason, and J. E. Furneaux, Phys. Rev. Lett. {\\bf 77}, 4938 (1996).\n\\bibitem{MPAFisher} M. P. A. Fisher, Phys. Rev. Lett. {\\bf 65}, 923\n(1990).\n\\bibitem{WeZe} X. G. Wen and A. Zee, Int. J. Mod. Phys. B {\\bf 4}, 437\n(1990).\n\\bibitem{BMO} M. R. Beasley, J. E. Mooij, and T. P. Orlando,\nPhys. Rev. Lett. {\\bf 42}, 1165 (1979).\n\\bibitem{SGCS} S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar,\nRev. Mod. Phys. {\\bf 69}, 315 (1997).\n\\bibitem{YaKa} A. Yazdani and A. Kapitulnik, Phys. Rev. Lett. {\\bf 74}, 3037\n(1995). \n\\bibitem{FGG} M. P. A. Fisher, G. Grinstein, and S. M. Girvin,\nPhys. Rev. Lett. {\\bf 64}, 587 (1990).\n\\bibitem{HPsu1} A. F. Hebard and M. A. Paalanen, Phys. Rev. Lett. {\\bf 65},\n927 (1990).\n\\bibitem{WET} H. P. Wei, L. W. Engel, and D. C. Tsui, Phys. Rev. B {\\bf\n50}, 14609 (1994).\n\\bibitem{NgSu} Numerical evidence was presented in: A. K. Nguyen and\nA. Sudb\\o, Phys. Rev. B {\\bf 60}, 15307 (1999), showing that even in the\nthermal 3-dimensional superconductor-normal transition, amplitude\nfluctuations are not critical at the transition temperature, and can be\nignored in the critical region.\n\\bibitem{reviews} S. M. Girvin, M. Wallin, M. C. Cha, M. P. A. Fisher,\nand A. P. Young, Prog. Theor. Phys. Suppl. {\\bf 107}, 135 (1992);\nA. M. J. Schakel in {\\it Correlations, Coherence, and Order} edited by\nD. N. Shopova und D. I. Uzunov (Plenum, New York, 1999), p. 295;\nJ. Phys. Studies {\\bf 3}, 337 (1999).\n\\bibitem{FF} D. S. Fisher and M. P. A. Fisher, Phys. Rev. Lett. {\\bf\n61}, 1847 (1988).\n\\bibitem{GFCM} H. Kleinert, {\\em Gauge Fields in Condensed Matter} (World\nScientific, Singapore, 1989).\n\\bibitem{KKS} M. Kiometzis and A. M. J. Schakel, Int. J. Mod. Phys. {\\bf B}\n7, 4271 (1993); M. Kiometzis, H. Kleinert, and A. M. J. Schakel,\nFortschr. Phys. {\\bf 43}, 697 (1995).\n\\bibitem{deGennes} P. G. de Gennes, {\\it Superconductivity of Metals\nand Alloys} (Addison-Wesley, New York, 1966), sec. 3.2.\n\\bibitem{Pearl} J. Pearl, Appl. Phys. Lett. {\\bf 5}, 65 (1964); in {\\it\nLow Temperature Physics---LT9}, edited by J. G. Danut, D. O. Edwards,\nF. J. Milford, and M. Yagub (Plenum, New York, 1965).\n\\bibitem{Goldenfeld} N. Goldenfeld, {\\it Lectures on Phase Transitions\nand the Renormalization Group} (Addison-Wesley, New York, 1992),\nsec. 7.2.\n\\bibitem{note} Note that only when the anomalous scaling dimension\n$d_{\\rho_{\\rm s}} = (d+z)-2$ vanishes, the form of the interaction\nbetween two vortices remains unchanged upon entering the critical\nregion. An example is given by the Berezinskii-Kosterlitz-Thouless\nphase transition where $d=2,z=0$. \n\\bibitem{OlTe} P. Olsson and S. Teitel, Phys. Rev. Lett. {\\bf 80}, 1964\n(1998). An independent confirmation of this result can be extracted\nfrom the Monte Carlo simulations by J. Hove and A. Sudb\\o,\nPhys. Rev. Lett. {\\bf 84}, 3426 (2000). \n\\bibitem{HeTe} I. F. Herbut and Z. Te\\v{s}anovi\\'{c}, Phys. Rev. Lett. {\\bf\n76}, 4588 (1996).\n\\bibitem{GWW} M. Greiter, F. Wilczek, and E. Witten, Mod. Phys. Lett.\nB {\\bf 3}, 903 (1989).\n\\bibitem{nr} A. M. J. Schakel, Mod. Phys. Lett. B {\\bf 4}, 927 (1990);\nInt. J. Mod. Phys. B {\\bf 8}, 2021 (1994); {\\it ibid} {\\bf 10}, 999\n(1996).\n\\bibitem{dualGL} M. Peskin, Ann. Phys. {\\bf 113}, 122 (1978);\nP. R. Thomas and M. Stone, Nucl. Phys. B {\\bf 144}, 513 (1978).\n\\end{thebibliography}\n\n\n\\end{multicols}\n\\end{document}\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002025.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{Delft} H. S. J. van der Zant, F. C. Fritschy, W. J. Elion,\nL. J. Geerligs, and J. E. Mooij, Phys. Rev. Lett. {\\bf 69}, 2971\n(1992).\n\\bibitem{IVQH} D. Shahar, D. C. Tsui, M. Shayegan, E. Shimshoni, and\nS. L. Sondhi, Science {\\bf 274}, 589 (1996).\n\\bibitem{KSSMF} S. V. Kravchenko, D. Simonian, M. P. Sarachik,\nW. E. Mason, and J. E. Furneaux, Phys. Rev. Lett. {\\bf 77}, 4938 (1996).\n\\bibitem{MPAFisher} M. P. A. Fisher, Phys. Rev. Lett. {\\bf 65}, 923\n(1990).\n\\bibitem{WeZe} X. G. Wen and A. Zee, Int. J. Mod. Phys. B {\\bf 4}, 437\n(1990).\n\\bibitem{BMO} M. R. Beasley, J. E. Mooij, and T. P. Orlando,\nPhys. Rev. Lett. {\\bf 42}, 1165 (1979).\n\\bibitem{SGCS} S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar,\nRev. Mod. Phys. {\\bf 69}, 315 (1997).\n\\bibitem{YaKa} A. Yazdani and A. Kapitulnik, Phys. Rev. Lett. {\\bf 74}, 3037\n(1995). \n\\bibitem{FGG} M. P. A. Fisher, G. Grinstein, and S. M. Girvin,\nPhys. Rev. Lett. {\\bf 64}, 587 (1990).\n\\bibitem{HPsu1} A. F. Hebard and M. A. Paalanen, Phys. Rev. Lett. {\\bf 65},\n927 (1990).\n\\bibitem{WET} H. P. Wei, L. W. Engel, and D. C. Tsui, Phys. Rev. B {\\bf\n50}, 14609 (1994).\n\\bibitem{NgSu} Numerical evidence was presented in: A. K. Nguyen and\nA. Sudb\\o, Phys. Rev. B {\\bf 60}, 15307 (1999), showing that even in the\nthermal 3-dimensional superconductor-normal transition, amplitude\nfluctuations are not critical at the transition temperature, and can be\nignored in the critical region.\n\\bibitem{reviews} S. M. Girvin, M. Wallin, M. C. Cha, M. P. A. Fisher,\nand A. P. Young, Prog. Theor. Phys. Suppl. {\\bf 107}, 135 (1992);\nA. M. J. Schakel in {\\it Correlations, Coherence, and Order} edited by\nD. N. Shopova und D. I. Uzunov (Plenum, New York, 1999), p. 295;\nJ. Phys. Studies {\\bf 3}, 337 (1999).\n\\bibitem{FF} D. S. Fisher and M. P. A. Fisher, Phys. Rev. Lett. {\\bf\n61}, 1847 (1988).\n\\bibitem{GFCM} H. Kleinert, {\\em Gauge Fields in Condensed Matter} (World\nScientific, Singapore, 1989).\n\\bibitem{KKS} M. Kiometzis and A. M. J. Schakel, Int. J. Mod. Phys. {\\bf B}\n7, 4271 (1993); M. Kiometzis, H. Kleinert, and A. M. J. Schakel,\nFortschr. Phys. {\\bf 43}, 697 (1995).\n\\bibitem{deGennes} P. G. de Gennes, {\\it Superconductivity of Metals\nand Alloys} (Addison-Wesley, New York, 1966), sec. 3.2.\n\\bibitem{Pearl} J. Pearl, Appl. Phys. Lett. {\\bf 5}, 65 (1964); in {\\it\nLow Temperature Physics---LT9}, edited by J. G. Danut, D. O. Edwards,\nF. J. Milford, and M. Yagub (Plenum, New York, 1965).\n\\bibitem{Goldenfeld} N. Goldenfeld, {\\it Lectures on Phase Transitions\nand the Renormalization Group} (Addison-Wesley, New York, 1992),\nsec. 7.2.\n\\bibitem{note} Note that only when the anomalous scaling dimension\n$d_{\\rho_{\\rm s}} = (d+z)-2$ vanishes, the form of the interaction\nbetween two vortices remains unchanged upon entering the critical\nregion. An example is given by the Berezinskii-Kosterlitz-Thouless\nphase transition where $d=2,z=0$. \n\\bibitem{OlTe} P. Olsson and S. Teitel, Phys. Rev. Lett. {\\bf 80}, 1964\n(1998). An independent confirmation of this result can be extracted\nfrom the Monte Carlo simulations by J. Hove and A. Sudb\\o,\nPhys. Rev. Lett. {\\bf 84}, 3426 (2000). \n\\bibitem{HeTe} I. F. Herbut and Z. Te\\v{s}anovi\\'{c}, Phys. Rev. Lett. {\\bf\n76}, 4588 (1996).\n\\bibitem{GWW} M. Greiter, F. Wilczek, and E. Witten, Mod. Phys. Lett.\nB {\\bf 3}, 903 (1989).\n\\bibitem{nr} A. M. J. Schakel, Mod. Phys. Lett. B {\\bf 4}, 927 (1990);\nInt. J. Mod. Phys. B {\\bf 8}, 2021 (1994); {\\it ibid} {\\bf 10}, 999\n(1996).\n\\bibitem{dualGL} M. Peskin, Ann. Phys. {\\bf 113}, 122 (1978);\nP. R. Thomas and M. Stone, Nucl. Phys. B {\\bf 144}, 513 (1978).\n\\end{thebibliography}"
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cond-mat0002026
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Thermodynamically consistent mesoscopic fluid particle models for a van der Waals fluid
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[
{
"author": "Mar Serrano and Pep Espa\\~nol"
}
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The GENERIC structure allows for a unified treatment of different discrete models of hydrodynamics. We first propose a finite volume Lagrangian discretization of the continuum equations of hydrodynamics through the Voronoi tessellation. We then show that a slight modification of these discrete equations has the GENERIC structure. The GENERIC structure ensures thermodynamic consistency and allows for the introduction of correct thermal noise. In this way, we obtain a consistent discrete model for Lagrangian fluctuating hydrodynamics. For completeness, we also present the GENERIC versions of the Smoothed Particle Dynamics model and of the Dissipative Particle Dynamics model. The thermodynamic consistency endorsed by the GENERIC framework allows for a coherent discussion of the gas-liquid phase coexistence of a van der Waals fluid.
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"name": "Generic-12jan00.tex",
"string": "\\documentstyle[aps,psfig,prl,twocolumn]{revtex}\n\n\n\\begin{document}\n\\wideabs{\n\\title{Thermodynamically consistent mesoscopic fluid particle models for\na van der Waals fluid}\n\\author{Mar Serrano and Pep Espa\\~nol}\n\\address{Departamento de F\\'{\\i}sica Fundamental, UNED,\nApartado 60141, 28080 Madrid, Spain}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nThe GENERIC structure allows for a unified treatment of different\ndiscrete models of hydrodynamics. We first propose a finite volume\nLagrangian discretization of the continuum equations of hydrodynamics\nthrough the Voronoi tessellation. We then show that a slight\nmodification of these discrete equations has the GENERIC structure.\nThe GENERIC structure ensures thermodynamic consistency and allows for\nthe introduction of correct thermal noise. In this way, we obtain a\nconsistent discrete model for Lagrangian fluctuating hydrodynamics.\nFor completeness, we also present the GENERIC versions of the Smoothed\nParticle Dynamics model and of the Dissipative Particle Dynamics\nmodel. The thermodynamic consistency endorsed by the GENERIC framework\nallows for a coherent discussion of the gas-liquid phase coexistence\nof a van der Waals fluid.\n\\end{abstract}\n}\n\\section{Introduction} \n\nThe behaviour of complex fluids like colloids, emulsions, polymers or\nmultiphasic fluids is affected by the strong coupling between the\nmicrostructure of these fluids with the macroscopic flow. The\ncomplexity of these systems requires the use of novel computer\nsimulations techniques and algorithms. Usual macroscopic approaches\nthat solve partial differential equations with constitutive equations\nare not very useful because the basic input, the constitutive equation\nis usually not known. Also, these approaches neglect the presence of\nthermal noise, which is the responsible for the Brownian motion of\nsmall suspended objects and, therefore, for the diffusive processes\nthat affect the microstructure of the fluid. In recent years, there has\nbeen a large effort in order to develop mesoscopic techniques in order\nto tackle the problems arising in the simulation of\ncomplex fluids.\n\nDissipative Particle Dynamics is a mesoscopic particle based\nsimulation method that allows one to model hydrodynamic behavior and\nthat it consistently includes thermal fluctuations. It was introduced\nby Hoogerbrugge and Koelman in 1991 under the motivation of designing\nan off-lattice algorithm inspired by the ideas behind the lattice-gas\nmethod \\cite{Hoogerbrugge92}. Since then, the model has received a\ngreat deal of attention. From a theoretical point of view, the model\nhas been given a solid background as a statistical mechanics model\n\\cite{Espanol95}. The hydrodynamic behavior has been analyzed\n\\cite{Espanol95,Espanol98} and the methods of kinetic theory have provided\nexplicit formulae for the transport coefficients in terms of the model\nparameters \\cite{Marsh97}. A generalization of DPD has also been\npresented in order to conserve energy \\cite{Bonet97}. From the side of\napplications, the method is very versatile and has proven to be useful\nin the simulation of flows in porous media \\cite{Koelman93}, colloidal\nsuspensions \\cite{Koelman93,Boek97}, polymer suspensions\n\\cite{Schlijper95}, microphase separation of copolymers\n\\cite{Groot97}, multicomponent flows \\cite{Coveney96} and thin-film\nevolution \\cite{Dzwinel99}.\n\n\\newpage\nThe physical picture behind the dissipative particles used in the\nmodel is that they represent mesoscopic portions of real fluid, say\nclusters of molecules moving in a coherent and hydrodynamic\nfashion. The interaction between these particles are postulated from\nsimplicity and symmetry principles that ensure the correct\nhydrodynamic behavior. DPD faces, however, a conceptual problem. The\nthermodynamic behavior of the model is determined by the conservative\nforces introduced in the model. This forces are assumed to be soft\nforces in counter distinction to the singular forces of the\nLennard-Jones type used in molecular dynamics. But there is no\nwell-defined procedure to relate the shape and amplitude of the\nconservative forces with a prescribed thermodynamic behavior (although\nattempts in that direction have been undertaken, see\nRef. \\cite{Groot97}). Also, it is not clear which physical time and\nlength scales the model actually describes, even though the presence\nof thermal noise suggests the foggy area of the mesoscopic realm. We\nwill see that both problems are closely related.\n\nDissipative Particle Dynamics is very similar in spirit to the popular\nmethod of Smoothed Particle Dynamics \\cite{Espanol98}. The method was\nintroduced in the context of astrophysics computation in the early\n70's \\cite{Monaghan92} and very recently it has been applied to the\nstudy of laboratory scale viscous \\cite{Takeda94} and thermal flows\n\\cite{Kum95} in simple geometries. SPD is essentially a Lagrangian\ndiscretization of the Navier-Stokes equations by means of a weight\nfunction. The procedure transforms the partial differential equations\nof continuum hydrodynamics into ordinary differential equations. These\nequations can be further interpreted as the equations of motion for a\nset of particles interacting with prescribed laws of force. The\ntechnique thus allows one to solve PDE's with molecular dynamics\ncodes. Again, these particles can be understood as physical portions\nof the fluid that evolve coherently along the flow. The problem with\nSmoothed Particle Hydrodynamics is that it does not include thermal\nfluctuations and, therefore, cannot be applied to the study of complex\nfluids at mesoscopic scales.\n\nWe have recently shown that the conceptual problems in DPD and the\ninclusion of thermal fluctuations in SPD can be resolved by\nformulating convenient generalizations of both methods under the\ngeneral framework of GENERIC \\cite{Espanol-prl99}. In the present\npaper, we take a further look at the problem of formulating consistent\nmodels for the simulation of hydrodynamic problems. Our point of view\nhere is to construct a finite volume algorithm for solving the\nNavier-Stokes equations in such a way that thermodynamic consistency\nis retained: The resulting algorithm conserves mass, momentum, and\nenergy, and the entropy is an increasing function of time. Most\nimportant, we show how to include thermal noise in a consistent way,\nthis is, producing the Einstein distribution function. We end up,\ntherefore, with an algorithm for simulating fluctuating hydrodynamics\nin a Lagrangian way \\cite{Eulerian}. This algorithm can be used as the\nbasis for simulating colloidal suspensions, where the Brownian motion\nof the colloidal particles is due to the thermal fluctuations on the\nsolvent \\cite{Bedeaux74}. We show also another potential application\nto multiphasic flow of the gas-liquid type. When the fluid is\ndescribed with a van der Waals equation of state, the Einstein\nequilibrium distribution allows to discuss the gas-liquid phase\ntransition in probabilistic terms. Such probabilistic approaches to\nequilibrium phase transitions have been studied in the past\n\\cite{Guemez88}. We should note, however, that the GENERIC finite\nvolume algorithm should allow us to study fully non-equilibrium\nsituations created by external boundary conditions that can drive the\nsystem out of equilibrium. Start up of boiling of water in a pot could\nbe addressed with the proposed GENERIC finite volume algorithm.\n\nOther very recent approaches to the study of the flow of liquid-vapor\ncoexisting fluids are the Lattice Boltzmann model introduced by Swift,\nOsborn and Yeomans \\cite{Swift95} and improved in Refs. \\cite{Luo98}\nin order to have a thermodynamically consistent model for a dense\nfluid that may exhibit liquid-vapor coexistence. However, only\nisothermal models have been considered up to now. A second very\npromising approach is the Direct Simulation Monte Carlo method\n\\cite{Bird94} that has been conveniently generalized to deal with\ndense liquids with liquid-vapor coexistence \\cite{Alexander97}.\n\nIn order to construct the finite volume algorithm we have been\nstrongly inspired by the work of Flekkoy and Coveney\n\\cite{Flekkoy99}. In that paper, the authors present a ``bottom-up''\nderivation of Dissipative Particle Dynamics. Physical space is divided\ninto Voronoi cells and explicit definitions for the mass, momentum and\nenergy of the cells in terms of the microscopic degrees of freedom\n(positions and momenta of the constituent molecules of the fluid) are\ngiven. The time derivatives of these phase functions have the\nstructure of ``microscopic balance equations'' in a discrete\nform. These equations are then divided into ``average'' and\n``stochastic'' parts. To further advance into the formulation of a\npractical algorithm, the authors then propose {\\em phenomenological},\nphysically sensible, expressions for the average part and require the\nfulfillment of the fluctuation-dissipation theorem for the stochastic\npart. Because of the use of the phenomenological expressions, we\ncannot consider this a ``bottom-up'' approach. Strictly speaking, a\nbottom-up approach would require the use of a projection operator\ntechnique or kinetic theory, in order to relate the transport\ncoefficients with the microscopic dynamics of the system (in the form\nof Green-Kubo formulae, for example).\n\nInstead, we propose in this paper a conspicuous ``top-down'' approach\nin which the deterministic continuum equations of hydrodynamics are\nthe starting point. By making intensive use of the smooth Voronoi\ntessellation discovered by Flekkoy and Coveney the form of the\ndiscrete equations is dictated by the very structure of the continuum\nequations. Our approach is similar to that in\nRef. \\cite{Hietel00}. However, we make a further requirement on the\nresulting finite volume discretization, which is that they must have\nthe GENERIC structure. This enforces the addition of a tiny bit into\nthe momentum equation. Having the GENERIC structure, it is trivial to\nobtain the stochastic part, which will be given by the\nfluctuation-dissipation theorem. In the concluding section we will\nshow the similarities and differences between our equations and those\nderived by Flekkoy and Coveney.\n\nThe approach presented in this work has also a strong resemblance with\nYuan and Doi simulation method which also uses the Voronoi tessellation\nin Lagrangian form \\cite{Yuan98} (see also \\cite{Yuan93}). They have\napplied the method to the simulation of concentrated emulsions under\nflow and several other applications to complex fluids are\nmentioned. The main difference between our work and that of\nRef. \\cite{Yuan98} is the special care we have taken in order to have\nthermodynamic consistency through the GENERIC formalism. This allows,\namong other things, to include correct thermal noise and describe\ndiffusive aspects produced by Brownian motion on mesoscopic\nobjects. Another difference is that we deal with a compressible fluid\nin which the pressure is given through the equation of state as a\nfunction of mass and entropy densities, in counterdistinction with\nRef. \\cite{Yuan98} where the pressure is obtained by satisfying the\nincompressibility condition. Compressibility is necessary if\ngas-liquid coexistence is to be described.\n\n\n\n\\newpage\n\\section{Review of GENERIC}\n\\label{review}\nFor the sake of completeness we review in this section the GENERIC\nformalism developed by \\\"Ottinger and Grmela \\cite{generic}. The\nformalism of GENERIC (acronym for General Equation for Non Equilibrium\nReversible Irreversible Coupling) states that all physically sounded\ntransport equations in non-equilibrium thermodynamics have the same\nstructure. A large body of evidence confirms this assertion: Linear\nirreversible thermodynamics, non-relativistic and relativistic\nhydrodynamics, Boltzmann's equation, polymer kinetic theory, and\nchemical reactions, just to mention a few, have all the GENERIC\nstructure \\cite{generic,gen-applyed}. The GENERIC formalism is not\nonly a way of rewriting known transport equations in a physically\ntransparent way, but it allows us to derive dynamical equations for\nnew systems not considered so far in an astonishing simple way. The\nvery structure of the formalism ensures thermodynamic consistency,\nenergy conservation and positive entropy production.\n\nThe essential assumption on which GENERIC is founded is that the\nrelevant variables $x$ used to describe the system at a certain level\nof description evolve in a time scale well-separated from the time\nscales of other variables in the system. In other words, the variables\nshould provide a closed description in which the present state of the\nsystem depends only on the very recent past and memory effects can be\nneglected. This is a recurrent theme in non-equilibrium statistical\nmechanics since the pioneering works of Zwanzig and Mori\n\\cite{Zwanzig60}. Actually, the GENERIC structure can be {\\em deduced}\nfrom first principles by using standard projection operator formalism\nunder a Markovian approximation \\cite{generic}.\n\nTwo basic building blocks in the GENERIC formalism are the energy\n$E(x)$ and entropy $S(x)$ functions of the variables $x$ describing\nthe state of the system at a particular level of description\n\\cite{generic}. The GENERIC dynamic equations are given then by\n\n\\begin{equation}\n\\frac{dx}{dt}=L\\!\\cdot\\!\\frac{\\partial E}{\\partial x}\n+M\\!\\cdot\\!\\frac{\\partial S}{\\partial x}.\n\\label{gen1}\n\\end{equation}\nThe first term in the right hand side is named the {\\em reversible}\npart of the dynamics and the second term is named the {\\em\nirreversible} part. The predictive power of GENERIC relies in the fact\nthat very strong requirements exists on the matrices $L,M$ leaving\nsmall room for the physical input about the system. First, $L$ is\nantisymmetric whereas $M$ is symmetric and positive semidefinite. Most\nimportant, the following {\\em degeneracy} conditions should hold\n\n\\begin{equation}\nL\\!\\cdot\\!\\frac{\\partial S}{\\partial x}=0,\\quad\\quad\\quad\nM\\!\\cdot\\!\\frac{\\partial E}{\\partial x}=0.\n\\label{gen2}\n\\end{equation}\nThese properties ensure that the energy is a dynamical invariant,\n$\\dot{E}=0$, and that the entropy is a non-decreasing function of\ntime, $\\dot{S}\\ge 0$, as can be proved by a simple application of the\nchain rule and the equations of motion (\\ref{gen1}). In the case that\nother dynamical invariants $I(x)$ exist in the system (as, for\nexample, linear or angular momentum), then further\nconditions must be satisfied by $L,M$. In particular,\n\n\\begin{equation}\n\\frac{\\partial I}{\\partial x}\\!\\cdot\\!\nL\\!\\cdot\\!\\frac{\\partial E}{\\partial x}=0,\\quad\\quad\\quad\n\\frac{\\partial I}{\\partial x}\\!\\cdot\\!\nM\\!\\cdot\\!\\frac{\\partial S}{\\partial x}=0.\n\\label{il}\n\\end{equation}\nwhich ensure that $\\dot{I}=0$. \n\n\nThe deterministic equations (\\ref{gen1}) are, actually, an\napproximation in which thermal fluctuations are neglected. If thermal\nfluctuations are not neglected, the dynamics is described by\nstochastic differential equations or, equivalently, by a Fokker-Planck\nequation that governs the probability distribution function\n$\\rho=\\rho(x,t)$. This FPE has the form \\cite{generic}\n\\begin{equation}\n\\partial_t\\rho =\n-\\frac{\\partial}{\\partial x}\\!\\cdot\\!\n\\left[\\rho\\left[ \n L\\!\\cdot\\!\\frac{\\partial E}{\\partial x}\n+ M\\!\\cdot\\!\\frac{\\partial S}{\\partial x} \\right]\n- k_B M\\!\\cdot\\!\\frac{\\partial \\rho}{\\partial x}\\right],\n\\label{FPE}\n\\end{equation}\nwhere $k_B$ is Boltzmann's constant. \n\nThe distribution function of the variables of a given system at\nequilibrium is given by the Einstein distribution function. This\nassertion can be proved under quite general hypothesis on the mixing\ncharacter of the microscopic dynamics of the system \\cite{mixing}. If\nthe microscopic dynamics ensures the existence of dynamical invariants\nlike the energy $E(x)$ and, perhaps, other invariants $I(x)$, then the\nEinstein distribution function takes the form \\cite{mixing}\n\n\\begin{equation}\n\\rho^{\\rm eq}(x) = g(E(x),I(x))\\exp \\{ S(x)/k_B\\},\n\\label{einst}\n\\end{equation}\nwhere the function $g$ is completely determined by the arbitrary\ninitial distribution of dynamical invariants. For example, if at\nan initial time the value of the invariants $E(x),I(x)$ are known\nwith high precision to be $E_0,I_0$, then the Einstein distribution\nfunction takes the form\n\n\\begin{equation}\n\\rho^{\\rm eq}(x)= \\frac{\\delta(E(x)-E_0)\\delta(I(x)-I_0)}\n{\\Omega(E_0,I_0)}\\exp\\{k_B^{-1} S(x)\\},\n\\label{ein0}\n\\end{equation}\nwhere $\\Omega(E_0,I_0)$ is the normalization. Given the general\nargument behind the Einstein distribution function \\cite{mixing}, it\nis sensible to demand that the Fokker-Planck equation (\\ref{FPE}) has\nas its (unique) equilibrium distribution function the Einstein\ndistribution. This can be achieved, actually, if the following\nfurther conditions on the form of the matrices $L,M$ hold,\n\n\\begin{equation}\n\\frac{\\partial}{\\partial x}\\!\\cdot\\!\\left[\nL \\!\\cdot\\!\\frac{\\partial E}{\\partial x}\\right]=0,\\quad\\quad\\quad\nM\\!\\cdot\\!\\frac{\\partial I}{\\partial x}=0.\n\\label{add}\n\\end{equation}\nThe first property can be derived independently with projection\noperator techniques \\cite{generic} whereas the second property implies\nthat the last equation in (\\ref{il}) is automatically satisfied. \n\nWhen fluctuations are present, the entropy function $S(x)$ might be a\n{\\em decreasing} function of time. However, if one considers the entropy\n{\\em functional}\n\n\\begin{equation}\n{\\cal S}[\\rho_t]=\\int\nS(x)\\rho(x,t)dx -k_B\\int \\rho(x,t)\\ln \\rho(x,t) dx,\n\\end{equation}\nit is possible to prove by using the Fokker-Planck equation (\\ref{FPE}) \nthat $\\partial_t {\\cal S}[\\rho_t]\\ge 0$. In other words, the entropy\nfunctional plays the role of a Lyapunov function. \n\nThe stochastic differential equations that\nare mathematically equivalent to the above Fokker-Planck equation\nare given, with It\\^o interpretation, by \\cite{Gardiner83}\n\n\\begin{equation}\ndx = \\left[\nL\\!\\cdot\\!\\frac{\\partial E}{\\partial x}\n+M\\!\\cdot\\!\\frac{\\partial S}{\\partial x}\n+k_B\\frac{\\partial }{\\partial x}\\!\\cdot\\!M\\right]dt\n+d\\tilde{x},\n\\label{sde1}\n\\end{equation}\nto be compared with the deterministic equations (\\ref{gen1}). The\nstochastic term $d\\tilde{x}$ in Eqn. (\\ref{sde1}) is a linear\ncombination of independent increments of the Wiener process. It\nsatisfies the mnemotechnical It\\^o rule\n\n\n\\begin{equation}\nd\\tilde{x}d\\tilde{x}^T=2k_BM dt,\n\\label{F-D}\n\\end{equation}\nwhich means that $d\\tilde{x}$ is an infinitesimal of order $1/2$\n\\cite{Gardiner83}. Eqn. (\\ref{F-D}) is a compact and formal statement of\nthe fluctuation-dissipation theorem.\n\nWhen formulating new models it might be convenient to specify\n$d\\tilde{x}$ directly instead of $M$. This ensures that $M$ through\n(\\ref{F-D}) automatically satisfies the symmetry and positive definite\ncharacter. In order to guarantee that the total energy and dynamical\ninvariants do not change in time, a strong requirement on the form of\n$d\\tilde{x}$ holds,\n\\begin{equation}\n\\frac{\\partial E}{\\partial x}\\!\\cdot\\! d\\tilde{x}=0,\\quad\\quad\n\\frac{\\partial I}{\\partial x}\\!\\cdot\\! d\\tilde{x}=0,\n\\label{consinv}\n\\end{equation}\nimplying the last equations in (\\ref{gen2}) and (\\ref{add}). The\ngeometrical meaning of (\\ref{consinv}) is clear. The random kicks\nproduced by $d\\tilde{x}$ on the state $x$ are orthogonal to the\ngradients of $E,I$. These gradients are perpendicular vectors\n(strictly speaking they are one forms) to the hypersurface $E(x)=E_0,\nI(x)=I_0$. Therefore, the kicks let the state $x$ always within the\nhypersurface of dynamical invariants.\n\nWe finally close this section by noting that, formally, the size of\nthe fluctuations is governed by the Boltzmann constant $k_B$. If we\ntake $k_B\\rightarrow 0$, the stochastic differential equation\n(\\ref{sde1}) becomes the deterministic equations (\\ref{gen1}) and the\nFokker-Planck equation (\\ref{FPE}) has only first derivatives in the\nsame way as the Liouville equation. In this limit, the distribution\nfunction $\\rho(x,t)$ does not show dispersion and it is essentially a\nDirac delta function evaluated on the solution of the deterministic\nequation. In this case, the entropy functional ${\\cal S}[\\rho]$\nreduces to the entropy $S(x)$ and the entropy is a non-decreasing\nfunction of time.\n\n\\section{Finite volume method with Voronoi cells}\n\\label{l-finite-volume}\nAs a first step for deriving the GENERIC equations for a model of fluid\nparticles, we consider the method of finite volumes for the numerical\nintegration of the equations of continuum hydrodynamics. The basic\nmotivation for this is to have a set of reference equations that serve\nas guidelines for the modeling in the GENERIC equations in such a way\nthat we can consider the GENERIC equations as reasonable\napproximations to the equations of continuum hydrodynamics.\n\nThe finite volume method consists on integrating the continuum\nequations of hydrodynamics in a finite region of space (or {\\em finite\nvolume}) in such a way that ordinary differential equations for the\naverage fields over the finite regions emerge. In this section we\npresent a finite volume method that uses the Voronoi construction as a\nconceptually and mathematically elegant method for discretizing the\ncontinuum equations of hydrodynamics.\n\nFollowing Flekkoy and Coveney \\cite{Flekkoy99}, we introduce\nthe smoothed characteristic function of the Voronoi cell $\\mu$\n\\begin{equation}\n\\chi_\\mu({\\bf r}) = \\frac{\\Delta(|{\\bf r}-{\\bf R}_\\mu|)}\n{\\sum_\\nu \\Delta(|{\\bf r}-{\\bf R}_\\nu|)},\n\\label{chi}\n\\end{equation}\nwhere the function $\\Delta(r)=\\exp\\{-r^2/2\\sigma^2\\}$ is a Gaussian\nof width $\\sigma$. When $\\sigma\\rightarrow0$, the smoothed\ncharacteristic function tends to the actual characteristic function\nof the Voronoi cell, this is\n\\begin{equation}\n\\lim_{\\sigma\\rightarrow 0}\\chi_\\mu({\\bf r}) = \\prod_\\nu\n\\theta(|{\\bf r}-{\\bf R}_\\nu|-|{\\bf r}-{\\bf R}_\\mu|),\n\\label{chi2}\n\\end{equation}\nwhere $\\theta(x)$ is the Heaviside step function. The Voronoi\ncharacteristic function (\\ref{chi2}) takes the value 1 if ${\\bf r}$ is\nnearer to ${\\bf R}_\\mu$ than to any other ${\\bf R}_\\nu$ with\n$\\nu\\neq\\mu$. Note that the characteristic function produces a\ncovering of all space (i.e., a partition of unity), this is,\n\\begin{equation}\n\\sum_\\mu\\chi_\\mu({\\bf r}) = 1.\n\\label{cov}\n\\end{equation}\nWe introduce the volume of the Voronoi cell through\n\\begin{equation}\n{\\cal V}_\\mu=\\int_{V_T} d{\\bf r} \\chi_\\mu({\\bf r}),\n\\label{volume}\n\\end{equation}\nwhich satisfies the closure condition\n\\begin{equation}\n\\sum_\\mu{\\cal V}_\\mu=V_T,\n\\label{clo}\n\\end{equation}\nwhere $V_T$ is the total volume.\nIn Fig. \\ref{tess} we show the Voronoi tessellation corresponding\nto 15 particles seeded at random in a periodic box. For an introduction\nto the Voronoi tessellation see \\cite{Voronoi}.\n\n\\begin{figure}[ht] \n\\begin{center} \n\\psfig{figure=figure1.ps,width=10cm,height=10cm}\n\\caption{\\label{tess} Contour line at $\\chi_\\mu({\\bf r})=0.5$ \nfor the set of all characteristic functions of the 15 particles \nlocated at random in a periodic box of size $L=100$. The value \nof $\\sigma$ is $ 0.03$.}\n\\psfig{figure=figure2.ps,width=10cm,height=10cm}\n\\caption{\\label{tess1}Contour lines of the function $\\chi_\\mu({\\bf r})$ for\nparticle $\\mu=1$ of the previous tessellation. Inside the closed region\nthe value of $\\chi_\\mu({\\bf r})$ is 1 and outside is zero.}\n\\end{center}\n\\end{figure}\n\nWe mention now some other useful properties of the smoothed characteristic \nfunction that will be needed later. First, due to the Gaussian form\nof $\\Delta(r)$, \n\\begin{equation}\n\\nabla \\Delta(r)=-\\frac{1}{\\sigma^2}\\Delta(r){\\bf r}.\n\\label{nade}\n\\end{equation}\nTherefore,\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial {\\bf r}}\\chi_\\mu({\\bf r})&=&\n-\\frac{1}{\\sigma^2}\\chi_\\mu({\\bf r})({\\bf r}-{\\bf R}_\\mu)\n\\nonumber\\\\\n&+&\\frac{1}{\\sigma^2}\\chi_\\mu({\\bf r})\n\\sum_\\nu\\chi_\\nu({\\bf r})({\\bf r}-{\\bf R}_\\nu).\n\\label{prop1}\n\\end{eqnarray}\nBy using the following property\n\\begin{equation}\n\\chi_\\mu({\\bf r})(1- \\chi_\\mu({\\bf r}))=\\sum_{\\nu\\neq\\mu}\\chi_\\mu({\\bf r})\\chi_\\nu({\\bf r}),\n\\label{p2}\n\\end{equation}\nwhich can be proved by using the definition (\\ref{chi}), one\ncan rewrite Eqn. (\\ref{prop1}) as\n\\begin{equation}\n\\frac{\\partial }{\\partial {\\bf r}}\\chi_\\mu({\\bf r})=\n\\frac{1}{\\sigma^2}\n\\sum_\\nu\\chi_\\mu({\\bf r})\\chi_\\nu({\\bf r})({\\bf R}_\\mu-{\\bf R}_\\nu).\n\\label{prop1b}\n\\end{equation}\nAnother useful relation is,\n\\begin{eqnarray}\n\\frac{\\partial }{\\partial {\\bf R}_\\nu}\\chi_\\mu({\\bf r})\n&=&\n\\delta_{\\mu\\nu}\\frac{1}{\\sigma^2}\\chi_\\mu({\\bf r})({\\bf r}-{\\bf R}_\\mu)\n\\nonumber\\\\\n&-&\\frac{1}{\\sigma^2}\\chi_\\mu({\\bf r})\\chi_\\nu({\\bf r})({\\bf r}-{\\bf R}_\\nu).\n\\label{prop2}\n\\end{eqnarray}\n\nWe now introduce the following two quantities\n\\begin{eqnarray}\nA_{\\mu\\nu}\n&\\equiv &R_{\\mu\\nu}\\int \\frac{d{\\bf r}}{\\sigma^2} \\chi_\\mu({\\bf r})\\chi_\\nu({\\bf r}),\n\\nonumber\\\\\n{\\bf c}_{\\mu\\nu}\n&\\equiv&\\frac{R_{\\mu\\nu}}{A_{\\mu\\nu}}\\int \\frac{d{\\bf r}}{\\sigma^2}\n\\chi_\\mu({\\bf r})\\chi_\\nu({\\bf r}) \\left({\\bf r}-\\frac{{\\bf R}_\\mu+{\\bf R}_\\nu}{2}\\right),\n\\label{area}\n\\end{eqnarray}\nwhere $R_{\\mu\\nu}=|{\\bf R}_\\mu-{\\bf R}_\\nu|$. In appendix\n\\ref{ap-vol} we show that in the limit $\\sigma\\rightarrow 0$ the\nquantity $A_{\\mu\\nu}$ is actually the area of the contact face\n$\\mu\\nu$ between Voronoi cells $\\mu$ and $\\nu$, whereas the vector\n${\\bf c}_{\\mu\\nu}$ is the position of the center of mass of the face\n$\\mu\\nu$ with respect to the ``center'' of the face $({\\bf R}_\\mu+{\\bf\nR}_\\nu)/2$.\n\n\n\\subsection{Balance equations}\n\nWe introduce now the cell average ${[\\phi]}_\\mu(t)$ over the Voronoi\ncell $\\mu$ of an arbitrary density field $\\phi({\\bf r},t)$\n\\begin{equation}\n{[\\phi]}_\\mu(t) =\\frac{1}{{\\cal V}_\\mu}\\int d{\\bf r}\\phi({\\bf r},t)\\chi_\\mu({\\bf r}).\n\\label{phimu}\n\\end{equation}\nWe will refer to ${[\\phi]}_\\mu (t)$ as a cell variable and we will\nsee that it is an approximation for the value of the field\nat the discrete points given by the cell centers.\n\n\nIn principle, the Voronoi cell centers are allowed to move in an\narbitrary way, this is, ${\\bf R}_\\mu(t)$ are prescribed functions of\ntime. The time derivative of the cell averages is given by\n\\begin{eqnarray}\n\\frac{d}{dt} {[\\phi]}_\\mu (t) &=& -\\frac{\\dot{\\cal V}_\\mu}{{\\cal V}_\\mu}{[\\phi]}_\\mu (t)\n\\nonumber\\\\\n&+&\\frac{1}{{\\cal V}_\\mu}\\int d{\\bf r}\\phi({\\bf r},t)\\frac{d}{dt}\\chi_\\mu({\\bf r})\n\\nonumber\\\\\n&+&\\frac{1}{{\\cal V}_\\mu}\\int d{\\bf r}\\chi_\\mu(t)\\partial_t \\phi({\\bf r},t),\n\\label{td1}\n\\end{eqnarray}\nwhere the dot means the time derivative. We see that ${[\\phi]}_\\mu (t)$ changes\ndue to both, the motion of the cells and the intrinsic dependence of the\nfield $\\phi({\\bf r},t)$ on time.\n\nNow, we will assume that the field $\\phi({\\bf r},t)$ obeys a \nbalance equation, this is \n\\begin{equation}\n\\partial_t\\phi({\\bf r},t) = -\\nabla\\!\\cdot\\!{\\bf J}({\\bf r},t),\n\\label{bal}\n\\end{equation}\nwhere ${\\bf J}({\\bf r},t)$ is an appropriate current density. By\nintegration by parts of the nabla operator and use of\nEqns. (\\ref{prop1b}) and (\\ref{p2}) one arrives easily at the following\nexpression\n\n\\begin{eqnarray}\n\\frac{d}{dt}{[\\phi]}_\\mu (t) &=& -\\frac{\\dot{\\cal V}_\\mu(t)}{{\\cal V}_\\mu}{[\\phi]}_\\mu (t)\n\\nonumber\\\\\n&+&\\frac{1}{{\\cal V}_\\mu}\\sum_\\nu A_{\\mu\\nu}\n{\\bf e}_{\\mu\\nu}\\!\\cdot\\!\\left([{\\bf J}]_{\\mu\\nu}\n-[\\phi]_{\\mu\\nu}\\frac{\\dot{{\\bf R}}_\\mu+\\dot{{\\bf R}}_\\nu}{2}\\right)\n\\nonumber\\\\\n&+&\\frac{1}{{\\cal V}_\\mu}\\sum_\\nu \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\n[\\mbox{\\boldmath $\\phi$}]^{||}_{\\mu\\nu}\\!\\cdot\\!\\dot{{\\bf R}}_{\\mu\\nu},\n\\label{l1}\n\\end{eqnarray}\nwhere \n\n\\begin{eqnarray}\n{\\bf e}_{\\mu\\nu} &=& \\frac{{\\bf R}_{\\mu\\nu}}{R_{\\mu\\nu}},\n\\nonumber\\\\\n{\\bf R}_{\\mu\\nu}&=&{\\bf R}_\\mu-{\\bf R}_\\nu,\n\\nonumber\\\\\nR_{\\mu\\nu}&=&|{\\bf R}_\\mu-{\\bf R}_\\nu|,\n\\label{defis1}\n\\end{eqnarray}\nand we have introduced the face averages\n\\begin{eqnarray}\n[\\cdots]_{\\mu\\nu}&=& \\frac{R_{\\mu\\nu}}{A_{\\mu\\nu}}\n\\int \\frac{d{\\bf r}}{\\sigma^2}\\chi_\\mu({\\bf r})\\chi_\\nu({\\bf r}) \\cdots\n\\nonumber\\\\\n{[\\cdots ]}_{\\mu\\nu}^{||}&=& \\frac{R_{\\mu\\nu}}{A_{\\mu\\nu}}\\int\n \\frac{d{\\bf r}}{\\sigma^2}\\chi_\\mu({\\bf r})\\chi_\\nu({\\bf r}) \n\\left({\\bf r}-\\frac{{\\bf R}_\\nu+{\\bf R}_\\mu}{2}\\right)\\cdots\n\\label{defis2}\n\\end{eqnarray}\nNote that in the limit of sharp boundaries $\\sigma\\rightarrow 0$,\n$\\mbox{\\boldmath $\\phi$}^{||}_{\\mu\\nu}$ is a vector which is parallel\nto the face $\\mu\\nu$, whereas ${\\bf e}_{\\mu\\nu}$ is perpendicular to\nthe face.\n\nWe can write Eqn. (\\ref{l1}) in the form\n\\begin{eqnarray}\n\\frac{d}{dt} \\left({\\cal V}_\\mu{[\\phi]}_\\mu \\right) &=&\n\\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\left([{\\bf J}]_{\\mu\\nu}-[\\phi]_{\\mu\\nu}\n\\frac{\\dot{{\\bf R}}_\\mu+\\dot{{\\bf R}}_\\nu}{2}\\right)\n\\nonumber\\\\\n&+&\\sum_\\nu \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\n[\\mbox{\\boldmath $\\phi$}]^{||}_{\\mu\\nu}\\!\\cdot\\!\\dot{\\bf R}_{\\mu\\nu},\n\\label{cons}\n\\end{eqnarray}\nwhich satisfies\n\\begin{equation}\n\\frac{d}{dt} \\left(\\sum_\\mu{\\cal V}_\\mu{[\\phi]}_\\mu \\right)=0,\n\\label{cons2}\n\\end{equation}\ndue to the symmetries $[\\cdots]_{\\mu\\nu}=[\\cdots]_{\\nu\\mu}$,\n$[\\cdots]^{||}_{\\mu\\nu}=[\\cdots]^{||}_{\\nu\\mu}$ of the face averages.\nEquation (\\ref{cons2}) shows that the Voronoi discretization of the\nbalance equation (\\ref{bal}) conserves {\\em exactly} the {\\em\nextensive} variables (which are of the form density$\\times$volume).\n\n\n\\subsection{Finite volumes for an inviscid fluid}\nIn what follows we will apply the method of finite volumes\nto the continuum equations of hydrodynamics. For the sake\nof clarity, we first consider the reversible part of\nthe equations, which correspond the the usual Euler equations\nfor an inviscid fluid. In the next subsection we consider\nthe irreversible part.\n\nThe Euler equations for an inviscid fluid are the continuity \nequation \n\\begin{equation}\n\\partial_t \\rho({\\bf r},t) =\n -\\nabla\\!\\cdot\\!{\\bf g}({\\bf r},t),\n\\label{cont}\n\\end{equation}\nthe momentum balance equation\n\\begin{equation}\n\\partial_t {\\bf g}({\\bf r},t) \n= -\\nabla P({\\bf r},t) - \\nabla\\!\\cdot\\!{\\bf g}({\\bf r},t){\\bf v}({\\bf r},t),\n\\label{mombal}\n\\end{equation}\nand the entropy equation\n\\begin{equation}\n\\partial_t s({\\bf r},t) = -\\nabla\\!\\cdot\\!s({\\bf r},t){\\bf v}({\\bf r},t).\n\\label{content}\n\\end{equation}\nIn these equations, $\\rho({\\bf r},t)$ is the mass density field, ${\\bf\ng}({\\bf r},t)=\\rho({\\bf r},t){\\bf v}({\\bf r},t)$ is the momentum\ndensity field, with ${\\bf v}({\\bf r},t)$ the velocity field, and\n$s({\\bf r},t)$ is the entropy density field (entropy per unit\nvolume). The pressure field $P({\\bf r},t)$ is given, according to the\nlocal equilibrium assumption, by $P({\\bf r},t)=P^{\\rm eq}(\\rho({\\bf\nr},t),s({\\bf r},t))$ where $P^{\\rm eq}(\\rho,s)$ is the equilibrium\nequation of state that gives the macroscopic pressure in terms of the\nmass and entropy densities.\n\nWe now write Eqn. (\\ref{cons}) for the case that $\\phi=\\rho$. \nIf the Voronoi cells do not move, then $\\dot{{\\bf R}}_\\mu=0$ and\nexpression (\\ref{cons}) simplifies considerably,\n\\begin{equation}\n\\frac{d}{dt}M_\\mu(t) = \\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\n\\!\\cdot\\![\\rho{\\bf v}]_{\\mu\\nu},\n\\label{eul}\n\\end{equation}\nwhere $M_\\mu={\\cal V}_\\mu {[\\rho}]_\\mu$ is the mass of the Voronoi\ncell $\\mu$. This corresponds to a {\\em Eulerian} discretization of the\ncontinuity equation (\\ref{cont}). The physical meaning of\nEqn. (\\ref{eul}) is clear: $A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\!\\cdot\\!\n[\\rho{\\bf v}]_{\\mu\\nu}$ is the total mass per unit time that crosses\nface $\\mu\\nu$ and the rate of change of the mass of the cell $\\mu$ is\nthe sum of this quantity for each face $\\mu\\nu$.\n\nThe {\\em Lagrangian} discretization of the continuity equation in\nEqn. (\\ref{cont}) is obtained by specifying the motion of the cells\naccording to\n\\begin{equation}\n\\dot{\\bf R}_\\mu(t)={[{\\bf v}]}_{\\mu}(t).\n\\label{dl}\n\\end{equation}\nIn this way, the Voronoi cells ``follow'' (the best they can) the\nflow field. We can write Eqn. (\\ref{cons}) for the case (\\ref{cont})\nas\n\\begin{eqnarray}\n\\left.\\dot{M}_\\mu\\right|_{\\rm rev} &=&\n\\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\!\\cdot\\!\n\\left({[\\rho{\\bf v}]}_{\\mu\\nu}-\n{[\\rho]}_{\\mu\\nu}\\frac{{[{\\bf v}]}_\\mu+{[{\\bf v}]}_\\nu}{2}\\right)\n\\nonumber\\\\\n&+&\\sum_\\nu\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}{[\\rho]}^{||}_{\\mu\\nu}\\!\\cdot\\!\n({[{\\bf v}]}_\\mu-{[{\\bf v}]}_\\nu).\n\\label{mass1}\n\\end{eqnarray}\nThe momentum balance equation (\\ref{mombal}) can similarly \nbe treated and the following Lagrangian finite volume equation\narise\n\\begin{eqnarray}\n\\left.\\dot{\\bf P}_\\mu\\right|_{\\rm rev} &=&\n\\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}{[P]}_{\\mu\\nu}\n\\nonumber\\\\\n&&+\\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\!\\cdot\\!\\left(\n{[{\\bf g}{\\bf v}]}_{\\mu\\nu}-{[{\\bf g}]}_{\\mu\\nu}\n\\frac{{[{\\bf v}]}_\\mu+{[{\\bf v}]}_\\nu}{2}\\right)\n\\nonumber\\\\\n&&+\\sum_\\nu \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}{[{\\bf g}]}^{||}_{\\mu\\nu}\\!\\cdot\\!({[{\\bf v}]}_\\mu-{[{\\bf v}]}_\\nu),\n\\label{mombal2}\n\\end{eqnarray}\nwhere we have introduced the total momentum of cell $\\mu$ ${\\bf\nP}_{\\mu} = {\\cal V}_\\mu {\\bf g}_\\mu$.\n\nFinally, the entropy equation Eqn. (\\ref{content}) takes\nthe following Lagrangian discretization \n\\begin{eqnarray}\n\\left.\\dot{S}_\\mu\\right|_{\\rm rev} &=&\n\\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\!\\cdot\\!\n\\left({[s{\\bf v}]}_{\\mu\\nu}-\n{[s]}_{\\mu\\nu}\\frac{{[{\\bf v}]}_\\mu+{[{\\bf v}]}_\\nu}{2}\\right)\n\\nonumber\\\\\n&+&\\sum_\\nu\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}{[s]}^{||}_{\\mu\\nu}\\!\\cdot\\!\n({[{\\bf v}]}_\\mu-{[{\\bf v}]}_\\nu),\n\\label{cdent}\n\\end{eqnarray}\nwhere $S_\\mu={\\cal V}_\\mu s_\\mu$ is the total entropy of cell $\\mu$.\n\nIn all these evolution equations, the subscript $|_{\\rm rev}$ denotes\nthat the equations are actually the reversible part of the dynamics\nof a truly viscous fluid. \n\n\n\\subsection{Gradient expansion}\nThe above equations (\\ref{mass1}), (\\ref{mombal2}), and (\\ref{cdent})\nare rigorous and exact and do not depend on the typical size of the\nVoronoi cells. By taking sufficiently small cells, we can approximate\nthe equations and transform them into a {\\em closed} set of equations\nfor the cell variables. In this way, a computationally feasible\nalgorithm can be proposed for the updating of cell variables.\n\nLet us assume that the hydrodynamic fields have a typical length\nscale of variation $\\lambda_H$ and that the typical distance\nbetween cell centers is $\\lambda_c$. We introduce the {\\em resolution}\nparameter as $r=\\lambda_c/\\lambda_H$ and will assume that this\nparameter is very small. Therefore the dimensionless quantity\n\\begin{equation}\n\\frac{\\lambda_c }{\\phi({\\bf R}_\\mu)}\\nabla \\phi({\\bf R}_\\mu)\\sim r,\n\\label{re}\n\\end{equation}\nwill be very small. By denoting with ${\\cal O}(\\nabla)$ those terms\nwhich are of relative size $r$, we can write the cell average as\n\n\\begin{eqnarray}\n{[\\phi]}_\\mu &\\equiv&\\frac{1}{{\\cal V}_\\mu}\n\\int d{\\bf r} \\chi_\\mu({\\bf r}) \\phi({\\bf r})\n\\nonumber\\\\\n&=&\\frac{1}{{\\cal V}_\\mu}\n\\int d{\\bf r} \\chi_\\mu({\\bf r}) \\phi({\\bf r}-{\\bf R}_\\mu +{\\bf R}_\\mu)\n\\nonumber\\\\\n&=&\\phi({\\bf R}_\\mu)+\\frac{1}{{\\cal V}_\\mu}\n\\int d{\\bf r} \\chi_\\mu({\\bf r}) ({\\bf r}-{\\bf R}_\\mu)\n\\!\\cdot\\!\\nabla\\phi({\\bf R}_\\mu)\n+{\\cal O}(\\nabla^2)\n\\nonumber\\\\\n&=&\\phi({\\bf R}_\\mu)+{\\cal O}(\\nabla).\n\\label{phimub}\n\\end{eqnarray}\nPerforming similar Taylor expansions we obtain easily\n\n\\begin{equation}\n{[\\phi]}_{\\mu\\nu}=\\phi\\left(\\frac{{\\bf R}_\\mu+{\\bf R}_\\nu}{2}\\right)\n+{\\cal O}(\\nabla).\n\\label{phimunu}\n\\end{equation}\nAlso\n\\begin{equation}\n\\phi\\left(\\frac{{\\bf R}_\\mu+{\\bf R}_\\nu}{2}\\right)\n=\\frac{\\phi({\\bf R}_\\mu)+\\phi({\\bf R}_\\nu)}{2}\n+{\\cal O}(\\nabla^2),\n\\label{phi3}\n\\end{equation}\nand, therefore,\n\\begin{equation}\n{[\\phi]}_{\\mu\\nu}=\n\\frac{{[\\phi]}_{\\mu}+{[\\phi]}_{\\nu}}{2}\n+{\\cal O}(\\nabla).\n\\label{phi4}\n\\end{equation}\nAfter some algebra it is easy to show that\n\\begin{equation}\n{[\\phi\\psi]}_{\\mu\\nu}={[\\phi]}_{\\mu\\nu}\n{[\\psi]}_{\\mu\\nu}+{\\cal O}(\\nabla^2).\n\\label{phiprod}\n\\end{equation}\nFinally,\n\\begin{equation}\n{[{\\bf \\phi}]}^{||}_{\\mu\\nu}\n=\\frac{{[\\phi]}_{\\mu}+{[\\phi]}_{\\nu}}{2}\n{\\bf c}_{\\mu\\nu}+{\\cal O}(\\nabla).\n\\label{phipar}\n\\end{equation}\nBy using these Taylor approximations in Eqns. (\\ref{mass1}),\n(\\ref{mombal2}), and (\\ref{cdent}) we obtain the final Voronoi\nfinite volume discrete equations for the inviscid fluid,\n\\begin{eqnarray}\n\\left.\\dot{M}_\\mu\\right|_{\\rm rev}&=&\\sum_\\nu\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\n\\frac{{[\\rho]}_{\\mu}+{[\\rho]}_{\\nu}}{2}\n{\\bf c}_{\\mu\\nu}\\!\\cdot\\!({[{\\bf v}]}_\\mu-{[{\\bf v}]}_\\nu),\n\\nonumber\\\\\n\\left.\\dot{\\bf P}_\\mu\\right|_{\\rm rev} &=&\\sum_\\nu A_{\\mu\\nu}\n{\\bf e}_{\\mu\\nu}\\frac{{[P]}_{\\mu}+{[P]}_{\\nu}}{2}\n\\nonumber\\\\\n&+&\\sum_\\nu \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\n\\frac{{[\\rho]}_{\\mu}+{[\\rho]}_{\\nu}}{2}\n\\frac{{[{\\bf v}]}_{\\mu}+{[{\\bf v}]}_{\\nu}}{2}\n{\\bf c}_{\\mu\\nu}\\!\\cdot\\!({[{\\bf v}]}_\\mu-{[{\\bf v}]}_\\nu),\n\\nonumber\\\\\n\\left.\\dot{S}_\\mu\\right|_{\\rm rev} &=&\\sum_\\nu\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\n\\frac{{[s]}_{\\mu}+{[s]}_{\\nu}}{2}\n{\\bf c}_{\\mu\\nu}\\!\\cdot\\!({[{\\bf v}]}_\\mu-{[{\\bf v}]}_\\nu).\n\\label{fvap}\n\\end{eqnarray}\nThese equations become closed equations for $M_\\mu,{\\bf P}_\\mu, S_\\mu$\nby using\n\\begin{eqnarray}\n{[\\rho]}_\\mu&=& \\frac{M_\\mu}{{\\cal V}_\\mu},\n\\nonumber\\\\\n{[s]}_\\mu&=& \\frac{S_\\mu}{{\\cal V}_\\mu},\n\\nonumber\\\\\n{[P]}_\\mu&=& P^{\\rm eq}({[\\rho]}_\\mu,{[s]}_\\mu),\n\\label{rsp}\n\\end{eqnarray}\nwhere in the last equation for the pressure we have used again a\nTaylor expansion and neglected terms of order ${\\cal O}(\\nabla)$.\n\n\\subsection{Finite volumes for a viscous fluid}\nHaving studied the inviscid fluid, in which there are no\ndissipative contributions to the motion of the fluid, we turn\nnow to the general viscous fluid. The continuum equations\nare given by \\cite{deGroot84}\n\n\\begin{eqnarray}\n\\partial_t \\rho({\\bf r},t) &=& -\\nabla\\!\\cdot\\!\\rho({\\bf r},t){\\bf v}({\\bf r},t),\n\\nonumber\\\\\n\\partial_t {\\bf g}({\\bf r},t) \n&=& -\\nabla P({\\bf r},t) - \\nabla\\!\\cdot\\!{\\bf g}({\\bf r},t){\\bf v}({\\bf r},t)\n-\\nabla\\!\\cdot\\!(\\overline{\\bf \\Pi}+\\Pi {\\bf 1}),\n\\nonumber\\\\\n\\partial_t s({\\bf r},t) &=& -\\nabla\\!\\cdot\\!s({\\bf r},t){\\bf v}({\\bf r},t)\n\\nonumber\\\\\n&-&\\frac{1}{T}\\nabla\\!\\cdot\\!{\\bf J}^q \n+ \\frac{2\\eta}{T} \\overline{\\nabla {\\bf v}}:\\overline{\\nabla {\\bf v}}\n+\\frac{\\zeta}{T} (\\nabla\\!\\cdot\\!{\\bf v})^2,\n\\label{viscous}\n\\end{eqnarray}\nwhere, by comparison with Eqns. (\\ref{cont}), (\\ref{mombal}),\n(\\ref{content}) we can recognize the purely irreversible terms in the\nmomentum and entropy equations. Here, $T$ is the temperature field\n(which, as the pressure, is a function of $\\rho,s$ through the local\nequilibrium assumption). The double dot implies double\ncontraction. These equations have to be supplemented with the\nconstitutive equations for the traceless symmetric part $\\overline{\\bf\n\\Pi}$ of the viscous stress tensor, the trace $\\Pi$ of the viscous\nstress tensor, and the heat flux ${\\bf J}^q$. They are\n\\begin{eqnarray}\n\\overline{\\bf \\Pi} &=& -2\\eta \\overline{\\nabla {\\bf v}},\n\\nonumber\\\\\n\\Pi &=& -\\zeta\\nabla\\!\\cdot\\!{\\bf v},\n\\nonumber\\\\\n{\\bf J}^q &=& -\\kappa \\nabla T = \\kappa T^2\\nabla\\frac{1}{T},\n\\label{constitut}\n\\end{eqnarray}\nwhere the traceless symmetric part of the velocity gradient tensor is\n\\begin{equation}\n\\overline{\\nabla {\\bf v}}\n=\\frac{1}{2}\\left(\\nabla {\\bf v}+(\\nabla {\\bf v})^T\\right)\n-\\frac{1}{D}\\nabla\\!\\cdot\\!{\\bf v}.\n\\label{on}\n\\end{equation}\nHere, $D$ is the dimension of physical space.\nIn principle, the shear viscosity $\\eta$, the bulk viscosity $\\zeta$\nand the thermal conductivity $\\kappa$ might depend on the state of the\nfluid through $\\rho,s$.\n\nFollowing identical steps as for the inviscid fluid, we see that the\nviscous (irreversible) term in the momentum balance equation\ntranslates into\n\\begin{equation}\n\\left.\\dot{\\bf P}_\\mu \\right|_{\\rm irr}=\n \\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\!\\cdot\\![\\eta \\overline{\\nabla {\\bf v}}]_{\\mu\\nu}\n+\\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}[\\zeta \\nabla \\!\\cdot\\!{\\bf v}]_{\\mu\\nu}.\n\\label{irrp}\n\\end{equation}\nWe now consider the gradient expansion on each term. \nFor example,\n\n\\begin{eqnarray}\n\\left[\\eta \\overline{\\nabla {\\bf v}}\\right]_{\\mu\\nu}\n&=&\\frac{\\left[\\eta \\overline{\\nabla {\\bf v}}\\right]_{\\mu}\n +\\left[\\eta \\overline{\\nabla {\\bf v}}\\right]_{\\nu}}{2}+{\\cal O}(\\nabla),\n\\nonumber\\\\\n\\left[\\zeta \\nabla \\!\\cdot\\!{\\bf v}\\right]_{\\mu\\nu}\n&=&\\frac{[\\zeta \\nabla \\!\\cdot\\!{\\bf v}]_{\\mu}+[\\zeta \\nabla \\!\\cdot\\!{\\bf v}]_{\\nu}}{2}+{\\cal O}(\\nabla).\n\\label{t1}\n\\end{eqnarray}\nTherefore, Eqn. (\\ref{irrp}) becomes\n\n\\begin{equation}\n\\left.\\dot{\\bf P}_\\mu \\right|_{\\rm irr}=\n \\sum_\\nu {\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\![\\eta \\overline{\\nabla {\\bf v}}]_{\\nu}\n+\\sum_\\nu {\\bf \\Omega}_{\\mu\\nu}[\\zeta \\nabla \\!\\cdot\\!{\\bf v}]_{\\nu}+{\\cal O}(\\nabla),\n\\label{irrp2}\n\\end{equation}\nwhere we have introduced\n\\begin{equation}\n{\\bf \\Omega}_{\\mu\\nu} = \\frac{1}{2}A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\n\\label{aeom}\n\\end{equation}\nNote that\nfor any quantity $\\phi$ we have\n\\begin{equation}\n{[\\nabla \\phi]}_\\mu = -\\frac{1}{{\\cal V}_\\mu}\n\\sum_\\nu {\\bf \\Omega}_{\\mu\\nu} {[\\phi]}_\\nu+{\\cal O}(\\nabla)\n\\label{gdisc}\n\\end{equation}\nTherefore, we see that ${\\bf \\Omega}_{\\mu\\nu}$ is a sort of discrete\nversion of the gradient operator. This discrete gradient satisfies\n\\begin{equation}\n\\sum_{\\nu\\neq\\mu}{\\bf \\Omega}_{\\mu\\nu}\n=\\sum_{\\nu\\neq\\mu} \\frac{1}{2}A_{\\mu\\nu}{\\bf e}_{\\mu\\nu} = 0\n\\label{green}\n\\end{equation}\nwhich is essentially the statement of the divergence theorem as can\nbe shown from the identity\n\\begin{eqnarray}\n0&=&\\int d{\\bf r} \\chi_\\mu({\\bf r}) \\frac{\\partial}{\\partial {\\bf r}}1\n=-\\int d{\\bf r} \\frac{\\partial}{\\partial {\\bf r}}\\chi_\\mu({\\bf r}) \n\\nonumber\\\\\n&=&-\\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\n\\label{diverthe}\n\\end{eqnarray}\nwhere Eqn. (\\ref{prop1b}) has been used in the last equality.\n\n\nThe $\\alpha,\\beta$ component of the tensor in the first\nterm in the lhs of (\\ref{irrp2}) becomes\n\\begin{eqnarray}\n[\\eta\\overline{\\nabla {\\bf v}}^{\\alpha\\beta}]_{\\mu} &=&\n[\\eta]_{\\mu}[\\overline{\\nabla {\\bf v}}^{\\alpha\\beta}]_{\\mu} +{\\cal O}(\\nabla)\n\\nonumber\\\\\n&=&\n-\\frac{{[\\eta]}_\\mu}{{\\cal V}_\\mu}\\left(\n\\frac{1}{2}\\sum_\\nu \\left({\\bf \\Omega}_{\\mu\\nu}^\\alpha{[{\\bf v}^\\beta]}_{\\nu}\n+{\\bf \\Omega}_{\\mu\\nu}^\\beta{[{\\bf v}^\\alpha]}_{\\nu}\\right)\\right.\n\\nonumber\\\\\n&-&\\left.\\frac{1}{D}\\delta^{\\alpha\\beta}\\sum_\\nu\n{\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!{[{\\bf v}]}_{\\nu}\\right) +{\\cal O}(\\nabla).\n\\label{gradterm}\n\\end{eqnarray}\nIn a similar way,\n\\begin{equation}\n[\\zeta\\nabla \\!\\cdot\\!{\\bf v}]_\\mu =\n-\\frac{{[\\zeta]}_\\mu}{{\\cal V}_\\mu}\n\\sum_\\nu {\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!{[{\\bf v}]}_{\\nu}.\n\\label{divterm}\n\\end{equation}\nBy introducing the following discrete versions of the quantities\n$\\overline{\\bf \\Pi}, \\Pi, \\overline{\\nabla {\\bf v}},\\nabla\\!\\cdot\\!{\\bf v}$\nin Eqns. (\\ref{constitut})\n\n\\begin{eqnarray}\n\\overline{\\bf \\Pi}_\\mu &=& -\\frac{2{[\\eta]}_\\mu}{{\\cal V}_\\mu}\n\\overline{\\bf G}_\\mu,\n\\quad\\quad\\quad\\quad\n\\Pi_\\mu = -\\frac{{[\\zeta]}_\\mu}{{\\cal V}_\\mu}\nD_\\mu,\n\\nonumber\\\\\n\\overline{\\bf G}_\\mu^{\\alpha\\beta} &= &-\n\\left[\n\\frac{1}{2}\\sum_\\nu[\n{\\bf \\Omega}_{\\mu\\nu}^{\\alpha}{\\bf v}_\\nu^\\beta+\n{\\bf \\Omega}_{\\mu\\nu}^{\\beta}{\\bf v}_\\nu^\\alpha]-\n\\frac{1}{D}\\delta^{\\alpha\\beta}\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!{\\bf v}_\\nu\\right],\n\\nonumber\\\\\nD_\\mu &=&-\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!{\\bf v}_\\nu,\n\\label{grad}\n\\end{eqnarray}\nwe can write the irreversible part of the momentum equation\nas\n\n\\begin{equation}\n\\left.\\dot{\\bf P}_\\mu \\right|_{\\rm irr}=\n \\sum_\\nu {\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!({\\bf \\Pi}_\\nu+\\Pi_\\nu {\\bf 1})\n+{\\cal O}(\\nabla).\n\\label{irrmom}\n\\end{equation}\n\nAfter a very similar procedure, the irreversible part\nof the dynamics in the entropy equation can be cast also\nin the form\n\n\\begin{eqnarray}\nT_\\mu\\left.\\dot{S}_\\mu\\right|_{\\rm irr} &=& \n\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!{[{\\bf J}^q]}_\\nu\n\\nonumber\\\\\n&+&\\frac{2\\eta_\\mu}{{\\cal V}_\\mu}\n\\overline{{\\bf G}}_{\\mu}:\\overline{{\\bf G}}_{\\mu}\n+\\frac{\\zeta_\\mu}{{\\cal V}_\\mu}D^2_\\mu,\n\\label{irrent}\n\\end{eqnarray}\nwhere \n\\begin{equation}\n{[{\\bf J}^q]}_\\mu = -\\frac{{[\\kappa]}_\\mu}{{\\cal V}_\\mu}T_\\mu^2\n\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}\\frac{1}{T_\\nu}.\n\\label{hf}\n\\end{equation}\n\nAddition of the irreversible parts Eqn. (\\ref{irrmom}), (\\ref{irrent})\nto the reversible part Eqns. (\\ref{fvap}) leads to the final\nLagrangian finite volume discretization of continuum hydrodynamics.\nThe numerical solution of the resulting set of ordinary differential\nequations would produce results which are accurate to order $r$.\n\nAdmittedly, there are many other possibilities for approximating\nequation (\\ref{irrp}) to first order in gradients. The presentation\nabove is just a convenient form which has the particular GENERIC\nstructure, as will be shown in the following sections. \n\n\n\\section{GENERIC model of fluid particles}\n\\label{GEN-model}\nWe have presented in Ref. \\cite{Espanol-prl99} a description of a Newtonian\nfluid in terms of discrete {\\em fluid particles}. In this section, we\npresent a slight modification of the fluid particle model presented in\nRef. \\cite{Espanol-prl99} inspired by the results of the previous section on\nthe finite Voronoi volume discretization.\n\nIn the fluid particle model of Ref. \\cite{Espanol-prl99}, the fluid particles\nare understood as thermodynamic subsystems which move with the\nflow. The state of the system was described by the set of variables\n$x=\\{{\\bf R}_\\mu, {\\bf P}_\\mu,{\\cal V}_\\mu,S_\\mu,\\;\\;i=1,\\ldots,M\\}$,\nwhere $M$ is the number of fluid particles and ${\\bf R}_\\mu$ is the\nposition, ${\\bf P}_\\mu$ is the momentum, ${\\cal V}_\\mu$ is the volume,\nand $S_\\mu$ is the entropy of the $\\mu$-th fluid particle. Because\neach fluid particle is understood as a thermodynamic subsystem, it has\na well-defined thermodynamic fundamental equation. The fundamental\nequation relates the internal energy ${\\cal E}_\\mu$ of the fluid\nparticle with its mass $M_\\mu$, volume ${\\cal V}_\\mu$ and entropy\n$S_\\mu$, this is ${\\cal E}_\\mu = {\\cal E}(M_\\mu,{\\cal V}_\\mu,S_\\mu)$.\nThe {\\em local equilibrium hypothesis} assumes that the fundamental\nequation for the fluid particles has the same functional form as the\nfundamental equation for the whole system at equilibrium.\n\nThe volume ${\\cal V}_\\mu$ was considered in Ref. \\cite{Espanol-prl99} as an\nindependent variable to be included in $x$. In the appendix\n\\ref{ap-dep} we show that, despite our intention of considering the\nvolume as an independent variable, due to the particular form of the\nmatrix $L$ selected in Ref. \\cite{Espanol-prl99}, the volume is\nactually a function of the positions. For this reason, in this paper\nwe consider from the very beginning that the volume ${\\cal V}_\\mu$ of\nthe fluid particles is a function of the positions of the particles,\ni.e., ${\\cal V}_\\mu={\\cal V}_\\mu({\\bf R}_1,\\ldots,{\\bf R}_M)$. We will\nactually assume that the volume of particle $\\mu$ is the volume of the\nVoronoi cell of this particle, this is, Eqn. (\\ref{volume}).\n\nUnfortunately, the fact that the volume is not an actual independent\nthermodynamic variable limits the applicability of the model in\nRef. \\cite{Espanol-prl99}. In order to recover the thermodynamic versatility\nrequired, it is necessary to define a model in which the {\\em mass} of\nthe particles changes. From the point of view of the finite volume\nmethod of the previous section, it is fairly clear that the mass of\nthe Voronoi cells should be taken as a dynamical variable.\n\n\nThe state of the fluid is given, therefore, by $x=\\{{\\bf R}_\\mu,{\\bf\nP}_\\mu,M_\\mu,S_\\mu\\}$. The energy and entropy functions are\npostulated to have the form\n\n\\begin{eqnarray}\nE(x) &=& \\sum_\\mu\\frac{{\\bf P}_\\mu^2}{2M_\\mu}\n+{\\cal E}(M_\\mu,S_\\mu,{\\cal V}_\\mu),\n\\nonumber\\\\\nS(x) &=& \\sum_\\mu S_\\mu.\n\\label{ES}\n\\end{eqnarray}\nwhere ${\\cal V}_\\mu$ is an implicit function of the positions of \nthe fluid particles.\nRegarding the dynamical invariants of the system, we require that the\ntotal mass $M(x)=\\sum_\\mu M_\\mu$ and total momentum ${\\bf P}(x)=\n\\sum_\\mu{\\bf P}_\\mu$ are the only dynamical invariants. Conservation\nof angular momentum would require the introduction of spin variables\nin this discrete model \\cite{Espanol98}.\n\n\nThe gradients of energy and entropy are given by\n\n\\begin{equation}\n\\frac{\\partial E}{\\partial x} =\n\\left(\n\\begin{array}{c} -\\sum_\\gamma \\frac{\\partial {\\cal V}_\\gamma}{\\partial {\\bf R}_\\nu} P_\\gamma\n\\\\\n\\\\{\\bf v}_\\nu\n\\\\\n\\\\-\\frac{{\\bf v}_\\nu^2}{2}+\\mu_\\nu \n\\\\\n\\\\T_\\mu\n\\end{array}\\right),\n\\quad\\quad\\quad\n\\frac{\\partial S}{\\partial x} =\n\\left(\n\\begin{array}{c} {\\bf 0}\\\\ \\\\ {\\bf 0}\\\\ \\\\ 0\\\\ \\\\ 1\n\\end{array}\\right),\n\\label{derES}\n\\end{equation}\nwhere we have introduced the velocity, pressure, chemical potential per unit mass, and temperature \naccording to the usual definitions,\n\\begin{eqnarray}\n{\\bf v}_\\nu&=&\\frac{{\\bf P}_\\nu}{M_\\nu},\n\\nonumber\\\\\n-P_{\\nu}&=&\\frac{\\partial {\\cal E}_\\nu}{\\partial {\\cal V}_\\nu},\n\\nonumber\\\\\n\\mu_{\\nu}&=&\\frac{\\partial {\\cal E}_\\nu}{\\partial M_\\nu},\n\\nonumber\\\\\nT_{\\nu}&=&\\frac{\\partial {\\cal E}_\\nu}{\\partial S_\\nu}.\n\\label{intdef}\n\\end{eqnarray}\n\n\\section{Reversible dynamics}\n\\label{reversible}\n\nIn this section we consider the reversible part of the dynamics for\nthe fluid particle model. The\nmatrix $L$ is made of $M\\times M$ blocks ${\\bf L}_{\\mu\\nu}$ of size\n$8\\times 8$. The antisymmetry of $L$ translates into ${\\bf\nL}_{\\mu\\nu}=-{\\bf L}_{\\nu\\mu}$. \n\nWe have a first strong requirements for the form of $L$. We wish that the\nreversible part of the dynamics produces the following equations of\nmotion for the positions \n\\begin{equation}\n\\dot{\\bf R}_\\mu = {\\bf v}_\\mu.\n\\label{require}\n\\end{equation}\nThe simplest non-trivial reversible part that produces the\nabove equation has the following form\n\n\\begin{equation}\n\\left(\n\\begin{array}{c} \n\\dot{\\bf R}_\\mu\\\\\n\\\\\n\\dot{\\bf P}_\\mu\\\\\n\\\\\n\\dot{M}_\\mu\\\\\n\\\\\n\\dot{S}_\\mu\n\\end{array}\\right)\n=\\sum_\\nu {\\bf L}_{\\mu\\nu}\n\\left(\n\\begin{array}{c} -\\sum_\\gamma \\frac{\\partial {\\cal V}_\\gamma}{\\partial {\\bf R}_\\nu} P_\\gamma\n\\\\\n\\\\{\\bf v}_\\nu\n\\\\\n\\\\-\\frac{{\\bf v}_\\nu^2}{2}+\\mu_\\nu \n\\\\\n\\\\T_\\nu\n\\end{array}\\right),\n\\label{revgen}\n\\end{equation}\nwhere the block ${\\bf L}_{\\mu\\nu}$ has the structure\n\\begin{equation}\n{\\bf L}_{\\mu\\nu} = \n\\left(\n\\begin{array}{ccccccccc}\n {\\bf 0} && {\\bf 1}\\delta_{\\mu\\nu} && {\\bf 0} && {\\bf 0} \\\\\n\\\\\n-{\\bf 1}\\delta_{\\mu\\nu} && {\\bf \\Lambda}_{\\mu\\nu} && {\\bf \\Delta}_{\\mu\\nu} && {\\bf \\Gamma}_{\\mu\\nu} \\\\\n\\\\\n{\\bf 0} &&-{\\bf \\Delta}_{\\nu\\mu}&& 0 && 0 \\\\\n\\\\\n{\\bf 0} &&-{\\bf \\Gamma}_{\\nu\\mu} && 0 && 0 \\\\\n\\end{array}\\right).\n\\label{lij}\n\\end{equation}\nThe first row of ${\\bf L}_{\\mu\\nu}$ ensures the equation of motion\n(\\ref{require}). The first column is fixed by antisymmetry of\n$L$. Note that in order to have antisymmetry of ${\\bf L}_{\\mu\\nu}$\n(which, in turn, ensures energy conservation), it is necessary that\n${\\bf \\Lambda}^T_{\\mu\\nu}=-{\\bf \\Lambda}_{\\nu\\mu}$. Performing the\nmatrix multiplication in Eqn. (\\ref{revgen}), the reversible part of\nthe dynamics takes the form\n\n\\begin{eqnarray}\n\\dot{\\bf R}_\\mu &=& {\\bf v}_\\mu,\n\\nonumber\\\\\n\\dot{\\bf P}_\\mu &=& \n\\sum_\\nu \\frac{\\partial {\\cal V}_\\nu}{\\partial {\\bf R}_\\mu} P_\\nu\n+\\sum_\\nu {\\bf \\Lambda}_{\\mu\\nu}\\!\\cdot\\!{\\bf v}_\\nu \n\\nonumber\\\\\n&+&\\sum_\\nu{\\bf \\Delta}_{\\mu\\nu}\\left(-\\frac{{\\bf v}_\\nu^2}{2}+\\mu_\\nu\\right)\n+{\\bf \\Gamma}_{\\mu\\nu} T_\\nu,\n\\nonumber\\\\\n\\dot{M}_\\mu &=&- \\sum_\\nu {\\bf \\Delta}_{\\nu\\mu} {\\bf v}_\\nu,\n\\nonumber\\\\\n\\dot{S}_\\mu &=&- \\sum_\\nu {\\bf \\Gamma}_{\\nu\\mu} {\\bf v}_\\nu.\n\\label{revdes}\n\\end{eqnarray}\nWe now develop the pressure term by using Eqn. (\\ref{fin1}) of\nappendix \\ref{ap-vol}\n\\begin{eqnarray}\n\\sum_{\\nu} \\frac{\\partial {\\cal V}_\\nu}{\\partial {\\bf R}_\\mu} P_\\nu\n&=&\\sum_{\\nu\\neq\\mu} \\frac{\\partial {\\cal V}_\\nu}{\\partial {\\bf R}_\\mu} (P_\\nu-P_\\mu)\n\\nonumber\\\\\n&=&\\sum_{\\nu\\neq\\mu} \n\\int \\frac{d{\\bf r}}{\\sigma^2}\\chi_\\mu({\\bf r})\\chi_\\nu({\\bf r})({\\bf r}-{\\bf R}_\\mu)(P_\\mu-P_\\nu)\n\\nonumber\\\\\n&=&\\sum_{\\nu\\neq\\mu} A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\frac{P_\\mu+P_\\nu}{2}\n\\nonumber\\\\\n&&+\\sum_{\\nu\\neq\\mu} \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}{\\bf c}_{\\mu\\nu}(P_\\mu-P_\\nu),\n\\label{volpres}\n\\end{eqnarray}\nwhere use has been made of the property (\\ref{green}).\n\nThe momentum equation becomes\n\\begin{eqnarray}\n\\dot{{\\bf P}}_\\mu&=&\n\\sum_{\\nu} A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\frac{P_\\mu+P_\\nu}{2}\n+\\sum_\\nu\\left({\\bf \\Lambda}_{\\mu\\nu}\\!\\cdot\\!{\\bf v}_\\nu-\n{\\bf \\Delta}_{\\mu\\nu}\\frac{{\\bf v}_\\nu^2}{2}\\right)\n\\nonumber\\\\\n&+&\n\\sum_\\nu\\left(\n \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}{\\bf c}_{\\mu\\nu}(P_\\mu-P_\\nu)\n+{\\bf \\Delta}_{\\mu\\nu}\\mu_\\nu +{\\bf \\Gamma}_{\\mu\\nu}T_\\nu\\right).\n\\label{m2}\n\\end{eqnarray}\n\nNow, the basic question to answer is, What forms for ${\\bf\n\\Lambda}_{\\mu\\nu}$, ${\\bf \\Delta}_{\\mu\\nu}$ and ${\\bf\n\\Gamma}_{\\mu\\nu}$ should we use in order to consider\nEqns. (\\ref{revdes}) as a discrete version of hydrodynamics? In what\nfollows we will propose forms for these quantities in such a way that\nEqns. (\\ref{revdes}) and (\\ref{mass1}), (\\ref{mombal2}), (\\ref{cdent})\ncoincide as much as possible.\n\nThe vectors ${\\bf \\Delta}_{\\mu\\nu}$ and ${\\bf \\Gamma}_{\\mu\\nu}$ are\neasily identified by comparing the mass and entropy equations in\n(\\ref{fvap}) and (\\ref{revdes}). The matrix ${\\bf \\Lambda}_{\\mu\\nu}$\nis obtained by inspection from the comparison between the momentum\nequation in (\\ref{fvap}) and (\\ref{revdes}). Our proposals are\n\n\\begin{eqnarray}\n{\\bf\\Delta}_{\\mu\\nu} &=&\n\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\\frac{\\rho_\\mu+\\rho_\\nu}{2}{\\bf\nc}_{\\mu\\nu} -\\delta_{\\mu\\nu}\\sum_\\sigma\n\\frac{A_{\\mu\\sigma}}{R_{\\mu\\sigma}}\\frac{\\rho_\\mu+\\rho_\\sigma}{2}{\\bf\nc}_{\\mu\\sigma},\n\\nonumber\\\\\n{\\bf\\Gamma}_{\\mu\\nu} &=&\n\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\\frac{s_\\mu+s_\\nu}{2}{\\bf\nc}_{\\mu\\nu} -\\delta_{\\mu\\nu}\\sum_\\sigma\n\\frac{A_{\\mu\\sigma}}{R_{\\mu\\sigma}}\\frac{s_\\mu+s_\\sigma}{2}{\\bf\nc}_{\\mu\\sigma},\n\\nonumber\\\\\n{\\bf \\Lambda}_{\\mu\\nu}&=&\n\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\\frac{\\rho_\\mu+\\rho_\\nu}{2}\n\\left[\\frac{{\\bf v}_\\mu+{\\bf v}_\\nu}{2}{\\bf c}_{\\mu\\nu}\n-{\\bf c}_{\\mu\\nu}\\frac{{\\bf v}_\\mu+{\\bf v}_\\nu}{2}\\right]\n\\nonumber\\\\\n&-&\\delta_{\\mu\\nu}\\sum_\\sigma\n\\frac{A_{\\mu\\sigma}}{R_{\\mu\\sigma}}\\frac{\\rho_\\mu+\\rho_\\sigma}{2}\n\\left[\\frac{{\\bf v}_\\mu+{\\bf v}_\\sigma}{2}{\\bf c}_{\\mu\\sigma}\n-{\\bf c}_{\\mu\\sigma}\\frac{{\\bf v}_\\mu+{\\bf v}_\\sigma}{2}\\right].\n\\nonumber\\\\\n&&\\label{proposal}\n\\end{eqnarray}\nNote that ${\\bf \\Lambda}_{\\mu\\nu}^T=-{\\bf \\Lambda}_{\\mu\\nu}\n={\\bf \\Lambda}_{\\nu\\mu}$ and,\ntherefore, the antisymmetry of $L$ is ensured. Note also that\n$\\sum_\\nu{\\bf \\Gamma}_{\\mu\\nu}=0$ and, therefore, the degeneracy\ncondition $L\\!\\cdot\\!\\partial S/\\partial x=0$ is satisfied.\n\nBy substitution of these forms into the mass and entropy equations\nin (\\ref{revdes}) one obtains the mass and entropy equations\nobtained in the finite volume method Eqns. (\\ref{fvap}). Substitution\ninto the momentum equation (\\ref{m2}) leads to the finite volume momentum\nequation (\\ref{fvap}), with an additional term which is\n\n\\begin{eqnarray}\n&&\n\\sum_\\nu\n \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}{\\bf c}_{\\mu\\nu}\n\\left((P_\\mu-P_\\nu)-\\frac{\\rho_\\mu+\\rho_\\nu}{2}(\\mu_\\mu -\\mu_\\nu )\\right.\n\\nonumber\\\\\n&&-\\left.\\frac{s_\\mu+s_\\nu}{2}(T_\\mu- T_\\nu)\\right).\n\\label{gd}\n\\end{eqnarray}\nThis term is strongly reminiscent of the Gibbs-Duhem relation\nwhich, in differential forms is $dP-\\rho d\\mu-sdT=0$. For this\nreason, we expect that this term, although not exactly zero, will\nbe very small.\n\nIn summary, the proposed GENERIC equations for the reversible part of the\nevolution of the variables ${\\bf R}_\\mu,{\\bf P}_\\mu,M_\\mu,S_\\mu$ are\n\n\\begin{eqnarray}\n\\dot{\\bf R}_\\mu &=& {\\bf v}_\\mu,\n\\nonumber\\\\\n\\dot{\\bf P}_\\mu &=&\\sum_\\nu A_{\\mu\\nu}{\\bf e}_{\\mu\\nu}\\frac{P_{\\mu}+P_{\\nu}}{2}\n\\nonumber\\\\\n&+&\\sum_\\nu \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\\frac{\\rho_{\\mu}+\\rho_{\\nu}}{2}\n\\frac{{\\bf v}_{\\mu}+{\\bf v}_{\\nu}}{2}\n{\\bf c}_{\\mu\\nu}\\!\\cdot\\!({\\bf v}_\\mu-{\\bf v}_\\nu)\n\\nonumber\\\\\n&+&\\sum_\\nu\n \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}{\\bf c}_{\\mu\\nu}\n\\left((P_\\mu-P_\\nu)-\\frac{\\rho_\\mu+\\rho_\\nu}{2}(\\mu_\\mu -\\mu_\\nu )\\right.\n\\nonumber\\\\\n&&-\\left.\\frac{s_\\mu+s_\\nu}{2}(T_\\mu- T_\\nu)\\right),\n\\nonumber\\\\\n\\dot{M}_\\mu &=&\\sum_\\nu\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\n\\frac{\\rho_{\\mu}+\\rho_{\\nu}}{2}\n{\\bf c}_{\\mu\\nu}\\!\\cdot\\!({\\bf v}_\\mu-{\\bf v}_\\nu),\n\\nonumber\\\\\n\\dot{S}_\\mu &=&\\sum_\\nu\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\n\\frac{s_{\\mu}+s_{\\nu}}{2}\n{\\bf c}_{\\mu\\nu}\\!\\cdot\\!({\\bf v}_\\mu-{\\bf v}_\\nu).\n\\label{REVER}\n\\end{eqnarray}\nHere, ${\\bf v}_\\mu = {\\bf P}_\\mu/M_\\mu$, $\\rho_\\mu = M_\\mu/{\\cal\nV}_\\mu$ and $s_\\mu = S_\\mu/{\\cal V}_\\mu$. These GENERIC equations for\nthe reversible part of the dynamics Eqns. (\\ref{revdes}) are identical\n(except for the small Gibbs-Duhem term) to the finite volume\ndiscretization of the continuum equations of inviscid hydrodynamics\nEqns. (\\ref{fvap}) and can, therefore, be considered as a proper\ndiscretization of the continuum equations of hydrodynamics. Total\nmass, momentum, and energy are conserved exactly and the total entropy\ndoes not change in time due to this reversible motion.\n\n\\section{Irreversible dynamics}\n\\label{irreversible}\n\nIn this section we consider the irreversible part of the dynamics\n$M\\!\\cdot\\!\\partial S/\\partial x$. We will postulate the random terms\n$d\\tilde{x}$ for the discrete equations and will construct, through\nthe fluctuation-dissipation theorem (\\ref{F-D}), the matrix $M$ and\nthe irreversible part of the dynamics. If we guess correctly the\nrandom terms, the resulting discrete equations should consistently\nproduce the correct dissipative part of the dynamics.\n\nThermal fluctuations are introduced into the continuum equations of\nhydrodynamics through the divergence of a random stress tensor and a\nrandom heat flux \\cite{Landau59},\\cite{Espanol-PA98}. In principle,\none could think about a random {\\em mass} flux that would, according\nto the fluctuation-dissipation theorem, produce an irreversible term\nin the mass balance equation. Such term is absent in simple fluids but\nnot in mixtures (it produces the diffusion terms). The reason why\nthere is no such a random mass flux in the continuous description of a\nsimple fluid can be understood with the method of projection\noperators. As it is well-known, the method produces Green-Kubo\nexpressions for the transport coefficients. This Green-Kubo forms\ninvolve the correlation of the {\\em projected} currents. Since, in\nthe continuum case, the time derivative of the microscopic density\nfield is precisely (minus) the divergence of the microscopic momentum\ndensity field (which is itself a relevant variable), it turns out that\nthe projected current vanish exactly and there is no Green-Kubo\ntransport coefficient in the mass equation. The situation is different\nwhen one considers discrete variables. The discrete variables are the\nmass, momentum and internal energy of the Voronoi cells as functions\nof the position and momenta of the fluid molecules. Even though a\nprojection operator derivation of the equations of motion for these\nvariables is extremely involved, it is possible to show that the\nprojected mass current does not strictly vanish. This amounts to\naccept that the mass in a given cell fluctuates not only due to the\nindirect action of the random stress and heat flux but also through\nthe direct effect of a random mass flux. For the time being and for\nthe sake of simplicity, however, we assume that this random mass flux\ncan be neglected. In this case, the noise term in the equation of\nmotion (\\ref{sde1}) has the form $d\\tilde{x}^T\\rightarrow\n\\left( {\\bf 0},d\\tilde{\\bf P}_\\mu,0,d\\tilde{S}_\\mu\\right)$.\n\nIn the following subsections we consider two different implementations\nof the noise terms. The first one, through a random stress tensor and\nrandom heat flux, can be considered as the natural way of constructing\ndiscrete equations that, in the continuum limit, converge towards the\nequations of continuum hydrodynamics. The second implementation is a\ncartoon of the first one and leads to the Dissipative Particle\nDynamics algorithm.\n\n\\subsection{Finite volume hydrodynamic}\n\\label{fin-vol}\nBy analogy with the continuum fluctuating hydrodynamics we construct\nthe random terms $d\\tilde{\\bf P}_\\mu,d\\tilde{S}_\\mu $ as the discrete\ndivergences of a random flux\n\n\n\\begin{eqnarray}\nd\\tilde{\\bf P}_\\mu &=& \n\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!d\\tilde{\\mbox{\\boldmath $\\sigma$}}_\\nu,\n\\nonumber\\\\\nd\\tilde{S}_\\mu &=& \n\\frac{1}{T_\\mu}\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!d\\tilde{\\bf J}^q_\\nu\n-\\frac{1}{T_\\mu}d\\tilde{\\mbox{\\boldmath $\\sigma$}}_\\mu\n:\\sum_\\nu{\\bf \\Omega}_{\\nu\\mu}{\\bf v}_\\nu^T.\n\\label{ran2}\n\\end{eqnarray}\nWe will select the form (\\ref{aeom}) for ${\\bf \\Omega}_{\\mu\\nu}$,\nbut the particular form is not important for the time being.\nThe random stress $d\\tilde{\\mbox{\\boldmath $\\sigma$}}_\\mu\n$ and random heat flux $d\\tilde{\\bf J}^q_\\mu $ are defined by\n\n\\begin{eqnarray}\nd\\tilde{\\mbox{\\boldmath $\\sigma$}}_\\mu &=&\na_\\mu\\overline{d{\\bf W}}^{S}_{\\mu}\n+b_\\mu\\frac{{\\bf 1}}{D}{\\rm tr}[d{\\bf W}_{\\mu}],\n\\nonumber\\\\\nd\\tilde{\\bf J}^q_\\mu &=& c_\\mu d{\\bf V}_\\mu.\n\\label{ran3}\n\\end{eqnarray}\nThe coefficients $a_\\mu,b_\\mu,c_\\mu$ are given by\n\\begin{eqnarray}\na_\\mu &=& \\left(4k_BT_\\mu\\frac{\\eta_\\mu}{{\\cal V}_\\mu}\\right)^{1/2},\n\\nonumber\\\\\nb_\\mu &=& \\left(2Dk_BT_\\mu\\frac{\\zeta_\\mu}{{\\cal V}_\\mu}\\right)^{1/2},\n\\nonumber\\\\\nc_\\mu &=& T_\\mu\\left(2k_B\\frac{\\kappa_\\mu}{{\\cal V}_\\mu}\\right)^{1/2}.\n\\label{abc}\n\\end{eqnarray}\nHere, $D$ is the physical dimension of space, $\\eta_\\mu$ is the shear\nviscosity, $\\zeta_\\mu$ is the bulk viscosity, and $\\kappa_\\mu$ is the\nthermal conductivity. These transport coefficient might depend in\ngeneral on the thermodynamic state of the fluid particle $\\mu$. The\nparticular form of the coefficients in Eqn. (\\ref{abc}) might appear\nsomehow arbitrary. Actually, it is only after writing up the final\ndiscrete equations and comparing them with the finite volume equations\n(\\ref{irrmom}), (\\ref{irrent}) that we could extract the particular\nfunctional form of these coefficients.\n\nThe traceless symmetric\nrandom matrix $\\overline{d{\\bf W}}^{S}_\\mu$ is given by\n\\begin{equation}\n\\overline{d{\\bf W}}^S_{\\mu}=\n\\frac{1}{2}\\left[d{\\bf W}_{\\mu}+d{\\bf W}^T_{\\mu}\\right]\n-\\frac{1}{D}{\\rm tr}[d{\\bf W}_{\\mu}]{\\bf 1}.\n\\label{decomp}\n\\end{equation}\n$d{\\bf W}_{\\mu}$ is a matrix\nof independent Wiener increments. The vector $d{\\bf V}_\\mu$ is also a\nvector of independent Wiener increments. They satisfy the It\\^o\nmnemotechnical rules\n\\begin{eqnarray}\nd{\\bf W}^{ii'}_{\\mu}d{\\bf W}^{jj'}_{\\nu}&=&\n\\delta_{\\mu\\nu}\\delta_{ij}\\delta_{i'j'}dt,\n\\nonumber\\\\\nd{\\bf V}^{i}_\\mu d{\\bf V}^{j}_\\nu&=&\\delta_{\\mu\\nu}\\delta_{ij}dt,\n\\nonumber\\\\\nd{\\bf V}^{i}_\\mu d{\\bf W}^{jj'}_\\nu&=&0,\n\\label{ran3b}\n\\end{eqnarray}\nwhere latin indices denote tensorial components.\nNote that the postulated forms for $d\\tilde{{\\bf P}}_\\mu,d\\tilde{S}_\\mu$ in\nEqn. (\\ref{ran2}) satisfy \n\\begin{eqnarray}\n\\sum_\\mu{\\bf\nv}_\\mu\\!\\cdot\\!d\\tilde{{\\bf P}}_\\mu+T_\\mu d\\tilde{S}_\\mu&=&0,\n\\nonumber\\\\\n\\sum_\\mu d\\tilde{{\\bf P}}_\\mu&=&0,\n\\label{de0}\n\\end{eqnarray}\n and, therefore, Eqns. (\\ref{consinv}) are\nsatisfied. This means that the postulated noise terms conserve\nmomentum and energy exactly. It is now a matter of algebra to\nconstruct the dyadic $d\\tilde{x}d\\tilde{x}^T$ and from\nEqn. (\\ref{F-D}) extract the matrix $M$. \nThe procedure is rather cumbersome but standard.\n\nOnce $M$ is constructed, the terms $M\\!\\cdot\\!\\partial S/\\partial x$\nin the equation of motion (\\ref{gen1}) can be written up. By\nassuming that the transport coefficients do not depend on the entropy\ndensity (but they might depend on the mass density), the resulting\nequations of motion are\n\n\\begin{eqnarray}\n\\left.d{\\bf P}_\\mu\\right|_{\\rm irr} &=& \n\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!\n(\\mbox{\\boldmath $\\Pi$}_\\nu +\\Pi_\\nu{\\bf 1})dt \n+ d\\tilde{\\bf P}_\\mu,\n\\nonumber\\\\\n\\left.T_\\mu d{S}_\\mu \\right|_{\\rm irr}&=& \\left(1-\\frac{k_B}{C_{Vi}}\\right)\n\\left[\\frac{2\\eta_\\mu}{{\\cal V}_\\mu}\n\\overline{\\bf G}_\\mu:\\overline{\\bf G}_\\mu \n+\\frac{\\zeta_\\mu}{{\\cal V}_\\mu} D_\\mu^2 \\right]dt\n\\nonumber\\\\\n&+&\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu} \\!\\cdot\\!{\\bf J}^q_\\nu dt\n\\nonumber\\\\\n&-&\\frac{k_B}{T_\\mu C_\\mu}\\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}^2\n\\frac{\\kappa_\\nu}{{\\cal V}_\\nu}T_\\nu^2 dt\n\\nonumber\\\\\n&-&\\frac{k_BT_\\mu}{m}\n\\left(\\left(\\frac{D^2+D-2}{2D}\\right)\n\\frac{2\\eta_\\mu}{{\\cal V}_\\mu}+\\frac{\\zeta_\\mu}{{\\cal V}_\\mu}\\right)\n\\sum_\\nu{\\bf \\Omega}_{\\nu\\mu}^2dt\n\\nonumber\\\\\n&+&T_\\mu d\\tilde{S}_\\mu.\n\\label{eqmot}\n\\end{eqnarray}\nIn these equations, we have introduced the same quantities\nas in Eqn. (\\ref{grad}). The heat flux ${\\bf J}^q_\\mu$ is defined by\n\\begin{equation}\n{\\bf J}^q_\\mu=-T_\\mu^2\\frac{\\kappa_\\mu}{{\\cal V}_\\mu}\\sum_\\nu{\\bf \\Omega}_{\\nu\\mu}\n\\frac{1}{T_\\nu}\\left(1-\\frac{k_B}{C_\\nu}\\right).\n\\label{disc}\n\\end{equation}\nFinally, the heat capacity at constant volume of particle $\\mu$ is defined by\n\\begin{equation}\nC_\\mu=T_\\mu\\left(\\frac{\\partial T_\\mu}{\\partial s_\\mu}\\right)_{\\cal V}^{-1}.\n\\label{cv}\n\\end{equation}\n\nWe observe that, quite remarkably, the above equations are in the\nlimit $k_B\\rightarrow0$ identical to the irreversible part of the\nparticular finite volume discretization of the continuum hydrodynamic\nequations presented in section \\ref{l-finite-volume}. We have,\ntherefore, shown that these equations (\\ref{eqmot}) are a proper\ndiscretization of the irreversible part of hydrodynamics with thermal\nnoise included consistently.\n\nBy collecting the reversible part $L\\!\\cdot\\!\\partial E/\\partial x$ in\nEqn. (\\ref{revgen}) and the irreversible part $M\\!\\cdot\\!\\partial\nS/\\partial x$ in Eqn. (\\ref{eqmot}), the final equations of motion for\nthe discrete hydrodynamic variables could be finally written.\n\n\\subsection{Irreversible part of DPD}\n\\label{dpd}\n\n\nIn this section we show how, by postulating a different form for the\nnoise $d\\tilde{x}$, one can obtain an irreversible part which is\nclosely related to the irreversible part of the Dissipative Particle\nDynamics model. The Dissipative Particle Dynamics model that we\npresent here, then, is a natural generalization of the classical DPD\nmodel \\cite{Hoogerbrugge92} in which not only an internal energy (or\nentropy) variable is included as in Refs. \\cite{Bonet97} but also a\nmass density variable is introduced. From the GENERIC point of view,\nthis DPD model differs from the finite volume hydrodynamics in section\n\\ref{fin-vol} model only in the form of the dissipative and random\nterms.\n\nInstead of (\\ref{ran2}) the postulated structure of the random terms\nis $d\\tilde{x}\\rightarrow({\\bf 0}, d\\tilde{\\bf\nP}_\\mu,0,d\\tilde{S}_\\mu)$ with the following definitions\n\n\\begin{eqnarray}\nd\\tilde{\\bf P}_\\mu&=&\\sum_\\nu {\\bf B}_{\\mu\\nu} dW_{\\mu\\nu},\n\\nonumber\\\\\nd\\tilde{S}_\\mu &=& -\\frac{1}{2T_\\mu}\\sum_\\nu {\\bf B}_{\\mu\\nu}\\!\\cdot\\!{\\bf v}_{\\mu\\nu}dW_{\\mu\\nu}\n+\\frac{1}{T_\\mu}\\sum_\\nu A_{\\mu\\nu}dV_{\\mu\\nu},\n\\label{noisedpd}\n\\end{eqnarray}\nwhere ${\\bf B}_{\\mu\\nu}=-{\\bf B}_{\\nu\\mu}$, ${\\bf B}_{\\mu\\mu}=0$, and\n$A_{\\mu\\nu}=A_{\\nu\\mu}$ are suitable functions of position and,\nperhaps, other state variables. The independent Wiener processes\nsatisfy $dW_{\\mu\\nu}=dW_{\\nu\\mu}$ and $dV_{\\mu\\nu}=-dV_{\\nu\\mu}$ and\nthe following It\\^o mnemotechnical rules\n\\begin{eqnarray}\ndW_{\\mu\\mu'}dW_{\\nu\\nu'} &=& [\\delta_{\\mu\\nu}\\delta_{\\mu'\\nu'}+\\delta_{\\mu\\nu'}\\delta_{\\mu'\\nu}]dt,\n\\nonumber\\\\\ndV_{\\mu\\mu'}dV_{\\nu\\nu'} &=& [\\delta_{\\mu\\nu}\\delta_{\\mu'\\nu'}-\\delta_{\\mu\\nu'}\\delta_{\\mu'\\nu}]dt,\n\\nonumber\\\\\ndW_{\\mu\\mu'}dV_{\\nu\\nu'} &=& 0.\n\\label{wie}\n\\end{eqnarray}\nNote that the noise terms in Eqn. (\\ref{noisedpd}) satisfy the requirements\n(\\ref{consinv}) which take the form\n\\begin{eqnarray}\n\\sum_\\mu d\\tilde{\\bf P}_\\mu&=&0,\n\\nonumber\\\\\n\\sum {\\bf v}_\\mu\\!\\cdot\\!d\\tilde{\\bf P}_\\mu+T_\\mu d\\tilde{S}_\\mu&=&0.\n\\label{dpds}\n\\end{eqnarray}\nThe first equation ensures momentum conservation while the second\nequation ensures energy conservation. Note that the\nrandom force $d\\tilde{\\bf P}_\\mu$ provides ``kicks'' to the particles\nalong the line joining the particles, and it satisfies Newton's third\nlaw. The first term of the random term $d\\tilde{S}_\\mu$ is suggested by\nthe last equation in Eqn. (\\ref{dpds}) whereas the last\nterm is dictated by our wish of modeling heat conduction\n\\cite{Bonet97}.\n\n\n\nWe note that the DPD noise terms (\\ref{noisedpd}) can be viewed as a\ncartoon of the noise terms (\\ref{ran2}) in the finite Voronoi volume\nmodel. For this reason, we will assume the following form for\n$A_{\\mu\\nu},{\\bf B}_{\\mu\\nu}$ to remain as close as possible to\n(\\ref{ran2}) while retaining the correct symmetries for\n$A_{\\mu\\nu},{\\bf B}_{\\mu\\nu}$,\n\n\n\\begin{eqnarray}\nA_{\\mu\\nu} &=& \\left(2k_BT_\\mu T_\\nu\\frac{\\kappa}{\\overline{\\cal V}}\n\\right)^{1/2}|{\\bf \\Omega}_{\\mu\\nu}|,\n\\nonumber\\\\\n{\\bf B}_{\\mu\\nu}&=&\\left(2k_B\\frac{T_\\mu T_\\nu}{T_\\mu+T_\\nu}\n\\frac{\\gamma}{\\overline{\\cal V}}\n\\right)^{1/2}\\mbox{\\boldmath $\\Omega$}_{\\mu\\nu}.\n\\label{bij}\n\\end{eqnarray}\nWe have introduced $\\kappa$ as a parameter with dimensions of a\nthermal conductivity and a coefficient $\\gamma$ with dimensions of a\nviscosity. This DPD model has only a single viscosity, instead of two\nviscosities (shear and bulk) appearing in the finite volume model of\nthe previous subsection. We denote with ${\\overline {\\cal V}}=V_T/M$\nthe average volume per particle.\n\nIt is now a matter of simple algebra to construct the irreversible\nmatrix $M$ for the DPD algorithm with the\nfluctuation-dissipation theorem (\\ref{F-D}). The final irreversible\npart of the equations of motion are\n\\begin{eqnarray}\n\\left. d{\\bf P}_\\mu\\right|_{\\rm irr} &=&\n-\\sum_\\nu\\frac{\\gamma_{\\mu\\nu}}{\\overline{\\cal V}}\n({\\bf v}_{\\mu\\nu}\\!\\cdot\\!{\\bf \\Omega}_{\\mu\\nu}){\\bf \\Omega}_{\\mu\\nu}dt\n+ d\\tilde{\\bf P}_\\mu,\n\\nonumber\\\\\n\\left.T_\\mu d{S}_\\mu\\right|_{\\rm irr} &=& \n\\sum_\\nu\\frac{\\gamma_{\\mu\\nu}}{2\\overline{\\cal V}}\n({\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!{\\bf v}_{\\mu\\nu})^2dt\n\\nonumber\\\\\n&+&\\frac{\\kappa}{\\overline{\\cal V}}\n\\sum_\\nu {\\bf \\Omega}^2_{\\mu\\nu}(T_\\nu-T_\\mu)dt\n\\nonumber\\\\\n&-&\\frac{\\gamma}{2\\overline{\\cal V}}\n\\sum_\\nu\\frac{T_\\nu}{T_\\mu+T_\\nu}\\frac{k_B}{C_\\mu}\n({\\bf \\Omega}_{\\mu\\nu}\\!\\cdot\\!{\\bf v}_{\\mu\\nu})^2dt\n\\nonumber\\\\\n&-&\\frac{2\\gamma k_B T_\\mu}{m\\overline{\\cal V}}\n\\sum_\\nu\\frac{T_\\nu}{T_\\mu+T_\\nu}{\\bf \\Omega}_{\\mu\\nu}^2 dt\n\\nonumber\\\\\n&-&\\frac{k_B}{C_{\\mu}}\\frac{\\kappa}{\\overline{\\cal V}}\n\\sum_{\\nu\\neq \\mu}{\\bf \\Omega}^2_{\\mu\\nu}T_\\nu dt\n+T_\\mu d\\tilde{S}_\\mu.\n\\label{eqmotdpd}\n\\end{eqnarray}\nwe have defined the pair viscosity as\n\\begin{equation}\n\\gamma_{\\mu\\nu}=\\gamma\n\\left(1-\n\\frac{T_\\mu T_\\nu}{(T_\\mu+T_\\nu)^2}\n\\left(\\frac{k_B}{C_\\mu}+\\frac{k_B}{C_\\nu}\\right)\\right).\n\\label{gij}\n\\end{equation}\nWe discuss each term of these equations now. The momentum of\na particle changes irreversibly due to a friction force that\ndepends on the velocity differences between particles and\ndue to a random noise. This is the conventional form of\nthe DPD forces except for the fact that the friction coefficient\ndepends on the temperatures of the particles (although with a \nsmall prefactor of order $k_B/C_\\mu$). \nConcerning the equation for the evolution of the entropy, the first\nterm models the process of viscous heating, the fact that the motion\nof the particles creates an internal friction that increases the\ninternal energy of the particles. This term is dictated basically by\nthe requirement of energy conservation. The second term takes into\naccount the process of heat conduction and it can be understood as a\nsimple discretization of the heat conduction equation. Note again that\nthis form of the conduction term is physically more reasonable that\nthe expressions given in \\cite{Bonet97}. The next three terms\nare proportional to $k_B$ and ensure the exact conservation of energy,\nas can be seen by explicitly computing $dE=\\frac{\\partial E}{\\partial x}dx\n+\\frac{1}{2}\\frac{\\partial^2 E}{\\partial x\\partial x}d\\tilde{x}d\\tilde{x}$.\n\nAt this point, we would like to make a close comparison between the\nmodel presented in Ref. \\cite{Flekkoy99} and the one presented in this\npaper. In Ref. \\cite{Flekkoy99} the variables used to describe the\nsystem of fluid particles are the positions, momentum and energy of\nthe particles. The mass is assumed to be a constant. We will focuse on\nthe momentum equation because in order to compare the energy equation\nin both works we should transform our system of variables with the\nentropy to a system with the energy as independent variable. The\nmomentum equation in \\cite{Flekkoy99} is (changing their notation\nto our notation)\n\\begin{eqnarray}\n\\dot{\\bf P}_\\mu&=&\n-\\sum_\\nu A_{\\mu\\nu}\\left\\{\n\\frac{P_\\mu-P_\\nu}{2}{\\bf e}_{\\mu\\nu}\\right.\n\\nonumber\\\\\n&-&\\left.\\frac{\\eta}{R_{\\mu\\nu}}[{\\bf v}_{\\mu\\nu}+\n({\\bf v}_{\\mu\\nu}\\!\\cdot\\!{\\bf e}_{\\mu\\nu}){\\bf e}_{\\mu\\nu}]\\right\\}\n+M_\\mu {\\bf g}+\\tilde{\\bf F}_\\mu\n\\label{mofle}\n\\end{eqnarray}\nwhere ${\\bf g}$ is a body force like gravity and $\\tilde{\\bf F}_\\mu$\nis a stochastic force with a form fixed by the fluctuation-dissipation\ntheorem. We note that the reversible part of this equation, the\npressure term, is identical to ours except for the small Gibbs-Duhem\nterm required for thermodynamic consistency when the mass and entropy\nare not conserved by the reversible part of the dynamics as happens in\nour case. The pressure difference instead of the sum is used in\nEqn. (\\ref{mofle}), but both are equivalent in view of\nEqn. (\\ref{diverthe}). The irreversible term proportional to the\n``viscosity'' $\\eta$, is of a form similar to the DPD irreversible\nterm $d{\\bf P}_\\mu|_{\\rm irr}$ in Eqns. (\\ref{eqmotdpd}), with ${\\bf\n\\Omega}_{\\mu\\nu}$ given by Eqn. (\\ref{aeom}). A term proportional to\n${\\bf v}_{\\mu\\nu}$, which breaks angular momentum conservation is\nincluded \\cite{Espanol98} but poses no conceptual problems. However,\nwe note that dimensionally $\\eta$ is not a true viscosity, a volume\nfactor is missing. Actually, we expect that a simulation of\nEqns. (\\ref{mofle}) with a fixed value of $\\eta$ at different\nresolutions (this is, the same external length scale, channel width\nfor example, and different number of fluid particles which have,\nconsequently, different typical volumes) would produce different\nvalues for the actual viscosity of the fluid being modeled.\n\n\n\\section{The SPH model}\n\\label{sphmodel}\nIn this section we show that the Smoothed Particle Hydrodynamics\nalgorithm for an inviscid fluid is a particular case of the GENERIC\nEqs. (\\ref{revdes}). For a viscous fluid, the SPH algorithm involves\nthe irreversible part (\\ref{eqmot}) with no fluctuation effects,\ni.e. $k_B=0$.\n\n\nIn SPH, a density variable $d_\\mu$ associated to each fluid particle is\ndefined through\n\\begin{equation}\nd_\\mu =\\sum_\\nu W(|{\\bf R}_\\mu-{\\bf R}_\\nu|),\n\\label{density}\n\\end{equation}\nwhere the weight function $ W(r)$ is a bell-shaped function with\nfinite support $h$ and normalized to unity. If particles are close\ntogether, then the density defined by Eqn. (\\ref{density}) is higher\nin that region of space. From the density one can define a volume\nassociated to each fluid particle\n\n\\begin{equation}\n{\\cal V}_\\mu({\\bf R}_1,\\ldots,{\\bf R}_M) = \\frac{1}{d_\\mu}.\n\\label{volsph}\n\\end{equation}\nThe SPH algorithm for an inviscid fluid is obtained from Eqns. (\\ref{revdes})\nby using ${\\bf\n\\Lambda}_{\\mu\\nu}$= ${\\bf \\Delta}_{\\mu\\nu}={\\bf\n\\Gamma}_{\\mu\\nu}=0$. The derivative of the volume (\\ref{volsph}) \nwith respect to the positions of the particles is easily \ncomputed as\n\\begin{equation}\n\\frac{\\partial {\\cal V}_\\nu}{\\partial {\\bf R}_\\mu} =\n-\\frac{1}{d^2_\\nu}(\\mbox{\\boldmath $\\omega$}_{\\mu\\nu}+\n\\delta_{\\mu\\nu}\\sum_k\\mbox{\\boldmath $\\omega$}_{\\mu\\gamma}),\n\\label{omsph}\n\\end{equation}\nwhere \n\\begin{equation}\n\\mbox{\\boldmath $\\omega$}_{\\mu\\nu}= W'(r_{\\mu\\nu}){\\bf e}_{\\mu\\nu}.\n\\end{equation}\nThe prime here denotes the derivative and ${\\bf e}_{\\mu\\nu}$ is\nthe unit vector joining particles $i,j$. \n\nSubstitution of Eqn. (\\ref{omsph}) into the reversible\nequations (\\ref{revdes}) leads to\n\n\\begin{eqnarray}\n\\dot{\\bf R}_\\mu &=& {\\bf v}_\\mu,\n\\nonumber\\\\\nM_\\mu\\dot{\\bf v}_\\mu &=& -\\sum_\\nu\n\\mbox{\\boldmath $\\omega$}_{\\mu\\nu}\\left[\n\\frac{P_\\nu}{d_\\nu^2}+\\frac{P_\\mu}{d_\\mu^2}\\right],\n\\nonumber\\\\\n\\dot{M}_\\mu &=& 0,\n\\nonumber\\\\\n\\dot{S}_\\mu &=& 0.\n\\label{revsph}\n\\end{eqnarray}\nThis is the form of the discretization preferred by Monagan\n\\cite{Monaghan92}. It is remarkable that this form is forced solely by\nthe selection of the volume function in Eqn. (\\ref{volsph}) and the\nGENERIC structure. For a viscous fluid, the irreversible part of the\nSPH equations are simply obtained from Eqns. (\\ref{eqmot}) by using as\n${\\bf \\Omega}_{\\mu\\nu}$ the function $-\\partial {\\cal V}_\\nu/\\partial\n{\\bf R}_\\mu$ given in Eqn. (\\ref{omsph}).\n\nIn the SPH equations (\\ref{revsph}), the mass of the particles is a\nconstant. We will show in section \\ref{equilibrium}, when we discuss\nthe equilibrium distribution function, that this constancy makes the\nalgorithm unsuitable for studying gas-liquid coexistence. On the other\nhand, the SPH algorithm suffers from an unphysical feature: if the\npressures of the neighbors of particle $\\mu$ are equal to the pressure\nof this particle $\\mu$, there is still a remnant force on this\nparticle, which is physically unacceptable. A possible correction of\nthe above defect is as follows.\n\nInstead of the volume defined as in (\\ref{volsph}) we define the\nvolume as\n\n\\begin{equation}\n{\\cal V}_\\mu = \\frac{1}{d_\\mu}\\frac{{\\cal V}_T}{\\sum_k d_k^{-1}}\n\\label{volsphcorr}\n\\end{equation}\nwhich satisfies $\\sum_\\mu {\\cal V}_\\mu ={\\cal V}_T$. Note that the\ncorrection factor ${\\cal V}_T/\\sum_k d_k^{-1}$ is expected to be close\nto 1. Note also that now the volume of a given particle depends on the\ncoordinates of all the particles in the system. This breaks the local\ndefinition of the volume of a particle in terms of the positions of\nthe neighbours.\n\nNow, by using this expression for the corrected volume of a particle\n we obtain\n\n\\begin{eqnarray}\n{\\bf \\Omega}_{\\mu\\nu} &=& \\left[\n\\frac{{\\cal V}_\\nu}{d_\\nu}(\\delta_{\\mu\\nu}\\sum_\\gamma\n\\mbox{\\boldmath $\\omega$}_{\\mu\\gamma}+\\mbox{\\boldmath $\\omega$}_{\\mu\\nu})\n\\right.\n\\nonumber\\\\\n&-&\\left.\\frac{{\\cal V}_\\nu}{{\\cal V}_T}\\sum_\\gamma\n\\left(\\frac{{\\cal V}_\\nu}{d_\\nu}+\\frac{{\\cal V}_\\mu}{d_\\mu}\\right)\n\\mbox{\\boldmath $\\omega$}_{\\mu\\gamma}\\right],\n\\end{eqnarray}\nand, therefore,\n\\begin{equation}\nM_\\mu\\dot{\\bf v}_\\mu = \n-\\sum_\\nu\\left[(P_\\mu -\\overline{P})\\frac{{\\cal V}_\\mu}{d_\\mu}\n+(P_\\nu -\\overline{P})\\frac{{\\cal V}_\\nu}{d_\\nu}\\right]\n\\mbox{\\boldmath $\\omega$}_{\\mu\\nu},\n\\label{neq}\n\\end{equation}\nwhere $\\overline{P}$ is a sort of ``spatial average'' of the\npressures of the particles, i.e.,\n\\begin{equation}\n\\overline{P}=\\frac{1}{{\\cal V}_T}\\sum_\\mu {\\cal V}_\\mu P_\\mu.\n\\label{spave}\n\\end{equation}\nEquation (\\ref{neq}) has the following good features: Total momentum\nand total volume are conserved variables. If the pressure of all the\nparticles is exactly the same, there is no force on the particles. And\nthe following drawbacks: The force on particle $\\mu$ depends on the\nstate of {\\em all} the particles of the systems. Therefore, the change\nin momentum of a bulk of fluid particles takes place non-locally, not\nthrough the neighbourhood of this bulk. This non-local effect breaks\nthe local transport of momentum and therefore the macroscopic\nhydrodynamic behaviour. We expect that this effect is small,\nparticularly when the range of the weight function is much larger than\nthe interparticle distance. The second drawback is that the remnant\nforce is zero only if the pressures of {\\em all} the particles of the\nsystem are equal. If in a local region of the system the pressures are\nequal for the particles in that region but different to the pressure\nof other regions, then there still exists a remnant force on the\nparticles of that region.\n\nA final word on the {\\em overlapping coefficient} is in order. We\ndefine the overlapping coefficient $s$ as the ratio of the range $h$\nof the weight function $W(r)$ to the typical interparticle distance\n$\\lambda$ between fluid particles, this is $s=h/\\lambda$. When $s\\gg\n1$ a given particle has typically many neigbhours. It is clear that if\nthe overlapping coefficient is much larger than 1 then, for\nhomogeneously distributed particles all the volumes of all the\nparticles will be typically the same. In the limit $s$ large, then,\nthe pressures of the particles, which depend only on the volumes of\nthe particles, will be the same and the forces on the particles will\nbe negligible. In this limit, the equations of motion (\\ref{revsph})\nlose their sense. Actually, we expect that the volume defined through\n(\\ref{volsphcorr}) will have sense only if the overlapping coefficient\nis slightly larger than 1. However, the usual derivations of SPH by\nconvoluting the continuum equations of hydrodynamics with the weight\nfunction $W(r)$ make sense only in the limit of large overlapping, in\nsuch a way that the integrals can be reasonably approximated by\nsums. Because of this inconsistency and the problems discussed above,\nwe prefer the formulation of discrete hydrodynamics in terms of finite\nVoronoi volumes rather than in terms of weight functions as it is done\nin SPH. From a computational point of view, we note that the large cpu\ntime required in order to update the Voronoi mesh (i.e. compute the\nquantities $A_{\\mu\\nu},{\\bf c}_{\\mu\\nu}$) may be compensated by the\nfact that typically only six neighbours need to be considered in the\nfinite Voronoi volume simulation whereas 30-40 neighbours are needed\nin a SPH simulation in 2D.\n\nWe would like to comment finally on the approach proposed in\nRef. \\cite{Hietel00} where a finite volume approach similar to the one\npresented here is advocated. However, these authors do not consider\nthe Voronoi limit $\\sigma\\rightarrow0$ but rather take a finite width\nfor the function $\\Delta(r)$, providing an overlapping coefficient of\ntypically 1.4. Because they do not consider the Voronoi construction,\nthey encounter the difficulty of evaluating integrals similar to those\nin Eqns. (\\ref{area}). In one dimension, as is the case considered in\nRef. \\cite{Hietel00}, this can be achieved with a numerical integration\nmethod, but this becomes readily infeasible in higher dimensions.\n\n\\section{Equilibrium distribution function}\n\\label{equilibrium}\nIn this section we discuss the equilibrium distribution function\n$\\rho^{\\rm eq}(x)$ corresponding to the equations of motion\n(\\ref{REVER}) plus the irreversible terms (\\ref{eqmot}). Note that the\nequilibrium distribution function of Eqns. (\\ref{eqmot}) and\n(\\ref{eqmotdpd}) is the same, irrespective of the actual form of the\nirreversible part of the dynamics. This is because both sets of\nequations have the GENERIC structure. The GENERIC structure of the\nequations of motion ensures that the equilibrium distribution function\nfor these variables is given by Einstein distribution function in the\npresence of dynamical invariants, Eqn. (\\ref{einst}). Because total\nmass $M(x)$ total energy $E_0$ and momentum ${\\bf P}_0$ are conserved\nby the dynamics, the Einstein distribution function will be given by\n\\begin{eqnarray}\n\\rho^{\\rm eq}(x)&=& \n\\frac{1}{\\Omega}\n\\delta(M(x)-{\\cal M}_0)\\delta(E(x)-E_0)\\delta({\\bf P}(x)-{\\bf P}_0),\n\\nonumber\\\\\n&\\times &\\exp\\{k_B^{-1} S(x)\\},\n\\label{ein}\n\\end{eqnarray}\nwhere we have assumed that we know with absolute precision the values\nof the total mass ${\\cal M}_0$, energy $E_0$ and momentum ${\\bf P}_0$\nat the initial time. This is the situation in a computer\nsimulation. $\\Omega$ is a normalization factor that ensures the\nnormalization of $\\rho^{\\rm eq}(x)$. A word is in order about the\ntotal volume. We note that due to the definition of the volume in\nEqn. (\\ref{volume}), {\\em any} configuration of positions ${\\bf\nR}_\\mu$ of the particles gives that the total volume has the same\nvalue $\\sum_\\mu {\\cal V}_\\mu = {\\cal V}_0$. Therefore, even though\ntotal volume is conserved, it does not produce a restriction in the\nform of a delta function in Eqn. (\\ref{ein}).\n\n\n\\subsection{The most probable state}\nThe most probable state $x^*$ at equilibrium according to (\\ref{ein})\nis the one that maximizes the entropy $S(x)$ subject to the\nconstraints $M(x)=M_0$, $E(x)=E_0$, and ${\\bf P}(x)= {\\bf P}_0$. By\nintroducing Lagrange multipliers $\\beta,\\lambda$ and ${\\bf V}$,\nthe most probable state $x^*$ is the state that maximizes\n$k_B^{-1}S(x) - \\beta (E(x) - \\mbox{\\boldmath $V$}\\!\\cdot\\!{\\bf\nP}(x)-\\lambda M(x))$ without constraints. By equating the partial\nderivatives with respect to every variable to zero, one obtains the\nfollowing implicit equations for the most probable values $x^*=\\{{\\bf\nR}_\\mu^*,{\\bf P}_\\mu^*,M^*_\\mu,S^*_\\mu\\}$\n\\begin{eqnarray}\n\\sum_\\nu\\frac{\\partial {\\cal V}_{\\nu}}{\\partial {\\bf R}_\\mu}P_\\nu(x^*)&=&0,\n\\nonumber\\\\\n\\frac{{\\bf P}^*_\\mu}{M^*_\\mu}&=&\\mbox{\\boldmath $V$},\n\\nonumber\\\\\n\\mu_\\mu(x^*)&=&\\lambda+\\frac{3}{2}{\\bf V}^2,\n\\nonumber\\\\\nT_\\mu(x^*)&=&\\frac{1}{k_B\\beta }.\n\\label{*}\n\\end{eqnarray}\nThe second equation states that in the most probable state all\nparticles move at the same velocity $\\mbox{\\boldmath $V$}$ which might\nbe set to zero without loss of generality. The two last equations\nstate, then, that the temperature and chemical potential per unit mass\nof all the fluid particles are equal at the most probable value of the\ndiscrete hydrodynamic variables. This implies that the pressure is\nalso the same for all the fluid particles (in a simple fluid the\nintensive parameters are not independent \\cite{callen}). The first\nequation is, therefore, trivially satisfied (because $\\sum_\\nu{\\cal\nV}_\\nu={\\rm ctn}$).\n\n\\subsection{Marginal distribution functions}\n\nIn this subsection we will integrate out the momentum variables in\nEqn. (\\ref{ein}) in order to have more specific information about the\ndistribution of different variables at equilibrium. To this end, we\ndenote the state $x=(y,\\{{\\bf P}\\})$ where $y=(\\{{\\bf\nR}\\},\\{M\\},\\{S\\})$ is the set of positions, masses, and entropies of\nall particles. Note that the total entropy and internal energy in\nEqn. (\\ref{ES}) do not depend on momentum variables.\n\nBy integrating the distribution function $\\rho^{\\rm eq}(x)$\nover momenta we will have the probability $\\rho^{\\rm eq}(y)$ \nof a realization of $y$\n\n\\begin{eqnarray}\n\\rho^{\\rm eq}(y)&=&\\exp \\left\\{S(y)/k_B\\right\\}\n\\frac{1}{\\Omega_0}\\delta\\left(\\sum^M_\\mu M_\\mu - {\\cal M}_0\\right)\n\\nonumber\\\\\n&\\times&\n\\int d{\\bf P}_1\\ldots d{\\bf P}_M\n\\delta\\left(\\sum_\\mu^M{\\bf P}_\\mu-{\\bf P}_0\\right)\n\\nonumber\\\\\n&\\times&\n\\delta\\left(\\sum_\\mu^M\n\\left(\\frac{{\\bf P}^2_\\mu}{2M_\\mu}+{\\cal E}_\\mu(y)\\right)-E_0\\right)\n\\nonumber\\\\\n&=&\\exp \\left\\{S(y)/k_B\\right\\}\n\\frac{1}{\\Omega_0}\\delta\\left(\\sum^M_\\mu M_\\mu - {\\cal M}_0\\right)\n\\nonumber\\\\\n&\\times&\n\\prod_\\mu^M(2M_\\mu)^{D/2}\\frac{\\omega_{D(M-1)}}{2}\n\\left[E_0-\\sum_\\mu{\\cal E}_\\mu(y)\\right]^{\\frac{D(M-1)}{2}-1},\n\\nonumber\\\\\n\\label{pne}\n\\end{eqnarray}\nwhere we have used Eqn. (\\ref{11}) of appendix \\ref{ap-mol}.\n\nWe find now a convenient approximation to Eqn. (\\ref{pne}) by noting\nthat this probability is expected to be highly peaked around the most\nprobable state. Therefore, for those values of the variables\n${\\cal E}_\\mu(y)$ for which $\\rho^{\\rm eq}(y)$ is appreciably different\nfrom zero we can approximate\n\n\\begin{eqnarray}\n&&\\left[E_0-\\sum_\\mu{\\cal E}_\\mu\\right]^P\n= \\left[E_0-\\sum_\\mu{\\cal E}_\\mu^*\\right]^P\n\\nonumber\\\\\n&\\times&\\left[1+\\frac{1}{P}\\beta^*\\sum_\\mu({\\cal E}_\\mu^*-{\\cal E}_\\mu)\\right]^P\n\\nonumber\\\\\n&\\approx&\n\\left[E_0-\\sum_\\mu{\\cal E}_\\mu^*\\right]^P\n\\exp\\{\\beta^*\\sum_\\mu \\left({\\cal E}_\\mu^*-{\\cal E}_\\mu\\right)\\}\n\\nonumber\\\\\n&=&{\\rm ctn.} \\exp \\{-\\beta^*\\sum_\\mu{\\cal E}_\\mu\\},\n\\label{approx}\n\\end{eqnarray}\nwhere ${\\cal E}_\\mu^*$ is the most probable value of ${\\cal E}_\\mu$\nand $P=D(M-1)/2-1$ is a very large number and we have introduced\n\\begin{equation}\n\\beta^*=\\frac{D(M-1)/2-1}{E_0-\\sum_\\nu{\\cal E}_\\nu^*}\\approx\n\\frac{DM/2}{E_0-\\sum_\\nu{\\cal E}_\\nu^*}.\n\\label{beta}\n\\end{equation}\n\nFinally, we can write Eqn. (\\ref{pne}) as\n\\begin{eqnarray}\n\\rho^{\\rm eq}(y)\n&=&\\frac{1}{\\Omega'_0}\n\\exp \\left\\{S(y)/k_B-\\beta^*\\sum^M_\\mu{\\cal E}_\\mu(y)\\right\\}\n\\nonumber\\\\\n&\\times&\\delta\\left(\\sum^M_\\mu M_\\mu - {\\cal M}_0\\right),\n\\label{pne2}\n\\end{eqnarray}\nwhere $\\Omega'_0$ is the corresponding normalization function.\nIn Eqn. (\\ref{pne2}) we have neglected a term\n$\\sum_\\mu \\log M_\\mu$ in front of $\\sum_\\mu{\\cal E}_\\mu (y)$. Note\nthat ${\\cal E}_\\mu$ is a first order function of its arguments\n$S_\\mu,M_\\mu,{\\cal V}_\\mu$ and, therefore, it is of order $M_\\mu$.\n\nChanging back to the original notation we write Eqn. (\\ref{pne2}) in\nthe form\n\n\\begin{eqnarray}\n\\rho^{\\rm eq}(\\{{\\bf R},M,S\\})\n&=&\\frac{1}{\\Omega'_0}\n\\exp \\left\\{\\sum^M_\\mu S_\\mu/k_B-\\beta^*{\\cal E}_\\mu(M_\\mu,S_\\mu,{\\cal V}_\\mu)\\right\\}\n\\nonumber\\\\\n&\\times&\\delta\\left(\\sum^M_\\mu M_\\mu - {\\cal M}_0\\right).\n\\label{pne3}\n\\end{eqnarray}\nWe see, therefore, that by integrating the momenta the\n``microcanonical'' form Eqn. (\\ref{ein}) becomes the ``canonical''\nform (\\ref{pne3}). \n\nWe are interested now on the distribution function $P(M_\\mu,S_\\mu)$ that\nthe particular cell $\\mu$ has the values $M_\\mu,S_\\mu$ for its mass\nand entropy, irrespective of the values of the rest of the variables\nin the system. We integrate (\\ref{pne3}) over the variables\nof all cells except $M_\\mu,S_\\mu$. \n\n\\begin{eqnarray}\nP(M_\\mu,S_\\mu) &=&\n\\frac{1}{\\Omega'_0} \\int d\\{{\\bf R}\\}d^{(M-1)}\\{M\\}d^{(M-1)}\\{S\\}\n\\nonumber\\\\\n&\\times&\n\\exp \\left\\{\\sum^M_{\\nu} S_\\nu/k_B\n-\\beta^*{\\cal E}_\\nu(M_\\nu,S_\\nu,{\\cal V}_\\nu)\\right\\}\n\\nonumber\\\\ \n&\\times&\n\\delta\\left(\\sum^M_\\mu M_\\mu - {\\cal M}_0\\right)\n\\nonumber\\\\\n&=&\n\\frac{{\\cal V}_0^M}{\\Omega'_0} \\int d\\{V\\}d^{(M-1)}\\{M\\}d^{(M-1)}\\{S\\}\n\\nonumber\\\\\n&\\times&\n\\exp \\left\\{\\sum^M_{\\nu} S_\\nu/k_B\n-\\beta^*{\\cal E}_\\nu(M_\\nu,S_\\nu, V_\\nu)\\right\\}\n\\nonumber\\\\ \n&\\times&\n\\delta\\left(\\sum^M_\\mu M_\\mu - {\\cal M}_0\\right)F(V_1,\\cdots,V_M)\n\\nonumber\\\\ \n\\label{pms}\n\\end{eqnarray}\nwhere we have introduced the identity\n\\begin{equation}\n\\int d\\{V\\}\\prod_\\mu^M\\delta(V_\\mu-{\\cal V}_\\mu(\\{{\\bf R}\\}))=1,\n\\label{iden}\n\\end{equation}\nand the function\n\\begin{equation}\nF(V_1,\\ldots,V_M)=\\frac{1}{V_T^M}\n\\int d\\{{\\bf R}\\}\\prod_\\mu^M\\delta(V_\\mu-{\\cal V}_\\mu(\\{{\\bf R}\\})).\n\\label{fvv}\n\\end{equation}\nThis function is the probability density that the the particles have\nthe particular distribution $V_1,\\ldots,V_M$ of volumes provided that\nthe distribution function of the positions is uniform. The\ncalculation of this function is difficult and we do not attempt to do\nit. Rather, we will assume that this function is highly peaked around\n$(\\overline{V},\\cdots, \\overline{V})$ in such a way that all the cells\nhave approximately the same volume $\\overline{V}={V}_T/M$. Under this\napproximation, Eqn. (\\ref{pms}) becomes\n\n\n\n\\begin{eqnarray}\nP(M_\\mu,S_\\mu) \n&=&\\frac{1}{\\Omega'_0}\n\\exp \\left\\{S_\\mu/k_B-\\beta^*{\\cal E}_\\mu(M_\\mu,S_\\mu, \\overline{V})\\right\\}\n\\nonumber\\\\\n&\\times&\\Phi({\\cal M}_0-M_\\mu),\n\\label{pm3}\n\\end{eqnarray}\nwhere we have introduced the function\n\n\\begin{eqnarray}\n\\Phi(X)&=&\\int d^{(M-1)}\\{M\\}d^{(M-1)}\\{S\\}\n\\nonumber\\\\\n&\\times&\n\\exp \\left\\{\\sum^M_{\\nu\\neq\\mu} S_\\nu/k_B\n-\\beta^*{\\cal E}_\\nu(M_\\nu,S_\\nu, \\overline{V})\\right\\}\n\\nonumber\\\\ \n&\\times&\n\\delta\\left(\\sum^M_{\\nu\\neq\\mu} M_\\nu - X\\right).\n\\nonumber\\\\ \n\\label{phix}\n\\end{eqnarray}\nThe functional form of $\\Phi(X)$ is very well approximated\nby an exponential. This can be seen by taking the derivative\n\n\n\\begin{eqnarray}\n\\Phi'(X)&=&-\\int d^{(M-1)}\\{M\\}d^{(M-1)}\\{S\\}\n\\nonumber\\\\\n&\\times&\n\\exp \\left\\{\\sum^M_{\\nu\\neq\\mu} S_\\nu/k_B\n-\\beta^*{\\cal E}_\\nu(M_\\nu,S_\\nu, \\overline{V})\\right\\}\n\\nonumber\\\\ \n&\\times&\n\\frac{\\partial}{\\partial M_\\sigma}\\delta\\left(\\sum^M_{\\nu\\neq\\mu} M_\\nu - X)\\right)\n\\nonumber\\\\ \n&=&-\\beta^*\\int d^{(M-1)}\\{M\\}d^{(M-1)}\\{S\\}\\mu(M_\\sigma,S_\\sigma, \\overline{V})\n\\nonumber\\\\\n&\\times&\n\\exp \\left\\{\\sum^M_{\\nu\\neq\\mu} S_\\nu/k_B\n-\\beta^*{\\cal E}_\\nu(M_\\nu,S_\\nu, \\overline{V})\\right\\}\n\\nonumber\\\\ \n&\\times&\n\\delta\\left(\\sum^M_{\\nu\\neq\\mu} M_\\nu - X)\\right),\n\\nonumber\\\\ \n\\label{phix2}\n\\end{eqnarray}\nwhere we have integrated by parts in the second equality and $\\sigma$\nis the label of any cell except $\\mu$. The most probable value of the\nintegrand in Eqn. (\\ref{phix2}) is the solution of\n\\begin{equation}\n\\mu(M_\\nu,S_\\nu,\\overline{V}) = \\overline{\\lambda},\n\\end{equation}\nwhere the chemical potential per unit mass equates $\\overline{\\lambda}$, a\nsuitable Lagrange multiplier that accounts for the mass conserving\ndelta function. We expect that when the number of variables $M$ is\nvery large, the integrand in Eqn. (\\ref{phix2}) becomes highly peaked\naround this most probable value. In this case, we have\n\n\\begin{equation}\n\\Phi'(X)\\approx -\\beta^*\\overline{\\lambda} \\Phi(X),\n\\end{equation}\nand therefore $\\Phi(X)$ is the exponential\n$\\exp\\{-\\beta^*\\overline{\\lambda}X\\}$.\n\nReturning back to Eqn. (\\ref{pm3}) we have finally,\n\\begin{equation}\nP(M_\\mu,S_\\mu) \n=\\frac{1}{Z}\n\\exp \\left\\{S_\\mu/k_B-\\beta^*\\left({\\cal E}(M_\\mu,S_\\mu, \\overline{V})\n-\\overline{\\lambda} M_\\mu\\right)\\right\\},\n\\label{pmsap}\n\\end{equation}\nwhere $Z$ is the appropriate normalization. Note that the argument of\nthe exponential is the Gibbs free energy with fixed values for the\ninverse temperature $\\beta^*$ and chemical potential per unit\nmass $\\overline{\\lambda}$.\n\n\n\\subsection{van der Waals fluid}\n\\label{f=0}\nWe will particularize the discussion of $P(M_\\mu,S_\\mu)$ by\nconsidering the fundamental equation ${\\cal E}(M_\\mu,S_\\mu,\n\\overline{V})$ for a van der Waals fluid. First we note that the\ninternal energy is a first order function of its variables and,\ntherefore,\n\\begin{equation}\n{\\cal E}(M_\\mu,S_\\mu, \\overline{V})\n=\\overline{V}\\epsilon(n_\\mu,s_\\mu),\n\\label{for}\n\\end{equation}\nwhere $n_\\mu = M_\\mu/(m_0\\overline{V})$ is the number density,\n$s_\\mu=S_\\mu/\\overline{V}$ is the entropy density, and $\\epsilon={\\cal\nE}/\\overline{V}$ is the internal energy density of cell\n$\\mu$. Therefore, from Eqn. (\\ref{pmsap}) we can obtain the\nprobability density that cell $\\mu$ has the values $n,s$ for its\nnumber density and entropy density. It is given by \n\\begin{equation}\nP(n,s) \n=\\frac{1}{Z}\n\\exp \\overline{V}\\left\\{s/k_B-\\beta^*(\\epsilon(n,s)\n-\\overline{\\lambda} m_0 n)\\right\\}.\n\\label{pden}\n\\end{equation}\n\nIt is convenient to use reduced units for the van der Waals fluid (see\nappendix \\ref{ap-vdW} for details of notation). In reduced units the\ndistribution function becomes\n\\begin{equation}\nP(\\tilde{n},\\tilde{s})=\n\\frac{1}{Z}\n\\exp \\{\\tilde{V}\\left\\{\n\\tilde{s}-\\tilde{\\beta}^{\\rm ext}(\\tilde{\\epsilon}(\\tilde{n},\\tilde{s})-\n\\tilde{\\mu}^{\\rm ext}\\tilde{n})\\right\\}\n\\label{ptil}\n\\end{equation}\n\nThe most probable value of this distribution function occurs\nat $\\tilde{n}^*,\\tilde{s}^*$ which are solutions of the equations\n\\begin{eqnarray}\n\\frac{\\partial\\tilde{\\epsilon}(\\tilde{n}^*,\\tilde{s}^*)}{\\partial \\tilde{n}}\n&=& \\tilde{\\mu}(\\tilde{n}^*,\\tilde{s}^*) = \\tilde{\\mu}^{\\rm ext}\n\\nonumber\\\\\n\\frac{\\partial\\tilde{\\epsilon}(\\tilde{n}^*,\\tilde{s}^*)}{\\partial \\tilde{s}}\n&=& \\tilde{T}(\\tilde{n}^*,\\tilde{s}^*) =\\frac{1}{\\tilde{\\beta}^{\\rm ext}}\n\\label{**}\n\\end{eqnarray}\nFor fixed $\\tilde{n}$, the relation between $\\tilde{T}$ and \n$\\tilde{s}$ is monotonic, whereas the chemical potential\nhas the form \n\\begin{equation}\n\\tilde{\\mu}=\\tilde{T}\n\\left(\\ln\\left(\\frac{\\tilde{n}}{3-\\tilde{n}}\\right)+\\frac{\\tilde{n}}{3-\\tilde{n}}\\right)\n-\\frac{9}{4}\\tilde{n} \n-\\frac{D}{2}\\tilde{T}\\ln\\left(\\frac{\\tilde {T}}{c}\\right).\n\\label{mu}\n\\end{equation}\n\n\n\\begin{figure}[ht]\n\\begin{center}\n%\\input{./FIGURAS/mu.tex}\n% GNUPLOT: LaTeX picture\n\\setlength{\\unitlength}{0.240900pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\sbox{\\plotpoint}{\\rule[-0.200pt]{0.400pt}{0.400pt}}%\n\\begin{picture}(1049,629)(0,0)\n\\font\\gnuplot=cmr10 at 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Chemical potential\n$\\tilde{\\mu}(\\tilde{n},\\tilde{T})$ as a function of $\\tilde{n}$ for\ndifferent $\\tilde{T}$. In descending order, $T=0.85,1.0, 1.15$. Observe\nthat for $\\tilde{T}<1$ the equation\n$\\tilde{\\mu}(\\tilde{n},\\tilde{T})=\\tilde{\\mu}^{\\rm ext}$ might have\nthree solutions for $\\tilde{n}$, depending on the actual value of\n$\\tilde{\\mu}^{\\rm ext}$. }\n\\end{center}\n\\end{figure}\n\\begin{figure}[ht]\n\\begin{center}\n%\\input{./FIGURAS/mu1.tex}\n% GNUPLOT: LaTeX picture\n\\setlength{\\unitlength}{0.240900pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\begin{picture}(1049,629)(0,0)\n\\font\\gnuplot=cmr10 at 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\nZoom of the previous figure for the isotherm $T=0.85$. The equal\narea construction gives a \nvalue $\\tilde{\\mu}^{\\rm ext}=-14.8971$ which produces two humps of\nequal height in $P(\\tilde{n},\\tilde{T})$.}\n\\end{center}\n\\end{figure}\nIn Fig. \\ref{figmu} we show the chemical potential for different\nvalues of the temperature. We observe that for $\\tilde{T}<1$ there are\nthree solutions for $n^*$ in the first equation in (\\ref{**}). This\nmeans that depending on $\\tilde{\\beta}^{\\rm ext}$ and\n$\\tilde{\\mu}^{\\rm ext}$, the distribution function\n$P(\\tilde{n},\\tilde{s})$ can present a bimodal form.\n\nDue to the particular functional form of the fundamental equation for\nthe van der Waals gas, it is more convenient to study the distribution\nfunction $P(\\tilde{n},\\tilde{T})$ instead of $P(\\tilde{n},\\tilde{s})$.\nThis function is computed in the appendix \\ref{ap-vdW}. \n\n\n\\begin{figure}[ht] \n\\begin{center} \n\\psfig{figure=figure3.ps,width=8cm,height=8cm}\n\\caption{\\label{bimodal1} The distribution function $P(\\tilde{n},\\tilde{T})$\nfor an external temperature $1/\\tilde{\\beta}^{\\rm ext}=0.85$. The external\nchemical potential is $\\tilde{\\mu}^{\\rm ext}=-14.85$. The typical volume\nof the cells is $\\tilde{V}=20$.}\n\\psfig{figure=figure4.ps,width=8cm,height=8cm}\n\\caption{\\label{bimodal2} The distribution function $P(\\tilde{n},\\tilde{T})$\nfor an external temperature $1/\\tilde{\\beta}^{\\rm ext}=0.85$. The external\nchemical potential is $\\tilde{\\mu}^{\\rm ext}=-14.8971$. The typical volume\nof the cells is $\\tilde{V}=20$. The two maxima have equal height.}\n\\psfig{figure=figure5.ps,width=8cm,height=8cm}\n\\caption{\\label{bimodal3} The distribution function $P(\\tilde{n},\\tilde{T})$\nfor an external temperature $1/\\tilde{\\beta}^{\\rm ext}=0.85$. The external\nchemical potential is $\\tilde{\\mu}^{\\rm ext}=-14.95$. The typical volume\nof the cells is $\\tilde{V}=20$.}\n\\end{center}\n\\end{figure}\n\n\n\nIn Figs. \\ref{bimodal1},\\ref{bimodal2},\\ref{bimodal3} we show the\ndistribution of number density and temperature\n$P(\\tilde{n},\\tilde{T})$ for a value of the external temperature\n$1/\\tilde{\\beta}^{\\rm ext}=0.85$ (below the unit critical temperature)\nand three different values of the external chemical potential\n$\\tilde{\\mu}^{\\rm ext}= -14.85,-14.8971,-14.95$. We observe the\npresence of two humps at the same value of the temperature\n$\\tilde{T}=0.85$ and different values of the density $\\tilde{n}$. The\nexistence of a bimodal structure in the distribution function\n$P(\\tilde{n},\\tilde{s})$ is a reflection of the gas-liquid transition,\nbecause a cell of size $\\tilde{V}$ can have a non-vanishing\nprobability of having two different values of the density. The\nexternal chemical potential $\\tilde{\\mu}^{\\rm ext}$ controls the\nrelative magnitude of the two maxima. For the particular value\n$\\tilde{\\mu}^{\\rm ext}=-14.8971$, at this value of the external\ntemperature, the two heights are equal. This value is called the\nchemical potential of coexistence and satisfies the equal area rule,\nsee Fig. \\ref{figmu1}.\n\nThe size $\\tilde{V}$ of the typical volume of a cell controls the\nsharpness of the distribution function. In Fig. \\ref{voldep1},\n\\ref{voldep2}, \\ref{voldep3} we show the distribution function\n$P(\\tilde{n},\\tilde{T}=0.85)$ for different values of $\\tilde{V}$.\nThe fluctuations become much smaller as $\\tilde{V}$ becomes larger,\nconsistent with the fact that in the thermodynamic limit fluctuations\nvanish. Another interesting property of the thermodynamic limit is\nthat the relative height of the to peaks in the distribution function\nof the density increases as $\\tilde{V}\\rightarrow \\infty$. Eventually\nonly one of the two peaks survives. This is true, whenever the two\npeaks are not equal in height. When they are exactly equal both spikes\ncoexist in the thermodynamic limit.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n%\\input{./FIGURAS/vol1.tex}\n% GNUPLOT: LaTeX picture\n\\setlength{\\unitlength}{0.240900pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\begin{picture}(1049,629)(0,0)\n\\font\\gnuplot=cmr10 at 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The distribution function\n$P(\\tilde{n},\\tilde{T})$ along the isotherm $\\tilde{T}=0.85$ for two\ndifferent values of $\\tilde{V}=20$ (solid line) and $60$ (dotted\nline). Here, the external chemical potential is $\\tilde{\\mu}^{\\rm\next}=-14.85$. Observe that the distribution becomes more peaked as the\ntypical volume $\\tilde{V}$ of the cells increases. The relative height\nincreases and, eventually, in the thermodynamic limit\n$\\tilde{V}\\rightarrow\\infty$ only one peak survives.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n%\\input{./FIGURAS/vol2.tex}\n% GNUPLOT: LaTeX picture\n\\setlength{\\unitlength}{0.240900pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\begin{picture}(1049,629)(0,0)\n\\font\\gnuplot=cmr10 at 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The distribution function\n$P(\\tilde{n},\\tilde{T})$ along the isotherm $\\tilde{T}=0.8971$ for two\ndifferent values of $\\tilde{V}=20$ (solid line) and $60$ (dotted\nline). The chemical potential is that\nof coexistence, $\\tilde{\\mu}^{\\rm ext}=-14.8971$. Because the heights\nare equal, as the typical volume $\\tilde{V}$ of the cells increases,\nthey remain equal at later times. In the thermodynamic limit two\nspikes localized at the gas and liquid densities remain. }\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\begin{center}\n%\\input{./FIGURAS/vol0.tex}\n% GNUPLOT: LaTeX picture\n\\setlength{\\unitlength}{0.240900pt}\n\\ifx\\plotpoint\\undefined\\newsavebox{\\plotpoint}\\fi\n\\begin{picture}(1049,629)(0,0)\n\\font\\gnuplot=cmr10 at 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The distribution function\n$P(\\tilde{n},\\tilde{T})$ at the critical point.\nThe critical isotherm $\\tilde{T}=1$ is plotted for\n$\\tilde{\\mu}^{\\rm ext}=-17.36$ for two\ndifferent values of $\\tilde{V}=20$ (solid line) and $60$ (dotted\nline). Note the broad\ndistribution of densities, which remain broad even in\nthe thermodynamic limit.}\n\\end{center}\n\\end{figure}\n\n\n\\section{Discussion}\n\\label{discusion}\nWe have presented a Lagrangian finite Voronoi volume discretization of\ncontinuum hydrodynamics and have shown that the discrete equations\nhave almost the GENERIC structure. The GENERIC structure can be\nrestored by simply adding a small term into the reversible part of the\ndynamics. Therefore, the obtained equations conserve mass, momentum,\nand energy, and the entropy is an strictly increasing function of time\nin the absence of fluctuations. Thermal fluctuations are consistently\nincluded which lead to the strict increase of the entropy functional\nand to the correct Einstein distribution function. The size of the\nfluctuations is given by the typical size of the volumes of the\nparticles, arguably scaling like the square root of this volume. The\nneed of incorporating thermal fluctuations in a particular system will\nbe determined by the external length scales that need to be\nresolved. For example, if sub-micron colloidal particles are\nconsidered, we need to resolve the size of the colloidal particle with\nfluid particles of size, say, an order of magnitude or two smaller\nthan the diameter of the colloidal particle. For these small volumes,\nfluctuations are important and lead to the Brownian motion of the\nparticle. A ping-pong ball, on the other hand requires fluid particles\nmuch larger, for which thermal fluctuations are negligible. Of course,\none could use a very large number of small fluid particles to deal\nwith the ping-pong ball, but in this case the (large) thermal\nfluctuations on each fluid particle average out among the (large)\nnumber of fluid particles.\n\nThe original formulations of Dissipative Particle Dynamics lack this\neffect of switching-off thermal fluctuations depending on the size of\nthe fluid particles. This is due to the fact that early formulations\ndid not include the volume and/or the mass of the particles as a\nrelevant dynamical variable. In this paper, we have extended the range\nof variables and have shown that a thermodynamically consistent\nDissipative Particle Dynamics model can be formulated. This model has\nthe same reversible part as the finite Voronoi volume model. Hence,\nthe usual conservative forces between dissipative particles are\nsubstituted by truly pressure forces. Therefore, prescribed\nthermodynamics can be given, without the need of ``reverse inference''\n(find out which conservative force will produce the desired equation\nof state \\cite{Groot97}). Even though the present DPD model has clear\nadvantages over previous DPD models, it is yet inferior to the finite\nVoronoi volume model presented in this paper. Note that the finite\nvolume is closely related to a discretization of Navier-Stokes\nequations, whereas the DPD model in this paper does not converge to\nthe Navier-Stokes equations, even though it displays correct\nhydrodynamic behaviour. This means that in order to simulate a fluid\nof a given viscosity, one has to tune the viscosity parameter of the\nDPD model by previous simulation runs. With the finite volume model\nthe true viscosity is given as input.\n\nWe have shown also that the Smoothed Particle Dynamics method has the\nGENERIC structure. However, we have pointed out several problems that\nfavour the use of the finite volume method of the present paper. For\nexample, in the Smoothed Particle Dynamics method we have noted that\nremnant forces appear even for the situation in which all particles\nare at rest with identical pressures. This effect can be made\narbitrarily small by increasing the overlapping coefficient. But this\nhas two drawbacks. The first one is of practical nature: when the\noverlapping coefficient is increased the number of interacting\nneighbours increases and, therefore, the computational time increases\ntoo. A second, more conceptual drawback is that in the large\noverlapping limit which, strictly speaking, is needed in order to\n``deduce'' the SPH equations for Navier-Stokes, the volume (or\ndensity) of the particles becomes a constant and the equations start\nloosing its sense. The pressures, which depend essentially on the\nvolumes, become a constant and the resulting equations cannot be\nclaimed to be a faithful discretization of Navier-Stokes equations.\nEven though hydrodynamic behaviour is displayed (momentum is conserved\nlocally) we expect that the thermodynamics and the transport\nproperties of the simulated fluid differ from the actual desired\nvalues.\n\nConcerning the GENERIC finite volume model of fluid particles\npresented in this paper, one of the most surprising realizations has\nbeen the fact that the mass of the fluid particles changes due to the\nreversible part of the dynamic. In Refs. \\cite{Espanol-prl99},\n\\cite{Flekkoy99} this was overlooked. Of course, we could, in\nprinciple, formulate a model with ${\\bf c}_{\\mu\\nu}= 0$ in the final\ndiscrete equations (\\ref{REVER}). This would result in a great\nsimplification of the equations without losing the desired GENERIC\nproperties. This approximation would imply that the mass of the\nparticles are constant. This spoils the equilibrium distribution\nfunction that would not be given by the Einstein distribution function\n(\\ref{ein}) but rather by\n\n\\begin{eqnarray}\n\\rho^{\\rm eq}(x)&=& \n\\frac{1}{\\Omega}\\prod_\\mu^M\n\\delta(M_\\mu(x)-M_\\mu^0)\\delta(E(x)-E_0)\\delta({\\bf P}(x)-{\\bf P}_0),\n\\nonumber\\\\\n&\\times &\\exp\\{k_B^{-1} S(x)\\}\n\\label{einmd0}\n\\end{eqnarray}\nwhere $M_\\mu^0$ is the initial value of the mass of particle $\\mu$.\nIn this case, not only total mass but also the individual masses are\ndynamical invariants of the discrete equations when ${\\bf\nc}_{\\mu\\nu}=0$. We note that we could not follow the same arguments\nthat lead to (\\ref{pmsap}) and which describe the liquid-gas\ncoexistence of a van der Waals fluid. This very same argument can be\napplied to the case of the SPH model discussed in section\n\\ref{sphmodel} which renders the SPH model unsuitable to discuss\ngas-liquid coexistence. From a theoretical point of view, the\npossibility of simulating liquid-gas coexistence with a particle model\nrepresents a stringent condition on the thermodynamic consistency of\nthe model proposed. Actually, we have learnt that the mass, and not\nthe volume, of the fluid particles should be included as an\nindependent dynamic variable if coexistence is to be described.\n\nFinally, we would like to note that the model presente does not\nexibit the phenomena of surface tension. Surface tension can\nbe easily included as an extra conservative contribution to the\nenergy function \\cite{Preprint}.\n\n\\section*{Acknowledgments}\nWe are grateful to E.G. Flekkoy and P.V. Coveney for exchange of\npreprints and useful comments. Helpful conversations and comments by\nH.C. \\\"Ottinger are greatly acknowledged. 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Espa\\~{n}ol, Physica A {\\bf 248}, 77 (1998).\n\\bibitem{callen} H.B. Callen {\\em Thermodynamics} (John Wiley \\& sons, New York 1960).\n\\bibitem{Preprint} {\\em Entropy for complex fluids} M. Serrano and\nP. Espa\\~nol, preprint.\n\\bibitem{handbook} Handbook of Chemistry and Physics, 59th Edition, \n R.C. Wheast Ed. (CRC Press Inc, 1978).\n\n\\end{references}\n\n\n\\section{Appendix: Voronoi properties}\n\\label{ap-vol}\nIn this appendix we explicitly compute the derivative of the volume of\ncell $\\mu$ with respect to the position ${\\bf R}_\\nu$ of cell $\\nu$,\nthis is\n\n\n\\begin{equation}\n{\\bf G}_{\\mu\\nu}=\\frac{\\partial{{\\cal V}_\\mu}}{{\\partial {\\bf R}_\\nu}}\n=\\frac{1}{\\sigma^2}\n\\int_{V_T}d{\\bf R}\n\\chi_\\mu({\\bf R})(\\delta_{\\mu\\nu}-\\chi_\\nu({\\bf R}))({\\bf R}-{\\bf R}_\\nu).\n\\label{omvor}\n\\end{equation}\nIts worth considering the cases $\\mu\\neq \\nu$ and $\\mu=\\nu$ explicitly.\n\\begin{eqnarray}\n\\mbox{\\boldmath $G$}_{\\mu\\nu}\n&=&-\\frac{1}{\\sigma^2}\n\\int_{V_T}d{\\bf R}\n\\chi_\\mu({\\bf R})\\chi_\\nu({\\bf R})({\\bf R}-{\\bf R}_\\nu),\n\\quad\\quad \\nu\\neq \\mu\n\\nonumber\\\\\n\\mbox{\\boldmath $G$}_{\\mu\\mu}\n&=&\n\\frac{1}{\\sigma^2}\n\\int_{V_T}d{\\bf R}\n\\chi_\\mu({\\bf R})(1-\\chi_\\mu({\\bf R}))({\\bf R}-{\\bf R}_\\mu)\n\\nonumber\\\\\n&=&\n\\sum_{\\nu\\neq \\mu}\\frac{1}{\\sigma^2}\n\\int_{V_T}d{\\bf R}\n\\chi_\\mu({\\bf R})\\chi_\\nu({\\bf R})({\\bf R}-{\\bf R}_\\mu)\n\\nonumber\\\\\n&=&-\\sum_{\\nu\\neq\\mu}\\mbox{\\boldmath $G$}_{\\nu\\mu},\n\\label{omvor2}\n\\end{eqnarray}\n\nIt is convenient to rewrite Eqn. (\\ref{omvor2}) for $\\nu\\neq\\mu$ as\n\n\\begin{eqnarray}\n\\mbox{\\boldmath $G$}_{\\mu\\nu}\n&=&-\\frac{1}{\\sigma^2}\n\\int_{V_T}d{\\bf R}\n\\chi_\\mu({\\bf R})\\chi_\\mu({\\bf R})\n\\left({\\bf R}-\\frac{{\\bf R}_\\mu+{\\bf R}_\\nu}{2}\\right)\n\\nonumber\\\\\n&&-{\\bf R}_{\\mu\\nu}\\frac{1}{2\\sigma^2}\n\\int_{V_T}d{\\bf R}\n\\chi_\\mu({\\bf R})\\chi_\\mu({\\bf R})\n\\label{omvor3}\n\\end{eqnarray}\nwith ${\\bf R}_{\\mu\\nu}={\\bf R}_\\mu-{\\bf R}_\\nu$. The first task is to\ncompute the limit $\\sigma\\rightarrow0$ for these two integrals. For\nthis reason, it is instructive to work out the actual forms of\n$\\chi_\\mu({\\bf R})$ and $\\chi_\\mu({\\bf R})\\chi_\\nu({\\bf R})$ for the\ncase that only two particles are present in the system, as has been\ndone by Flekkoy and Coveney \\cite{Flekkoy99}. Simple algebra leads to\n\\begin{eqnarray}\n\\chi_\\mu({\\bf R})\n&=&\\frac{1}{1+\\exp\\{-{\\bf R}_{\\mu\\nu}\n\\!\\cdot\\!({\\bf R}-({\\bf R}_\\mu+{\\bf R}_\\nu)/2)/\\sigma^2\\}}\n\\nonumber\\\\\n\\chi_\\mu({\\bf R})\\chi_\\nu({\\bf R})\n&=&\\frac{1}{4\\cosh^2({\\bf R}_{\\mu\\nu}\n\\!\\cdot\\!({\\bf R}-({\\bf R}_\\mu+{\\bf R}_\\nu)/2)/2\\sigma^2\\})}\n\\nonumber\\\\\n&&\\label{explicit}\n\\end{eqnarray}\nNote that $\\chi_{\\mu}({\\bf R})\\chi_{\\nu}({\\bf R})$ is different from\nzero only around the boundary of the Voronoi cells of particles\n$\\mu,\\nu$. In the limit of small $\\sigma$ this is even more true. The\nintegrals in (\\ref{omvor3}) therefore can be performed not over the\nfull volume ${V}_T$ but only over a region $\\partial_{\\mu\\nu}$\n``around'' the boundary of the $\\mu,\\nu$ cells. In this region, we can\nfurther substitute the expression of $\\chi_{\\mu}({\\bf\nr})\\chi_{\\nu}({\\bf R})$, which depend on the positions of all the\ncenter cells, by Eqn. (\\ref{explicit}), which depends only on the\nposition of the centers of cells $\\mu,\\nu$. Actually, we can make a\ntranslation from ${\\bf R}$ to ${\\bf R}'= {\\bf R}-({\\bf R}_\\mu+{\\bf\nR}_\\nu)/2$ (we put the origin exactly at the boundary between\ncells). We can also make a rotation in such a way that the $x$ axis is\nalong the line joining the cell centers. In this way, we can write\n\\begin{eqnarray}\n&&\\frac{1}{\\sigma^2}\n\\int_{V_T} d{\\bf R} \\chi_\\mu({\\bf R})\\chi_\\nu({\\bf R})\n\\nonumber\\\\\n&=&\n\\frac{1}{4\\sigma^2}\\int_{\\partial_{\\mu\\nu}}d{\\bf R}'\n\\frac{1}{\\cosh^2({\\bf R}'\\!\\cdot\\!{\\bf R}_{\\mu\\nu}/2\\sigma^2)}\n\\nonumber\\\\\n&=&\n\\frac{1}{4\\sigma^2}A_{\\mu\\nu}\\int_{-\\infty}^{\\infty}dx\n\\frac{1}{\\cosh^2(x R_{\\mu\\nu}/2\\sigma^2)}\n\\nonumber\\\\\n&=&\\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}\n\\label{area1}\n\\end{eqnarray}\nNote that $\\int_0^\\infty \\cosh^{-2}(x)dx=1$. Here, $A_{\\mu\\nu}$ is\nthe actual area of the boundary between Voronoi cells of particles\n$\\mu,\\nu$.\n\nIn a similar way, one computes the first integral in Eqn. (\\ref{omvor3})\nwith the result\n\n\\begin{equation}\n\\frac{1}{\\sigma^2}\n\\int_{V_T}d{\\bf R}\n\\chi_\\mu({\\bf R})\\chi_\\mu({\\bf R})\n\\left({\\bf R}-\\frac{{\\bf R}_\\mu+{\\bf R}_\\nu}{2}\\right)\n= \\frac{A_{\\mu\\nu}}{R_{\\mu\\nu}}{\\bf c}_{\\mu\\nu}\n\\label{area2}\n\\end{equation}\nwhere the vector ${\\bf c}_{\\mu\\nu}$ is, by definition, the position of\nthe center of mass of the face between Voronoi cells $\\mu,\\nu$ with\nrespect to the point $({\\bf R}_\\mu+{\\bf R}_\\nu)/2$. Collecting\nEqns. (\\ref{omvor3}),(\\ref{area2}) and (\\ref{area1}) one finally\nobtains\n\n\\begin{equation}\n\\mbox{\\boldmath $G$}_{\\mu\\nu} =-A_{\\mu\\nu}\\left(\\frac{{\\bf\nc}_{\\mu\\nu}}{R_{\\mu\\nu}}+\\frac{{\\bf e}_{\\mu\\nu}}{2}\\right)\n\\label{fin1}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n{\\bf e}_{\\mu\\nu}& =& \\frac{{\\bf R}_{\\mu\\nu}}{R_{\\mu\\nu}}\n\\nonumber\\\\\n{\\bf R}_{\\mu\\nu}&=&{\\bf R}_{\\mu}-{\\bf R}_\\nu\n\\nonumber\\\\\nR_{\\mu\\nu}&=& |{\\bf R}_{\\mu\\nu}|\n\\label{defis3}\n\\end{eqnarray}\n\nNote that $\\sum_\\nu {\\bf G}_{\\mu\\nu}=0$ due to Eqn. (\\ref{omvor2}).\nAlso $\\sum_\\mu {\\bf G}_{\\mu\\nu}=0$ because of Eqn. (\\ref{omvor})\nand the fact that total volume is a constant, independent of positions.\n\n\\section{Appendix: Dependent vs. independent variables in GENERIC}\n\\label{ap-dep}\nWhat happens if in the description of the state of a system one\nintroduces, perhaps without knowing it, variables which are not\nindependent of each other? We provide in this appendix the \nanswer to this question.\n\nLet us assume that the state of a system is described by\na set of {\\em independent} variables $x$. Assume, for the\nsake of simplicity, that the dynamics is purely\nreversible. The case $M\\neq 0$ follows along a similar way as\nthe one presented here for the reversible part.\n\nNow assume that the basic building blocks of GENERIC have\nthe following structure\n\n\\begin{eqnarray}\nE(x) &=& \\overline{E}(x,y(x))\n\\nonumber\\\\\nS(x) &=& \\overline{S}(x,y(x))\n\\nonumber\\\\\nL(x) &=& \\overline{L}(x,y(x))\n\\end{eqnarray}\nwhere $y(x)$ is a prescribed (vector) function of the sate $x$.\nFrom the chain rule one has\n\n\\begin{eqnarray}\n\\nabla E(x) = \\nabla_x\\overline{E}(x,y(x))+J^T(x)\\nabla_y\n\\overline{E}(x,y(x))\n\\label{derove}\n\\end{eqnarray}\nand similarly for $S(x)$ and $L(x)$. Here, the matrix $J(x)$ is\ndefined by $J(x)=\\partial y/\\partial x$.\n\nThe GENERIC reversible part of the dynamics is given by\n\\begin{eqnarray}\n\\dot{x}&=&L(x)\\nabla E(x)\n\\nonumber\\\\\n&=&\n\\overline{L}(x,y(x))\\nabla_x\\overline{E}(x,y(x))\n\\nonumber\\\\\n&+&\\overline{L}(x,y(x))J^T(x)\\nabla_y\\overline{E}(x,y(x))\n\\label{eq1}\n\\end{eqnarray}\nWe can consider the time derivative of the functions $y(x(t))$\nwhich, again through the chain rule, is given by\n\n\\begin{equation}\n\\dot{y}(x) = J(x)\\dot{x}\n\\label{eq2}\n\\end{equation}\nBy using now Eqn. (\\ref{eq1}) into (\\ref{eq2}) we can finally\ngroup both equation into the form\n\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\\dot{x}\\\\\n\\\\\n\\dot{y}\\\\\n\\end{array}\\right)\n=\n\\left(\n\\begin{array}{ccc}\n\\overline{L}&& \\overline{L}J^T \\\\\n\\\\\nJ\\overline{L}&& J\\overline{L}J^T \\\\\n\\end{array}\\right)\n\\left(\n\\begin{array}{c}\n\\nabla_x\\overline{E}\\\\\n\\\\\n\\nabla_y\\overline{E}\\\\\n\\end{array}\\right)\n\\label{compact0}\n\\end{equation}\nor, by introducing the obvious notation $z=(x,y(x))$,\n\\begin{equation}\n\\dot{z}={\\cal L}(z)\\nabla_z E(z)\n\\label{compact}\n\\end{equation}\nA very striking nicety is that the matrix ${\\cal L}(z)$ {\\em has all\nthe required properties for being a proper GENERIC reversible\nmatrix}. This is\n\\begin{eqnarray}\n{\\cal L}^T(z)&=& -{\\cal L}(z)\n\\nonumber\\\\\n{\\cal L}(z)\\nabla_z S(z) &=& 0\n\\nonumber\\\\\n\\nabla_z[{\\cal L}(z)\\nabla_z E(z)] &=& 0\n\\label{allprop}\n\\end{eqnarray}\nas can be easily checked from the properties of $L(x)$.\n\n\nNow, let us assume that we start describing the state of\na system by a vector $z=(x,y)$ and that the matrix $L(z)$\nhas the structure\n\n\\begin{equation}\n\\left(\n\\begin{array}{ccc}\n\\overline{L}(x,y)&& \\overline{L}(x,y)J^T(x) \\\\\n\\\\\nJ(x)\\overline{L}(x,y)&& J(x)\\overline{L}(x,y)J^T(x) \\\\\n\\end{array}\\right)\n\\label{compact2}\n\\end{equation}\nwith $J(x) = \\partial h/\\partial x$ for some set of functions\n $h(x)$. Then, we will have\n\n\\begin{equation}\n\\frac{d}{dt}y \n= \\frac{\\partial h}{\\partial x}(x) \\dot{x}=\\frac{d}{dt}h(x)\n\\label{148}\n\\end{equation}\nand, therefore, $y=h(x)$. Therefore, if in a given model the matrix\n $L(z)$ happens to have the structure given in (\\ref{compact2}), then,\n necessarily, the variables $y$ are dependent on the variables $x$.\n\nIn our formulation in Ref. \\cite{Espanol-prl99} we have assumed that \nthe volume evolves according to\n\\begin{equation}\n\\dot{\\cal V}_\\mu = \\sum_\\nu{\\bf \\Omega}_{\\mu\\nu}\\dot{\\bf R}_\\nu\n\\end{equation}\nwhich for most reasonable forms of ${\\bf \\Omega}_{\\mu\\nu}$ has\nthe form of Eqn. (\\ref{148}). Therefore, the dynamical equation\nfor the volume is nothing else than application of the chain\nrule and the volume cannot be considered as a truly independent\nvariable of positions.\n\n\n\\section{Appendix: Molecular ensemble}\n\\label{ap-mol}\nIn this appendix we want to compute explicitly the following integral\n\n\\begin{eqnarray}\n\\Phi({\\bf P}_0,E_0,M)&=&\\int d^{DM}{\\bf P}\n\\,\\,\\delta\\!\\left(\\sum_\\mu^M \\frac{{\\bf P}_\\mu^2}{2M_\\mu}-E_0\\right)\n\\nonumber\\\\\n&\\times&\n\\delta^D\\!\\left(\\sum_\\mu^M{\\bf P}_\\mu-{\\bf P}_0\\right)\n\\nonumber\\\\\n&=&\n\\prod_\\mu^M(2M_\\mu)^{D/2}\\int d^{DM}{\\bf P}\n\\,\\,\\delta\\!\\left(\\sum_\\mu^M {\\bf P}_\\mu^2-E_0\\right)\n\\nonumber\\\\\n&\\times&\n\\delta^D\\!\\left(\\sum_\\mu^M(2M_\\mu)^{1/2}\n{\\bf P}_\\mu-{\\bf P}_0\\right)\n\\label{4}\n\\end{eqnarray}\nwhich appears repeatedly when computing molecular averages. The\nequation $\\sum_\\mu^M(2M_\\mu)^{1/2}{\\bf P}_\\mu={\\bf P}_0$ are actually D\nequations (one for each component of the momentum) which define D\nplanes in $R^{DM}$. The integral in (\\ref{4}) is actually over a\nsubmanifold which is the intersection of the D planes with the surface\nof a $DM$ dimensional sphere of radius $E_0^{1/2}$. This\nintersection will be also a sphere, which will be now of smaller\nradius and also of smaller dimension, $D(M-1)$.\n\nIn order to compute (\\ref{4}), we change to the following notation\n\n\n\\begin{eqnarray}\n\\mbox{\\boldmath ${\\cal P}$}\n&=&(p_1^x,\\ldots,p_M^x,p_1^y,\\ldots,p_M^y,p_1^z,\\ldots,p_M^z)\n\\nonumber\\\\\n\\mbox{\\boldmath ${\\cal C}$}_x &=&((2M_1)^{1/2},\\ldots,(2M_M)^{1/2},0,\\ldots,0,0,\\ldots,0)\n\\nonumber\\\\\n\\mbox{\\boldmath ${\\cal C}$}_y &=&(0,\\ldots,0,(2M_1)^{1/2},\\ldots,(2M_M)^{1/2},0,\\ldots,0)\n\\nonumber\\\\\n\\mbox{\\boldmath ${\\cal C}$}_z &=&(0,\\ldots,0,0,\\ldots,0,(2M_1)^{1/2},\\ldots,(2M_M)^{1/2})\n\\label{defis}\n\\end{eqnarray}\nNote that these vectors satisfy\n $\\mbox{\\boldmath ${\\cal C}$}_x\\!\\cdot\\!\\mbox{\\boldmath ${\\cal C}$}_y=0$\n $\\mbox{\\boldmath ${\\cal C}$}_y\\!\\cdot\\!\\mbox{\\boldmath ${\\cal C}$}_z=0$\n $\\mbox{\\boldmath ${\\cal C}$}_z\\!\\cdot\\!\\mbox{\\boldmath ${\\cal C}$}_x=0$. \nWith these vectors so defined, Eqn. (\\ref{4}) becomes\n\\begin{eqnarray}\n\\Phi({\\bf P}_0,E_0,M)\n&=&\\int d^{DM}\\mbox{\\boldmath ${\\cal P}$}\n\\delta\\!\\left(\\mbox{\\boldmath ${\\cal P}$}^2-E\\right)\n\\,\\,\\delta\\!\\left(\n\\mbox{\\boldmath ${\\cal C}$}_x\\!\\cdot\\!\n\\mbox{\\boldmath ${\\cal P}$}-P_0^x\\right)\n\\nonumber\\\\\n&\\times&\n\\,\\,\\delta\\!\\left(\n\\mbox{\\boldmath ${\\cal C}$}_y\\!\\cdot\\!\n\\mbox{\\boldmath ${\\cal P}$}-P_0^y\\right)\n\\,\\,\\delta\\!\\left(\n\\mbox{\\boldmath ${\\cal C}$}_z\\!\\cdot\\!\n\\mbox{\\boldmath ${\\cal P}$}-P_0^z\\right)\n\\nonumber\\\\\n&\\times& \\prod_\\mu^M(2M_\\mu)^{D/2}\n\\label{4b}\n\\end{eqnarray}\n\n\nNow we consider the following change of variables\n\n\\begin{equation}\n\\mbox{\\boldmath ${\\cal P}$}'=\\mbox{\\boldmath ${\\cal P}$}-\n\\left(\n\\frac{\\mbox{\\boldmath ${\\cal C}$}_x}{|\\mbox{\\boldmath ${\\cal C}$}_x|^2}\n P_0^x\n+\n\\frac{\\mbox{\\boldmath ${\\cal C}$}_y}{|\\mbox{\\boldmath ${\\cal C}$}_y|^2} P_0^y\n+\n\\frac{\\mbox{\\boldmath ${\\cal C}$}_z}{|\\mbox{\\boldmath ${\\cal C}$}_z|^2} P_0^z\n\\right)\n\\label{tra}\n\\end{equation}\nwhich is simply a translation and has unit Jacobian. Simple algebra \nleads to\n\n\\begin{eqnarray}\n\\Phi({\\bf P}_0,E_0,M)&=& \\prod_\\mu^M(2M_\\mu)^{D/2}\n\\int d^{DM}\\mbox{\\boldmath ${\\cal P}$}'\n\\,\\,\\delta\\!\\left(\n\\frac{{\\mbox{\\boldmath ${\\cal P}$'}^2}}{2m}\n-U_0\\right)\n\\nonumber\\\\\n&\\times&\n\\delta\\!\\left(\\mbox{\\boldmath ${\\cal\nC}$}_x\\!\\cdot\\!\\mbox{\\boldmath ${\\cal P}$}'\\right)\n\\,\\,\\delta\\!\\left(\\mbox{\\boldmath ${\\cal\nC}$}_y\\!\\cdot\\!\\mbox{\\boldmath ${\\cal P}$}'\\right)\n\\,\\,\\delta\\!\\left(\\mbox{\\boldmath ${\\cal\nC}$}_z\\!\\cdot\\!\\mbox{\\boldmath ${\\cal P}$}'\\right)\n\\label{4c}\n\\end{eqnarray}\nwhere we have introduced the total internal energy\n\n\\begin{equation}\nU_0=\\left(E_0-\\frac{{\\bf P}_0^2}{2{\\cal M}_0}\\right)\n\\label{u}\n\\end{equation}\nwhere ${\\cal M}_0=\\sum_\\mu M_\\mu$ is the total mass.\n\nWe now consider a second change of variables $\\mbox{\\boldmath ${\\cal\nP}$}''={\\bf \\Lambda}\\!\\cdot\\!\\mbox{\\boldmath ${\\cal P}$}'$ through a\nrotation ${\\bf \\Lambda}$ such that\n\n\\begin{eqnarray}\n{\\bf \\Lambda}\\mbox{\\boldmath ${\\cal C}$}_x \n&=&(2{\\cal M}_0)^{1/2}(1,\\ldots,0,0,\\ldots,0,0,\\ldots,0)\n\\nonumber\\\\\n{\\bf \\Lambda}\\mbox{\\boldmath ${\\cal C}$}_y \n&=&(2{\\cal M}_0)^{1/2}(0,\\ldots,0,1,\\ldots,0,0,\\ldots,0)\n\\nonumber\\\\\n{\\bf \\Lambda}\\mbox{\\boldmath ${\\cal C}$}_z \n&=&(2{\\cal M}_0)^{1/2}(0,\\ldots,0,0,\\ldots,0,1,\\ldots,0)\n\\label{defis2b}\n\\end{eqnarray}\nIt is always possible to find a matrix ${\\bf \\Lambda}$ that satisfies\nEqns. (\\ref{defis2b}). For example, consider a block diagonal matrix\nmade of three identical blocks of size $M\\times M$. Then assume that\neach block is the same orthogonal matrix which transforms the vector\n$((2M_1)^{1/2},(2M_2)^{1/2},\\ldots,(2M_M)^{1/2})$ into $(2{\\cal\nM}_0)^{1/2}(1,0,\\ldots,0)$. After the rotation (which has unit\nJacobian and leaves the modulus of a vector invariant) the integral\n(\\ref{4c}) becomes\n\n\n\\begin{eqnarray}\n\\Phi&=&\n\\prod_\\mu^M(2M_\\mu)^{D/2}\n\\int d^{DM}\\mbox{\\boldmath ${\\cal P}$}''\n\\delta\\!\\left(\\mbox{\\boldmath ${\\cal P}$}''^2-U_0\\right)\n\\nonumber\\\\\n&\\times&\n\\,\\,\\delta((2{\\cal M}_0)^{1/2}p_1''^x)\n\\,\\,\\delta((2{\\cal M}_0)^{1/2}p_1''^y)\n\\,\\,\\delta((2{\\cal M}_0)^{1/2}p_1''^z)\n\\nonumber\\\\\n&=&\\prod_\\mu^M(2M_\\mu)^{D/2}\n\\int d^{D(M-1)}\\mbox{\\boldmath ${\\cal P}$}''\n\\delta\\!\\left(\\mbox{\\boldmath ${\\cal P}$}''^2-U_0\\right).\n\\label{10}\n\\end{eqnarray}\n\n\nWe compute now the integral over the sphere in Eqn. (\\ref{10}) by\nusing that the integral of an arbitrary function $F({\\bf x}) = f(|{\\bf\nx}|) $ that depends on ${\\bf x}$ only through its modulus $|{\\bf x}|$\ncan be computed by changing to polar coordinates\n\\begin{equation}\n\\int F({\\bf x})d^M{\\bf x}=\\omega_M\\int_0^{\\infty} f(r) r^{M-1}dr.\n\\label{0}\n\\end{equation}\nThe numerical factor $\\omega_M$, which comes from the integration of\nthe angles, can be computed by considering the special case when\n$f(r)$ is a Gaussian. The result is\n\\begin{equation}\n\\omega_M = 2 \\frac{\\pi^{M/2}}{\\Gamma(M/2)}\n\\label{1b}\n\\end{equation}\nBy using Eqn. (\\ref{0}), Eqn. (\\ref{10}) becomes\n\n\\begin{eqnarray}\n\\Phi({\\bf P}_0,E_0,M)\n&=& \\prod_\\mu^M(2M_\\mu)^{D/2}\\omega_{D(M-1)}\n\\nonumber\\\\\n&\\times&\\int dp \\,\\,p^{D(M-1)-1}\n\\,\\,\\delta\\!\\left(p^2-U\\right)\n\\label{10b}\n\\end{eqnarray}\nWe need now the property\n\n\\begin{equation}\n\\delta(f(x))=\\sum_\\mu\\frac{\\delta(x-x_\\mu)}{|f'(x_\\mu)|}\n\\end{equation}\nwhere $x_\\mu$ are the zeros of $f(x_\\mu)=0$. For the case of Eqn. (\\ref{10b}) we\nhave $f(x)=x^2-U$, $x_\\mu =\\pm (U)^{1/2}$ and $f'(x)=2x$. Therefore,\n\n\\begin{equation}\n\\Phi({\\bf P}_0,E_0,M) = \\frac{1}{2}\\omega_{D(M-1)}\nU_0^{\\frac{D(M-1)-2}{2}}\\prod_\\mu^M(2M_\\mu)^{D/2}\n\\label{11}\n\\end{equation}\n\n\\section{Appendix: van der Waals fluid}\n\\label{ap-vdW}\nThe free energy $F$ of a system has as natural variables the number\nof particles $N$, the temperature $T$ and the volume $V$, this\nis $F=F(N,T,V)$. Its derivatives are given by the pressure, the\nentropy and the chemical potential\n\\begin{eqnarray}\n-P &=& \\left. \\frac{\\partial F}{\\partial V}\\right|_{T,N}\n\\nonumber\\\\\n-S &=& \\left. \\frac{\\partial F}{\\partial T}\\right|_{V,N}\n\\nonumber\\\\\n\\mu &=& \\left. \\frac{\\partial F}{\\partial N}\\right|_{T,V}\n\\label{if}\n\\end{eqnarray}\nBecause the free energy is a first order function of the\nextensive variables $N,V$, we have\n\\begin{equation}\nF(N,T,V) = V f(n,T)\n\\label{fex}\n\\end{equation}\nwhere $n$ is the number density and $f(n,T)$ is the free energy\ndensity. The property (\\ref{fex}) implies in Eqns. (\\ref{if})\n\\begin{eqnarray}\n-P(n,T) &=& f(n,T) -\\mu n\n\\nonumber\\\\\ns&=&-\\frac{\\partial f}{\\partial T}(n,T)\n\\nonumber\\\\\n\\mu &=& \\frac{\\partial f}{\\partial n}(n,T)\n\\label{ifd}\n\\end{eqnarray}\nwhere $s=S/V$ is the entropy density.\n\nFor a van der Waals gas, the free energy density is given by\n\\begin{equation}\nf(n,T) = -k_BTn\\left(1+\\ln\\left(\\frac{1-nb}{n\\Lambda^D(T)}\\right)\\right)-an^2\n\\label{fnt}\\end{equation}\nwhere the thermal wavelength is defined by\n\n\\begin{equation}\n\\Lambda(T) = \\frac{h}{(2\\pi m_0 k_BT)^{1/2}}\n\\label{lamb}\\end{equation}\nHere, $h$ is Planck's constant, $m_0$ is the mass of a molecule, and\n$k_B$ is Boltzmann constant. The constants $a,b$ are the attraction parameter\nand the excluded volume, respectively.\n\nWith Eqns. (\\ref{ifd}) plus the well-known relationship $f=\\epsilon - Ts$\nwhere $\\epsilon$ is the internal energy density, we easily arrive at\nthe following relations\n\n\\begin{eqnarray}\n\\epsilon(n,s) &=& \\frac{D}{2}k_B T(n,s)n - an^2\n\\nonumber\\\\\nk_BT(n,s) &=& \\left(\\frac{h^2}{2\\pi m_0}\\right)\n \\left(\\frac{n}{1-nb}\\right)^{2/D}\\exp \\left\\{\\frac{2s}{Dk_B n}\n-\\frac{D+2}{D}\\right\\}\n\\nonumber\\\\\nP(n,s) &=& \\frac{k_BT n}{1-nb}-an^2\n\\nonumber\\\\\n\\mu(n,s)&=&\nk_BT\\left(\\frac{nb}{1-nb}+ \\ln\\left(\\frac{nb}{1-nb}\\right)\\right)\n\\nonumber\\\\\n&-&k_BT\\ln\\left( \\frac{b}{\\Lambda^D(T)}\\right)-2an\n\\label{finden}\n\\end{eqnarray}\nwhich give the fundamental equation $\\epsilon(n,s)$ and the three\nequations of state in terms of the variables $n,s$.\n\nAs it is customary for the van der Waals fluid, we introduce reduced variables\n\\begin{eqnarray}\n\\tilde{T}&=& \\frac{27 b}{8a} k_B T,\n\\nonumber\\\\\n\\tilde{P}&=& \\frac{27 b^2}{a}P.\n\\nonumber\\\\\n\\tilde{\\mu}&=& \\frac{27 b}{8a}\\mu.\n\\nonumber\\\\\n\\tilde{n} &=& 3bn,\n\\nonumber\\\\\n\\tilde{s}&=& \\frac{3bs}{k_B}\n\\nonumber\\\\\n\\tilde{\\epsilon}&=& \\frac{27 b}{8a}3b\\epsilon\n\\label{redapp}\n\\end{eqnarray}\n\nIn reduced units we have that the fundamental equation becomes\n\n\\begin{equation}\n\\tilde{\\epsilon}(\\tilde{n},\\tilde{s}) \n= \\frac{D}{2}\\tilde{T}(\\tilde{n},\\tilde{s}) \\tilde{n}-\\frac{9}{8}\\tilde{n}^2\n\\label{ep}\n\\end{equation}\nand the three equations of state are\n\\begin{eqnarray}\n\\tilde{T}(\\tilde{n},\\tilde{s}) &=& c \\exp\\frac{2}{D}\n\\left\\{\\frac{\\tilde{s}}{\\tilde{n}}-\\frac{D+2}{2}+\n\\ln\\left(\\frac{\\tilde{n}}{3-\\tilde{n}}\\right)\\right\\}\n\\nonumber\\\\\n\\tilde{P}(\\tilde{n},\\tilde{s})\n&=& 8\\frac{\\tilde{T}\\tilde{n}}{3-\\tilde{n}}-3\\tilde{n}^2\n\\nonumber\\\\\n\\tilde{\\mu}(\\tilde{n},\\tilde{T})\n&=&\\tilde{T}\\left(\\ln\\left(\\frac{\\tilde{n}}{3-\\tilde{n}}\\right)+\n\\frac{\\tilde{n}}{3-\\tilde{n}}\\right)-\\frac{9}{4}\\tilde{n}\n-\\tilde{T}\\ln\\left(\\frac{\\tilde{T}}{c}\\right)^{D/2}\n\\label{eqest}\n\\end{eqnarray}\nIn this equation, we have introduced the the constant $c$ which\n depends on microscopic parameters through\n\n\\begin{equation}\nc=\\frac{27 b^{(D-2)/D}h^2}{16\\pi m_0 a}=4.836 \\times 10^{-5}\n\\label{c}\n\\end{equation}\nwhere we have used the values corresponding to water in $D=3$\ndimensions, $a = 1.5262 \\times 10^{-48} m^5 kg/s^2$, $b= 5.0622 \\times\n10^{-29} m^3$, $m_0=2.991\\times 10^{-26} kg $, $h=1.054 \\times\n10^{-34} Js$ \\cite{handbook}. This constant appears in the the\ndimensionless quantity\n\\begin{equation}\n\\frac{b}{\\Lambda^D(T)} = \\left(\\frac{\\tilde{T}}{c}\\right)^{D/2}\n\\label{l}\n\\end{equation}\n\nNote that in reduced units we still have the relationships\n\\begin{eqnarray}\n\\frac{\\partial\\tilde{\\epsilon}(\\tilde{n},\\tilde{s})}{\\partial \\tilde{n}}\n&=& \\tilde{\\mu}(\\tilde{n},\\tilde{s})\n\\nonumber\\\\\n\\frac{\\partial\\tilde{\\epsilon}(\\tilde{n},\\tilde{s})}{\\partial \\tilde{s}}\n&=& \\tilde{T}(\\tilde{n},\\tilde{s})\n\\label{intred}\n\\end{eqnarray}\n\nLet us focus now on the distribution function (\\ref{pden})\nwhich in reduced units becomes\n\n\\begin{equation}\nP(\\tilde{n},\\tilde{s})=\n\\frac{1}{Z}\n\\exp \\{\\tilde{V}\\left\\{\n\\tilde{s}-\\tilde{\\beta}^{\\rm ext}(\\tilde{\\epsilon}(\\tilde{n},\\tilde{s})-\n\\tilde{\\mu}^{\\rm ext}\\tilde{n}\\right\\}\n\\label{apptil}\n\\end{equation}\nwhere $\\tilde{V}=\\overline{V}/3b$, $\\tilde{\\beta}^{\\rm\next}=8a\\beta/27b$, is the reduced inverse external temperature and\n$\\tilde{\\mu}^{\\rm ext}= \\overline{\\lambda}m_0 27b/8a$ is the reduced\nexternal chemical potential.\n\nDue to the particular functional form of the fundamental equation for\nthe van der Waals gas, it is more convenient to study the distribution\nfunction of $\\tilde{n},\\tilde{T}$ instead of $\\tilde{n},\\tilde{s}$.\nThe Jacobian of the transformation is $\\frac{D\\tilde{n}}{2\\tilde{T}}$ and,\ntherefore, the new distribution function is\n\\begin{equation}\nP(\\tilde{n},\\tilde{T})=\\frac{\\tilde{n}}{Z\\tilde{T}}\n\\exp\\tilde{V}\\{\\tilde{s}(\\tilde{n},\\tilde{s})-\n\\tilde{\\beta}^{\\rm ext}\n(\\tilde{\\epsilon}(\\tilde{n},\\tilde{s})-\\tilde{\\mu}^{\\rm ext}\\tilde{n})\\}\n\\label{appnt}\n\\end{equation}\n\n\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002026.extracted_bib",
"string": "\\bibitem{Hoogerbrugge92} \nP.J. Hoogerbrugge and J.M.V.A. Koelman, Europhys. Lett. {\\bf 19}, 155\n(1992)\n\n\\bibitem{Espanol95}\nP. Espa{\\~{n}}ol and P. Warren, Europhys. Lett. {\\bf 30}, 191 (1995).\nP. Espa{\\~{n}}ol, Phys. Rev. E, {\\bf 52}, 1734 (1995).\n\n\\bibitem{Espanol98} \nP. Espa\\~{n}ol, Phys. Rev. E, {\\bf 57}, 2930 (1998).\n\n\\bibitem{Marsh97} \nC. Marsh, G. Backx, and M.H. Ernst, Europhys. Lett. {\\bf 38}, 411\n(1997). C. Marsh, G. Backx, and M.H. Ernst, Phys. Rev. E {\\bf 56},\n1976 (1997).\n\n\\bibitem{Bonet97}\nJ. Bonet Aval\\'os and A.D. Mackie, Europhys. Lett. {\\bf 40}, 141\n(1997). P. Espa\\~nol, Europhysics Letters {\\bf 40}, 631\n(1997). M. Ripoll, P. Espa\\~nol, and M.H. Ernst, International Journal\nof Modern Physics C, {\\bf 9}, 1329 (1998).\n\n\\bibitem{Koelman93} \nJ.M.V.A. Koelman and P.J. Hoogerbrugge, Europhys. Lett. {\\bf 21}, 369\n(1993).\n\n\\bibitem{Boek97}\nE.S. Boek, P.V. Coveney, H.N.W. Lekkerkerker, and P. van der Schoot,\nPhys. Rev. E {\\bf 55}, 3124 (1997).\n\n\\bibitem{Schlijper95}\nA.G. Schlijper, P.J. Hoogerbrugge, and\nC.W. Manke, J. Rheol. {\\bf 39}, 567 (1995).\n\n\\bibitem{Groot97}\nR.D. Groot and P.B. Warren, J. Chem. Phys. {\\bf 107}, 4423\n(1997). R.D. Groot and T.J. Madden, J. Chem. Phys. {\\bf 108}, 8713\n(1998). R.D. Groot, T.J. Madden, and D.J. Tildesley,\nJ. Chem. Phys. {\\bf 110}, 9739, (1999).\n\n\\bibitem{Coveney96}\nP.V. Coveney and K. Novik, Phys. Rev. E {\\bf 54}, 5134\n(1996). S.I. Jury, P. Bladon, S. Krishna, and M.E. Cates,\nPhys. Rev. {\\bf 59} R2535 (1999). K. Novik and P.V. Coveney,\nPhys. Rev. E {\\bf 61}, 435 (2000).\n\n\\bibitem{Dzwinel99} \nW. Dzwinel and D.A. Yuen, Molecular Simulations, {\\bf 22}, 369 (1999).\n\n\\bibitem{Monaghan92} \nL.B. Lucy, Astron. J. {\\bf 82}, 1013 (1977).\nJ.J. Monaghan, Annu. Rev. Astron. Astrophys. {\\bf 30}, 543 (1992).\n\n\\bibitem{Takeda94}\nH. Takeda, S.M. Miyama, and M. Sekiya, Prog. Theor. Phys. {\\bf 92},\n939 (1994). H.A. Posch, W.G. Hoover, and O. Kum, Phys. Rev. E {\\bf\n52}, 1711 (1995).\n\n\\bibitem{Kum95}\nO. Kum, W.G. Hoover, and H.A. Posch, Phys. Rev. E {\\bf 52}, 4899\n(1995).\n\n\\bibitem{Espanol-prl99}\nP. Espa\\~nol, M. Serrano, and H.C. \\\"Ottinger, Phys. Rev. Lett. \n{\\bf 83}, 4542 (1999).\n\n\\bibitem{Eulerian} {\\em Eulerian} implementations of fluctuating\nhydrodynamics have been considered by A.L. Garcia, M.M. Mansour,\nG.C. Lie, and E. Clementi, J. Stat. Phys. {\\bf 47}, 209 (1987) and\nH.-P. Breuer and F. Petruccione, Physica {\\bf A 192}, 569 (1993).\n\n\\bibitem{Bedeaux74} \nP. Mazur and D. Bedeaux, Physica {\\bf 76}, 235 (1974).\n\n\\bibitem{Guemez88} J. G\\\"u\\'emez, S. Velasco, and A.C. Hern\\'andez,\nPhysica A {\\bf 152}, 226, (1988), Physica A {\\bf 152}, 243, (1988),\nPhysica A {\\bf 153}, 299, (1988). M. Rovere, D.W. Hermann, and K. Binder,\nEurophys. Lett. {\\bf 6} 585 (1988).\n\n\\bibitem{Swift95}M.R. Swift, W.R. Osborn, and J.M. Yeomans, Phys. Rev. \nLett. {\\bf 75}, 830 (1995). M.R. Swift, E. Orlandini, W.R. Osborn, and\nJ.M. Yeomans, Phys. Rev. E {\\bf 54 }, 5041 (1996).\n\n\\bibitem{Luo98}Li-Shi Luo, Phys. Rev. Lett. {\\bf 81}, 1618 (1998). \nA.D. ANgelopoulos, V.N. Paunov, V.N. Burganos, and A.C. Payatakes,\nPhys. Rev. E {\\bf 57}, 3237 (1998).\n\n\\bibitem{Bird94} G.A. Bird, \n{\\em Molecular Gas Dynamics and the Direct Simulation\nof Gas flows}, (Clarendon, Oxford 1994).\n\n\\bibitem{Alexander97} F. Alexander, A. Garcia and B. Alder, Physica A\n{\\bf 240} 196 (1997). N. Hadjiconstantinou, A. Garcia, and B. Alder, preprint\n``The Surface Properties of a van der Waals Fluid''.\n\n\\bibitem{Flekkoy99} E.G. Flekkoy and P.V. Coveney, \nPhys. Rev. Lett. {\\bf 83}, 1775 (1999). \nE.G. Flekkoy, P.V. Coveney, and G. De Fabritiis, preprint\n\n\\bibitem{Hietel00} D. Hietel, K. Steiner and J. Strucktmeier, Mathematical \nModels and Methods in Applied Sicence, in press.\n\n\\bibitem{Yuan98} X.-F. Yuan and M. Doi, Colloid and Surfaces A: \nPhysicochemical and Engineering Aspects {\\bf 144}, 305 (1998).\n\n\\bibitem{Yuan93}\nX.-F. Yuan, R.C. Ball, and S.F. Edwards, J. Non-Newtonian Fluid Mech. {\\bf 46}, 331 (1993).\nX.-F. Yuan, R.C. Ball, and S.F. Edwards, J. Non-Newtonian Fluid Mech. {\\bf 54}, 423 (1994).\nX.-F. Yuan, R.C. Ball, J. Chem. Phys. {\\bf 101}, 9016 (1994),\nX.-F. Yuan, S.F. Edwards, J. Non-Newtonian Fluid Mech. {\\bf 60}, 335 (1995).\n\n\\bibitem{Zwanzig60} R. Zwanzig, J. Chem. Phys. {\\bf 33}, 1338 (1960),\nH. Mori, Prog. theor. Phys. {\\bf 33}, 423 (1965).\n\n\\bibitem{generic} M. Grmela and H.C. \\\"Ottinger, Phys. Rev. E {\\bf\n56}, 6620 (1997). H.C. \\\"Ottinger and M. Grmela, Phys. Rev. E {\\bf\n56}, 6633 (1997). H.C. \\\"Ottinger, Phys. Rev. E {\\bf 57}, 1416 (1998).\n\n\\bibitem{gen-applyed} H.C.~\\\"Ottinger, J. Non-Equilib.\\ Thermodyn.\\\n{\\bf 22}, 386 (1997). H.C.~\\\"Ottinger, Physica A {\\bf 254}, 433\n(1998).\n\n\\bibitem{mixing} J. Espa\\~{n}ol and F.J. de la Rubia,\nPhysica {\\bf A 187}, 589 (1992).\n\n\\bibitem{Gardiner83} C.W. Gardiner, {\\em Handbook of Stochastic Methods},\n(Springer Verlag, Berlin, 1983). H. C. \\\"{O}ttinger, {\\em Stochastic\nProcesses in Polymeric Fluids}, Springer-Verlag, 1996.\n\n\n\\bibitem{Voronoi} An introduction to Voronoi tessellations and related\ncomputational geometry concepts can be found in \nhttp://www.voronoi.com. A particularly useful Java applet can be found in \nhttp://wwwpi6.fernuni-hagen.de/Geometrie-Labor/ VoroGlide/index.html.en .\n\n\\bibitem{deGroot84} S.R. de Groot and P. Mazur, {\\em Non-equilibrium\nthermodynamics} (North-Holland Publishing Company, Amsterdam, 1962).\n\n\\bibitem{Landau59} L.D. Landau and E.M. Lifshitz, {\\em Fluid Mechanics}\n(Pergamon Press, 1959).\n\n\n\\bibitem{Espanol-PA98} P. Espa\\~{n}ol, Physica A {\\bf 248}, 77 (1998).\n\n\\bibitem{callen} H.B. Callen {\\em Thermodynamics} (John Wiley \\& sons, New York 1960).\n\n\\bibitem{Preprint} {\\em Entropy for complex fluids} M. Serrano and\nP. Espa\\~nol, preprint.\n\n\\bibitem{handbook} Handbook of Chemistry and Physics, 59th Edition, \n R.C. Wheast Ed. (CRC Press Inc, 1978).\n\n"
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cond-mat0002027
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Multi-scaling properties of truncated L\'evy flights
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[
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"author": "Hiroya Nakao"
}
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Multi-scaling properties of one-dimensional truncated L\'evy flights are studied. %% Due to the broken self-similarity of the distribution of jumps, they are expected to possess multi-scaling properties in contrast to the ordinary L\'evy flights. %% We argue this fact based on a smoothly truncated L\'evy distribution, and derive the functional form of the scaling exponents. %% Specifically, they exhibit bi-fractal behavior, which is the simplest case of multi-scaling.
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[
{
"name": "paper.tex",
"string": "\\documentstyle[pre,aps,epsf,floats]{revtex}\n\n\\begin{document}\n\n\\draft\n\n\\title{Multi-scaling properties of truncated L\\'evy flights}\n\n\\author{Hiroya Nakao}\n\n\\address{Graduate School of Mathematical Sciences, University of\n Tokyo,\\\\ 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\n Multi-scaling properties of one-dimensional truncated L\\'evy flights\n are studied.\n %%\n Due to the broken self-similarity of the distribution of jumps, they\n are expected to possess multi-scaling properties in contrast to the\n ordinary L\\'evy flights.\n %%\n We argue this fact based on a smoothly truncated L\\'evy\n distribution, and derive the functional form of the scaling\n exponents.\n %%\n Specifically, they exhibit bi-fractal behavior, which is the\n simplest case of multi-scaling.\n\\end{abstract}\n\n\\section{Introduction}\n\nL\\'evy flights are random processes based on L\\'evy stable\ndistributions\\cite{Feller}. They have been utilized successfully in\nmodeling various complex spatio-temporal behavior of non-equilibrium\ndissipative systems such as fluid turbulence, anomalous diffusion, and\nfinancial markets\\cite{Shlesinger}.\n\nThe L\\'evy stable distribution is self-similar to its convolutions.\nIt has a long power-law tail that decays much more slowly compared\nwith an exponential decay, which gives rise to infinite variance.\n%%\nHowever, in practical situations, it is usually truncated due to\nnonlinearity or finiteness of the system.\n%%\nIn order to incorporate this fact, Mantegna and Stanley\\cite{Mantegna}\nintroduced the notion of truncated L\\'evy flight.\n%%\nIt is based on a truncated L\\'evy stable distribution with a sharp\ncut-off in its power-law tail. Therefore, the distribution is not\nself-similar when convoluted, and has finite variance.\n%%\nThus, the truncated L\\'evy flight converges to a Gaussian process due\nto the central limit theorem in contrast to the ordinary L\\'evy stable\nprocess.\n%%\nHowever, as they pointed out, its convergence to a Gaussian is very\nslow and the process exhibits anomalous behavior in a wide range\nbefore the convergence.\n%%\nKoponen\\cite{Koponen} reproduced their result analytically using a\ndifferent type of truncated L\\'evy distribution with a smoother\ncut-off.\n\nDubrulle and Laval\\cite{Dubrulle} applied their idea to the velocity\nfield of 2D fluid turbulence, and claimed that the truncation is\nessential for the multi-scaling property of the velocity field to\nappear.\n%%\nThey showed that the broken self-similarity of the distribution makes\na qualitative difference on the scaling property of the corresponding\nrandom process, i.e., the ordinary L\\'evy stable process exhibits mere\nsingle-scaling, while the truncated L\\'evy process exhibits\nmulti-scaling, although their analysis was mostly based on numerical\ncalculation and the obtained scaling exponents were rather inaccurate.\n%%\nThe idea of truncated L\\'evy flights has also been applied to the\nanalysis of financial data such as stock market prices and foreign\nexchange rates\\cite{Cont,Bouchaud}. Multi-scaling analyses of the\nfinancial data have also been attempted\\cite{Vandewalle,Ivanova}.\n\nIn this paper, we treat the truncated L\\'evy flights analytically\nbased on the smooth truncation introduced by Koponen, and clarify\ntheir multi-scaling properties.\n%%\nThey exhibit the simplest form of multi-scaling, i.e., bi-fractality,\ndue to the characteristic shape of the truncated L\\'evy distribution.\n%%\nOur results may have some relevance to the multi-scaling properties of\nthe velocity field in 2D fluid turbulence and of the fluctuation of\nthe stock market prices.\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode\n \\epsfxsize=0.5\\textwidth\n \\epsfbox{Fig01.eps}\n \\caption{Truncated L\\'evy distribution $P_{TL}(x ; \\lambda)$ \n and its $n$-times convolutions for the symmetric case. The\n parameters are $\\lambda=0.001$, $\\alpha=0.75$, and\n $q-p=0$. Corresponding ordinary L\\'evy stable distribution\n $P_{L}(x)$ is also shown for comparison. Each convoluted\n distribution is rescaled as $n^{-1/\\alpha} P(n^{-1/\\alpha} x)$.}\n \\label{Fig:01}\n \\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode\n \\epsfxsize=0.5\\textwidth\n \\epsfbox{Fig02.eps}\n \\caption{Truncated L\\'evy distribution $P_{TL}(x ; \\lambda)$ \n and its $n$-times convolutions for the one-sided case. The\n parameters are $\\lambda=0.001$, $\\alpha=0.75$, and\n $q-p=-1$. Corresponding ordinary L\\'evy stable distribution\n $P_{L}(x)$ is also shown for comparison. Each convoluted\n distribution is rescaled as $n^{-1/\\alpha} P(n^{-1/\\alpha} x)$.}\n \\label{Fig:02}\n \\end{center}\n\\end{figure}\n\n\\section{Truncated L\\'evy distribution}\n\nLet $P(x)$ an probability distribution and $e^{\\psi(\\zeta)}$ its\ncharacteristic function, i.e.,\n%%\n\\begin{equation}\n P(x) = \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} e^{\\psi(\\zeta)} e^{-i\n \\zeta x} d\\zeta, \\;\\;\\;\\; e^{\\psi(\\zeta)} =\n \\int_{-\\infty}^{\\infty} P(x) e^{i \\zeta x} dx.\n \\label{Eq:Fourier}\n\\end{equation}\n%%\nAs explained in Feller\\cite{Feller}, (the argument of) the\ncharacteristic function for a L\\'evy stable distribution is given by\nthat of a compound Poisson process:\n%%\n\\begin{equation}\n \\psi(\\zeta) = \\int_{-\\infty}^{\\infty} \\left( e^{i \\zeta x} - 1 - i\n \\zeta \\tau(x) \\right) f(x) \\ dx,\n\\end{equation}\n%%\nwhere $f(x)$ is a probability distribution of increments and $\\tau(x)$\nis a certain centering function.\n%%\n$f(x)$ is assumed to be\n%%\n\\begin{equation}\n f_{L}(x) = \\left\\{\n \\begin{array}{ll}\n C q \\ |x|^{-1-\\alpha} \\;\\;&\\;\\; (x<0), \\cr\n C p \\ x^{-1-\\alpha} \\;\\;&\\;\\; (x>0), \\cr\n \\end{array}\n \\right.\n \\label{Eq:increments}\n\\end{equation}\n%%\nwhere $C > 0$ is a scale constant, $0 < \\alpha < 2$, $p \\geq 0$, $q\n\\geq 0$, and $p+q=1$. The function $\\tau(x)$ is chosen as $0$ for $0 <\n\\alpha < 1$ and $x$ for $1 < \\alpha < 2$.\n%%\nBy integration, we obtain\n%%\n\\begin{equation}\n \\psi_L(\\zeta) = C \\Gamma(-\\alpha) |\\zeta|^{\\alpha} \\left( \\cos\n \\frac{\\alpha \\pi}{2} \\pm i (q-p) \\sin \\frac{\\alpha \\pi}{2} \\right)\n \\;\\;\\;\\; (\\alpha \\neq 1),\n \\label{Eq:Levy}\n\\end{equation}\n%%\nwhere the upper sign applies when $\\zeta > 0$, and the lower for\n$\\zeta < 0$.\n%%\nThis is a well-known form of the L\\'evy stable characteristic\nfunction, and we obtain a L\\'evy stable distribution $P_{L}(x)$\nthrough an inverse Fourier transform (see Figs.~\\ref{Fig:01} and\n\\ref{Fig:02}).\n\nThis characteristic function satisfies $n \\psi_L(\\zeta) =\n\\psi_L(n^{1/\\alpha} \\zeta)$, which means that the corresponding\nprobability distribution is stable to convolution, i.e.,\n%%\n\\begin{equation}\n P_{L}^{n}(x) = n^{-1/{\\alpha}} P_{L}^{1}(n^{-1/{\\alpha}} x).\n\\end{equation}\n%%\nHere $P^{n}(x)$ denotes a $n$-times convoluted distribution of\n$P^{1}(x) \\equiv P(x)$, i.e., $P^{n}(x) = (2 \\pi)^{-1} \\int e^{n\n \\psi(\\zeta)} e^{-i \\zeta x} d\\zeta$.\n%%\nThe L\\'evy stable distribution $P_{L}(x)$ is symmetric when $q-p = 0$,\nand one-sided when $q-p = \\pm 1$ and $0 < \\alpha < 1$.\\footnote{The\n distribution is one-sided to the right ($x>0$) when $q-p = -1$ and\n to the left ($x<0$) when $q-p = 1$.}\n%%\nIt has a power-law tail of the form $|x|^{-1-\\alpha}$, and the\nabsolute moment $\\langle x^q \\rangle := \\int_{-\\infty}^{\\infty} |x|^q\nP_{L}(x) dx$ does not exist for $q \\geq \\alpha$.\n\nNow, let us truncate this L\\'evy stable distribution following\nKoponen\\cite{Koponen}. We introduce a cut-off parameter $\\lambda > 0$\nand truncate the original $f_{L}(x)$ in Eq.~(\\ref{Eq:increments}) as\n%%\n\\begin{equation}\n f_{TL}(x) = \\left\\{\n \\begin{array}{ll}\n C q \\ |x|^{-1-\\alpha} e^{-\\lambda |x|} \\;\\;&\\;\\; (x<0), \\cr\n C p \\ x^{-1-\\alpha} e^{-\\lambda x} \\;\\;&\\;\\; (x>0). \\cr\n \\end{array}\n \\right.\n \\label{Eq:increments_cutoff}\n\\end{equation}\n%%\nFor the case $0 < \\alpha < 1$, $\\tau(x)$ can be omitted, and we obtain\nby integration\n%%\n\\begin{equation}\n \\psi_{TL}(\\zeta ; \\lambda) = C \\Gamma(-\\alpha) \\left\\{ q (\\lambda +\n i \\zeta)^{\\alpha} + p (\\lambda - i \\zeta)^{\\alpha} -\n \\lambda^{\\alpha} \\right\\},\n \\label{Eq:truncated_Levy}\n\\end{equation}\n%%\nor, by expanding the first two terms\n%%\n\\begin{equation}\n \\psi_{TL}(\\zeta ; \\lambda) = C \\Gamma(-\\alpha) \\left\\{ \\left(\n \\lambda^2 + \\zeta^2 \\right)^{\\alpha / 2} \\left( \\cos \\alpha\n \\theta + i (q-p) \\sin \\alpha \\theta \\right) - \\lambda^{\\alpha}\n \\right\\},\n\\end{equation}\n%%\nwhere $\\theta = \\arctan \\left( \\zeta / \\lambda \\right)$.\n%%\nApart from the scale constant, this is the characteristic function of\ntruncated L\\'evy distribution given by Koponen.\\footnote{Note the\n misprint of Eq.~(3) in Ref.~\\cite{Koponen}. It should read $\\ln\n {\\hat P}(k) = c \\left\\{c_0 - (k^2+{\\nu}^2)^{\\nu/2} / \\cos(\\pi \\nu /\n 2) ... \\right\\}$.}\n%%\nFor the case $1 < \\alpha < 2$, we use a centering function $\\tau(x) =\nx$ and obtain\n%%\n\\begin{equation}\n \\psi_{TL}(\\zeta ; \\lambda) = C \\Gamma(-\\alpha) \\left\\{ q (\\lambda +\n i \\zeta)^{\\alpha} + p (\\lambda - i \\zeta)^{\\alpha} -\n \\lambda^{\\alpha} - i \\alpha \\lambda^{\\alpha-1} (q-p) \\zeta\n \\right\\}.\n \\label{Eq:truncated_Levy_2}\n\\end{equation}\n%%\nIn this case, an extra term appears in addition to\nEq.~(\\ref{Eq:truncated_Levy}), which induces a drift of the\nprobability distribution when $q-p \\neq 0$.\n%%\nIt can easily be seen that in the limit $\\lambda \\to +0$, these\ncharacteristic functions go back to the L\\'evy stable characteristic\nfunction Eq.~(\\ref{Eq:Levy}).\n\nWe obtain a truncated L\\'evy distribution $P_{TL}(x ; \\lambda)$\nthrough an inverse Fourier transform from\nEq.~(\\ref{Eq:truncated_Levy}) or Eq.~(\\ref{Eq:truncated_Levy_2}).\n%%\nThe parameter $\\lambda$ modifies the behavior of $\\psi_{TL}(\\zeta ;\n\\lambda)$ when $\\zeta$ is comparable to $\\lambda$ and removes the\nsingularity of the characteristic function at the origin.\n%%\nIt thus changes the behavior of $P_{TL}(x ; \\lambda)$ when $x \\sim\n\\lambda^{-1}$ and introduces an exponential cut-off to the power-law\ntail of $P_{TL}(x ; \\lambda)$.\n%%\nTherefore, all absolute moments $\\langle x^q \\rangle$ of $P_{TL}(x ;\n\\lambda)$ are finite in contrast to the case of the L\\'evy stable\ndistribution $P_{L}(x)$.\n\nThe characteristic function $\\psi_{TL}(\\zeta ; \\lambda)$ is infinitely\ndivisible but no longer stable. The convolutions of its corresponding\nprobability distribution cannot be collapsed by scaling and shifting\nthe variable $x$.\n%%\nHowever, they can still be collapsed by scaling both $x$ and $\\lambda$\nappropriately; the characteristic function $\\psi_{TL}(\\zeta ;\n\\lambda)$ satisfies $ n \\psi_{TL}(\\zeta ; \\lambda) =\n\\psi_{TL}(n^{1/{\\alpha}} \\zeta ; n^{1/{\\alpha}} \\lambda) $, which\nmeans that the corresponding $n$-times convoluted probability\ndistribution satisfies\n%%\n\\begin{equation}\n P_{TL}^{n}(x ; \\lambda) = n^{-1/{\\alpha}} P_{TL}^{1} (\n n^{-1/{\\alpha}} x ; n^{1/{\\alpha}} \\lambda).\n \\label{Eq:collapse}\n\\end{equation}\n%%\nWe utilize this fact for calculating the scaling exponents of the\ntruncated L\\'evy flights.\\footnote{In the context of random processes\n we discuss below, this operation can be considered as a\n renormalization-group transformation. The limiting L\\'evy stable\n process corresponds to a fixed point, and the exponent $1/\\alpha$\n can be regarded as a sort of critical exponent.}\n\nIn Figs.~\\ref{Fig:01} and \\ref{Fig:02}, truncated L\\'evy distributions\n$P_{TL}(x ; \\lambda)$ and their convolutions are displayed for\nsymmetric ($q-p = 0$) and one-sided ($q-p = -1$) cases in comparison\nwith the corresponding ordinary L\\'evy stable distributions $P_{L}(x)$\nin a log-log scale.\n%%\nAs expected, each truncated L\\'evy distribution has a cut-off at $x\n\\sim \\lambda^{-1} = 10^3$ after a power-law decay. The cut-off\nposition gradually approaches the origin with the convolution, and the\nself-similarity of the convoluted distribution is broken in the tail\npart.\n\nWe can extend the power-law decaying part arbitrarily longer by making\n$\\lambda$ smaller.\n%%\nHereafter, we assume the cut-off parameter $\\lambda$ to be very small,\ni.e., the cut-off is far away from the origin, since we are interested\nin the transient anomalous behavior of the corresponding random\nprocess before it converges to a Gaussian due to the central limit\ntheorem.\n\n\\section{Truncated L\\'evy flights}\n\nThe truncated L\\'evy flight\\cite{Mantegna,Koponen,Dubrulle} is a\ntemporally discrete stochastic process characterized by the truncated\nL\\'evy distribution $P_{TL}(x ; \\lambda)$.\n%%\nAt each time step $i$, a particle jumps a random distance $x(i)$\nchosen independently from $P_{TL}(x ; \\lambda)$.\n%%\nThe position $y(i)$ of the particle started from the origin is given\nby $y(i) = \\sum_{j=1}^{i} x(j)$.\n%%\nHere we consider two representative cases of truncated L\\'evy flights,\ni.e., the symmetric case ($q-p = 0$, $0<\\alpha<2$) and the one-sided\ncase ($q-p = -1$, $0<\\alpha<1$).\n\nFigure~\\ref{Fig:03} shows typical realizations of the jump $x(i)$ and\nthe position of the particle $y(i)$ for the symmetric case.\n%%\nThe time sequence of the jump $x(i)$ is intermittent; $x(i)$ mostly\ntakes small values but sometimes takes very large values.\n%%\nCorrespondingly, the movement of the particle $y(i)$ is also\nintermittent.\n%%\nThis intermittency gives rise to the anomalous scaling property of the\ntrace of $y(i)$ in which we are interested.\n\nFigure~\\ref{Fig:04} displays typical realizations of $x(i)$ and $y(i)$\nfor the one-sided case. Since the probability distribution $P_{TL}(x ;\n\\lambda)$ vanishes for $x < 0$, each jump takes a positive value and\nthe position of the particle increases monotonically. As in the\nsymmetric case, their time sequences are intermittent.\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode\n \\epsfxsize=0.5\\textwidth\n \\epsfbox{Fig03.eps}\n \\caption{Typical realization of random sequence $x(i)$ (top) and \n corresponding truncated L\\'evy flights $y(i)$ (bottom) for\n $\\lambda=0.001$, $\\alpha=0.75$, and $q-p=0$ (symmetric case).}\n \\label{Fig:03}\n \\end{center}\n\\end{figure}\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode\n \\epsfxsize=0.5\\textwidth\n \\epsfbox{Fig04.eps}\n \\caption{Typical realization of random sequence $x(i)$ (top) and\n corresponding truncated L\\'evy flights $y(i)$ (bottom) for\n $\\lambda=0.001$, $\\alpha=0.75$, and $q-p=-1$ (one-sided case).}\n \\label{Fig:04}\n \\end{center}\n\\end{figure}\n\n\\section{Multi-scaling properties}\n\nIn order to characterize the intermittent time sequences shown in\nFigs.~\\ref{Fig:03} and \\ref{Fig:04}, we introduce multi-scaling\nanalysis. It has been employed successfully in characterizing velocity\nand energy dissipation fields of fluid turbulence and rough interfaces\nof surface growth phenomena\\cite{Frisch,Bohr}.\n\nMulti-scaling analysis concerns a partition function of the measure\ndefined suitably for the field under consideration.\n%%\nHere we focus on the apparent similarity of the intermittent time\nsequences of truncated L\\'evy flights to those of fluid turbulence,\ni.e., the similarity of $y(i)$ in Fig.~\\ref{Fig:03} to the velocity\nfield, and that of $x(i)$ in Fig.~\\ref{Fig:04} to the energy\ndissipation field.\n\nThus, we apply the ``multi-affine'' analysis to the trace of $y(i)$\nfor the symmetric sequence. The measure $h(n)$ is defined as the\nabsolute hight difference of $y(i)$ between two points separated by a\ndistance $n$, i.e., $h(n) = |y(i+n) - y(i)|$. The distribution of\n$h(n)$ does not depend on $i$, since the increment of this process is\nstatistically stationary.\n%%\nThe partition function is then defined as $ Z_h(n ; q) = \\langle\nh(n)^{q} \\rangle = \\langle \\left| \\sum_{j=i+1}^{i+n} x(j) \\right|^q\n\\rangle $, where $\\langle ... \\rangle$ denotes a statistical average.\nThis function is called ``structure function'' in the context of fluid\nturbulence.\n\nOn the other hand, for the one-sided case, we focus on the trace of\n$x(i)$ and apply the ``multi-fractal'' analysis. The measure $m(n)$ is\ndefined as the area below the trace of $x(i)$ between two points\nseparated by a distance $n$, i.e., $m(n) = \\sum_{j=i+1}^{i+n} x(j)$,\nand the partition function is defined as $ Z_m(n ; q) = \\langle m(n)^q\n\\rangle = \\langle \\left( \\sum_{j=i+1}^{i+n} x(j) \\right)^q \\rangle $.\n\nThese partition functions are expected to scale with $n$ as $Z_h(n ;\nq) \\sim n^{\\zeta(q)}$ and $Z_m(n ; q) \\sim n^{\\tau(q)}$ for small $n$.\nFurther, if these scaling exponents $\\zeta(q)$ and $\\tau(q)$ exhibit\nnonlinear dependence on $q$, the corresponding measures $h(n)$ and\n$m(n)$ are called multi-affine and multi-fractal, respectively.\n\\footnote{The multi-fractal partition function $Z_m(n ; q)$ defined\n here is different from the traditional one that is defined as\n $N(n)^{-1} \\langle m(n)^q \\rangle$, where $N(n)$ is a number of\n boxes of size $n$ that are needed to cover the whole sequence. This\n makes a difference of $-1$ to the scaling exponent $\\tau(q)$ for the\n one-dimensional case considered here.}\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode\n \\epsfxsize=0.5\\textwidth\n \\epsfbox{Fig05.eps}\n \\caption{Partition functions $Z_m(n ; q)$ for $q=0.0$ - $3.0$\n at intervals of $0.2$. The bottom line corresponds to $q=0.0$,\n and the top one to $q=3.0$.}\n \\label{Fig:05}\n \\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{Fig:05}, we display the partition functions $Z_m(n ; q)$\nfor several values of $q$ for the one-sided case in a log-log\nscale. As can be seen from the figure, each partition function\nexhibits power-law dependence on $n$ for small $n$. We obtain a\nsimilar figure for the partition functions $Z_h(n ; q)$ for the\nsymmetric case.\n%%\nThe corresponding scaling exponents $\\zeta(q)$ and $\\tau(q)$ are shown\nin Fig.~\\ref{Fig:06}. Each curve exhibits strong non-linear dependence\non $q$; it is linear for small $q$ and constant for large $q$. Thus,\nthe sequences generated by truncated L\\'evy flights clearly possess\nmulti-scaling properties.\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\leavevmode\n \\epsfxsize=0.5\\textwidth\n \\epsfbox{Fig06.eps}\n \\caption{Multi-scaling exponents $\\zeta(q)$ and $\\tau(q)$\n for the symmetric (top) and one-sided (bottom) cases. The\n cut-off parameter $\\lambda$ is set at $0.00001$.}\n \\label{Fig:06}\n \\end{center}\n\\end{figure}\n\n\\section{Derivation of the scaling exponents}\n\nNow let us derive the scaling exponents $\\zeta(q)$ and $\\tau(q)$ from\nthe characteristic function Eq.~(\\ref{Eq:truncated_Levy}). It is clear\nfrom the definition of the partition functions that they can be\ncalculated once we know the probability distribution of the sum of\nrandom variables $z(n) := \\sum_{j=1}^{n} x(j)$.\n%%\nSince the jumps are independent from each other, the probability\ndistribution of $z(n)$ is given by a $n$-times convolution of the\ntruncated L\\'evy distribution $P_{TL}(x ; \\lambda)$, i.e., by\n$P_{TL}^{n}(x ; \\lambda)$.\n\nAs we explained previously, $P_{TL}^{n}(x ; \\lambda)$ can easily be\nobtained from the original $P_{TL}(x ; \\lambda)$ by scaling $x$ and\n$\\lambda$.\n%%\nMaking use of this fact, the $q$-th absolute moment $\\langle z(n)^q\n\\rangle$ of $z(n)$ can be calculated as\n%%\n\\begin{eqnarray}\n \\langle z(n)^q \\rangle_{\\lambda}\n &=& A \\int_{0}^{\\infty} z^q P_{TL}^{n}(z ; \\lambda) dz\n = A \\int_{0}^{\\infty} z^q n^{-1/{\\alpha}}\n P_{TL}(n^{-1/{\\alpha}} z ; n^{1/{\\alpha}} \\lambda) dz \\cr \\cr\n &=& n^{q / \\alpha} A \\int_{0}^{\\infty} z^q P_{TL}(z ; n^{1/{\\alpha}} \\lambda) dz\n = n^{q / \\alpha} \\langle z(1)^q \\rangle_{n^{1/{\\alpha}} \\lambda},\n\\end{eqnarray}\n%%\nwhere the constant $A$ is $2$ for the symmetric case and $1$ for the\none-sided case. Here we explicitly indicated the parameter $\\lambda$\nof the distribution $P_{TL}(x ; \\lambda)$ as the subscript of the\naverage.\n%%\nNote that if $\\langle z(1)^q \\rangle_{\\lambda}$ does not depend on\n$\\lambda$, the scaling exponent is simply given by a linear function\n$q / \\alpha$, and the process does not exhibit multi-scaling. This is\nthe case for the ordinary L\\'evy stable distribution.\n\nThus, all we need to calculate is the moment $\\langle z(1)^q\n\\rangle_{\\lambda}$. However, of course, an analytical expression for\n$P_{TL}(x ; \\lambda)$ is not attainable except a few specific cases.\n%%\nHere we adopt an approximation which utilizes the facts that the\ntruncated L\\'evy distribution is different from the ordinary L\\'evy\nstable distribution only in the tail part, and that it has a power-law\ndecaying part with an exponent $-1-\\alpha$ in the middle (see\nFigs.~\\ref{Fig:01} and \\ref{Fig:02}).\n%%\nTherefore, it is expected that the moment whose degree $q$ is lower\nthan $\\alpha$ is almost the same as that obtained from the ordinary\nL\\'evy stable distribution, and the moment for $q > \\alpha$ is mostly\ndetermined by the asymptotic tail of the truncated L\\'evy\ndistribution. Note that the moment for $q > \\alpha$ does not exist for\nthe ordinary L\\'evy stable distribution.\n\n(i) Lower moments ($0 < q < \\alpha$).\n\nWe can approximate $P_{TL}(x ; \\lambda)$ by $P_{L}(x)$ in this case,\nsince they are different only in their tail parts, which does not\ncontribute to the moments lower than $\\alpha$ significantly.\n%%\nTherefore, $\\langle z(1)^q \\rangle_{\\lambda}$ does not depend on\n$\\lambda$ approximately, and we obtain $\\langle z(n)^q \\rangle \\simeq\nconst. \\ n^{q / {\\alpha}}$. Thus, the scaling exponents $\\zeta(q)$ and\n$\\tau(q)$ are given by $q / {\\alpha}$ for $0 < q < \\alpha$.\n%%\nThe broken self-similarity of the distribution is not important in\nthis regime.\n\n(ii) Higher moments ($q > \\alpha$).\n\nSince the tail of the distribution mainly contributes to the higher\nmoments, we can calculate $\\langle z(1)^q \\rangle_{\\lambda}$\napproximately if we know the asymptotic form of the distribution.\n%%\nFor this purpose, we expand the characteristic function as in the case\nof the series expansion of ordinary L\\'evy stable\ndistribution\\cite{Feller}.\n%%\nBy expanding the integrand, the truncated L\\'evy distribution can be\nexpressed as\n%%\n\\begin{eqnarray}\n P_{TL}(x ; \\lambda) &=& e^{-C \\Gamma(-\\alpha) \\lambda^{\\alpha}}\n \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} \\exp \\left[ C\n \\Gamma(-\\alpha) \\left\\{ q(\\lambda + i \\zeta)^{\\alpha} + p(\\lambda -\n i \\zeta)^{\\alpha} \\right\\} \\right] e^{-i \\zeta x} d\\zeta \\cr &=&\n Re \\frac{-i}{\\pi x} e^{-C \\Gamma(-\\alpha) \\lambda^{\\alpha}}\n \\sum_{k=0}^{\\infty} \\frac{1}{k!} \\left\\{ C \\Gamma(-\\alpha)\n \\lambda^{\\alpha} \\right\\}^{k} \\int_0^{\\infty} \\left[ q\\left(\n 1+\\frac{z}{\\lambda x} \\right)^{\\alpha} + p\\left(\n 1-\\frac{z}{\\lambda x} \\right)^{\\alpha} \\right]^{k} e^{-z} dz\n \\;\\;\\;\\;\\;\\;\\;\\;\n \\label{Eq:expansion}\n\\end{eqnarray}\n%%\nfor the case $0 < \\alpha < 1$.\n%%\nAt the lowest order ($k=1$), we recover the original distribution of\nthe increments:\n%%\n\\begin{equation}\n P_{TL}(x ; \\lambda) \\sim e^{-C \\Gamma(-\\alpha) \\lambda^{\\alpha}} C p\n x^{-1-\\alpha} e^{-\\lambda x} \\;\\;\\;\\; (x \\gg 1).\n\\end{equation}\n%%\nWith this approximation, the moment $\\langle z(1)^{q}\n\\rangle_{\\lambda}$ is calculated as\n%%\n\\begin{equation}\n \\langle z(1)^{q} \\rangle_{\\lambda} \\simeq e^{-C \\Gamma(-\\alpha)\n \\lambda^{\\alpha}} C p \\int_{0}^{\\infty} z^{q-1-\\alpha} e^{-\\lambda\n z} dz = e^{-C \\Gamma(-\\alpha) \\lambda^{\\alpha}} C p \\Gamma(1+q)\n \\lambda^{-q+\\alpha},\n\\end{equation}\n%%\nand we obtain\n%%\n\\begin{equation}\n \\langle z(n)^{q} \\rangle_{\\lambda} = n^{q / \\alpha} \\langle z(1)^{q}\n \\rangle_{n^{1/{\\alpha}} \\lambda} \\sim n^{q / \\alpha} \\left (\n n^{1/{\\alpha}} \\lambda \\right)^{-q+\\alpha} \\sim n^{1}.\n\\end{equation}\n%%\nThus, the scaling exponents $\\zeta(q)$ and $\\tau(q)$ are given by $1$\nfor $q > \\alpha$.\n\nIn summary, we approximately derived the following expression for the\nmulti-scaling exponents $\\zeta(q)$ and $\\tau(q)$:\n%%\n\\begin{equation}\n \\zeta(q), \\tau(q) = \\left\\{\n \\begin{array}{cc}\n q/{\\alpha} & \\;\\;\\;\\; (0 < q < \\alpha), \\cr \\cr\n 1 & \\;\\;\\;\\; (q > \\alpha).\n \\end{array}\n \\right.\n \\label{Eq:bifractal}\n\\end{equation}\n%%\nNote that the above approximation becomes more and more accurate as we\nextend the power-law decaying part by decreasing $\\lambda$, and this\nresult is exact in the asymptotic limit.\n%%\nIn Fig.~\\ref{Fig:06}, these theoretical curves are compared with the\nexperimental results. Except for small deviations near the transition\npoints, the theoretical results well reproduce the experimental\nresults.\n%%\nAlthough our estimation here is done for the case $0 < \\alpha < 1$,\nthe theoretical result Eq.~(\\ref{Eq:bifractal}) seems to be also\napplicable for $1 < \\alpha < 2$.\\footnote{This implies that\n Eq.~(\\ref{Eq:expansion}) still has its meaning as an asymptotic\n expansion for $1 < \\alpha < 2$. This is proved for the case of\n ordinary L\\'evy stable distribution\\cite{Bergstrom}.}\n%%\nThis type of simple multi-scaling is sometimes called\n``bi-fractality'' and is known, for example, in randomly forced\nBurgers' equation\\cite{Frisch}.\n\n\\section{Discussion}\n\nWe analyzed the multi-scaling properties of the truncated L\\'evy\nflights based on the smooth truncation introduced by Koponen.\n%%\nAs Dubrulle and Laval claimed, truncation is essential for the\nmulti-scaling properties to appear.\n%%\nWe clarified this fact and derived the functional form of the scaling\nexponents for both symmetric and one-sided cases.\n\nAs we mentioned previously, the cut-off parameter $\\lambda$ may\nrepresent the finiteness of the system under consideration. Then it\nwould be natural to assume $\\lambda$ as a decreasing function of the\nsystem size $L$, for example $\\lambda = L^{-1}$, and we can think of\nthe system-size dependence of the truncated L\\'evy flight. Of course,\ndistribution functions $P_{TL}$ for different system sizes cannot be\ncollapsed simply by rescaling $x$, i.e., finite-size scaling does not\nhold.\\footnote{We may also take a different viewpoint, where $\\lambda$\n is another tunable parameter independent of the system size\n $L$. Then Eq.~(\\ref{Eq:collapse}) may be viewed as a finite-size\n scaling relation between systems of size $L/n$ and $L$, similar to\n that in statistical mechanics.}\n%%\nHowever, approximate finite-size scaling relations still hold\nseparately, if we divide $P_{TL}$ into two parts at the power-law\ndecaying part, i.e., into the self-similar part and the tail part.\n%%\nAs we explained, the self-similar part is insensitive to $\\lambda =\nL^{-1}$, and $P_{TL}$ for different system sizes collapse to a single\ncurve in this region without rescaling.\n%%\nOn the other hand, the tail part is asymptotically given by $P_{TL}(x ;\nL^{-1}) \\sim x^{-1-\\alpha} e^{-x/L}$, which can be rescaled as\n$L^{1+\\alpha} P_{TL}(x L ; L^{-1})$ to give a universal curve.\n%%\nThus, we have two different approximate finite-size scaling relations\nin different regions of $x$, which is separated by the power-law\ndecaying part.\n%%\nOf course, this is closely related to the bi-fractal behavior of the\nscaling exponents.\n%%\nSimilar asymptotic finite-size scaling is also reported in the\nsandpile models of self-organized criticality\\cite{Chessa}.\n\nRecently, Chechkin and Gonchar\\cite{Chechkin} discussed the finite\nsample number effect on the scaling properties of a stable L\\'evy\nprocess. (They treated only the symmetric case using a different\nargument from ours, which was more qualitative and therefore more\ngeneral in a sense.)\n%%\nThey claimed that, due to the finite sample number effect,\n``spurious'' multi-affinity is observed in the numerical simulation\nand derived the ``spurious'' multi-scaling exponent.\n%%\nInterestingly, or in some sense obviously, their ``spurious''\nmulti-scaling exponent is the same as our multi-scaling exponent,\nsince the truncation of the power-law by $\\lambda$ can also be\ninterpreted as mimicking the finite sample number effect of\nexperiments.\n\nOur calculation presented in this paper is similar to our previous\nwork\\cite{Nakao} on the multi-scaling properties of the amplitude and\ndifference fields of anomalous spatio-temporal chaos found in systems\nof non-locally coupled oscillators.\n%%\nThe distribution treated there was not the truncated L\\'evy type but\nhad a form like $(1+x^2)^{\\alpha/2} e^{-\\lambda |x|}$. Since this form\nis easily generated by a simple multiplicative stochastic process,\nsome attempts have been made to model the economic activity using this\ntype of distribution\\cite{Sato,Sornette}.\n\nThe bi-fractal behavior of the scaling exponent is the simplest case\nof multi-scaling, while experimentally observed scaling exponents\nusually exhibit more complex behavior.\n%%\nActually, it has long been discussed in the context of fluid\nturbulence what the shape of the distribution should be to reproduce\nthe experimentally observed scaling exponent\\cite{Frisch,Bohr,Benzi}.\n%%\nIn order to reproduce the behavior of the scaling exponent more\nrealistically in the framework of the truncated L\\'evy flights\ndiscussed here, introduction of correlations to the random variables\nwill be necessary. Studies in this direction are expected.\n\n\\acknowledgments\n The author gratefully acknowledges Professor Michio Yamada and\n University of Tokyo for warm hospitality. He also thanks\n Dr. A. Lema\\^{\\i}tre, Dr. H. Chat\\'e, and anonymous referees for\n useful comments. This work is supported by the JSPS Research\n Fellowships for Young Scientists.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Feller} W. Feller, {\\it An Introduction to Probability Theory\n and Its Applications vol.II} (John Wiley \\& Sons, Inc., New York,\n 1971).\n\n\\bibitem{Shlesinger} M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch(Eds.),\n {\\it L\\'evy Flights and Related Topics In Physics}\n (Springer, Berlin, 1995).\n\n\\bibitem{Mantegna}\n R. N. Mantegna and H. E. Stanley,\n %%Stochastic process with ultraslow convergence to a Gaussian:\n %%The truncated L\\'evy flight\n Phys. Rev. Lett. {\\bf 73} (1994), 2946.\n\n\\bibitem{Koponen}\n I. Koponen,\n %%Analytic approach to the problem of convergence of truncated L\\'evy flight\n %%towards the Gaussian stochastic process\n Phys. Rev. E {\\bf 52} (1995), 1197.\n\n\\bibitem{Dubrulle}\n B. Dubrulle and J. -Ph. Laval,\n %%Truncated L\\'evy laws and 2D turbulence.\n Eur. Phys. J. B {\\bf 4} (1998), 143.\n\n\\bibitem{Cont}\n R. Cont, M. Potters, and J-P. Bouchaud,\n %%Scaling in stock market data: Stable laws and beyond\n cond-mat/9705087.\n \n\\bibitem{Bouchaud}\n J-P. Bouchaud and M. Potters,\n %%Theory of Financial Risk: Basic Notions in Probability\n cond-mat/9905413.\n\n\\bibitem{Vandewalle}\n N. Vandewalle and M. Ausloos,\n %%Multi-affine analysis of typical currency exchange rates\n Eur. Phys. J. B {\\bf 4} (1998), 257.\n\n\\bibitem{Ivanova}\n K. Ivanova and M. Ausloos,\n %%Low $q$-moment multifractal analysis of Gold price, Dow Jones Industrial Average\n %%and BGL-USD exchange rate\n Eur. Phys. J. B {\\bf 8} (1999), 665.\n\n\\bibitem{Frisch}\n U. Frisch,\n {\\it Turbulence, the legacy of A. N. Kolmogorov}\n (Cambridge University Press, Cambridge, 1995).\n\n\\bibitem{Bohr}\n T. Bohr, M. H. Jensen, G. Paladin, and A. Vulpiani,\n {\\it Dynamical Systems Approach to Turbulence}\n (Cambridge University Press, Cambridge, 1998).\n\n\\bibitem{Bergstrom}\n H. Bergstr\\\"om,\n %%On some expansions of stable destribution functions\n Arkiv F\\\"or Matematik {\\bf 2} 1952, 375.\n\n\\bibitem{Chessa}\n A. Chessa, H. E. Stanley, A. Vespignani, and S. Zapperi,\n %%Universality in sandpiles\n Phys. Rev. E. {\\bf 59} (1999), 12.\n\n\\bibitem{Chechkin}\n A. V. Chechkin and V. Yu. Gonchar,\n %%Self-affinity of ordinary Levy motion, spurious multi-affinity\n %%and pseudo-Gaussian relations.\n cond-mat/9907234.\n\n\\bibitem{Nakao}\n %%Multi-affinity and multi-fractality in systems of chaotic elements\n %%with long-wave forcing\n H. Nakao and Y. Kuramoto,\n Eur. Phys. J. B {\\bf 10} (1999), 345.\n\n\\bibitem{Sato}\n A-H. Sato and H. Takayasu,\n %%Dynamic numerical models of stock amrket price:\n %%from microscopic determinism to macroscopic randomness\n Physica A {\\bf 250} (1998), 231.\n\n\\bibitem{Sornette}\n D. Sornette,\n %%Linear stochastic dynamics with nonlinear fractal properties\n Physica A {\\bf 250} (1998), 295.\n\n\\bibitem{Benzi}\n R. Benzi, L. Biferale, G. Paladin, A. Vulpiani, and M. Vergassola,\n %%Multifractality in the statistics of the velocity gradients in turbulence\n Phys. Rev. Lett. {\\bf 67} (1991), 2299.\n\n\\end{thebibliography}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002027.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Feller} W. Feller, {\\it An Introduction to Probability Theory\n and Its Applications vol.II} (John Wiley \\& Sons, Inc., New York,\n 1971).\n\n\\bibitem{Shlesinger} M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch(Eds.),\n {\\it L\\'evy Flights and Related Topics In Physics}\n (Springer, Berlin, 1995).\n\n\\bibitem{Mantegna}\n R. N. Mantegna and H. E. Stanley,\n %%Stochastic process with ultraslow convergence to a Gaussian:\n %%The truncated L\\'evy flight\n Phys. Rev. Lett. {\\bf 73} (1994), 2946.\n\n\\bibitem{Koponen}\n I. Koponen,\n %%Analytic approach to the problem of convergence of truncated L\\'evy flight\n %%towards the Gaussian stochastic process\n Phys. Rev. E {\\bf 52} (1995), 1197.\n\n\\bibitem{Dubrulle}\n B. Dubrulle and J. -Ph. Laval,\n %%Truncated L\\'evy laws and 2D turbulence.\n Eur. Phys. J. B {\\bf 4} (1998), 143.\n\n\\bibitem{Cont}\n R. Cont, M. Potters, and J-P. Bouchaud,\n %%Scaling in stock market data: Stable laws and beyond\n cond-mat/9705087.\n \n\\bibitem{Bouchaud}\n J-P. Bouchaud and M. Potters,\n %%Theory of Financial Risk: Basic Notions in Probability\n cond-mat/9905413.\n\n\\bibitem{Vandewalle}\n N. Vandewalle and M. Ausloos,\n %%Multi-affine analysis of typical currency exchange rates\n Eur. Phys. J. B {\\bf 4} (1998), 257.\n\n\\bibitem{Ivanova}\n K. Ivanova and M. Ausloos,\n %%Low $q$-moment multifractal analysis of Gold price, Dow Jones Industrial Average\n %%and BGL-USD exchange rate\n Eur. Phys. J. B {\\bf 8} (1999), 665.\n\n\\bibitem{Frisch}\n U. Frisch,\n {\\it Turbulence, the legacy of A. N. Kolmogorov}\n (Cambridge University Press, Cambridge, 1995).\n\n\\bibitem{Bohr}\n T. Bohr, M. H. Jensen, G. Paladin, and A. Vulpiani,\n {\\it Dynamical Systems Approach to Turbulence}\n (Cambridge University Press, Cambridge, 1998).\n\n\\bibitem{Bergstrom}\n H. Bergstr\\\"om,\n %%On some expansions of stable destribution functions\n Arkiv F\\\"or Matematik {\\bf 2} 1952, 375.\n\n\\bibitem{Chessa}\n A. Chessa, H. E. Stanley, A. Vespignani, and S. Zapperi,\n %%Universality in sandpiles\n Phys. Rev. E. {\\bf 59} (1999), 12.\n\n\\bibitem{Chechkin}\n A. V. Chechkin and V. Yu. Gonchar,\n %%Self-affinity of ordinary Levy motion, spurious multi-affinity\n %%and pseudo-Gaussian relations.\n cond-mat/9907234.\n\n\\bibitem{Nakao}\n %%Multi-affinity and multi-fractality in systems of chaotic elements\n %%with long-wave forcing\n H. Nakao and Y. Kuramoto,\n Eur. Phys. J. B {\\bf 10} (1999), 345.\n\n\\bibitem{Sato}\n A-H. Sato and H. Takayasu,\n %%Dynamic numerical models of stock amrket price:\n %%from microscopic determinism to macroscopic randomness\n Physica A {\\bf 250} (1998), 231.\n\n\\bibitem{Sornette}\n D. Sornette,\n %%Linear stochastic dynamics with nonlinear fractal properties\n Physica A {\\bf 250} (1998), 295.\n\n\\bibitem{Benzi}\n R. Benzi, L. Biferale, G. Paladin, A. Vulpiani, and M. Vergassola,\n %%Multifractality in the statistics of the velocity gradients in turbulence\n Phys. Rev. Lett. {\\bf 67} (1991), 2299.\n\n\\end{thebibliography}"
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cond-mat0002028
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Time inhomogeneous Fokker-Planck equation for wave distributions in the abelian sandpile model
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"author": "L. Anton\\cite{emailaddress}"
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The time and size distribution of the waves of topplings in the Abelian sandpile model are expressed as the first arrival at the origin distribution for a scale invariant, time inhomogeneous Fokker-Planck equation. Assuming a linear conjecture for the the time inhomogeneity exponent as function of loop erased random walk (LERW) critical exponent, suggested by numerical results, this approach allows one to estimate the lower critical dimension of the model and the exact value of the critical exponent for LERW in three dimension. The avalanche size distribution in two dimensions is found to be the difference between two closed power laws.
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"name": "paperxxxv5.tex",
"string": "%&latex209 \n\\documentstyle[aps,epsfig]{revtex}\n\\begin{document}\n\\wideabs{\n\\draft \\title{Time inhomogeneous Fokker-Planck equation for wave\ndistributions in the abelian sandpile model} \\author{L.\nAnton\\cite{emailaddress}} \\address{Institute for Theoretical Physics,\nUniversity of Stellenbosch, Private Bag X1, 7602 Matieland, South\nAfrica \\\\ and\\\\ Institute of Atomic Physics, INFLPR, Lab 22, PO Box\nMG-36 R76900, Bucharest, Romania} \\date{\\today} \\maketitle\n\\begin{abstract}\nThe time and size distribution of the waves of topplings in the\nAbelian sandpile model are expressed as the first arrival at the\norigin distribution for a scale invariant, time inhomogeneous\nFokker-Planck equation. Assuming a linear conjecture for the the time\ninhomogeneity exponent as function of loop erased random walk (LERW)\ncritical exponent, suggested by numerical results, this approach\nallows one to estimate the lower critical dimension of the model and the\nexact value of the critical exponent for LERW in three dimension. The\navalanche size distribution in two dimensions is found to be the\ndifference between two closed power laws.\n\\end{abstract}\n\\pacs{05.10.Gg,04.40.-a} \n \n}\n%\\narrowtext\n\nThe abelian sandpile model (ASM) was introduced by Bak, Tang, and\nWiesenfeld \\cite{Bak87} as a minimal description for natural phenomena\ncharacterized by intermittent time evolution through events called\navalanches which have scale invariant properties. The model is defined\non a hypercubic lattice whose sites can accommodate a variable,\npositive number of grains. With a uniform distribution a site is\nchosen and its number of grains is increased by one. If its total\nnumber of grains exceeds a given critical threshold $z_{max}$, the\nnearest neighbor sites increase their number of grains by one and the\ninitial site loses the corresponding number of grains. Then the newly\nupdated sites are checked for stability until there are no more\nunstable states. This event is an avalanche and the analytical\nproperties of its distribution is still an open question\n\\cite{Ktitarev00,Menech98,Menech00,Drossel99,Tebaldi99}. Analytical\napproaches rely on the algebraic properties of the toppling rules and\nthe decomposition of avalanches into simpler events called waves which\nare related to spanning trees on a\nlattice\\cite{Ktitarev00,Dhar90,Ivashkevich94}. A wave of topplings is\nsimply obtained by restraining the initial site, from where an\navalanche was initiated, to topple again only after all the other\nunstable sites have relaxed. Recently it was shown that the wave\ndistribution satisfies the finite size scaling ansatz, with critical\nexponents deduced from the equivalence between waves and spanning trees\n\\cite{Ktitarev00}.\n\nIn this Letter we present numerical evidence that the critical\nexponents of the wave size and time distribution, which were deduced\nfrom geometrical considerations in\n\\cite{Ktitarev00,Dhar90,Ivashkevich94}, can be related to the\nparameters of a scale invariant Fokker-Planck equation (FPE) in any\ndimension. In this way we make a connection between the geometrical\nproperties of the waves and an evolution equation. As further\nconfirmation of the validity of a FPE description for the abelian\nsandpile model, we show that the scaling behavior of the average\nnumber of unstable sites as function of time is predicted by the\nFPE. Using the expressions for the critical exponents deduced in\n\\cite{Ktitarev00} together with those inferred form the FPE approach\nwe are able to find the lower critical dimension of the abelian\nsandpile model and, as an extra benefit, the exact value for the loop\nerased random walk (LERW) critical exponent $\\nu$ in three\ndimensions. In two dimensions, the case in which the distribution\nof the last wave is known \\cite{Ivashkevich94}, we can compute the\nasymptotic behavior of the avalanche distribution with the result that\nit has the form of a difference between two close power laws. This has\nbeen previously proposed as an explanation for the failure of the\nfinite size scaling approach \\cite{Menech98,Dhar99}.\n\nThe abelian sandpile model is, from its definition, a Markovian\nprocess whose states are specified by the lattice configuration. Once\nthe initial point has been chosen randomly the dynamics of relaxation is\n deterministic, with the evolution determined by the\ninitial configuration and the toppling rule.\n \nWe consider a coarse grain description of the sandpile\nevolution. Instead of a complete description, involving the number of\ngrains at each site of the lattice, we use as variable the total\nnumber of unstable sites which exist at a given time, after an\navalanche has started, irrespective of the configuration in which the\nsandpile is. This description is stochastic since the transition\nbetween two states with a given number of unstable sites depends on\nthe configuration of the sandpile which is now taken randomly. Let us\nconsider for this process the evolution equation describing the\ntransition between states with different numbers of unstable sites: $\nP(t+1, n)=\\sum_{n'} W(t;n,n')P(t,n')$; where $ W(n,n';t) $ is obtained\nby averaging over all the transitions between the configurations with\nthe same number of unstable sites $n,n'$ at a given time $t$. The\ntransition probability $W(t;n,n')$ may depend on time since the\nconfiguration of the unstable sites also depends on time; i.e. at the\nfirst step of a wave the unstable sites are some of the nearest\nneighbors of the initial site, eventually moving away. In this\nformulation the wave is equivalent to a particle performing a discrete\nrandom walk on the positive semi-axis with the transition\nprobabilities depending on the particle position. An wave event\nis a first arrival at the origin problem, for the wave stops when all\nthe sites, except the initial one, are stable. This random walk is\nclose to a diffusion process since the number of unstable sites varies\nwith bounded steps, $x(t+1)<2Dx(t)$, where $x(t)$ is the number of\nunstable sites and $D$ is the lattice dimension. From this analogy we\nexpect that the distribution of first arrival at the origin, that is\nthe wave duration, has a power law distribution as it has for the\nfirst return distribution of the simple diffusion process.\n\nIf we take the lattice size to infinity and the time unit to zero in\nan appropriate way the discrete Markov chain can be cast into a\ndiffusion equation via the Kramer-Moyal expansion \\cite{Risken96}.\nHere we shall not propose an explicit way to construct the FPE for the\nASM, instead we use the fact that in the stationary state the sandpile\nis at criticality and we shall investigate the general FPE which\nyields critical behavior and has diffusion and drift coefficients\nbehaving similarly to the sandpile model. Generically the FPE has the\nform $ \\partial_t p =-\\partial_x [v(x,t)p]\n+(1/2)\\partial^{2}_{xx}[D_2(x,t)p] $, where $p(x,t)$ is the\nprobability density of the number of unstable sites, $x$ is the number\nof unstable sites, $t$ is the time since the wave has started. The\ndrift coefficient $v(x,t)$ and the diffusion coefficient $D_2(x,t) $\nare obtained by taking the continuum limit of the local first order\nmoment $\\sum_j(x_i-x_j)W(i,j;t)$ and of the local second order moment\n$\\sum_j(x_i-x_j)^2W(i,j;t)$ \\cite{Risken96}. Numerically we have\nfound that the discrete diffusion coefficient $D_2(x,t)$ depends\nlinearly on the number of unstable sites $x$. The slope depends on the\ndimension of the lattice but does not depend on the lattice size and\nthe geometric condition (bulk or boundary wave)( see\nFig. \\ref{fig1}). Also we have found that the discrete drift\ncoefficient $v(x)$ tends to a constant as the size of lattice grows,\nFig.\\ (\\ref{fig1}). The finite size effects affect the drift\ncoefficient for the bulk waves at any value of the number of unstable\nsites, since the transition to states with larger number of unstable\nsites is smaller when the wave takes place near the boundary, while\nthe statistics collects all the waves.\n\nThe simplest Fokker-Planck equation satisfying the scale invariance\nassumption and the numerical behavior of the discrete coefficients\n$D_2(x,t)$ and $v(x,t)$ is\n\\begin{equation}\\label{FPinit}\n\\partial_t p(x,t)=\n-\\partial_x[v t^{-\\alpha} p(x,t)]\n+\\partial^{2}_{xx}[D_2 x t^{-\\alpha} p(x,t)]\n\\end{equation} \nwhere $v,D_2,\\alpha$ are constants. The initial condition for the\n above equation is $p(x,t=t_0)=\\delta(x-x_0)$. Since we are interested\n in the time and size distribution for waves, which are first arrival\n events, we set an absorbing boundary condition at the origin\n $p(x=0,t)=0$, for the wave stops when the number of unstable sites,\n except for the initial one, is zero. The above differential equation\n is invariant under a scale transformation $x\\rightarrow bx,\\quad\n t\\rightarrow b^{1/(1-\\alpha)}t$. We observe that we can eliminate the\n parameter $D_2$ by variable change $x\\rightarrow D_2x$ and we change\n to $v\\rightarrow v/D_2$.\n\nUsing a standard approach \\cite{Feller71b} one can find the asymptotic\nbehavior of the first arrival at the origin for Eq. (\\ref{FPinit}):\n $P_t(t)\\approx t^{-\\tau_t},\\quad t\\gg 1$ with\n\\begin{eqnarray}\n\\tau_t&=&1+(1-\\alpha)|1-v|\\label{eqtaut}.\n\\end{eqnarray}\n\nThe second critical exponent we are interested in is the size\ndistribution of waves. The size of the wave is the sum of the\nnumber of unstable sites until the wave stops; in the continuous\nformulation we have $ s(t)=\\int_{t_0}^{t}dt' x(t')+x(t_0) $. \n\n We make the observation that the variable $s$ is a monotonic function\n of time as $\\dot{s}=x(t) > 0$. The relation between the two\n variables can be found using the average relation $ \\langle \\dot\n s(t)\\rangle=\\langle x(t)\\rangle$. Multiplying Eq. (\\ref{FPinit}) by\n $x$ and integrating over $x$ we obtain that $ \\langle x(t)\\rangle\n \\approx t^{1-\\alpha},\\quad t \\gg 1 $, where the average was\n normalized to the probability of surviving $ \\int dx p(x,t)$ until\n the moment $t$. Now we can use the variable $s$ in the time\n distribution of waves. We have $t^{-\\tau_t}dt\\approx\n s^{\\frac{1}{2-\\alpha}}ds^{\\frac{1}{2-\\alpha}}=s^{-\\tau_a}ds$ for\n large $t$ and $s$, where\n\\begin{equation}\n\\tau_a=1+\\frac{(1-\\alpha)}{2-\\alpha}|1-v|\\label{eqtaua}.\n\\end{equation}\nThis result can be checked easily by Monte Carlo simulation of a\nrandom walk constructed to be the discrete version of\nEq. (\\ref{FPinit}) or equivalent ones.\n\nIn Table \\ref {table} we show the values of the critical exponents\n$\\tau_a,\\tau_t$ taken from \\cite{Ktitarev00}, using\nEqs.~(\\ref{eqtaut},\\ref{eqtaua}) we have computed the values of the\nparameters $v$ and $\\alpha$. We observe that $\\alpha$ has the same\nvalue for bulk and boundary waves and the time inhomogeneity\ndisappears at the critical dimension, $\\alpha=0$ for $D=4$. Thus the\nexponent $\\alpha$ can be interpreted as a measure of the correlation\namong the unstable sites bellow critical dimension. \n\nAlso, for $D=2$ we note that $\\alpha$ has the same numerical value as\nthe critical exponent characterizing the decay of the autocorrelation\nfunction of waves found in \\cite{Menech00}. One more hint that the\nexponent $\\alpha$ is related to the correlation on the lattice is that\nit depends linearly on the erased random critical exponent $\\nu$ which\nhas the values $\\nu=4/5\\,( D=2),\\, \\nu=0.616\\, (D=3),\\, \\nu=1/2\\,\n(D\\ge 4)$ \\cite{Ktitarev00,Bradley95}. Indeed one can see easily that\nthe relation\n\\begin{equation}\\label{eqaofnu}\n\\alpha=\\frac{4}{3}\\Bigl(\\nu-\\frac{1}{2}\\Bigr)\n\\end{equation}\nholds exactly in $D=2$ and $D=4$ and has an error of $0.003$ for\n$D=3$, case in which the critical exponent $\\nu$ is only known from\nnumerical simulation \\cite{Bradley95}. \n\nIn the following we shall use Eq. (\\ref{eqaofnu}), which holds for\nboth bulk and boundary waves, as a conjecture for further exploration\nof the Table I. The fact that $\\alpha$ has the same value for both\nbulk and boundary waves can be checked numerically using the first two\nmoments $m_n(t)=\\int x^np(x,t)$ of the solution of\nEq. (\\ref{FPinit}). Integrating Eq.\\ (\\ref{FPinit}) over $x$ and using\nthe absorbing boundary conditions at the origin, we obtain\n$m_1(t)/m_0(t)\\approx t^{1-\\alpha}$ which is independent of $v$.\nNumerical estimation of the above ratio in $D=2,3,4$, presented in\nFig. (\\ref{fig2}), shows an excellent agreement with the predicted\nvalues (the error is less than $0.01$).\n\n\nIn \\cite{Ktitarev00} it was shown that for waves the critical\nexponents for size and time distribution are given by\n\\begin{equation}\\label{tauktit}\n\\begin{array}{l}\n\\tau_{a}=2-\\frac{1+\\sigma}{d_f}\\\\\n\\tau_{t}=1+(d_f-(1+\\sigma))\\nu\n\\end{array}\n\\end{equation} \nwhere $d_f$ is the fractal dimension of the wave, which has the values of\nthe Euclidean dimension for $D=2,3,4$ and value $4$ for higher\ndimension; $\\sigma=1,0$ for bulk and boundary\nwave respectively. \n\nUsing the above proposed conjecture, Eq.~(\\ref{eqaofnu}), and\nEqs.~(\\ref{eqtaut},\\ref{eqtaua},\\ref{tauktit}), from\n$(2-\\alpha)(\\tau_a-1) =\\tau_t-1$ we obtain\n\\begin{equation}\\label{eq:d_f}\nd_f=\\frac{8}{3}\\frac{1}{\\nu}-\\frac{4}{3}.\n\\end{equation}\nThis relation shows that that the minimum critical dimension of the\nabelian sandpile model is $4/3$ since the maximum value of $\\nu$ is\n$1$. This result is in agreement with the observation that in $D=1$\nthe scaling in the abelian sandpile model breaks down \\cite{Dhar99}.\n\nAn additional benefit of the above relation is that it gives the value\nof the critical exponent for the loop erased random walk in three\ndimension. Indeed if we put $d_f=3$, and keep in mind that $d_f=D$\nfor $D<4$, we get $\\nu_{D=3}=8/13=0.615384...$ in perfect agreement\nwith the numerical value found in \\cite{Bradley95}. In fact, assuming\nthe conjecture Eq. (\\ref{eqaofnu}) and identifying $D=d_f$,\nEq. (\\ref{eq:d_f}) yields a relation between the LERW critical exponent\nand space dimensionality.\n\nKtitarev {\\it et al} \\cite{Ktitarev00} argue that all critical\nexponents of the abelian sandpile are determined by the critical\nexponent $\\nu$. In order to complete this program we need the relation\nbetween the drift coefficient $v$ and $\\nu$ and a relation between the\nthe drift coefficients $v$ for the bulk and boundary waves. Using the\nEq. (\\ref{tauktit}) for $\\tau_t$ together with\nEqs. (\\ref{eqtaut},\\ref{eqtaua}) we obtain \n\\begin{equation}\\label{vofv}\n\\frac{|1-v_{\\text{bulk}}|}\n{|1-v_{\\text{boundary}}|}\n=\\frac{d_f-2}{d_f-1}.\n\\end{equation} \n\nWe need one more relation to connect the coefficients $\\alpha$ and $v$\n(bulk or boundary) with $\\nu$. This can be obtained by using for\nexample Eqs. (\\ref{eqtaua},\\ref{eqaofnu},\\ref{tauktit}) from which we\nextract\n\\begin{equation}\\label{vofnu}\nv_{\\text{bulk}}=\\left\\{\n\\begin{array}{ll}\n\\frac{6\\nu-3}{-4\\nu+5}&\\quad \\mbox{if}\\quad \\nu \\le 4/5\\\\\n\\frac{-14\\nu+13}{-4\\nu+5}&\\quad \\mbox{if}\\quad \\nu > 4/5\\\\\n\\end{array}.\n\\right .\n\\end{equation} \n\nWe make the observation that the\nEqs.~(\\ref{eqaofnu},\\ref{vofv},\\ref{vofnu}) hold exactly in $D=3$ if\nwe use the deduced value $\\nu=8/13$, see Table I. At this point, we\ncan conclude that we have found a self-consistent description of the\ncritical properties for the time and size distribution of waves in\nASM. The exponent $\\alpha$ captures the lattice correlation and the\ndrift $v$ controls the boundary condition of the wave, both of them\nbeing a function of the erased loop random walk critical exponent\n$\\nu$ through the Eqs.~(\\ref{eqaofnu},\\ref{vofv},\\ref{vofnu}).\n\nNow we are in the position to compute the asymptotic behavior for the\navalanches in $D=2$ using the FPE. In this description an avalanche is\nthe sum of a random number of waves. The waves are statistically\nindependent being recurrent events \\cite{Feller71a}. This assumption\nmight appear to be in contradiction with analysis of \\cite{Menech00}\nbut in fact in this approach the correlation is included already in\nthe time inhomogeneity. When a wave of size $s$ touches the origin it\nhas the the probability $p_d(s)$ to die, thus also concluding the\navalanche, or a new wave can start with the probability\n$(1-p_d(s))$. We choose the probability for the wave to die as\n$p_d(s)=s^{-3/8}\\ln s$, so as to recover asymptotically the\nprobability for the last wave: $p_w(s)p_d(s)= p_{lw}(s)\\sim\ns^{-\\frac{11}{8}}$ \\cite{Dhar94,Ivashkevich94}. The avalanche\ndistribution can then be written as\n%\\begin{equation}\n%\\begin{array}{lcl}\n$\np_{a}(s)=p_w(s)p_d(s)\n+\\int_{0}^{s}ds' p_{w}(s')(1-p_d(s'))\np_{w}(s-s')p_d(s-s')+\\dots\n$.\n%\\end{array}\n%\\end{equation}\nWe can sum the previous series after a Laplace transform in $s$ and we\nhave\n\\begin{equation}\np_a(\\lambda)=\\frac{p_{lw}(\\lambda)}{1-((1-p_d)p_w)_\\lambda}.\n\\end{equation}\nApplying again the Tauberian theorem \\cite{Feller71b} we find that\nasymptotically the avalanche size distribution behaves like\n\\begin{equation}\np_a(s)\\approx C_1(s\\ln s)^{-1}+C_2 s^{-\\frac{11}{8}}.\n\\end{equation}\nThis kind of behavior has been proposed previously in the literature\n\\cite{Dhar99}. The fact that $1< 11/8 <2$ makes it difficult to obtain\nthe 'pure' dominant behavior. From a numerical fit we obtain that\n$C_2/C_1\\approx -0.25$, therefore $C_1(s\\ln s)^{-1} \\gg C_2\ns^{-\\frac{11}{8}}$ for $s \\ge 10^6$. Thus, the FPE approach predicts\nthat the avalanche distribution in the bulk must have the same\nasymptotic behavior as the waves for very large values of $s$,\nprovided that the statistics excludes the avalanches which are\naffected by the boundary.\n\nIn conclusion, we have used numerical hints to propose a FPE for the\ntime and size distributions of the waves in the ASM. In this approach\na wave is a first return event and the asymptotic properties of its\ndistributions (time and size) are described by the first return\nprobabilities of a time inhomogeneous FPE; the time inhomogeneity\nappears below the critical dimension $D=4$. Furthermore, this\napproach yields an analytical expression for the asymptotic behavior\nof the avalanche distribution in $D=2$ which goes beyond the finite size\nscaling hypothesis and in agreement with recent results\n\\cite{Menech98,Dhar99}.\n\nUsing the relation for the critical exponents $\\tau_a$, $\\tau_t$\ndeduced in \\cite{Ktitarev00} together with the the relations found\nthrough FPE approach and the conjecture (\\ref{eqaofnu}) we the propose\nexplicit dependence of the critical exponents $\\tau_a$, $\\tau_t$ of\nthe critical exponent of LERW $\\nu$ (via $\\alpha$ and $v$). A bonus of\nthis approach is that it yields the value of the lower critical\ndimension, $4/3$, for the ASM and the exact value of $\\nu=8/13$ in\n$D=3$ for LERW.\n\nThe author thanks H.~B.~Geyer, F.~Scholtz, L.~Boonaazier and A.~van\n~Biljon for useful discussions and H.~B.~Geyer for a critical reading\nof the manuscript.\n\n\\begin{references}\n\n\\bibitem[*]{emailaddress}\nemail: anton@physics.sun.ac.za\n\n\\bibitem{Bak87}\nP. Bak, C. Tang, and K. Wiesenfeld, Phy. Rev. Lett. {\\bf 59}, 381\n (1987).\n\n\\bibitem{Ktitarev00}\nD. Ktitarev, S. L\\\"{u}beck, P. Grassberger, and V. Priezzhev, \nPhys. Rev. E {\\bf 61}, 81 (2000), cond-mat/9907157.\n\n\\bibitem{Menech98}\nM. De~Menech, A.~L. Stella, and C. Tebaldi, Phy. Rev. E {\\bf 58}, 2677\n (1998), cond-mat/9805045.\n\n\\bibitem{Menech00}\nM. De~Menech, and A.~L. Stella, unpublished,cond-mat/0002310.\n\n\\bibitem{Drossel99}\nB. Drossel, cond-mat/9904075 v2 (unpublished).\n\n\\bibitem{Tebaldi99}\nC. Tebaldi, M. De~Menech, and A.~L. Stella, Phy. Rev, Lett.\n{\\bf 83}, 3952,(1999), cond-mat/99032270.\n\n\\bibitem{Dhar90}\nD.~ Dhar, Phy. Rev. Lett. {\\bf 64}, 1613 (1990).\n\n\\bibitem{Dhar94}\nD.~ Dhar and S.~.S.~ Manna, Phy. Rev. E {\\bf 49}, 2684 (1994).\n\n\\bibitem{Ivashkevich94} E.~V.~ Ivashkevich, D.~V.~ Ktitarev and, V.~B.~\nPriezzhev, Physica A {\\bf 209}, 347 (1994).\n\n\\bibitem{Dhar99} D.~ Dhar, Physica A {\\bf 263}, 4 (1999).\n\n\\bibitem{Bradley95} R.~E.~Bradley, S.~Windwer, Phys. Rev. E {\\bf 51}, 241 (1995)\n\n\\bibitem{Risken96}\nH. Risken, {\\em The Fokker-Planck Equation} (Springer-Verlang, Berlin\n Heidelberg, 1996).\n\n\\bibitem{Feller71a}\nW. Feller, {\\em An Introduction to Probability Theory and Its Application}\n (Wiley, New York, 1971), Vol.~I.\n\n\\bibitem{Feller71b} W. Feller, {\\em An Introduction to Probability\nTheory and Its Application} (Wiley, New York, 1971), Vol.~II;\nA.~J. Bray, Phys. Rev. E {\\bf 62}, 103 (2000), cond-mat/9910135.\n\n%\\bibitem{Gradsteyn80}\n%I. Gradshteyn and I. Ryzhik, {\\em Table of Integrals, Series and Products}\n% (Academic Press, New York, 1980).\n\n\\end{references}\n\n\\begin{figure}\n%\\epsfxsize=5cm\n%\\epsfysize=10cm\n\\vspace*{-15mm}\n\\centering\n\\epsfig{file=fig1.ps,%\nheight=100mm, width=75mm,bbllx=100pt,bblly=100pt,bburx=495pt,bbury=742pt}\n\\vspace*{5mm}\n\\caption{The local drift coefficient for bulk and boundary wave in\na)$D=2$; $L=1024\\,(+)$, $L=512\\,(\\times)$ for bulk waves and\n$L=1024\\,(\\Box)$, $L=512(\\circ)$ for boundary waves; b) $D=3$;\n$L=128\\,(+)$, $L=64(\\times)$ for bulk waves and $L=128\\,(\\Box)$,\n$L=64(\\circ)$ for boundary waves; c) $D=4$; $L=32\\,(+)$, \n$L=16\\,(\\times)$ for bulk waves and $L=32\\,(\\Box)$, $L=16(\\circ)$ for\nboundary waves; d)the local second order moment for bulk waves $(+)$\n($D=2(L=1024)$, $D=3(L=128)$, $D=4(L=32)$) and for boundary waves\n$\\Box$ ($D=2(L=512)$, $D=3(L=64)$, $D=4(L=16)$). \n%Note that the\n%slope of the second moment is independent of the geometrical condition\n%of the wave and also of the system size.\n}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n%\\vspace*{-13mm}\n\\centering\n\\epsfig{file=fig2.ps,%\nheight=80mm, width=75mm,bbllx=100pt,bblly=100pt,bburx=495pt,bbury=742pt}\n\\vspace*{5mm}\n\\caption{Finite size scaling for the moment ratio $m_{10}(t)=m_1(t)/m_0(t)$\nin: $D=2$(a), $D=3$ (b), $D=4$ (c) with continuous line for bulk\nwaves and with dashed line for boundary waves. There is a good fit of the\ndata with the exponents taken from Table (\\protect\\ref{table}). The\nlogarithmic correction to scaling at $D=4$ has the same exponent as\nin Ref.\\ \\protect\\cite{Ktitarev00}.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{table}\n\\caption{The values for the critical exponent $\\tau_t,\\,\\,\\tau_a$ for\nthe time and size distribution of waves taken from\n\\protect\\cite{Ktitarev00} together with the values of the exponent\n$\\alpha$ and the drift coefficient $v$ computed from Eqs.\n(\\protect\\ref{eqtaut},\\protect\\ref{eqtaua}). The last line shows the\nvalues of the erased loop random walk critical exponent $\\nu$. For\n$D=3$ we show in parenthesis the values computed with the exact value\n $\\nu=8/13$.}\n\\label{table}\n\\begin{tabular}{llll}\nD & 2 & 3 & 4\\\\\n\\tableline\nbulk& $\\tau_a=1$& $\\tau_a=4/3$ & $\\tau_a=3/2$\\\\\n & $\\tau_t=1$& $\\tau_t=1.616\\;(19/13)$ & $\\tau_t=2$\\\\\n & $\\alpha=2/5$& $\\alpha=0.152\\;(2/13)$ & $\\alpha=0$\\\\\n & $v=1$ & $v=0.274\\;(3/11)$ & $v= 0$\\\\\nboundary& $\\tau_a=3/2$& $\\tau_a=5/3$ & $\\tau_a=7/4$ \\\\\n & $\\tau_t=9/5$& $\\tau_t=2.232\\;(29/13)$ & $\\tau_t=5/2$ \\\\\n & $\\alpha=2/5$& $\\alpha=0.152\\;(2/13)$ & $\\alpha=0$ \\\\\n & $v=-1/3$ & $v=-0.453\\;(-5/11)$ & $v=-1/2$\\\\\n$\\nu$ & $4/5$ & $0.616\\;(8/13)$ & $1/2$\n\\end{tabular}\n\\end{table}\n\n\n%\\bibliographystyle{/home/anton/revtex/prsty}\n%\\bibliography{/home/anton/papers/biblio}\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n% LocalWords: Fokker LERW abelian ASM Bak Wiesenfeld hypercubic FPE Eq\n% LocalWords: topplings\n"
}
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[
{
"name": "cond-mat0002028.extracted_bib",
"string": "\\bibitem[*]{emailaddress}\nemail: anton@physics.sun.ac.za\n\n\n\\bibitem{Bak87}\nP. Bak, C. Tang, and K. Wiesenfeld, Phy. Rev. Lett. {\\bf 59}, 381\n (1987).\n\n\n\\bibitem{Ktitarev00}\nD. Ktitarev, S. L\\\"{u}beck, P. Grassberger, and V. Priezzhev, \nPhys. Rev. E {\\bf 61}, 81 (2000), cond-mat/9907157.\n\n\n\\bibitem{Menech98}\nM. De~Menech, A.~L. Stella, and C. Tebaldi, Phy. Rev. E {\\bf 58}, 2677\n (1998), cond-mat/9805045.\n\n\n\\bibitem{Menech00}\nM. De~Menech, and A.~L. Stella, unpublished,cond-mat/0002310.\n\n\n\\bibitem{Drossel99}\nB. Drossel, cond-mat/9904075 v2 (unpublished).\n\n\n\\bibitem{Tebaldi99}\nC. Tebaldi, M. De~Menech, and A.~L. Stella, Phy. Rev, Lett.\n{\\bf 83}, 3952,(1999), cond-mat/99032270.\n\n\n\\bibitem{Dhar90}\nD.~ Dhar, Phy. Rev. Lett. {\\bf 64}, 1613 (1990).\n\n\n\\bibitem{Dhar94}\nD.~ Dhar and S.~.S.~ Manna, Phy. Rev. E {\\bf 49}, 2684 (1994).\n\n\n\\bibitem{Ivashkevich94} E.~V.~ Ivashkevich, D.~V.~ Ktitarev and, V.~B.~\nPriezzhev, Physica A {\\bf 209}, 347 (1994).\n\n\n\\bibitem{Dhar99} D.~ Dhar, Physica A {\\bf 263}, 4 (1999).\n\n\n\\bibitem{Bradley95} R.~E.~Bradley, S.~Windwer, Phys. Rev. E {\\bf 51}, 241 (1995)\n\n\n\\bibitem{Risken96}\nH. Risken, {\\em The Fokker-Planck Equation} (Springer-Verlang, Berlin\n Heidelberg, 1996).\n\n\n\\bibitem{Feller71a}\nW. Feller, {\\em An Introduction to Probability Theory and Its Application}\n (Wiley, New York, 1971), Vol.~I.\n\n\n\\bibitem{Feller71b} W. Feller, {\\em An Introduction to Probability\nTheory and Its Application} (Wiley, New York, 1971), Vol.~II;\nA.~J. Bray, Phys. Rev. E {\\bf 62}, 103 (2000), cond-mat/9910135.\n\n%\n\\bibitem{Gradsteyn80}\n%I. Gradshteyn and I. Ryzhik, {\\em Table of Integrals, Series and Products}\n% (Academic Press, New York, 1980).\n\n"
}
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cond-mat0002029
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Electron-phonon scattering in quantum wires exposed to a normal magnetic field
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[
{
"author": "Mher M. Aghasyan"
}
] |
A theory for the relaxation rates of a test electron and electron temperature in quantum wires due to deformation, piezoelectric acoustical and polar optical phonon scattering is presented. We represent intra- and inter-subband relaxation rates as an average of rate kernels weighted by electron wave functions across a wire. We exploit these expressions to calculate phonon emission power for electron intra- and inter-subband transitions in quantum wires formed by a parabolic confining potential. In a magnetic field free case we have calculated the emission power of acoustical (deformation and piezoelectric interaction) and polar optical phonons as a function of the electron initial energy for different values of the confining potential strength. In quantum wires exposed to the quantizing magnetic field normal to the wire axis, we have calculated the polar optical phonon emission power as a function of the electron initial energy and of the magnetic field.
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[
{
"name": "manuscript.tex",
"string": "\n\\documentstyle[prb,preprint,aps]{revtex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%TCIDATA{Created=Tue May 11 17:49:04 1999}\n%TCIDATA{LastRevised=Fri Jan 07 15:54:11 2000}\n%TCIDATA{Language=American English}\n\n\\newcommand{\\MF}{{\\large{\\manual META}\\-{\\manual FONT}}}\n\\newcommand{\\manual}{rm}\n\\newcommand\\bs{\\char '134 }\n\n\\begin{document}\n\\title{Electron-phonon scattering in quantum wires exposed to a normal magnetic\nfield}\n\\author{Mher M. Aghasyan}\n\\address{Department of Physics, Yerevan State University, 375049 Yerevan, Armenia}\n\\author{Samvel M. Badalyan}\n\\address{Department of Radiophysics, Yerevan State University, \\\\\n375049 Yerevan, Armenia}\n\\author{Garnett W. Bryant}\n\\address{National Institute of Standards and Technology, \\\\\nGaithersburg, MD 20899 USA}\n\\maketitle\n\n\\begin{abstract}\nA theory for the relaxation rates of a test electron and electron\ntemperature in quantum wires due to deformation, piezoelectric acoustical\nand polar optical phonon scattering is presented. We represent intra- and\ninter-subband relaxation rates as an average of rate kernels weighted by\nelectron wave functions across a wire. We exploit these expressions to\ncalculate phonon emission power for electron intra- and inter-subband\ntransitions in quantum wires formed by a parabolic confining potential. In a\nmagnetic field free case we have calculated the emission power of acoustical\n(deformation and piezoelectric interaction) and polar optical phonons as a\nfunction of the electron initial energy for different values of the\nconfining potential strength. In quantum wires exposed to the quantizing\nmagnetic field normal to the wire axis, we have calculated the polar optical\nphonon emission power as a function of the electron initial energy and of\nthe magnetic field.\n\\end{abstract}\n\n\\date{\\today}\n\n%\\narrowtext\n\\newpage\n\n\\section{Introduction}\n\nSemiconductor quantum wires attract considerable interest both for\nunraveling novel fundamental phenomena and for possible device applications.\nRapid carrier relaxation is crucial for many of technological applications\nof these systems, therefore, understanding and characterizing carrier\nscattering in quantum wires are important for controlling carrier dynamics\nin thermalization, optical, and transport processes.\n\nTheoretically, electron-phonon relaxation in quantum wires has been\naddressed in several works \\cite\n{leburton,fishman,ridley,gold,sarma,shik,constan,mitin,band,peet,pevzner}.\nScattering by optical phonons has been investigated in rectangular wires \n\\cite{leburton,sarma}. In cylindrical wires, a simple model with constant\nelectron wave function inside the wire \\cite{fishman}, an infinite and a\nfinite well confining potential \\cite{ridley,constan} have been considered.\nInter-subband scattering has been treated \\cite{peet}. Acoustical phonon\nrelaxation has been studied in wires with parabolic (in one direction) \\cite\n{shik} and infinite well confining potentials \\cite{mitin,band}. Optical\nphonon generation has been investigated due to electrophonon resonances\\cite\n{pevzner}. Although significant progress has been achieved, the problem\nstill cannot be considered as solved.\n\nIn this work we present a theory for calculations of the relaxation rates of\na test electron and electron temperature in quantum wires exposed to a\nnormal magnetic field. We represent the intra- and inter-subband relaxation\nrates as an average of rate kernels weighted by the electron wave functions\nacross the wire. Exploiting these expressions and the appropriate forms of\nthe electron subband wave functions, we evaluate the relaxation rates in\nquantum wires under different environments. \n% (for instance, the electron wave functions in T- or V-shaped quantum wires \n%can be used to evaluated the electron-phonon relaxation rates). \nWe discuss the scattering rates in quantum wires in zero and quantizing\nmagnetic fields. In the magnetic field free case we present calculations of\nthe electron scattering rates due to emission of deformation (DA) and\npiezoelectric (PA) acoustical, and polar optical (PO) phonons in quantum\nwires with a parabolic confining potential as a function of the electron\ninitial energy for different values of the inter-subband separation. In the\nquantizing magnetic field applied normal to the wire axis we study electron\nscattering rate due to PO phonon emission as a function of electron initial\nenergy and of the magnetic field.\n\n\\section{Relaxation of a test electron}\n\nIn quantum wires, particle motion is described by eigenfunctions $%\n|\\lambda\\rangle\\equiv |n l k\\rangle=|n l\\rangle|k\\rangle$ which factor into\nsubband functions $|nl\\rangle=\\chi _{nl}({\\bf R})$ (${\\bf R}=(x,y)$) labeled\nby indices $n$ and $l$ corresponding to the lateral quantization across the\nwire and into plane waves $|k\\rangle=e^{i k z}$ labeled by a wave vector $k$\ncorresponding to the free translation electron motion along the wire axes $z$%\n. The single-particle energy is given by $\\varepsilon_{nl}(k)=%\n\\varepsilon(k)+\\varepsilon_{nl}$ where the kinetic energy is $%\n\\varepsilon(k)=\\hbar^2k^2/2m^*$ ($m^*$ is the electron effective mass) and $%\n\\varepsilon_{nl}$ is the subband energy.\n\nThe energy-loss power ${\\cal Q}$ for a test electron \\cite{gantlev} between\nsubbands $n,l$ and $n^{\\prime },l^{\\prime }$ is defined as \n\\begin{eqnarray}\n{\\cal Q}_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^\\Upsilon \\left(\n\\varepsilon (k)\\right) &=&{\\cal Q}_{n,l\\rightarrow n^{\\prime },l^{\\prime\n}}^{+\\Upsilon }(\\varepsilon )-{\\cal Q}_{n,l\\rightarrow n^{\\prime },l^{\\prime\n}}^{-\\Upsilon }(\\varepsilon ), \\label{eq1} \\\\\n{\\cal Q}_{nl\\rightarrow n^{\\prime }l^{\\prime }}^{\\pm \\Upsilon }\\left(\n\\varepsilon \\right) &=&\\sum_{\\lambda ^{\\prime }}^{(\\pm )}\\hbar \\omega\nW_{\\lambda \\rightarrow \\lambda ^{\\prime }}^{\\pm {\\bf q}\\Upsilon }{\\frac{%\n1-f_T\\left( \\varepsilon \\mp \\hbar \\omega \\right) }{1-f_T\\left( \\varepsilon\n\\right) }} \\label{eq2}\n\\end{eqnarray}\nwhere $W_{nlk\\rightarrow n^{\\prime }l^{\\prime }k^{\\prime }}^{\\pm {\\bf q}%\n\\Upsilon }$ is the scattering probability at which one phonon of the mode $%\n\\Upsilon $ with the wave vector ${\\bf q}=(q_z,{\\bf q_{\\perp }})$ and the\nfrequency $\\omega =\\omega _\\Upsilon $ is emitted or absorbed by an electron, \n$f_T$ is the Fermi factor at crystal temperature $T$. The summation $(+)$\nand $(-)$ over the final states $\\varepsilon (k^{\\prime })<\\varepsilon (k)$\nand $\\varepsilon (k^{\\prime })>\\varepsilon (k)$ corresponds to the phonon\nemission and absorption processes, respectively. In the Born approximation\nusing the explicit form of the transition probability $W_{nlk\\rightarrow\nn^{\\prime }l^{\\prime }k^{\\prime }}^{\\pm {\\bf q}\\Upsilon }$, we represent the\nenergy-loss power ${\\cal Q}$ in the following general form\n\n\\begin{eqnarray}\n{\\cal Q}_{{n,l\\rightarrow n^{\\prime },l^{\\prime }}}^{\\pm \\Upsilon }\\left(\n\\varepsilon \\right) &=&{\\cal Q}_0^\\Upsilon \\!\\int \\!\\!d^2\\!R\\!\\!\\int\n\\!\\!d^2\\!R^{\\prime }\\chi _{n^{\\prime }l^{\\prime }}^{*}\\left( {\\bf R^{\\prime }%\n}\\right) \\chi _{nl}\\left( {\\bf R^{\\prime }}\\right) \\label{eq3} \\\\\n&\\times &\\chi _{n^{\\prime }l^{\\prime }}\\left( {\\bf R}\\right) \\chi\n_{nl}^{*}\\left( {\\bf R}\\right) K_{n,l\\rightarrow n^{\\prime },l^{\\prime\n}}^{\\pm \\Upsilon }\\left( \\varepsilon (k),\\left| {\\bf R}-{\\bf R}^{\\prime\n}\\right| \\right) \\nonumber\n\\end{eqnarray}\nwhere ${\\cal Q}_0^\\Upsilon $ is the nominal power and $K_{n,l\\rightarrow\nn^{\\prime },l^{\\prime }}^{\\pm \\Upsilon }$ is a rate kernel which depends on\nthe type of electron-phonon interaction. By considering different\ninteraction mechanisms, we obtain the rate kernels $K_{n,l\\rightarrow\nn^{\\prime },l^{\\prime }}^{\\pm \\Upsilon }$ and the nominal powers ${\\cal Q}%\n_0^\\Upsilon $. For $\\Upsilon =$ PO phonons: ${\\cal Q}_0^{PO}=\\hbar \\omega\n_{PO}/{\\overline{\\tau }}_{PO}$ and \n\\begin{eqnarray}\nK_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^{\\pm PO}\\left( \\varepsilon\n(k),\\left| {\\bf R}-{\\bf R}^{\\prime }\\right| \\right) &=&\\frac{\\sqrt{\\hbar\n\\omega _{PO}}}{2\\sqrt{\\varepsilon +\\Delta _{nl,n^{\\prime }l^{\\prime }}\\mp\n\\hbar \\omega _{PO}}} \\label{eq4} \\\\\n&\\times &K_0\\left( q_z^{\\pm }\\left| {\\bf R}-{\\bf R}^{\\prime }\\right| \\right)\n\\Psi ^{\\pm }\\left( \\varepsilon _{l,n}(k),\\hbar \\omega _{PO}\\right) . \n\\nonumber\n\\end{eqnarray}\nFor $\\Upsilon =$ DA and $\\Upsilon =$ PA phonons: ${\\cal Q}_0^\\Upsilon =\\hbar\nsp_{_{PO}}/{\\overline{\\tau }}_\\Upsilon $ and \n\\begin{eqnarray}\nK_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^{\\pm \\Upsilon }\\left( \\varepsilon\n,\\left| {\\bf R}-{\\bf R}^{\\prime }\\right| \\right) &=&\\frac{\\sqrt{2ms^2}}{%\n2(sp_{_{PO}})^{2+\\sigma _\\Upsilon }} \\label{eq5} \\\\\n&\\times &\\int\\limits_{\\omega _1^{\\pm }}^{\\omega _2^{\\pm }}\\frac{d\\omega\n\\omega ^{1+\\sigma _\\Upsilon }J_0\\left( q_{\\perp }^{\\pm }\\left| {\\bf R}-{\\bf R%\n}^{\\prime }\\right| \\right) }{\\sqrt{\\varepsilon +\\Delta _{_{ln,l^{\\prime\n}n^{\\prime }}}\\mp \\hbar \\omega }}\\Psi ^{\\pm }\\left( \\varepsilon\n_{ln}(k);\\hbar \\omega \\right) , \\nonumber\n\\end{eqnarray}\nwith $\\sigma _{_{DA}}=2$ and $\\sigma _{_{PA}}=0$. Here the nominal\nscattering times are given by \\cite{gantlev} \n\\begin{equation}\n\\frac 1{\\overline{\\tau }_{_{PO}}}=2\\alpha _{_{PO}}\\omega _{_{PO}},\\frac 1{%\n\\overline{\\tau }_{_{DA}}}=\\frac{\\Xi ^2p_{_{PO}}^3}{2\\pi \\hbar \\varrho s^2},%\n\\frac 1{\\overline{\\tau }_{_{PA}}}=\\frac{(e\\beta )^2p_{_{PO}}}{2\\pi \\hbar\n\\varrho s^2} \\label{eq6}\n\\end{equation}\nwhere $\\alpha _{_{PO}}$ is the Fr\\\"{o}hlich coupling, $\\Xi $ and $e\\beta $\nare the deformation and piezoelectric potential constants, $\\varrho _0$ is\nthe crystal mass density, $s$ the sound velocity, $\\hbar p_{_{PO}}=\\sqrt{%\n2m\\hbar \\omega _{_{PO}}}$, and $\\omega _{_{PO}}$ the polar optical phonon\nfrequency. In Eqs.\\ (\\ref{eq4}) and (\\ref{eq5}) $\\Delta _{nl,n^{\\prime\n}l^{\\prime }}=\\varepsilon _{l,n}-\\varepsilon _{l^{\\prime },n^{\\prime }}$, $%\nJ_0$ is the Bessel function of the first kind and $K_0$ the modified Bessel\nfunction of the second kind, the function $\\Psi ^{\\pm }$ is \n\\begin{equation}\n\\Psi ^{\\pm }\\left( x,y\\right) =\\left( N_T(y)+\\frac 12\\pm \\frac 12\\right) \n\\frac{1-f_T\\left( x\\mp y\\right) }{1-f_T\\left( x\\right) } \\label{eq7}\n\\end{equation}\nwhere $N_T$ is the Bose factor. The phonon momenta are given by \n\\begin{eqnarray}\nq_z^{\\pm } &=&\\sqrt{\\frac{2m}{\\hbar ^2}}\\left| \\sqrt{\\varepsilon (k)}-r\\sqrt{%\n\\varepsilon (k)+\\Delta _{nl,n^{\\prime }l^{\\prime }}\\mp \\hbar \\omega _{_{PO}}}%\n\\right| , \\label{eq8} \\\\\nq_{\\perp }^{\\pm } &=&\\left| \\frac{\\omega ^2}{s^2}-\\left( q_z^{\\pm }\\right)\n^2\\right| ^{1/2}. \\label{eq9}\n\\end{eqnarray}\nIn Eq.\\ (\\ref{eq9}), $q_z$ should be taken from Eq.\\ (\\ref{eq8}) with $%\n\\omega _{_{PO}}$ replaced by $\\omega $, $r=+1$ ( $r=-1$) corresponds to\nscattering (backscattering) processes when the electron momentum $k$ does\nnot (does) change its direction. The limits of integration in Eq.\\ (\\ref{eq5}%\n) are $\\omega _2^{-}=\\infty ,\\omega _2^{+}=\\varepsilon (k_z)+\\Delta\n_{nl,n^{\\prime }l^{\\prime }}/{\\hbar }$ . $\\omega _1^{^{\\pm }}$ are solutions\nof the equation $q_{\\perp }^{\\pm }=0$. For intra-subband scattering $\\omega\n_1^{^{\\pm }}=0$ while inter-subband scattering always has a threshold, $%\n\\omega _1^{^{\\pm }}\\neq 0$. The existence of this threshold is important\nespecially at low temperatures when typical electron energies in scattering\nare $\\varepsilon (k)\\sim \\omega _1^{^{\\pm }}$. If either $\\varepsilon (k)$\nor $\\Delta _{nl,n^{\\prime }l^{\\prime }}$ are much larger than $\\omega\n_1^{^{\\pm }}$ then one can find the following analytical approximation \n\\begin{equation}\n\\omega _1^{^{\\pm }}\\approx {\\frac 1\\hbar }{\\frac{\\left| \\sqrt{\\varepsilon (k)%\n}-r\\sqrt{\\varepsilon (k)+\\Delta _{nl,n^{\\prime }l^{\\prime }}}\\right| }{%\n\\left( \\sqrt{2ms^2}\\right) ^{-1}\\mp r\\left( 2\\sqrt{\\varepsilon (k)+\\Delta\n_{nl,n^{\\prime }l^{\\prime }}}\\right) ^{-1}}.} \\label{eq10}\n\\end{equation}\nThus, Eqs. (\\ref{eq1})-(\\ref{eq3}) with the rate kernels given by Eqs.\\ (\\ref\n{eq4}) and (\\ref{eq5}) provide a new approach to calculate the energy-loss\npower due to PO, PA, and DA phonon scattering in quantum wires with an\narbitrary cross section and under different environments.\n\n\\section{Electron temperature relaxation}\n\nIf the distribution of hot electrons can be described by an electron\ntemperature $T_e>T$, we can determine the energy relaxation rate for the\nwhole electron gas. In this case electron temperature relaxation between\nsubbands $n,l$ and $n^{\\prime },l^{\\prime }$ can be described by the\nenergy-loss power per electron \\cite{gantlev} which is given by \n\\begin{eqnarray}\n\\overline{Q}_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^\\Upsilon (T_e,T) &=&%\n\\overline{Q}_{_{n,l\\rightarrow n^{\\prime },l^{\\prime }}}^{+\\Upsilon }-%\n\\overline{Q}_{_{n,l\\rightarrow n^{\\prime },l^{\\prime }}}^{-\\Upsilon },\n\\label{eq11} \\\\\n\\overline{{\\cal Q}}_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^\\Upsilon\n(T_e,T) &=&{\\frac 1{N_1L}}\\sum_{{n,l,k}}\\,f_{T_e}(\\varepsilon )\\sum_{{%\nn^{\\prime },l^{\\prime }k^{\\prime }}}^{(\\pm )}\\,\\hbar \\omega \\, \\label{eq12}\n\\\\\n&\\times &W_{nlk\\rightarrow n^{\\prime }l^{\\prime }k^{\\prime }}^{\\pm {\\bf q}%\n\\Upsilon }[1-f_{T_e}(\\varepsilon \\mp \\hbar \\omega )]. \\nonumber\n\\end{eqnarray}\nHere $N_1$ is the electron linear concentration. Direct calculations show\nthat to obtain $\\overline{Q}_{_{n,l\\rightarrow n^{\\prime },l^{\\prime\n}}}^{\\pm \\Upsilon }$ one can use Eq.\\ (\\ref{eq3}) but with the kernel $%\nK_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^{\\pm \\Upsilon }$ replaced by the\naverage rate kernel $\\overline{K}_{n,l\\rightarrow n^{\\prime },l^{\\prime\n}}^{\\pm \\Upsilon }$. For $\\Upsilon =$ PO phonons, we obtain \n\\begin{eqnarray}\n\\overline{K}_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^{\\pm PO}\\left(\nT_e,T;\\left| {\\bf R}-{\\bf R}^{\\prime }\\right| \\right) &=&\\frac{p_{_{PO}}}{%\n2\\pi N_1}\\int\\limits_0^\\infty \\frac{d\\varepsilon (k)}{\\sqrt{\\varepsilon (k)}}\n\\label{eq13} \\\\\n&\\times &\\frac{K_0\\left( q_z^{\\pm }\\left| {\\bf R}-{\\bf R}^{\\prime }\\right|\n\\right) \\Phi ^{\\pm }\\left( \\varepsilon _{l,n}(k),\\hbar \\omega _{PO}\\right) }{%\n\\sqrt{\\varepsilon (k)+\\Delta _{_{ln,l^{\\prime }n^{\\prime }}}\\mp \\hbar \\omega\n_{PO}}}. \\nonumber\n\\end{eqnarray}\nFor $\\Upsilon =$ DA and $\\Upsilon =$ PA phonons \n\\begin{eqnarray}\n\\overline{K}_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^{\\pm \\Upsilon }\\left(\nT_e,T;\\left| {\\bf R}-{\\bf R}^{\\prime }\\right| \\right) &=&\\frac{ms}{2\\pi\n\\hbar N_1(sp_{_{PO}})^{2+\\sigma _\\Upsilon }}\\int\\limits_0^\\infty \\frac{%\nd\\varepsilon (k)}{\\sqrt{\\varepsilon (k)}}\\int\\limits_{\\omega _1^{\\pm\n}}^{\\omega _2^{\\pm }}\\frac{\\omega ^3d\\omega }{\\sqrt{\\varepsilon (k)+\\Delta\n_{_{ln,l^{\\prime }n^{\\prime }}}\\mp \\hbar \\omega }} \\label{eq14} \\\\\n&\\times &J_0\\left( q_{\\perp }^{\\pm }\\left| {\\bf R}-{\\bf R}^{\\prime }\\right|\n\\right) \\Phi ^{\\pm }(\\varepsilon _{l,n}(k);\\hbar \\omega ) \\nonumber\n\\end{eqnarray}\nwhere $\\Phi $ is the following function\n\n\\begin{eqnarray}\n\\Phi ^{\\pm }(x;y) &=&\\left( N_T(y)+\\frac 12\\pm \\frac 12\\right) \\label{eq15}\n\\\\\n&\\times &f_{T_e}\\left( x\\right) \\left( 1-f_{T_e}\\left( x\\mp y\\right) \\right) \n\\nonumber\n\\end{eqnarray}\n\n\\section{Scattering in the zero magnetic field}\n\nIn quantum wires with confining potential of cylindrical symmetry, the\nelectron wave functions in the absence of the magnetic field in the plane\nperpendicular to the wire axis are represented in the form. \n\\begin{equation}\n\\chi _{nl}\\left( {\\bf R}\\right) =\\frac 1{\\sqrt{2\\pi }}e^{il\\varphi }\\chi\n_{nl}\\left( R\\right) . \\label{eq16}\n\\end{equation}\nSubstituting this into Eq.\\ (\\ref{eq3}) and integrating over $\\varphi $, we\nrepresent the energy-loss power in the form \n\\begin{eqnarray}\n{\\cal Q}_{{n,l\\rightarrow n^{\\prime },l^{\\prime }}}^{\\pm \\Upsilon }\\left(\n\\varepsilon \\right) &=&{\\cal Q}_0^\\Upsilon \\!\\int \\!RdR\\!\\!\\int \\!R^{\\prime\n}dR^{\\prime }\\chi _{n^{\\prime }l^{\\prime }}^{*}\\left( R^{\\prime }\\right)\n\\chi _{nl}\\left( R^{\\prime }\\right) \\label{eq17} \\\\\n&\\times &\\chi _{n^{\\prime }l^{\\prime }}\\left( R\\right) \\chi _{nl}^{*}\\left(\nR\\right) \\left( K_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^{\\pm \\Upsilon\n}\\left( \\varepsilon (k);R,{R}^{\\prime }\\right) \\right) _{cyl} \\nonumber\n\\end{eqnarray}\nwhere the rate kernel factors into a product of two functions, each of them\ndepends only on the modulus $R$ or $R^{\\prime }$. We find for PO interaction\nthat the rate kernels $\\left( K_{n,l\\rightarrow n^{\\prime },l^{\\prime\n}}^{\\pm PO}\\right) _{cyl}$ and $\\left( \\overline{K}_{n,l\\rightarrow\nn^{\\prime },l^{\\prime }}^{\\pm PO}\\right) _{cyl}$ for a test electron and for\nelectron temperature relaxation can be obtained by the replacement \n\\[\nK_0\\left( q_z^{\\pm }\\left| {\\bf R}-{\\bf R}^{\\prime }\\right| \\right) \\to\nK_{l-l^{\\prime }}\\left( q_z^{\\pm }R\\right) I_{l-l^{\\prime }}\\left( q_z^{\\pm\n}R^{\\prime }\\right) \n\\]\nin Eqs.\\ (\\ref{eq4}) and (\\ref{eq13}), respectively. Here $I_l$ is the\nmodified Bessel function of the first kind and $R>R^{\\prime }$. To obtain\nthe kernels $\\left( K_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^{\\pm\nPA,DA}\\right) _{cyl}$ and $\\left( \\overline{K}_{n,l\\rightarrow n^{\\prime\n},l^{\\prime }}^{\\pm PA,DA}\\right) _{cyl}$, the replacement \n\\[\nJ_0\\left( q_z^{\\pm }\\left| {\\bf R}-{\\bf R}^{\\prime }\\right| \\right) \\to\nJ_{l-l^{\\prime }}\\left( q_z^{\\pm }R\\right) J_{l-l^{\\prime }}\\left( q_z^{\\pm\n}R^{\\prime }\\right) \n\\]\nshould be done in Eqs.\\ (\\ref{eq5}) and (\\ref{eq14}), respectively.\n\nFor the parabolic confining potential $V(R)=m^{*}\\omega _0^2R^2/2$, the\nelectron wave functions are given by \\cite{fock} \n\\[\n\\chi _{nl}\\left( R\\right) =\\sqrt{\\frac{2n!}{(n+|l|)!}}\\frac 1{a_0}e^{-\\frac{%\nr^2}{2a_0^2}}\\left( \\frac r{a_0}\\right) ^{|l|}L_n^{|l|}\\left( \\frac{r^2}{%\na_0^2}\\right) . \n\\]\nThe subband energy $\\varepsilon _{n,l}=\\left( 2n+|l|+1\\right) \\hbar \\omega\n_0 $ where $\\omega _0$ is confining potential strength, $a_0=\\sqrt{\\hbar\n/(m^{*}\\omega _0)}$, $L_n^{|l|}(x)$ gives the generalized Laguerre\npolynomial. Using these functions we have calculated PO, PA, and DA emission\npower for electron transitions between subbands $n,l$ and $n^{\\prime\n},l^{\\prime }$ with $n,l=0,1$ as a function of the electron initial energy\nfor different values of $\\omega _0$. In Figs.\\ \\ref{fg1} and \\ref{fg2} we\npresent the results of calculations. It is seen from Fig.\\ \\ref{fg1} that\nthe PO phonon emission rate diverges at $\\varepsilon =\\Delta _{nl,n^{\\prime\n}l^{\\prime }}+\\hbar \\omega _{PO}$ due to transitions with the electron final\nstates at the subband bottom where the 1D density of states has a\nsquare-root singularity. For energies far from $\\Delta _{nl,n^{\\prime\n}l^{\\prime }}+\\hbar \\omega _{PO}$ we find a weak dependence of the emission\nrate on $\\varepsilon (k)$ while there is a strong dependence on the subband\nseparation $\\omega _0$ and the quantum numbers $n,l$. The intra-subband\nacoustical phonon emission rate has a peak at small energies (Fig.\\ \\ref{fg2}%\n) while inter-subband emission is finite even at $\\varepsilon (k)=0$. The\npeak position decreases with an increase of $a_0$. We find that\nintra-subband DA phonon emission is weaker than PA phonon emission. This\ndifference is less pronounced at inter-subband emission.\n\n\\section{Scattering in the magnetic field normal to the wire axis}\n\nWhen the magnetic field is applied perpendicular to the wire axis, the\nelectron energy and wave functions in the parabolic confining potential $%\nV(x)=m\\omega _x^2x^2/2$ and $V(y)=m\\omega _y^2y^2/2$ are given by\\cite\n{childers}\n\n\\begin{equation}\n\\chi _{n,l}({\\bf R})=\\frac{\\exp \\left[ -\\frac{(x-x_0)^2}{2a_x^2}\\right]\nH_n\\left( \\frac{x-x_0}{a_x}\\right) }{\\sqrt{2^nn!a_x\\sqrt{\\pi }}}\\frac{\\exp\n\\left[ -\\frac{y^2}{2a_y^2}\\right] H_l\\left( \\frac y{a_y}\\right) }{\\sqrt{%\n2^ll!a_y\\sqrt{\\pi }}}, \\label{eq18}\n\\end{equation}\n\n\\begin{equation}\n\\varepsilon _{n,l}(k)=\\varepsilon _B(k)+\\hbar \\Omega _x\\left( n+\\frac 12%\n\\right) +\\hbar \\omega _y\\left( l+\\frac 12\\right) ,\\varepsilon _B(k)=\\frac{%\n\\hbar ^2k^2}{2m_B} \\label{eq19}\n\\end{equation}\nwhere $a_x=\\sqrt{\\hbar /m^{*}\\Omega _x}$, $a_y=\\sqrt{\\hbar /m^{*}\\omega _y}$%\n, $\\Omega _x^2=\\omega _x^2+\\omega _B^2$, $m_B=m^{*}\\Omega _x^2/\\omega _x^2$, \n$\\omega _B=eB/m^{*}c$, and $H_n$ gives the Hermite polynomials.\n\nIn this case to obtain the rate kernels $K^{\\pm \\Upsilon }$ we should\nmultiply Eqs.\\ (\\ref{eq4}), (\\ref{eq5}), (\\ref{eq13}), and (\\ref{eq14}) by a\nfactor $\\Omega _x/\\omega _x$ and replace $\\varepsilon \\left( k\\right) $ by $%\n\\varepsilon _B\\left( k\\right) $ in these equations. Below we will discuss\nonly electron PO phonon scattering. Scattering by acoustical phonons in the\nnormal magnetic field has been studied by Shik and Challis \\cite{shik}.\n\nSubstituting the kernel $K^{PO}$ and wave functions (\\ref{eq18}) in Eq. (\\ref\n{eq3}), we represent the PO phonon emission power in the form\n\n\\begin{equation}\nQ_{n,l\\rightarrow n^{\\prime },l^{\\prime }}^{PO}=Q_0^{PO}\\frac{\\sqrt{\\hbar\n\\omega _{PO}}}{\\sqrt{\\varepsilon _B+\\Delta _{n,l,n^{\\prime },l^{\\prime\n}}-\\hbar \\omega _{PO}}}\\frac{\\Omega _x}{\\omega _x}I_{nl}^{n^{\\prime\n}l^{\\prime }} \\label{eq20}\n\\end{equation}\nwhere the form factors $I_{nl}^{n^{\\prime }l^{\\prime }}$ for the most\nimportant intra-subband $00\\rightarrow 00$ and inter-subband $10\\rightarrow\n00$ transitions are reduced to the following one-dimensional integrals\n\n\\begin{equation}\nI_{00}^{00}=\\int_0^\\infty \\frac{e^{-\\zeta }d\\zeta }{\\left( 2\\zeta\n+q_z^2a_x^2\\right) \\left( 2\\zeta +q_z^2a_y^2\\right) }, \\label{eq21}\n\\end{equation}\n\n\\begin{equation}\nI_{10}^{00}=I_{00}^{00}+\\frac 2{q_z^2a_x^2-q_z^2a_y^2}\\int_0^\\infty \\left(\n1-\\zeta \\right) e^{-\\zeta }\\sqrt{\\frac{2\\zeta +q_z^2a_y^2}{2\\zeta +q_z^2a_x^2%\n}}d\\zeta \\label{eq22}\n\\end{equation}\nwhich we calculate numerically. The results of calculation are shown in\nFigs. \\ref{fg3}-\\ref{fg7}. The diagrams of Figs. \\ref{fg3} and \\ref{fg4}\nrepresent the intra-subband PO phonon emission power dependencies on the\nelectron initial energy and on the magnetic field, respectively, for several\nvalues of the confining potential strengths $\\omega _x$ and $\\omega _y$. It\nis seen from both figures that, as it was in the magnetic field free case,\nthe PO phonon emission power diverges at the phonon emission threshold which\nis given in this case by $\\varepsilon _B\\left( k\\right) =\\hbar \\omega _{PO}$%\n. Because of the electron mass dependence on the magnetic field, the\nthreshold values are trivially shifted to higher electron energies with an\nincrease of the magnetic field. At energies far from the threshold, the\neffect of the magnetic field is weak. At inter-subband transitions, $\\Delta\n_{10,00}$ differs from zero and depends on the magnetic field. In this case\nthere is a threshold line given by $\\varepsilon _B\\left( k\\right) =\\hbar\n\\omega _{PO}-\\Delta _{10,00}$ and shown in Fig. \\ref{fg5}. The electron\nthreshold energy increases in the magnetic field from $\\varepsilon _0=\\hbar\n(\\omega _{PO}-\\omega _x)$ at $B=0$ up to the value $\\varepsilon _1$ at $%\nB=B_1 $ ($B_1$ is determined from threshold conditions $\\Omega _x=2\\omega\n_{PO}/3$). For magnetic fields larger than $B_1$ the threshold energy\ndecreases and vanishes at $B=B_{PO}$ ($B_{PO}$ is determined from the\nresonance $\\Omega _x=\\omega _{PO}$). According to this the emission power\ndependence on the initial energy has no divergence for $B=21,22$ and $25$ T\n(see Fig. \\ref{fg6}). For magnetic fields larger but near $B_{PO}$, $Q^{PO}$\nhas a peak for small values of $\\varepsilon $ while for magnetic fields far\nfrom $B_{PO}$, $Q^{PO}$ increases monotonically in $\\varepsilon $. For a\ngiven value of $\\varepsilon <\\varepsilon _1$, there is an interval of\nmagnetic fields where PO phonon emission is not possible (Fig. \\ref{fg5})\nwhile $Q^{PO}$diverges at the edges of this interval (Fig. \\ref{fg7}). This\ninterval vanishes at $\\varepsilon =\\varepsilon _1$ so that at energies\nlarger but not far from $\\varepsilon _1$, $Q^{PO}$ as a function of the\nmagnetic field has a peak at $B=B_1$ (Fig. \\ref{fg7}). The second peak in\nthe magnetic field dependence of the PO phonon emission power occurs at the\nresonance field $B_{PO}$ and corresponds to the vertical electron\ntransitions with the phonon momentum $q_z=0$.\n\nIn conclusion, we have presented a theory for the test carrier and carrier\ntemperature relaxation rates in quantum wires. This theory has been\nexploited to calculate the PO, PA and DA phonon emission power for electron\nintra- and inter-subband transitions in quantum wires with the parabolic\nconfining potential for different values of the potential strength. We have\ndiscussed the phonon emission power in quantum wires in the zero and\nquantizing magnetic field normal to the wire axis.\\bigskip\n\n\\acknowledgements\nS.M.B. and M.M.A. would like to thank A. A. Kirakossyan for for useful\ndiscussion. The research described in this publication was made possible in\npart by Award No. 375100 of the U. S. Civilian Research \\& Development\nFoundation for Independent States of the Former Soviet Union (CRDF). S.M.B.\nacknowledge partial support from the Belgian Federal Office for Scientific,\nTechnical, and Cultural Affairs.\n\n\\begin{references}\n\\bibitem{leburton} J.P. Leburton, J. Appl. Phys. {\\bf 56} (1984) 2850.\n\n\\bibitem{fishman} Guy Fishman, Phys. Rev. B {\\bf 36} (1987) 7448.\n\n\\bibitem{ridley} N.C. Constantinou and B.K. Ridley, J. Phys.: Condens.\nMatter {\\bf 1} (1989) 2283.\n\n\\bibitem{gold} A. Gold and A. Ghazali, Phys. Rev. B {\\bf 41} (1990) 7626.\n\n\\bibitem{sarma} V.B. Campos and S. Das Sarma, Phys. Rev. B {\\bf 45} (1992)\n3898.\n\n\\bibitem{shik} A.Y. Shik and L.J. Challis, Phys. Rev. B {\\bf 47} (1993)\n2082.\n\n\\bibitem{constan} M. Masale and N.C. Constantinou, Phys. Rev. B {\\bf 48}\n(1993) 11 128.\n\n\\bibitem{mitin} R. Mickevicius and V. Mitin, Phys. Rev. B {\\bf 48} (1993)\n17 194.\n\n\\bibitem{band} N. Telang and S. Bandyopadhyay, Phys. Rev.B {\\bf 48} (1993)\n18 002.\n\n\\bibitem{peet} A.A. Leao {\\it et al.}, Superlatt. \\& Microstruct. {\\bf 13}\n(1993) 37.\n\n\\bibitem{pevzner} V. L. Gurevich, V.B. Pevzner and G. J. Iafrate, J. Phys.:\nCondens. Matter, {\\bf 7} (1995) L445.\n\n\\bibitem{gantlev} V.F. Gantmakher and Y.B. Levinson, Scattering of Carriers\nin Metals and Semiconductors (North-Holland, Amsterdam, 1987).\n\n\\bibitem{fock} V. Fock, Zeitschrift f\\\"{u}r Physik {\\bf 47} (1928) 446.\n\n\\bibitem{childers} D. Childers and P. Pincus, Phys. Rev. {\\bf 177} (1969)\n1036.\n\\end{references}\n\n\\begin{figure}[tbp]\n\\caption{The PO phonon emission power versus the electron initial kinetic\nenergy in zero magnetic field at the intra- and inter-subband electron\ntransitions and for different values of the subband separation $%\n\\hbar\\omega_0 $.}\n\\label{fg1}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{The PA and DA phonon emission power versus the electron initial\nkinetic energy in zero magnetic field at the intra- and inter-subband\nelectron transitions and for different values of the subband separation $%\n\\hbar\\omega_0$.}\n\\label{fg2}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{The PO phonon emission power dependence on the electron initial\nkinetic energy in the quantizing magnetic field at the intra-subband\nelectron transitions for various values of the subband separations $%\n\\hbar\\omega_x$ and $\\hbar\\omega_y$.}\n\\label{fg3}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{The PO phonon emission power dependence on the magnetic field at\nthe intra-subband electron transitions for various values of the subband\nseparations $\\hbar\\omega_x$ and $\\hbar\\omega_y$.}\n\\label{fg4}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{The threshold line in the ($\\varepsilon,B$)-plane which separates\nregions where PO phonon emission is and is not possible. $%\n\\varepsilon_0/\\hbar \\omega_{PO}=0.84$, $\\varepsilon_1/\\hbar \\omega_{PO}=5.97$%\n, $B_1=13.64$ T, $B_{PO}=20.67$ T.}\n\\label{fg5}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\caption{The PO phonon emission power dependence on the electron initial\nkinetic energy at the inter-subband electron transitions for various values\nof the magnetic field.}\n\\label{fg6}\n\\end{figure}\n\\begin{figure}[tbp]\n\\caption{The PO phonon emission power dependence on the magnetic field at\nthe inter-subband electron transitions for various values of electron\ninitial kinetic energy.}\n\\label{fg7}\n\\end{figure}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002029.extracted_bib",
"string": "\\bibitem{leburton} J.P. Leburton, J. Appl. Phys. {\\bf 56} (1984) 2850.\n\n\n\\bibitem{fishman} Guy Fishman, Phys. Rev. B {\\bf 36} (1987) 7448.\n\n\n\\bibitem{ridley} N.C. Constantinou and B.K. Ridley, J. Phys.: Condens.\nMatter {\\bf 1} (1989) 2283.\n\n\n\\bibitem{gold} A. Gold and A. Ghazali, Phys. Rev. B {\\bf 41} (1990) 7626.\n\n\n\\bibitem{sarma} V.B. Campos and S. Das Sarma, Phys. Rev. B {\\bf 45} (1992)\n3898.\n\n\n\\bibitem{shik} A.Y. Shik and L.J. Challis, Phys. Rev. B {\\bf 47} (1993)\n2082.\n\n\n\\bibitem{constan} M. Masale and N.C. Constantinou, Phys. Rev. B {\\bf 48}\n(1993) 11 128.\n\n\n\\bibitem{mitin} R. Mickevicius and V. Mitin, Phys. Rev. B {\\bf 48} (1993)\n17 194.\n\n\n\\bibitem{band} N. Telang and S. Bandyopadhyay, Phys. Rev.B {\\bf 48} (1993)\n18 002.\n\n\n\\bibitem{peet} A.A. Leao {\\it et al.}, Superlatt. \\& Microstruct. {\\bf 13}\n(1993) 37.\n\n\n\\bibitem{pevzner} V. L. Gurevich, V.B. Pevzner and G. J. Iafrate, J. Phys.:\nCondens. Matter, {\\bf 7} (1995) L445.\n\n\n\\bibitem{gantlev} V.F. Gantmakher and Y.B. Levinson, Scattering of Carriers\nin Metals and Semiconductors (North-Holland, Amsterdam, 1987).\n\n\n\\bibitem{fock} V. Fock, Zeitschrift f\\\"{u}r Physik {\\bf 47} (1928) 446.\n\n\n\\bibitem{childers} D. Childers and P. Pincus, Phys. Rev. {\\bf 177} (1969)\n1036.\n"
}
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cond-mat0002030
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Step wise destruction of the pair correlations in micro clusters by a magnetic field
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[
{
"author": "N.K. Kuzmenko"
},
{
"author": "194021"
},
{
"author": "St.-Petersburg"
},
{
"author": "Russia"
},
{
"author": "V.M. Mikhajlov"
},
{
"author": "and S. Frauendorf"
},
{
"author": "Department of Physics"
},
{
"author": "Notre Dame"
},
{
"author": "IN 46556"
},
{
"author": "USA and IKH"
},
{
"author": "Research Center Rossendorf"
},
{
"author": "Germany"
}
] |
The response of $nm$-size spherical superconducting clusters to a magnetic field is studied for the canonical ensemble of electrons in a single degenerate shell. For temperatures close to zero, the discreteness of the electronic states causes a step like destruction of the pair correlations with increasing field strength, which shows up as peaks in the susceptibility and heat capacity. At higher temperatures the transition becomes smoothed out and extends to field strengths where the pair correlations are destroyed at zero temperature.
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[
{
"name": "supermag3.tex",
"string": "\\documentstyle[aps,twocolumn,psfig]{revtex}\n% \\input /usr/lib/teTeX/texmf/tex/generic/misc/psfig.sty\n%\\input psfig\n\\begin{document}\n\\author{N.K. Kuzmenko\\\\V.G. Khlopin Radium Institute, 194021,\nSt.-Petersburg, Russia\\\\\nV.M. Mikhajlov\\\\\nInstitute of Physics St.--Petersburg State\nUniversity 198904, Russia\\\\\nand S. Frauendorf \\\\\nDepartment of Physics, University of Notre Dame,\nNotre Dame, IN 46556, USA and IKH, Research Center\nRossendorf, Germany \n}\n\\title{Step wise destruction of the pair correlations in micro clusters\nby a magnetic field}\n\\maketitle\n\n\\begin{abstract}\nThe response of $nm$-size spherical superconducting clusters\nto a magnetic field is studied for the canonical ensemble of\nelectrons in a single degenerate shell. For temperatures close to zero,\nthe discreteness of the electronic\nstates causes a step like destruction of the pair correlations\n with increasing field strength, which shows up as\n peaks in the susceptibility and heat capacity.\n At higher temperatures the transition becomes\nsmoothed out and extends to field strengths where the pair correlations\nare destroyed at zero temperature.\n\\end{abstract}\n\nThe electron pair correlations in small systems\nwhere the single-particle spectrum is discrete and the mean\nlevel spacing is comparable with the pairing gap have recently\nbeen studied by means of electron transport through\n$nm$-scale Al clusters \\cite{3}. The pair correlations were found to\nsustain an external magnetic field of several Tesla, in contrast to\nmuch weaker critical field of $H_c=99$ Gauss of bulk Al. A step wise\ndestruction of the pair correlations was suggested \\cite{3}. It is\ncaused by the subsequent excitation of quasi particle levels, which gain\nenergy due to the interaction of their spin with the external field.\nThis mechanism is very different from the transition to the normal state\ncaused by a magnetic field applied to a macroscopic superconductor.\nHence it is\nexpected that physical quantities, as the susceptibility $\\chi$ and the\nspecific heat capacity $C$, which indicate the transition, behave very\ndifferently in the micro cluster. The present letter addresses this\nquestion.\n\nFor the micro clusters the energy to remove an electron is much larger than \nthe temperature $T$. The fixed number of the electrons on the cluster was\ndemonstrated by the tunneling experiments \\cite{3}.\nHence, one must study the transition from the paired to the unpaired state\nin the frame of the canonical ensemble. The small number\nof particles taking part in superconductivity causes\n considerable fluctuations of the order parameter, which modify the\ntransition \\cite{muehl,kuzmenko}. Consequences of particle number\nconservation for the pair correlations in micro clusters have also been\ndiscussed recently in \\cite{5,6,golubev,mastellone,braun}, where a more\ncomplete list of references to earlier work can be found.\n\nIn order to elucidate the qualitative features we consider the highly\nidealized model of pair correlations between electrons in a degenerate\nlevel, which permits to calculate the canonical partition function.\nA perfect Al-sphere of radius $R=(1-5)nm$ confines\n$N\\approx 2\\cdot 10^3- 3\\cdot10^5$ free electrons. \nIts electron levels have good angular\nmomentum $l$. Taking the electron spin into account, each of these levels\nhas a degeneracy of $2M$, where $M=2l+1$. \nFor a spherical oscillator potential, the average angular momentum at the \nFermi surface is $\\l\\approx N^{1/3}-1.4$, where \n $N$ is the number \nof free electrons. The distance\nbetween these levels is $\\Delta e \\sim 10$ $meV$ is much larger than\nthe BCS gap parameter $\\Delta$, which is less than 1$meV$. Therefore, \n it is sufficient to\nconsider the pair correlations within the last incompletely filled level.\nThis single-shell model also applies to a hemisphere, because its\nspectrum consists of the spherical levels with odd $l$, and\nto clusters with a\nsuperconducting layer covering an insulating sphere (cf. \\cite{layer})\nor hemisphere.\n\nThe single-shell model Hamiltonian\n\\begin{eqnarray}\\label{ham}\nH=H_{pair}-\\omega (L_z+2S_z), \\\\\nH_{pair}=-GA^+A,\\; \\;\nA^+=\\sum_{k>0} a^+_ka^+_{\\bar {k}}, \\nonumber\n\\end{eqnarray}\nconsists of the pairing interaction $H_{pair}$, which\nacts between the electrons in the last shell\nwith the effective strength $G$,\nand the interaction with the magnetic field. We introduced the\nLarmour frequency $\\hbar\\omega=\\mu_B B$, the Bohr magneton $\\mu_B$,\n the $z$-components of the total\norbital angular momentum and spin $L_z$ and $S_z$.\nThe label $k=\\{\\lambda ,\\sigma \\}$ denotes the\n$z$ -projections of orbital momentum and spin of the electrons,\nrespectively, and\n $A^+$ creates\nan electron pair on states\n$(k,\\bar {k})$, related by the time reversal.\n\n The magnetic susceptibility and\nheat capacity of the electrons are\n\\begin{equation}\\label{chi}\n\\chi=-\\frac{\\mu_B^2}{\\hbar^2V}\\frac{\\partial^2F(T,\\omega)}{\\partial\\omega^2},\n~~C=-\\frac{\\partial^2F(T,\\omega)}{\\partial T^2},\n\\end{equation}\nThe free energy $F$ derived from the Hamiltonian\n(\\ref{ham}) gives only the\nparamagnetic part $\\chi_P$ of the susceptibility, \nbecause we left out the term quadratic\nin $B$. For the fields we are\ninterested in (magnetic length is small as compared to the cluster size),\nthe latter can be treated in first order perturbation theory,\n generating the diamagnetic part of the susceptibility\n\\begin{equation}\\label{chidia}\n\\chi_D=-\\frac{m\\mu_B^2<x^2+y^2>}{\\hbar^2 V}\\sim-4\\cdot 10^{-6}N^{2/3},\n\\end{equation}\nwhere $m$ is the electron mass is and $V$ the volume of the cluster.\nIt is nearly temperature and field independent \\cite{kuzmenko}.\nThe numerical estimate for Al assumes constant electron density. \nFor $nm$ scale clusters \n$\\chi_D\\sim 10^{-3}$. It is much smaller than $\\chi_D\\sim -1$ for \n macroscopic superconductors, which show the Meissner effect. \nSince the magnetic field penetrates\nthe cluster it can sustain a very high field of $B\\sim Tesla$.\n On the other hand, the $\\chi_D$ is three orders of magnitude\nlarger than the Landau diamagnetic susceptibility observed in normal bulk\nmetals. \n\nThe exact solutions to the pairing problem of particles in a degenerate shell\nwere found in nuclear physics \\cite{9}\nin terms of representations of the group $SU_2$.\nThe eigenvalues $E_{\\nu}$ of $H_{pair}$ are\n\\begin{equation}\\label{Enu}\nE_{\\nu}=-\\frac{G}{4}(N_{sh}-\\nu)(2M+2-N_{sh}-\\nu).\n\\end{equation}\nwhere $N_{sh}$ is the number of particles in the shell.\nThe seniority, which is the number of\nunpaired particles, is constrained by $0 \\le \\nu \\le N_{sh}$ and $\\nu\\leq M$.\nThe degenerate states $\\{\\nu,i\\}$ of given seniority $\\nu$\n differ by their magnetic moments $\\mu_Bm_{\\nu,i}$, where $i=\\{L\\Lambda\nS\\Sigma\\}$ takes all values\nof the total orbital $(L)$ and total spin $(S)$ momenta and their total\n$z$-projections $(\\Lambda ,\\Sigma)$ that\nare compatible with the Pauli principle\nfor $\\nu$ electrons.\nIn presence of a magnetic field\n the states have the energy\n\\begin{equation}\\label{Unu}\nU_{\\nu,i}(\\omega)=E_{\\nu} - \\omega m_{\\nu,i},\n~~m_{\\nu,i}=(\\Lambda+2\\Sigma)_{\\nu,i},\n\\end{equation} %4\nand the canonical partition function \n becomes\n\\begin{eqnarray}\\label{Z}\nZ=\\sum_{\\nu,i}\\exp(-\\beta U_{\\nu,i})\\nonumber\\\\\n=\\sum_{\\nu} \\exp(-\\beta E_{\\nu})\n[\\Phi_{\\nu}-\\Phi_{\\nu-2}(1-\\delta_{\\nu .0})], \\\\\n\\beta=1/T,~~~~\\Phi_{\\nu}=\\sum_{i}\n\\exp(-\\beta \\omega m_{\\nu ,i}).\\label{Znu}\n\\end{eqnarray}\nTo evaluate the sums\nwe take into account symmetry of\nthe wave functions of the $\\nu$ unpaired electrons and reduce the sums\n (\\ref{Znu})\n to products of sums over orbital projections of completely\nantisymmetric states (one column Young diagram, cf. \\cite{Hammermesh})\nwith $\\nu /2+\\Sigma$ and $\\nu /2 - \\Sigma$ electrons.\n\\begin{eqnarray}\\label{Phi}\n\\Phi_{\\nu}=\\sum_{\\Sigma=\\Sigma_{min}}^{\\nu /2}\n2(1+\\delta_{\\Sigma .0})^{-1}\n\\tilde\\Phi_{\\nu/2+\\Sigma}\\tilde\\Phi_{\\nu/2-\\sigma}\n\\cosh(2\\beta\\omega\\Sigma), \\\\\n\\Sigma_{min}=[1-(-)^{\\nu}]/4 \\nonumber\\\\\n\\tilde\\Phi_k=\\delta_{k.0}+(1-\\delta_{k.0})\n\\prod_{\\mu=1}^k\\frac{\\sinh\\beta\\omega\n\\frac{2l+2-\\mu}{2}}{\\sinh\\beta\\omega \\frac{\\mu}{2}}.\n\\label{Phit}\n\\end{eqnarray}\nThe derivation of (\\ref{Z} - \\ref{Phit}) will be published separately\n\\cite{oddeven}.\n\nThe pair correlation energy is\n\\begin{eqnarray}\\label{epair}\nE_c(T,\\omega)=\\frac{1}{Z}\\sum_{\\nu}E_{\\nu}exp(-\\beta E_{\\nu})\n[\\Phi_{\\nu}-\\Phi_{\\nu-2}(1-\\delta_{\\nu .0})] \\nonumber\\\\\n\\equiv-\\Delta_c^2(T,\\omega)/G.\n\\end{eqnarray}\nHere we have defined the parameter $\\Delta_c$, which measures the amount\nof pair correlations.\nApplying the mean field approximation and the grand canonical ensemble\nto our model, the thus introduced $\\Delta_c$ becomes the familiar BCS gap\nparameter $\\Delta$. Accordingly\nwe also refer to $\\Delta_c$ as the \"canonical gap\". However, $\\Delta_c$\n must be clearly distinguished from $\\Delta$ because it incorporates the\ncorrelations caused by the fluctuations of the order parameter $\\Delta$.\nFor the case of a half filled shell and even $N_{sh}$,\nthe BCS gap is $\\Delta(0)\\equiv\\Delta(T=0,\\omega=0)=GM/2$. \n Ref. \\cite{3} found $\\Delta(0)=0.3-0.4~meV$\nfor Al-clusters with $R=5-10~nm$,\nwhich sets the energy scale.\n\n \\begin{figure}\n\\noindent\n\\mbox{\\psfig{file=sumagdelom.eps,width=8cm}}\n\\caption{\\label{f:delom}\n Canonical gap $\\Delta_c(T,\\omega)$ (full lines) and BCS gap\n$\\Delta(T,\\omega)$ (\ndotted lines) v.s. the Larmour frequency $\\omega$.\n}\n\\end{figure}\n\n\nLet us first consider the destruction of the pair correlations at $T=0$.\n The lowest state of each seniority multiplet\nhas the maximal magnetic moment\n\\begin{equation}\\label{mnu}\nm_{\\nu}=\\frac{\\mu_B}{4} \\{ \\nu(2M-\\nu)-\n\\frac{1}{2}[1-(-)^{\\nu}] + 4(1-\\delta_{\\nu.0} ) \\}.\n\\end{equation}\n According to (\\ref{Enu}), (\\ref{Unu})\nand (\\ref{mnu}) the state of\nlowest energy changes from $\\nu$ to $\\nu +2$ at\n\\begin{eqnarray}\n\\omega_{\\nu+2}=\\frac{2\\Delta (0)}{M}\\left[\\delta_{\\nu.0} +\n\\frac{M-\\nu}{M-\\nu-1}(1-\\delta_{\\nu.0})\\right].\n\\end{eqnarray} %7\nAt each such step $m_{\\nu}$ increases according to (\\ref{mnu}).\nThe pair correlations\n are reduced because two electron states are blocked.\nAt the last step leading to the maximum seniority $\\nu_{max}$\n all electron states are blocked. Hence the\nfield $B_{c}$ corresponding to $\\omega_{c}=\\omega_{\\nu_{max}}$ can be\nregarded as\nthe critical one, which destroys the pairing completely at $T=0$.\nFor a half filled shell \n$ \\omega_{c}=3\\Delta(0)/M$\nfor even electron number and $4\\Delta(0)/M$ for odd \n($\\nu_{max}=M-1$ and $M$, respectively).\n\n\n\nFig. \\ref{f:delom} illustrates the step wise destruction of pairing by\nblocking for the half filled shell $M=11$ ($l=5$).\nThis mechanism was discussed in\n\\cite{3}, where the crossing of states with different seniority could be\nobserved. It is a well established effect in nuclear physics, where the\nstates of maximum angular momentum, are observed as \"High-K isomers\"\n\\cite{highk}. Fig. \\ref{f:delom} also shows results for the mean field\n(BCS) approximation (cf. \\cite{kuzmenko,9}) to the single shell model.\nThe pair correlations are more rapidly destroyed. The\nquantum fluctuations of the order parameter stabilize the pairing.\n\n\n\n\n\\begin{figure}[t]\n\\noindent\n\\mbox{\\psfig{file=sumagdelt.eps,width=8cm}}\n\\caption{\\label{f:delt}\n Canonical gap $\\Delta_c(T,\\omega)$ (full lines) and BCS gap\n$\\Delta(T,\\omega)$ (\ndotted lines) v.s.\n the temperature $T$}\n\\end{figure}\n\n\n\nWe introduce $T_c$\nas the temperature at which the {\\em mean field\n pair gap} $\\Delta(T_c,\\omega=0)$\n takes the value zero when the magnetic field is absent.\nFor the half filled shell $T_c=\\Delta(0)/2$.\nFig. \\ref{f:delom} shows\nthe mean field gap $\\Delta(T,\\omega)$, It\n behaves as expected from macroscopic\nsuperconductors: The frequency where $\\Delta=0$\nshifts towards smaller values with increasing $T$.\nFig. \\ref{f:delt} shows that the temperature where\n $\\Delta=0$ shifts from $T_c$ to lower values for $\\omega>0$.\n\nHowever, fig. \\ref{f:delom} also demonstrates that the canonical gap\n$\\Delta_c$ behaves differently. For\n$T=0.8T_c$ there is a region above $\\omega_{c}$ where there are still\npair correlations. For $T=2T_c$ this region\nextends to $2\\omega_{c}$. The pair correlations fall off\nvery gradually with $\\omega$. Fig. \\ref{f:delt} shows how these\n\"temperature induced\" pair correlations manifest themselves with increasing\n$T$. For $\\omega=0$ there is a pronounced drop of of $\\Delta_c$ around\n$T_c$, which signalizes the break down of the static pair field. Above this\ntemperature there is a long tail of dynamic pairing.\nFor $\\omega\\geq \\omega_c$ the dynamic pair correlations only built up with\nincreasing $T$.\n\nThe temperature induced pairing can be understood in the following way:\nAt $T=0$, all electrons are unpaired when the\nstate of maximum seniority becomes the ground state for\n$\\omega>\\omega_{crit}$. At $T>0$ excited states with lower seniority enter\nthe canonical ensemble, reintroducing the pair correlations.\n\n Here we have adopted the\nterminology of nuclear physics, calling \"static\" the mean field\n(BCS) part of the pair correlations and \"dynamic\" the quantal and\nstatistical\nfluctuations of the mean field (or equivalently of the order parameter).\nThe \"pair vibrations\", which are oscillations of\nthe pair field around\nzero \\cite{bm2}, are well established in nuclei.\n Fluctuation induced superconductivity was\ndiscussed before \\cite{fluctuation}.\nThe fluctuations play a particularly important role \nin the systems the size of which is smaller than the\ncoherence length,\n\n\n\n \\begin{figure}[t]\n\\noindent\n\\mbox{\\psfig{file=sumagdelom723.eps,width=8cm}}\n\\caption{\\label{f:delom723}\nCanonical gap $\\Delta_c (T,\\omega)$ v.s. $\\omega$. }\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\noindent\n\\hspace{.5cm}\t\n\\mbox{\\psfig{file=sumagchiom.eps,width=7cm}}\n%\\noindent\n%\\mbox{\\psfig{file=sumagchiom23.eps,width=8cm}}\n\\caption{\\label{f:chiom723} Susceptibility $\\chi(T,\\omega)$ v.s. $\\omega$. }\n\\end{figure}\n\n\n\n\n\nFig. \\ref{f:delom723} shows a very small cluster ($M=7$) with very\npronounced steps. Already at $T=0.1T_c$ the steps are noticeably washed out.\nIn the single-shell model the step length is $\\omega_{\\nu}-\\omega_{\\nu-2}\\sim\n\\Delta(0)/M$. Accordingly,\nno individual steps are recognizable at $T=0.1T_c$\nfor the large cluster ($M=23$) shown. Yet there is\nsome irregularity around $\\omega=0.7\\omega_{c}$, which is a residue of\nthe discreteness of the electronic states. It is\nthermally averaged out for $T=0.2T_c$.\nHence only for $M<50$ i. e. $N<2\\cdot10^4$ and $T<0.1T_c$\nthe step wise change of $\\Delta_c$ is observable.\n\n\n\n\n\n\n\nThe discreteness of the electronic levels has dramatic consequences for the\nthe susceptibility at low temperatures. As shown in the upper panel of\nfig. \\ref{f:chiom723}, $\\chi_P$ has pronounced peaks at the frequencies where\nthe states with higher seniority and magnetic moment take over. The\nparamagnetic contribution is very sensitive to the temperature and\n to the fluctuations of the order parameter.\nUsing the BCS mean field approximation we find much more narrow peaks,\nwhich are one to two orders of magnitude higher. For the larger cluster in\nthe lower panel the individual steps are no longer resolved, resulting in a\npeak of $\\chi_P$ near $\\omega=0.7\\omega_{c}$.\n Since for the considered temperatures it\nis unlikely to excite states with finite magnetic moment,\n$\\chi_P$ is small at low $\\omega$ .\nIt grows\nwith $\\omega$ because these states come down.\nIt falls off at large $\\omega$ when approaching\n the maximum magnetic moment of the electrons in the shell.\nThe curve $T=0.1T_c$ shows still a double peak structure, which is residue\nof the discreteness of the electron levels.\n\n\\begin{figure}\n\\noindent\n\\mbox{\\psfig{file=sumagcom723.eps,width=8cm}}\n\\caption{\\label{f:com723} Heat capacity $C(T,\\omega)$ v.s. $\\omega$. }\n\\end{figure}\n\n\n\nThe heat capacity is displayed in fig. \\ref{f:com723}.\nIs has a double peak structure for the $M=7$ cluster at\n $T=0.02T_c$. In this case $C$ is very small because the spacing between the\nstates of different $\\nu$ is much larger than $T$.\nNear a crossing the spacing becomes small and $C$ goes up. The dip\nappears because at the crossing frequency the two states are degenerate.\n Then they do not contribute to $C$ because their\nrelative probability does not depend on $T$.\n For $T=0.1T_c$ the probability\nto excite states with different seniority has increased and\n$C$ takes substantial values between the crossings. The dips due to the\ndegeneracy at the crossings remain.\nFor the $M=23$ cluster $C$ shows only two wiggles, which are\n the residue of the discreteness of the electronic levels.\n\nThe deviation of real clusters from sphericity will attenuate the orbital\npart of $\\chi_P$ and round the steps of $\\Delta_c$\nalready at $T=0$. The back-bending phenomenon\nobserved in deformed rotating nuclei \\cite{highk} is an example.\nHow strongly the orbital angular momentum is suppressed needs to be addressed\nby a more sophisticated model than the present one.\nIn any case, there will be steps caused by the reorientation of\nthe electron spin, if the spin orbit coupling is small as in Al \\cite{3}.\nMost of the findings of the present paper are expected to hold qualitatively\nfor these spin flips.\n\n\nIn summary, at a temperature $T<0.1T_c$ an increasing external magnetic\nfield causes the magnetic moment of small spherical\nsuperconducting clusters ($R<5nm$) to grow in a step like manner.\nEach step reduces the pair correlations until they are destroyed.\nThe steps manifest themselves as peaks in the magnetic susceptibility\nand the heat capacity. The steps are washed out at $T>0.2T_c$.\nFor $T\\sim T_c$, reduced but substantial pair correlations\npersist to a higher field strength than for $T=0$.\nThis phenomenon of the temperature-induced pairing in a strong\nmagnetic field is only found for the canonical ensemble.\n\nSupported by the grant INTAS-93-151-EXT.\n\n\\begin{references}\n%\\bibitem{1} D.C.Ralf,\n%C.T.Black, and M.Tinkham, Phys. Rev. Lett. {\\bf 74},3241(1995).\n%\\bibitem{2} C.T.Black, D.C.Ralph, and M.Tinkham, Phys.Rev.Lett.\n%{\\bf 76},688(1996).\n\\bibitem{3} D.C.Ralf,\nC.T.Black, and M.Tinkham, Phys. Rev. Lett. {\\bf 78},4087(1997) and\nealier works cited therein.\n\\bibitem{muehl} B. M\\\"uhlschlegel, D. J. Scalapino and R. Denton,\nPhys. Rev. {\\bf B 6}, 1767 (1972)\n\\bibitem{kuzmenko} N. K. Kuzmenko, V. M. Mikhajlov, S. Frauendorf,\nJ. Cluster Sci., {\\bf 10}, 195 (1999); cond-mat 9807011 v2 10 Aug 1998.\n\\bibitem{5} B.Janko, A.Smith, and V.Ambegaokar, Phys.Rev.\n{\\bf B50},1152(1994).\n\\bibitem{6} F.Braun, J.von Delft,\nand M.Tinkham, Phys. Rev. Lett.\n{\\bf 79},921(1997).\n\\bibitem{golubev}\nD. S. Golubev, A. D. Zaikin, Phys. Lett. {\\bf A 195}, 380 (1994)\n\\bibitem{mastellone}\nA. Mastellone, G. Falci, and R. Fazio, Phys. Rev. Lett. {\\bf 80}, 4542 (1998)\n\\bibitem{braun} F. Braun and J. v. Delft, Phys. Rev. Lett. {\\bf 81}, 4712\n(1998)\n\\bibitem{layer} R. Neuendorf, M. Quinten, and U. Kreibig, J. Chem. Phys. {\\bf\n104}, 6348 (1996)\n%\\bibitem{7} A.K.Kerman, Ann. Phys. {\\bf 12},300(1961).\n%\\bibitem{8} A.L.Goodman, Nucl. Phys. {\\bf A352},30(1981).\n\\bibitem{9} C.Esebag, J.L.Edigo, Nucl. Phys. {\\bf A552},205(1993)\nand earlier work cited therein.\n\\bibitem{Hammermesh} M.Hammermesh, Group theory and its applicationM\nto physical problem, Addison-Wesley (1964)\n\\bibitem{oddeven} N. K. Kuzmenko, V. M. Mikhajlov, S. Frauendorf,\n Phys. Rev. {\\bf B}, in preparation\n\\bibitem{highk} Z. Szymanski,\n Fast Nuclear Rotation, Claredon, (1983)\n\\bibitem{bm2} Bohr, A., and Mottelson, B., Nuclear Structure II,\nW.A. Benjamin, (1975)\n\\bibitem{fluctuation} W. J. Skocpol and M. Tinkham, Rep. Prog. Phys. {\\bf 38}\n1049, (1975)\n%\\bibitem{12} P.Ring, P.Schuck,\n% The Nuclear Many Body Problem (Springer,Berlin,1980).\n%\\bibitem{13} A.I.Larkin,Zh.Eksp.Teor.Fiz.\n%{\\bf 48},323(1965).[Sov.Phys.-JETP{\\bf 21},153(1965).\n%\\bibitem{14} J.A.A.J. Perenbom, P.Wyder, and F.Myer, Phys. Rep.\n%{\\bf 78},174(1981).\n%\\bibitem{15} P.M. Tedrow, R.Mersvey, Phys. Rev.\n%{\\bf B8},5098(1973).\n%\\bibitem{16} S. Str\\\"assler, P.Wyder, Phys. Rev.\n{\\bf 158},319(1967).\n\\end{references}\n\n\\end{document}\nIndependently of details of spherical\nsymmetric potentials the electron spectra of small\nclusters must possess high orbital degeneration on the Fermi\nsurface where the single-particle orbital momentum\n. (Hereafter we do not\ntake into account the spin-orbital interaction.)\nConsequently, the level bunching in a spherical mean field\nhas to broaden out the energy intervals between $\\l$-states\nto such an extent that the pairing gap typical for a given\n$N$ can become markedly less than the level spacing. Moreover,\nfor such systems even the maximum Zeeman splitting,\ncreated by the critical mafnetic field destroying\nsuperconductivity, does not lead to crossing\nlevel with different $\\l$. Limiting ourselves to low\ntemperatures to ensure prevaling the level spacing\nover $kT$ we thus arrive at the possibility to employ\nthe single-shell model to construct the canonical\nthermodynamics of small superconducting clusters.\\\\\n\nThese values are maximimum for both systems and with\ndecreasing $N_{sh}<M$ (increasing $N_{sh}>M$) $\\omega_{crit}$\ngoes down. Evidently, $\\omega_{crit}$ is independent\nof $\\l$ $(M=2\\l +1)$ and determined only by $\\mid M-N_{sh}\\mid$.\nEqs.(12),(13), obtained in the single-shell model, show\nthat the experimentally observed size-dependence of $B_{crit}$\n(see the review in Ref.[14] and Ref[15]) originates from the\nsize-dependence of $G$.\\\\\n\\indent\nThe ratio $B_g/B_{crit}$ predicted by the single-shell\nmodel is less than that in Ref.[13]:\n$B_g/B_{crit}=exp(7/3)/4\\simeq2.57$.\nThis quantity is independent of physical parameters of\nconsidered systems. Therefore the difference\nin the estimates of $B_g/B_{crit}$ may stem from the\ndifferent mathematical approaches to the problem.\\\\\n\n\nSimultaneously there takes place lowering the gap parameters.\nEq.(6) means that the difference in magnetic energies of the $(\\nu+2)$-\n and $\\nu$- states, owing to the greater $(\\nu+2)$ magnetic moment,\nEq.(3), prevails over the difference in their pairing energies.\nAs consequence of Eqs.(2),(4),(6) one gains\n\nThe field $B_g$, as the first seniority jump happens or,\nthat is the same, the first electron pair is broken,\ncan be considered the point of arising the gapless\nsuperconductivity:\n\n\\begin{eqnarray}\n\\omega^{(even)}_g=G, \\\\\n\\nonumber \\\\\n\\omega^{(odd)}_g=G(1+\\frac{1}{M-2}).\n\\end{eqnarray} %8,9\n\nThe values of $\\omega_g$, practically coinciding for $M\\gg 1$,\nare in agreement with that given in Refs.[13,14] where rather large\nspherical systems ($N>10^6$) with almost continuos spectrum were\nconsidered and $\\omega_g$ was found to be equal\n\n\\begin{equation}\n\\omega_g=\\Delta\\frac{\\hbar}{p_FR}.\n\\end{equation} %10\n\nIn Eq.(10) $\\Delta$ is the $BCS$ pairing gap at $B=0$ and $T=0$,\n$p_F$ is the Fermi momentum and $R$ is the cluster radius, i.e.\n$p_FR/\\hbar\\simeq\\l\\simeq M/2$.\nIn the single-shell model the quasiparticle energy\nat $T=0$ $E(0)$ is equal to $GM/2$ and independent of $N_{sh}$.\nFor a half-filled single-shell it coincides with\nthe $BCS$ pairing gap at $T=0$.\nThus, Eqs.(8),(9) and Eq.(10) have the same meaning.\\\\\n\\indent\n\\begin{eqnarray}\nE(T,\\omega>\\omega_{crit})\\simeq E_{\\nu_{max}-2}\nexp\\{-\\beta\\left[ E_{\\nu_{max}-2}+\\omega\n(\\mid M-N_{sh}\\mid+1)\\right]\\},\\\\\nE_{\\nu_{max}-2}+\\omega(\\mid M-N_{sh}\\mid +1)>0 \\nonumber\n\\end{eqnarray} %21\n\nindicates that increasing temperature gives rise to growth of the\npairing. Thus, there appears a tendency towards restoration of\nthe pairing correlations at $T>0$ whereas at\n$B\\geq B_{crit}$ and $T=0$ they are absent. Thereby the\nexact solution in the framework of the single-shell model\nreveals a phenomenon of the temperature-induced\npair correlation reentrance at $B\\geq B_{crit}$ (Fig.1).\nAt $B=0$ a similar phenomenon, considered in Ref.[17],\narises in an odd system with equal level spacing model\nwhen the blocking effect breaks pairing at $T=0$.\\\\\n\\indent\nIn the low temperature limit\n($\\beta E_0$ and $\\beta\\omega\\gg 1$)\nthe breaks in $\\Delta$ and the free energy mentioned above\ncause oscillations in the $B$-dependence of the heat capacity\n$C$. In fact, at small fields when the first terms of $Z$,\n$exp[-\\beta E_0]$ or $exp[-\\beta (E_1-\\omega m_1)]$\nare leading\nfor even or odd systems respectively $C$ takes almost\nzero values. With increasing $\\omega$ the second terms in\n$Z$ begin to play a role and $C$, e.g. for even systems,\nbecomes equal to\n\n\n\n\\begin{equation}\nC/k\\simeq\\beta^2(E_0-E_2+\\omega m_2)^2\nexp\\left[-\\beta(E_2-E_0-\\omega m_2)\\right],\n\\end{equation} %22\n\ni.e. also increases with $B$. But at $\\omega m_2=E_2-E_0$\nthe leading terms\n$exp[-\\beta E_0]$ and $exp[-\\beta (E_2-\\omega m_2)]$\nare equal and do not therefore contribute to $C$.\nThen the process is repeated with growth of $B$, i.e.\nwhen $\\omega$ passes by points of seniority changing\nit results in lowering $C$.\\\\\n\\indent\nInstead of minima in the $B$-dependence of $C$ the paramagnetic\nsusceptibility $\\chi_P$ shows maxima in the same points.\nAt small $T$ and $\\omega\\sim\\omega_{\\nu+2}$, Eq.(3),\nthe two leading terms in $Z$ are\n\n\\begin{displaymath}\nZ\\simeq exp\\left[-\\beta(E_{\\nu}-\\omega m_{\\nu})\\right]+\nexp\\left[-\\beta(E_{\\nu+2}-\\omega m_{\\nu+2})\\right].\n\\end{displaymath}\n\nThey give the main component of $\\chi_P$\n\n\\begin{displaymath}\n\\chi_P(\\omega\\simeq\\omega_{\\nu+2})\\simeq\n\\frac{\\beta}{V}\\mu_B^2\n\\left(\\frac{m_{\\nu+2}-m_{\\nu}}{2}\\right)^2=\n\\frac{\\beta}{V}\\mu_B^2\n\\left(\\frac{M-\\nu-1}{2}\\right)^2,\n\\end{displaymath}\n\ni.e. the highest values of $\\chi_P$ correspond to\n$\\omega_2$ for even systems and $\\omega_3$ for odd ones.\\\\\n\\indent\nFor weak fields and low temperatures one can keep only first\ntwo terms in $Z$ that, e.g. for odd systems, leads to\n\n\\begin{displaymath}\n\\chi_P(\\beta\\omega\\gg 1)\\simeq\n\\frac{\\beta}{V}\\mu_B^2\n\\left(m_1-m_3\\right)^2\nexp\\{-\\beta\\left[E_3-E_1-\\omega(m_3-m_1)\\right]\\}\n\\end{displaymath}\n\n\n\nHence, for weak fields, that $E_3-E_1-\\omega(m_3-m_1)>0$\none arrives at the zero limit of $\\chi_P$ at $T\\rightarrow 0$.\nObviously, the same result takes place for even systems as well.\\\\\n\\indent\nThe exact partition function , Eqs.(16),(17), in the zero field\nlimit enables to find an analytical expression of\n$\\chi_P(\\omega=0)$ for the single $\\l$-level\n\n\\begin{eqnarray}\n\\chi_P(\\omega=0)=\\frac{\\beta\\mu_B^2}{V}\n\\frac{l(l+1)+3}{3(4l+1)} \\nonumber \\\\\n\\frac{1}{Z}\n\\sum_{\\nu}\\{exp[-\\beta E_{\\nu}]-(1-\\delta_{\\nu .\\nu_{max}})\nexp\\left[-\\beta E_{\\nu+2}\\right]\n\\left(\n\\begin{array}{c}\n2M \\\\\n\\nu\n\\end{array} \\right)\n\\nu (2M-\\nu).\n\\end{eqnarray} %23\n\nEq.(23) indicates that for low $T$ the zero field\nsusceptibility of odd systems is much larger than for even\nones since in this case the main term in Eq.(23) with\n$\\nu=0$ falls out of the sum. Nevertheless, this\nequation (established for $\\beta\\omega\\ll 1$) cannot be\nused in the limit $T\\rightarrow 0$ for which we showed just\nabove that $\\chi_P\\rightarrow 0$ irrespective of the particle\nnumber parity.\\\\\n\\indent\nResults of the calculations of the pairing gap, Eq.(5),\nthe heat capacity\nand susceptibility with $Z$, Eqs.(16),(17) are presented in Fig.2\nfor an even system. The left panel of the figure with $\\l$ as small\nas $3$, $M=7$, is especially introduced to demonstrate\ndistinctly the correlations in changing $\\Delta$ and\noscillations\nof $C$ and $\\chi_P$ in the range $B_g<B<B_{crit}$. One can see\nthat minima of $C$ and maxima of $\\chi$\nare situated just between steps in the $B$-dependence of $\\Delta$.\nIn Fig.2 $\\chi=\\chi_P + \\chi_D$ is plotted in units of $\\chi_D$.\nThe diamagnetic contribution, $\\chi_D\\propto R^2$,\ndoes not practically depend on $T$ and $B$\nand is calculated for $Al$ with $N=10^3$.\nIn the right panel the same quantities are drawn for a\nlarger $\\l=11$. In this case the steps in $\\Delta$ are smoothed\nout and the extremes in $C$ and $\\chi$ are visible only between\nthe first seniority changes wich are divided by the distance\n$\\sim 1/M$, whereas other intervals are proportional\nto $\\sim 1/M^2$, Eq.(14).\\\\\n\\indent\nIt should be mentioned that our calculations of $\\chi_P$\ncarried out in the famework of the grand canonical\napproach ($GCA$) show that in the points of breaks of\n$\\Delta$ (or the points of jumping seniority)\ngrand canonical $\\chi_P$ exceeds the canonical one\nby two orders of magnititude. This fact is connected\nwith the behavior of the canonical pairing gap which\nhas much more smoothed character than the GCA one\nin the region of the gapless superconductivity.\\\\\n\\indent\n"
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[
{
"name": "cond-mat0002030.extracted_bib",
"string": "\\bibitem{1} D.C.Ralf,\n%C.T.Black, and M.Tinkham, Phys. Rev. Lett. {\\bf 74},3241(1995).\n%\n\\bibitem{2} C.T.Black, D.C.Ralph, and M.Tinkham, Phys.Rev.Lett.\n%{\\bf 76},688(1996).\n\n\\bibitem{3} D.C.Ralf,\nC.T.Black, and M.Tinkham, Phys. Rev. Lett. {\\bf 78},4087(1997) and\nealier works cited therein.\n\n\\bibitem{muehl} B. M\\\"uhlschlegel, D. J. Scalapino and R. Denton,\nPhys. Rev. {\\bf B 6}, 1767 (1972)\n\n\\bibitem{kuzmenko} N. K. Kuzmenko, V. M. Mikhajlov, S. Frauendorf,\nJ. Cluster Sci., {\\bf 10}, 195 (1999); cond-mat 9807011 v2 10 Aug 1998.\n\n\\bibitem{5} B.Janko, A.Smith, and V.Ambegaokar, Phys.Rev.\n{\\bf B50},1152(1994).\n\n\\bibitem{6} F.Braun, J.von Delft,\nand M.Tinkham, Phys. Rev. Lett.\n{\\bf 79},921(1997).\n\n\\bibitem{golubev}\nD. S. Golubev, A. D. Zaikin, Phys. Lett. {\\bf A 195}, 380 (1994)\n\n\\bibitem{mastellone}\nA. Mastellone, G. Falci, and R. Fazio, Phys. Rev. Lett. {\\bf 80}, 4542 (1998)\n\n\\bibitem{braun} F. Braun and J. v. Delft, Phys. Rev. Lett. {\\bf 81}, 4712\n(1998)\n\n\\bibitem{layer} R. Neuendorf, M. Quinten, and U. Kreibig, J. Chem. Phys. {\\bf\n104}, 6348 (1996)\n%\n\\bibitem{7} A.K.Kerman, Ann. Phys. {\\bf 12},300(1961).\n%\n\\bibitem{8} A.L.Goodman, Nucl. Phys. {\\bf A352},30(1981).\n\n\\bibitem{9} C.Esebag, J.L.Edigo, Nucl. Phys. {\\bf A552},205(1993)\nand earlier work cited therein.\n\n\\bibitem{Hammermesh} M.Hammermesh, Group theory and its applicationM\nto physical problem, Addison-Wesley (1964)\n\n\\bibitem{oddeven} N. K. Kuzmenko, V. M. Mikhajlov, S. Frauendorf,\n Phys. Rev. {\\bf B}, in preparation\n\n\\bibitem{highk} Z. Szymanski,\n Fast Nuclear Rotation, Claredon, (1983)\n\n\\bibitem{bm2} Bohr, A., and Mottelson, B., Nuclear Structure II,\nW.A. Benjamin, (1975)\n\n\\bibitem{fluctuation} W. J. Skocpol and M. Tinkham, Rep. Prog. Phys. {\\bf 38}\n1049, (1975)\n%\n\\bibitem{12} P.Ring, P.Schuck,\n% The Nuclear Many Body Problem (Springer,Berlin,1980).\n%\n\\bibitem{13} A.I.Larkin,Zh.Eksp.Teor.Fiz.\n%{\\bf 48},323(1965).[Sov.Phys.-JETP{\\bf 21},153(1965).\n%\n\\bibitem{14} J.A.A.J. Perenbom, P.Wyder, and F.Myer, Phys. Rep.\n%{\\bf 78},174(1981).\n%\n\\bibitem{15} P.M. Tedrow, R.Mersvey, Phys. Rev.\n%{\\bf B8},5098(1973).\n%\n\\bibitem{16} S. Str\\\"assler, P.Wyder, Phys. Rev.\n{\\bf 158},319(1967).\n"
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cond-mat0002031
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Charge-density waves in the Hubbard chain: evidence for 4$k_F$ instability
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"author": "Thereza Paiva$^{\\, (1)}\\!$ and Raimundo R.\\ dos Santos$^{\\, (2)}$"
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Charge density waves in the Hubbard chain are studied by means of finite-temperature Quantum Monte Carlo simulations and Lanczos diagonalizations for the ground state. We present results both for the charge susceptibilities and for the charge structure factor at densities $\rho=1/6$ and 1/3; for $\rho=1/2$ (quarter filled) we only present results for the charge structure factor. The data are consistent with a $4k_F$ instability dominating over the $2k_F$ one, at least for sufficiently large values of the Coulomb repulsion, $U$. This can only be reconciled with the Luttinger liquid analyses if the amplitude of the $2k_F$ contribution vanishes above some $U^*(\rho)$.
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"name": "v4-3.tex",
"string": "\\documentstyle[aps,prb,epsf,epsfig,twocolumn]{revtex}\n\n\\begin{document}\n\n\\draft\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname\n\n\\title{Charge-density waves in the Hubbard chain: evidence for 4$k_F$ instability}\n\\author{Thereza Paiva$^{\\, (1)}\\!$ and Raimundo R.\\ dos Santos$^{\\, (2)}$}\n\\address{$^{(1)}$Department of Physics, University of California, Davis, California, \n 95616-8677\\\\ \n $^{(2)}$Instituto de F\\' \\i sica, Universidade Federal do Rio de Janeiro,\n Cx.P.\\ 68.528, 21945-970 Rio de Janeiro RJ, Brazil\\\\}\n \n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\nCharge density waves in the Hubbard chain are studied\nby means of finite-temperature Quantum Monte Carlo simulations and Lanczos\ndiagonalizations for the ground state.\nWe present results both for the charge susceptibilities and for the charge\nstructure factor at densities $\\rho=1/6$ and 1/3; \nfor $\\rho=1/2$ (quarter filled)\nwe only present results for the charge structure factor. \nThe data are consistent with a $4k_F$ instability dominating over the \n$2k_F$ one, at least for sufficiently large values of the Coulomb \nrepulsion, $U$.\nThis can only be reconciled with the Luttinger liquid analyses if the\namplitude of the $2k_F$ contribution vanishes above some $U^*(\\rho)$.\n\\end{abstract}\n\n\\pacs{PACS: \n 71.27.+a, % Strongly correlated electronic systems; h.f.'s\n 71.10.-w, % Theories and models of many-electron systems\n 71.45.Lr, % Charge-density-wave systems\n 72.15.Nj, % Collective modes (e.g., in one-dimensional conductors)\n 73.20.Mf. % Collective excitations (including plasmons and other \n % charge-density excitations)\n}\n\\vskip2pc]\n\n\nCharge-density waves (CDW's) are present in a variety of strongly \ncorrelated electron systems, ranging from quasi-one-dimensional organic \nconductors\\cite{JS82,Pouget89,Jerome94} to the more recently discovered \nmanganites.\\cite{Cheong}\nIn order to understand the influence of CDW formation on magnetic order and \ntransport properties, a crucial issue is to establish the \nperiod of charge modulation in the ground state. \nThis period, in turn, is primarily determined by the interplay between \nelectron-phonon and electron-electron couplings. \nFor the specific example of quasi-one dimensional organic conductors at \nquarter-filled band, both period-2 and period-4 modulations have \nbeen observed;\\cite{Pouget89}\nthese correspond, respectively, to $4k_F$ and $2k_F$, where \n$k_F=\\pi\\rho/2$ is the Fermi wave vector for a density $\\rho$ of free \nelectrons on a periodic lattice.\nThe use of simplified effective models capturing the basic physical ingredients \nshould therefore be extremely helpful in predicting the dominant \ninstability.\\cite{Ung94}\nIn this context, the Hubbard model can be thought of as a limiting case \n(of vanishing electron-phonon interaction), in which the influence of \nelectronic correlations on CDW modulation can be monitored.\nHowever, even for this simplest possible model there has been a disagreement\nbetween analyses of the continuum (Luttinger liquid) \nversion,\\cite{Solyom79,Schulz90,Frahm90,Voit94} and\nearly finite temperature (world-line) quantum Monte Carlo (QMC) \nsimulations.\\cite{HS}\nAccording to the former, the large-distance behaviour of the charge\ndensity \ncorrelation function is given by\\cite{Schulz90}\n\\begin{equation}\n\\langle n(x)n(0)\\rangle = {K_\\rho\\over (\\pi x)^2} \n + A_1 { \\cos (2 k_F x) \\over x^{1+K_\\rho} \\ln^{3/2} x}\\\\\n + A_2 { \\cos (4 k_F x) \\over x^{4K_\\rho}},\n\\label{nn}\n\\end{equation}\nwhere the amplitudes $A_1$ and $A_2$, and the exponent $K_\\rho$ are \ninteraction- and density-dependent parameters; \nfor repulsive interactions\\cite{Schulz90} ${1\\over 2} \\leq K_\\rho < 1$, \nso that charge correlations are expected to be dominated by the $2k_F$ term.\nBy contrast, the simulations pointed towards $4k_F$ being the\nmain correlations. \nNonetheless, based on a Renormalization Group (RG) analysis, it was \nargued\\cite{HS} that the $2k_F$ instability should\neventually dominate over the $4k_F$ one for sufficiently low temperatures.\nFor infinite coupling the system becomes effectively a spinless fermion\nproblem with a $4k_F$ instability,\\cite{Schulz90,HS,Ogata90} and there\nis no disagreement in this case.\n\nSince present day computational capabilities allow one to reach much lower \ntemperatures and larger system sizes than before, a numerical reanalysis of \nthe model is certainly in order. \nOur purpose here is to present the results of such a study.\nThe Hubbard Hamiltonian reads\n\\begin{equation}\n{\\cal H}=-t\\sum_{i,\\sigma} \n c_{i\\sigma}^{\\dagger}c_{i+1\\sigma}^{\\phantom{\\dagger}}+\n U\\sum_i n_{i\\uparrow}^{\\phantom{\\dagger}} \n n_{i\\downarrow}^{\\phantom{\\dagger}},\n\\label{c-ham}\n\\end{equation}\nwhere, in standard notation, $U$ is the on-site Coulomb repulsion;\nthe hopping integral sets the energy scale, so we take $t=1$ throughout this\npaper. \nWe probe finite temperature properties through\ndeterminantal QMC simulations\\cite{bss,Hirsch85,lg92,vdl92} for the \ngrand-canonical version of (\\ref{c-ham}), \n$\\hat{\\cal H}\\equiv {\\cal H} -\\mu \\hat{N}$, \nwhere \n$\\hat{N}\\equiv \\sum_{i\\sigma} n_{i\\sigma}$;\nthe chemical potential $\\mu$ is adjusted to yield the desired particle density.\nThis analysis is supplemented by zero temperature calculations:\nthe ground state of Eq.\\ (\\ref{c-ham}), for finite \nlattices of $N_s$ sites with periodic boundary conditions,\nis obtained through the Lanczos algorithm,\\cite{Roomany80,Gagliano86,Dagotto94}\nin the subspace of fixed particle-density (canonical ensemble). \n\nThe signature of a CDW instability is a {\\em peak} at $q=q^*$ in the \nzero-temperature limit of the charge-density susceptibility,\n\\begin{equation}\nN(q)={1\\over N_s}\\int_0^\\beta d\\tau \\sum_{i,\\ell} \n \\langle\\; n_i(\\tau) n_{i+\\ell}(0)\\; \\rangle\\; e^{iq\\ell},\n\\label{Nq}\n\\end{equation} \nwhere the imaginary-time dependence of the operators is given by \n$n_i(\\tau)\\equiv e^{\\tau\\hat{\\cal H}}\\; n_i\\; e^{-\\tau\\hat{\\cal H}}$,\nwith $n_i = n_{i\\uparrow} + n_{i\\downarrow}$.\nWe recall that in simulations `time' is discretized in intervals \n$\\Delta\\tau$, such that \nthe size along this direction is $L=\\beta/\\Delta\\tau$.\nThe CDW instability should also show up as a {\\em cusp}, again at $q=q^*,$ in the \nzero-temperature charge-density structure factor,\n\\begin{equation}\nC(q)={1\\over N_s}\\sum_{i,\\ell}\n\\langle 0|\\; n_i n_{i+\\ell}\\; |0\\rangle\\; e^{iq\\ell},\n\\label{Cq}\n\\end{equation}\nwhere $|0\\rangle$ is the ground state. \n\\begin{figure}\n\\begin{center}\n%\\leavevmode\n\\epsfxsize=8.5cm\n%\\epsfysize=4cm\n\\epsffile{nbeta16.eps}\n\\caption{QMC results for the charge susceptibility as a function of \nthe inverse temperature $\\beta$ for a chain with $N_s=36$ sites and\noccupation $\\rho=1/6$: (a) $U=2$, (b) $U=6$, and (c) $U=9$.\nCircles and squares represent data for $q=2k_F$ and $4k_F$, respectively. \nError bars are smaller than data points, and all lines are guides to \nthe eye only. \n}\n\\label{nbeta16} % Fig. 1\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\epsfxsize=8.5cm\n%\\epsfysize=6cm\n\\epsffile{fss.eps}\n\\caption{QMC results for the charge susceptibility as a function of \n$\\ln N_s$ for $U=9$; each point is obtained at $\\beta=N_s/4$.\nCircles and squares represent data for $q=2k_F$ and $4k_F$, respectively.\nThe straight line through data for $4k_F$ is a least squares fit.}\n\\label{nn16} % Fig 2\n\\end{center}\n\\end{figure}\n\nLet us first discuss results for an electronic density $\\rho=1/6$,\nfor which we found that the fermionic determinants in QMC \nsimulations do not suffer from the `minus-sign problem';\\cite{lg92,vdl92}\nthis allowed us to reach inverse temperatures as large as $\\beta=25$.\n>From Fig. \\ref{nbeta16} we see that the $4k_F$ charge susceptibility \nappears to be increasing with decreasing temperature, at a \nrate faster than that at $2k_F$, especially for $U=6$ and 9.\nThe data in Fig.\\ \\ref{nbeta16} were obtained for $\\Delta\\tau=0.125$,\nbut we have explicitly tested other values to ensure they do not \nchange significantly as $\\Delta\\tau\\to 0$;\neach datum point involves typically 20,000 QMC sweeps \nover all time slices.\nFurther, in order to check if this increase is limited by finite-size or \nfinite-temperature effects, we performed additional simulations on a\n`square space-time lattice',\\cite{HS} i.e., we set $N_s=L$; \nthe result is displayed in Fig.\\ \\ref{nn16} for $N_s=L\\leq 96$. \nWhile the charge susceptibility at $2k_F$ seems to saturate as \n$N_s$ increases, the one at $4k_F$ still scales with $\\ln N_s$, \nup to the largest sizes considered. \nThus, in spite of the very low temperatures reached, we were still \nunable to find indications of a crossover temperature below which the \nsystem is dominated by the $2k_F$ instability.\nAlso, since $N_s\\propto 1/T$ for the data in Fig.\\ \\ref{nn16}, \nthe $4k_F$ charge susceptibility grows logarithmically with the temperature in \nthis range, similarly to the infinite coupling limit.\\cite{HS}\n\\begin{figure}[t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsffile{cq16.eps}\n\\caption{Charge structure factor for electron density $\\rho=1/6$ and \n$U=0$ (up triangles), 2 (left triangles), \n4 (squares), 6 (down triangles), 9 (diamonds), 12 (circles),\nand 20 (right triangles). The system size is $N_s=24$, and \nsuccessive vertical shifts by -0.02 have been imposed on the\ncurves, for clarity.\n}\n\\label{cq16} % Fig 3\n\\end{center}\n\\end{figure}\n\nWe now discuss the charge structure factor at zero temperature, as\nobtained from Lanczos diagonalizations, still for $\\rho=1/6$. \nFigure \\ref{cq16} shows $C(q)$ for several values of $U$;\nfor clarity, the curves are shown after suffering successive displacements. \nFor the free case, $U=0$, we see a sharp plateau beginning at \n$q=2k_F=\\pi/6$, which is the signature of the Peierls instability. \nThis behaviour is quite different from the one observed for $U \\neq 0$:\nthough somewhat rounded for $U=2$ and 4,\nthe {\\it plateaux} now start at $q=4k_F=\\pi/3$.\nIt is instructive to examine how these roundings evolve\nwith system size. \nIn Fig.\\ \\ref{cqfss}, we single out data for $C(q)$ with\n$U=3$ and 12, and sizes $N_s=12$ and 24.\nFor each value of $U$, the data below and above $4k_F$ \nrespectively move down and up as $N_s$ increases, thus sharpening the cusp; \nin addition, the position of the latter shows no tendency of shifting \nfrom $4k_F$.\nThus, the Lanczos results are consistent with those from \nQMC simulations, in the sense that a $4k_F$ instability\nis dominant, already for moderate values of $U$.\nAt this point, it is worth pointing out that the charge structure factor\nfor the Hubbard chain with second neighbour hopping has been calculated through\ndensity matrix renormalization group (DMRG).\\cite{Daul98}\nThough their interest was to extract $K_\\rho$ from the slope of $C(q)$\nat $q=0,$\nif one reinterprets those data along the lines discussed here, a\npredominance of the $4k_F$ instability can be clearly inferred for the\nlargest values of $U$ shown in Fig.\\ 11 of Ref.\\ \\onlinecite{Daul98}.\nDMRG calculations for the two-leg Hubbard ladder have also led\nto $4k_F$-like charge correlations.\\cite{Noack96}\n\n\\begin{figure}[t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsffile{cqfss.eps}\n\\caption{Charge structure factor for electron density $\\rho=1/6$. \nEmpty squares: $U=3,\\ N_s=12$;\nfilled squares: $U=3,\\ N_s=24$; \nempty circles: $U=12,\\ N_s=12$;\nfilled circles: $U=12,\\ N_s=24$. \nA vertical shift by -0.025 has been imposed on the curves for $U=12$, for clarity.\n}\n\\label{cqfss} % Fig 4\n\\end{center}\n\\end{figure}\n\n\nWe now change the band filling to $\\rho=1/3$; the average sign of the\nfermionic determinant is $\\sim0.9$ in the worst cases, thus posing no problems\nto the resulting averages. \nFigure \\ref{nbeta13} shows QMC data for the charge susceptibilities\nwith $U=6$ and $U=8$.\nFor $U=6$ an upturn at lower temperatures seems to be setting in\nfor $4k_F$, while the $2k_F$ data show no noticeable change in growth rate.\nOn the other hand, for $U=8$ the $4k_F$ susceptibility grows unequivocally faster\nwith $\\beta$ than the one at $2k_F$; see Fig.\\ \\ref{nbeta13}. \nThe corresponding Lanczos data for the charge structure factor are shown in Fig.\\ \\ref{cq13}\nand, similarly to Fig.\\ \\ref{cq16}, \nthe cusp at $q=4k_F=2\\pi/3$ gets visibly sharper as $U$ increases. \n\n\\begin{figure}%[t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=8.5cm\n%\\epsfysize=6cm\n\\epsfbox{nbeta13.eps}\n\\caption{QMC results for the charge susceptibility as a function of \nthe inverse temperature $\\beta$ for a chain with $N_s=36$ sites and \noccupation $\\rho=1/3$: (a) $U=6$ and (b) $U=8$.\nCircles and squares represent data for $q=2k_F$ and $4k_F$, respectively.\nError bars are smaller than data points, and all lines are guides to \nthe eye only.}\n\\label{nbeta13} % Fig 5\n\\end{center}\n\\end{figure}\n\n\\begin{figure}%[t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{cq13.eps}\n\\caption{Charge structure factor for electron density $\\rho=1/3$ and \n$U=0$ (up triangles), 4 (squares), 6 (down triangles), 8 (left triangles),\n9 (diamonds), 12 (circles),\nand 20 (right triangles). The system size is $N_s=18$, and \nsuccessive vertical shifts by -0.03 have been imposed on the\ncurves, for clarity.}\n\\label{cq13} % Fig 6\n\\end{center}\n\\end{figure}\n\nUnfortunately, for larger band fillings the `minus-sign problem' prevents us from \nreaching very low temperatures. \nNonetheless, down to the lowest temperatures probed with acceptable \naverage signs of the fermionic determinant \n(i.e., $\\langle {\\rm sign}\\rangle\\sim0.7$ at \n$T\\sim 1/20$), no indications of a $2k_F$ peak dominating \nthe charge susceptibility were found for $\\rho=1/2$ or 3/4.\nAccordingly, the charge structure factor at $T=0$, \ncalculated through Lanczos diagonalizations on a 16-site chain\nat quarter filling, shown in Fig.\\ \\ref{cq12}, confirms the\nprevious patterns: there is some rounding near \n$q=4k_F=\\pi$, which sharpens as $U$ increases, \nconsistently with a $4k_F$ instability setting in for finite $U$'s. \n\n\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{cq12.eps}\n\\caption{Charge structure factor for electron density $\\rho=1/2$ and \n$U=0$ (up triangles), 3 (squares), 6 (down triangles), \n9 (diamonds), 12 (circles). \nThe system size is $N_s=16$, and \nsuccessive vertical shifts by -0.03 have been imposed on the\ncurves, for clarity.\n}\n\\label{cq12} % Fig 7\n\\end{center}\n\\end{figure}\n \n\nIn summary, for all band fillings examined, the charge instability seems\nto be characterized by a $4k_F$ modulation, rather than by \n$2k_F$, at least for $U$ greater than some $U^*(\\rho)$.\nThe question of how can these findings be reconciled with the analyses of\nthe continuum model still remains.\nSince Lanczos data have been obtained at zero temperature, and\nQMC simulations reached much lower temperatures than before,\\cite{HS} \nthe scenario of a temperature-driven crossover seems now unlikely;\nit should be recalled that this crossover was predicted \nbased on a {\\em weak coupling} RG analysis.\nWe therefore envisage the following scenario.\nWhile analyses of the Luttinger liquid have so far provided detailed \ninsight into the behaviour of the exponent $K_\\rho$, little is known\nabout the dependence of the amplitude $A_1$ of the $2k_F$ contribution\n[see Eq.\\ (\\ref{nn})] with $\\rho$ and with the coupling constant. \nOur results may be indicating that, for fixed density $\\rho$,\n$A_1\\to 0$ very fast with increasing $U$, either exponentially or, \nless likely, as \n$\\left(U^*(\\rho)-U\\right)^\\psi,$ for $U\\leq U^*(\\rho)$, with $\\psi>1$;\na crude examination of the roundings near $4k_F$ in the \nstructure factors is consistent with $U^*$ growing with $\\rho$.\nAn alternative scenario could be that $2k_F$ charge \ncorrelations in the lattice model would suffer from \nunusually slow finite-size effects, thus hindering any present day\nnumerical calculations to detect their predominance over the $4k_F$\nones; however, since one would need effects slower than those suggested \nby the logarithmic `correction' in Eq.\\ (\\ref{nn}), this scenario\nis less appealing.\nTherefore, the presence of a $4k_F$ instability can be made compatible \nwith Luttinger liquid picture through the behaviour of the amplitude\nof the $2k_F$ contribution. \nWe hope our results stimulate more extensive work both on the\nLuttinger liquid and on the \nlattice model in order to extract a quantitative behaviour for\nthe amplitude $A_1(\\rho,U)$. \n\n\\acknowledgments\nThe authors are grateful to H.\\ Ghosh, A.\\ L.\\ Malvezzi and \nD.\\ J.\\ Scalapino for useful discussions, \nto S.\\ L.\\ A.\\ de Queiroz and R.\\ T.\\ Scalettar for\ncomments on the manuscript, and to R.\\ Bechara Muniz for computational \nassistance.\nFinancial support from the Brazilian Agencies FAPERJ, FINEP, CNPq \nand CAPES \n%Financiadora de Estudos e Projetos (FINEP), \n%Conselho Nacional de Desenvolvimento Cient\\'\\i fico e\n%Tecnol\\'ogico (CNPq) and Coordena\\c c\\~ao de Aperfei\\c coamento do\n%Pessoal de Ensino Superior (CAPES), \nis also gratefully acknowledged.\nThe authors are also grateful to Centro de Supercomputa\\c c\\~ao\nda Universidade Federal do Rio Grande do Sul for the use of\nthe Cray T94, and to the Instituto de F\\'\\i sica at Universidade\nFederal Fluminense, where this work initiated.\n\n\\begin{references}\n\n\\bibitem{JS82} D.\\ J\\'erome and H.\\ J.\\ Schulz, Adv.\\ Phys.\\ {\\bf 31}, 299 (1982).\n\n\\bibitem{Pouget89} see, e.g., J.\\ P.\\ Pouget and R.\\ Comes, in {\\it Charge \n Density Waves in Solids,} edited by L.\\ P.\\ Gorkov and \n G.\\ Gr\\\"uner (North-Holland, Amsterdam, 1989), pp.\\ 85--136.\n \n\\bibitem{Jerome94} D.\\ J\\'erome, {\\it Organic Superconductors,} (Dekker, New York, 1994).\n\n\\bibitem{Cheong} C.\\ H.\\ Chen and S.-W.\\ Cheong, \\prl {\\bf 76}, 4042 (1996);\n Nature {\\bf 392}, 473 (1998).\n\n\n\n\\bibitem{Ung94} K.\\ C.\\ Ung, S.\\ Mazumdar, and D.\\ Toussaint,\n \\prl {\\bf 73}, 2603 (1994).\n\n\\bibitem{Solyom79} J.\\ Solyom, Adv.\\ Phys.\\ {\\bf 28}, 201 (1979).\n\n\\bibitem{Schulz90} H.\\ J.\\ Schulz, \\prl {\\bf 64}, 2831 (1990). \n \n\\bibitem{Frahm90} H.\\ Fr\\\"ahm and V.\\ Korepin, \\prb {\\bf 42}, 10 553 (1990).\n\n\\bibitem{Voit94} J.\\ Voit, Rep.\\ Prog.\\ Phys.\\ {\\bf 57}, 977 (1994).\n \n\\bibitem{HS} J.\\ E.\\ Hirsch and D.\\ J.\\ Scalapino, \\prl {\\bf 50}, 1168 (1983);\n \\prb {\\bf 27}, 7169 (1983); {\\it ibid.} {\\bf 29}, 5554.\n\n\\bibitem{Ogata90} M.\\ Ogata and H.\\ Shiba, \\prb {\\bf 41} 2326 (1990).\n \n\\bibitem{bss} R.~Blankenbecler,\nD.~J.~Scalapino, and R.~L.~Sugar,\n\\prd {\\bf 24}, 2278 (1981).\n\n\\bibitem{Hirsch85} J.~E.~Hirsch, \\prb {\\bf 31}, 4403 (1985);\n{\\it ibid.} {\\bf 38}, 12023 (1988).\n\n\\bibitem{lg92} E.~Y.~Loh, Jr.\\ and J.~E.~Gubernatis, in {\\it Electronic\nPhase Transitions}, edited by W.~Hanke and Yu.~V.~Kopaev (Elsevier, Amsterdam,\n1992).\n\n\\bibitem{vdl92} W.~von der Linden, Phys.~Rep.~{\\bf 220}, 53 (1992).\n\n\\bibitem{Roomany80} H.~H.~Roomany,\n% {\\it et al.,}\nH.~W.~Wyld and L.~E.~Holloway, \n\\prd {\\bf 21}, 1557 (1980).\n\n\\bibitem{Gagliano86} E.~Gagliano, % {\\it et al.,}\n E.~Dagotto, A.~Moreo, and F.~C.~Alcaraz,\n\\prb {\\bf 34}, 1677 (1986); (E) {\\bf 35}, 5297 (1987).\n\n\\bibitem{Dagotto94} E.~Dagotto, \\rmp {\\bf 66}, 763 (1994).\n\n\\bibitem{Daul98} S.\\ Daul and R.\\ M.\\ Noack, \\prb {\\bf 58}, 2635 (1998).\n\n\\bibitem{Noack96} R.\\ M.\\ Noack, S.\\ R.\\ White, and D.\\ J.\\ Scalapino,\nPhysica C {\\bf 270}, 281 (1996).\n\n\n\\end{references}\n\n\\end{document}\n\n"
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[
{
"name": "cond-mat0002031.extracted_bib",
"string": "\\bibitem{JS82} D.\\ J\\'erome and H.\\ J.\\ Schulz, Adv.\\ Phys.\\ {\\bf 31}, 299 (1982).\n\n\n\\bibitem{Pouget89} see, e.g., J.\\ P.\\ Pouget and R.\\ Comes, in {\\it Charge \n Density Waves in Solids,} edited by L.\\ P.\\ Gorkov and \n G.\\ Gr\\\"uner (North-Holland, Amsterdam, 1989), pp.\\ 85--136.\n \n\n\\bibitem{Jerome94} D.\\ J\\'erome, {\\it Organic Superconductors,} (Dekker, New York, 1994).\n\n\n\\bibitem{Cheong} C.\\ H.\\ Chen and S.-W.\\ Cheong, \\prl {\\bf 76}, 4042 (1996);\n Nature {\\bf 392}, 473 (1998).\n\n\n\n\n\\bibitem{Ung94} K.\\ C.\\ Ung, S.\\ Mazumdar, and D.\\ Toussaint,\n \\prl {\\bf 73}, 2603 (1994).\n\n\n\\bibitem{Solyom79} J.\\ Solyom, Adv.\\ Phys.\\ {\\bf 28}, 201 (1979).\n\n\n\\bibitem{Schulz90} H.\\ J.\\ Schulz, \\prl {\\bf 64}, 2831 (1990). \n \n\n\\bibitem{Frahm90} H.\\ Fr\\\"ahm and V.\\ Korepin, \\prb {\\bf 42}, 10 553 (1990).\n\n\n\\bibitem{Voit94} J.\\ Voit, Rep.\\ Prog.\\ Phys.\\ {\\bf 57}, 977 (1994).\n \n\n\\bibitem{HS} J.\\ E.\\ Hirsch and D.\\ J.\\ Scalapino, \\prl {\\bf 50}, 1168 (1983);\n \\prb {\\bf 27}, 7169 (1983); {\\it ibid.} {\\bf 29}, 5554.\n\n\n\\bibitem{Ogata90} M.\\ Ogata and H.\\ Shiba, \\prb {\\bf 41} 2326 (1990).\n \n\n\\bibitem{bss} R.~Blankenbecler,\nD.~J.~Scalapino, and R.~L.~Sugar,\n\\prd {\\bf 24}, 2278 (1981).\n\n\n\\bibitem{Hirsch85} J.~E.~Hirsch, \\prb {\\bf 31}, 4403 (1985);\n{\\it ibid.} {\\bf 38}, 12023 (1988).\n\n\n\\bibitem{lg92} E.~Y.~Loh, Jr.\\ and J.~E.~Gubernatis, in {\\it Electronic\nPhase Transitions}, edited by W.~Hanke and Yu.~V.~Kopaev (Elsevier, Amsterdam,\n1992).\n\n\n\\bibitem{vdl92} W.~von der Linden, Phys.~Rep.~{\\bf 220}, 53 (1992).\n\n\n\\bibitem{Roomany80} H.~H.~Roomany,\n% {\\it et al.,}\nH.~W.~Wyld and L.~E.~Holloway, \n\\prd {\\bf 21}, 1557 (1980).\n\n\n\\bibitem{Gagliano86} E.~Gagliano, % {\\it et al.,}\n E.~Dagotto, A.~Moreo, and F.~C.~Alcaraz,\n\\prb {\\bf 34}, 1677 (1986); (E) {\\bf 35}, 5297 (1987).\n\n\n\\bibitem{Dagotto94} E.~Dagotto, \\rmp {\\bf 66}, 763 (1994).\n\n\n\\bibitem{Daul98} S.\\ Daul and R.\\ M.\\ Noack, \\prb {\\bf 58}, 2635 (1998).\n\n\n\\bibitem{Noack96} R.\\ M.\\ Noack, S.\\ R.\\ White, and D.\\ J.\\ Scalapino,\nPhysica C {\\bf 270}, 281 (1996).\n\n\n"
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cond-mat0002032
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Phase ordering and roughening on growing films
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"author": "Barbara Drossel$^{1,2}$ and Mehran Kardar$^3$"
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We study the interplay between surface roughening and phase separation during the growth of binary films. Already in 1+1 dimensions, we find a variety of different scaling behaviors, depending on how the two phenomena are coupled. In the most interesting case, related to the advection of a passive scalar in a velocity field, nontrivial scaling exponents are obtained in simulations. \noindent{PACS numbers: 68.35.Rh, 05.70.Jk, 05.70.Ln, 64.60.Cn}
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"name": "prl4.tex",
"string": "\\documentstyle[multicol,prl,aps,epsf]{revtex}\n\\begin{document}\n\n\\title{Phase ordering and roughening on growing films}\n\n\\author{Barbara Drossel$^{1,2}$ and Mehran Kardar$^3$}\n\\address{${}^1$ Department of Physics, \n University of Manchester, Manchester M13 9PL, England}\n\\address{${}^2$ School of Physics and Astronomy, Raymond and Beverley Sackler Faculty of Exact Sciences, Tel Aviv 69978, Israel}\n\\address{${}^3$ Department of Physics, Massachussetts Institute of Technology, \nCambridge, MA02139, USA}\n \n\\maketitle\n\n\\begin{abstract}\nWe study the interplay between surface roughening \nand phase separation during the growth of binary films. \nAlready in 1+1 dimensions, we find a variety of different scaling behaviors, \ndepending on how the two phenomena are coupled. \nIn the most interesting case, related to the advection of a \npassive scalar in a velocity field, \nnontrivial scaling exponents are obtained in simulations. \n\n\\noindent{PACS numbers: 68.35.Rh, 05.70.Jk, 05.70.Ln, 64.60.Cn}\n\\end{abstract}\n\n\n\\begin{multicols}{2}\n\nThin solid films are grown for a variety of technological applications, \nusing molecular beam epitaxy (MBE) or vapor deposition. \nIn order to create materials with specific electronic, optical, \nor mechanical properties, often more than one type of particle is deposited. \nWhen the particle mobility in the bulk is small, surface configurations \nbecome frozen in the bulk, leading to anisotropic structures that reflect the\ngrowth history, and are different from bulk equilibrium phases\\cite{hel95}. \nCharacterizing structures generated during composite film growth is not\nonly of technological importance, but represents also an interesting\nand challenging problem in statistical physics. \n\nIn this paper, we examine the growth of binary films through vapor deposition, \nand study some of the rich phenomena resulting from the interplay of {\\it phase \nseparation} and {\\it surface roughening}. \nSimple models for {\\it layer by layer} growth assume either that the \nprobability that an incoming atom sticks to a given surface site depends \non the state of the neighboring sites in the layer below \\cite{kan90}, \nor that the top layer is fully thermally equilibrated \\cite{dro97}. \nAssuming that the bulk mobility is zero, once a site is occupied, \nits state does not change any more. \nIf the growth rules are invariant under the exchange of the two particle types, \nthe phase separation is in the universality class of an \nequilibrium Ising model. \nCorrelations perpendicular to the growth direction are\ncharacterized by the critical exponent $\\nu$ of the Ising model, and\nthose parallel to the growth direction by the exponent $\\nu z_m$, with $z_m$\nbeing the dynamical critical exponent of the Ising model.\n\nHowever, the {\\it layer by layer} growth mode underlying these simple\nmodels is unstable, and the growing surface becomes {\\it rough}. \nIn many cases the fluctuations in the height $h({\\bf x},t)$, at position\n${\\bf x}$ and time $t$ are {\\it self-affine}, with correlations\n\\begin{equation}\\label{Chh}\n\\langle \\left[h({\\bf x},t)-h({\\bf x'},t')\\right]^2\\rangle \n\\sim |{\\bf x-x'}|^{2\\chi} g\\left(t/|{\\bf x-x'}|^{z_h}\\right),\n\\end{equation}\nwhere $\\chi$ is the roughness exponent, \nand $z_h$ is a dynamical scaling exponent.\nA computer model with local sticking probabilities that allows for a rough \nsurface was introduced in \\cite{kot98}.\nIn 1+1 dimensions, the authors find phase separation into domains\n(with sizes consistent with the Ising model), and a very rough surface\nprofile with sharp minima at the domain boundaries.\nWe may ask the following questions:\n{\\bf (1)} Are the roughness exponents different at the phase transition point?\n{\\bf (2)} Are the critical exponents modified on a rough surface?\nWe shall demonstrate that the coupling of roughening and phase \nseparation leads to a rich phase diagram, and to nontrivial critical exponents \nalready in 1+1 dimensions.\n\nTo characterize phase separation, we introduce an {\\it order parameter}\n$m({\\bf x},t)$, which is the difference in the densities of the two\nparticle types at the surface at position ${\\bf x}$ and time $t$. \nThe interplay between the fluctuations in $m$, and the height $h$ is\ncaptured phenomenologically by the coupled Langevin equations, \n\\begin{eqnarray}\n\\partial_t h &=& \\nu \\nabla^2 h + {\\lambda\\over 2} (\\nabla h)^2 +\n{\\alpha\\over 2} m^2 + \\zeta_h, \\label{langevin1}\\\\ \\partial_t m &=& K\n(\\nabla^2m+rm-um^3) + a\\nabla h \\cdot \\nabla m+bm\\nabla^2 h\\nonumber \\\\\n&& + {c\\over 2} m (\\nabla h)^2 + \\zeta_m .\\label{langevin2}\n\\end{eqnarray}\nHere, we have included the lowest order (potentially relevant) terms\nallowed by the symmetry $m \\to -m$. \nEquation (\\ref{langevin1}) is the Kardar-Parisi-Zhang (KPZ) equation\\cite{kar86} \nfor surface growth, plus a coupling to the order parameter. \nEquation (\\ref{langevin2}) is the time dependent Landau--Ginzburg equation \nfor a (non-conserved) Ising model, with three different couplings to the\nheight fluctuations. \nThe Gaussian, delta-correlated noise terms, $\\zeta_h$ and $\\zeta_m$, \nmimic the effects of faster degrees of freedom. \nA different set of equations was proposed by \nL\\'eonard and Desai\\cite{leo97} for phase separation during MBE. \nTheir equations reflect the MBE conditions of random particle\ndeposition (in contrast to sticking probabilities that depend on \nthe local environment), and a conserved order parameter which\nevolves by surface diffusion. \nThey do not include the KPZ nonlinearity. \nComputer simulations of corresponding 1+1 dimensional systems \nare presented in \\cite{leo97,leo97a}.\n\nDimensional analysis indicates that the couplings appearing in\nEqs.~(\\ref{langevin1}-\\ref{langevin2}) are relevant, and may lead to \nnew universality classes.\nWe shall leave the renormalization group analysis of these equations\nto a more technical paper, and focus here instead on computer\nsimulations in 1+1 dimensions. \nThe quantities evaluated in the computer simulations are the height \ncorrelation function in Eq.~(\\ref{Chh}), and the order parameter correlation \nfunctions perpendicular and parallel to the growth direction.\nAllowing for the possibility of different dynamic exponents,\n$z_m$ and $z_h$, for the order parameter and the height variables,\nwe fit to the scaling forms\n\\begin{eqnarray}\nG_m^{(x)}(x-x')&\\equiv& \\langle m(x,t) m( x',t) \\rangle \\nonumber\\\\\n&=&|x-x'|^{\\eta-1}g_m^{\\perp}(| x-x'|/\\xi)\\nonumber\\\\ \nG_m^{(t)}(t-t') &\\equiv& \\langle m(x,t)m(x,t')\\rangle\\nonumber\\\\\n&=& |t-t'|^{(\\eta-1)/z_m} g_m^{\\parallel}(|t-t'|/\\xi^{z_m})\\, . \\label{corr}\n\\end{eqnarray}\n\nOur simulations were done using a ``brick wall'' restricted\nsolid-on-solid model (see Fig.~\\ref{fig1}). \nStarting from a flat surface, particles are added such that no overhangs \nare formed, and with the center of each particle atop the edge\nof two particles in the layer below. \nWe use two types of particles, $A$ and $B$ (black and grey in the figures). \nThe probability for adding a particle to a given surface site, and the rule\nfor choosing its color, depend on the local neighborhood. \nWhen $A$ particles are more likely to be added to $A$ dominated regions,\nand vice versa, the particles tend to phase separate and form domains. \nIn this case, the order parameter correlation length $\\xi$ is of the order of \nthe average domain width. \nBy changing the growth rules, it is possible to study cases in which \nsome (or all) of the couplings $a$, $b$, $c$, and $\\alpha$ vanish,\nand thus to gain a complete picture of the different ways in \nwhich the height and the order parameter influence each other. \n\\begin{figure}\n\\centerline{\\epsfysize=0.06\\columnwidth{{\\epsfbox{wall.eps}}}}\n\\narrowtext{\\caption{The ``brick wall'' model used in the simulations.\\label{fig1} \n}}\n\\end{figure} \n\nThe decoupled case, $\\alpha=a=b=c=0$, is implemented using the following\nupdating rules: A surface site is chosen at random, and a particle is \nadded if it does not generate overhangs. \nIts color is then chosen depending on the colors of its two\nneighbors in the layer below. \nIf both neighbors have the same color, the newly added particle takes this\ncolor with probability $1-p$, and the other color with probability $p$ \n(where $p$ is much smaller than 1). \nIf the two neighbors have different colors, the new particle takes\neither color with probability 1/2. \nNeighbors within the same layer are not considered.\n\nSince the probability of adding a particle to a given surface site\ndoes not depend on its color, the surface grows exactly\n as with only one particle type, and is characterized by\nthe KPZ exponents $\\chi=1/2$, and $z_h=3/2$. \nSimilarly, the choice of particle color at a given\nsite is not affected by the height profile. \nThe height profile determines only the moment at which a site is added, \nsince the no-overhang condition requires both neighbors in the previous \nlayer to be occupied. \nIf we equate layer number with time, domain walls move to\nthe right or left with probability 1/2 during one time unit, and a\npair of new domain walls is created with probability $p$. \nThis is identical to the Glauber model for a \none-dimensional Ising chain with coupling $J$ and at temperature $T$, \nwith $p=\\exp(4J/kT)$. \nThe correlation length $\\xi$ perpendicular to the\ngrowth direction is consequently $\\xi=\\exp(-2J/kT)=1/\\sqrt{p}$, \nand the correlation time is $\\tau=\\exp(-4J/kT)=1/p$. \nThe dynamical critical exponent for the order parameter is thus $z_m=2$. \nNote that the ``time'' used for the order parameter (namely layer number)\nis different from real time, which is for each particle the moment\nwhen it is added to the growing surface. \nHowever, this difference becomes negligible for sufficiently small $p$ \nsince the thickness of the surface over the correlation length, \n$\\sqrt{\\xi}$, is much smaller than the characteristic time, $\\xi^2$, for\norder parameter fluctuations. \nSimulations indeed confirm that the order parameter and height evolve \ncompletely independently. \nA typical profile is shown in Fig.~\\ref{snaps}a; the corresponding scaling\nanalysis conforms to expectations, and is not presented here.\n%A snapshot of a system of width $L$ is\n%shown in Fig.~\\ref{fig2}.\n%\\begin{figure}\n%\\centerline{\\epsfysize=0.55\\columnwidth{{\\epsfbox{snapshotb.eps}}}}\n%\\narrowtext{\\caption{Snapshot of the last 400 layers for the decoupled\n%case, with $L=200$, $p=1/90$. \\label{fig2}}}\n%\\end{figure} \n\nThe situation $\\alpha>0$ with $a=b=c=0$ can be implemented by updating\nsites on top of particles of different colors less often by a factor $r<1$ \ncompared to sites above particles of the same color. \nAs the order parameter is not affected by the height variable, \nits dynamics is still the same as that of an Ising model, with $z_m=2$. \nThe height profile now has domain boundaries sitting preferentially\nat its local minima, with mounds forming over domains (see Fig.~\\ref{snaps}b). \nThis leads to a surface roughness exponent of $\\chi=1$ on length scales $\\xi$, \nwhich is the case studied in\\cite{kot98}. \nAt these scales, changes in the height profile are slaved to domain wall motion,\nand the dynamic exponent is $z_h=2$.\nHowever, on length scales much larger than $\\xi$, the KPZ exponents of\n$\\chi=1/2$ and $z_h=3/2$ are regained.\nThe crossover in the roughness can be described by a scaling form\n$$ \\langle[ h(x,t)-h(x',t)]^2\\rangle = |x-x'|^2 g(|x-x'|/\\xi),$$ \nwith a constant $g(y)$ for $y\\ll 1$, and $g(y) \\sim 1/y$ for $y \\gg 1$. \n%\\begin{figure}\n%\\centerline{\\epsfysize=0.5\\columnwidth{{\\epsfbox{snapshotd.eps}}}}\n%\\narrowtext{\\caption{Snapshot of the last 400 layers for $\\alpha>0$, with\n%$L=200$, $p=1/200$, and $r=1/20$. \\label{fig3}}}\n%\\end{figure} \n\nTo mimic the influence of surface roughness on the order parameter\n(nonzero $a$, $b$, or $c$ in Eqs.(\\ref{langevin2})),\nthe color of a newly added particle is made dependent not only\non those of its two neighbors in the layer below, \nbut also on the colors of its two nearest neighbors on the same layer, \nif these sites are already occupied. \nWith probability $1-p$, the newly added particle takes the\ncolor of the majority of its 2, 3, or 4 neighbors, and with\nprobability $p$ it assumes the opposite color. \nIf there is a tie, the color is chosen at random with equal probability.\nThe height variable now affects the order parameter in two ways: \n{\\bf (1)} {\\it Domain walls are driven downhill.}\nThe reason is that the neighbor on the hillside of a site being updated \nis more likely to be occupied than the one on the valley side. \nThe newly added particle is thus more likely to\nhave the color on the hillside. \n(This corresponds to $a>0$ in Eq.~(\\ref{langevin2}).)\n{\\bf (2)} {\\it New domains are predominantly formed on hilltops.} \nThis is because domains on hilltops can expand more easily \nthan those on slopes or in valleys, indicating $b>0$ in Eq.~(\\ref{langevin2}). \nAnother consequence is that for the same $p$,\nthe correlation length $\\xi$ is much larger than in the \ndecoupled case, as is apparent in Figs.\\ref{snaps}c,d.\n\n\\end{multicols}\\widetext\n\\begin{figure}\n\\centerline{\\epsfxsize=.97\\columnwidth{{\\epsfbox{snapshots.eps}}}}\n{\\caption{Snapshot of the last 400 layers of simulations, for $L=200$ sites.\n{\\bf (a)} The decoupled case with $p=1/90$, and $r=1$.\n{\\bf (b)} For $p=1/200$, and $r=1/20$, the height is coupled to the domains, \nbut not vice versa.\n{\\bf (c)} The fully coupled case, using the same parameters as (b), \nbut with updating rules that include neighbors in the same layer. \n{\\bf (d)} With $r=1$, and the updating rules of (c), the domains are\ninfluenced by the height, but not vice versa. (Note that the profiles in\n(a) and (d) are identical since we used the same random numbers.)\n \\label{snaps}}}\n\\end{figure} \n\\begin{multicols}{2}\\narrowtext\n\n\\begin{figure}\n\\centerline{\\epsfxsize=0.7\\columnwidth{{\\epsfbox{gmxt.eps}}}}\n\\narrowtext{\\caption{Scaling collapse of correlations $G_m^{(x)}$ and\n$G_m^{(t)}$ in Fig.~\\protect{\\ref{snaps}}d. \nFor each $p$, the data is an average over\n7500 widely separated layers, and for systems of size up to 8192.\n\\label{Gcollapse}}}\n\\end{figure} \n\nFor the fully coupled case depicted in Fig.\\ref{snaps}c we find essentially \nthe same scaling behavior as in Fig.\\ref{snaps}b, i.e. a height profile slaved\nto the Glauber dynamics of the domains. \nThe most interesting case, shown in Fig.\\ref{snaps}d, is when the height\nprofile is independent of the domains ($\\alpha=0$), evolving with KPZ dynamics,\nwhile the order parameter is influenced by the roughness. \nThe dynamic exponent $z_m$ for the order parameter was first \nobtained by collapsing the correlation functions using Eqs.~(\\ref{corr}), \nas shown in Fig.\\ref{Gcollapse}. %\\ref{fig5} and \\ref{fig6}. \nThese curves imply that $\\eta = 1$, $\\xi \\propto p^{-0.542}$, \nand $\\tau \\propto \\xi^{z_m} \\propto 1/p$, \ngiving $z_m \\simeq 1/0.542 \\simeq 1.85$. \n\nThe same non-trivial value for $z_m$ is obtained by a completely\nindependent measurement of the dynamics of domain coarsening \nfollowing a quench from a ``high temperature'' ($p$ close to 0.5) \nto zero temperature ($p$=0). \nFig.~\\ref{fig7} shows the domain density as function of time \nfor a system of size $L=16384$.\nThe resulting $z_m \\simeq 1.85$, is in agreement\nwith the value from the scaling collapse.\n\n\\begin{figure}\n\\centerline{\\epsfysize=0.5\\columnwidth{{\\epsfbox{coarsening.eps}}}}\n\\narrowtext{\\caption{Domain density as a function of time for $L=16384$,\naveraged over 100 samples. The dotted line is a power-law fit\n(slightly shifted for better visibility) with exponent of $1/z_m = 0.542$. \nFor comparison, a power law with exponent $-0.5$ is also shown\n(dashed line).\n\\label{fig7}}}\n\\end{figure} \n\nThe following simple argument fails to provide the exponent \n$z_m \\simeq 1.85$.\nConsider a Langevin equation, $\\dot x = \\eta(t)$, for the position $x$\nof a single domain wall at time $t$. \nSince the motion of the domain wall is strongly influenced by\nthe height profile, the noise $\\eta(t)$ must have long-range correlations\n$\\langle \\eta(t)\\eta(t')\\rangle = D |t-t'|^\\alpha,$\nreflecting the dynamics of surface.\nThis choice leads to $z_m=2$ for $\\alpha >1$, \nand $z_m=2/(2-\\alpha)$ for $\\alpha <1$. \nFor a colored noise dominated by the slope fluctuations,\n$\\alpha=2/3$ and $z_m=3/2$, i.e. the height imposes its characteristic \ntime scale on the order parameter. \nThis would presumably be the case if the domain walls were\nuniformly distributed along the surface.\nHowever, due to their tendency to move downhill, \nthey are preferentially found near valleys.\nA different scaling of the slope fluctuations in the valleys \nmay be the reason for the nontrivial value of $z_m$. \nIndeed, for short times, before the domain walls have moved \nto their preferred positions, the exponent $3/2$ is seen.\n\nThe dynamics of domain walls on a growing KPZ surface bears some\nresemblance to the advection of a passive scalar in a turbulent\nvelocity field, which is characterized by nontrivial\nexponents and multiscaling \\cite{kra94}. \nIf we neglect interactions between domain walls, and treat them as \nindependent ``dust particles'' floating on the KPZ surface, \nthe Langevin equation for the particle density $\\rho$ is\n\\begin{equation}\n\\partial_t \\rho = K \\nabla^2\\rho+ a(\\nabla h \\cdot \\nabla \\rho+\\rho\\nabla^2 h) + \n\\zeta_\\rho .\\label{langevin3}\n\\end{equation}\nThe second term describes the {\\it advection} of particles along a velocity\nfield $\\vec v=\\nabla h$.\nIndeed this transformation maps the KPZ equation into the Burgers equation \nfor a vorticity-free, compressible fluid flow \\cite{kar86}. \nEquation (\\ref{langevin3}) is a special case of Eq.~(\\ref{langevin2}) for $m$,\nwith $r=u=c=0$, $b=a$, and with a conserved noise $\\zeta_\\rho$. \n(Together with Eq.~(\\ref{langevin1}) for the height profile, \nit is also a special case of the equations used to \ndescribe the dynamic relaxation of drifting polymers\\cite{ert93}.) \nIn the remainder, we give the results of computer simulations for this case. \nThe rules for the motion of ``dust particles'' are\nidentical to those for domain walls.\nHowever, each particle is treated as if the others were not present. \nThis means in particular that\nthere is no creation or annihilation of particles. \n\n\\begin{figure}\n\\centerline{\\epsfysize=0.55\\columnwidth{{\\epsfbox{diffusion.eps}}}}\n\\narrowtext{\\caption{Mean square displacement of a single domain wall in a \nsystem of size $L=4096$. The power law fit (dotted line) has an exponent 1.1467, \ncorresponding to $z_\\rho \\simeq 1.74$. \n\\label{fig8}}}\n\\end{figure} \n\n Fig.~\\ref{fig8} shows the mean square displacement of a single ``dust\nparticle'' in a system of size $L=4096$. \nTo obtain good statistics, we averaged over 512 independent \nand noninteracting particles, and used more than 40 runs. \nThe best fit is obtained for $z_\\rho \\simeq 1.74$, distinct from the \nprevious $z_m \\simeq 1.85$, implying that the exponents depend\non whether or not the domain walls (or ``dust particles'') are conserved. \nIn contrast to the advection of a passive scalar in a turbulent velocity field, \nwe find no sign of multiscaling.\n Fig.~\\ref{fig9} shows the positions of 1024 independent ``dust particles'' in \na system of length $L=512$. \nWhile there is some correlation between minima of the surface profile \nand wall positions, there are also clusters of particles at higher elevations, \nindicating that particle diffusion is not sufficiently fast to fully adjust the\ndensity to the faster changing height profile.\nA fit of the density-density correlation function to\n$\\langle \\rho(x)\\rho(0)\\rangle\\sim 1/x^{2(1-\\chi_\\rho)}$,\ngives an exponent $\\chi_\\rho \\simeq 0.85$.\n\n\\begin{figure}\n\\centerline{\\epsfysize=0.5\\columnwidth{{\\epsfbox{snap512.eps}}}}\n\\narrowtext{\\caption{\nHistogram of the positions of 1024 domain walls\nalong a surface profile (indicated in grey) of size $L=512$. \n\\label{fig9}}}\n\\end{figure} \n\nIn summary, the interplay between surface roughening and phase \nseparation leads to a variety of novel critical scaling behaviors.\nAt one extreme, the height profile adapts to the dynamics of\ncritical domain ordering.\nAt the other, the dynamics of domain wall motion is influenced by\nthe roughness, exhibiting new and nontrivial scaling behaviors. \n\n\n% \\acknowledgements \nThis work was supported by EPSRC\n(grant No.~GR/K79307, for BD), and\nthe National\nScience Foundation (Grant No. DMR-98-05833, for MK).\n\n\\begin{references} \n\\bibitem{hel95} P.W. Rooney, A.L. Shapiro, M.Q. Tran, and F. Hellman, Phys. Rev. \nLett. {\\bf 75}, 1843 (1995).\n\\bibitem{kan90} Y. Bar-Yam, D. Kandel, and E. Domany, \n Phys. Rev. B {\\bf 41}, 12869 (1990).\n\\bibitem{dro97} B. Drossel and M. Kardar, Phys. Rev. E {\\bf 55}, 5026 (1997).\n\\bibitem{kot98} M. Kotrla and M. Predota, Europhys. Lett. {\\bf 39}, 251 (1997); \nM. Kotrla, M. Predota, and F. Slanina, Surface Science {\\bf 404}, 249 (1998); \nM. Kotrla, F. Slanina and M. Predota, Phys. Rev. B {\\bf 58}, 10 003 (1998).\n\\bibitem{kar86} M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. {\\bf \n56}, 889 (1986).\n\\bibitem{leo97} F. L\\'eonard and R.C. Desai, Phys. Rev. B {\\bf 55}, 9990 (1997). \n\\bibitem{leo97a} F. L\\'eonard, M. Laradji, and R.C. Desai, Phys. Rev. B {\\bf \n55}, 1887 (1997). \n\\bibitem{kra94} R.H. Kraichnan, Phys. Rev. Lett. {\\bf 72}, 1016 (1994).\n\\bibitem{ert93} D. Ertas and M. Kardar, Phys. Rev. Lett. {\\bf 69}, 929 (1992).\n\\end{references} \n\n\\end{multicols}\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002032.extracted_bib",
"string": "\\bibitem{hel95} P.W. Rooney, A.L. Shapiro, M.Q. Tran, and F. Hellman, Phys. Rev. \nLett. {\\bf 75}, 1843 (1995).\n\n\\bibitem{kan90} Y. Bar-Yam, D. Kandel, and E. Domany, \n Phys. Rev. B {\\bf 41}, 12869 (1990).\n\n\\bibitem{dro97} B. Drossel and M. Kardar, Phys. Rev. E {\\bf 55}, 5026 (1997).\n\n\\bibitem{kot98} M. Kotrla and M. Predota, Europhys. Lett. {\\bf 39}, 251 (1997); \nM. Kotrla, M. Predota, and F. Slanina, Surface Science {\\bf 404}, 249 (1998); \nM. Kotrla, F. Slanina and M. Predota, Phys. Rev. B {\\bf 58}, 10 003 (1998).\n\n\\bibitem{kar86} M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. {\\bf \n56}, 889 (1986).\n\n\\bibitem{leo97} F. L\\'eonard and R.C. Desai, Phys. Rev. B {\\bf 55}, 9990 (1997). \n\n\\bibitem{leo97a} F. L\\'eonard, M. Laradji, and R.C. Desai, Phys. Rev. B {\\bf \n55}, 1887 (1997). \n\n\\bibitem{kra94} R.H. Kraichnan, Phys. Rev. Lett. {\\bf 72}, 1016 (1994).\n\n\\bibitem{ert93} D. Ertas and M. Kardar, Phys. Rev. Lett. {\\bf 69}, 929 (1992).\n"
}
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cond-mat0002033
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A microscopic approach to phase transitions in quantum systems
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We present a new theoretical approach for the study of the phase diagram of interacting quantum particles: bosons, fermions or spins. In the neighborhood of a phase transition, the expected renormalization group structure is recovered both near the upper and lower critical dimension. Information on the microscopic hamiltonian is also retained and no mapping to effective field theories is needed. A simple approximation to our formally exact equations is studied for the spin-$S$ Heisenberg model in three dimensions where explicit results for critical exponents, critical temperature and coexistence curve are obtained.
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"name": "lett.tex",
"string": "%\\documentstyle[preprint,aps,psfig]{revtex}\n\\documentstyle[prl,aps,psfig]{revtex}\n%\n\\def \\bp {{\\bf p}}\n\\def \\bg {{\\bf g}}\n\\def \\bR {{\\bf R}}\n%\n\\begin{document}\n%\\draft\n\n%%%%%%%%%%%%%%%%%%\n%\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname\n%@twocolumnfalse\\endcsname\n%%%%%%%%%%%%%%%%%%\n\n\\widetext\n\\title{ A microscopic approach to phase transitions in quantum systems}\n\\author { Pietro Gianinetti$^{1,2}$ and Alberto Parola$^{1,3}$\n\\footnote{Fax: +39-02-2392482, e-mail: parola@mi.infm.it} }\n\\address{ \n$^1$ Istituto Nazionale per la Fisica della Materia \\\\\n$^{2}$ Dipartimento di Fisica, Universit\\'a di Milano, Via Celoria 16,\nMilano, Italy \\\\\n$^3$ Dipartimento di Scienze Fisiche, Universit\\'a dell'Insubria, Via Lucini 3\nComo, Italy } \n\n\\maketitle\n\\begin{abstract}\nWe present a new theoretical approach for the study of \nthe phase diagram of interacting quantum particles: bosons, fermions or spins.\nIn the neighborhood of a phase transition, \nthe expected renormalization group structure is recovered\nboth near the upper and lower critical dimension. \nInformation on the microscopic hamiltonian is also retained and\nno mapping to effective field theories is needed.\nA simple approximation to our formally exact equations is\nstudied for the spin-$S$ Heisenberg model in three dimensions\nwhere explicit results for critical exponents, critical temperature and \ncoexistence curve are obtained. \n\\end{abstract}\n\\pacs{75.10.Jm 05.70.Jk 75.40.Cx}\n\n%%%%%%%%%%%%%%%%%%\n%]\n%%%%%%%%%%%%%%%%%%\n%\\narrowtext\nSeveral physical systems, ranging from magnets to superfluids and \nsuperconductors,\ndisplay rich phase diagrams in a temperature regime where quantum effects\ncannot be neglected. Different scenarios, characterized by competing\norder parameters and zero temperature phase transitions have been \nrecently advocated also in the framework of high temperature superconductivity\nwhere antiferromagnetic order, Cooper pairing and, possibly, phase separation \nare at play in the same region of the phase diagram \\cite{htc}.\nA satisfactory understanding of phase transitions in\nquantum models has been attained years ago through the seminal work\nby Hertz \\cite{quantum} who showed that, at low energy and long \nwavelengths, quantum models may be described by a suitable classical\naction. However, a quantitative theory\nof the thermodynamic behavior is still lacking and we mostly rely on \nmean field approaches or weak coupling renormalization group\n(RG) calculations \\cite{mfrg}, applied to quantum systems via the mapping to \nthe appropriate effective\nfield theory. In particular, the interplay between thermal and quantum\nfluctuations is expected to give rise to crossover phenomena\nwhose extent strongly depends on the microscopic features of the system.\nEven for the most extensively studied models, like the Heisenberg\nantiferromagnet, our knowledge of the phase diagram is in fact limited, \nand the first precise finite temperature simulation attempting to fill this gap\nhas become available only recently \\cite{sandvik}.\nBy contrast, in classical models, numerical simulations are quite efficient\neven in the neighborhood of critical points \\cite{mc} and,\nfrom the analytical side, microscopic approaches especially devised for\nthe quantitative description of the phase diagram of classical fluids\nand magnets are available. For instance, the hierarchical reference\ntheory of fluids (HRT) \\cite{hrt} has proven quite accurate in\nlocating the phase transition lines both in lattice and in continuous\nmodels. \n\nIn this Letter we sketch the derivation of the quantum hierarchical \nreference theory of fluids (QHRT) which we then apply to the \nHeisenberg antiferromagnet. We will\ndemonstrate that the known renormalization group equations near four and\nnear two dimensions are naturally recovered within our approach, which\ntherefore unifies two complimentary techniques. On approaching the\ncritical point, the spin velocity vanishes according to the\nexpected dynamical critical exponent for an antiferromagnet. \nFinally, the phase diagram of this model in three dimensions \nis computed by numerical integration of a simple approximation to the\nthe QHRT equations, providing a concrete application of our general approach.\n\nThe starting point is a microscopic, many body hamiltonian $H$ written\nas the sum of a reference part $H_0$ and an interaction term $V$. \nThe interaction is assumed to be bilinear in some operator $\\rho(r)$,\nwhich is assumed either linear in bosonic operators or quadratic in fermionic\nones:\n\\begin{equation}\nV={1\\over 2}\\int dx dy \\rho(x) w(x-y) \\rho(y) \n\\end{equation}\nwith a non singular (i.e. Fourier transformable) two body potential $w$.\nThe properties of the reference system under the action of\nan external field $h$ coupled to the order parameter $\\rho(r)$ are \nsupposed known. No specific assumption on the reference system is\nmade: in particular we do not need that $H_0$ corresponds to\na non interacting system, where Wick theorem applies (such a feature is\ncrucial in setting up the QHRT equations).\nThese requirements are indeed rather general and include several models \nof current interest in many body physics: quantum magnets (where\n$\\rho(r)$ represents the local spin variable), fermionic systems, like the\nHubbard model, or even the Holstein model for the\nelectron-phonon problem.\n\nThe first task is to build up a formal perturbative\nexpansion of the partition function of the model: $Z={\\rm Tr}\\exp(-\\beta H)$.\nFollowing a standard procedure \\cite{fetter}, $Z/Z_0$ can be written as \nthe average over the reference distribution function of an imaginary \ntime evolution operator $U(\\beta)$. When this operator is \nwritten as a power series of \nthe interaction $w(r)$, we formally recover a perturbative\nexpansion identical to that of a classical partition function for a \n$(d+1)$ dimensional model. The additional ``temporal\" dimension is \nlimited to the interval $(0,\\beta)$ and $w(r)\\delta(t)/\\beta$ plays \nthe role of classical two body interaction $w_c(r,t)$.\nThe reference system of the associated classical\nmodel is implicitly defined by requiring that its \ncorrelation functions coincide with those of the quantum reference \nhamiltonian $H_0$. \n{\\sl Approximate} mappings \nbetween a quantum model and an effective classical system have \nbeen proposed and studied in the literature \\cite{tognetti}\nin order to clarify the role of thermal and quantum \nfluctuations. The novelty of our approach is that $i)$ it is {\\sl exact} \nand $ii)$ it applies to all temperature regimes, including the $T\\to 0 $\nlimit. Having reduced the quantum problem to a\nclassical one, we can directly apply the techniques developed in that \nframework. In particular, the already mentioned HRT is an \nimplementation of the momentum space renormalization method \nwhich preserves information on the details of the microscopic hamiltonian. \nIn HRT, different Fourier components of the two body interaction\nare included gradually, starting from the shortest wavelength:\nphysically this corresponds to a smooth turning on of fluctuations over\nlarger and larger lengthscales. This procedure can be carried out exactly \nby defining a sequence of auxiliary systems interacting via\na potential whose Fourier components coincide with $w(k)$ for\n$k>Q$ and vanish elsewhere. As a result, we obtain\na set of coupled differential equations\nexpressing the change in the free energy and in the correlation\nfunctions of the model when a given Fourier component $Q$ of the\npotential is included. The initial condition represents the system\nin which fluctuations are frozen, and in fact coincides with\nthe known mean field result. As the {\\sl cut-off} wavevector\n$Q\\to 0$, the fully interacting system\nis recovered. Mean field {\\sl approximation} therefore\ncorresponds to neglecting the change in the properties of the\nmodel as described by our differential equations. Details can be\nfound in Ref. \\cite{hrt}.\n\nAs an example, we study the the spin-$S$ antiferromagnetic \nHeisenberg model on a hypercubic lattice in $d$ dimension:\n\\begin{equation}\nH=H_0+V=h\\sum_\\bR e^{i\\bg\\cdot\\bR} S^z_\\bR+\nJ \\sum_{<\\bR,\\bR^\\prime>} {\\bf S_R}\\cdot {\\bf S}_{{\\bf R}^\\prime}\n\\end{equation}\nwhere the sum is restricted to nearest neighbors and ${\\bf g}$ is\nthe antiferromagnetic wavevector of components $g_i=\\pi$.\nIn applying the HRT approach to a quantum model, we decided to\nimpose the cut-off $Q$ only to the spatial Fourier\ncomponents of $w_c(r,t)$. Different physical models might require\nother choices of the cut-off, whose only role is to\ncontinuously connect the reference system\nto the fully interacting one, which is recovered in the $Q\\to 0$ limit.\nThe exact evolution equation for the Helmholtz free energy \ndensity $a$ of the system describes how $a$ is modified due to \na change in the cut-off $Q$:\n\\begin{eqnarray}\n\\frac{d \\, a^Q}{dQ}=\\frac{1 }{2\\beta} \n\\int_{\\bp\\in\\Sigma_Q} \\sum_{\\omega}\n&\\Big \\{& 2\\,\\ln \\left [ (1-F^Q_{xx}(\\bp,\\omega)w(\\bp))(1+F^Q_{xx}\n(\\bp^{\\prime},\\omega)w(\\bp))\n-F_{xy}^Q(\\bp,\\omega)F_{xy}^Q(\\bp^{\\prime},\\omega) w(\\bp)^2 \\right ]\\nonumber\\\\\n&+& \\ln\\left [(1-F^Q_{zz}(\\bp,\\omega)w(\\bp))(1+F^Q_{zz}(\\bp^{\\prime},\\omega)\nw(\\bp))\\right]\\Big \\} \n\\label{qhrt}\n\\end{eqnarray}\nHere $\\bp^{\\prime}=\\bp+\\bg$, \n$w(\\bp)=2J\\gamma(\\bp)=2J\\sum_i \\cos(p_i)$ is the Fourier transform of the\ninteraction, the summation is over the Matsubara frequencies\n$\\omega_n=2\\pi n/\\beta$, the $(d-1)$ dimensional integral \nis restricted to the\nsurface $\\Sigma_Q$ defined by $\\gamma(\\bp)=-\\sqrt{d^2-Q^2}$ \nand the functions $F^Q_{ij}(\\bp,\\omega)= <S^i(\\bp,\\omega)\nS^j(-\\bp,-\\omega)>$ \nare the Fourier transforms of the spin-spin dynamical correlation functions\n(in imaginary time) for a system where only fluctuations of\nwavevector $\\bp$ such that $|\\gamma(\\bp)|<\\sqrt{d^2-Q^2}$ are included.\nThe isotropy of the model implies $F_{yy}=F_{xx}$ and $F_{yx}=-F_{xy}$.\nAnalogous equations can be derived for the many spin correlation\nfunctions of the model. The (infinite) set of differential equations\nforms the QHRT hierarchy.\nWhen $Q=d$, fluctuations are neglected and the exact initial condition\nfor the first QHRT equation (\\ref{qhrt}) coincides with the mean field\nfree energy density. The magnetic structure factors at $Q=d$ \ncan be explicitly written as:\n\\begin{eqnarray}\nF^Q_{xx}(\\bp,\\omega)&=&{\\mu_\\perp-w(\\bp)\\over m^{-2}\\omega^2+\n\\mu_\\perp^2-w(\\bp)^2};\\nonumber \\\\\nF^Q_{xy}(\\bp,\\omega)&=&{m^{-1}\\omega\\over\nm^{-2}\\omega^2+\\mu_\\perp^2-w(\\bp)^2};\\nonumber \\\\\nF^Q_{zz}(\\bp,\\omega)&=&{\\delta_{\\omega,0}\\over\\mu_{||}+w(\\bp)}\n\\label{fij}\n\\end{eqnarray}\nwhere $m$ is the staggered magnetization and $\\mu_\\perp,\\,\\,\n\\mu_{||}$ are known functions of $m$. For $S=1/2$: \n$\\mu_\\perp=2T\\,m^{-1}\\tanh^{-1}(2m)$ and $\\mu_{||}=4T(1-4m^2)^{-1}$. \nNote that the dependence of\nthe transverse magnetic structure factors in (\\ref{fij}) on \nfrequency and momentum is consistent with the \nfirst order spin wave result at zero temperature and reproduces the\nknown single mode approximation which well represents\nantiferromagnetic correlations at low temperatures\n\\cite{stringari,singh}. The longitudinal\ncorrelations in equation (\\ref{fij}) are instead purely classical\nand, as such, satisfy the relationship $T\\,\\chi_{||}(k)=S_{||}(k)$\nbetween longitudinal susceptibility $\\chi_{||}(k)$ and the corresponding \nstatic structure factor $S_{||}(k)$. \n\nEquation (\\ref{qhrt}), although\nformally exact, is not closed because the evolution of the free energy\ndepends on the unknown magnetic structure factors of the model \n$F_{ij}^Q(\\bp,\\omega)$. Therefore, we have to introduce some approximate\nparametrization of the structure factors in terms of the free energy.\nThe simple approximation we have studied is to retain the form \nof $F_{ij}^Q(\\bp,\\omega)$ as given in Eq. (\\ref{fij}) but imposing \nthermodynamic\nconsistency in order to determine the two scalar parameters $\\mu_\\perp$\nand $\\mu_{||}$. More precisely, for every $Q$ we related the transverse\nand longitudinal staggered susceptibilities to the free energy \nvia the exact sum rules: \n\\begin{eqnarray}\n(\\mu_\\perp-2dJ)^{-1}&=&F_{xx}^Q(\\bg,\\omega=0)=\nm(\\partial a^Q/\\partial m)^{-1} \\nonumber \\\\\n(\\mu_{||}-2dJ)^{-1}&=&F_{zz}^Q(\\bg,\\omega=0)=\n(\\partial^2 a^Q/\\partial m^2 )^{-1} \n\\label{sum}\n\\end{eqnarray}\nFrom the adopted structure of the dynamical correlation functions,\nwe also obtain the relationship between the parameters entering\n$F_{ij}^Q$ and the zero temperature non linear sigma model coupling constants:\nuniform transverse susceptibility $\\chi_0=1/(4d)$, spin wave velocity\n$c=\\sqrt{4d}\\,m$ and spin stiffness $\\rho_s=m^2$. The hydrodynamic \nrelation $\\chi_0 c^2=\\rho_s$ is automatically satisfied by our ansatz\nfor arbitrary spontaneous magnetization $m$.\nIt is interesting to note that from our parametrization,\nthe scaling of the spin wave velocity $c$ on approaching the \ncritical temperature ($t=(T_c-T)/T_c\\to 0$)\ngives $c\\propto m$ that is $c\\propto t^\\beta$ along the coexistence curve. \nThe dynamic scaling hypothesis instead predicts $c\\propto t^{\\nu(z-1)}$\nwhere $z$ is the dynamical critical exponent. By use of scaling laws\nand recalling that in our approximation the correlation critical\nexponent vanishes ($\\eta=0$), we get $z=d/2$ which is the expected\nresult for an antiferromagnet (i.e. model G) \\cite{hohenberg}.\nEquation (\\ref{qhrt}), together with (\\ref{fij}) and (\\ref{sum}) \ngive rise to a partial differential equation for the free energy \ndensity of the Heisenberg model $a^Q(m)$ as a function of the \ncut-off $Q$ and of the\nmagnetization $m$. The frequency sum can be carried out analytically\ngiving the final equation:\n\\begin{equation}\n\\frac{d \\, a^Q}{dQ}=\\frac{1 }{2\\beta} \n\\int_{\\bp\\in\\Sigma_Q} \n\\left \\{ 4\\,\\ln \\left [{\\sinh\\left ({1\\over 2}\\beta m\\mu_\\perp\\right )\\over \n\\sinh\\left ({1\\over 2}\\beta m\\sqrt{\\mu_\\perp^2-w(\\bp)^2}\\right )}\\right ]\n+ \\ln\\left [{\\mu_{||}^2\\over\\mu_{||}^2-w(\\bp)^2} \\right]\\right \\} \n\\label{qhrt2}\n\\end{equation}\nWe numerically solved this partial differential equation for several \nvalues of the\nspin $S$ and different temperatures in order to study the phase diagram of\nthis system. Note that $S$ just enters the theory through the \ninitial condition $a^Q(m)$ at $Q=d$, while the form of the differential \nequation is unaffected by $S$. Before showing the numerical results, however, \nit is useful to discuss the behavior of Eq. (\\ref{qhrt2}) near\na phase transition. In particular, we studied the neighborhood of the\ncritical point (in $d>2$) and the low temperature region. \nBoth at the critical point and along the coexistence curve the \nsusceptibilities diverge due to the presence of critical fluctuations\nand Goldstone bosons, respectively. From Eq. (\\ref{sum}) we conclude that \nat long wavelengths (i.e. $Q\\to 0$) and near a phase\ntransition we have $\\mu_\\perp\\sim\\mu{||}\\sim 2dJ$. In this region\nequation (\\ref{qhrt2}) simplifies and, by rescaling the free energy\nas $a^Q Q^{-d}$ and the magnetization as $m Q^{(2-d)/2}$, it reduces\nto the RG equation obtained by Stanley {\\it et al.}\n\\cite{stanley} for a $O(3)$ symmetric $\\phi^4$ hamiltonian. Such an\nequation has been analyzed near four dimension and proved to give\nthe correct critical exponents to first order in the $\\epsilon=4-d$\nexpansion. In three dimensions, the numerical solution of the \nuniversal fixed point equation gives for the correlation length\ncritical exponent the result $\\nu=0.826$, to be compared with the\naccepted value $\\nu=0.71$. The other critical exponents follow from\nthe scaling laws, noting that our analytical form of the two point\nfunctions (\\ref{fij}) forces the anomalous dimension exponent to vanish\n$\\eta=0$. We therefore find non classical exponents in three dimensions.\nSpecial care must be paid when dealing with the $T\\to 0 $ limit of\nour equation. In this case, the asymptotic form of the equation changes\nand it can be shown to give rise, near a hypothetical quantum critical\npoint, to critical exponents appropriate for\na $O(3)$ model in $d+1$ dimensions as expected. \nA separate analysis should be carried out in the low temperature phase.\nIf symmetry is spontaneously broken, following Chakravarty {\\it et al.}\n\\cite{chn}, we may ask how quantum and thermal\nfluctuations modify the zero field magnetization. In order to \nanswer this question we perform a Legendre transform on \nour equation (\\ref{qhrt2}): we first derive it with respect to the \nmagnetization $m$ obtaining an evolution equation for the magnetic\nfield $h^Q(m)$ at fixed $m$. \nThen we find the equation governing the evolution of \nthe spontaneous magnetization $m^Q$ implicitly defined by the \nrequirement $h^Q(m^Q)=0$ \nat every $Q$. This procedure gives rise to a differential\nequation for $m^Q$. In the $Q\\to 0$ limit, taking into account that \nthe longitudinal susceptibility diverges more slowly than $Q^{-2}$, \nQHRT reduces to a simple ordinary differential equation:\n\\begin{equation}\n{d m^Q\\over dQ}=K_d \\left({Q \\over \\sqrt{d}}\\right)^{d-2} \n\\left [ \\tanh \\left (Q \\beta \\, m^Q\\right)\\right ]^{-1}\n\\label{mag}\n\\end{equation}\nwhere $K_d$ is a geometrical factor (ratio between the solid angle and\nvolume of the Brillouin zone). By introducing the rescaled variable\n$g=\\sqrt{4d}(Q/\\sqrt{d})^{d-1}/m^Q$, equation (\\ref{mag}) becomes identical,\nto order $g^2$, to the known weak coupling RG\nequations for the non linear sigma model applied\nby Chakravarty {\\it et al.} \\cite{chn} to the analysis of the \nantiferromagnetic Heisenberg model at long wavelengths and low\ntemperatures. As an example, we plot in Fig. 1 the RG flux of $g^Q$\nobtained by the integration of the {\\sl full} QHRT equation (\\ref{qhrt2}).\nWe clearly see the effect of the unstable zero temperature weak\ncoupling fixed point while, for the nearest neighbor Heisenberg model, \nthe other fixed point ($g_c$), governing the quantum critical regime, has no\neffect on the RG trajectories.\nThis analysis shows that the single mode approximation to \nQHRT reproduces the correct long wavelength structure both near four and two\ndimensions. Furthermore, we expect QHRT to be superior to the weak coupling\nrenormalization group equations because our non perturbative approach \nalso describes the critical region and the high temperature regime\nwhere $m^Q\\to 0$ as $Q\\to 0$, corresponding to $g\\to \\infty$ i.e. to\nthe strong coupling phase of the non linear sigma model.\n\nFinally, we present few results of the numerical integration of Eq.\n(\\ref{qhrt2}) in three dimensions. As already pointed out in the\nclassical case \\cite{hrt}, the HRT approach is able to correctly\nimplement Maxwell construction at first order phase transitions\nand in fact the free energy density\n$a^Q(m)$ at the end of the integration, i.e. in the $Q\\to 0$ limit,\nbecomes rigorously flat in a finite region of the magnetization\naxis for $T<T_c$. Therefore it is easy to extract from the numerical output \nthe critical temperature and the coexistence curve, shown \nin Fig. 2 for several values of the spin $S$. The zero temperature \nlimits of this curve agree within about $2\\%$ with the accepted \nestimates based on spin wave theory, Monte Carlo simulations or \nseries expansions. Regrettably, for the spontaneous magnetization \nat finite temperature there are just few available results going\nbeyond mean field approaches. Simulation data for the classical\n$S\\to\\infty$ case \\cite{binder} and recent series expansion for the $S=1/2$\nmodel \\cite{kok} seem to give somewhat larger coexistence regions. \nHowever, we believe that a more systematic analysis of these models by \naccurate numerical techniques is necessary before reaching a definite\nconclusion on the accuracy in the determination of the coexistence\ncurve. The critical temperature for the classical model is \nknown by several methods \\cite{mc,domb} to be $T_c=1.443 \\,J$ while\nfor the $S=1/2$ case it has been recently estimated as $T_c=0.946 \\,J$\n\\cite{sandvik} by use of a newly developed Quantum Monte Carlo method\nand $T_c=0.93 \\,J$ \\cite{kok} by series expansions.\nOur results are a few percent lower, being $T_c=1.419 \\,J$ for\n$S=\\infty$ and $T_c=0.90 \\,J$ for $S=1/2$. From the solution of the QHRT\nequation we also obtain other important information on the model, for \ninstance, the equation of state, the specific heat and also the temperature\ndependent dynamical structure factors, via analytic continuation of \nthe adopted expressions (\\ref{fij}). In order to improve \nthe QHRT results we have just discussed, other approximate \nexpressions for the magnetic structure factors should be \nexamined, possibly keeping the same form (\\ref{fij}) but allowing for a\nnon trivial renormalization factor for the uniform susceptibility.\nThis method can be applied in a straightforward way to other models\nof interest in quantum many body physics, like the Hubbard model,\nand may help to determine the location of the magnetic phase transitions\nand the possible occurrence of phase separation in a purely repulsive \nelectron system.\n\n\n\\begin{references}\n\\bibitem{htc} P. Montoux {\\it et al.} \\prb {\\bf 46}, 14803 (1992);\nC. Castellani {\\it et al.} \\prl {\\bf 75}, 4650 (1995).\n\\bibitem{quantum} J.Hertz, \\prb {\\bf 14}, 1165 (1976).\n\\bibitem{mfrg} See for instance E.Fradkin in {\\it Field theory of\ncondensed matter systems} (Addison-Wesley, 1991).\n\\bibitem{sandvik} A.W.Sandvik, Phys.Rev.Lett.{\\bf 80},5196 (1998).\n\\bibitem{mc} C.Holm, W.Janke, Phys.Rev.B {\\bf 48}, 936 (1993); \nK.Chen {\\it et al.}, Phys.Rev.B {\\bf 48}, 3249 (1993).\n\\bibitem{hrt} A. Parola, L. Reatto, Adv.Phys {\\bf 44},211 (1995). \n\\bibitem{fetter} L.Fetter, J.D.Walecka, \n{\\it Quantum Theory of Many particle\nSystems} (McGraw-Hill, New York,1971).\n\\bibitem{tognetti} A.Cuccoli {\\it et al} Phys.Rev.Lett.{\\bf 77},3439 (1996). \n\\bibitem{stringari} S.Stringari, Phys.Rev.B {\\bf 49}, 6710 (1994). \n\\bibitem{singh} R.R.P.Singh, Phys.Rev.B {\\bf 47}, 12337 (1993).\n\\bibitem{hohenberg} P.C.Hohenberg, B.I.Halperin, \nRev.Mod.Phys. {\\bf 49},435 (1977).\n\\bibitem{stanley} J.F.Nicoll, T.S.Chang, and H.E.Stanley\nPhys.Rev.Lett.{\\bf 33},540 (1974). \n\\bibitem{chn} S.Chakravarty, B.I.Halperin, and D.R.Nelson, \nPhys.Rev.Lett.{\\bf 60},1057 (1988); Phys.Rev.B.{\\bf 39},2344 (1989).\n\\bibitem{binder} K.Binder, H.M\\\"uller-Krumbhaar, Phys.Rev.B {\\bf 7}, 3297\n(1973).\n\\bibitem{kok}Kok-Kwei Pan,Phys.Rev.B {\\bf 59}, 1168 (1999). \n\\bibitem{domb} G.S.Rushbrooke {\\it et al.} in {\\it Phase\nTransitions and Critical Phenomena},Vol.3,\nC.Domb,M.S.Green eds., (Academic, 1974).\n\\end{references}\n\n\\begin{figure}[htbp]\n\\protect\n\\centerline{\\psfig{figure=fig1.ps,width=15cm}}\n \\caption{RG trajectories for the two dimensional Heisenberg model\ncomputed via numerical integration of the QHRT equation.}\n\\end{figure}\n\n\\newpage\n\n\\begin{figure}[htbp]\n\\protect\n\\centerline{\\psfig{figure=fig2.ps,width=15cm}}\n \\caption{Reduced spontaneous magnetization as a function of temperature\nfor different values of the spin: $S=1/2$ (triangles) $S=1$ (squares)\n$S=5/2$ (circles) and $S=\\infty$ (full line).}\n\\end{figure}\n\\end{document}\n\n"
}
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[
{
"name": "cond-mat0002033.extracted_bib",
"string": "\\bibitem{htc} P. Montoux {\\it et al.} \\prb {\\bf 46}, 14803 (1992);\nC. Castellani {\\it et al.} \\prl {\\bf 75}, 4650 (1995).\n\n\\bibitem{quantum} J.Hertz, \\prb {\\bf 14}, 1165 (1976).\n\n\\bibitem{mfrg} See for instance E.Fradkin in {\\it Field theory of\ncondensed matter systems} (Addison-Wesley, 1991).\n\n\\bibitem{sandvik} A.W.Sandvik, Phys.Rev.Lett.{\\bf 80},5196 (1998).\n\n\\bibitem{mc} C.Holm, W.Janke, Phys.Rev.B {\\bf 48}, 936 (1993); \nK.Chen {\\it et al.}, Phys.Rev.B {\\bf 48}, 3249 (1993).\n\n\\bibitem{hrt} A. Parola, L. Reatto, Adv.Phys {\\bf 44},211 (1995). \n\n\\bibitem{fetter} L.Fetter, J.D.Walecka, \n{\\it Quantum Theory of Many particle\nSystems} (McGraw-Hill, New York,1971).\n\n\\bibitem{tognetti} A.Cuccoli {\\it et al} Phys.Rev.Lett.{\\bf 77},3439 (1996). \n\n\\bibitem{stringari} S.Stringari, Phys.Rev.B {\\bf 49}, 6710 (1994). \n\n\\bibitem{singh} R.R.P.Singh, Phys.Rev.B {\\bf 47}, 12337 (1993).\n\n\\bibitem{hohenberg} P.C.Hohenberg, B.I.Halperin, \nRev.Mod.Phys. {\\bf 49},435 (1977).\n\n\\bibitem{stanley} J.F.Nicoll, T.S.Chang, and H.E.Stanley\nPhys.Rev.Lett.{\\bf 33},540 (1974). \n\n\\bibitem{chn} S.Chakravarty, B.I.Halperin, and D.R.Nelson, \nPhys.Rev.Lett.{\\bf 60},1057 (1988); Phys.Rev.B.{\\bf 39},2344 (1989).\n\n\\bibitem{binder} K.Binder, H.M\\\"uller-Krumbhaar, Phys.Rev.B {\\bf 7}, 3297\n(1973).\n\n\\bibitem{kok}Kok-Kwei Pan,Phys.Rev.B {\\bf 59}, 1168 (1999). \n\n\\bibitem{domb} G.S.Rushbrooke {\\it et al.} in {\\it Phase\nTransitions and Critical Phenomena},Vol.3,\nC.Domb,M.S.Green eds., (Academic, 1974).\n"
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cond-mat0002034
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Hole-burning experiments within solvable glassy models
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[
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"author": "Leticia F. Cugliandolo$^*$ and Jos\\'e Luis Iguain$^{**}$"
}
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We reproduce the results of non-resonant spectral hole-burning experiments with fully-connected (equivalently infinite-dimensional) glassy models that are generalizations of the mode-coupling approach to nonequilibrium situations. We show that an ac-field modifies the integrated linear response and the correlation function in a way that depends on the amplitude and frequency of the pumping field. We study the effect of the waiting and recovery-times and the number of oscillations applied. This calculation will help descriminating which results can and which cannot be attributed to dynamic heterogeneities in real systems.
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[
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"name": "hole_burning.tex",
"string": "\\documentstyle[prl,epsfig,aps]{revtex}\n\\begin{document}\n\\preprint{LPTENS-97/18}\n%\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname\n\n\\title{Hole-burning experiments within solvable glassy models}\n\n\\author{Leticia F. Cugliandolo$^*$ and \nJos\\'e Luis Iguain$^{**}$ }\n\\address{ $^*$\n\\it Laboratoire de Physique Th\\'eorique de l'\\'Ecole Normale \nSup\\'erieure,\n\\cite{add3} \n24 rue Lhomond, 75231 Paris Cedex 05, France and \\\\\n Laboratoire de Physique Th\\'eorique et Hautes Energies, Jussieu, \n5\\`eme \\'etage, Tour 24, 4 Place Jussieu, 75005 Paris France\n\\\\\n$^{**}$\n\\it Departamento de F\\'{\\i}sica, Universidad Nacional de Mar del Plata, \nDe\\'an Funes 3350, \n7600, Mar del Plata, Argentina}\n\n\\date\\today\n\\maketitle\n\\begin{abstract}\nWe reproduce the results of non-resonant spectral hole-burning experiments \nwith fully-connected (equivalently infinite-dimensional) glassy models \nthat are generalizations of the mode-coupling approach \nto nonequilibrium situations. \nWe show that an ac-field modifies the integrated linear response and the \ncorrelation function in a way that depends on the amplitude and frequency of \nthe pumping field. We study the effect of the waiting and \nrecovery-times and the number of oscillations applied. This calculation \nwill help descriminating which results can and which cannot be attributed to \ndynamic heterogeneities in real systems.\n\\end{abstract}\n\\vspace{.2cm}\n\\hspace{1.8cm} PACS Numbers: 64.70.Pf, 75.10Nr\n\\twocolumn \n\\vskip.5pc]\n\\narrowtext\n \n\nOne of the most interesting questions in glassy physics is whether\n{\\it localized spatial heterogenities} are generated \nin supercooled liquids and glasses. \\cite{Sillescu}\n\nIn most supercooled liquids, the linear response to small external \nperturbations is nonexponential in the time-difference $\\tau$. \nWithin the ``heteregeneous scenario'',\nthe stretching is due to the existence of dynamically\ndistinguishable entities in the sample,\neach of them relaxing exponentially with its\nown characteristic time. \nA different interpretation is that the macroscopic response is\nintrinsically nonexponential. \nIn the glass phase, the relaxation is nonstationary and the dependence\nin $\\tau$ is also much slower than exponential. \n\nThe heterogeneous regions, if they exist, \nare expected to be nanoscopic.\nThe development of experimental techniques capable of giving evidence \nfor the existence of such distinguishable spatial regions \nhas been a challenge for experimentalists. \n\nWith non-resonant spectral hole-burning (NSHB) \ntechniques one expects to probe,\nselectively, the microscopic responses.\\cite{Bohmer1} \nThe method is based on a wait, pump, recovery and \nprobe scheme depicted in Fig.~\\ref{schema}.\nThe amplitude of the ac perturbation is \nsufficiently large to pump energy in the sample, \nmodifying the response as a \nlinear function of the absorbed energy.\nThe step-like perturbation $\\delta$ is very weak and serves \nas a probe to measure the integrated linear response of the full system.\nThe large ac and small dc fields can be \nmagnetic, electric, or any \nother perturbation relevant for the sample studied. \nThe idea behind the method is that the comparison of the \nmodified (perturbed by the oscillation) and unmodified\n(unperturbed) integrated responses yield information \nabout the microscopic structure of the sample.\nOn the one hand, a spatially homogeneous sample will absorb energy uniformly \nand its modified integrated response\nis expected to be a simple translation towards shorter time-differences $\\tau$ \nof the unmodified one. On the other hand, in a heterogeneous sample,\nthe degrees of freedom that respond near the pump frequency\n$\\Omega$ are expected to absorb an important amount of energy and a maximum \ndifference in the relaxation (equivalently, a spectral\nhole) is expected to generate around $t\\sim 1/\\Omega$. \n\n\\begin{figure}\n\\centerline{\\hbox{\n \\epsfig{figure=schema.ps,width=7cm}}\n }\n\\caption{Wait, pump, recovery and probe scheme. \n}\n\\label{schema}\n\\end{figure}\n\nThe NSHB technique has been first applied \nto the study of supercooled liquids.\nThe polarization response of dielectric samples, glycerol and \npropylene carbonate, \nwas measured after being modified by an ac electric field.\\cite{Bohmer1}\nMore recently, ion-conducting glasses like CKN \\cite{Bohmer2}, \nrelaxor ferroelectrics (90PMN-10PT ceramics) \\cite{Bohmer3}\nand spin-glasses (5\\% Au:Fe) \\cite{Chamberlin}\nwere studied with similar methods. \nThe results have been interpreted \nas evidence for the existence of \nspatial heterogeneities.\nWe show here that their main features\ncan be reproduced by a system with {\\it no spatial structure}. \nWe use one model, out of a family, that captures\nmany of the experimentally observed features of super-cooled \nliquids and glasses as, for instance, \na two-step equilibrium relaxation close and above\n$T_c$ \\cite{Gotze}, aging effects below $T_c$ \\cite{Cuku}, etc. \nThe model is the $p$ spherical spin-glass \\cite{Crso}, that is intimately related to the \n$F_{p-1}$ mode-coupling model \\cite{Kith}. It can be interpreted as \na system of $N$ fully-connected continuous spins\nor as a model of a particles in an infinite\ndimensional random environment. \\cite{review} \nIn both cases, no reference to a geometry in real space nor \nany identification \nof spatially distinguishable regions can be made. \n\nIn the presence of a uniform field, the model \nis\n\\begin{equation}\nH_J[{\\bf s}] = \\sum_{i_1 \\leq \\dots \\leq i_p} J_{i_1\\dots i_p} \\; s_{i_1} \\dots s_{i_p}\n+\nh \\; \\sum_{i=1}^N s_i \n\\; .\n\\label{ham_pspin}\n\\end{equation}\nThe interactions \n$J_{i_1\\dots i_p}$ are quenched independent random variables taken from\na Gaussian distribution with zero mean and \nvariance $[J^2_{i_1\\dots i_p}]_J={\\tilde J}^2 p!/(2 N^{p-1})$. \n$p$ is a parameter and we take $p=3$.\nHereafter $[\\;\\;]_J$ represents an average over $P[J]$ and $\\tilde J=1$.\nThe continuous variables $s_i$ are constrained spherically $\\sum_{i=1}^N s_i^2=N$.\nA stochastic evolution is given to ${\\bbox s}$, \n$\n\\dot s_i(t) = -\\delta_{s_i(t)} H_J[{\\bbox s}] + \\xi_i(t) \n$\nwith $\\xi_i$ a white noise with $\\langle \\xi_i\\rangle =0$ and \n$\n\\langle \\xi_i(t) \\xi_i(t') \\rangle = 2 T \\delta(t-t') \n$. When $N\\to \\infty$, standard techniques\nlead to a set of coupled integro-differential equations \nfor the autocorrelation $\nNC(t,t') \\equiv \\sum_{i=1}^N [\\langle s_i(t) s_j(t') \\rangle]_J\n$ and the linear response \n$\nR(t,t') \\equiv \\left. \\sum_{i=1}^N \n\\delta [\\langle s_i(t) \\rangle]_J /\\delta\n\\delta_i(t') \\right|_{\\delta=0}\n$, with\n$\\delta_i(t')$ an infinitesimal\nperturbation modifying the energy at time \n$t'$ according to $H \\to H - \\sum_i \\delta_i s_i$. \nThe dynamic equations read \\cite{Cuku3}\n\\begin{eqnarray} \n& & \\partial_t C(t,t')\n=\n -z(t) \\, C(t,t') +\n\\frac{p}{2} \n\\int_0^{t'} dt'' C^{p-1}(t,t'') R(t',t'') \n\\nonumber\\\\\n& & \n\\;\\;\\;\\;\\;\\;\\;\n+\n\\frac{p (p-1)}{2}\\int_0^t dt'' C^{p-2}(t,t'') R(t,t'') C(t'',t')\n\\nonumber\\\\\n& &\n\\;\\;\\;\\;\\;\\;\\;\n+ 2T R(t',t) + h(t) \\int_0^{t'} dt'' h(t'') R(t',t'')\n\\; ,\n\\label{eqC}\n\\\\\n& & \n\\partial_t R(t,t') \n=\n -z(t) \\, R(t,t')\n\\nonumber \\\\\n& & \n\\;\\;\\;\\;\\;\\;\\;+ \\frac{p (p-1)}{2}\n \\int_{t'}^t dt'' C^{p-2}(t,t'') R(t,t'') R(t'',t')\n\\; ,\n\\label{eqR}\n%\\\\\n%& & \n%d_t m(t)\n%=\n%-z(t) m(t) + h(t) \n%\\nonumber\\\\\n%& & \n%\\;\\;\\;\\;\\;\\;\\;+ \\frac{p (p-1)}{2}\n% \\int_{0}^t dt'' C^{p-2}(t,t'') R(t,t'') m(t'')\n%\\; ,\n%\\nonumber\\\\\n%& & \n%z(t)=\n%T + \n%\\frac{p^2}{2}\n%\\int_0^t dt'' C^{p-1}(t,t'') R(t,t'') \n%+\n%h(t) m(t)\n%\\; .\n%\\nonumber\n\\end{eqnarray}\nThe Lagrange multiplier $z(t)$ enforces the spherical constraint\nand an integral equation for it follows from Eq.~(\\ref{eqC}) and \nthe condition $C(t,t)=1$.\nIn deriving these equations, a random initial condition \nat $t_0=0$ has been used.\nIt corresponds to an infinitely fast quench from equilibrium \nat $T=\\infty$\nto the working temperature $T$. The evolution continues in \nisothermal conditions. \n\nIn the absence of energy pumping, these models have \na dynamic phase transition at a ($p$-dependent) \ncritical temperature $T_c$, $T_c \\sim 0.61$ for $p=3$. \n When an external ac-field is applied, it drives \nthe system out-of-equilibrium and stationarity and \nFDT do not necessarily hold at {\\it any} temperature. \nThe question as to whether the clearcut dynamic transition survives \nunder an oscillatory field is open and we do not address it here. \nWe simply\nstudy the dynamics close to the critical temperature in \nthe absence of the field by\nconstructing a numerical solution to Eqs.~(\\ref{eqC}) and (\\ref{eqR}) \nwith a constant grid algorithm of spacing $\\epsilon$. \nWe present data for small spacings, typically $\\epsilon=0.02$, \nto minimize the numerical errors.\nDue to the fact that Eqs.~(\\ref{eqC}) and (\\ref{eqR}) include integrals\nranging from $t_0=0$ to present time $t$,\nthe algorithm is limited to a maximum number of iterations \nof the order of $8000$ that imposes a lower limit \n$\\Omega \\sim 2\\pi/(8000 \\epsilon) \\sim 0.1$\nto the frequencies we use.\n\n\nA word of caution concerning the scheme in \nFig.~\\ref{schema} and the times involved \nis in order. For the purpose of collecting the data for each reference \nunmodified integrated response, \nthe sample is prepared at the working temperature $T$ at $t_0=0$ and let freely evolve during a total waiting time $t_w+t_1+t_r$. Depending on $T$, \nthis interval may or may not be enough to equilibrate the sample.\n($t_1$ is chosen as $t_1=2\\pi n_c/\\Omega$ with \n$\\Omega$ the angular velocity of \nthe field that will be used to record the modified curve.) \nA constant infinitesimal probe $\\delta$ is applied after $t_w+t_1+t_r$\nto measure\n\\begin{eqnarray*}\n\\Phi(\\tau) \\equiv \\int_0^\\tau d\\tau' \\, R(t_w+t_1+t_r+\\tau, t_w+t_1+t_r+\\tau')\n\\; .\n\\end{eqnarray*}\nAs an abuse of notation we explicitate only the $\\tau$ dependence\nand eliminate the possible $t_w+t_1+t_r$ dependence.\nThe modified integrated response $\\Phi^*$ is measured after\nwaiting $t_w$, applying $n_c$ oscillations of duration \n$t_1=2\\pi n_c/\\Omega$, further waiting $t_r$, and only then \napplying the probe $\\delta$. \nThe effect of the ac perturbation is then quantified by \nstudying the difference:\n\\begin{equation}\n\\Delta \\Phi \\equiv \\Phi^* - \\Phi\n\\; .\n\\end{equation} \n\nWe have examined $\\Delta \\Phi$ at $T=0.8 > T_c$ and $T=0.59< T_c$.\nWe pump one oscillation with $h_F=0.1$\nand later check that this field is small enough to provoke a \nspectral modification that is linear in the absorbed energy (see\nFig.~\\ref{checklinearity} below). \nFor simplicity, we start by choosing $t_w=t_r=0$. \nIn Fig.~\\ref{Delta_Phi_1} we show $\\Delta \\Phi$ against $\\log \\tau$ \nfor different $\\Omega$ at $T=0.8$.\nAll the curves are bell-shaped and vanish both at short and long times. \nIn panel a, the $\\Omega$s are larger than a threshold value\n$\\Omega_c \\sim 1$. The height of the\npeak $\\Delta\\Phi_m \\equiv \\max ( \\Delta\\Phi)$ \ndecreases with increasing frequency reaching the limit $\\Delta\\Phi_m= 0$\nfor $\\Omega\\to\\infty$. In addition, the location of the peak \n$t_m$ moves towards longer times when $\\Omega$ decreases. \nIn panel b, $\\Omega< \\Omega_c$ and the behaviour of the \nheight of the peak is the opposite, it decreases when $\\Omega$ decreases and, \nwithin numerically errors, \nits position is either independent of $\\Omega$ or it very \nsmoothly moves towards shorter times for increasing $\\Omega$. \nThe nonmonotonic behaviour of $\\Phi_m$ with $\\Omega$\nis a consequence of the interplay between $t_\\alpha$,\nthe $\\alpha$ relaxation time, and $2\\pi/\\Omega$ the period of \nthe oscillation. The term $\\int_0^{\\min(t',t_1)} dt'' h(t'') R(t',t'')$\nin Eq.~(\\ref{eqC}) controls the effect of the field and, clearly, \nvanishes in the limits $\\Omega\\to\\infty$ and $\\Omega\\to 0$. \nThe inversion then occurs\nat a frequency $\\Omega_c$ that is of the order of $2\\pi/t_\\alpha$.\nThese results qualitatively coincide with the measurements \nof the electric relaxation\nin CKN at $T<T_g$ in Fig~1 a and b of Ref.~\\cite{Bohmer2}. \nIn Fig.~\\ref{Delta_Phi_2} we show $\\Delta \\Phi$ against \n$\\log \\tau$ for different $\\Omega$ \nat $T=0.59$. For all $\\Omega$ we reproduced the \nsituation of panel a in Fig.~\\ref{Delta_Phi_1}, as if $\\Omega > \\Omega_c$. \nWe have not found a threshold $\\Omega_c$, that has gone \nbelow the minimum $\\Omega$ reachable with the algorithm.\n\n\\vspace{-.25cm}\n\n\\begin{figure}\n\\centerline{\\hbox{\n \\epsfig{figure=fig1.ps,width=8cm}}\n }\n\\caption{\nTime-difference dependence of the distortion \n$\\Delta\\Phi$ due to a single oscillation in a \nlog-linear scale. $T=0.8> T_c$, $h_F=0.1$, $t_w=t_r=0$. \nAt high pumping frequencies $\\Omega > \\Omega_c\\sim 1$, shown in panel a, \nboth the height of the peak $\\Delta\\Phi_m$ and its position $t_m$\ndecrease with increasing frequency. \nThe dotted (blue), solid (red) and dashed (black) \ncurves correspond to $\\Omega=1,2,5$ respectively. \nIn panel b, $\\Omega < \\Omega_c\\sim 1$ and $\\Delta\\Phi_m$\nincreases with increasing $\\Omega$ while $t_m$ is almost unchanged. \nThe dotted (blue) and solid (red) curves correspond to $\\Omega=0.1$ and $0.2$, \nrespectively. }\n\\label{Delta_Phi_1}\n\\centerline{\\hbox{\n \\epsfig{figure=fig2.ps,width=8cm}}\n }\n\\caption{\nThe same plot as in Fig.~\\ref{Delta_Phi_1} for $T=0.59 < T_c$.\nFor all pumping frequencies explored, the peak moves towars shorter \ntimes and its height decreases for increasing frequencies. The dashed (black), \nsolid (red) and dotted (blue) curves correspond to $\\Omega=0.1,0.5$ and $1$,\nrespectively.}\n\\label{Delta_Phi_2}\n\\end{figure}\n\nThe maximum modification of the relaxation $\\Delta\\Phi_m$ \nincreases quadratically with the square of the amplitude of the pumping field \n$h_F$, and hence linearly in the absorbed energy, \nas long as $h_F \\leq 1$. In Fig.~\\ref{checklinearity} we display the \nrelation $\\Delta\\Phi_m\\propto h_F^2$ in a log-log scale for the two \ntemperatures explored. The amplitude $h_F=0.1$ used in Figs.~\\ref{Delta_Phi_1} \nand \\ref{Delta_Phi_2} is in the linear regime. \n\n\\begin{figure}\n\\centerline{\\hbox{\n \\epsfig{figure=inset.ps,width=8cm}}\n }\n\\caption{\nCheck of $\\Delta\\Phi_m\\propto h_F^2$ in a log-log scale.\nSquares (pink) correspond to $T=0.8$ and circles (blue) \nto $T=0.59$. In both cases\none cycle of an ac-field with $\\Omega=1$ was\napplied. The line has a slope equal to $2$ and is a guide-to-the-eye. \nThe linear relation breaks down beyond $h_F \\sim 1$.\n}\n\\label{checklinearity}\n\\centerline{\\hbox{\n \\epsfig{figure=refilling.ps,width=8cm}}\n }\n\\caption{\n The normalized maximum distortion for several \nrecovery times $t_r$. In panel a $T=0.59$ and \ncrosses, diamonds and pluses correspond\nto $\\Omega=1,2$ and $3$, respectively.\nIn panel b $T=0.8$ and\ncrosses, pluses and diamonds correspond\nto $\\Omega=1,5,10$, respectively.}\n\\label{recovery}\n\\end{figure}\n\nThe effect of the pump diminishes with increasing recovery time $t_r$.\nA convenient way of displaying this result is to plot the normalized\nmaximum deviation $\\Delta\\Phi_m(t_r)/\\Delta\\Phi_m(0)$ {\\it vs} $\\Omega t_r$.\nUsing several frequencies and recovery times, \nwe verified that this scaling holds for $T=0.59$ but does not hold \nfor $T=0.8$, as shown in Fig.~\\ref{recovery}. This simple saling \nholds very nicely in the relaxor ferroelectric \\cite{Bohmer3}\nand in the spin-glass \\cite{Chamberlin} but it is very different \nfrom the $\\Omega$-independence of the propylene carbonate \\cite{Bohmer1}. \n\nUp to now, the effect of a single cycle of different frequencies \nhas been studied. Another procedure can be envisaged.\nSince $t_1=2\\pi n_c/\\Omega$, we can change $t_1$ \nby applying different numbers of cycles $n_c$ while keeping $\\Omega$ fixed. \nIn Fig.~\\ref{ncyc} we show the distortion due to \n$n_c=10,2,1$ cycles with $\\Omega=10$ at $T=0.8$.\nThe qualitative dependence on $n_c$ is indeed the same as \nthe dependence on $1/\\Omega$:\nthe peaks are displaced towards longer times with increasing $n_c$\n(longer $t_1$).\nThis behaviour is similar to the results\nobtained for propylene carbonate in Fig.~11 of \nRef.~\\cite{Bohmer1}b. though we do not reach the expected saturation within \nour accessible time window.\n\n\\begin{figure}\n\\centerline{\\hbox{\n \\epsfig{figure=figncyc.ps,width=8cm}}\n }\n\\caption{\nEffect of several cycles at \n$T=0.8$ for $\\Omega=10$. The dashed (blue), solid (red) and \ndotted (pink) curves correspond to \n$n_c=10,2,1$ respectively. \n}\n\\label{ncyc}\n\\centerline{\\hbox{\n \\epsfig{figure=tw-dep.ps,width=8cm}}\n }\n\\caption{\nDistortion for $t_w=8$ with \ndots (pink) compared with the one for $t_w=0$ with full line (red)\nat $T=0.59$.\n}\n\\label{tw-dep}\n\\end{figure}\nBelow $T_c$ the nonperturbed model never equilibrates and the \nrelaxation depends on $t_w$. Indeed, $t_\\alpha$ is an approximately \nlinear function of $t_w$\\cite{Cuku,review} and the distortion \nmight depend on $t_w$. We compare $\\Delta\\Phi$ vs $\\log\\tau$ \nfor two $t_w$'s in Fig.~\\ref{tw-dep}. \n \nFinally, we checked that the effect of one \nor many pump oscillations \non the difference $\\Delta C \n\\equiv C^*(t_w+t_1+t_r+\\tau,t_w+t_1+t_r)-C(t_w+t_1+t_r+\\tau,t_w+t_1+t_r)$ \nis very similar to the one observed in $\\Delta\\Phi$. \nThis observation is interesting since it is easier to compute numerically\ncorrelations than responses. \nFigure~\\ref{corr} shows the modification observed at \n$T=0.8$ and $\\Omega > \\Omega_c$ (to be compared to Fig.~\\ref{Delta_Phi_1}).\n\nWe conclude by stressing that we\ndo not claim that spatial heterogeneities do not exist \nin real glassy systems. We just wish to stress that the \nambiguities in the interpretation of\nexperimental results have to be eliminated in order to have \nunequivocal evidence for them. The detailed comparison of the \nexperimental measurements to the behaviour of glassy models \n{\\it with and without space} will certainly help us refine the experimental \ntechniques. Numerical simulations can play an important role in this \nrespect. \n\n\\begin{figure}\n\\centerline{\\hbox{\n \\epsfig{figure=figC.ps,width=8cm}}\n }\n\\caption{\nChange in the autocorrelation. $T=0.8$ and $\\Omega=1$ (dashed blue), \n$\\Omega=2$ (solid red) and $\\Omega=5$ (dotted black).\n}\n\\label{corr}\n\\end{figure}\n\nLFC and JLI thank the Dept. of Phys. (UNMDP) and LPTHE (Jussieu) \nfor hospitality, and ECOS-Sud, CONICET and UNMDP for financial support. \nWe thank R. B\\\"ohmer, H. Cummins, G. Diezemann, M. Ediger, \nJ. Kurchan and G. Mc Kenna for very useful discussions and \nT. Grigera, N. Israeloff and E. Vidal-Russel for introducing \nus to the hole-burning experiments. \n\n\\begin{thebibliography}{99}\n\n\\bibitem{Sillescu}\nH. Sillescu, J. Non-Cryst. Solids, {\\bf 243}, 81 (1999).\nM. T. Cicerone and M. D. Ediger, J. Chem. Phys. {\\bf 103}, 5684 (1995). \nR. B\\\"ohmer {\\it et al}, J. Non-Cryst. Solids {\\bf 235-237} I-9\n(1998). \n\n\\bibitem{Bohmer1}\na. B. Schiener {\\it et al}\nScience {\\bf 274}, 752 (1996). \nb. B. Schiener{\\it et al}\nJ. Chem. Phys. {\\bf 107}, 7746 (1997). \n\n\\bibitem{Bohmer2} R. Richert and R B\\\"ohmer, Phys. Rev. Lett. {\\bf 83}, 4337 (1999). \n\n\\bibitem{Bohmer3} O. Kircher, B. Schiener and R. B\\\"ohmer, Phys. Rev. Lett. {\\bf 20}, 4520 (1998).\n\n\\bibitem{Chamberlin} R. V. Chamberlin, Phys. Rev. Lett. {\\bf 24}, 5134 (1999).\n\n\\bibitem{Gotze} W. G\\\"otze, J. Phys. {\\bf C11}, A1 (1999).\n\n\\bibitem{Cuku} L. F. Cugliandolo and J. Kurchan, \nPhys. Rev. Lett. {\\bf 71}, 173 (1993).\n \n%\\bibitem{Chakra} B. K. Chakravarti and M. Acharyya, cond-mat/9811086. \n\n\\bibitem{Crso} \nA. Crisanti and H.-J. Sommers, Z. Phys. {\\bf B87}, 341 (1992).\n\n\\bibitem{Kith} T. R. Kirkpatrick and D. Thirumalai, \nPhys. Rev. {\\bf B36}, 5388 (1987).\n\n\\bibitem{review}\nFor a review see\nJ-P Bouchaud, L. F. Cugliandolo, J. Kurchan and M. M\\'ezard, \nin Sping glasses and random fields, A. P. Young ed. (World Scientific, 1998).\n\n\\bibitem{Cuku3}\nL. F. Cugliandolo and J. Kurchan, Phys. Rev. {\\bf B60}, 922 (1999).\n\n\n\\end{thebibliography}\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002034.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Sillescu}\nH. Sillescu, J. Non-Cryst. Solids, {\\bf 243}, 81 (1999).\nM. T. Cicerone and M. D. Ediger, J. Chem. Phys. {\\bf 103}, 5684 (1995). \nR. B\\\"ohmer {\\it et al}, J. Non-Cryst. Solids {\\bf 235-237} I-9\n(1998). \n\n\\bibitem{Bohmer1}\na. B. Schiener {\\it et al}\nScience {\\bf 274}, 752 (1996). \nb. B. Schiener{\\it et al}\nJ. Chem. Phys. {\\bf 107}, 7746 (1997). \n\n\\bibitem{Bohmer2} R. Richert and R B\\\"ohmer, Phys. Rev. Lett. {\\bf 83}, 4337 (1999). \n\n\\bibitem{Bohmer3} O. Kircher, B. Schiener and R. B\\\"ohmer, Phys. Rev. Lett. {\\bf 20}, 4520 (1998).\n\n\\bibitem{Chamberlin} R. V. Chamberlin, Phys. Rev. Lett. {\\bf 24}, 5134 (1999).\n\n\\bibitem{Gotze} W. G\\\"otze, J. Phys. {\\bf C11}, A1 (1999).\n\n\\bibitem{Cuku} L. F. Cugliandolo and J. Kurchan, \nPhys. Rev. Lett. {\\bf 71}, 173 (1993).\n \n%\\bibitem{Chakra} B. K. Chakravarti and M. Acharyya, cond-mat/9811086. \n\n\\bibitem{Crso} \nA. Crisanti and H.-J. Sommers, Z. Phys. {\\bf B87}, 341 (1992).\n\n\\bibitem{Kith} T. R. Kirkpatrick and D. Thirumalai, \nPhys. Rev. {\\bf B36}, 5388 (1987).\n\n\\bibitem{review}\nFor a review see\nJ-P Bouchaud, L. F. Cugliandolo, J. Kurchan and M. M\\'ezard, \nin Sping glasses and random fields, A. P. Young ed. (World Scientific, 1998).\n\n\\bibitem{Cuku3}\nL. F. Cugliandolo and J. Kurchan, Phys. Rev. {\\bf B60}, 922 (1999).\n\n\n\\end{thebibliography}"
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cond-mat0002035
|
Effect of Magnetic Impurities on Suppression of the Transition Temperature in Disordered Superconductors
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[
{
"author": "Robert A Smith"
}
] |
We calculate the first-order perturbative correction to the transition temperature $T_c$ in a superconductor with both non-magnetic and magnetic impurities. We do this by first evaluating the correction to the effective potential, $\Om(\De)$, and then obtain the first-order correction to order parameter, $\De$, by finding the minimum of $\Om(\De)$. Setting $\De=0$ finally enables $T_c$ to be evaluated. $T_c$ is now a function of both the resistance per square, $R_{\square}$, a measure of the non-magnetic disorder, and the spin-flip scattering rate, $1/\tau_s$, a measure of magnetic disorder. We find that the effective pair-breaking rate per magnetic impurity is virtually independent of the resistance per square of the film, in agreement with an experiment of Chervenak and Valles. This conclusion is supported both by the perturbative calculation, and by a non-perturbative re-summation technique.
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[
{
"name": "magtcprb2.tex",
"string": "%\\documentstyle{article}\n%\\documentstyle[preprint,aps]{revtex}\n%\\def\\mathcal{\\cal}\n\\documentstyle[prb,aps,multicol,epsf]{revtex}\n%\\documentstyle[12pt,prb,aps,epsf]{revtex}\n%\\input bookrefs\n\\input psfig\n\\parindent=20pt\n\\raggedbottom\n\\def\\ct{\\centerline}\n\\def\\ub{\\underbar}\n\\def\\ni{\\noindent}\n\\def\\di{\\displaystyle}\n\\def\\psiuppl{\\psi^{+}_{\\uparrow}}\n\\def\\psiup{\\psi_{\\uparrow}}\n\\def\\psidnpl{\\psi^{+}_{\\downarrow}}\n\\def\\psidn{\\psi_{\\downarrow}}\n\\def\\de{\\delta}\n\\def\\De{\\Delta}\n\\def\\Db{\\overline{\\Delta}}\n\\def\\zb{\\overline{z}}\n\\def\\Dp{\\overline{\\Delta}'}\n\\def\\zp{\\overline{z}'}\n\\def\\om{\\omega}\n\\def\\omb{\\overline{\\omega}}\n\\def\\Om{\\Omega}\n\\def\\ep{\\varepsilon}\n\\def\\la{\\lambda}\n\\def\\apb{\\overline{\\alpha}_{+}}\n\\def\\amb{\\overline{\\alpha}_{-}}\n\\def\\apm{\\alpha_{\\pm}}\n\\def\\bpm{\\beta_{\\pm}}\n\\def\\apmb{\\overline{\\alpha}_{\\pm}}\n\\def\\ap{\\alpha_{+}}\n\\def\\am{\\alpha_{-}}\n\\def\\bp{\\beta_{+}}\n\\def\\bm{\\beta_{-}}\n\\def\\Ga{\\Gamma}\n\\def\\ta{\\tau_1\\sigma_3}\n\\def\\tb{\\tau_2\\sigma_3}\n\\def\\tc{\\tau_3\\sigma_0}\n\\def\\pa{\\partial}\n\\def\\Wb{\\overline{W}}\n\\def\\Wp{\\overline{W}'}\n\\def\\sqr#1#2{{\\vcenter{\\vbox{\\hrule height .#2pt\n \\hbox{\\vrule width .#2pt height#1pt \\kern#1pt\n \\vrule width.#2pt}\n \\hrule height.#2pt}}}}\n\\def\\square{\\mathchoice\\sqr34\\sqr34\\sqr{2.1}3\\sqr{1.5}3}\n \n \n\n\\begin{document}\n\\bibliographystyle{simpl1}\n\n\\title{Effect of Magnetic Impurities on Suppression of the\nTransition Temperature in Disordered Superconductors}\n\n\\author{Robert A Smith}\n\n\\address{School of Physics and Astronomy, University of Birmingham,\nEdgbaston, Birmingham B15 2TT, England\\hfil\\break\nand Laboratory of Atomic and Solid State Physics,\nCornell University, Ithaca, New York 14853}\n\n\\author{Vinay Ambegaokar}\n\n\\address{Laboratory of Atomic and Solid State Physics,\nCornell University, Ithaca, New York 14853}\n\n\\maketitle\n%\\centerline{April 9, 1999}\n\\bigskip\n\n\\begin{abstract}\nWe calculate the first-order perturbative correction to the transition\ntemperature $T_c$ in a superconductor with both non-magnetic and\nmagnetic impurities. We do this by first evaluating \nthe correction to the effective\npotential, $\\Om(\\De)$, and then obtain the first-order correction to\norder parameter, $\\De$, by finding the minimum of $\\Om(\\De)$.\nSetting $\\De=0$\nfinally enables $T_c$ to be evaluated. $T_c$ is now a function of both\nthe resistance per square, $R_{\\square}$, a measure of the non-magnetic\ndisorder, and the spin-flip scattering rate, $1/\\tau_s$, a measure of\nmagnetic disorder. We find that the effective pair-breaking rate per\nmagnetic impurity is virtually independent of the resistance per square\nof the film, in agreement with an experiment of Chervenak and Valles.\nThis conclusion is supported both by the perturbative calculation, and by\na non-perturbative re-summation technique.\n\\end{abstract}\n\n\\section{Introduction}\n\nMany experiments\\cite{Fink94} performed on homogeneous disordered thin film\nsuperconductors have shown that superconductivity is suppressed\nby increasing disorder, as measured by the normal state resistance\nper square, $R_{\\square}$. The majority of the data is of the\ntransition temperature as a function of the resistance per square, \n$T_c(R_{\\square})$\\cite{RLCM,GB,HLG,LK}, although there is some data on the\nupper critical field, $H_{c2}(T,R_{\\square})$\\cite{GB,HP,OKOK}, \nand the order parameter $\\De(R_{\\square})$\\cite{VDG,Vall94}. The main data\nto be explained thus consists of $T_c(R_{\\square})$ curves for\ndifferent materials. To see how disorder might affect $T_c$, consider\nthe mean field equation,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nT_{c0}=1.13\\om_D\\exp{\\left[-{1\\over N(0)(\\la-\\mu^*)}\\right]},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $\\la$ is the attractive BCS interaction mediated by\nphonons of energy less than the Debye frequency, $\\om_D$,\n$\\mu^*$ is the Coulomb pseudopotential, the effective strength of\nthe Coulomb repulsion, and $N(0)$ is the single particle density\nof states at the Fermi surface. Obviously disorder could affect $T_c$\nby changing $\\la$, $\\mu^*$, and $N(0)$. In particular the diffusive\nmotion of electrons caused by the disorder is known to lead to an\nincreased effective strength of the Coulomb interaction, as the screening\nis less efficient than with ballistic electrons, and this leads to\nan increase in $\\mu^*$, and a decrease in $N(0)$. Calculating the \nfirst-order perturbative correction caused by the disorder shows that\nwe must consider all these processes together\\cite{Fink87,SRW}. \nThis is because the\ndisorder-screened Coulomb interaction has a low-momentum singularity\nwhich leads to the separate effects being large; however, when they are\nadded together this singularity is cancelled, and the actual effect is \nmuch smaller than might be naively expected. The final result has the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\ln{\\left({T_c\\over T_{c0}}\\right)}=-{1\\over 3}{R_{\\square}\\over R_0}\n\\ln^3{\\left({1\\over 2\\pi T_{c0}\\tau}\\right)},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $R_0=2\\pi h/e^2\\approx 162k\\Omega$ and $\\tau$ is the elastic \nscattering time. We see that this curve is essentially ``universal'',\ndepending on only a single fitting parameter, \n$\\beta=\\ln{(1/2\\pi T_{c0}\\tau)}$. Experimentally it is found that\n$T_c(R_{\\square})$ curves from a wide variety of materials fit well\nto this equation, or extensions of it that allow for stronger disorder.\nThe simplest extension simply consists of replacing $T_{c0}$ by $T_c$\non the right-hand side of Eq. (2), which leads to the cubic equation\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nx={t\\over 3}(\\beta+x)^3,\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $x=\\ln{(T_{c0}/T_c)}$ and $t=R_{\\square}/R_0$. This equation shows\nunphysical reentrance at strong disorder, an artefact which is removed\nby either a renormalization group treatment\\cite{Fink87}, or the use of \na non-perturbative resummation technique\\cite{OF} to yield the formula\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\ln{\\left({T_c\\over T_{c0}}\\right)}={1\\over\\lambda}\n-{1\\over 2\\sqrt{t}}\\ln{\\left({1+\\sqrt{t}/\\lambda\\over\n1-\\sqrt{t}/\\lambda}\\right)}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\par\nThe fact that most data can be fit to a single curve is pleasing in\nthat it shows that the basic ingredients of our theory -- disorder,\nBCS attraction and Coulomb repulsion -- are correct. However it does\nnot allow analysis of the sensitivity of the theory or experiment to \nchanges in details of the system, such as the exact form of the \nphonon-mediated attraction. Moreover there are other theories\\cite{Belitz}\nwhich posit the importance of such details which give predictions that are\nequally in agreement with experiment. We see that the $T_c(R_{\\square})$\ncurves alone are not enough to allow consideration of the relative\nmerits of different theories.\nWhat we would like to do is to add some additional parameter\nto the experimental system to give a whole new set of data -- for\nexample a family of $T_c(R_{\\square})$ curves for a single material\nas this new parameter is altered. Chervenak and Valles\\cite{CV} have recently\nperformed an experiment of this type in which magnetic $Gd$ impurities\nare added to thin films of $Pb_{0.9}Bi_{0.1}$. This introduces the\nnew feature of spin-flip scattering to the system, which is measured\nby the spin-flip scattering rate, $1/\\tau_s$. The task of the theorist\nis to now make predictions for $T_c$ as a function of\nboth $R_{\\square}$ (the measure of non-magnetic disorder), and $1/\\tau_s$,\n(the measure of magnetic disorder), and to compare these to experiment.\n\\par\nIn this paper we calculate the first-order perturbative correction to\nthe transition temperature, $T_c$, of a superconductor with both\nnon-magnetic and magnetic impurities. The model used consists of a\nfeatureless BCS attraction, $-\\lambda$, and a Coulomb repulsion, $V_C(q)$,\nbetween electrons which scatter off non-magnetic and magnetic impurities.\nThe model is the simplest one that contains the essential physics, and\nits shortcoming of not considering the details of the attractive\ninteraction is offset by the fact that we can consider all processes to\na given order of perturbation theory. This is an important consideration\nin view of the cancellation of low-momentum singularities in the screened\nCoulomb potential discussed in the opening paragraph. In fact, an obvious\nquestion is whether this cancellation persists in the presence of magnetic \nimpurities. We find that this is indeed the case, and so the details of\nthe screened Coulomb interaction are removed, leading to a ``universal''\nform for $T_c(R_{\\square},1/\\tau_s)$. \n\\par\nThe main result of the paper is that\nthe pair-breaking rate per magnetic impurity, $\\alpha'(R_{\\square})$,\ndefined by \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\alpha'(R_{\\square})={T_c(R_{\\square},0)-T_c(R_{\\square},1/\\tau_s)\n\\over 1/\\tau_s}\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nis roughly independent of $R_{\\square}$ except near the\nsuperconductor-insulator transition, in agreement with experiment.\nThis is confirmed both by first-order perturbation theory, and also\nby a non-perturbative resummation technique which we introduce to \nremove concerns about reentrance problems at stronger disorder. This\nagreement of the two theoretical approaches with each other and the\nexperimental data gives us confidence in our results.\n\\par\nTo calculate the correction to $T_c$ we use a collective mode formalism\nderived in a previous paper\\cite{SRW} (which we refer to as {\\bf I} from now\non) on the suppression of $\\De$ by non-magnetic disorder.\nThe introduction of\nmagnetic impurities means that we have to modify the formalism\nsomewhat, and so we include most of the derivation in this paper.\nThe method used in this paper to evaluate the correction to $T_c$\nproceeds in three stages. First we find the first-order correction\nto the grand canonical potential, $\\Om_1(\\De)$, of the superconductor\ndue to fluctuations of its collective modes. Then by minimizing the\ntotal grand canonical potential, $\\Om_0(\\De)+\\Om_1(\\De)$ with respect\nto the order parameter, $\\De$, we obtain the first-order correction to the\norder parameter self-consistency equation. Finally by setting $\\De=0$\nwe obtain the first-order correction to transition temperature $T_c$. \nThe method we use has\nthe advantage that it is impossible to ``miss diagrams'' since there\nis only one diagram in the $\\Om_1(\\De)$ calculation, and we also obtain\nthe equation for $\\De$ at no extra cost. The equations for $T_c$ and\n$\\De$ must reduce to those of {\\bf I} when we set spin-flip scattering\nto zero, providing a useful consistency check.\nA key result of the calculation in {\\bf I} was that the\nsingularity in the screened Coulomb potential persists below $T_c$,\nand that this singularity is cancelled in the formula for the\nsuppression of $\\De$ in a similar manner to the cancellation in the\nformula for $T_c$. Therefore an important question\nis whether this singularity and cancellation remains when magnetic\nimpurities are added. We show that this is indeed the case, and moreover\nthat the cancellation is due to gauge invariance, and will occur in\nfirst-order perturbation theory in the presence of any kind of impurity\nscattering. In other words, it is not possible to obtain stronger suppression\nof the transition temperature by introducing some exotic scattering\nmechanism. It is also reassuring to know that an otherwise mysterious\ncancellation between diagrams has its physical origin in gauge invariance,\nand we hope that similar arguments may be applied to show that the\nresult holds to all orders in perturbation theory.\n\\par\nThe outline of the rest of the paper is as follows. In section\n\\uppercase\\expandafter{\\romannumeral2} we derive the matrix formalism\nfor superconductors with magnetic impurities, and the collective\nmode approach we will use. We derive the RPA screened bosonic\npropagators, and show that low-momentum singularities persist in\nthe screened Coulomb propagator below $T_c$. In section\n\\uppercase\\expandafter{\\romannumeral3} we derive the first order\nperturbative correction to the grand canonical potential\n$\\Om_1(\\De)$, and from this the correction to the order parameter $\\De$.\nIn section \\uppercase\\expandafter{\\romannumeral4} we set $\\De=0$ to\nobtain the correction to transition temperature $T_c$. In section\n\\uppercase\\expandafter{\\romannumeral5} we calculate $T_c$ numerically\nusing both the perturbative results of section \n\\uppercase\\expandafter{\\romannumeral4} and a recently developed\nnon-pertubative technique, and compare to experiment.\n\n\n\\section{Derivation of the $4\\times 4$ Matrix Formalism}\n\n\\noindent{\\bf Superconductivity with Magnetic Impurities:}\n\\medskip\n\nWe consider a system of electrons that scatter off static non-magnetic\nand magnetic impurities, and interact with each other via the long-range\nCoulomb interaction and the BCS attraction. The scattering from static\nimpurities is described by Hamiltonian\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nH_{e-i} = \\sum_{\\alpha\\beta}\\int d{\\bf x}\\psi^{+}_{\\alpha}({\\bf x})\n\\left\\{\\left[-{\\nabla^2_x\\over 2m}+\\sum_i u_0({\\bf x}-{\\bf x_i})\\right]\n\\delta_{\\alpha\\beta} + \\sum_j J({\\bf x-x_j}){\\bf S_j\\cdot\\sigma_{\\alpha\\beta}}\n\\right\\}\\psi_{\\beta}({\\bf x}),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $\\psi^{+}_{\\alpha}$, $\\psi_{\\alpha}$ are the electron creation\nand annhilation operators, $u_0({\\bf x}-{\\bf x_i})$ is the\nimpurity potential at ${\\bf x}$ due to a non-magnetic impurity at\n${\\bf x_i}$, ${\\bf S_j}$ is a magnetic impurity spin moment at ${\\bf x_j}$,\nand $J({\\bf x})$ is the electron-impurity exchange coupling. The potential\nand spin-flip scattering rates are then given by\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n{1\\over\\tau_0}&=&2\\pi N(0)n_i |u_0|^2\\nonumber\\\\\n{1\\over\\tau_s}&=&2\\pi N(0)n_j J^2 S(S+1),\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $u_0$ is the Fourier transform of $u_0({\\bf x})$, and $J$ is the\nFourier transform of $J({\\bf x})$, both assumed\nindependent of momentum, $n_i$ is the non-magnetic impurity density,\nand $n_j$ the magnetic impurity density.\n\\par\nThe Coulomb repulsion between electrons is described by Hamiltonian\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nH_C=\\sum_{\\alpha\\beta}\\int d{\\bf x} \\int d{\\bf x'}\n\\psi^{+}_{\\alpha}({\\bf x})\\psi_{\\alpha}({\\bf x})\n{e^2\\over |{\\bf x}-{\\bf x'}|}\n\\psi^{+}_{\\beta}({\\bf x'})\\psi_{\\beta}({\\bf x'}),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nleading to a bare Coulomb propagator that is just the Fourier transform\nof the above potential.\n\\par\nThe BCS attraction is described by the Hamiltonian\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nH_{BCS}=-\\lambda\\sum_{\\alpha\\beta}\\int d{\\bf x}\n\\psi^{+}_{\\alpha}({\\bf x})\\psi_{\\alpha}({\\bf x})\n\\psi^{+}_{\\beta}({\\bf x})\\psi_{\\beta}({\\bf x}),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\ncorresponding to an instantaneous contact interaction\n$-\\lambda\\delta({\\bf x}-{\\bf x'})$.\n\nHaving introduced the model Hamiltonian we need to describe the system,\nwe discuss the standard four-dimensional matrix representation\\cite{Grif}\nneeded\nto describe a superconductor with magnetic impurities. We need four\ncomponents to describe the two spin degrees of freedom, and the two\ntypes of correlation -- the usual particle-hole correlation\n$<\\psiup\\psiuppl>$, amd the anomalous pairing correlation\n$<\\psiup\\psidn>$. We introduce\nthe four-dimensional vector operator\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Psi = \\left(\\matrix{\\psiup\\cr \\psidn\\cr \\psidnpl\\cr \\psiuppl\\cr}\n \\right) \\quad;\\quad\n \\Psi^{+}= \\left(\\matrix{\\psiuppl&\\psidnpl&\\psidn&\\psiup\\cr}\\right),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwith matrix propagator\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n<\\Psi\\Psi^{+}> = \\left[\\matrix{\n<\\psiup\\psiuppl>&<\\psiup\\psidnpl>&<\\psiup\\psidn>&<\\psiup\\psiup>\\cr\n<\\psidn\\psiuppl>&<\\psidn\\psidnpl>&<\\psidn\\psidn>&<\\psidn\\psiup>\\cr\n<\\psidnpl\\psiuppl>&<\\psidnpl\\psidnpl>&<\\psidnpl\\psidn>&<\\psidnpl\\psiup>\\cr\n<\\psiuppl\\psiuppl>&<\\psiuppl\\psidnpl>&<\\psiuppl\\psidn>&<\\psiuppl\\psiup>\\cr\n}\\right].\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nIn the normal state the temperature Green function is\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nG(k,i\\om)={1\\over z-\\ep_k\\tau_3\\sigma_0},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $z=i\\om$, $\\om=(2n+1)\\pi T$ is a Fermi Matsubara frequency,\nand the $\\tau_i$ and $\\sigma_i$ are Pauli matrices operating on different\nspaces. The $\\sigma_i$ operate in the usual spin space, whilst the\n$\\tau_i$ operate in the Nambu (electron-hole) space. The diagrammatic\nrules are then the same as in the normal state except for the matrix\nstructure of the Green function, and the presence of matrix\n$\\tau_3\\sigma_0$ at each interaction or impurity vertex due to the\nelectron density operator being written in the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\rho=\\psiuppl\\psiup+\\psidnpl\\psidn=\n{1\\over 2}\\Psi^{+}\\tau_3\\sigma_0\\Psi.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\par\nThe pairing correlations in the clean superconductor can then be taken\ninto account self-consistently as shown in Fig. (1a). Making the ansatz\n$\\Sigma=\\De\\tau_1\\sigma_3$ for the self-energy, the Green function for\nthe pure superconductor becomes\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nG_0(k,z)={1\\over z-\\ep_k\\tau_3\\sigma_0-\\De\\tau_1\\sigma_3}\n={z+\\ep_k\\tau_3\\sigma_0+\\De\\tau_1\\sigma_3\\over z^2-\\ep_k^2-\\De^2},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand the diagram of Fig. (1a) gives self-energy\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\Sigma\\displaystyle&=&-\\lambda T\\sum_\\om N(0)\\int d\\ep_k\n{\\tau_3\\sigma_0(z+\\ep_k\\tau_3\\sigma_0+\\De\\tau_1\\sigma_3)\\tau_3\\sigma_0\n\\over \\ep_k^2-z^2+\\De^2}\\nonumber\\\\\n&=&\\displaystyle N(0)\\lambda\\De\\tau_1\\sigma_3 \nT\\sum_\\om {1\\over\\sqrt{\\om^2+\\De^2}},\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhich gives us the usual BCS self-consistency equation\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n1=N(0)\\lambda T\\sum_\\omega {1\\over\\sqrt{\\omega^2+\\De^2}}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe can treat the presence of non-magnetic and magnetic impurities by\nincluding an extra self-energy diagram to describe the dressing of the\nelectron line by impurities as shown in Fig. (1b). We then make the\nansatz that the pairing energy has the form $\\Sigma_p=\\De\\tau_1\\sigma_3$,\nand the impurity self-energy has the form\n$\\Sigma_{imp}=-(\\zb-z)+(\\Db-\\De)\\tau_1\\sigma_3$, so that the Green\nfunction for the dirty superconductor is\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nG_0(k,z)={\\zb+\\ep_k\\tau_3\\sigma_0+\\Db\\tau_1\\sigma_3\\over\n\\zb^2-\\ep_k^2-\\Db^2},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhich is just the Green function for the clean superconductor with\n$z$, $\\De$, replaced by $\\zb$, $\\Db$, respectively. Since the\nimpurity line has the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{imp}\n\\Ga_0={1\\over 2\\pi N(0)\\tau_0}\\tau_3\\sigma_0\\otimes\\tau_3\\sigma_0\n+{1\\over 6\\pi N(0)\\tau_s}\\left[\\tau_0\\sigma_1\\otimes\\tau_0\\sigma_1\n+\\tau_0\\sigma_2\\otimes\\tau_0\\sigma_2\n+\\tau_3\\sigma_3\\otimes\\tau_3\\sigma_3\\right],\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwe obtain the self-consistency equation for $\\zb=i\\omb$ and $\\Db$,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{selfcons}\n\\omb-\\om=\\left({1\\over 2\\tau_0}+{1\\over 2\\tau_s}\\right)\n{\\omb\\over\\sqrt{\\omb^2+\\Db^2}}\\quad;\\quad\n\\Db-\\De=\\left({1\\over 2\\tau_0}-{1\\over 2\\tau_s}\\right)\n{\\Db\\over\\sqrt{\\omb^2+\\Db^2}}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe diagrammatic definition of the pairing energy, $\\Sigma_p$, leads to\nthe same self-consistency equation for $\\De$ as in the pure case except\nthat $\\om$, $\\De$, are replaced by $\\omb$, $\\Db$. In the absence of\nmagnetic impurities -- i.e. $1/\\tau_s=0$ -- we see that\n$\\omb/\\Db=\\om/\\De$, and the equation for $\\De$ is unchanged. This is\nAnderson's theorem\\cite{PWA} \nthat superconductivity is unaffected by non-magnetic\nimpurities at mean-field level. In the presence of magnetic impurities\nwe see that $\\omb/\\Db\\ne\\om/\\De$, and if we define $u=\\omb/\\Db$,\n$\\zeta=1/\\tau_s\\De$, the problem reduces to solving the equation\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{transc}\n{\\om\\over\\De} = u\\left(1-\\zeta{1\\over\\sqrt{u^2+1}}\\right).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe self-consistency equation for $\\De$ then takes the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{mfeqn}\n1=N(0)\\lambda T\\sum_\\om {1\\over\\sqrt{u^2+1}},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand in particular if we set $\\De=0$ we get for $T_c$,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n1=N(0)\\lambda T_c\\sum_\\om {1\\over |\\om|+1/\\tau_s}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nSubtracting off the equation for $T_{c0}$, the transition temperature\nin the absence of magnetic impurities leads to the famous \nresult\\cite{AG}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\log{\\left({T_c\\over T_{c0}}\\right)} =\n\\psi\\left({1\\over 2}\\right) -\n\\psi\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}\\right).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\null\n\\medskip\n\\noindent{\\bf Collective Mode Formalism and RPA:}\n\nThe idea of the collective mode formalism is to treat the screened\ninteractions in the system as bosonic collective modes. The relevant\nbosonic operators are order parameter amplitude and phase, and electron\ndensity. These are the only modes that are coupled by a bare interaction\n-- the BCS interaction for order parameter amplitude and phase,\nthe Coulomb interaction for electron density. The main advantage of this\napproach is that we are able to treat order parameter fluctuations and\nthe Coulomb interaction on an identical footing.\nThis procedure may be formally carried out within the path integral\ntheory of superconductivity by decoupling the two four-fermion interaction\nterms with the introduction of appropriate collective variables.\nThis is discussed\nin detail in the paper of Eckern and Pelzer\\cite{EP88}. The end result is that\nthere are three effective bosonic modes, order parameter amplitude and\nphase and electronic density, and each can be written in the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\hat{O}_i={1\\over 2}\\Psi^+ M_i\\Psi,\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwith matrices $M_{\\Delta}=\\tau_1\\sigma_3$ for order parameter amplitude,\n$M_{\\phi}=\\tau_2\\sigma_3$ for order parameter phase, and\n$M_{\\rho}=\\tau_3\\sigma_0$ for electronic density. Interactions occur by\nexchange of these collective modes, and so the effective interaction\npotential is now a $3\\times 3$ matrix. The screened interaction is found\nfrom the equation\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nV_{ij}=V^0_{ij}+\\sum_{kl}V^0_{ik}\\Pi_{kl}V_{lj},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nas shown in Fig. (2). Here $\\Pi_{ij}$ is the polarization operator\nand $V^0_{ij}$ is the bare interaction matrix which is given by\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nV^0=\\left(\\matrix{-\\lambda/2&0&0\\cr 0&-\\lambda/2&0\\cr \n0&0&{4\\pi e^2/q^2}\\cr}\\right),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nthe BCS attraction being split equally between the two order parameter\nmodes. The only new diagrammatic feature is that any of the matrices\n$M_i$ can now appear at an interaction vertex, corresponding to\ninteraction with the collective variable described by that matrix.\n\nIn order to carry out the calculations in section\n\\uppercase\\expandafter{\\romannumeral3}, we need the impurity dressed\nRPA polarization bubbles, $\\Pi_{ij}$, shown in Fig. (3). To evaluate the\npolarization bubble $\\Pi_{ij}$ we must first evaluate the geometric\nseries\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Pi=S+S\\Ga_0 S+S\\Ga_0 S\\Ga_0 S+\\dots,\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere\n$\\Ga_0$ is the impurity line defined in Eqn. ({\\ref{imp}), \nand $S$ is the momentum sum of a direct product of Green functions\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nS=\\sum_k G(k,i\\om)\\otimes G(k+q,i\\om+i\\Om).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nSince we do not need the complete matrix structure of $\\Pi$, but just\nits traces with two matrices from the set $\\ta$, $\\tb$, $\\tc$, we\nactually evaluate the impurity dressed vertices $\\Pi_j$ which have one \nmatrix from the above set inserted between the two terms of the direct \nproduct in $\\Pi$. These satisfy the equations\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Pi_j=S_j+S\\Ga_0\\Pi_j,\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $S_j$ is obtained by inserting the matrix $M_j$ between the two\nterms of the direct product in $S$.\nThese equations are solved in Appendix A to obtain the results\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\Pi_{\\De}&=&-{\\pi N(0)\\over D_+}\n\\left[{UU'+uu'-1\\over UU'}-{i(u'+u)\\over UU'}\\ta\\right]\\ta\\nonumber\\\\\n\\Pi_{\\phi}&=&-{\\pi N(0)\\over D_-}\n\\left[{UU'+uu'+1\\over UU'}-{i(u'-u)\\over UU'}\\ta\\right]\\tb\\nonumber\\\\\n\\Pi_{\\phi}&=&{\\pi N(0)\\over D_-}\n\\left[{UU'-uu'-1\\over UU'}+{i(u'-u)\\over UU'}\\ta\\right]\\tc,\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $U=\\sqrt{u^2+1}$, $u'=u(\\omega+\\Omega)=u(\\omega')$,\n$U'=\\sqrt{u'^2+1}$ and\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nD_{\\pm}=\n\\left[Dq^2+\\De U+\\De U'+{1\\over\\tau_s}\n\\left({uu'\\mp 1\\over UU'}-1\\right)\\right].\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWe can finally obtain the non-zero polarization bubbles $\\Pi_{ij}$\nby inserting the second matrix from the set $\\ta$, $\\tb$, $\\tc$ into\n$\\Pi_j$, taking the trace, and recalling the factor $-1$ for a fermion\nloop. This yields\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{piall}\n\\Pi_{\\De\\De}(q,\\Om) &=&\\displaystyle\\pi N(0)T\\sum_\\om \\left[\n{UU'+uu'-1\\over UU'}\\right]{1\\over\n\\left(Dq^2+\\De U+\\De U'-\\di{1\\over\\tau_s}\\left[{UU'-uu'+1\\over UU'}\\right]\n\\right)}\\nonumber\\\\\n\\Pi_{\\phi\\phi}(q,\\Om) &=&\\displaystyle \\pi N(0)T\\sum_\\om \\left[\n{UU'+uu'+1\\over UU'}\\right]{1\\over\n\\left(Dq^2+\\De U+\\De U'-\\di{1\\over\\tau_s}\\left[{UU'-uu'-1\\over UU'}\\right]\n\\right)}\\nonumber\\\\\n\\Pi_{\\rho\\rho}(q,\\Om) &=&\\displaystyle-\\pi N(0)T\\sum_\\om \\left[\n{UU'-uu'-1\\over UU'}\\right]{1\\over\n\\left(Dq^2+\\De U+\\De U'-\\di{1\\over\\tau_s}\\left[{UU'-uu'-1\\over UU'}\\right]\n\\right)} + N(0)\\nonumber\\\\\n\\Pi_{\\phi\\rho}(q,\\Om) &=&\\displaystyle-\\pi N(0)T\\sum_\\om \\left[\n{u'-u\\over UU'}\\right]{1\\over\n\\left(Dq^2+\\De U+\\De U'-\\di{1\\over\\tau_s}\\left[{UU'-uu'-1\\over UU'}\\right]\n\\right)}=-\\Pi_{\\rho\\phi}(q,\\Om).\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWe note that if we set $1/\\tau_s=0$ in Eqn. (\\ref{piall}) we will obtain\nexactly the results found in {\\bf I}, as of course we must. The screened\npotentials $V_{ij}$ are then given by\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{Veqn}\nV=\\left[\\matrix{\n(-\\la^{-1}+\\Pi_{\\De\\De})^{-1}&0&0\\cr\n0&[(2V_C(q))^{-1}+\\Pi_{\\rho\\rho}]/{\\cal{D}}&-\\Pi_{\\phi\\rho}/{\\cal{D}}\\cr\n0&\\Pi_{\\phi\\rho}/{\\cal{D}}&(-\\la^{-1}+\\Pi_{\\phi\\phi})/{\\cal{D}}\\cr}\\right],\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{\\cal{D}}\\equiv(-\\la^{-1}+\\Pi_{\\phi\\phi})[(2V_C(q))^{-1}+\\Pi_{\\rho\\rho}]\n+\\Pi_{\\phi\\rho}^2.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe coupling between phase and density fluctuations caused by the non-zero\nvalue of $\\Pi_{\\phi\\rho}=-\\Pi_{\\rho\\phi}$ is a manifestation\nof gauge invariance.\n\nWe can next show that the propagators $V_{\\phi\\phi}$, $V_{\\phi\\rho}$\nand $V_{\\rho\\rho}$ all have a low-momentum singularity of the form\n$1/q^{d-1}$ for all non-zero frequencies and all temperatures. In other\nwords, these propagators have the same long-range behavior as the unscreened\nCoulomb potential. An analogous situation occurs for the \ndisorder screened potential in the normal metal, where the singularity \nis known to\nstrongly affect the properties of the system. To show the existence of\nthe singularity we simply need to show that the denominator ${\\cal{D}}$\nvanishes at $q=0$ for all $\\Om\\ne 0$ and all $T$. Since\n$V_C(q)\\sim 1/q^{d-1}$, we need only prove that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n[-\\lambda^{-1}\\Pi_{\\phi\\phi}(0,\\Om)]\\Pi_{\\rho\\rho}(0,\\Om)+\n\\Pi_{\\phi\\rho}(0,\\Om)^2 = 0.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThis is proved in Appendix B where we show that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n[-\\lambda^{-1}+\\Pi_{\\phi\\phi}(0,\\Om)]={\\Om\\over 2\\De}\\Pi_{\\phi\\rho}(0,\\Om)\n\\quad,\\quad \\Pi_{\\phi\\rho}(0,\\Om)=-{\\Om\\over\\De}\\Pi_{\\rho\\rho}(0,\\Om),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nby direct calculation. We also show that this result is guaranteed\nby gauge invariance and is therefore true no matter which scattering\nmechanisms we include.\n\n\\section{First Order Correction to Grand Potential and Order-Parameter\nSelf-Consistency Equation}\n\nIn this section we evaluate the first-order perturbation correction to\nthe grand potential, $\\Om_1(\\De)$. By minimising the sum\n$\\Om_0(\\De)+\\Om_1(\\De)$ with respect to $\\De$ we obtain the corresponding\ncorrection to the order parameter self-consistency equation.\nThis method was first used for the system with only non-magnetic\nimpurities by Eckern and Pelzer\\cite{EP88}, and we choose to use it as it\ninvolves the smallest number of diagrams. The same result for $T_c$\ncan also be obtained using the Eliashberg diagrams for $\\De$ shown\nin Fig. (5), or the pair propagator diagrams shown in Fig. (6).\n\\par\nThe diagram for the first order correction to the grand potential\nsimply consists of the ``string of bubbles'' diagram shown in Fig. (4)\nSince the polarization bubbles in this diagram are just those\nevaluated in the previous section, we have all the information we\nneed to derive $\\Om_1(\\De)$. The only thing we need to remember is\nthe extra symmetry factor of $1/n$ required for the diagram with\n$n$ bubbles. So whereas previously the RPA equation involved the series\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nV = V_0+V_0\\Pi V_0+V_0\\Pi V_0\\Pi V_0+\\dots\n= (V_0^{-1}-\\Pi)^{-1},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nit now becomes\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Om_1 = V_0\\Pi+{1\\over 2}V_0\\Pi V_0\\Pi+{1\\over 3}V_0\\Pi V_0\\Pi V_0\\Pi\n+\\dots= -\\log{[V_0^{-1}-\\Pi]}+\\log{[V_0^{-1}]}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nAfter summing over all the internal variables -- momentum $q$, Bose\nMatsubara frequency $\\Om$ and the three bosonic modes -- we end up\nwith the final expression for $\\Om_1$,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{gpcorr}\n\\Om_1=-T\\sum_\\Om\\sum_q\\left\\{\\log{(-\\la^{-1}+\\Pi_{\\De\\De}(q,\\Om))}\n+\\log{\\big[(-\\la^{-1}+\\Pi_{\\phi\\phi}(q,\\Om))\n[(2V_C(q))^{-1}+\\Pi_{\\rho\\rho}(q,\\Om)]+\n\\Pi_{\\phi\\rho}(q,\\Om)^2\\big]}\\right\\}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nTo proceed we need to minimise the total grand potential,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{minpot}\n{\\pa\\over\\pa\\De}\\left\\{\\Om_0(\\De)+\\Om_1(\\De)\\right\\}=0.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe mean-field grand potential, $\\Om_0(\\De)$, is given by\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Om_0(\\De)=N(0){\\De^2\\over\\la}+T\\sum_\\om\\sum_k\n\\hbox{Tr}\\left[\\log{(\\zb-\\xi_k\\tc-\\Db\\ta)}\\right],\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand after taking the derivative of $\\Om_0(\\De)$ with respect to $\\De$,\nwe see that Eqn. (\\ref{minpot}) takes the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{1\\over\\la}-\\pi N(0)T\\sum_\\om {1\\over U} = -{1\\over\\De}\n{\\pa\\Om_1\\over\\pa\\De}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe next step in the procedure is to evaluate $\\pa\\Om_1(\\De)/\\pa\\De$.\nFrom Eqn. (\\ref{gpcorr}) we see that $\\pa/\\pa\\De$ will be acting upon the\n$\\Pi_{ij}$ to give\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{newsc}\n-{\\pa\\Om_1\\over\\pa\\De} = T\\sum_\\Om \\sum_q \\left\\{\n{\\pa\\Pi_{\\De\\De}\\over\\pa\\De}V_{\\De\\De}+\n{\\pa\\Pi_{\\phi\\phi}\\over\\pa\\De}V_{\\phi\\phi}-\n2{\\pa\\Pi_{\\phi\\rho}\\over\\pa\\De}V_{\\phi\\rho}+\n{\\pa\\Pi_{\\rho\\rho}\\over\\pa\\De}V_{\\rho\\rho}\\right\\}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nFrom Eqn. (\\ref{piall}) we see that the $\\pa/\\pa\\De$ can act either on the\ncoherence factor or on the denominator in the expression for $\\Pi_{ij}$.\nActing on the denominator gives a result proportional to the\ndenominator squared, corresponding to the two-ladder diagrams of\nFig. (5a) in the Eliashberg approach. Similarly acting on the coherence\nfactor leads to the one-ladder diagrams of Fig. (5b). We note that the\nexplicit evaluation of the three-ladder diagrams of Fig. (5c) will give\na zero result.\n\\par\nThe only difficulty in taking the derivatives of the polarization\nbubbles, $\\Pi_{ij}$, with respect to $\\De$ is that the quantity\n$u(\\om)$ present in all these equations satisfies the transcendental\nEqn. (\\ref{transc}). In Appendix C we evaluate the derivatives of\nthe $\\Pi_{ij}$ to obtain the results\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n{\\pa\\Pi_{\\De\\De}\\over\\pa\\De}&=&-2\\pi N(0)T\\sum_\\om \\left\\{\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times \n{1\\over\\De}{u(u'+u)\\over U^3U'}{1\\over D_{+}}+\n\\left(1+{uu'-1\\over UU'}\\right)\\left\\{{1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'-u)\\over UU'}\\right)\\right\\}{1\\over D_{+}^2}\\right\\}\n\\nonumber\\\\\n{\\pa\\Pi_{\\phi\\phi}\\over\\pa\\De} &=&-2\\pi N(0)T\\sum_\\om \\left\\{\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times \n{1\\over\\De}{u(u'-u)\\over U^3U'}{1\\over D_{-}}+\n\\left(1+{uu'+1\\over UU'}\\right)\\left\\{{1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'+u)\\over UU'}\\right)\\right\\}{1\\over D_{-}^2}\\right\\}\n\\nonumber\\\\\n{\\pa\\Pi_{\\rho\\rho}\\over\\pa\\De} &=&-2\\pi N(0)T\\sum_\\om \\left\\{\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times\n{1\\over\\De}{u(u'-u)\\over U^3U'}{1\\over D_{-}}-\n\\left(1-{uu'+1\\over UU'}\\right)\\left\\{{1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'+u)\\over UU'}\\right)\\right\\}{1\\over D_{-}^2}\\right\\}\n\\nonumber\\\\\n{\\pa\\Pi_{\\phi\\rho}\\over\\pa\\De} &=& -2\\pi N(0)T\\sum_\\om \\left\\{\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times \n{1\\over\\De}{u(uu'+1)\\over U^3U'}{1\\over D_{-}}-\n\\left({u'-u\\over UU'}\\right)\\left\\{{1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'+u)\\over UU'}\\right)\\right\\}{1\\over D_{-}^2}\\right\\}.\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThese formulas together with Eqn. (\\ref{newsc}) lead to our final result for\nthe first order correction to the order parameter self-consistency\nequation:\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{finalsc}\n&&\\di{1\\over N(0)\\la}-\\pi T\\sum_\\om {1\\over U}={\\pi\\over 2}T\\sum_\\om\nT\\sum_\\Om \\sum_q \\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times\\nonumber\\\\\n&&\\di\\left\\{\\left[{1\\over\\De^2}{1\\over D_{+}}{u(u'+u)\\over U^3U'}\n+{1\\over\\De}{1\\over D_{+}^2}{1\\over U}\\left\\{1-{\\zeta\\over U}\\left(\n1+{u(u'-u)\\over UU'}\\right)\\right\\}\\left(1+{uu'-1\\over UU'}\\right)\n\\right]V_{\\De\\De}(q,\\Om)\\right.\\nonumber\\\\\n&&\\di+{1\\over\\De^2}{1\\over D_{-}}\n\\left[{u(u'-u)\\over U^3U'}V_{\\phi\\phi}(q,\\Om)-\n{2u(uu'+1)\\over U^3U'}V_{\\phi\\rho}(q,\\Om)+\n{u(u'-u)\\over U^3U'}V_{\\rho\\rho}(q,\\Om)\\right]\\nonumber\\\\\n&&\\di\\left.+{1\\over\\De}{1\\over D_{-}^2}\\left\\{1-{\\zeta\\over U}\n\\left(1+{u(u'-u)\\over UU'}\\right)\\right\\}\n\\left[\\left(1+{uu'+1\\over UU'}\\right)V_{\\phi\\phi}(q,\\Om)+\n{2(u'-u)\\over UU'}V_{\\phi\\rho}(q,\\Om)-\n\\left(1-{uu'+1\\over UU'}\\right)V_{\\rho\\rho}\\right]\\right\\}.\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\par\nThe above formula is valid for all temperatures $0\\le T\\le T_c$, \nbut we are usually interested\nin the special cases $T=0$ and $\\De=0$ (i.e. $T=T_c$). In these two\ncases the sum over $\\om$ on the LHS can be performed analytically to yield\nthe two simple forms\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\log{\\left({\\De(0)\\over\\De_0(0)}\\right)} &=&{\\pi\\over 2}\nT\\sum_\\Om \\sum_q T\\sum_\\om \\dots \\nonumber\\\\\n\\log{\\left({T_c\\over T_{c0}}\\right)} &=&{\\pi\\over 2}\nT\\sum_\\Om \\sum_q T\\sum_\\om \\dots\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\par\nHaving noted the presence of the $1/q^{d-1}$ singularities in the\npotentials $V_{\\phi\\phi}$, $V_{\\phi\\rho}$ and $V_{\\rho\\rho}$, we should\nnow see whether the terms in Eqn. (\\ref{finalsc}) containing these\nsingularities cancel out. If we go back to Eqn. (\\ref{gpcorr}) for the\ncorrection to the grand potential, we see that the term ${\\cal{D}}$\nthat goes as $q^{d-1}$ is inside a logarithm. Since $q^{d-1}$ occurs\nas a product, we can simply take off the term $\\log{(q^{d-1})}$, and\nupon differentiating with respect to $\\De$ should get zero. In other\nwords we naively expect no singular term in Eqn. (\\ref{finalsc}). However\nthis is not quite correct since to prove that the denominator\n${\\cal{D}}$ vanished at $q=0$ we needed to replace $\\la^{-1}$ using\nthe mean-field self-consistency equation, Eqn. (\\ref{mfeqn}). We note that\nalthough the two sides of Eqn. (\\ref{mfeqn}) are numerically equal in\nthe mean-field case, their dependences on $\\De$ differ -- $\\la^{-1}$\ngives zero under $\\pa/\\pa\\De$, whilst $1/U$ does not. This discrepancy\nthen leads to the only singular term in Eqn. (\\ref{finalsc}), which may be\nwritten\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\ln{\\left({\\De\\over\\De_0}\\right)}_{mf}=\n{1\\over 4}\\pi T\\sum_\\om {1\\over\\De^2 U^3}\n\\left[1-{\\zeta\\over U^3}\\right]^{-1} \nT\\sum_\\Om\\sum_q V_{\\phi\\phi}(q,\\Om).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nSince this term tends to half the pair propagator contribution to\nthe suppression of $T_c$ when we let $\\De\\rightarrow 0$, we interpret\nit as the phase fluctuation contribution. It is singular because of the\nMermin-Wagner-Hohenberg theorem\\cite{MWH} \nwhich tells us that we cannot have broken\nsymmetry states in 2D systems at finite temperature. In the following we\nwill be mainly interested in the correction to $T_c$ due to Coulomb\ninteraction and so will ignore this term.\n\n\n\\section{First Order Correction to the Transition Temperature}\n\nWe can now evaluate the first order correction to the transition\ntemperature by linearizing the order parameter self-consistency\nequation with respect to $\\De$. The former can also be obtained\ndirectly from the normal state by calculating the pair propagator\n$L(q,\\Om)$ to first order, and looking for the instability at\n$q=\\Om=0$. $L$ is given by\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nL^{-1}(q,\\Om)=\\lambda^{-1}+P(q,\\Om),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $P(q,\\Om)$ is the pair polarization bubble. The zeroth order\npolarization bubble $P_0(q,\\Om)$ is shown in Fig. (6b) and leads\nto the mean-field result\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\nL_0^{-1}(q,\\Om)&=&N(0)\\left[\\log{\\left({T\\over T_{c00}}\\right)}\n+\\psi\\left({1\\over 2}+{1\\over 2\\pi T\\tau_s}+\n{Dq^2+|\\Om|\\over 4\\pi T}\\right)-\\psi\\left({1\\over 2}\\right)\\right]\n\\nonumber\\\\\n&=&N(0)\\left[\\log{\\left({T\\over T_{c0}}\\right)}\n+\\psi\\left({1\\over 2}+{1\\over 2\\pi T\\tau_s}+\n{Dq^2+|\\Om|\\over 4\\pi T}\\right)\n-\\psi\\left({1\\over 2}+{1\\over 2\\pi T\\tau_s}\\right)\\right],\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $T_{c00}$ is the BCS transition temperature (the mean field\nvalue in the absence of magnetic impurities), and $T_{c0}$ is the\nmean field value for the system with magnetic impurities. A\ncorrection to the polarization operator $\\delta P(0,0)$ will lead\nto a change in the transition temperature, which is defined as the\ntemperature at which the denominator of $L$ becomes zero, given by\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\log{\\left({T_c\\over T_{c0}}\\right)} =\n{\\delta P(0,0)\\over N(0)}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\par\nIf we look at Fig. (6) we see that there are 7 diagrams which\ncontribute to the first order correction to $T_c$. We will set\n$\\De=0$ in the order parameter result of Eqn. (\\ref{finalsc})\nto get the transition\ntemperature equation, and we will be able to identify the contribution\nthat comes from each of the $P_i$ diagrams. When we set\n$\\De\\rightarrow 0$, then $\\De u\\rightarrow (|\\om|+1/\\tau_s)\\hbox{sgn}(\\om)$;\n$\\De U\\rightarrow |\\om|+1/\\tau_s$; $V_{\\rho\\rho}\\rightarrow 2V_C$;\n$V_{\\De\\De}$ and $V_{\\phi\\phi}\\rightarrow -L$;\n$V_{\\phi\\rho}\\rightarrow 2\\Pi_{\\phi\\rho}LV_C$. The coherence factors\nthen become Heaviside functions that set the relative signs of the\nfrequencies\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n1-{uu'\\pm 1\\over UU'} &\\rightarrow&2\\theta(-\\om(\\om+\\Om))\\nonumber\\\\\n1+{uu'\\pm 1\\over UU'} &\\rightarrow&2\\theta(\\om(\\om+\\Om)).\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe two denominators, $D_{\\pm}$, both become $Dq^2+|\\Om|$ for $\\om$, $\\om+\\Om$\nof opposite sign; $Dq^2+|2\\om+\\Om|+2/\\tau_s$ for $\\om$, $\\om+\\Om$\nof the same sign. Making all these substitutions leads to\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\nP_1&=&-\\pi N(0)T\\sum_\\om T\\sum_\\Om \\sum_q \\left[\n{1\\over (|\\om|+1/\\tau_s)^2}{1\\over (Dq^2+|\\Om|)}\n+{2\\over (|\\om|+1/\\tau_s)}{1\\over (Dq^2+|\\Om|)^2}\\right]\nV_C(q,\\Om)\\theta(-\\om(\\om+\\Om))\\nonumber\\\\\nP_2&=&\\pi N(0)T\\sum_\\om T\\sum_\\Om \\sum_q\n{1\\over (|\\om|+1/\\tau_s)^2}{1\\over Dq^2+|2\\om+\\Om|+2/\\tau_s}\nV_C(q,\\Om)\\theta(\\om(\\om+\\Om))\\nonumber\\\\\nP_3&=&-\\pi N(0)T\\sum_\\om T\\sum_\\Om \\sum_q\n{1\\over (|\\om|+1/\\tau_s)(|\\om+\\Om|+1/\\tau_s)}\n\\left[{1\\over (Dq^2+|\\Om|)}+{2/\\tau_s\\over(Dq^2+|\\Om|)^2}\\right]\nV_C(q,\\Om)\\theta(-\\om(\\om+\\Om))\\nonumber\\\\\nP_4&=&-\\pi N(0)T\\sum_\\om T\\sum_\\Om \\sum_q\n{1\\over (|\\om|+1/\\tau_s)(|\\om+\\Om|+1/\\tau_s)}\n{1\\over (Dq^2+|2\\om+\\Om|+2/\\tau_s)}\nV_C(q,\\Om)\\theta(\\om(\\om+\\Om))\\nonumber\\\\\nP_5&=&-\\pi N(0)T\\sum_\\om T\\sum_\\Om \\sum_q{1\\over (|\\om|+1/\\tau_s)^2}\\left[\n{1\\over (Dq^2+|2\\om+\\Om|+2/\\tau_s)}+\n{|\\om|-1/\\tau_s\\over (Dq^2+|2\\om+\\Om|+2/\\tau_s)^2}\n\\right]L(q,\\Om)\\theta(\\om(\\om+\\Om))\\nonumber\\\\\nP_6&=&\\pi N(0)T\\sum_\\om T\\sum_\\Om \\sum_q\n{1\\over (|\\om|+1/\\tau_s)^2 (Dq^2+|\\Om|)}\nL(q,\\Om)\\theta(-\\om(\\om+\\Om))\\nonumber\\\\\nP_7&=&4\\pi^2 N(0)^2 T\\sum_\\Om \\sum_q\n\\left[T\\sum_\\om {\\hbox{sgn}(\\om+\\Om)\\over (|\\om|+1/\\tau_s)\n(Dq^2+|\\om|+|\\om+\\Om|+2/\\tau_s)\\theta(\\om(\\om+\\Om)))}\\right]^2\nV_C(q,\\Om)L(q,\\Om).\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe assignment of terms to the polarization bubble diagram they\nwould arise from if we had done the calculation by that method is\nunique, and can be summarised below:\n\\begin{itemize}\n\\item[] $P_1$ : term proportional to $V_C$, $\\theta(-\\om(\\om+\\Om))$\nwith no $|\\om+\\Om|+1/\\tau_s$ denominator.\n\\item[] $P_2$ : term proportional to $V_C$, $\\theta(\\om(\\om+\\Om))$\nwith no $|\\om+\\Om|+1/\\tau_s$ denominator.\n\\item[] $P_3$ : term proportional to $V_C$, $\\theta(-\\om(\\om+\\Om))$\nwith $|\\om+\\Om|+1/\\tau_s$ denominator.\n\\item[] $P_4$ : term proportional to $V_C$, $\\theta(\\om(\\om+\\Om))$\nwith no $|\\om+\\Om|+1/\\tau_s$ denominator.\n\\item[] $P_5$ : term proportional to $L$ and $\\theta(-\\om(\\om+\\Om))$.\n\\item[] $P_6$ : term proportional to $L$ and $\\theta(\\om(\\om+\\Om))$.\n\\item[] $P_7$ : term proportional to $LV_C$.\n\\end{itemize}\n\nWe find that these reduce to the results of I when we set\n$1/\\tau_s\\rightarrow 0$, providing a useful consistency check on the \npresent calculation.\n\\par\nTo evaluate the correction to $T_c$ we split it into two parts:\nthe Coulomb part consisting of those parts that contain a Coulomb\npropagator, ($P_1-P_4$, $P_7$), and consequently require special\nattention at $q=0$, and the fluctuation part consisting of those\nterms that contain only a fluctuation propagator, ($P_5$, $P_6$).\nPerforming the $\\om$-sum first we get for the Coulomb part\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{finaltc}\n&&\\log{\\left({T_c\\over T_{c0}}\\right)}=\n-T_c\\sum_\\Om \\sum_q \\left\\{ {1\\over 2\\pi T_c}{Dq^2\\over\\Om^2-(Dq^2)^2}\n\\psi'\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}+{|\\Om|\\over 2\\pi T_c}\n\\right)\\right.\\nonumber\\\\\n&+&\\left.\\left[{2Dq^2[\\Om^2+(Dq^2)^2]\\over |\\Om|[\\Om^2-(Dq^2)^2]^2}\n-{\\displaystyle\\left({2\\over\\tau_s}\\right)Dq^2\\over \n\\displaystyle|\\Om|\\left(|\\Om|+{2\\over\\tau_s}\\right)\\left(|\\Om|+Dq^2\\right)}\n\\right]\\left[\\psi\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}\n+{|\\Om|\\over 2\\pi T_c}\\right)-\\psi\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}\n\\right)\\right]\\right.\\nonumber\\\\\n&-&\\left.\\di{4(Dq^2)^2\\over [\\Om^2-(Dq^2)^2]^2}\n{\\displaystyle\\left[\\psi\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}\n+{|\\Om|\\over 2\\pi T_c}\\right)\n-\\psi\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}\\right)\\right]^2\\over\n\\displaystyle\\left[\\psi\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}\n+{Dq^2+|\\Om|)\\over 4\\pi T_c}\\right)-\n\\psi\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}\\right)\\right]}\\right\\}V_C(q,\\Om).\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nSince the worst singularity possible in $V_C(q,\\Om)$ at $q=0$ goes\nas $1/q^2$, the overall $q^2$ factor multiplying $V_C$\nin the above expression means that this singularity is removed.\nIt follows that the removal of the $q=0$ singularity in the Coulomb\npart is unaffected by the addition of magnetic impurities -- this\nis because it is a general feature enforced by gauge invariance, as\nwe will show in Appendix B.\n\\par\nTo calculate the Coulombic suppression term of Eqn. (\\ref{finaltc}), \nwe change variables to\n$m=\\Om/2\\pi T$ and $y=Dq^2/2\\pi T$, noting that $\\sum_q=\\int dy/(8\\pi^2 DT)$,\nand\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nN(0)V_C(q,\\Om)=N(0)\\left[{q^2\\over 4\\pi e^2}\n+{2N(0)Dq^2\\over Dq^2+|\\Om|}\\right]^{-1}\n\\approx {Dq^2+|\\Om|\\over 2Dq^2}={m+y\\over 2y}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThis leads to the result\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{finaltc1}\n\\log{\\left({T_c\\over T_{c0}}\\right)}=\n\\displaystyle -{1\\over 8\\pi^2N(0)D}\\sum_{m=1}^M\\int_0^M dy&&\n\\left\\{{1\\over m-y}\\psi'\\left({1\\over 2}+\\alpha\\right)\\right.\\nonumber\\\\\n&+&\\left[{2y(m^2+y^2)\\over m(m^2-y^2)^2}\n-{2\\alpha y\\over m(m+2\\alpha)(m+y)}\\right]\n\\left[\\psi\\left({1\\over 2}+\\alpha+m\\right)-\\psi\\left({1\\over 2}\n+\\alpha\\right)\\right]\\nonumber\\\\\n&-&\\left.{4y\\over (m-y)(m^2-y^2)}\n{\\left[\\psi\\left({1\\over 2}+\\alpha+m\\right)-\n\\left({1\\over 2}+\\alpha\\right)\\right]^2\\over\n\\left[\\psi\\left({1\\over 2}+\\alpha+{m+y\\over 2}\\right)-\n\\psi\\left({1\\over 2}+\\alpha\\right)\\right]}\\right\\},\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $\\alpha=1/2\\pi T_c\\tau_s$ and the upper cutoff $M=1/2\\pi T_c\\tau$.\nThe leading order term is that which goes like $1/y$ at large $y$, leading\nto logarithmic behavior. To isolate this, we add and subtract the term\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n&&\\sum_{m=1}^{M}\\int_0^M {dy\\over (m+y)}\n\\left\\{\\left({2\\over m}-{2\\alpha\\over m(m+2\\alpha)}\\right)\n\\left[\\psi\\left({1\\over 2}+\\alpha+m\\right)\n-\\psi\\left({1\\over 2}+\\alpha\\right)\\right]\n-\\psi'\\left({1\\over 2}+\\alpha\\right)\\right\\}\\nonumber\\\\\n=&&\\sum_{m=1}^{M} \\ln{\\left({M+m\\over m}\\right)}\n\\left\\{{2(m+\\alpha)\\over m(m+2\\alpha)}\n\\left[\\psi\\left({1\\over 2}+\\alpha+m\\right)\n-\\psi\\left({1\\over 2}+\\alpha\\right)\\right]\n-\\psi'\\left({1\\over 2}+\\alpha\\right)\\right\\},\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nto give the result\n\\begin{eqnarray}\n\\label{onept}\n\\ln{\\left({T_c\\over T_{c0}}\\right)}=&-&{R_{\\square}\\over R_0}\\left\\{\n\\sum_{m=1}^M \\ln{\\left({M+m\\over m}\\right)}\n\\left[{2(m+\\alpha)\\over m(m+2\\alpha)} \n\\left[\\psi\\left({1\\over 2}+\\alpha+m\\right)\n-\\psi\\left({1\\over 2}+\\alpha\\right)\\right]\n-\\psi'\\left({1\\over 2}+\\alpha\\right)\\right]\\right.\\nonumber\\\\\n&-&\\sum_{m=1}^M\\int_0^M dy{4y\\over (m-y)(m^2-y^2)}\n\\left[{\\left[\\psi\\left({1\\over 2}+\\alpha+m\\right)-\n\\psi\\left({1\\over 2}+\\alpha\\right)\\right]^2\\over\n\\left[\\psi\\left({1\\over 2}+\\alpha+{m+y\\over 2}\\right)\\right]}\\right.\\nonumber\\\\\n&-&\\left.\\left.\\left[\\psi\\left({1\\over 2}+\\alpha+m\\right)-\n\\psi\\left({1\\over 2}+\\alpha\\right)\\right]\n+{y-m\\over 2}\\psi'\\left({1\\over 2}+\\alpha\\right)\\right]\\right\\},\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere we have noted that $1/8\\pi^2N(0)D=R_{\\square}/R_0$. \nWe could now proceed to evaluate this expression, but before we do\nso, let us consider the domain of validity of the first-order perturbative\nresult.\n\n\\section{Beyond Perturbation Theory}\n\nSince we now have the full first-order perturbative correction\nto the transition temperature due to the effect of disorder on\nthe Coulomb interaction, we could in principle plot curves of \n$T_c(R_{\\square},1/\\tau_s)$ and compare to experiment. \nHowever the curves of $T_c$ vs\n$R_{\\square}$ for different values of $1/\\tau_s$ would simply\nbe exponential decays with different initial slopes. First-order\nperturbation theory is unable to treat the strong disorder region,\nand so cannot lead to the complete destruction of the superconductivity\nby non-magnetic disorder.\nIf we are to consider the effects of arbitrary disorder strength, we\nmust work beyond perturbation theory. In what follows we discuss two\nmethods of doing this, and compare the results we obtain from them.\n\nThe simplest way to proceed is to ``self-consistently'' solve the\nfirst-order perturbative expression of Eq. (\\ref{onept}). This simply\nmeans that we replace $T_{c0}$ by $T_c$ on the right-hand side of\nEq. (\\ref{onept}), and solve the implicit equation we obtain for $T_c$\nwhich has the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{oneptsc}\n\\ln{\\left({T_c\\over T_{c0}}\\right)}=\\psi\\left({1\\over 2}\\right)\n-\\psi\\left({1\\over 2}+{1\\over 2\\pi T_c\\tau_s}\\right)\n-{R_{\\square}\\over R_0}f(T_c,1/\\tau_s).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nHere $f(T_c,1/\\tau_s)$ is the complicated expression on the right-hand side\nof Eq. (\\ref{onept}),\nwhilst the first term is just the mean-field suppression of $T_c$ by the\nmagnetic impurities. From our knowledge of the situation without magnetic\nimpurities, we know that unphysical re-entrance problems may arise with\nthe solution of this equation, and we should not take it too seriously\nin the region where superconductivity is strongly suppressed.\n\nThe fact that we cannot trust the results obtained from this \n``self-consistent'' theory leads us to ask the\nquestion of how to correctly go beyond first-order perturbation theory.\nThe best approach is to derive the effective field theory from which the\nperturbation series may be deduced -- in this case a non-linear sigma\nmodel\\cite{Fink83} -- and treat this using the renormalization group. \nThis has been done by Finkel'stein\\cite{Fink87}\nfor the system without magnetic impurities, but\nhas the problem that it is very difficult, and would become even more\nso if magnetic impurities were added. Recently Oreg and Finkel'stein\\cite{OF}\ndemonstrated that the same results could be obtained using a much\nsimpler non-perturbative resummation technique, which we show\ndiagrammatically in Fig. (7). The method uses a featureless Coulomb\ninteraction of magnitude $N(0)V_C=1/2$, consistent with the cancellation\nof the $1/q^{d-1}$ divergence discussed earlier, and keeps only diagrams\n3 and 4 of Fig. (6), since they give the greatest contribution. This \nleads to the equation for the pair scattering amplitude, \n$\\Gamma(\\om_n,\\om_l)$,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{of}\n\\Gamma(\\om_n,\\om_l)=-|\\lambda|+t\\Lambda(\\om_n,\\om_l)\n-\\pi T\\sum_{m=-(M+1)}^M [-|\\lambda|+t\\Lambda(\\om_n,\\om_m)]\n{1\\over |\\om_m|+1/\\tau_s}\n\\Gamma(\\om_m,\\om_l),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $\\om_n=2\\pi T(n+1/2)$ is a Fermi Matsubara frequency, and the upper\ncut-off $M=1/2\\pi T\\tau$. The amplitude $\\Lambda(\\om_n,\\om_l)$ is given by \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Lambda(\\om_n,\\om_l)=\\cases{\n\\di\\ln{\\left[{1\\over (|\\om_n|+|\\om_l|)\\tau}\\right]}&$\\quad\\om_n\\om_l<0$\\cr\n\\di\\ln{\\left[{1\\over (|\\om_n|+|\\om_l|+2/\\tau_s)\\tau}\\right]}\n&$\\quad\\om_n\\om_l>0$\\cr}\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere the breaking of time-reversal invariance by the spin-flip scattering\nmeans that $\\Lambda$ has a different form depending upon the relative signs\nof its two Matsubara frequencies. If we treat the $\\Gamma(\\om_n,\\om_m)$ as\nelements of a matrix $\\hat{\\Gamma}$, the matrix equation for $\\hat{\\Gamma}$\ncan be solved to yield\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\hat{\\Gamma}=\\hat{\\om}^{1/2}(\\hat{I}-|\\lambda|\\hat{\\Pi})^{-1}\n\\hat{\\om}^{-1/2}(-|\\lambda|\\hat{1}+t\\hat{\\Lambda}),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\hat{\\Pi}={1\\over 2}\\hat{\\om}^{-1/2}\n[\\hat{1}-|\\lambda|^{-1}t\\hat{\\Lambda}]\\hat{\\om}^{-1/2},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n$\\hat{\\om}_{nm}=(n+1/2+\\alpha)\\delta_{nm}$,\n$\\hat{\\Lambda}_{nm}=\\Lambda(\\om_n,\\om_m)$, $\\hat{1}_{nm}=1$ and\n$\\hat{I}_{nm}=\\delta_{nm}$. The matrix $\\hat{\\Gamma}$ becomes singular when\nan eigenvalue of $\\hat{\\Pi}$ equals $1/|\\lambda|$, and this signals the onset\nof superconductivity. Note that the matrix $\\hat{\\Pi}$ depends on temperature\nboth through the temperature dependence of its elements, and also through its\nrank $2M$. To find $T_c$, we start at the BCS value $T_{c0}$, which corresponds\nto a value of $M$ given by $M_0=1/2\\pi T_{c0}\\tau$. We decrease the \ntemperature $T$ by increasing the upper cut-off $M$ successively by one.\nFor each value of $M$, we construct the matrix $\\hat{\\Pi}$, and diagonalise\nit. When its lowest eigenvalue equals $1/|\\lambda|$, we have reached the\ntransition temperature $T_c$, which is given by $T_c/T_{c0}=M_0/M$. This\nmethod allows us to go to as low a temperature as we like, provided that\nwe are prepared to diagonalize large enough matrices.\n\nWe will now plot curves of $T_c$ vs $R_{\\square}$ for fixed $1/\\tau_s$,\nand $T_c$ vs $1/\\tau_s$ for fixed $R_{\\square}$, derived both from the\nself-consistent perturbation theory of Eqn. (\\ref{oneptsc}), and from the\nnon-perturbative resummation approach of Eqn. (\\ref{of}). This is done\nin Fig.~(8), and we see that the two approaches are in rough agreement. \nThe resummation technique is seen to remove the re-entrance problem which \noccurs in the $\\alpha=0$ curve at large $R_{\\square}$, but surprisingly\nthis re-entrance seems to be partially cured by the presence of magnetic\nimpurities.\n\nThe above curves are fine from the theorist's point of view, but \nexperimentally what is measured is the suppression of $T_c$ by a\ncertain fixed amount of magnetic impurities as the thickness of the\nsuperconductor is altered. The data is then presented in the form of\nthe pair-breaking per magnetic impurity which can be written as\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\alpha'(R_{\\square})={T_c(R_{\\square},0)-T_c(R_{\\square},1/\\tau_s)\\over\n1/\\tau_s},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhich we can also generate from our theoretical expressions.\nThe result will of course depend upon the magnitude of the value of\n$\\alpha$ we choose: we would like to choose $\\alpha$ as small as possible\nso that we are always in the linear regime of pair-breaking, but not too\nsmall so that the difference is very sensitive to the discrete sums used\nin the numerical calculation. A typical plot is shown in Fig. (9). If we\nignore the numerical noise we see that $\\alpha'$ is roughly constant and\nequal to its mean field value of $\\pi^2/2$. It only appears to increase\nas we approach the region where superconductivity is destroyed, and its\ntotal variation is only about $10\\%$ even if we include this region. This\nis in agreement with the experimental data of Chervenak and Valles\\cite{CV}.\n\n\\section{Discussion and Conclusions}\n\nThe main conclusion of this paper is that the effects of localization and\ninteraction do not lead to an appreciable change of pair-breaking rate per\nmagnetic impurity in disordered superconducting films provided that we are \nnot too close to the \nsuperconductor-insulator transition. The experimental data agrees with\nthis theoretical prediction, and thus confirms the validity of the basic\nmodel of $T_c$ suppression in disordered superconductors which consists of\nthe BCS interaction, Coulomb repulsion and static disorder. The fact that\nthe theoretical\nprediction is obtained both from first-order perturbation theory, and from\na non-perturbative resummation technique, gives us increased confidence in\nits validity. Our calculations demonstrate that the resummation technique is\na very powerful tool for going beyond perturbation theory which can be\nadapted to a variety of situations. Moreover we find that the ad hoc\n``self-consistent'' extension of first-order perturbation theory can give\nsensible results even at values of $R_{\\square}$ near the \nsuperconductor-insulator transition, at least in the presence of\npair-breaking. \n\nThe effect of nonmagnetic disorder on pair-breaking in superconducting films\nhas previously been considered by Devereaux and Belitz\\cite{Dev96} using a\nmodel in which strong coupling effects are considered. Good agreement with\nexperiment is also obtained with this approach, although more fitting\nparameters are required in this model. We note that only a single fitting\nparameter -- the initial slope of the $T_c(R_{\\square})$ curve -- is\nrequired in our approach. Unfortunately we see that the experimental data\nis unable to determine which, if any, of the two approaches is correct.\nIn support of our approach we note that it is a ``minimal model'' in the\nsense that it contains the minimal physics to describe the system, and\nrequires the input of a single fitting parameter. However, this is not to\nsay that strong-coupling effects are not important in this system. \n\nAnother important result which emerges from the approach based on the\ngrand-canonical potential is that the $1/q^2$ singularity of the disorder\nscreened Coulomb potential is always cancelled in first-order perturbation\ntheory. This removes the possibility of changing some experimental parameter\nto obtain a strong suppression of $T_c$ from this singularity. We have shown\nthat this cancellation is enforced by gauge invariance, and leads us to\nsuspect that it occurs to all orders in perturbation theory. It is this\ncancellation which makes it legitimate to use a featureless interaction in\nthe resummation technique.\n\n \n\n\\bigskip\n\\centerline {\\bf ACKNOWLEDGEMENTS}\n\\medskip\n\nWe thank I. Aleiner, A.M. Finkel'stein and Y. Oreg for helpful discussions.\nR.A.S. acknowledges support from the Nuffield Foundation.\nV.A. is supported by the U.S. National Science Foundation under grant DMR-9805613.\n\n\\medskip\n\\appendix\n\\section{Calculation of Polarization Bubbles}\n\\medskip\n\nIn this appendix we give a detailed derivation of the polarization\nbubbles, $\\Pi_{ij}$, shown in Fig. (3). To evaluate these we must\nfirst calculate the impurity ladder, $\\Pi$, which is given by the\ngeometric series\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Pi=S+S\\Ga_0 S+S\\Ga_0 S\\Ga_0 S+\\dots,\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere\n$\\Ga_0$ is the impurity line\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Ga_0={1\\over 2\\pi N(0)\\tau}\\left[\n\\lambda_1\\tau_3\\sigma_0\\otimes\\tau_3\\sigma_0+\n\\lambda_2 \\left( \\tau_0\\sigma_1\\otimes\\tau_0\\sigma_1+\n\\tau_0\\sigma_2\\otimes\\tau_0\\sigma_2+\\tau_3\\sigma_3\\otimes\\tau_3\\sigma_3\n\\right)\\right],\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand $1/\\tau=1/\\tau_0+1/\\tau_s$ is the total impurity scattering\nrate, $\\lambda_1=\\tau/\\tau_0$, and $\\lambda_2=\\tau/3\\tau_s$.\n$S$ is the momentum sum of a direct product of Green functions\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\nS&=&\\sum_k G(k,i\\om)\\otimes G(k+q,i\\om+i\\Om)\\nonumber\\\\\n&=&\\displaystyle\\pi N(0)\\tau I\\left[\\tau_3\\sigma_0\\tau_3\\sigma_0-\n{(\\zb-\\Db\\tau_1\\sigma_3)(\\zp-\\Dp\\tau_1\\sigma_3)\\over\\ep\\ep'}\\right],\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand $I$ is the integral\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{Iint}\nI={1\\over\\pi\\tau}\\int d\\xi_k d\\hat{\\Om}\n{\\xi_k(\\xi_k-{\\bf q.v_F})\\over (\\xi_k^2-\\ep^2)\n[(\\xi_k-{\\bf q.v_F})^2-\\ep'^2]}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nSince we do not need the complete matrix structure of $\\Pi$, but just\nits traces with two matrices from the set $\\ta$, $\\tb$, $\\tc$, we\nactually evaluate the impurity dressed vertices $\\Pi_j$ which have one\nmatrix from the above set inserted between the two terms of the direct\nproduct in $\\Pi$. These satisfy the equation\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{pijeqn}\n\\Pi_j=S_j+S\\Ga_0\\Pi_j.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nStarting with $\\Pi_{\\De}$ we see that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\nS_\\De &=&\\displaystyle 2\\pi N(0)\\tau I\\left[\\tc\\ta\\tc-\n{(\\zb+\\Db\\ta)\\ta(\\zp+\\Dp\\ta)\\over\\ep\\ep'}\\right]\\nonumber\\\\\n&=&\\displaystyle 2\\pi N(0)\\tau I\\left[-1-\n{(\\zb+\\Db\\ta)(\\zp+\\Dp\\ta)\\over\\ep\\ep'}\\right]\\ta\\nonumber\\\\\n&=&\\displaystyle 2\\pi N(0)\\tau I(\\apb-\\bp\\ta)\\ta,\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere the $\\alpha$ and $\\beta$ terms are coherence factors\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\apm=1-{\\zb\\zp\\pm\\Db\\Dp\\over\\ep\\ep'}\\quad;\\quad\n\\apmb=\\apm-2\\quad;\\quad\n\\bpm={\\zp\\Db\\pm\\zb\\Dp\\over\\ep\\ep'}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nBy inspection we see that $\\Pi_{\\De}$ must have the matrix form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Pi_\\De=2\\pi N(0)\\tau I[A+B\\ta]\\ta,\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand we now substitute this into Eqn. (\\ref{pijeqn}) to deduce the\ncoefficients $A$ and $B$. To derive the second term on the RHS of\nEqn. (\\ref{pijeqn}) we see that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Ga_0\\Pi_\\De=-I\\la_1(A-B\\ta)\\ta+3I\\la_2(A+B\\ta)\\ta,\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand thus\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\nS\\Ga_0\\Pi_\\De &=&\\displaystyle2\\pi N(0)\\tau I\\left\\{\\la_1(A+B\\ta)\\ta+\n{(\\zb+\\Db\\ta)\\la_1(A+B\\ta)\\ta(\\zp+\\Dp\\ta)\\over\\ep\\ep'}\\right.\\nonumber\\\\\n&&\\left.\\qquad\\qquad\\qquad\\displaystyle -3\\la_2(A-B\\ta)\\ta-\n{(\\zb+\\Db\\ta)\\la_2(A-B\\ta)\\ta(\\zp+\\Dp\\ta)\\over\\ep\\ep'}\\right\\}\\nonumber\\\\\n&=&\\displaystyle 2\\pi N(0)\\tau I\\left\\{(\\la_1-3\\la_2)A\\left[1+\n{(\\zb+\\Db\\ta)(\\zp+\\Dp\\ta)\\over\\ep\\ep'}\\right]\\ta\\right.\\nonumber\\\\\n&&\\displaystyle\\left.\\qquad\\qquad\\qquad +(\\la_1+3\\la_2)B\\ta\\left[1-\n{(\\zb+\\Db\\ta)(\\zp+\\Dp\\ta)\\over\\ep\\ep'}\\right]\\ta\\right\\}\\nonumber\\\\\n&=&\\displaystyle 2\\pi N(0)\\tau I[(\\la_1-3\\la_2)A(-\\apb+\\bp\\ta)\\ta\n+(\\la_1+3\\la_2)B(\\ap-\\bp\\ta)].\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWe can now equate the coefficients of $1$ and $\\ta$ on the LHS and\nRHS of Eqn. (\\ref{pijeqn}) to obtain the linear equations for $A$ and $B$,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\left[\\matrix{\n1+I(\\la_1-3\\la_2)\\apb&I(\\la_1+3\\la_2)\\bp\\cr\n-I(\\la_1-3\\la_2)\\bp&1-I(\\la_1+3\\la_2)\\ap\\cr}\\right]\n\\left[\\matrix{A\\cr B\\cr}\\right]=\n\\left[\\matrix{\\apb\\cr -\\bp\\cr}\\right].\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThis matrix equation can then be inverted by inverting the $2\\times 2$\nmatrix and using the identity $\\ap\\apb=\\bp^2$ to obtain\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\left[\\matrix{A\\cr B\\cr}\\right]={1\\over D_+}\\left[\\matrix{\n1-I(\\la_1+3\\la_2)\\ap& -I(\\la_1+3\\la_2)\\bp\\cr\nI(\\la_1-3\\la_2)\\bp&1+I(\\la_1-3\\la_2)\\apb\\cr}\\right]\n\\left[\\matrix{\\apb\\cr -\\bp\\cr}\\right]\n= {1\\over D_+}\\left[\\matrix{\\apb\\cr -\\bp\\cr}\\right],\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere the determinant $D_+$ can be written\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\nD_+&=&[1+I(\\la_1-3\\la_2)\\apb][1-I(\\la_1+3\\la_2)\\ap]\n+I^2(\\la_1+3\\la_2)(\\la_1-3\\la_2)\\bp^2\\nonumber\\\\\n&=& 1+I(\\la_1-3\\la_2)\\apb-I(\\la_1+3\\la_2)\\ap\n=\\displaystyle 1-2I\\la_1+6I\\la_2\\left({\\zb\\zp+\\Db\\Dp\\over\\ep\\ep'}\\right).\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nSimilar results are obtained for $\\Pi_{\\phi}$ and $\\Pi_{\\rho}$,\nleading to the results\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\Pi_{\\De} &=&2\\pi N(0)\\tau{I\\over D_+}(\\apb-\\bp\\ta)\\ta\\nonumber\\\\\n\\Pi_{\\phi}&=&2\\pi N(0)\\tau{1\\over D_-}(\\amb-\\bm\\ta)\\tb\\nonumber\\\\\n\\Pi_{\\rho}&=&2\\pi N(0)\\tau{I\\over D_-}(\\am-\\bm\\ta)\\tc,\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{denom}\nD_{\\pm} = 1-2I\\la_1+6I\\la_2\\left({\\zb\\zp\\pm\\Db\\Dp\\over\\ep\\ep'}\\right).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nIf we evaluate the integral in Eqn. (\\ref{Iint}) we find that $I$ is given by\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{Ival}\n2I\\tau={1\\over\\Wb+\\Wp}-{q^2v_F^2\\over 2(\\Wb+\\Wp)^3},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Wb=\\sqrt{\\omb^2+\\Db^2}\\quad;\\quad \\Wb=\\sqrt{\\om'^2+\\Dp^2}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nFrom the second part of Eqn. (\\ref{selfcons}) we see that we can write\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{Wval}\n\\Wb+\\Wp = {1\\over\\tau_0}-{1\\over\\tau_s}+\\De U+\\De U',\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand substituting Eqns. (\\ref{Wval}) and (\\ref{Ival})\ninto Eqn. (\\ref{denom}) gives the result\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{denomval}\nD_{\\pm}=\\di 1-{\\di\\left[{1\\over\\tau_0}-{1\\over\\tau_s}\n\\left({uu'\\mp 1\\over UU'}\\right)\\right]\n\\over\\di\\left[{1\\over\\tau_0}-{1\\over\\tau_s}+\\De U+\\De U'\\right]}\n+Dq^2\\tau\\nonumber\n\\approx\\left[Dq^2+\\De U+\\De U'+{1\\over\\tau_s}\n\\left({uu'\\mp 1\\over UU'}-1\\right)\\right]\\tau.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWe can finally obtain the non-zero polarization bubbles $\\Pi_{ij}$\nby inserting the second matrix from the set $\\ta$, $\\tb$, $\\tc$ into\n$\\Pi_j$ and taking the trace. This yields\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{pialla}\n\\Pi_{\\De\\De}(q,\\Om) &=&\\displaystyle\\pi N(0)T\\sum_\\om \\left[\n{UU'+uu'-1\\over UU'}\\right]{1\\over\n\\left(Dq^2+\\De U+\\De U'-\\di{1\\over\\tau_s}\\left[{UU'-uu'+1\\over UU'}\\right]\n\\right)}\\nonumber\\\\\n\\Pi_{\\phi\\phi}(q,\\Om) &=&\\displaystyle \\pi N(0)T\\sum_\\om \\left[\n{UU'+uu'+1\\over UU'}\\right]{1\\over\n\\left(Dq^2+\\De U+\\De U'-\\di{1\\over\\tau_s}\\left[{UU'-uu'-1\\over UU'}\\right]\n\\right)}\\nonumber\\\\\n\\Pi_{\\rho\\rho}(q,\\Om) &=&\\displaystyle-\\pi N(0)T\\sum_\\om \\left[\n{UU'-uu'-1\\over UU'}\\right]{1\\over\n\\left(Dq^2+\\De U+\\De U'-\\di{1\\over\\tau_s}\\left[{UU'-uu'-1\\over UU'}\\right]\n\\right)} + N(0)\\nonumber\\\\\n\\Pi_{\\phi\\rho}(q,\\Om) &=&\\displaystyle-\\pi N(0)T\\sum_\\om \\left[\n{u'-u\\over UU'}\\right]{1\\over\n\\left(Dq^2+\\De U+\\De U'-\\di{1\\over\\tau_s}\\left[{UU'-uu'-1\\over UU'}\\right]\n\\right)}=-\\Pi_{\\rho\\phi}(q,\\Om).\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n\n\n\\medskip\n\\section{Low-Momentum Singularities in Density and Phase Propagators}\n\\medskip\n\nThe identities\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{finalident}\n-\\lambda^{-1}+\\Pi_{\\phi\\phi}(0,\\Om) = x\\Pi_{\\phi\\rho}(0,\\Om)~,~~~~\n\\Pi_{\\phi\\rho}(0,\\Om) = -x\\Pi_{\\rho\\rho}(0,\\Om),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $x=\\Om/2\\De$, play a central role in the present paper and in I. As\nwe have seen, their consequence\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n[-\\lambda^{-1}+\\Pi_{\\phi\\phi}(0,\\Om)]\\Pi_{\\rho\\rho}(0,\\Om)\n+\\Pi_{\\phi\\rho}(0,\\Om)^2 = 0\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nleads to the potentials\n$V_{\\phi\\phi}$, $V_{\\phi\\rho}$, and $V_{\\rho\\rho}$ having $1/q^{d-1}$\nsingularities at low momentum $q$ for all temperatures $0\\le T\\le T_c$\nand all non-zero frequencies $\\Om\\ne 0$. The importance of\nthese identities suggests that they embody an underlying invariance\nprinciple. In this appendix we show that they are Ward identities\nconnected to charge conservation, which is very reasonable since the\nimpossibility of instantaneously moving the conserved screening charge\na finite distance is at the root of these singularites, The ideas\nat work here go back to Nambu's 1960 paper and its \nelaborations\\cite{Namb60,Amb61,Schr}\nand they are only included here for completeness. It is unfortunate\n%---and somewhat scandalous---\nthat the physical basis for\nthese identities was left obscure in I.\n\nTo avoid irrelevant notational complications, we shall work within the\n$2\\times 2$ Nambu space. The $4\\times 4$ space needed to deal with spin\nflip scattering does not affect the general argument, and we shall in any\ncase explicitly verify the identities for this case later in this\nappendix.\n\nThe `proper polarization parts', $\\Pi$, are calculated within\na mean field approximation, in which the interactions are replaced\naccording to\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{vmf}\nV\\rightarrow V_{MF} = \\De Tr{\\Psi^\\dagger \\tau_1 \\Psi},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhich implies a choice of phase for the order parameter. It is known that\nthe quasiparticles obtained in this approximation do not conserve charge,\nbecause $V_{MF}$ does not commute with the electron density. Since\nthe only other non-commuting part of the Hamiltonian is the kinetic energy,\nthe operator equation of motion for the density $\\rho\\equiv\nTr{\\Psi^\\dagger\\tau_3\\Psi}$ is\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{eqmot1}\n{\\pa\\rho\\over\\pa t} + \\nabla\\cdot\\vec j = i [V_{MF},\\rho] =\n2\\De Tr{\\Psi^\\dagger\\tau_2\\Psi},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $\\vec j = Tr [\\Psi^\\dagger \\vec\\nabla \\Psi - \\Psi\\vec\\nabla\\Psi^\\dagger]\n$ is the current density operator.\nEq. (\\ref{eqmot1}) leads to the identity\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{eqmot2}\n&~~~~~~~~~~~~{\\pa\\over\\pa t_2} T \\langle\\Psi_i (\\vec x_1,t_1) \\rho\n(\\vec x _2, t_2) \\Psi^\\dagger_j (\\vec x_3,t_3)\\rangle = \\nonumber\\\\\n&\\de (\\vec x _1 -\n\\vec x_2) \\de (t_1 - t_2) i [\\tau _3 G(\\vec x_2,t_2, \\vec x_3,\nt_3)]_{ij} -\\de (\\vec x_2 -\\vec x_3) \\de (t_2 - t_3) i [G(\\vec\nx_1,t_1, \\vec x_2,t_2)\\tau_3 ]_{ij}\\nonumber\\\\ \n&-T \\langle\\Psi_i (\\vec x_1,t_1)\n\\nabla_2\\cdot\\vec j (\\vec x_2, t_2) \\Psi^\\dagger_j (\\vec x_3,t_3)\\rangle \n+ T\\langle\\Psi_i (\\vec x_1,t_1) \\rho_\\phi\n(\\vec x_2, t_2) \\Psi^\\dagger_j (\\vec x_3,t_3)\\rangle,\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere $T$ is the time ordering operator and we have defined $\\rho_\\phi\\equiv\nTr{\\Psi^\\dagger\\tau_2\\Psi}$. [The first two terms on the right of Eq.\n(\\ref{eqmot2})\ncome from the derivative of the time ordering operator.]\nMultiplying Eq. (\\ref{eqmot2}) by $\\tau_3$, taking the trace, and Fourier\ntransforming in space and time leads in the limit of zero wave vector to\nthe identity\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Om~\\Pi_{\\rho \\rho}(0, \\Om) = 2 \\De~\\Pi_{\\rho \\phi} (0, \\Om).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nOn the other hand, multiplying by $\\tau_2$ and performing these\nsame operations yields\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\Om~\\Pi_{\\phi \\rho}(0,\\Om)&=& - {i\\over\\beta} Tr [\\tau_1 G] + 2\\De\n~\\Pi_{\\phi \\phi}\\nonumber\\\\ \n&=& 2 \\De [-\\lambda^{-1} + \\Pi_{\\phi \\phi}].\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nIn the second line above the self consistency equation within the mean\nfield approximation has been used. To obtain Eq. (\\ref{finalident}) \nwe must note\nthat $\\Pi_{\\rho \\phi}$ is antisymmetric in its indices\nbecause of time reversal invariance---under which $\\rho$ is symmetric and\n$\\rho_\\phi$ antisymmetric.\n\nSince the impurity interaction commutes with the charge density these\nidentities survive in any `conserving' approximation\\cite{Kad61}\nand, in particular, in our sum of all non-overlapping graphs.\nIn the main body of this paper, the mean field approximation was used as\na way station on the road to the single loop approximation of Section III.\nThere the phase of the order parameter is not fixed as in Eq.\n(\\ref{vmf}) but determined self consistently, which restores charge\nconservation.\\cite{PWA58,Rick58}\n\nFinally, we shall verify explicitly that the identities\n(\\ref{finalident}) are satisfied by our calculated expressions.\nWe start with the equations for $-\\lambda^{-1}+\\Pi_{\\phi\\phi}$, $\\Pi_{\\phi\\rho}$,\nand $\\Pi_{\\rho\\rho}$,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{piforms}\n-\\lambda^{-1}+\\Pi_{\\phi\\phi}(0,\\Om) &=& \\di T\\sum_{\\om} \\left[\n{UU'+uu'+1\\over UU'\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\n\\right)} -{1\\over U}\\right]\\nonumber\\\\\n\\Pi_{\\phi\\rho}(0,\\Om) &=&\\di T\\sum_{\\om}\n{u'-u\\over UU'\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)}\n\\nonumber\\\\\n\\Pi_{\\rho\\rho}(0,\\Om) &=&\\di {1\\over\\pi} - T\\sum_{\\om}\n{UU'-uu'-1\\over UU'\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\n\\right)},\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere we have removed the common factor\n$\\pi N(0)$ from each $\\Pi$, and set $\\De=1$ for algebraic convenience.\nThese factors can, of course, be replaced when we have finished.\n\\par\nWe will first prove the relationship between $-\\lambda^{-1}+\\Pi_{\\phi\\phi}$ \nand $\\Pi_{\\phi\\rho}$, namely\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{piident1}\n-\\lambda^{-1}+\\Pi_{\\phi\\phi}(0,\\Om) = {\\Om\\over 2}\\Phi_{\\phi\\rho}(0,\\Om).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nTo proceed note that we can write\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nuu' = {1\\over 2}[u^2+u'^2-(u'-u)^2]\n={1\\over 2}[U^2+U'^2-2-(u'-u)^2],\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nfrom which it follows that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{ident1}\nUU'+uu'+1 &=& {1\\over 2}[(U+U')^2-(u'-u)^2]\\nonumber\\\\\nUU'-uu'-1 &=& -{1\\over 2}[(U'-U)^2-(u'-u)^2].\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nIn the last term on the RHS of Eq. (\\ref{piforms}) for $\\Pi_{\\phi\\phi}$,\nwe can use the transformation $\\om\\leftrightarrow -\\om'$, under which\nthe sum over $\\om$ is invariant. This leads\nto $u\\leftrightarrow -u'$ and $U\\leftrightarrow U'$, so that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{intermed1}\n2T\\sum_{\\om} {1\\over U} = T\\sum_{\\om} \\left[ {1\\over U}\n+{1\\over U'}\\right] = T\\sum_{\\om} {U+U'\\over UU'}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWe can now write, using Eqs. (\\ref{piforms}), (\\ref{ident1}) and\n(\\ref{intermed1}),\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{intermed2}\n2(-\\lambda^{-1}+\\Pi_{\\phi\\phi})-\\Om\\Pi_{\\phi\\rho}&=&\nT\\sum_{\\om} \\left[{(U+U')^2-(u'-u)^2+\\Om(u'-u)\\over UU'\n\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)}\n-{U+U'\\over UU'}\\right]\\nonumber\\\\ &=&\nT\\sum_{\\om}{(U+U')^2-(u'-u)(u'-u-\\Om)-(U+U')\n\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)\\over\nUU'\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)}.\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nFrom the definition of $u$ and $u'$ in Eqn. (\\ref{transc}) we obtain the \nidentity\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{ident2}\nu'-u-\\Om&=&(u'-\\om')-(u-\\om)=\\zeta\\left({u'\\over U'}-{u\\over U}\\right)\n\\nonumber\\\\\n\\Rightarrow\\quad\n(u'-u)(u'-u-\\Om)&=&\\zeta(u'-u)\\left({u'\\over U'}-{u\\over U}\\right)\n\\nonumber\\\\\n&=&\\zeta\\left[{u'^2\\over U'}+{u^2\\over U}-{uu'\\over U'}-{uu'\\over U}\\right]\n\\nonumber\\\\\n&=&\\zeta\\left[U+U'-{1\\over U}-{1\\over U'}-{uu'\\over U}-{uu'\\over U'}\\right]\n\\nonumber\\\\\n&=&\\zeta(U+U')\\left[{UU'-uu'-1\\over UU'}\\right].\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nIt follows that the numerator in Eqn. (\\ref{intermed2}) is zero,\nand hence we have proved the required result (\\ref{piident1}).\n\\par\nWe next prove the relation between $\\Pi_{\\rho\\rho}$ and\n$\\Pi_{\\phi\\rho}$, namely\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{piident2}\n\\Pi_{\\phi\\rho}(0,\\Om)=-{\\Om\\over 2}\\Pi_{\\rho\\rho}(0,\\Om).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWe start by considering the sum\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\lim_{\\om_0\\rightarrow\\infty} T\\sum_{\\om=-(\\om_0+\\Om)}^{\\om_0}\n{u\\over\\Om U}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nAs $|\\om|\\rightarrow\\infty$, $u\\rightarrow (|\\om|+\\zeta)\\hbox{sgn}(\\om)$\nand $u/U\\rightarrow \\hbox{sgn}(\\om)$. Since there are $\\Om/2\\pi T$\nmore negative terms than positive, the sum becomes\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\lim_{\\om_0\\rightarrow\\infty} T\\sum_{\\om=-(\\om_0+\\Om)}^{\\om_0}\n{u\\over\\Om U} = -\\left({\\Om\\over 2\\pi T}\\right)\\left({T\\over\\Om}\\right)\n=-{1\\over 2\\pi}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nWe have chosen the limits so that we are able to make the usual\n$\\om\\leftrightarrow -\\om'$ transformation in this sum. It follows that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{1\\over\\pi} = -2T\\sum_{\\om} {u\\over U\\Om}\n =T\\sum_{\\om}{1\\over\\Om}\\left({u'\\over U'}\n -{u\\over U}\\right)\n = T\\sum_{\\om} {U+U'\\over \\Om(u'-u)}\n \\left({UU'-uu'-1\\over UU'}\\right),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhere we first make the $\\om\\leftrightarrow -\\om'$\ntransformation, and then use\n(\\ref{ident2}). We can then rewrite Eq. (\\ref{piforms}) for $\\Pi_{\\rho\\rho}$\nin the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{intermed3}\n\\Pi_{\\rho\\rho} &=& T\\sum_{\\om}\\left({UU'-uu'-1\\over UU'}\\right)\n\\left[{U+U'\\over\\Om(u'-u)}-{1\\over\n\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)}\\right]\\nonumber\\\\\n&=& {T\\over\\Om}\\sum_{\\om}\\left({UU'-uu'-1\\over UU'}\\right)\n{(U+U')\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)\n-(u-u')\\Om\\over (u'-u)\n\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)}.\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nFrom the identity (\\ref{ident2}) we see that the \nnumerator of (\\ref{intermed3}) can be rewritten as\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n&&(U+U')\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)\n-(u'-u)\\Om\\nonumber\\\\\n&=& (U+U')^2-(u'-u)(u'-u-\\Om)-(u'-u)\\Om\\nonumber\\\\\n&=& (U+U')^2-(u'-u)^2,\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand inserting the identity (\\ref{ident2}) into the first factor in\n(\\ref{intermed3}), we get\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Pi_{\\rho\\rho}= -{T\\over 2\\Om}\\sum_{\\om}\n{[(U'-U)^2-(u'-u)^2][(U+U')^2-(u'-u)^2]\\over (u'-u)UU'\n\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nMultiplying out the numerator yields\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n&&[(U'-U)^2-(u'-u)^2][(U+U')^2-(u'-u)^2]\\nonumber\\\\\n&=& (u'^2-u^2)^2-2(u'-u)^2(U^2+U'^2)+(u'-u)^4\\nonumber\\\\\n&=& (u'-u)^2[(u+u')^2-2u^2-2u'^2-4+(u'-u)^2]\\nonumber\\\\\n&=& -4(u'-u)^2,\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nfrom which it follows that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\Pi_{\\rho\\rho}(0,\\Om)=\n-{2T\\over\\Om}\\sum_{\\om}{u'-u\\over UU'\n\\left(U+U'-\\zeta\\left[{UU'-uu'-1\\over UU'}\\right]\\right)}\n=-{2\\over\\Om}\\Pi_{\\phi\\rho}(0,\\Om),\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\ncompleting our proof of the result (\\ref{piident2}).\n\n\\medskip\n\\section{Evaluating Derivatives of $\\Pi_{ij}$ with respect to $\\De$}\n\\medskip\n\nIn this appendix we evaluate the derivatives of the polarization bubbles\n$\\Pi_{ij}$ with respect to the order parameter $\\De$ so that we may\nevaluate the first order correction to the order parameter\nself-consistency equation. The formulas for the $\\Pi_{ij}$ are given in\nEqn. (\\ref{pialla}), and we see that the derivative can operate either on the\ncoherence factor or the denominator present in these expressions.\n\\par\nThe difficulty in evaluating these derivatives arises because $u(\\om)$\nsatisfies the transcendental equation\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{\\om\\over\\De}=u\\left[1-{1\\over\\De\\tau_s}{1\\over (u^2+1)^{1/2}}\\right],\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nfrom which it follows that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n-{\\om\\over\\De^2}={\\pa u\\over\\pa\\De}\\left[1-{1\\over\\De\\tau_s}\n{1\\over (u^2+1)^{3/2}}\\right]+{1\\over\\De^2\\tau_s}{u\\over (u^2+1)^{1/2}},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nand thus\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{\\pa u\\over\\pa\\De}=-{u\\over\\De}\\left[1-{1\\over\\De\\tau_s}\n{1\\over (u^2+1)^{3/2}}\\right]^{-1},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwith a similar result for $u'$.\n\\par\nWe first consider the effect of $\\pa/\\pa\\De$ on the coherence factors\npresent in the $\\Pi_{ij}$. We see that it suffices to evaluate the\nderivatives\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{diffs}\n{\\pa\\over\\pa\\De}\\left\\{ {uu'\\over UU'}\\quad;\\quad {1\\over UU'}\n\\quad;\\quad {u'-u\\over UU'} \\right\\}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe first term in Eqn. (\\ref{diffs}) gives\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n\\label{diff1}\n{\\pa\\over\\pa\\De}\\left[{uu'\\over UU'}\\right]=\n{u'\\over U'}\\left[{1\\over U}-{u^2\\over U^3}\\right]{\\pa u\\over\\pa\\De}\n+(u\\leftrightarrow u')\n= -{1\\over\\De}{uu'\\over U^3U'}\\left[1-{\\zeta\\over U^3}\\right]^{-1}\n+(u\\leftrightarrow u').\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nSince this expression will occur inside a sum over $\\om$, and will\nmultiply an expression that is invariant under the transformation\n$\\om\\leftrightarrow -\\om'$, we see that the two terms in Eqn. (\\ref{diff1})\nwill give equal results. Thus\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{\\pa\\over\\pa\\De}\\left[{uu'\\over UU'}\\right]\\equiv\n-{2\\over\\De}{uu'\\over U^3U'}\\left[1-{\\zeta\\over U^3}\\right]^{-1}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe second term in Eqn. (\\ref{diffs}) gives\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{\\pa\\over\\pa\\De}\\left[{1\\over UU'}\\right] = -{1\\over U'}{u\\over U^3}\n{\\pa u\\over\\De}+(u\\leftrightarrow u')\n\\equiv {2\\over\\De}{u^2\\over UU'}\\left[1-{\\zeta\\over U^3}\\right]^{-1},\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nwhilst the third term in Eqn. (\\ref{diffs}) gives\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{\\pa\\over\\pa\\De}\\left[u'-u\\over UU'\\right] =\n-{u'\\over U'}{u\\over U^3}{\\pa u\\over\\pa\\De} -\n{1\\over U'}{1\\over U^3}{\\pa u\\over\\pa\\De}+(u\\leftrightarrow u')\n\\equiv {2\\over\\De}{u(uu'+1)\\over U^3U'}\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe effect of $\\pa/\\pa\\De$ on the coherence factors can then be\nsummarised in the form\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n\\label{coheqn}\n{\\pa\\over\\pa\\De}\\left[{uu'\\pm 1\\over UU'}\\right]\n&\\equiv& -{2\\over\\De}{u(u'\\mp u)\\over U^3U'}\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\nonumber\\\\\n{\\pa\\over\\pa\\De}\\left[{u'-u\\over UU'}\\right]\n&\\equiv& {2\\over\\De}{u(uu'+1)\\over U^3U'}\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}.\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\par\nNext we must consider the effect of $\\pa/\\pa\\De$ on the two\ndenominators\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\nD_{\\pm}=Dq^2+\\De U+\\De U'-{1\\over\\tau_s}\n\\left(1-{uu'\\mp 1\\over UU'}\\right).\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe last term on the RHS is a coherence factor, and its derivative can\nbe read off from Eqn. (\\ref{coheqn}) above. The only terms then to consider\nare $\\De U$ and $\\De U'$, which, of course, will give identical results\nafter summation over $\\om$. We see that\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{\\pa\\over\\pa\\De}(\\De U) = U + \\De {u\\over U^3}{\\pa u\\over\\pa\\De}\n= U - {u^2\\over U^3}\\left[1-{\\zeta\\over U^3}\\right]^{-1}\n= {1\\over U}\\left[1-{\\zeta\\over U}\\right]\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nFrom this we obtain the final result\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{equation}\n{\\pa\\over\\pa\\De}D_{\\pm} = \\left\\{ {1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'-u)\\over UU'}\\right)\\right\\}\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}.\n\\end{equation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\par\nHaving now evaluated the action of $\\pa/\\pa\\De$ on all the components of\nthe polarization bubbles, $\\Pi_{ij}$, we can now write down the results for\nthe $\\pa\\Pi_{ij}/\\pa\\De$,\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eqnarray}\n{\\pa\\Pi_{\\De\\De}\\over\\pa\\De} &=& -2\\pi N(0)T\\sum_\\om \\left\\{\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times\n{1\\over\\De}{u(u'+u)\\over U^3U'}{1\\over D_{+}}+\n\\left(1+{uu'-1\\over UU'}\\right)\\left\\{{1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'-u)\\over UU'}\\right)\\right\\}{1\\over D_{+}^2}\\right\\}\\nonumber\\\\\n{\\pa\\Pi_{\\phi\\phi}\\over\\pa\\De} &=& -2\\pi N(0)T\\sum_\\om \\left\\{\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times\n{1\\over\\De}{u(u'-u)\\over U^3U'}{1\\over D_{-}}+\n\\left(1+{uu'+1\\over UU'}\\right)\\left\\{{1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'+u)\\over UU'}\\right)\\right\\}{1\\over D_{-}^2}\\right\\}\\nonumber\\\\\n{\\pa\\Pi_{\\rho\\rho}\\over\\pa\\De} &=& -2\\pi N(0)T\\sum_\\om \\left\\{\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times\n{1\\over\\De}{u(u'-u)\\over U^3U'}{1\\over D_{-}}-\n\\left(1-{uu'+1\\over UU'}\\right)\\left\\{{1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'+u)\\over UU'}\\right)\\right\\}{1\\over D_{-}^2}\\right\\}\\nonumber\\\\\n{\\pa\\Pi_{\\phi\\rho}\\over\\pa\\De} &=& -2\\pi N(0)T\\sum_\\om \\left\\{\n\\left[1-{\\zeta\\over U^3}\\right]^{-1}\\times\n{1\\over\\De}{u(uu'+1)\\over U^3U'}{1\\over D_{-}}-\n\\left({u'-u\\over UU'}\\right)\\left\\{{1\\over U}-{\\zeta\\over U^2}\n\\left(1+{u(u'+u)\\over UU'}\\right)\\right\\}{1\\over D_{-}^2}\\right\\}.\n\\end{eqnarray}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{references}\n\n\\bibitem{Fink94} A useful review of the whole area can be found in:\nA.M. Finkel'stein, Physica {\\bf 197B}, 636 (1994).\n\n\\bibitem{RLCM} H.R. Raffy, R.B. Laibowitz, P. Chaudhari and S. Maekawa,\nPhys. Rev. B {\\bf 26}, 6607 (1983).\n\n\\bibitem{GB} J.M. Graybeal and M.R. Beasley, Phys. Rev. B {\\bf 29},\n4167 (1984).\n\n\\bibitem{HLG} D.B. Haviland, Y. Liu and A.M. Goldman, Phys. Rev. Lett.\n{\\bf 62}, 2180 (1989).\n\n\\bibitem{LK} S.J. Lee and J.B. Ketterson, Phys. Rev. Lett. {\\bf 64},\n3078 (1990)\n\n\\bibitem{HP} A.F. Hebard and M.A. Paalanen, Phys. Rev. B {\\bf 30},\n4063 (1984).\n\n\\bibitem{OKOK} S. Okuma, F. Komori, Y. Ootuka and S. Kobayashi,\nJ. Phys. Soc. Jpn. {\\bf 52}, 3269 (1983).\n\n\\bibitem{VDG} J.M. Valles Jnr, R.C. Dynes and J.P. Garno, Phys. Rev. B\n{\\bf 40}, 6680 (1989); Phys. Rev. Lett. {\\bf 69}, 3567 (1992).\n \n\\bibitem{Vall94} J.M. Valles Jnr, S-Y Hsu, R.C. Dynes and J.P. Garno,\nPhysica {\\bf 197B}, 522 (1994).\n\n\\bibitem{Fink87} A.M. Finkel'stein, Pis'ma Zh. Eksp. Teor. Fiz. {\\bf 45}, 37\n(1987) [JETP Lett. {\\bf 45}, 46 (1987)].\n\n\\bibitem{SRW} R.A. Smith, M.Y. Reizer and J.W. Wilkins, Phys. Rev. B\n{\\bf 51}, 6470 (1995).\n\n\\bibitem{OF} Y. Oreg and A.M. Finkel'stein, Phys. Rev. Lett.\n{\\bf 83}, 191 (1999).\n\n\\bibitem{Belitz} D. Belitz, Phys. Rev. B {\\bf 35}, 1636 (1987); {\\bf 35},\n1651 (1987); {\\bf 40}, 111 (1989).\n\n\\bibitem{CV} J.A. Chervenak and J.M. Valles Jnr, Phys. Rev. B\n{\\bf 51}, 11977 (1995).\n\n\\bibitem{Grif} V. Ambegaokar and A. Griffin, Phys. Rev.\n{\\bf 137}, A1151 (1964), Appendix A.\n\n\\bibitem{PWA} P.W. Anderson, J. Phys. Chem. Solids {\\bf 11}, 26 (1959).\n\n\\bibitem{AG} A.A. Abrikosov and L.P. Gor'kov, Zh. Eksp. Teor. Fiz. {\\bf 39},\n1781 (1961) [Sov. Phys. JETP {\\bf 12}, 1243 (1961)].\n\n\\bibitem{EP88} U. Eckern and F. Pelzer, J. Low. Temp. Phys. {\\bf 73},\n433 (1988).\n\n\\bibitem{MWH} N.D. Mermin and H. Wagner, Phys. Rev. Lett. {\\bf 17},\n1133 (1966); P.C. Hohenberg, Phys. Rev. {\\bf 158}, 383 (1967).\n\n\\bibitem{Fink83} A.M. Finkel'stein, Zh. Eksp. Teor. Fiz. {\\bf 84}, 168 (1983)\n[Sov. Phys. JETP {\\bf 57}, 97 (1983)]; Z. Phys. B: Condens. Matter {\\bf 56},\n189 (1984).\n\n\\bibitem{Dev96} T.P. Devereaux and D. Belitz, Phys. Rev. B {\\bf 53}, 359 (1996)\n.\n\n\\bibitem{Namb60} Y. Nambu, Phys. Rev. {\\bf 117}, 648 (1960).\n\n\\bibitem{Amb61} V. Ambegaokar and L.P. Kadanoff, Nuovo Cimento {\\bf 22},\n914 (1961).\n\n\\bibitem{Schr} J.R. Schrieffer, {\\it Theory of Superconductivity}\n(Perseus, Reading MA, 1964), Chap 8.\n\n\\bibitem{Kad61} G. Baym and L.P. Kadanoff, Phys. Rev. {\\bf 124}, 287 (1961).\n\n\\bibitem{PWA58} P.W. Anderson, Phys. Rev. {\\bf 112}, 1900 (1958).\n\n\\bibitem{Rick58} G. Rickayzen, Phys. Rev. {\\bf 111}, 817 (1958).\n\n\n\\end{references}\n\n\\newpage\n\n\\centerline{\\bf FIGURES}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig1.eps,width=16cm}}\n\\medskip\n\\caption{The electron Green function in a superconductor.\n(a) Self-energy for a clean superconductor. The wiggly line is\nthe BCS interaction. (b) Extra self-energy diagram needed for\ndirty superconductor.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig2.eps,width=14cm}}\n\\medskip\n\\caption{Definition of the screened potential $V_{ij}$\nin terms of the polarization bubble $\\Pi_{ij}$ and bare\npotential $V^0_{ij}$.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig3.eps,width=8cm}}\n\\medskip\n\\caption{Definition of the polarization bubbles $\\Pi_{ij}$.\n(a) The geometric series for the ladder $\\Pi$.\n(b) the geometric series for the vertex function $\\Pi_j$ which\nis obtained from $\\Pi$ by taking the trace at one end with a Pauli\nmatrix.\n(c) The polarization bubble $\\Pi_{ij}$ is obtained from the vertex\noperator $\\Pi_j$ by taking the trace with a Pauli matrix at the open\nend.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig4.eps,width=12cm}}\n\\medskip\n\\caption{The first-order correction to the grand canonical potential.\nThis has the form of a ``string of bubbles'', where the wiggly lines\ncan be either the bare Coulomb or BCS interaction, and the bubbles are\nany of the non-zero polarization bubbles.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig5.eps,width=16cm}}\n\\medskip\n\\caption{The equivalent Eliashberg-like self-energy diagrams for the\ncorrection to the order parameter $\\De$. (a) The two-ladder diagrams\nare obtained by differentiating the diffusion propagator term in the\n$\\Pi_{ij}$ with respect to $\\De$. (b) The one-ladder diagrams are obtained\nby differentiating the coherence factor term in $\\Pi_{ij}$ with respect\nto $\\De$. (c) No three-ladder terms are obtained by differentiating\n$\\Om_1(\\De)$ with respect to $\\De$, and direct calculation of these\ndiagrams shows that they equal zero.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig6.eps,width=12cm}}\n\\medskip\n\\caption{The first-order correction to the pair propagator. The value\nof temperature at which this first diverges is the transition temperature\n$T_c$.\n(a) Definition of pair propagator in terms of BCS interaction and\npair polarization bubble.\n(b) Zeroth-order (mean field) pair polarization bubble.\n(c) The 7 diagrams which contribute to the first-order correction to\nthe pair polarization bubble. The wiggly line is the screened Coulomb\ninteraction, whilst the spring-like line is the pair propagator, as defined\nin (a).}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig9.eps,width=16cm}}\n\\medskip\n\\caption{Diagrammatic equation for the scattering amplitude matrix\n$\\Gamma(\\om_n,\\om_l)$. Block $\\gamma$ is the BCS interaction.\nBlock $t\\Lambda$ is the correction to the effective interaction\ncaused by the interplay of Coulomb interaction and disorder.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig7a.eps,width=9cm}\\hskip 0.25truein\n\\psfig{figure=fig7b.eps,width=9cm}}\n\\medskip\n\\caption{Plots of transition temperature as a function of resistance\nper square and spin-flip scattering rate. The plot on the left shows\n$T_c$ as a function of $R_{\\square}$ for values of (top to bottom)\n$\\alpha=1/2\\pi T_{co}\\tau_s$ equal to $0$, $0.2$, $0.4$, $0.6$ and $0.8$\ntimes the critical value $\\alpha_0$. We see that the $\\alpha=0$ curve\nhas a re-entrance problem, but that the situation improves for finite\n$\\alpha$. The circles are the results from the non-perturbative resummation\ntechnique. We see that they are roughly in agreement with the perturbation\ntheory. The plot on the right is of $T_c$ as a function of spin-flip\nscattering measured in the dimensionless form $\\alpha/\\alpha_0$ for values\nof $R_{\\square}$ equal to (top to bottom)\n$0\\Om$, $500\\Om$, $1000\\Om$, $1500\\Om$ and $2000\\Om$.\nThe circles are the non-perturbative resummation results, and are again\nin good agreement with perturbation theory except for the $2000\\Om$ curve.\nThis might be expected since this curve is very close to the\nsuperconductor-insulator transition.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\centerline{\\psfig{figure=fig8a.eps,width=8cm}\\hskip 0.25truein\n\\psfig{figure=fig8b.eps,width=8cm}}\n\\medskip\n\\caption{Plot of pair-breaking rate per impurity versus resistance\nper square of film. We see that this is roughly constant, only increasing\nvery near to the superconductor-insulator transition, with a variation of\nonly 10\\% over the whole range. The curve on the left is from perturbation\ntheory; the curve on the right from the non-perturbative resummation.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\end{document}\n\n\n"
}
] |
[
{
"name": "cond-mat0002035.extracted_bib",
"string": "\\bibitem{Fink94} A useful review of the whole area can be found in:\nA.M. Finkel'stein, Physica {\\bf 197B}, 636 (1994).\n\n\n\\bibitem{RLCM} H.R. Raffy, R.B. Laibowitz, P. Chaudhari and S. Maekawa,\nPhys. Rev. B {\\bf 26}, 6607 (1983).\n\n\n\\bibitem{GB} J.M. Graybeal and M.R. Beasley, Phys. Rev. B {\\bf 29},\n4167 (1984).\n\n\n\\bibitem{HLG} D.B. Haviland, Y. Liu and A.M. Goldman, Phys. Rev. Lett.\n{\\bf 62}, 2180 (1989).\n\n\n\\bibitem{LK} S.J. Lee and J.B. Ketterson, Phys. Rev. Lett. {\\bf 64},\n3078 (1990)\n\n\n\\bibitem{HP} A.F. Hebard and M.A. Paalanen, Phys. Rev. B {\\bf 30},\n4063 (1984).\n\n\n\\bibitem{OKOK} S. Okuma, F. Komori, Y. Ootuka and S. Kobayashi,\nJ. Phys. Soc. Jpn. {\\bf 52}, 3269 (1983).\n\n\n\\bibitem{VDG} J.M. Valles Jnr, R.C. Dynes and J.P. Garno, Phys. Rev. B\n{\\bf 40}, 6680 (1989); Phys. Rev. Lett. {\\bf 69}, 3567 (1992).\n \n\n\\bibitem{Vall94} J.M. Valles Jnr, S-Y Hsu, R.C. Dynes and J.P. Garno,\nPhysica {\\bf 197B}, 522 (1994).\n\n\n\\bibitem{Fink87} A.M. Finkel'stein, Pis'ma Zh. Eksp. Teor. Fiz. {\\bf 45}, 37\n(1987) [JETP Lett. {\\bf 45}, 46 (1987)].\n\n\n\\bibitem{SRW} R.A. Smith, M.Y. Reizer and J.W. Wilkins, Phys. Rev. B\n{\\bf 51}, 6470 (1995).\n\n\n\\bibitem{OF} Y. Oreg and A.M. Finkel'stein, Phys. Rev. Lett.\n{\\bf 83}, 191 (1999).\n\n\n\\bibitem{Belitz} D. Belitz, Phys. Rev. B {\\bf 35}, 1636 (1987); {\\bf 35},\n1651 (1987); {\\bf 40}, 111 (1989).\n\n\n\\bibitem{CV} J.A. Chervenak and J.M. Valles Jnr, Phys. Rev. B\n{\\bf 51}, 11977 (1995).\n\n\n\\bibitem{Grif} V. Ambegaokar and A. Griffin, Phys. Rev.\n{\\bf 137}, A1151 (1964), Appendix A.\n\n\n\\bibitem{PWA} P.W. Anderson, J. Phys. Chem. Solids {\\bf 11}, 26 (1959).\n\n\n\\bibitem{AG} A.A. Abrikosov and L.P. Gor'kov, Zh. Eksp. Teor. Fiz. {\\bf 39},\n1781 (1961) [Sov. Phys. JETP {\\bf 12}, 1243 (1961)].\n\n\n\\bibitem{EP88} U. Eckern and F. Pelzer, J. Low. Temp. Phys. {\\bf 73},\n433 (1988).\n\n\n\\bibitem{MWH} N.D. Mermin and H. Wagner, Phys. Rev. Lett. {\\bf 17},\n1133 (1966); P.C. Hohenberg, Phys. Rev. {\\bf 158}, 383 (1967).\n\n\n\\bibitem{Fink83} A.M. Finkel'stein, Zh. Eksp. Teor. Fiz. {\\bf 84}, 168 (1983)\n[Sov. Phys. JETP {\\bf 57}, 97 (1983)]; Z. Phys. B: Condens. Matter {\\bf 56},\n189 (1984).\n\n\n\\bibitem{Dev96} T.P. Devereaux and D. Belitz, Phys. Rev. B {\\bf 53}, 359 (1996)\n.\n\n\n\\bibitem{Namb60} Y. Nambu, Phys. Rev. {\\bf 117}, 648 (1960).\n\n\n\\bibitem{Amb61} V. Ambegaokar and L.P. Kadanoff, Nuovo Cimento {\\bf 22},\n914 (1961).\n\n\n\\bibitem{Schr} J.R. Schrieffer, {\\it Theory of Superconductivity}\n(Perseus, Reading MA, 1964), Chap 8.\n\n\n\\bibitem{Kad61} G. Baym and L.P. Kadanoff, Phys. Rev. {\\bf 124}, 287 (1961).\n\n\n\\bibitem{PWA58} P.W. Anderson, Phys. Rev. {\\bf 112}, 1900 (1958).\n\n\n\\bibitem{Rick58} G. Rickayzen, Phys. Rev. {\\bf 111}, 817 (1958).\n\n\n"
}
] |
cond-mat0002036
|
Dynamics of Activated Escape, and Its Observation in a Semiconductor Laser
|
[
{
"author": "J. Hales$^{(a)}$"
},
{
"author": "A. Zhukov$^{(b)}$"
},
{
"author": "R. Roy$^{(c)}$"
},
{
"author": "and M.I.~Dykman$^{(b)}$\\cite{byline1}"
}
] |
[
{
"name": "dropout1.tex",
"string": "%\\documentstyle[aps,preprint,epsf]{revtex}\n\\documentstyle[prl,aps,twocolumn,epsf]{revtex}\n\\hyphenation{pre-fac-tor}\n%\\topmargin 5pt\n\\begin{document}\n\\draft \n\\twocolumn[ \n\\hsize\\textwidth\\columnwidth\\hsize\\csname@twocolumnfalse\\endcsname \n\n\\title{Dynamics of Activated Escape, and Its Observation in a\nSemiconductor Laser}\n\n\\author{J. Hales$^{(a)}$,\nA. Zhukov$^{(b)}$, R. Roy$^{(c)}$, and M.I.~Dykman$^{(b)}$\\cite{byline1}} \n\n\\address{$^{(a)}$ CREOL, University of Central Florida,\nOrlando, Fl 32816\\\\\n$^{(b)}$Department of Physics and Astronomy,\nMichigan State University, East Lansing, Michigan 48824\\\\\n$^{(c)}$Department of Physics, University of Maryland, College\nPark, MD 20742}\n\n\n\\date{\\today} \n\\maketitle\n\\widetext \n\\begin{quote}\nWe report a direct experimental observation and provide a theory of\nthe distribution of trajectories along which a fluctuating system\nmoves over a potential barrier in escape from a metastable state. The\nexperimental results are obtained for a semiconductor laser with\noptical feedback. The distribution of paths displays a distinct peak,\nwhich shows how the escaping system is most likely to move. We argue\nthat the specific features of this distribution may give an insight\ninto the nature of dropout events in lasers.\n\\end{quote}\n\\pacs{PACS numbers: 05.40.-a, 42.60.Mi, 05.20.-y, 42.55.Px}\n] \n\\narrowtext \n\n%\\end{center}\n%\\section{Introduction}\n\nFluctuation-induced escape plays an important role in many physical\nphenomena, from traditionally studied diffusion in solids and protein\nfolding to\nswitching in lasers \\cite{Willemsen,Yacomotti}, resonantly driven\ntrapped electrons \\cite{Gabrielse}, and systems which display\nstochastic resonance \\cite{RPP,SR}. In the analysis of escape, it is\nimportant to be able not only to calculate, but also to control the\nescape probability. To do this one has to know how the system {\\it\nmoves} when it escapes.\n\nEscape is an example of a large fluctuation. If fluctuations are\nsmall on average, for most of the time the system wanders near the\ninitially occupied metastable state $q_a$ and only occasionally moves\nfar away from it. The central idea of the theory of large\nfluctuations is that paths to a remote state $q_f$ lie within a narrow\ntube centered at an {\\it optimal} path to this state $q_{\\rm\nopt}(t|q_f,t_f)$ \\cite{Onsager,Grareview}, where $t_f$ is the instant\nof reaching $q_f$. Optimal paths reveal determinism of motion in large\nfluctuations. They can be observed by analyzing the prehistory\nprobability density (PPD) $p_h(q,t|q_f,t_f)$ for a system to have\npassed through a point $q$ at time $t$ provided the system had been\nfluctuating about the stable state for a long time and reached $q_f$\nat time $t_f$. For given $t_f-t>0$, $p_h(q,t|q_f,t_f)$ should peak\nfor $q$ lying on $q_{\\rm opt}(t|q_f,t_f)$ \\cite{prehistory}. The\nsharply peaked PPDs have indeed been observed, but so far only in\nanalog and digital simulations \\cite{RPP}, and for points $q_f$ lying\ninside the attraction basin of $q_a$\n%, i.e. prior to escape will have occurred \n\\cite{private}.\n\n\nIn the present paper we analyze the dynamics of the system during\nescape and, using a semiconductor laser with optical feedback, provide\na direct experimental observation of the prehistory distribution. This\ndistribution displays a distinct peak, as seen from Fig.~2 below. We\nshow that such peak arises even for final states lying {\\it behind}\nthe boundary of the domain of attraction to the initially occupied\nmetastable state, e.g. behind the top of the potential barrier in\nFig.~1.\n% although there is {\\it no} uniquely defined\n%most probable overbarrier trajectory. \nWe reveal qualitative features of the PPD, relate them to escape\ndynamics, and compare the theoretical and experimental results.\n\nIn the analysis of escape dynamics we will use a simple model of an\none-variable overdamped system which performs Brownian motion in a\nmetastable potential $U(q)$, with equation of motion\n\n\\begin{equation}\n\\label{Langevin}\n\\dot q = -U'(q) + \\xi(t), \\; \\langle\n\\xi(t)\\xi(t')\\rangle = 2D\\delta (t-t').\n\\end{equation}\n\n\\noindent\nHere, $\\xi(t)$ is zero-mean white Gaussian noise. We assume that the\nnoise intensity is small compared to the height of the potential\nbarrier, $D\\ll \\Delta U$, where $\\Delta U=U(q_b)-U(q_a)$, see Fig.~1\n($q_a$ and $q_b$ are the positions of the local minimum and maximum of\n$U(q)$). In this case the escape rate $W\\propto \\exp(-\\Delta U/D)$ is\nsmall compared to the characteristic reciprocal relaxation time\n$t_r^{-1} = U^{\\prime\\prime}(q_a)$.\n%and in a broad time range $t_r\\ll t\\ll 1/W$\n%the intrawell distribution of the system is quasistationary.\n\n%%--------------------------------------------------------------------\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=3.0in %so many inches wide\n\\leavevmode\\epsfbox{fig1.eps}\n\\vspace{-0.1in}\n\\end{center}\n\n\\caption{The positions of the maxima of the prehistory probability\ndensity with respect to the coordinate $q$ for given time $t$ (solid\nline), and with respect to $t$ for given $q$ (triangles). The data of\nsimulations refer to a Brownian particle (\\ref{Langevin}),\n(\\ref{potential}), the noise intensity $D=1/60$ ($\\Delta U/D=10$),\n$q_f=1.2$, and $t_r=1$. The dashed line shows the asymptotic results\n(\\ref{harmonic}), (\\ref{inside_2}) for $t_m(q)$. Inset: escape from a\npotential well; the motion in the regions A-D is discussed in the\ntext.}\n\\end{figure}\n%%--------------------------------------------------------------------\n\n\nOn its way to a point $q_f$ behind the barrier, the system is expected\nto move differently in the four regions shown in Fig.~1. In the region\nA behind the barrier top it should move nearly along the noise-free\ntrajectory $\\dot q = -U'$.\n%\nIn the region B near the barrier top, where $|q-q_b|\\alt l_D$ ($l_D =\n(2D/\\lambda)^{1/2}$ is the diffusion length, $\\lambda =\n|U^{\\prime\\prime}(q_b)|$), the influence of noise becomes\nsubstantial. The motion is diffusive and is controlled by\naverage-strength fluctuations. The system stays here for the Suzuki\ntime $ t_{_{\\rm S}}=\\lambda^{-1}\\ln |q_b-q_a|/l_D$ \\cite{Suzuki}.\n%\nIn the region C ($|q-q_b|\\gg l_D$) the system is driven by the noise\n$\\xi(t)$ against the regular force $-U^{\\prime}$, which requires a\nstrong outburst of noise. For Gaussian noise, the probabilities of\ndifferent appropriate realizations of $\\xi(t)$ differ from each other\nexponentially strongly. Therefore there is an optimal realization of\nnoise, which is much more probable than others. It corresponds to an\noptimal path of the system $q_{\\rm opt}$. For fluctuations from the\nattractor to an intrawell state such path is given by \\cite{Grareview}\n\n\\begin{equation}\n\\label{opt_path}\n\\dot q_{\\rm opt} = U^{\\prime}(q_{\\rm opt}).\n\\end{equation}\n%\n%In the present case it can be expected to be the same. \nIn the region D near the attractor, $|q-q_a|\\alt (Dt_r)^{1/2}$, the\nsystem performs small fluctuations before a large fluctuation leading\nto escape occurs.\n\nDiffusive motion near the barrier top $q_b$ gives rise to a strong\nbroadening of the distribution of fluctuational paths. If the\ndestination point $q_f$ approaches $q_b$ from inside the well, the\ndistribution width diverges in the bounce-type approximation\n\\cite{prehistory}. As we show, the divergence\ndisappears if one goes beyond this approximation. The analytic\nsolution will be obtained\n%using the standard assumption\n%(cf. \\cite{RPP,Grareview}) that $U(q)$ is parabolic near the fixed\n%points $q_{a,b}$ over distances exceeding the diffusion length,\n%$|U(q_b\\pm l_D) - U(q_b) - U''(q_{b})]l_D^2/2|\\ll D$ (and similarly\n%for $q_a$). In fact the matching requires a stronger inequality\nassuming that $\\ln[\\Delta U/D]\\gg 1$. This condition is {\\it not}\nneeded for the physical picture of the escape dynamics to apply, as we\ndemonstrate experimentally and through simulations.\n\nFor a Markov system (\\ref{Langevin}), the PPD can be written as\n\n\\begin{eqnarray}\n\\label{definition}\np_h(q,t|q_f,t_f)={\\rho(q_f,t_f|q,t)\\rho(q,t,|q_i,t_i)\n\\over \\rho(q_f,t_f|q_i,t_i)}\n\\end{eqnarray}\n\n\\noindent\nwhere $\\rho(q_1,t_1|q_2,t_2)$ is the probability density of the\ntransition from $q_2$ at the instant $t_2$ to $q_1$ at the instant\n$t_1$ ($t_1>t_2$). We choose the initial instant $t_i$ so that $\nW^{-1} \\gg t_f-t_i > t-t_i \\gg t_r$. In this time range the system\nforgets its the initial intrawell state $q_i$. The statistical\ndistribution inside and outside the well (not too far from the barrier\ntop) is quasistationary, $\\rho(q,t|q_i,t_i) = \\rho(q)$, and can be\neasily calculated.\n\n\n%Behind the barrier top, the distribution $\\rho(q)$ is determined by\n%the stationary outgoing current, $\\rho(q) = W/|U'(q)| \\;{\\rm for}\\;\n%q-q_b\\gg l_D$. Deep inside the potential well $\\rho(q)$ is given by\n%the Boltzmann distribution,\n%whereas near the barrier top\n%where $r(q)=\\exp(\\tilde q^2)\\left[1-{\\rm erf}(\\tilde q)\\right]/2,\\;\n%\\tilde q= (q-q_b)/l_D$, and \n%$Z=[2\\pi D/U^{\\prime\\prime}(q_a)]^{1/2}$ (we set $U(q_a)=0$).\n\nThe prehistory distribution $p_h$ has a simple form for $q$ and $q_f$\nlying behind the barrier top $q_b$ in the region A in Fig.~1. For\nbrevity, we give $p_h(q,t|q_f,t_f)$ in the case where $q,q_f$ are both\nin the range where $U(q)$ is {\\it parabolic} near $q_b$, but $q_f$ is\nfar enough behind $q_b$, $q_f-q_b\\gg l_D$. The transition\nprobability density $\\rho(q_f,t_f|q,t)$ for such $q,q_f$ is known\n\\cite{Kampen}, and from (\\ref{definition})\n\n\\begin{equation}\n\\label{harmonic}\np_h(q,t|q_f,0)= (2 z_f/l_D)r(q)e^{\\lambda t}\n\\exp\\left[-(z-z_fe^{\\lambda t})^2\\right],\n\\end{equation}\n\n\\noindent \n(we have set $t_f=0$). Here, $z \\equiv z(q)=[1-\\exp(2\\lambda\nt)]^{-1/2}\\tilde q$, $z_f \\equiv z(q_f)$ (note that $t< 0$),\n$r(q)=\\exp(\\tilde q^2)\\left[1-{\\rm erf}(\\tilde q)\\right]/2$, and\n$\\tilde q= (q-q_b)/l_D$.\n\n\nFor $|t|\\alt \\lambda^{-1}$, the distribution (\\ref{harmonic}) has a\nsharp Gaussian peak as a function of $q$, with width $\\propto\nl_D$. Behind the barrier, the peak lies on the noise-free trajectory\n$\\dot q = U^{\\prime}(q)=-\\lambda (q-q_b)$, which arrives at $q_f$ for\n$t=0$.\n\n%The shape of the PPD (\\ref{harmonic}) is qualitatively different on\n\nInterestingly, the PPD peak remains {\\it sharp}, with width $\\sim l_D$, even\nwhere its maximum reaches the barrier top, which happens for $t=\nt(q_b) = -\\lambda^{-1}\\ln[\\pi^{1/2}(q_f-q_b)/l_D]$.\n%However, it becomes non-Gaussian in $q$.\n\nFor earlier times $-t>-t(q_b)$,\n% before reaching $q_f$, \nthe system is mostly on the {\\it intrawell} side of the barrier, and\nfor large $|t/t(q_b)|$ the peak of $p_h$ as a function of $q$ moves\naway from the harmonic range. Of interest is the position $t_m(q)$ of\nthe peak of $p_h$ as a function of time for given $q$. It shows when\nthe particle was most likely to pass through the point $q$ before\narriving at $q_f$. Inside the well, for $q_b-q \\gg l_D$, the time\n$t_m$ and the integral width of the PPD $\\gamma (q)= 1/p_h(q,t_m)$ are\nof the form\n\n\\begin{eqnarray}\n\\label{int_2}\n\\lambda t_m= -\\ln[2(q_b-q)(q_f-q_b)/l_D^2],\\; \n\\gamma = e|q_b-q|.\n\\end{eqnarray}\n\n\\noindent\nFrom (\\ref{int_2}), $t_m$ depends on $q_b-q$ logarithmically. In\ncontrast, $\\gamma(q)$ grows linearly with $q_b-q$. It becomes {\\it\nparametrically} larger than the distribution width $\\gamma\\sim l_D\n\\propto D^{1/2}$ at the barrier top and outside the well.\n\nFar from the barrier top in the region C in Fig.~1, the motion of the\nsystem is determined by large fluctuations against the force\n$-U^{\\prime}(q)$. \n%Therefore $t_m(q)$ may be expected to change with\n%$q$ so as to stay on the optimal fluctuational path (\\ref{opt_path})\n%to the point $q$, $q_{\\rm opt}(t_m)=q$, in which case $dt_m/dq =\n%1/U'(q)$. From (\\ref{int_2}), close to the barrier top $t_m$ indeed\n%obeys this equation.\nIn this region, $p_h(q,t|q_f,t_f)$ can be obtained using the\nSmoluchowski equation which follows from (\\ref{definition}),\n\n\\begin{equation}\n\\label{convolution}\np_h(q,t|q_f,t_f)=\\int dq'\\,p_h(q,t|q',t')\\,\np_h(q',t'|q_f,t_f).\n\\end{equation}\n\n\\noindent\nIt is convenient to choose $t'$ in (\\ref{convolution}) so that the\nmajor contribution to the integral over $q'$ came from $q'$ lying on\nthe internal side of the barrier close to $q_b$ and yet away from the\ndiffusion region, $q_b-q_a \\gg q_b-q' \\gg l_D$. Then the second\nintegrand in (\\ref{convolution}) is given by (\\ref{harmonic}). \n%As a\n%function of $q,t$, the first integrand was discussed earlier\n%\\cite{prehistory}. It peaks on the optimal path (\\ref{opt_path}). The\n%duration of motion along the optimal path from $q$ to $q'$ is $\\sim\n%t_r\\sim 1/\\lambda$. This suggests that we choose $t'-t\\sim 1/\\lambda$,\n%and consider the range $\\exp[\\lambda(t_f-t')] \\gg \\exp[\\lambda\n%|t(q_b)|]$ where the peak of $p_h(q',t'|q_f,t_f)$ (\\ref{harmonic}) is\n%inside the well.\n\n\n\nThe distribution $p_h(q,t|q',t')$ as a function of $q'$ can be\nobtained from (\\ref{definition}) by solving, in the eikonal\napproximation, the Fokker-Planck equation for $\\rho(q',t'|q,t) =\n\\exp[-S(q',t'-t|q,0)/D]$. To zeroth order in $D$, $S$ satisfies a\nHamilton-Jacobi equation for the action of an auxiliary dynamical\nsystem \\cite{Grareview}. An appropriate Hamiltonian trajectory of this\nsystem gives the optimal path $q_{\\rm opt}(t'-t|q,0)$ for a\nfluctuation in which the original system (\\ref{Langevin}) starts at\nthe point $q$ and moves further away from the attractor\n\\cite{q_opt_ref}. This path is given by Eq.~(\\ref{opt_path}). The\nmajor contribution to the integral (\\ref{convolution}) comes from the\npoints $q'$ which lie close to this path. For small $\\delta q'=\nq'-q_{\\rm opt}(t'-t|q,0)$, it suffices to keep quadratic in $\\delta\nq'$ terms in $S$, and then $p_h(q,t|q',t')$ is Gaussian in $\\delta\nq'$. In the appropriately chosen parameter range,\n%$|q_b-q_{\\rm opt}(t'-t)| \\gg |q_f-q_b|\\exp[\\lambda (t'-t_f)]$, \nthe time $t'$ drops out from\n(\\ref{convolution}), and one obtains \\cite{to_be_published}\n\n\n\\begin{eqnarray}\n\\label{inside_2}\n&&p_h(q,t|q_f,0) = \\left|\\lambda/U^{\\prime}(q)\\right|M\\exp(-M),\n\\end{eqnarray}\n\n\\noindent\nwhere $M=M(q,t)= -\\lambda(q_f-q_b) \\left[q_{\\rm\nopt}(-t|q,0)-q_b\\right]/D$.\n%(it is assumed that $\\exp(-\\lambda t)\\sim \\lambda t_S $).\n\nEq.~(\\ref{inside_2}) describes the distribution of trajectories along\nwhich the escaping system moves inside the well. This distribution has\na distinct peak. For given $q$, the peak is located for $M(q,t_m(q))=\n1$. From (\\ref{opt_path}), \n%$\\partial M/\\partial t = \\lambda M$ for\n%characteristic times $-t\\sim t_S$. Therefore \nthe position of the peak\nobeys the equation $dt_m/dq = 1/U'(q)$. This means that, inside the\npotential well, the particle is most likely to move along the optimal\npath (\\ref{opt_path}). In a multi-dimensional system, the peak of\n$p_h$ will lie on the most probable escape path, which goes from the\nattractor to the saddle point. \n%We note that, in the direction\n%transverse to the path, the peak is narrow, with width $\\propto\n%D^{1/2}$, whereas along the path it becomes broad inside the well\n%\\cite{to_be_published}.\n\nThe distribution (\\ref{inside_2}) is strongly asymmetric, both in $q$\nand $t$. The integral width\n\n\\begin{equation}\n\\label{int_3}\n\\gamma(q) = 1/p_h(q,t_m|q_f,0) =e|U^{\\prime}(q)|/\\lambda\n\\end{equation}\n\n\\noindent\nis {\\it independent} of the noise intensity and is {\\it nonmonotonic}\nas a function of $q$. It is maximal for $U^{\\prime\\prime}(q)=0$ where\nthe velocity along the optimal path is maximal. \n%This happens because, although \nThe broadening of the tube of escape paths in time comes largely from\nthe area near the barrier top. However, it is ``amplified'' as it is\ncarried away by the trajectories flow, and therefore it is maximal where\nthe flow is most fast.\n\n%%--------------------------------------------------------------------\n\\begin{figure}%color\n\\begin{center}\n\\epsfxsize=2.9in %so many inches wide\n\\leavevmode\\epsfbox{fig2.eps}\n\\vspace{-0.1in}\n\\end{center}\n\\caption{(color) The prehistory probability distribution of the scaled\nradiation intensity $I$ for experimentally observed dropout events in\na semiconductor laser. Inset: the PPD for a Brownian particle,\nobtained from simulations for the same parameters as in Fig.~1.}\n\\end{figure}\n\n%%--------------------------------------------------------------------\n\n\nAs $-t$ increases further, the peak of the distribution\n(\\ref{inside_2}) approaches the diffusion region $|q-q_a|\\sim\n(Dt_r)^{1/2}$ near the potential minimum, and the peak width\n(\\ref{int_3}) again shrinks down. For large $-t$, the PPD\n(\\ref{definition}) goes over into the stationary distribution\n$\\rho(q)$, which has a nearly Gaussian peak at $q_a$ with variance\n$Dt_r/2$. \n%The crossover from (\\ref{inside_2}) to $\\rho(q)$ will be\n%discussed elsewhere \\cite{to_be_published}.\n\nWe note that, in the most interesting region C, the positions of the\nmaxima of $p_h$ (\\ref{inside_2}) with respect to $q$ for given $t$ and\nwith respect to $t$ for given $q$ are different. This indicates that\nthere is no well-defined most probable escape path in space and time,\nwhich would go from the metastable state all the way over the barrier\ntop. Still one can tell when the escaped system passed, most\nprobably, through a given point, and where the system was most\nprobably located at a given time.\n\n\nThe discussed qualitative features of the prehistory distribution can\nbe seen from the results of digital simulations for the model\npotential\n\n\\begin{equation}\n\\label{potential}\nU(q)=q^2/2 - q^3/3,\n\\end{equation}\n\n\\noindent\nThe data were obtained using a standard algorithm \\cite{Mannella}, and\nrefer to 8000 events.\n\nA distinct peak of the simulated $p_h$ is seen in Fig.~2. The peak of\n$p_h$ as a function of $q$ changes with increasing $|t|$ from a narrow\nGaussian near $q_f$ to a broad and asymmetric between $q_a$ and $q_b$,\nand then again to a comparatively narrow Gaussian near $q_a$. The\npositions of the peak of $p_h$ with respect to time, $t_m(q)$, and\ncoordinate, $q_m(t)$, are compared in Fig.~1. Both curves are close to\neach other. Outside the well they practically coincide and closely\nfollow the noise-free path of the system, $dt_m/dq = -1/U'(q)$. The\nmotion displays a characteristic slowing down near the barrier top.\nInside the well the peak moves close to the optimal fluctuational\npath $\\dot q = U'(q)$.\nThe distribution $p_h$ becomes time-independent for large $|t|$. Therefore\n$t_m(q)$ is well-defined only for $q$ not too close to the potential\nminimum $q_a$.% Respectively is shown only for not too small $|q-q_a|$.\n\nThe data of simulations in Figs.~1, 2 refer to the noise intensities\n$D/\\Delta U=0.1$, where the asymptotic analytical theory applies only\nqualitatively. In particular, the expressions (\\ref{harmonic}) and\n(\\ref{inside_2}) for $p_h$ in different ranges of $q$ do not merge\nwith each other smoothly, as seen from Fig.~1. However, there is good\nqualitative agreement between the analytical and numerical data,\nincluding the position of the peak and the integral width of the PPD.\n\n\nNumerical results on the standard deviation of the PPD $\\sigma$ for\ntwo noise intensities are shown in Fig.~3. \n%The data refer to the peak of $p_h$\n%as a function of $q$ for given $t$. \nAs expected, the distribution width reaches its maximum well inside\nthe well, near the inflection point $U''(q)=0$. For higher $D$, the\nmaximum is less pronounced.\n%, and the value of $\\sigma$ for large $|t|$ is higher.\n%We note that the width is {\\it smaller} than predicted by\n%the asymptotic theory based on linearization about an optimal\n%path.\n\n\n%%--------------------------------------------------------------------\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=2.6in %so many inches wide\n\\leavevmode\\epsfbox{fig3.eps}\n\\vspace{-0.1in}\n\\end{center}\n\\caption{Standard deviation and the position of the maximum of the\nprehistory distribution at given time, for simulated Brownian motion\n(left panel) and a semiconductor laser with optical feedback (right\npanel). Solid and dashed lines on the left panel refer to $\\Delta\nU/D=10$ and 3, respectively. The scales in the panels are different\n(see the text).}\n\\end{figure}\n\n%%--------------------------------------------------------------------\n\nThe experimental observation of the prehistory distribution was made\nusing a semiconductor laser with optical feedback. The setup was\nsimilar to that used before \\cite{Roy_Hohl} and consisted of a\ntemperature-stabilized laser diode and a remote flat-surface\nmirror. The feedback could be controlled by a variable attenuator\nbetween them. Near the solitary laser threshold, such a system is\nunstable: after some time of nearly steady operation the radiation\nintensity drops down, then it comparatively fast recovers to the\noriginal value, then drops down again, etc. In the experiment, the\nintensity output was digitized, with time resolution 1~ns. To obtain\nthe prehistory distribution, the intensity records were superimposed\nbackward in time, starting from the instant at which the intensity, on\nits way down, reached a certain level (10\\% above the extreme dropout\npoint). The PPD obtained from 1512 events for the feedback 15.63\\%\nis shown in Fig.~2.\n\n\n\nThe mechanism of power dropouts is vividly discussed in the literature\n\\cite{Roy_Hohl,Yacomotti,Huyet,Vaschenko,Hohl}. Most authors agree\nthat the role of noise in this effect is crucial. A simple model\n\\cite{Henry} describes the dropouts in terms of activation escape of\nthe light intensity $I$ over a potential barrier with shape\n(\\ref{potential}). Previous observations \\cite{Roy_Hohl} were in\nagreement with this model, which motivated us to measure the\nprehistory distribution for dropout events.\n\nIt is seen from Figs.~2 and 3 that \n%there is an agreement between \nthe results of the observations agree with major qualitative results\non noise-induced escape. The experimental PPD displays a distinct\npeak. The shape of this peak is similar to the shape of the PPD of a\nnoise-driven system, with the light intensity $I$ playing the role of\nthe coordinate $q$ ($I$ is scaled by its metastable value). The peak\nis narrow at small time $|t|$, and displays a characteristic\nbroadening at intermediate times. For larger $|t|$, the peak becomes\ntime-independent. From the data in Figs.~2 and 3, the relaxation time\nof the system is $t_r\\approx 2$~ns. From the value of the escape rate\n$W\\approx 5\\times 10^{-3}$~ns$^{-1}$ found in \\cite{Roy_Hohl}, it\nfollows that, for the model ~(\\ref{potential}), $\\Delta U/D \\approx\n3$. Using an estimate $\\sigma_0\\approx\n(D/U^{\\prime\\prime}(q_a))^{1/2}$ for $\\sigma$ at large $-t$, one can\nestimate the difference $6(\\Delta U/D)^{1/2}\\sigma_0$ in the light\nintensity $I$ at the minimum and maximum of the potential\n(\\ref{potential}). It then follows from Fig.~3b that the system goes\nthrough the potential maximum for $t\\sim -4$~ns, i.e. the width\n$\\sigma$ reaches its maximum near the potential maximum. In\ncombination with larger $\\sigma_{\\max}/\\sigma_0$ compared to that in\nFig.~3a for the same $\\Delta U/D$, this indicates that the model\n(\\ref{Langevin}), (\\ref{potential}) is oversimplified. However, the\noverall form of the PPD seen from the data provides an important\nargument in favor of the stochastic model of dropout events. We expect\nthat it will be possible to use high-resolution data on the prehistory\ndistribution in order to establish a quantitative model of the system.\n\n\nIn conclusion, we have analyzed the dynamics of a system in activated\nescape and revealed its distinctive features related to the occurrence\nof optimal paths and to the motion slowing down near a barrier\ntop. The escape trajectories lie within a well-defined tube, and the\nsystem is most likely to go through a cross-section of this tube at a\nwell-defined time before it is found behind the barrier. For the first\ntime, a tube of escape trajectories has been observed in experiment,\nby analyzing dropout events in a semiconductor laser.\n\nThe work at MSU was partly supported from the NSF funded MRSEC and the\nNSF grants no. PHY-9722057 and no. DMR-9809688.\n\n%\\input{dropout1.ref}\n\\begin{thebibliography}{99}\n\\bibitem[\\dagger]{byline1} e-mail: dykman@pa.msu.edu\n\n\\bibitem{Willemsen} R. Roy, R. Short, J. Durnin, and L. Mandel,\nPhys. Rev. Lett. {\\bf 45}, 1486 (1980); M. B. Willimsen {\\it et al.},\n%, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, \nPhys. Rev. Lett. {\\bf 82}, 4815 (1999).\n\n\n\n\\bibitem{Yacomotti} A. M. Yacomotti {\\it et al.},\n%, M. C. Eguia, J. Aliaga, O. E. Martinez, and G. B. Mindlin, \nPhys. Rev. Lett. {\\bf 83}, 292 (1999).\n\n\n\\bibitem{Gabrielse} L.J. Lapidus, D. Enzer, and G. Gabrielse,\nPhys. Rev. Lett. {\\bf 83}, 899 (1999); M.I. Dykman, C.M. Maloney,\nV.N. Smelyanskiy, and M. Silverstein, Phys. Rev. E {\\bf 57}, 5202\n(1998).\n\n\\bibitem{RPP} D.G. Luchinsky, P.V.E. McClintock, and M.I. Dykman,\nRep. Progr. Phys. {\\bf 61}, 889 (1998).\n\n\\bibitem{SR} M.I. Dykman {\\it et al.}, Nuovo Cimento D {\\bf 17}, 661\n(1995); L. Gammaitoni {\\it et al.}, Rev. Mod. Phys. {\\bf 70}, 223\n(1998); R.D. Astumian and F. Moss, Chaos {\\bf 8}, 533 (1998);\nK. Wiesenfeld and F. Jaramillo, Chaos {\\bf 8}, 539 (1998).\n\n\\bibitem{Onsager} L. Onsager and S. Machlup, Phys. Rev. {\\bf 91}, 1505\n(1953).\n\n\\bibitem{Grareview} M.~I. Freidlin and A.~D. Wentzel, {\\em Random\nPerturbations in Dynamical Systems} (Springer, New-York, 1984);\nR. Graham, in {\\it Noise in Nonlinear Dynamical Systems}, edited by\nF. Moss and P.V.E. McClintock (Cambridge University, Cambridge, 1989),\nv.~1, p.~225; \nM.I. Dykman, M.A. Krivoglaz, and S.M. Soskin, {\\it ibid.}, v.~2, p.~347.\n% R.S. Maier and D.L. Stein, J. Stat. Phys. {\\bf \n%83}, 291 (1996).\n\n\n\\bibitem{prehistory} M.I. Dykman {\\it et al.},\n%, P.V.E. McClintock, V.N. Smelyanskiy, N.D. Stein, and N.G.~Stocks, \nPhys. Rev. Lett. {\\bf 68}, 2718 (1992).\n\n\\bibitem{private} A peak in the prehistory distribution for $q_f$\nlying at the top of a potential barrier, was seen in analog\nsimulations (P.V.E. McClintock, private communication).\n\n\\bibitem{Suzuki} M. Suzuki, J. Stat. Phys. {\\bf 16}, 11, 477 (1977).\n\n\n\n%\\bibitem{more_prehistory} M.I. Dykman, D.G. Luchinksy, P.V.E. McClintock,\n%and V.N. Smelyanskiy, Phys. Rev. Lett. {\\bf 77}, 5229 (1996);\n%M. Morillo, J.M. Casado, and J. G\\'{o}mez-Ord\\'{o}\\~{n}ez,\n%Phys. Rev. E {\\bf 54}, 1 (1996); {\\bf 55}, 1521 (1997). \n%M.I. Dykman and V.N. Smelyanskiy, \n%Superlattices and Microstructures {\\bf 23}, 495 (1998).\n\n\\bibitem{Kampen} N.G. van Kampen, {\\it Stochastic Processes in Physics\nand Chemistry} (Elsevier, Amsterdam 1981).\n\n\\bibitem{q_opt_ref} In contrast to the present case, in the standard\nanalysis \\cite{RPP,Grareview} of interest are the optimal paths that start\nfrom the attractor and {\\it end} at a given state.\n\n\n\\bibitem{to_be_published} A. Zhukov and M.I. Dykman, in preparation.\n\n\n\n\\bibitem{Mannella} R. Mannella, in {\\em Supercomputation in Nonlinear\nand Disordered Systems}, edited by L. V\\'{a}zquez, F. Tirando, and\nI. Martin (World Scientific, Singapore, 1997), p.~ 100.\n\n\n\\bibitem{Roy_Hohl} A. Hohl, H.J.C. van der Linden, and R. Roy,\nOpt. Lett. {\\bf 20}, 2396 (1995).\n\n\\bibitem{Huyet} G. Huyet {\\it et al.}, \n%S. Hegarty, M. Giudici, B. de Bruyn, and J. G. McInerney,\nEurophys. Lett. {\\bf 40}, 619\n(1997); G. Huyet {\\it et al.}, Opt. Comm. {\\bf 149}, 341 (1998).\n\n\\bibitem{Vaschenko} G. Vaschenko {\\it et al.}, \n%M. Giudici, J. J. Rocca, C. S. Menoni, J. R. Tredicce, and S. Balle, \nPhys. Rev. Lett. {\\bf 81}, 5536 (1998).\n\n\\bibitem{Hohl} A. Hohl and A. Gavrielides, Phys. Rev. Lett. {\\bf 82},\n1148 (1999).\n\n\n\\bibitem{Henry} C.H. Henry and R.F. Kazarinov, IEEE J. Quantum\nElectron. {\\bf QE-22}, 294 (1986). \n\n\n\\end{thebibliography}\n\n%\\end{document}\n\n%\\bibitem{blowtorch} R. Landauer, J. Stat. Phys. {\\bf 53}, 233 (1988).\n\n\n\\end{document}\n\n\n"
}
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[
{
"name": "cond-mat0002036.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem[\\dagger]{byline1} e-mail: dykman@pa.msu.edu\n\n\\bibitem{Willemsen} R. Roy, R. Short, J. Durnin, and L. Mandel,\nPhys. Rev. Lett. {\\bf 45}, 1486 (1980); M. B. Willimsen {\\it et al.},\n%, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, \nPhys. Rev. Lett. {\\bf 82}, 4815 (1999).\n\n\n\n\\bibitem{Yacomotti} A. M. Yacomotti {\\it et al.},\n%, M. C. Eguia, J. Aliaga, O. E. Martinez, and G. B. Mindlin, \nPhys. Rev. Lett. {\\bf 83}, 292 (1999).\n\n\n\\bibitem{Gabrielse} L.J. Lapidus, D. Enzer, and G. Gabrielse,\nPhys. Rev. Lett. {\\bf 83}, 899 (1999); M.I. Dykman, C.M. Maloney,\nV.N. Smelyanskiy, and M. Silverstein, Phys. Rev. E {\\bf 57}, 5202\n(1998).\n\n\\bibitem{RPP} D.G. Luchinsky, P.V.E. McClintock, and M.I. Dykman,\nRep. Progr. Phys. {\\bf 61}, 889 (1998).\n\n\\bibitem{SR} M.I. Dykman {\\it et al.}, Nuovo Cimento D {\\bf 17}, 661\n(1995); L. Gammaitoni {\\it et al.}, Rev. Mod. Phys. {\\bf 70}, 223\n(1998); R.D. Astumian and F. Moss, Chaos {\\bf 8}, 533 (1998);\nK. Wiesenfeld and F. Jaramillo, Chaos {\\bf 8}, 539 (1998).\n\n\\bibitem{Onsager} L. Onsager and S. Machlup, Phys. Rev. {\\bf 91}, 1505\n(1953).\n\n\\bibitem{Grareview} M.~I. Freidlin and A.~D. Wentzel, {\\em Random\nPerturbations in Dynamical Systems} (Springer, New-York, 1984);\nR. Graham, in {\\it Noise in Nonlinear Dynamical Systems}, edited by\nF. Moss and P.V.E. McClintock (Cambridge University, Cambridge, 1989),\nv.~1, p.~225; \nM.I. Dykman, M.A. Krivoglaz, and S.M. Soskin, {\\it ibid.}, v.~2, p.~347.\n% R.S. Maier and D.L. Stein, J. Stat. Phys. {\\bf \n%83}, 291 (1996).\n\n\n\\bibitem{prehistory} M.I. Dykman {\\it et al.},\n%, P.V.E. McClintock, V.N. Smelyanskiy, N.D. Stein, and N.G.~Stocks, \nPhys. Rev. Lett. {\\bf 68}, 2718 (1992).\n\n\\bibitem{private} A peak in the prehistory distribution for $q_f$\nlying at the top of a potential barrier, was seen in analog\nsimulations (P.V.E. McClintock, private communication).\n\n\\bibitem{Suzuki} M. Suzuki, J. Stat. Phys. {\\bf 16}, 11, 477 (1977).\n\n\n\n%\\bibitem{more_prehistory} M.I. Dykman, D.G. Luchinksy, P.V.E. McClintock,\n%and V.N. Smelyanskiy, Phys. Rev. Lett. {\\bf 77}, 5229 (1996);\n%M. Morillo, J.M. Casado, and J. G\\'{o}mez-Ord\\'{o}\\~{n}ez,\n%Phys. Rev. E {\\bf 54}, 1 (1996); {\\bf 55}, 1521 (1997). \n%M.I. Dykman and V.N. Smelyanskiy, \n%Superlattices and Microstructures {\\bf 23}, 495 (1998).\n\n\\bibitem{Kampen} N.G. van Kampen, {\\it Stochastic Processes in Physics\nand Chemistry} (Elsevier, Amsterdam 1981).\n\n\\bibitem{q_opt_ref} In contrast to the present case, in the standard\nanalysis \\cite{RPP,Grareview} of interest are the optimal paths that start\nfrom the attractor and {\\it end} at a given state.\n\n\n\\bibitem{to_be_published} A. Zhukov and M.I. Dykman, in preparation.\n\n\n\n\\bibitem{Mannella} R. Mannella, in {\\em Supercomputation in Nonlinear\nand Disordered Systems}, edited by L. V\\'{a}zquez, F. Tirando, and\nI. Martin (World Scientific, Singapore, 1997), p.~ 100.\n\n\n\\bibitem{Roy_Hohl} A. Hohl, H.J.C. van der Linden, and R. Roy,\nOpt. Lett. {\\bf 20}, 2396 (1995).\n\n\\bibitem{Huyet} G. Huyet {\\it et al.}, \n%S. Hegarty, M. Giudici, B. de Bruyn, and J. G. McInerney,\nEurophys. Lett. {\\bf 40}, 619\n(1997); G. Huyet {\\it et al.}, Opt. Comm. {\\bf 149}, 341 (1998).\n\n\\bibitem{Vaschenko} G. Vaschenko {\\it et al.}, \n%M. Giudici, J. J. Rocca, C. S. Menoni, J. R. Tredicce, and S. Balle, \nPhys. Rev. Lett. {\\bf 81}, 5536 (1998).\n\n\\bibitem{Hohl} A. Hohl and A. Gavrielides, Phys. Rev. Lett. {\\bf 82},\n1148 (1999).\n\n\n\\bibitem{Henry} C.H. Henry and R.F. Kazarinov, IEEE J. Quantum\nElectron. {\\bf QE-22}, 294 (1986). \n\n\n\\end{thebibliography}"
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cond-mat0002037
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Disorder Induced Transitions in Layered Coulomb Gases and Superconductors
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[
{
"author": "Baruch Horovitz{$^1$} and Pierre Le Doussal{$^2$}"
}
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A 3D layered system of charges with logarithmic interaction parallel to the layers and random dipoles is studied via a novel variational method and an energy rationale which reproduce the known phase diagram for a single layer. Increasing interlayer coupling leads to successive transitions in which charge rods correlated in $N>1$ neighboring layers are nucleated by weaker disorder. For layered superconductors in the limit of only magnetic interlayer coupling, the method predicts and locates a disorder-induced defect-unbinding transition in the flux lattice. While $N=1$ charges dominate there, $N>1$ disorder induced defect rods are predicted for multi-layer superconductors.
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[
{
"name": "prl16.tex",
"string": "\n%\\documentstyle[prl,aps,epsf,multicol,amssymb]{revtex}\n%%%%%%%next gives one column but with errors%%%%%%%%\n%\\documentstyle[12pt,prl,aps]{revtex}\n%%%%%%%%%%%%%next gives two column%%%%%%%%%\n\\documentstyle[prl,aps,epsf,multicol]{revtex}\n%%%%%%%%%%%%next gives bigger spaces (can do 2.5 or 3.0 alternatively%%%%%%%\n%\\renewcommand{\\baselinestretch}{2.0}\n%\\draft\n\\begin{document}\n\n\n%%%%%%%%%%%FIGURES%%%%%%%%%%%%%%%%%%\n%\n\\newcommand{\\fig}[2]{\\epsfxsize=#1\\epsfbox{#2}}\n% \n%\n%%%%%%%%%DEUX COLONNES%%%%%%%%%%%%%%\n% \n\\newcommand{\\passage}{%%\n\\end{multicols}\\widetext\\noindent\\rule{8.8cm}{.1mm}%\n \\rule{.1mm}{.4cm}} \n \\newcommand{\\retour}{%%\n % \\hspace{.2cm}\n\\noindent\\rule{9.1cm}{0mm}\\rule{.1mm}{.4cm}\\rule[.4cm]{8.8cm}{.1mm}%\n \\begin{multicols}{2} }\n \\newcommand{\\unecol}{\\end{multicols}}\n \\newcommand{\\deuxcol}{\\begin{multicols}{2}}\n%\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\n%\n\\newcommand{\\beq}{\\begin{equation}}\n\\newcommand{\\eeq}{\\end{equation}}\n\\newcommand{\\beqa}{\\begin{eqnarray}}\n\\newcommand{\\eeqa}{\\end{eqnarray}}\n%\n%\\cst {\\rm Cst}\n\n%\\setcounter{page}{1}\n\n\n\\tolerance 2000\n\n\n\n\\author{Baruch Horovitz{$^1$} and Pierre Le Doussal{$^2$} } \n\\address{{$^1$} Department of Physics, Ben Gurion university, Beer Sheva\n84105 Israel}\n\\address{{$^2$}CNRS-Laboratoire de Physique Th{\\'e}orique de\nl'Ecole Normale Sup{\\'e}rieure,\n24 rue Lhomond,75231 Cedex 05, Paris France.}\n\n\\title{Disorder Induced Transitions in Layered Coulomb Gases \nand Superconductors}\n% \\date{\\today}\n\\maketitle\n\n\n\\begin{abstract}\nA 3D layered system of charges with logarithmic interaction parallel\nto the layers and random dipoles is studied via\na novel variational method and an energy rationale which reproduce the \nknown phase diagram for a single layer. Increasing interlayer coupling\nleads to \nsuccessive transitions in which charge rods correlated in $N>1$\nneighboring \nlayers are nucleated by weaker disorder. \nFor layered superconductors in the limit of\nonly magnetic interlayer coupling, the method \npredicts and locates a disorder-induced\ndefect-unbinding transition in the flux lattice.\nWhile $N=1$ charges dominate there, $N>1$ disorder induced \ndefect rods are predicted for multi-layer superconductors.\n\\end{abstract}\n\n%\\pacs{to be added}\n\n%\\narrowtext\n\n\n\\deuxcol\n\n%\\renewcommand{\\baselinestretch}{2.0}\n\nTopological phase transitions induced by quenched disorder\nare relevant for numerous physical systems. Such transitions\nare likely to shape the phase diagram of type II superconductors.\nIt was proposed \\cite{tgpldbragg} that the flux lattice (FL) remains\na topologically ordered Bragg glass at low field,\nunstable to the proliferation of dislocations \nabove a threshold disorder or field, providing one\nscenario for the controversial \"second peak\" line \n\\cite{Kes,speakexp}. Another scenario \\cite{H2}\nis based on a disorder-induced decoupling transition (DT)\nresponsible for a sharp drop in the FL tilt modulus.\nFurthermore, for the {\\it pure} system, it was shown recently\n\\cite{Dodgson} that in the absence of Josephson coupling,\npoint \"pancake\" vortices, i.e vacancies and interstitials in the FL,\nare nucleated at a temperature $T_{def}$, distinct from\nmelting above some field. It is believed that this\npure system topological transition merges with the \nthermal DT \\cite{Daemen,H1} once the Josephson coupling \nis finite, being two anisotropic limits of the\nsame transition \\cite{H3} (at which superconducting order\nis destroyed while FL positional correlations are\nmaintained). Thus an interesting possibility is that a similar, \nbut now disorder-induced, vacancy-interstitial unbinding\ntransition can be demonstrated in 3D layered \nsuperconductors, relevant to many layered \nand multilayer materials \\cite{Kes,Bruynseraede}.\n\nIn 2D recent progress was made to describe\ndisorder induced topological transitions,\nin terms of Coulomb gases of charges with\nlogarithmic long range interactions. It was shown\n\\cite{nattermann95,scheidl97,tang96,dcpld} that quenched \nrandom dipoles lead to a transition, via defect proliferation,\nat a finite threshold disorder, even at $T=0$.\n\nIn this Letter we develop a theory for a\n3D defect-unbinding transition in presence of disorder.\nIt is achieved for systems which can be mapped onto a\nlayered Coulomb gas with quenched random dipoles, in which the\ninteraction energy between two charges on layers $n$ and $n'$ is \n$2 J_{n-n'}\\ln r$ with $r$ the charge separation parallel to the layers.\nOne physical realization is the FL\nin layered superconductors \\cite{Kes,Bruynseraede,Blatter}\nwith only magnetic coupling, for which we predict \nand locate the vacancy-interstitial unbinding\ntransition. Indeed, as we argue, disorder induced deformations\nof the lattice result in random dipoles as seen by the defects.\nTo study this problem we develop an efficient variational\nmethod which allows for fugacity distributions,\nknown \\cite{dcpld} to be important in 2D as they become\nbroad at low $T$. We test the method on a\nsingle layer and reproduce the phase diagram, known from\nrenormalization group (RG) with a $T=0$ disorder threshold\n$\\sigma_{cr}=1/8$\n\\cite{footnote2}. For the 2-layer system we find that\nabove a critical anisotropy $\\eta \\equiv -J_1/J_0 = \\eta_c = 1-\n\\frac{1}{\\sqrt{2}}$\nthe single layer type transition is preempted by a transition induced\nby bound states of two pancake vortices on the two layers with\n$\\sigma_{cr}<1/8.$ We develop a $T=0$ energy rationale by an\napproximate mapping to a Cayley tree problem and find that it\nreproduces the 2-layer result. Extension to many\nlayers with only nearest layer coupling shows a\ncascade of transitions in which the number of correlated\ncharges on $N$ neighboring layers increases, while the critical\ndisorder decreases with $\\eta$, with $N\\rightarrow \\infty$,\n$\\sigma_{cr}\\rightarrow 0$ as $\\eta \\rightarrow 1/2$. \nFinally we consider arbitrary range $n _0$ for $J_n$ with the constraint\n$\\sum_n J_n =0$, as appropriate for layered superconductors.\nFor $N > n _0^2$ states with $\\sigma_{cr} \\sim n_0^2/ N \\rightarrow 0$\nare possible but only at exponentially large length scales for\n$n _0 \\gg 1$. Thus for layered\nsuperconductors we expect that the N=1 state dominates and find\nits phase diagram. Varying the system parameters by forming\nmultilayers reduces $n _0$ and allows for realization of the new\n$N>1$ phases.\n\nWe study the Hamiltonian:\n\n\\begin{eqnarray}\n&& {\\cal H}= - \\case{1}{2} \\sum_{{\\bf r} \\neq {\\bf r}'}\\sum_{n,n'}2J_{n-n'}\n s_n({\\bf r}) \\ln ({\\bf r}-{\\bf r}') s_{n'}({\\bf r}') \\\\\n&& -\n \\sum_{{\\bf r},n} V_n({\\bf r}) s_n({\\bf r}) \\label{H}\n \\end{eqnarray}\n where $s_n({\\bf r})=\\pm 1, 0$ define the positions ${\\bf r}$ of charges\non the $n$-th layer, $V_n({\\bf r})$ is a disorder potential with long\nrange correlations $\\overline{V_n({\\bf q}) V_{n'}({-\\bf q})}\n= 4 \\pi \\sigma J_0^2 \\Delta_{n-n'}/q^2$ with $\\Delta_{0}=1$\n(the short distance cutoff being set to unity).\nFor simplicity we start with uncorrelated disorder from layer to layer\n$\\Delta_{n-n'} = \\delta_{nn'}$ with\n\\begin{eqnarray}\n \\overline{[V_n({\\bf r})-V_{n}({\\bf r}')]^2}= 4 \\sigma J_0^2\n \\ln|{\\bf r}-{\\bf r}'| \\label{corr}\n\\end{eqnarray}\nrepresenting quenched dipoles on each layer.\nAt $T=0$ the problem amounts to find minimal energy configurations \nof charges in a logarithmically correlated random potential. For a\nsingle layer it was studied either using \\cite{nattermann95,rem}\na ``random energy model'' (REM)\napproximation \\cite{footnote1}, or more accurately using a\nrepresentation in terms of directed polymers on a Cayley tree (DPCT)\n\\cite{tang96} shown to emerge \\cite{dcpld} (as a continuum\nbranching process) from the Coulomb gas RG \nof the single layer problem. Schematically, the tree has independent \nrandom potentials (Fig. 1)\n$v_i$ on each bond with variance $\\overline{ v_i^2}=2 \\sigma J_0^2$.\nAfter $l$ generations one has $\\sim e^{2 l}$ sites which are mapped\nonto a 2D layer, i.e. two points separated by $r \\sim e^{l}$ have\na common ancestor at the previous $l \\approx \\ln r$ generation.\nEach point ${\\bf r}$ has a unique path on the tree (DP) with\n$v_1,...,v_l$ potentials and is assigned a potential $V({\\bf\nr})=v_1+...+v_l$. Since all bonds previous to the common ancestor\nare identical $\\overline{[V({\\bf r})-V({\\bf r}')]^2}=\n2 \\sum_{i=1}^{l}\\overline{v_i^2}$\nreproducing (\\ref{corr}) on each layer. Exact\nsolution of the DPCT \\cite{ct} yields the energy gained\nfrom disorder $V_{min}=min_{\\bf r} V({\\bf r}) \n\\approx - \\sqrt{8 \\sigma} J_0 \\ln L$ for a volume $L^2$,\nwith only $O(1)$ fluctuations \\cite{dcpld}, i.e\n$- \\sqrt{8 \\sigma} J_0$ per generation $l=\\ln L$.\n\n\\begin{figure}[htb]\n\\centerline{ \\fig{6cm}{fig1.eps} }\n\\caption{{\\narrowtext Critical disorder values with only\nnearest neighbor coupling $J_1$ vs. the anisotropy $\\eta=-J_1/J_0$.\nTransitions between different $N$ phases are marked with arrows.\nInset: the Cayley tree representation (for $N=3$ neighboring layers)\nwith $+$ charges (at the tree endpoints) separated by $L^\\epsilon$\nalong the layers, and separated by $L$ from the $N=3$ $-$ charges.}}\n\\label{fig1}\n\\end{figure}\n\nOptimal energy configurations for $M$ coupled layers\nare constructed considering $N$ neighboring layers\nwith a $+,-$ pair on each layer and no charges on the other layers.\nWe can take $J_0>0$ and $J_{n \\neq 0} \\leq 0$ \nso that {\\em equal}\ncharges on different layers attract. The DPCT representation\nnow involves, on a single tree, $N$ $+$ polymers (each seeing different \ndisorder) and $N$ $-$ polymers (each seeing opposite disorder $-v_i$\nto their $+$ partner). A plausible configuration is that\nthe $+$ charges bind\nwithin a scale $L^{\\epsilon}$\n($0 \\le \\epsilon \\le 1$), so do the $-$ charges, while the $+$ to\n$-$ charge separations define the scale $L$. Its tree representation\n(Fig. 1) has $2N$ branches with $\\epsilon \\ln L$ generations,\ni.e. an optimal energy of $-2 N \\sqrt{8\\sigma}J_0\\epsilon \\ln L$.\nOn the scale between $L^{\\epsilon}$ and $L$ the $+$ charges \nact as a single charge with a potential $\\sum_{n=1}^N V_n({\\bf r})$\n(the $N$ polymers share the same branch) of variance $N\\sigma$\nhence the optimal energy is $-2\\sqrt{8N\\sigma}J_0 (1-\\epsilon)\\ln L$.\nThe total disorder energy is\n\\cite{footnote3}:\n\\begin{equation}\nE_{dis} \\approx -2J_0\\sqrt{8\\sigma}[\\epsilon N\n+(1-\\epsilon)\\sqrt{N}]\\ln L \\,.\n\\end{equation}\n\nThe competing interaction energy $E_{int}$ is the sum of the \none for the $+-$ pairs, $[2J_0N+4 \\sum_{n=1}^{N}J_n(N-n)]\\ln L$\nand for the $++/--$ pairs, $-4\\sum_{n=1}^N J_n(N-n)\\epsilon \\ln L$.\nThe total energy\n$E_{tot}=E_{dis}+E_{int}$ being linear in $\\epsilon$, its\nminimum is at either $\\epsilon=1$ or $\\epsilon=0$.\nSince $\\epsilon =1$ implies \nthat the $+$ charges unbind, it is\nsufficient to consider $\\epsilon=0$ with all $N \\geq 1$, i.e. a \nrod with $N$ correlated charges has energy\n(with $\\eta_n =-J_n/J_0$):\n\\begin{equation}\nE_{tot} = 2J_0N[1-2\\sum_{n=1}^N \\eta_n(1-\\frac{n}{N})\n-\\sqrt{\\frac{8\\sigma}{N}}]\n\\ln L \\,. \\label{Etot}\n\\end{equation}\nDisorder induces the $N$ vortex state at the critical value:\n\\begin{equation}\n\\sigma_{cr}=\\frac{N}{8}[1-2\\sum_{n=1}^N\\eta_n (1-\\frac{n}{N})]^2\n\\,.\n\\end{equation}\n(i.e. $E_{tot}=0$). Consider first only nearest neighbor coupling\n$\\eta_l=\\eta_1 \\delta_{l 1}$. Then $\\sigma_{cr}$ is minimal \nat $N=1$ with $\\sigma_{cr}=1/8$ if $\\eta _1<1-1/\\sqrt{2}$.\nFor larger anisotropies successive $N$ states form at \n$1/(1-2\\eta _1) = 1 + \\sqrt{N(N-1)} \\sim N$\nwith diverging $N$ as $\\eta _1 \\to \\frac{1}{2}$ (Fig 1)\n\\cite{footnote9}.\n\nConsider now $J_n$ of range $n_0$ constrained by\n$\\sum_n J_n=0$ as for the superconductor, e.g.\n$\\eta_n=\\eta_1 e^{-(n-1)/n_0}$ for which\n$\\sigma_{cr}=(1-e^{-N/n_0})/8N(1-e^{-1/n_0})$.\nFor $n_0 \\gg 1$, each $\\eta _{n\\ne 0}$ is small:\nfor $N \\lesssim n_0$ the lowest $\\sigma_{cr}$ is at $N=1$.\nHowever, the combined strength of $N \\approx n_0$ vortices being\nsignificant $\\sigma_{cr}$ has a maximum\nand decreases back to zero for $N > n_0$ as $\\sigma_{cr} \\approx n_0^2/8N$. \nHence $\\sigma_{cr} \\rightarrow 0$ as $N \\rightarrow\n\\infty$ and any small disorder seems to nucleate such vortices.\nThis is because the perfect screening of the zero mode\n$\\sum_n J_n=0$ implies that an infinite \ncharge rod has a vanishing $\\ln r$ interaction; hence a\nlogarithmically correlated disorder is always dominant.\n\nThe realization of the large $N$ rods\ndepends, however, on the type of thermodynamic limit.\nAdding to (\\ref{Etot}) the core energy $E_c N$ and\nminimizing yields a $N$-vortex scale\n\\begin{equation}\nL\\approx \\exp\n\\{E_c\\sqrt{N}/[2J_0(\\sqrt{8\\sigma}-\\sqrt{8\\sigma_{cr}})]\\}\n\\label{L}\\,.\n\\end{equation}\nHence as $\\sigma\\rightarrow 0$ such states are only\nachievable when $L/N$ diverges\nexponentially. Using $\\sigma_{cr}\\approx n_0^2/8N$,\nfor $N> n_0^2/8\\sigma$ the\nlowest scale $L$ in this range is achieved at $N=n_0^2/2\\sigma$\nand leads to a lower bound\n$L_{min}\\approx \\exp [E_c n_0/4J_0 \\sigma]$ for observing\nlarge $N$ states with a given $\\sigma <\\case{1}{8}$.\nFor layered superconductors $E_c/J_0 \\gg 1$ \\cite{footnote7} and $n_0 \\gg 1$\nand this large $N$ instability occurs at unattainable\nscales, thus $N=1$ dominates. One needs $n_0\\approx 2-3$,\nas in multilayers,\nto realize the $N>1$ states.\n\nTo substantiate these results we develop a variational method\nfor $M$ layers which allows for fugacity distributions,\nan essential feature in the one-layer problem.\nDisorder averaging (\\ref{H}) in Fourier using replicas yields:\n\\begin{eqnarray}\n&& \\beta {\\cal H}_r= \\case{1}{2 d^2} \\int_k \\int_q\ns_a({\\bf q},k)(G_0)_{ab}({\\bf q},k)s_b^*({\\bf q},k) \\label{Hr} \\\\\n&& + \\beta E_c \\sum _{{\\bf r},n} s_{na}^2({\\bf r}) \\nonumber\n\\end{eqnarray}\nwhere $(G_0)_{ab}({\\bf q},k)=(4\\pi /q^2)[g(k) \\delta_{a b}- \n\\sigma J_0^2 \\beta^2\n\\Delta(k)]$, $g(k)=\\beta J(k) = \n\\beta d \\sum_n J_n \\exp(ikdn)$, $d$ the interlayer\nspacing \\cite{footnote5} (for uncorrelated layers \n$\\Delta(k) = d$), $a,b=1,...,m$ are replica indices\nand $m\\rightarrow 0$ is to be carefully taken.\nIn transforming to a sine-Gordon Hamiltonian \\cite{H3}\nit is crucial to keep {\\it all} charge fugacities \\cite{dcpld},\nwhich yields:\n\\begin{eqnarray}\n&& \\beta {\\cal H}_{SG} =\\case{1}{2} \\int_{k q}\n\\chi _a({\\bf q},k)(G_0)^{-1}_{ab}\\chi^*_b({\\bf q},k) \\nonumber \\\\\n&& - \\sum_{{\\bf r}} \\sum_{{\\bf s} \\neq {\\bf 0}} Y[{\\bf s}]\n\\exp( i {\\bf s} \\cdot {\\mbox{\\boldmath $\\chi$}}({\\bf r}) ) \\quad .\n\\label{sg}\n\\end{eqnarray}\n>From now on ${\\bf s} = \\{ s_{na} \\}_{n=1,.M,a=1,.m}$\nis an integer vector both in layer label and replica \nspace (i.e. of length m M)\nof entries $0, \\pm 1$ and the summation is over all such non null\nvectors (also $\\chi({\\bf r}) \\equiv \\{ \\chi_{n,a}({\\bf r}) \\}$, \n${\\bf s} \\cdot {\\mbox{\\boldmath $\\chi$}}\n=\\sum_{na} s_{na} \\chi_{na}$).\nWe now look for the best gaussian approximation of (\\ref{sg})\nwith propagator $G^{-1}_{ab}({\\bf q},k)=(G_0)^{-1}_{ab}({\\bf q},k) +\n\\sigma _c(k) \\delta _{ab} + \\sigma_0(k)$. The bare fugacity \nbeing $Y[{\\bf s}]=\\exp{(-\\beta E_c \\sum_{n,a} s_{n,a}^2)}$\nthe naive approach would be to restrict to charges ${\\bf s}$ with a\nsingle non zero entry, leading to a uniform fugacity term\n$ - y \\sum_{{\\bf r},n,a} \\cos({\\mbox{\\boldmath $\\chi$}_{na}}({\\bf r}))$\nand a diagonal $k$-independent replica mass term. Instead we\nkeep {\\it all} composite charges ${\\bf s}$, which allow for\nvariational solutions with off diagonal and $k$-dependent replica mass terms.\nThis corresponds respectively to fluctuations of fugacity \nand $N >1$ charge rods being generated and becoming relevant\nas also seen from RG.\nThe variational free energy is ${\\cal F}_{var}={\\cal F}\n_0+\\langle {\\cal H}_{SG}-{\\cal H}_0\\rangle_0$ where\n$\\langle...\\rangle$ is an average using $\\beta {\\cal H}\n_0=\\case{1}{2}\\int_{{\\bf q},k}\\chi_a({\\bf q},k)G_{ab}({\\bf\nq},k)\\chi^*_b({\\bf q},k)$ and \n$\\beta {\\cal F}_0=-\\case{1}{2} \\text{tr} \\ln G$.\nThe Gaussian average \n$F[{\\bf s}] \\equiv Y[{\\bf s}]\n\\langle \\exp{i{\\bf s} \\cdot {\\mbox{\\boldmath\n$\\chi$}}({\\bf r})}\\rangle_0$ yields:\n\\begin{eqnarray}\n&& F[{\\bf s}] = \\exp\\{-\\case{1}{2 d^2} \n\\int_k \\sum_{ab} (\\tilde{G}_c(k) \\delta_{ab} - A(k)) \ns_a(k) s^*_b(k)\\}\n\\nonumber \\\\\n&& G_c(k) = g(k)\\ln[\\Lambda/(4\\pi g(k)\\sigma_c(k))] \\\\\n&& A(k) = \\sigma \\beta^2 J_0^2 \n\\Delta(k) (G_c(k)/g(k) - 1) + g(k)\\sigma_0(k)/\\sigma_c(k) \\nonumber\n\\end{eqnarray}\nwhere $s_a(k)=d \\sum_n s_{na} e^{i k n d}$, \n$\\tilde{G}_c(k)=G_c(k)+2 \\beta E_c d$, $\\Lambda$ the UV cutoff on $q^2$.\n$F_{var}$ is minimized by $\\sigma_c(k)\\delta_{ab}+\\sigma_0(k)=\n\\Lambda d^{-2} \\sum_{{\\bf s}} s_{a}(k) s_{b}^*(k) F[{\\bf s}]$.\nWriting the $A(k)$ term as an average over\n$M$ random gaussian fugacities ${w_{k}}$:\n\\begin{equation}\n\\exp{\\{\\case{1}{2}|\\sum_{a} s_a(k)|^2 A(k)\\}}=\n\\langle \\exp{w_{k}\\sum_{a} s_a(k)} \\rangle_{w}\n\\end{equation}\nwhere $\\langle...\\rangle_{w}=\\prod_k \\int...\ne^{-|w_{k}|^2/2 A(k)} d^2 w_{k}/\\sqrt{2\\pi A(k)}$,\nallows to perform the exact sum on replicas yielding\n$\\sum_{{\\bf s}} F[{\\bf s}] = \\langle Z^m\\rangle_{w}$\nwith $Z=\\sum_{\\{s_n=0,\\pm 1\\}}\n\\exp(- \\frac{1}{2 d^2} \\int_k \\tilde{G}_c(k) |s(k)|^2 \n+ \\frac{1}{d} \\int_k w_{k} s^*(k))$.\nThe variational equations for $m\\rightarrow 0$ become\n\\cite{us_inprep}\n\\begin{equation}\n\\sigma_c(k)= \\Lambda \\langle \\frac{\\partial^2 \\ln Z}{\\partial w_{k}\n\\partial w^*_{k}} \\rangle_{w} \\,; {\\mbox {\\hspace{2mm}}}\n\\sigma_0(k)= \\Lambda \\langle |\\frac{\\partial \\ln Z}{\\partial \nw_{k} }|^2 \\rangle_{w} \\label{consistency}\n\\end{equation}\nFor a single layer $k=0$ and $Z=1+e^{u+w}+e^{u-w}$,\n$2 u=-\\tilde{G}_c(0)/d$, $w=w_0$ is a trinomial.\n(\\ref{consistency}) can be solved for the critical line\nwhere $\\sigma_c(0) \\rightarrow 0$.\nThe phase diagram shown in Fig. 2 (full line) reproduces\nprecisely recent RG results. The variational scheme, allowing for\nall replica charges ${\\bf s}$, therefore treats disorder\ncorrectly. For two layers $k d=0,\\pi$\nwe need two fugacity distributions\n$w_0,w_\\pi$ and $Z$ is a \"ninomial\",\ni.e. $Z=1+$ eight exponentials involving\n$G_c(0)$, $G_c(\\pi)$. Focusing on the low $T$\nboundary, where $\\sigma_c(\\pi)\\sim\n[\\sigma_c(0)]^{\\alpha}\\rightarrow 0$ we find \\cite{us_inprep} either (i) $\\alpha\n=1$ for $\\eta_1 < \\eta_c = 1-1/\\sqrt{2}$, representing decoupled layers, or\n(ii) $\\alpha \\rightarrow \\infty$ for $\\eta_1\n>\\eta_c$, representing a $++$ bound states on the two\nlayers. \nThe $T=0$ energy rationale is therefore reproduced. The phase\ndiagram for two layers with $\\eta_c<\\eta<1/2$ is shown\nin Fig. 2 \\cite{footnote2}\n\n\\begin{figure}[htb]\n\\centerline{ \\fig{6cm}{fig2.eps} }\n\\caption{{\\narrowtext Phase diagram for the onset of \nthe $N=1,2$ instabilities for anisotropy $\\eta=0.35$. At low $T$\ntwo distinct transitions are possible, the first being\nto the rod $N=2$ phase. At high $T$\nthe independent layer $N=1$ transition dominates}}\n\\label{fig1}\n\\end{figure}\n\n\nFor any number of layers one obtains a simple $N$ rod solution\nby restricting the sum over ${\\bf s}$ in (\\ref{sg}) to\na subclass of charges of the form \n$s_{na}= s_a \\sum_{j=1,N} \\delta_{n,n'+j-1}$. The variational\nsolution, of the form $\\sigma_c(k)=\\sigma_c \\phi_N(k)$, reduces\nto an effective one layer problem, in term of the\nstructure factor of the rod \n$s_a(k) s_b^*(k) = \\phi_N(k) \\equiv \\sin^2(N k d/2)/\\sin^2(k d/2)$.\nThe $N$ rod becomes critical at:\n\\begin{eqnarray}\n\\sigma_{cr} = (\\int_k \\phi_N(k) J(k))^2/(8 J_0^2 \\int_k \\phi_N(k) \\Delta(k))\n\\label{critical}\n\\end{eqnarray}\na formula which can equivalently be obtained within the\nCayley tree rationale. Indeed for any correlations $\\Delta_n$,\nthe energy of the $\\epsilon =0$ configurations is still\ngiven by (\\ref{Etot}) replacing \n$\\sigma \\to \\sigma (1 + 2 \\sum_{n=1}^N \\Delta_n (1-n/N))$).\n(\\ref{critical}) reproduces both single (using $\\phi(0)=N$) and two layer\nresults. Finally, for\nmany layers and weak interlayer coupling (e.g. \n$\\eta_1<1-1/\\sqrt 2$ in Fig.1) the $N=1$ transition dominates\nand occurs at $\\sigma_{cr}=1/8$ ((\\ref{critical}) using $\\phi_1(k)=1$).\n\n\nAs a direct application we consider a flux lattice in\na layered superconductor with no Josephson coupling and a magnetic\nfield $B$ perpendicular to the layers. \nThe FL is composed of pancake vortices\ndisplaced from the\n$p$-th line position ${\\bf R}_p$ at the $n$-th layer into \n${\\bf R}_p+{\\bf u}_p^n$. The\ndefects $s_n({\\bf r})$ couple to the lattice via ${\\cal\nH}_{vac}=\\sum_{{\\bf r},p,n,n'} s_{n'}({\\bf r})G_v({\\bf R}_p+{\\bf\nu}_p^n -{\\bf r}, n'-n)$ where, in Fourier \n\\cite{H3} $G_v({\\bf q},k)=(\\phi _0^2d^2/4\\pi\n\\lambda_{ab}^2q^2)/[1+f({\\bf q},k)]$ where $f({\\bf\nq},k)=(d/4\\lambda_{ab}^2q)\\sinh qd /[\\sinh ^2(qd/2)+ \\sin\n^2(kd/2)] $; $\\phi_0=B a^2$ is the flux quantum,\n$a$ the FL spacing, $\\lambda_{ab}$ the\npenetration length along the layers.\n To 0-th order in ${\\bf u}_p^n$ the defects feel\na periodic potential fixing their position in a unit\ncell, hence $s({\\bf q},k)$ involve only $|q|<1/a$.\n\nIn the limit $q \\to 0$ the longitudinal modes,\nto which defects couple,\nhave for (tilt) elastic energy \\cite{Goldin} \n${\\cal H}_{el}=\\case{1}{2 d^2 a^4} \\int_{kq} D(k)|{\\bf u}_L({\\bf q},k)|^2$\nwith $D(k)=\\case{1}{2}\\sum_{{\\bf Q}\\neq 0\n}[G_v({\\bf Q},k)-G_v({\\bf Q},0)]+G_v(k)$ where ${\\bf Q}$ are\nreciprocal wavevectors of the lattice and $G_v(k)=\\lim\n_{q\\rightarrow 0}G_v({\\bf q},k)q^2=\\phi_0^2d^2k_z^2/[4\\pi\n(1+\\lambda_{ab}^2k_z^2)]$ and $k_z=(2/d)\\sin (kd/2)$. The sum on\n${\\bf Q}$ is due to the high momentum components of the magnetic\nfield and is responsible for the non-perfect screening of the\ndefect interaction and to a finite $T_{def}$. Minimizing ${\\cal\nH}_{vac}+{\\cal H}_{el}$ yields \n${\\bf u}_{vac}({\\bf q},k)=i{\\bf q}s({\\bf q},k)G_v(k)a^2/D(k)q^2$\nand (\\ref{Hr}) with:\n\\begin{equation}\ng(k)=\\beta G_v(k)[1-G_v(k)/D(k)]/4\\pi \\label{g}\n\\end{equation}\nThus the long range interaction is $\\sim \\ln r$ and its\ncoefficient determines $T_{def}=2 J_0$ (via $\\int_k g(k)=2$).\nSince $\\int_k G_v(k) \\sim \\phi_0^2 d/\\lambda_{ab}^2$,\nthe scale of the melting transition\n\\cite{Blatter}, the defect transition occurs before melting and\ncan thus be consistently described only if $D(k)-G_v(k)\\ll D(k)$.\nThis is possible if\neither $d\\ll a \\ll \\lambda_{ab}$ where \n$g(k)=\\beta d\\tau' \\ln (1+a^2k_z^2/4\\pi)$ with \n$\\tau'=\\phi_0^2da^2/(128\\pi^3 \\lambda_{ab}^4)$ and \\cite{Dodgson}\n$T_{def}=\\tau' ln (a/d)$ or for $d >\na$ where $g(k)=\\beta (d^4/a)\\tau' k_z^2e^{-2\\pi\nd/a}$ leading to $T_{def}=4(d/a) \\tau'\ne^{-2\\pi d/a}$. Remarkably $D(k)-G_v(k)\\ll D(k)$ also \nyields that the long range response ${\\bf u}_{vac}({\\bf\nr})\\sim a^2 {\\bf r}/r^2$ to a vacancy \nat ${\\bf r}=0$ is confined to the same layer.\n\nPoint disorder deforms the flux lattice, producing quenched\ndipoles coupling to our defects. Expansion of the disorder \nenergy, valid below the Larkin length \n\\cite{tgpldbragg}, and minimization together with ${\\cal\nH}_{vac}+{\\cal H}_{el}$ yields readily (\\ref{H}). A more\ngeneral argument, valid at all scales, treats\n$u_{vac}$ as a small perturbation around the Bragg glass\nconfiguration. Systematic expansion of the free energy\n$F = F_{BG} + \\frac{1}{d^2 a^2} \\int_{q k} i q s(q,k) G_v(q,k) \n\\langle u(q,k) \\rangle_{s=0}\n+ O(s^2)$ in defect density in a given disorder configuration\nshows that a defect feels a logarithmically correlated\nrandom potential $V_l({\\bf r})$ as in (\\ref{H}, \\ref{Hr})\nwith $\\sigma J_0^2 \\Delta (k) = \nG_v(k)^2 \\lim_{q \\to 0} C_{BG}(q,k)/4 \\pi d^2 a^4$\nwhere $C_{BG}({\\bf r},l)=\\overline{\\langle u^L_0({\\bf 0}) \\rangle \n\\langle u^L_l({\\bf r}) \\rangle}$\nis the correlation in the unperturbed\nBragg glass\n$C_{BG}(0,k) \\sim 1/(c^2_{44} (k^4 + R_c^{-1} k^3)$,\n$R_c$ a Larkin length along $c$ \\cite{tgpldbragg}. It yields a $k$-independent $\\Delta (k)$\nfor $k>1/R_c$ while $\\Delta (k) \\sim k$ for $k < 1/R_c$.\n\nApplications to FL depends on the interlayer form of (\\ref{g})\nof range $n_0\\approx a/d$ for large $a/d$. Remarkably $g(k=0)=0$,\ni.e perfect screening holds as in 2D \\cite{Dodgson}.\nHence $\\sum _n J_n=0$ and as $n_0$ is reduced $J_0, J_1$ dominate the sum, i.e.\n$\\eta_1 \\rightarrow \\case{1}{2}$ when $d \\gg a$. One finds that\n$\\eta_1$ crosses the critical value $1-1/\\sqrt{2}$ \nwhen $d/a\\approx 1$, depending weakly on \n$a/\\lambda_{ab}$. We thus propose that FL in\nmultilayer superconductors, where $d>a$ can be achieved, can show\na rich phase diagram with $N>1$ phases. \nIn layered superconductors $a/d\\approx 10-100$ \\cite{Kes}\nand the $N=1$ transition at $\\sigma_{cr}=1/8$\ndominates for realistic sizes. The disorder-induced\ndecoupling transition, neglecting defects, predicted \\cite{H1}\nat $\\sigma_{dec}=2$ is thus above the defect transition\n(with $B \\sim \\sigma$) in the $B-T$ plane (similarly thermal \ndecoupling occurs at $T_{dec}= 8 T_{def}$ for $d\\ll a\\ll\n\\lambda$). A natural scenario is again of a single\ntransition at $\\sigma_c$ varying from\n$2$ to $1/8$ as the bare Josephson coupling is\nreduced, e.g. by increasing $d$ in multilayers.\n\nIn conclusion, we developed a variational method and a Cayley\ntree rationale to study layered Coulomb gas. The results are relevant\nto flux lattices where we find the phase boundaries and\npropose new $N>1$ phases for $d\\gtrsim a$. The present methods\nmay be useful for other 2D disordered systems, such as quantum Hall.\n\nThis work was supported by the French-Israeli program Arc-en-ciel\nand by the Israel Science Foundation.\n\n\\begin{thebibliography}{999}\n\n%\\begin{references}\n\n\\bibitem{tgpldbragg}\nT. Giamarchi, P. Le Doussal, Phys. Rev. B {\\bf 52} 1242 (1995)\nand Phys. Rev. B {\\bf 55} 6577 (1997).\n\\bibitem{Kes} P. H. Kes, J. Phys. I\nFrance {\\bf 6}\n2327 (1996).\n\\bibitem{speakexp} B. Kaykovich et al. Phys. Rev. Lett {\\bf 76}\n2555 (1996), K. Deligiannis et al. Phys. Rev. Lett {\\bf 79}\n2121 (1997).\n\\bibitem{H2} B. Horovitz, cond-mat/9903167, Phys. Rev. B {\\bf 60} R9939\n(1999).\n\\bibitem{Dodgson} M. J. W. Dodgson, V. B. Geshkenbein and G.\nBlatter Phys. Rev. Lett. {\\bf 83} 5358 (1999). \n\\bibitem{Daemen} L. Daemen et al., Phys. Rev. Lett. {\\bf 70}, 1167\n(1993)\n\\bibitem{H1} B. Horovitz and T. R. Goldin, Phys. Rev. Lett. {\\bf\n80},1734 (1998).\n\\bibitem{H3} B. Horovitz, Phys. Rev. B{\\bf 47}, 5947 (1993)\n\\bibitem{Bruynseraede} Y. Bruynseraede et al., Phys. Scr. {\\bf T42},\n37 (1992).\n\n\\bibitem{nattermann95} T. Nattermann et al., J. Phys. I (France) {\\bf 5}, 565 (1995)\n\n\\bibitem{scheidl97} S. Scheidl, Phys. Rev. {\\bf B 55}, 457 (1997)\n\n\\bibitem{tang96} L. H. Tang, Phys. Rev. {\\bf B 54}, 3350 (1996).\n\n\\bibitem{dcpld} D. Carpentier and P. Le Doussal, \nPhys. Rev. Lett. {\\bf 81} 2558 (1998), cond-mat/9908335\nand in preparation.\n\n\\bibitem{Blatter} G. Blatter et al. Rev. Mod. Phys.\n{\\bf 66} 1125 (1994).\n\n\\bibitem{footnote2} these phase diagrams are exact\nin terms of {\\it renormalized} parameters $\\sigma_R$, $g_R(k)$\nas seen from RG studies\\cite{dcpld,us_inprep}.\n\n\\bibitem{rem} B. Derrida Phys. Rev. B {\\bf 24} 2613 (1981)\n\n\\bibitem{footnote1} i.e. replacing the $V({\\bf r})$ by $L^2$ variables\n{\\it uncorrelated} in ${\\bf r}$, with the same on-site \nvariance $\\overline{V^2({\\bf r})} \\sim 2 \\sigma J_0^2 \\ln L$\nalso yielding \\cite{rem} $V_{min} \\sim - \\sqrt{8 \\sigma} J_0 \\ln L$.\n\n\\bibitem{ct} B. Derrida, H. Spohn, J. Stat. Phys. {\\bf 51} 817 (1988).\n\n\\bibitem{footnote3} an upper bound which \ncan be argued to be exact \\cite{us_inprep}.\n\n\\bibitem{us_inprep} B. Horovitz and P. Le Doussal in preparation.\n\n\\bibitem{footnote9} if $\\eta _1>\\case{1}{2}$ the defects form a lattice\neven without disorder.\n\n\\bibitem{footnote7} $E_c/J_0 \\approx (\\lambda_{ab}/a)^2 /\\ln(a/d)$ in that case.\n\n\\bibitem{Goldin} T. R. Goldin and B. Horovitz, Phys. Rev. B{\\bf\n58}, 9524 (1998).\n\n\\bibitem{footnote5} with $\\int_q \\equiv \\int\n\\frac{d^2q}{(2\\pi)^2}$, $\\int_k \\equiv \\frac{1}{M d}\n\\sum_k \\to \\int_{-\\pi/d}^{\\pi/d} \\frac{dk}{2 \\pi}$ for\nlarge $M$, $s_a({\\bf q},k)=d \\sum_{n,{\\bf r}} \ns_{na}({\\bf r}) e^{i {\\bf q} {\\bf r} + i k d n}$ and\n$\\beta=1/T$.\n\n\\end{thebibliography}\n\n%\\vspace{20cm}\n\\unecol\n%\\end{references}\n\n\\end{document}\n\n\n"
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[
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"name": "cond-mat0002037.extracted_bib",
"string": "\\begin{thebibliography}{999}\n\n%\\begin{references}\n\n\\bibitem{tgpldbragg}\nT. Giamarchi, P. Le Doussal, Phys. Rev. B {\\bf 52} 1242 (1995)\nand Phys. Rev. B {\\bf 55} 6577 (1997).\n\\bibitem{Kes} P. H. Kes, J. Phys. I\nFrance {\\bf 6}\n2327 (1996).\n\\bibitem{speakexp} B. Kaykovich et al. Phys. Rev. Lett {\\bf 76}\n2555 (1996), K. Deligiannis et al. Phys. Rev. Lett {\\bf 79}\n2121 (1997).\n\\bibitem{H2} B. Horovitz, cond-mat/9903167, Phys. Rev. B {\\bf 60} R9939\n(1999).\n\\bibitem{Dodgson} M. J. W. Dodgson, V. B. Geshkenbein and G.\nBlatter Phys. Rev. Lett. {\\bf 83} 5358 (1999). \n\\bibitem{Daemen} L. Daemen et al., Phys. Rev. Lett. {\\bf 70}, 1167\n(1993)\n\\bibitem{H1} B. Horovitz and T. R. Goldin, Phys. Rev. Lett. {\\bf\n80},1734 (1998).\n\\bibitem{H3} B. Horovitz, Phys. Rev. B{\\bf 47}, 5947 (1993)\n\\bibitem{Bruynseraede} Y. Bruynseraede et al., Phys. Scr. {\\bf T42},\n37 (1992).\n\n\\bibitem{nattermann95} T. Nattermann et al., J. Phys. I (France) {\\bf 5}, 565 (1995)\n\n\\bibitem{scheidl97} S. Scheidl, Phys. Rev. {\\bf B 55}, 457 (1997)\n\n\\bibitem{tang96} L. H. Tang, Phys. Rev. {\\bf B 54}, 3350 (1996).\n\n\\bibitem{dcpld} D. Carpentier and P. Le Doussal, \nPhys. Rev. Lett. {\\bf 81} 2558 (1998), cond-mat/9908335\nand in preparation.\n\n\\bibitem{Blatter} G. Blatter et al. Rev. Mod. Phys.\n{\\bf 66} 1125 (1994).\n\n\\bibitem{footnote2} these phase diagrams are exact\nin terms of {\\it renormalized} parameters $\\sigma_R$, $g_R(k)$\nas seen from RG studies\\cite{dcpld,us_inprep}.\n\n\\bibitem{rem} B. Derrida Phys. Rev. B {\\bf 24} 2613 (1981)\n\n\\bibitem{footnote1} i.e. replacing the $V({\\bf r})$ by $L^2$ variables\n{\\it uncorrelated} in ${\\bf r}$, with the same on-site \nvariance $\\overline{V^2({\\bf r})} \\sim 2 \\sigma J_0^2 \\ln L$\nalso yielding \\cite{rem} $V_{min} \\sim - \\sqrt{8 \\sigma} J_0 \\ln L$.\n\n\\bibitem{ct} B. Derrida, H. Spohn, J. Stat. Phys. {\\bf 51} 817 (1988).\n\n\\bibitem{footnote3} an upper bound which \ncan be argued to be exact \\cite{us_inprep}.\n\n\\bibitem{us_inprep} B. Horovitz and P. Le Doussal in preparation.\n\n\\bibitem{footnote9} if $\\eta _1>\\case{1}{2}$ the defects form a lattice\neven without disorder.\n\n\\bibitem{footnote7} $E_c/J_0 \\approx (\\lambda_{ab}/a)^2 /\\ln(a/d)$ in that case.\n\n\\bibitem{Goldin} T. R. Goldin and B. Horovitz, Phys. Rev. B{\\bf\n58}, 9524 (1998).\n\n\\bibitem{footnote5} with $\\int_q \\equiv \\int\n\\frac{d^2q}{(2\\pi)^2}$, $\\int_k \\equiv \\frac{1}{M d}\n\\sum_k \\to \\int_{-\\pi/d}^{\\pi/d} \\frac{dk}{2 \\pi}$ for\nlarge $M$, $s_a({\\bf q},k)=d \\sum_{n,{\\bf r}} \ns_{na}({\\bf r}) e^{i {\\bf q} {\\bf r} + i k d n}$ and\n$\\beta=1/T$.\n\n\\end{thebibliography}"
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cond-mat0002038
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\begin{flushright} {SU{-}4240{-}714} \end{flushright} The Statistical Mechanics of Membranes
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"author": "$^1$Physics Department"
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"author": "Syracuse"
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{
"author": "NY 13244-1130"
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The fluctuations of two-dimensional extended objects ({\em membranes}) is a rich and exciting field with many solid results and a wide range of open issues. We review the distinct universality classes of membranes, determined by the local order, and the associated phase diagrams. After a discussion of several physical examples of membranes we turn to the physics of {\em crystalline} (or {\em polymerized}) membranes in which the individual monomers are rigidly bound. We discuss the phase diagram with particular attention to the dependence on the degree of {\em self-avoidance} and {\em anisotropy}. In each case we review and discuss analytic, numerical and experimental predictions of critical exponents and other key observables. Particular emphasis is given to the results obtained from the renormalization group $\vap$-expansion. The resulting renormalization group flows and fixed points are illustrated graphically. The full technical details necessary to perform actual calculations are presented in the Appendices. We then turn to a discussion of the role of topological defects whose liberation leads to the {\em hexatic} and {\em fluid} universality classes. We finish with conclusions and a discussion of promising open directions for the future.
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"name": "memxxx.tex",
"string": "\\documentclass[12pt]{article}\n\\usepackage{epsf}\n\n\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\bea}{\\begin{eqnarray}}\n\\newcommand{\\eea}{\\end{eqnarray}}\n\\newcommand{\\vap}{\\varepsilon}\n\\newcommand{\\pary}{\\partial_{y}}\n\\newcommand{\\parp}{\\partial}\n\\newcommand{\\vet}{\\overline}\n\n\\begin{document}\n\n\\title{\n\\begin{flushright}\n{\\small SU{-}4240{-}714}\n\\end{flushright} \nThe Statistical Mechanics of Membranes}\n\n\\author{\\small \\\\ Mark~J. Bowick\\thanks{\\tt \nbowick@physics.syr.edu} \\, \nand Alex Travesset\\thanks{\\tt alex@suhep.phy.syr.edu}\n\\\\ $^1$Physics Department, Syracuse University,\\\\\nSyracuse, NY 13244-1130, USA \\\\ }\n\n\\date{}\n\n\\maketitle\n\n\n\n\\begin{abstract}\n\nThe fluctuations of two-dimensional extended objects ({\\em membranes}) is a\nrich and exciting field with many solid results and a wide range of \nopen issues. \nWe review the distinct universality classes of membranes, determined by\nthe local order, and the associated phase diagrams. \nAfter a discussion of several physical examples of membranes \nwe turn to the physics of {\\em crystalline} (or {\\em polymerized})\nmembranes in which the individual monomers are rigidly bound. \nWe discuss the phase diagram with particular attention to the\ndependence on the degree of {\\em self-avoidance} and {\\em\nanisotropy}. In each case we review and discuss\nanalytic, numerical and experimental predictions of critical\nexponents and other key observables. \nParticular emphasis is given to the results obtained from the \nrenormalization group $\\vap$-expansion.\nThe resulting renormalization group flows and fixed points are illustrated\ngraphically. The full technical details necessary to perform actual\ncalculations are presented in the Appendices. \nWe then turn to a discussion of the role of topological defects \nwhose liberation leads to the {\\em hexatic} and {\\em fluid} universality\nclasses. We finish with conclusions and a discussion of promising open\ndirections for the future. \n\n\n\\end{abstract}\n\n\\vfill\\newpage\n\n\\pagebreak\n\n\\tableofcontents\n\n\\newpage\n\n\\section{Introduction}\n\\label{SECT__Intro}\n\n\n\\bibliographystyle{unsrt}\n\n\nThe statistical mechanics of one-dimensional structures (polymers) is\nfascinating and has proved to be fruitful from the fundamental and\napplied points of view \\cite{deGennes,desCloiseaux}. \nThe key reasons for this success lie in the\nnotion of {\\em universality} and the relative simplicity of\none-dimensional geometry. Many features of the long-wavelength behavior\nof polymers are independent of the detailed physical and chemical\nnature of the monomers that constitute the polymer building blocks and\ntheir bonding into macromolecules. These microscopic details simply\nwash out in the thermodynamic limit of large systems and allow\npredictions of critical exponents that should apply to a wide class of\nmicroscopically distinct polymeric systems. Polymers are also\nsufficiently simple that considerable analytic and numerical progress\nhas been possible. Their statistical mechanics is essentially that of\nensembles of various classes of {\\em random walks} in some\n$d$-dimensional bulk or embedding space. \n\nA natural extension of these systems is to intrinsic two-dimensional\nstructures which we may call generically call {\\em membranes}. The\nstatistical mechanics of these {\\em random surfaces} is far more\ncomplex than that of polymers because two-dimensional geometry is far\nricher than the very restricted geometry of lines. Even planar\ntwo-dimensional {---} monolayers {---} are complex, as evidenced by\nthe KTNHY \\cite{KT,NH:79,Young} theory of defect-mediated melting of \nmonolayers with two distinct {\\em continuous} phase transitions \nseparating an intermediate\nhexatic phase, characterized by quasi-long-range bond orientational\norder, from both a low-temperature {\\em crystalline} phase and a\nhigh-temperature {\\em fluid} phase. But full-fledged membranes are\nsubject also to shape fluctuations and their macroscopic behavior is\ndetermined by a subtle interplay between their particular microscopic\norder and the entropy of shape and elastic deformations. For\nmembranes, unlike polymers, distinct types of microscopic order\n(crystalline, hexatic, fluid) will lead to distinct long-wavelength\nbehavior and consequently a rich set of universality classes. \n\nFlexible membranes are an important member of the enormous class of \n{\\em soft} condensed matter systems \\cite{deGennes2,Lub:97,GuPa:92,CL:95},\nthose which respond easily to external forces. Their physical\nproperties are to a considerable extent dominated by the entropy of thermal\nfluctuations. \n \nIn this review we will describe some of the presently understood behavior of\ncrystalline (fixed-connectivity), hexatic and fluid membranes,\nincluding the relevance of self-avoidance, intrinsic anisotropy and \ntopological defects. Emphasis will be given to the role\nof the renormalization group in elucidating the critical behavior of\nmembranes. The polymer pastures may be lovely but a dazzling world\nawaits those who wander into the membrane meadows. \n\nThe outline of the review is the following.\nIn sec.~\\ref{SECT__Examples} we describe a variety of important\nphysical examples of membranes, with representatives from the key\nuniversality classes. In sec.~\\ref{SECT__RG} we introduce basic\nnotions from the renormalization group and some formalism that we will\nuse in the rest of the review. In sec.~\\ref{SECT__POLYMEM} we review\nthe phase structure of crystalline membranes for both phantom and\nself-avoiding membranes, including a thorough discussion of the\nfixed-point structure, RG flows and critical exponents of each global\nphase. In sec.~\\ref{SECT__POLYMEM_ANI} we turn to the same issues\nfor intrinsically anisotropic membranes, with the new feature of the\n{\\em tubular} phase. In sec.~\\ref{SECT__Defects} we address the\nconsequences of allowing for membrane defects, leading to a discussion\nof the hexatic membrane universality class. We end with a\nbrief discussion of fluid membranes in sec.~\\ref{SECT__Fluid}\nand conclusions. \n\n\\section{Physical examples of membranes}\n\\label{SECT__Examples}\n\nThere are many concrete realizations of membranes in\nnature, which greatly enhances the significance of their study.\nCrystalline membranes, sometimes termed {\\em tethered} or {\\em polymerized}\nmembranes, are the natural generalization of linear polymer chains to\nintrinsically two-dimensional structures. They possess in-plane\nelastic moduli as well as bending rigidity and are characterized by\nbroken translational invariance in the plane and fixed connectivity\nresulting from relatively strong bonding. Geometrically speaking they\nhave a preferred two-dimensional metric. \nLet's look at some of the examples. \nOne can polymerize suitable chiral oligomeric precursors to form\nmolecular sheets \\cite{Stupp}. This approach is based directly on the\nidea of creating an intrinsically two-dimensional polymer. \nAlternatively one can permanently cross-link\nfluid-like Langmuir-Blodgett films or amphiphilic bilayers by adding\ncertain functional groups to the hydrocarbon tails and/or the \npolar heads \\cite{Fendler1,Fendler2} as shown schematically in \nFig.~\\ref{fig__polymerize}. \n\n\\begin{figure}[htb]\n\\epsfxsize=3in\n\\centerline{\\epsfbox{polymerize.eps}}\n\\caption{The polymerization of fluid-like membrane to a crystalline membrane.}\n\\label{fig__polymerize}\n\\end{figure}\n\nThe cytoskeletons of cell membranes are beautiful and naturally occurring \ncrystalline membranes that are essential to cell membrane stability\nand functionality. The simplest and most thoroughly studied example \nis the cytoskeleton of mammalian erythrocytes (red blood cells). \nThe human body has roughly $5 \\times 10^{13}$ red blood cells. \nThe red blood cell cytoskeleton is a fishnet-like\nnetwork of triangular plaquettes formed primarily by the proteins {\\em\nspectrin} and {\\em actin}. The links of the mesh are spectrin\ntetramers (of length approximately 200 nm) and the nodes are short\nactin filaments (of length 37 nm and typically 13 actin monomers long)\n\\cite{Skel:93,Branton}, as seen in Fig.\\ref{fig__spectrin1} and \nFig.\\ref{fig__spectrin2}. There are\nroughly 70,000 triangular plaquettes in the mesh altogether and the\ncytoskeleton as a whole is bound by ankyrin and other proteins \nto the cytoplasmic side of the fluid\nphospholipid bilayer which constitutes the other key component of the\nred blood cell membrane. \n\n\\begin{figure}[hp]\n\\epsfxsize=3in\n\\centerline{\\epsfbox{spectrinxxx.eps}}\n\\caption{An electron micrograph of a region of the erythrocyte\ncytoskeleton. The skeleton is negatively stained (magnification\n365,000) and has been artificially spread to a surface area nine to\nten times as great as in the native membrane \\cite{Byers}. }\n\\label{fig__spectrin1}\n\\epsfxsize=3in\n\\centerline{\\epsfbox{spectrin2.eps}}\n\\caption{An extended view of the crystalline spectrin/actin\nnetwork which forms the cytoskeleton of the red blood cell membrane \n\\cite{Branton2}. }\n\\label{fig__spectrin2}\n\\end{figure}\nThere are also inorganic realizations of crystalline membranes.\nGraphitic oxide (GO) membranes are micron size sheets of solid carbon\nwith thicknesses on the order of 10\\AA, formed by exfoliating carbon\nwith a strong oxidizing agent. \nTheir structure in an aqueous suspension has been examined by several\ngroups \\cite{Hwa,Wen,Zasa:94}.\nMetal dichalcogenides such as MoS$_2$ have also been observed to form\nrag-like sheets \\cite{Chianelli}.\nFinally similar structures occur in the large sheet molecules, shown \nin Fig.\\ref{fig__sheet_mol}, believed to be an ingredient in glassy\n$B_2O_3$. \n\n\\begin{figure}[htb]\n\\epsfxsize=3in\n\\centerline{\\epsfbox{sheet_mol.eps}}\n\\caption{The sheet molecule $B_2O_3$}\n\\label{fig__sheet_mol}\n\\end{figure}\n\nIn contrast to crystalline membranes, fluid membranes are\ncharacterized by vanishing shear modulus and dynamical connectivity.\nThey exhibit significant shape fluctuations controlled by an effective bending\nrigidity parameter. \n \n\\begin{figure}[htb]\n\\epsfxsize=3in\n\\centerline{\\epsfbox{black.eps}}\n\\caption{Schematic of experimental procedure to make a {\\em black}\nmembrane.}\n\\label{fig__black}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\epsfxsize=2in\n\\centerline{\\epsfbox{liposome.eps}}\n\\caption{The structure of a liposome with its pure lipid spherical bilayer}\n\\label{fig__liposome}\n\\end{figure}\n\nA rich source of physical realizations of fluid membranes is found in \n{\\em amphiphilic} systems \\cite{Lipowsky,GS:94,Peliti}. Amphiphiles are\nmolecules with a two-fold character {--} one part is hydrophobic and\nanother part hydrophilic. The classic examples are lipid molecules,\nsuch as phospholipids, which have polar or ionic head groups\n(the hydrophilic component) and hydrocarbon tails (the hydrophobic\ncomponent). Such systems are observed to self-assemble into a\nbewildering array of ordered structures, such as monolayers, planar\n(see Fig.\\ref{fig__black}) and spherical bilayers (vesicles or\nliposomes) (see Fig.~\\ref{fig__liposome}) as well as lamellar, hexagonal and\nbicontinuous phases \\cite{Gruner}. In each case the basic ingredients are thin\nand highly flexible surfaces of amphiphiles. The lipid bilayer of cell\nmembranes may itself be viewed as a fluid membrane with considerable\ndisorder in the form of membrane proteins (both peripheral and\nintegral) and with, generally, an attached crystalline cytoskeleton,\nsuch as the spectrin/actin mesh discussed above. \n\nA complete understanding of these biological membranes will require a\nthorough understanding of each of its components (fluid and\ncrystalline) followed by the challenging problem of the coupled system\nwith thermal fluctuations, self-avoidance, potential anisotropy and\ndisorder. The full system is currently beyond the\nscope of analytic and numerical methods but there has been\nconsiderable progress in the last fifteen years.\n\nRelated examples of fluid membranes arise when the surface tension \nbetween two normally immiscible substances, such as oil and water, \nis significantly lowered by the surface action of amphiphiles\n(surfactants), which preferentially orient with their polar heads \nin water and their hydrocarbon tails in oil. \nFor some range of amphiphile concentration both phases can span the\nsystem, leading to a bicontinuous complex fluid known as a {\\em\nmicroemulsion}. The oil-water interface of a microemulsion is a rather \nunruly fluid surface with strong thermal fluctuations \\cite{Safran} \n(see Fig.\\ref{fig__water_oil}).\n\\begin{figure}[hp]\n\\epsfxsize=3in\n\\centerline{\\epsfbox{water_oil.eps}}\n\\caption{The structure of a microemulsion formed by the addition of\nsurfactant to an oil-water mixture.}\n\\label{fig__water_oil}\n\\epsfxsize=3in\n\\centerline{\\epsfbox{tubule.eps}}\n\\caption{Metal-coated fluid microcylinders (tubules) formed by chiral lipids.}\n\\label{fig__tubule}\n\\end{figure}\n\nThe structures formed by membrane/polymer complexes are \nof considerable current theoretical, experimental and medical\ninterest. To be specific it has recently been found that mixtures of \ncationic liposomes (positively charged vesicles) and linear DNA chains\nspontaneously self-assemble into a coupled two-dimensional smectic\nphase of DNA chains embedded between lamellar lipid \nbilayers \\cite{Safinya1,Salditt}.\nFor the appropriate regime of lipid\nconcentration the same system can also form an inverted\nhexagonal phase with the DNA encapsulated by cylindrical columns of\nliposomes\\cite{Safinya2} (see Fig.\\ref{fig__Cyrusall}). \nIn both these structures the liposomes may\nact as non-viral carriers (vectors) for DNA with many \npotentially important applications in gene therapy \\cite{Crystal:95}. \nLiposomes themselves have long been studied and utilized \nin the pharmaceutical industry as drug carriers \\cite{Needham}.\n\\begin{figure}[hp]\n\\epsfxsize=3in\n\\centerline{\\epsfbox{DNA_MEM.eps}}\n\\caption{The lamellar and inverted hexagonal DNA-membrane complexes from the work of \\cite{Safinya2}}\n\\label{fig__Cyrusall}\n\\end{figure}\nOn the materials science side the self-assembling ability of membranes\nis being exploited to fabricate microstructures for advanced material\ndevelopment. One beautiful example is the use of chiral-lipid based\nfluid microcylinders (tubules) as a template for metallization.\nThe resultant hollow metal {\\em needles} may be half a micron in\ndiameter and as much as a millimeter in length\n\\cite{Schnur:93,SEJSSS:96}, as illustrated in Fig.\\ref{fig__tubule}. \nThey have potential applications as, for example, cathodes for vacuum\nfield emission and microvials for controlled release \\cite{Schnur:93}. \n\n\\section{The Renormalization Group}\\label{SECT__RG}\n\nThe Renormalization Group (RG) has provided an extremely general \nframework that has unified whole areas of physics and chemistry \\cite{WiKo:74}.\nIt is beyond the scope of this review to discuss the RG formalism in \ndetail but there is an ample literature to which we refer the reader\n(see the articles in this issue).\nIt is the goal of this review to apply the RG framework to the\nstatistical mechanics of membranes, and for this reason we briefly emphasize and review\nsome well known aspects of the RG and its related $\\vap$-expansion.\n\nThe RG formalism elegantly shows that the large distance properties\n(or equivalently low $p$-limit) of different models are actually \ngoverned by the properties of the corresponding Fixed Point (FP). \nIn this way one can compute observables in a variety of models,\nsuch as a molecular dynamics simulation or a continuum Landau \nphenomenological approach, and obtain the same\nlong wavelength result. The main idea is to encode the effects \nof the short-distance degrees of freedom in redefined couplings. \nA practical way to implement such a program \nis the Renormalization Group Transformation (RGT), which provides an explicit \nprescription for integrating out all the \nhigh $p$-modes of the theory. One obtains the large-distance universal term\nof any model by applying a very large ($\\infty$ to be rigorous)\nnumber of RGTs.\n\nThe previous approach is very general and simple but presents\nthe technical problem of the proliferation in the number of operators \ngenerated along the RG flow. There are established techniques to \ncontrol this expansion, one of the most successful ones being the\n$\\vap$-expansion. The $\\vap$-expansion may also performed via \na field theoretical approach using Feynman diagrams and dimensional \nregularization within a minimal subtraction scheme, which we briefly discuss\nbelow. Whereas it is true that this technique is rather abstract \nand intuitively not very close to the physics of the model, \nwe find it computationally much simpler. \n\nGenerally we describe a particular model by several fields \n$\\{ \\phi,\\chi,\\cdots \\}$ and we construct the Landau free energy by\nincluding all terms compatible with the symmetries and introducing \nnew couplings $(u,v,\\cdots)$ for each term. The Landau free energy\nmay be considered in arbitrary dimension $d$, and then, one usually\nfinds a Gaussian FP (quadratic in the fields) which is infrared stable \nabove a critical dimension ($d_U$). Below $d_U$ there are one or\nseveral couplings that define relevant directions. \nOne then computes all physical quantities as\na function of $\\vap\\equiv d_{U}-d$, that is, as perturbations of\nthe Gaussian theory.\n\nIn the field theory approach, we introduce a renormalization constant\nfor each field $(Z_{\\phi},Z_{\\chi},\\cdots)$ and a renormalization \nconstant $(Z_{u},Z_{v},\\cdots)$ for each relevant direction below $d_U$.\nIf the model has symmetries, there are some relations among observables\n(Ward identities) and some of these renormalization constants \nmay be related. This not only reduces their number but also has the\nadded bonus of providing cross-checks in practical calculations. \nWithin dimensional regularization,\nthe infinities of the Feynman diagrams appear as poles in $\\vap$,\nwhich encode the short-distance details of the model. If we use\nthese new constants ($Z$'s) to absorb the poles in $\\vap$, thereby producing\na complete set of finite Green's functions, we have \nsucceeded in carrying out the RG program of including the appropriate short-distance \ninformation in redefined couplings and fields. This particular\nprescription of absorbing only the poles in $\\vap$ in the $Z$'s is called the\nMinimal Subtraction Scheme (MS), and it considerably\nsimplifies practical calculations. \n\nAs a concrete example, we consider the theory of a single scalar field\n$\\phi$ with two independent coupling constants. The one-particle \nirreducible Green's function has the form\n\\be\\label{RG__Green}\n\\Gamma^{N}_R({\\bf k}_i;u_R,v_R,M)=Z^{N/2}_{\\phi} \n\\Gamma^N({\\bf k}_i;u,v; \\frac{1}{\\vap}) \\ ,\n\\ee\nwhere the function on the left depends on a new parameter $M$, which\nis unavoidably introduced in eliminating the poles in $\\vap$. The associated\ncorrelator also depends on redefined couplings $u_R$ and $v_R$. \nThe rhs depends on the poles in $\\vap$, but its only dependence on $M$ \narises through $Z_{\\phi}$. This observation allows one to write \n\\bea\\label{RG_CA}\n&&M\\frac{d}{d M}\\left( Z_{\\phi}^{-N/2} \\Gamma_R^{(N)}\\right)=\n\\\\\\nonumber\n&=&\n\\left( M \\frac{\\parp}{\\parp M}+\\beta(u_R)\\frac{\\parp}{\\parp u_R}\n+\\beta(v_R)\\frac{\\parp}{\\parp v_R}-\\frac{N}{2} \\Gamma_{\\phi} \\right)\n\\Gamma_R^{(N)}=0 \\ ,\n\\eea\nwhere\n\\bea\\label{define_set_RG}\nu_R=M^{-\\vap}F(Z_{\\phi},Z_{\\chi},\\cdots|Z_{u}) u & \\ , &\nv_R=M^{-\\vap}F(Z_{\\phi},Z_{\\chi},\\cdots|Z_{v}) v\n\\\\\\nonumber\n\\beta_u(u_R,v_R)=\\left.\\left(M\\frac{\\parp u_R}{\\parp M}\\right)\n\\right|_{u,v} \n& \\ , &\n\\beta_v(u_R,v_R)=\\left.\\left(M\\frac{\\parp v_R}{\\parp M}\\right)\n\\right|_{u,v} \n\\nonumber\\\\\\nonumber\n\\gamma_{\\phi}&=&\\left.\\left(M\\frac{\\parp \\ln Z_{\\phi} }{\\parp M}\\right)\n\\right|_{u,v} \\ .\n\\eea\nThe $\\beta$-functions control the running of the coupling by\n\\be\\label{run__FP}\nM\\frac{d u_R}{d M}=\\beta_u(u_R,v_R) \\ , \\\nM\\frac{d v_R}{d M}=\\beta_v(u_R,v_R)\n\\ee\nThe existence of a FP, at which couplings cease to flow, requires \n$\\beta(u_R^{\\ast},v_R^{\\ast})=0$ for all $\\beta$-functions of the model.\nThose are the most important aspects of the RG we wanted to review.\nIn Appendix~\\ref{APP_RGstuff} we derive more appropriate expressions\nof the RG-functions for practical convenience. For a detailed\nexposition of the $\\vap$-expansion within the field theory framework\nwe refer to the excellent book by Amit\\cite{Amit:84}.\n\n\n\\section{Crystalline Membranes}\\label{SECT__POLYMEM}\n\nA crystalline membrane is a two dimensional fish-net structure with\nbonds (links) that never break - the connectivity of the monomers\n(nodes) is fixed. It is useful to keep the discussion general and\nconsider $D$-dimensional objects embedded in $d$-dimensional space.\nThese are described by a $d$-dimensional vector \n${\\vec r}({\\bf x})$, with ${\\bf x}$ the $D$-dimensional internal\ncoordinates, as illustrated in Fig.\\ref{fig__quadr}. \nThe case $(d=3,D=2)$ corresponds to the physical crystalline membrane. \n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{quadrat.eps}}\n \\caption{Representation of a membrane.}\n \\label{fig__quadr}\n \\end{figure}\n\nTo construct the Landau free energy of the model, one must recall that the \nfree energy must be invariant under global translations, so the \norder parameter is given by derivatives of the embedding ${\\vec r}$, that is\n${\\vec t}_{\\alpha}=\\frac{\\parp{\\vec r}}{\\parp u_{\\alpha}}$,\nwith $\\alpha=1,\\cdots,D$. This latter condition, together with the\ninvariance under rotations (both in internal and bulk space), \ngive a Landau free energy \\cite{NP:87,Jer1,PKN:88} \n\\bea\\label{LAN_CR_VER}\nF({\\vec r})&=&\\int d^D{\\bf x} \\left[ \n\\frac{1}{2}\\kappa (\\parp_{\\alpha}^2 {\\vec r})^2+\n\\frac{t}{2}(\\parp_{\\alpha} {\\vec r})^2+u(\\parp_{\\alpha} {\\vec r} \n\\parp_{\\beta} {\\vec r} )^2+v(\\parp_{\\alpha} {\\vec r} \n\\parp^{\\alpha} {\\vec r})^2 \\right]\n\\nonumber\\\\\n&+&\\frac{b}{2}\n\\int d^D{\\bf x}\\, d^D{\\bf y} \\delta^d({\\vec r}({\\bf x})-\n{\\vec r}({\\bf y})) \\ ,\n\\eea\nwhere higher order terms may be shown to be irrelevant at\nlong wavelength, as discussed later. The physics in Eq.(\\ref{LAN_CR_VER}) \ndepends on five parameters,\n\n\\begin{itemize}\n\\item{$\\kappa$, \\underline{bending rigidity} :} This is the coupling\nto the extrinsic curvature (the square of the Gaussian mean\ncurvature). Since reparametrization invariance is broken for\ncrystalline membranes, this term may be replaced by its\nlong-wavelength limit. For large and positive bending \nrigidities flatter surfaces are favored.\n\n\\item{$t,u,v$, \\underline{elastic constants} :} These coefficients \nencode the microscopic elastic properties of the membrane. In a flat\nphase, they may be related to the Lam\\'e coefficients of Landau elastic\ntheory (see sect.~\\ref{SUB__subflat}).\n\n\\item{$b$, \\underline{Excluded volume or self-avoiding coupling} : } \nThis is the coupling that imposes an energy penalty for the membrane to\nself-intersect. The case $b=0$, i.\\ e.\\ no self-avoidance, \ncorresponds to a {\\em phantom} model.\n\n\\end{itemize}\n\n\\noindent We generally expand ${\\vec r}({\\bf x})$ as\n\\be\\label{mf_variable}\n{\\vec r}({\\bf x})=(\\zeta {\\bf x}+{\\bf u}({\\bf x}),h({\\bf x})) \\ ,\n\\ee\nwith ${\\bf u}$ the $D$-dimensional phonon in-plane modes, and\n$h$ the $d-D$ out-of-plane fluctuations. If $\\zeta=0$ the model \nis in a rotationally invariant crumpled phase, where the typical \nsurfaces have fractal dimension, and there is no real \ndistinction between the in-plane phonons and out-of plane modes. \nFor a pictorial view, see cases a) and b) in Fig.\\ref{fig__PHASES}.\n\nIf $\\zeta \\neq 0$ the membrane is flat up to small fluctuations\nand the full rotational symmetry is spontaneously broken. The fields \n$h$ are the analog of the Goldstone bosons and they have different naive \nscaling properties than ${\\bf u}$. See Fig.\\ref{fig__PHASES} for a\nvisualization of a typical configuration in the flat phase.\n\n \\begin{figure}[htb]\n \\epsfxsize=5 in \\centerline{\\epsfbox{confs3.eps}}\n \\caption{Examples of a) crumpled phase, b) crumpling transition (crumpled \n phase) and c) a flat phase. Results correspond to a numerical simulation\n of the phantom case \\cite{BCFTA:96} and gives a very intuitive physical picture \n of the different phases.}\n \\label{fig__PHASES}\n \\end{figure}\n\nWe will begin by studying the phantom case first.\nThis simplified model may even be relevant to physical systems since\none can envision membranes that self-intersect (at least over some\ntime scale). \nOne can also view the model as a fascinating toy model \nfor understanding the more physical self-avoiding case \nto be discussed later. Combined analytical and numerical studies have\nyielded a thorough understanding of the phase diagram of phantom\ncrystalline membranes. \n\n\\subsection{Phantom}\\label{SUBSECT__PHAN}\n\nThe Phantom case corresponds to setting $b=0$ in the free energy \nEq.(\\ref{LAN_CR_VER}):\n\\be\\label{LAN_CR_PH}\nF({\\vec r})=\\int d^D{\\bf x} \\left[ \n\\frac{1}{2}\\kappa (\\parp_{\\alpha}^2 {\\vec r})^2+\n\\frac{t}{2}(\\parp_{\\alpha} {\\vec r})^2+u(\\parp_{\\alpha} {\\vec r} \n\\parp_{\\beta} {\\vec r} )^2+v(\\parp_{\\alpha} {\\vec r} \n\\parp^{\\alpha} {\\vec r})^2 \\right] \\ .\n\\ee\n\nThe mean field effective potential, using the decomposition of \nEq.(\\ref{mf_variable}), becomes\n\\be\\label{mf_eq_ph}\nV(\\zeta)=D \\zeta^2(\\frac{t}{2}+(u+vD)\\zeta^2) \\ ,\n\\ee\nwith solutions\n\\be\\label{mf_zeta}\n\\zeta^2 = \\left\\{\n \\begin{array}{r@{\\quad:\\quad}l}\n\t 0 & t \\ge 0 \\\\\n\t -\\frac{t}{4(u+vD)} & t < 0 \\ .\n\t \\end{array}\n\t \\right. \n\\ee\nThere is, consequently, a flat phase for $t < 0$ and a crumpled phase for \n$t >0$, separated by a crumpling transition at $t=0$ (see Fig.\\ref{fig__mfsoln}).\n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{crumpling.eps}}\n \\caption{Mean field solution for crystalline membranes.}\n \\label{fig__mfsoln}\n \\end{figure}\n\nThe actual phase diagram agrees qualitatively with the phase \ndiagram of the model shown schematically in Fig.\\ref{fig__PHAN}.\nThe crumpled phase is described by a line of equivalent FPs(GFP). There is \na general hyper-surface, whose projection onto the $\\kappa-t$ plane \ncorresponds to a one-dimensional curve (CTH), which corresponds to \nthe crumpling transition. Within the CTH there is an infrared stable \nFP (CTFP) which describes the large distance properties \nof the crumpling transition. Finally,\nfor large enough values of $\\kappa$ and negative values of $t$, the system\nis in a flat phase described by the corresponding infra-red stable\nFP (FLFP) \\footnote{The FLFP is actually a line of equivalent fixed\npoints.}. Although the precise phase diagram turns out to be slightly \nmore complicated than the one depicted in Fig.\\ref{fig__PHAN}, the \nadditional subtleties do not modify the general picture.\n \n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{pd_phantom.eps}}\n \\caption{Schematic plot of the phase diagram for phantom\n membranes. GFP are the equivalent FPs describing the crumpled phase.\n The crumpling transition is described by the Crumpling transition \n critical line (CTH), which contains the Crumpling Transition FP (CTFP). \n The Flat phase is described by the (FLFP).}\n \\label{fig__PHAN}\n \\end{figure}\n\nThe evidence for the phase diagram depicted in Fig.\\ref{fig__PHAN} \ncomes from combining the results of a variety of analytical and\nnumerical calculations. We present in detail the results obtained\nfrom the $\\vap$-expansion since they have wide applicability and allow\na systematic calculation of the $\\beta$-function and the critical\nexponents. We also describe briefly results obtained with other\napproaches.\n\n\\subsubsection{The crumpled phase}\\label{SUBSUBSECT__isocrphase}\n\nIn the crumpled phase, the free energy Eq.(\\ref{LAN_CR_PH}) for \n$D \\ge 2$ simplifies to\n\\be\\label{LAN_CR_PH_IRR}\nF({\\vec r})=\\frac{t}{2} \\int d^D {\\bf x} (\\parp_{\\alpha} {\\vec r})^2\n+\\mbox{Irrelevant terms} \\ ,\n\\ee\nsince the model is completely equivalent to a linear sigma model in\n$D \\le 2$ dimensions having $O(d)$ symmetry, and therefore all derivative \noperators in ${\\vec r}$ are irrelevant by power counting. The parameter \n$t$ labels equivalent Gaussian FPs, as depicted in \nFig.\\ref{fig__PHAN}. In RG language, it defines a \ncompletely marginal direction. This is true provided the condition \n$t>0$ is satisfied. The large distance properties of this phase are\ndescribed by simple Gaussian FPs and therefore the connected Green's\nfunction may be calculated exactly with result\n\\be\\label{CR_gg}\nG({\\bf x}) \\sim \\left\\{\n \\begin{array}{c l}\n\t |{\\bf x}|^{2-D} & D\\neq 2 \\\\\n\t \\log |{\\bf x}| & D=2\n\t \\end{array}\n\t \\right. \n\\ee\nThe associated critical exponents may also be computed exactly. \nThe Hausdorff dimension $d_H$, or equivalently the size exponent $\\nu=D/d_H$, \nis given (for the membrane case $D=2$) by \n\\be\\label{CR_Haus}\nd_H=\\infty \\ (\\nu = 0) \\rightarrow R_G^2 \\sim \\log L \\ .\n\\ee\nThe square of the radius of gyration $R_G^2$ scales logarithmically \nwith the membrane size $L$. This result is in complete \nagreement with numerical simulations of tethered membranes in \nthe crumpled phase where the logarithmic behavior of the radius \nof gyration \nis accurately checked\n\\cite{KKN:86,KKN:87,BEW:89,ADJ:89,RK:90,HW:91,WS:93,BET:94,W:96,BCFTA:96}.\nReviews may be found in \\cite{GK1:97,GK2:97}.\n\n\\subsubsection{The Crumpling Transition}\n\nThe Free energy is now given by\n\\be\\label{LAN_CRTR_PH}\nF({\\vec r})=\\int d^D{\\bf x} \\left[ \n\\frac{1}{2}(\\parp_{\\alpha}^2 {\\vec r})^2+\nu(\\parp_{\\alpha} {\\vec r} \\parp_{\\beta} {\\vec r} )^2+\n\\hat{v}(\\parp_{\\alpha} {\\vec r} \\parp^{\\alpha} {\\vec r})^2 \\right] \\ ,\n\\ee\nwhere the dependence on $\\kappa$ may be included in the couplings \n$u$ and $\\hat{v}$. With the leading term having two derivatives, the directions \ndefined by the couplings $u$ and $\\hat{v}$ are relevant by naive power counting \nfor $D \\le 4$. This shows that the model is \namenable to an $\\vap$-expansion with $\\vap=4-D$. For practical purposes,\nit is more convenient to consider the coupling $v=\\hat{v}+\\frac{u}{D}$.\nWe provide the detailed derivation of the corresponding $\\beta$ functions in \nAppendix~\\ref{APP__CT__FP}. The result is \n\\be\\label{beta_FLAT_TR}\n\\begin{array}{c c r}\n\\beta_u(u_R,v_R) &=& -\\vap u_R+\\frac{1}{8 \\pi^2}\\left\\{\n(d/3+65/12)u^2_R+6u_R v_R+4/3 v^2_R \\right\\} \\\\\n\\beta_v(u_R,v_R) &=& -\\vap v_R+\\frac{1}{8 \\pi^2}\\left\\{\n21/16u^2_R+21/2 u_R v_R+(4d+5) v^2_R \\right\\} \n\\end{array}\n\\ee\nRather surprisingly, this set of $\\beta$ functions does not possess a FP, \nexcept for $d > 219$. This result would suggest that the crumpling \ntransition is first order for $d=3$. Other estimates, however, give \nresults which are consistent with the crumpling transition being \ncontinuous. These are \n\\begin{itemize}\n\\item{Limit of large elastic constants}\\cite{DG:88}: \nThe Crumpling transition is approached from the flat phase, \nin the limit of infinite elastic constants.\nThe model is\n\\be\\label{CR_LGD}\nH_{NL}=\\int d^D\\sigma \\frac{\\kappa}{2} (\\Delta {\\vec r})^2 \\ ,\n\\ee\nwith the further constraint \n${\\parp_{\\alpha} \\vec r} {\\parp_{\\beta} \\vec r}= \\delta_{\\alpha \\beta}$.\nRemarkably, the $\\beta$-function may be computed within a large $d$ expansion,\nyielding a continuous crumpling transition with size exponent at \nthe transition (for $D=2$)\n\\be\\label{nu_exp}\nd_H=\\frac{2d}{d-1} \\rightarrow \\nu=1-\\frac{1}{d} \\ .\n\\ee\n\\item{SCSA Approximation}\\cite{LDR:92}: The Schwinger-Dyson equations for the model \ngiven by Eq.(\\ref{LAN_CRTR_PH}) are truncated to include up to four \npoint vertices. The result for the Hausdorff dimension and size exponent is\n\\be\\label{CR_SCSA}\n d_H=2.73 \\rightarrow \\nu=0.732 \\ .\n\\ee\n\n\\item{MCRG Calculation} \\cite{ET1:96}: The crumpling transition is studied using \nMCRG (Monte Carlo Renormalization Group) techniques. Again, the \ntransition is found to be continuous with exponents\n\\be\\label{CR_MCRG}\n d_H=2.64(5) \\rightarrow \\nu=0.85(9) \\ .\n\\ee\n\\end{itemize}\n\nEach of these three independent estimates give a continuous crumpling \ntransition with a size exponent in the range $\\nu \\sim 0.7 \\pm .15$.\nIt would be interesting to understand how the $\\vap$-expansion must be\nperformed in order to reconcile it with these results.\n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{spec.eps}}\n \\caption{Plot of the specific heat observable \\cite{BCFTA:96}. \n The growth of the specific heat peak with system size \n\t indicates a continuous transition.}\n \\label{fig__CR__cont}\n \\end{figure}\n\nFurther evidence for the crumpling transition being continuous is \nprovided by numerical simulations \n\\cite{KKN:86,KKN:87,BEW:89,ADJ:89,RK:90,HW:91,WS:93,BET:94,W:96,BCFTA:96} \nwhere the analysis of observables\nlike the specific heat (see Fig.\\ref{fig__CR__cont}) or the radius of gyration \nradius give textbook continuous phase transitions, although the precise \nvalue of the exponents at the transition are difficult to pin down. \nSince this model has also been explored numerically with different \ndiscretizations on several lattices, there is clear evidence \nfor universality of the crumpling transition \\cite{BET:94}, again consistent with a \ncontinuous transition. In Appendix~\\ref{APP__Dis} we present more\ndetails of suitable discretizations of the energy for numerical\nsimulations of membranes. \n\n\\subsubsection{The Flat Phase}\\label{SUB__subflat}\n\n\\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{quadrat_u.eps}}\n \\caption{Coordinates for fluctuations in the flat phase}\n \\label{fig__quadr_u}\n \\end{figure}\n\nIn a flat membrane (see Fig.\\ref{fig__quadr_u}), we consider the strain tensor\n\\be\\label{def_strain}\nu_{\\alpha \\beta}= \\partial_{\\alpha} u_{\\beta}+\\partial_{\\beta} u_{\\alpha}\n+\\partial_{\\alpha} h \\partial_{\\beta} h \\ .\n\\ee\nThe free energy Eq.(\\ref{LAN_CR_PH}) becomes\n\\be\\label{LAN_FL_PH}\nF({\\bf u},h)=\\int d^D {\\bf x} \\left[\n\\frac{\\hat{\\kappa}}{2} (\\parp_{\\alpha} \\parp_{\\beta} h )^2+\n\\mu u_{\\alpha \\beta} u^{\\alpha \\beta} + \\frac{\\lambda}{2} (u^{\\alpha}_{\\alpha})^2\n\\right] \\ ,\n\\ee\nwhere we have dropped irrelevant terms. \nOne recognizes the standard Landau Free energy of elasticity theory,\nwith Lam\\'e coefficients $\\mu$ and $\\lambda$, plus an extrinsic\ncurvature term, with bending rigidity $\\hat \\kappa$. These \ncouplings are related to the original ones\nin Eq.(\\ref{LAN_CR_PH}) by $\\mu=u\\zeta^{4-D}$, $\\lambda=2 v \\zeta^{4-D}$, \n$\\hat{\\kappa}=\\kappa \\zeta^{4-D}$ and \n$t=-4(\\mu+\\frac{D}{2}\\lambda)\\zeta^{D-2}$. \n\nThe large distance properties of the flat phase for crystalline membranes are\ncompletely described by the Free energy Eq.(\\ref{LAN_FL_PH}).\nSince the bending rigidity may be scaled out at the crumpling\ntransition, the free energy becomes a function of $\\frac{\\mu}{\\kappa^2}$ and\n$\\frac{\\lambda}{\\kappa^2}$. The $\\beta$-function for the couplings $u,v$\nat $\\kappa=1$ may be calculated within an $\\vap$-expansion, which we \ndescribe in detail in Appendix~\\ref{APP__FP__FP}. Let us recall that \nthe dependence on $\\kappa$ may be trivially restored at any stage. The \nresult is\n\\bea\\label{beta_CR_TR}\n\\beta_{\\mu}(\\mu_R,\\lambda_R)&=&-\\vap \\mu_R\n +\\frac{\\mu_R^2}{8 \\pi^2}(\\frac{d_c}{3}+20A) \n \\\\ \\nonumber\n\\beta_{\\lambda}(\\mu_R,\\lambda_R) &=& -\\vap \\lambda_R+\n \\frac{1}{8 \\pi^2}(\\frac{d_c}{3} \\mu^2_R+2 (d_c+10A) \\lambda_R \\mu_R\n +2 d_c \\lambda^2_R) \\ ,\n\\eea\nwhere $d_c=d-D$, and $A=\\frac{\\mu_R+\\lambda_R}{2 \\mu_R+\\lambda_R}$.\nThese $\\beta$ functions show four fixed points whose actual \nvalues are shown in Table~\\ref{TAB__FL_EXP}. \n\n \\begin{figure}[htb]\n \\epsfxsize=3 in \\centerline{\\epsfbox{pd_flat.eps}}\n \\caption{Phase diagram for the phantom flat phase. There are three\n infra-red unstable FPs, labelled by FLP1, FLP2 and FLP3, but\n the physics of the flat phase is governed by the infra-red stable FP\n (FLFP).}\n \\label{fig__PD_FLAT}\n \\end{figure}\n\nAs apparent from Fig.\\ref{fig__PD_FLAT}, the phase diagram of the \nflat phase turns out to be slightly more involved than the one\nshown in Fig.\\ref{fig__PHAN}, as there are three FPs \nin addition to the FLFP already introduced. These additional FPs\nare infra-red unstable, however, and can only be reached for very specific values \nof the Lam\\'e coefficients, so for any practical situation we can \nregard the FLFP as the only existing FP in the flat phase.\n\n \\begin{table}[htb]\n \\centerline{\n \\begin{tabular}{|c|c|c|c|c|}\\hline\n FP & $\\mu^\\ast_R$ & $\\lambda_R^\\ast$ & $\\eta$ & $\\eta_u$\\\\\\hline\n FP1 & $0$ & $0$ & $0$ & $0$ \\\\\\hline \n FP2 & $0$ & $2\\vap/d_c$ & $0$ & $0$ \\\\\\hline \n FP3 & $\\frac{12 \\vap}{20+d_c}$ & $\\frac{-6 \\vap}{20+d_c}$ \n & $\\frac{\\vap}{2+d_c/10} $ & $\\frac{\\vap}{1+20/d_c}$ \\\\\\hline \n FLFP & $\\frac{12 \\vap}{24+d_c}$ & $\\frac{-4 \\vap}{24+d_c}$ \n & $\\frac{\\vap}{2+d_c/12}$ & $\\frac{\\vap}{1+24/d_c}$ \\\\\\hline \n \\end{tabular}}\n \\caption{The FPs and critical exponents of the flat phase.}\n \\label{TAB__FL_EXP}\n \\end{table}\n\n\\bigskip\n\n\\noindent {\\bf The properties of the flat phase}\n\\medskip\n\nThe flat phase is a very important phase as will be apparent once we\nstudy the full model, including self-avoidance. For that reason we turn\nnow to a more detailed study of its most important properties.\n\nFig.\\ref{fig__PHASES} (c) gives an intuitive visualization of\na crystalline membrane in the flat phase. The membrane is essentially a \nflat two dimensional object up to fluctuations in the \nperpendicular direction. The rotational symmetry of the model \nis spontaneously broken, being reduced from $O(d)$ to $O(d-D)\\times\nO(D)$. The remnant rotational symmetry is realized in \nEq.(\\ref{LAN_FL_PH}) as\n\\bea\\label{sym__trans}\nh_i({\\bf x}) &\\rightarrow & h_i({\\bf x})+A^{i \\alpha} {\\bf x}_{\\alpha}\n\\\\\\nonumber\nu_{\\alpha}({\\bf x}) &\\rightarrow& u_{\\alpha} - A^{i \\alpha} h_i \n-\\frac{1}{2} \\delta^{i j} (A^{i \\alpha} A^{\\beta j} {\\bf x}_{\\beta})\n\\ ,\n\\eea\nwhere $A^{i \\alpha}$ is a $D \\times (d-D)$ matrix. This relation\nis very important as it provides Ward identities which simplify \nenormously the renormalization of the theory.\n\nLet us first study the critical exponents of the model. There are\ntwo key correlators, involving the in-plane and the out-of-plane phonon\nmodes. Using the RG equations, it is easy to realize that at any\ngiven FP, the low-$p$ limit of the model is given by\n\\bea\\label{low_q_mode}\n\\Gamma_{u u}({\\vec p}) &\\sim & |{\\vec p}|^{2+\\eta_u}\n\\\\\\nonumber\n\\Gamma_{h h}({\\vec p}) &\\equiv& |{\\vec p}|^4 \\kappa({\\vec p}) \\sim\n|{\\vec p}|^{4-\\eta} \\ ,\n\\eea\nwhere the last equation defines the anomalous elasticity \n$\\kappa({\\vec p})$ as a function of momenta ${\\vec p}$. These two\nexponents are not independent, since they satisfy the scaling \nrelation \\cite{AL:88} \n\\be\\label{FLAT__scaling}\n\\eta_u=4-D-2\\eta \\ ,\n\\ee\nwhich follows from the Ward identities associated with the remnant\nrotational symmetry (Eqn.(\\ref{sym__trans}). Another important exponent is the \nroughness exponent $\\zeta$, which measures the fluctuations transverse to\nthe flat directions. It can be expressed as $\\zeta=\\frac{4-D-\\eta}{2}$.\n\nThe long wavelength properties of the flat phase are described by the \nFLFP (see Fig.\\ref{fig__PD_FLAT}). Since the FLFP occurs at non-zero renormalized\nvalues of the Lam\\'e coefficients, the associated critical exponents\ndiscussed earlier are clearly non-Gaussian. Within an $\\vap$-expansion, the values for\nthe critical exponents are given in Table~\\ref{TAB__FL_EXP}. There are\nalternative estimates available from different methods. These are\n\\begin{itemize}\n\\item{Numerical Simulation:} In \\cite{BCFTA:96} a large scale simulation of\nthe model was performed using very large meshes. The results obtained for the \ncritical exponents are very reliable, namely \n\\be\\label{Sim__exponents}\n \\begin{array}{l l l}\n\t \\eta_u=0.50(1) & \\eta=0.750(5) & \\zeta=0.64(2) \n\t \\end{array}\n\\ee\nFor a review of numerical results see \\cite{GK1:97,GK2:97}.\n\\item{SCSA Approximation:} This consists of suitably truncating the \nSchwinger-Dyson equations to include up to four-point correlation\nfunctions \\cite{LDR:92}. The result for general $d$ is\n\\be\\label{SCSA_flat}\n\\eta(d)=\\frac{4}{d_c+(16-2d_c+d^2_c)^{1/2}} \\ ,\n\\ee\nwhich for $d=3$ gives\n\\be\\label{SCSA_flat_d3}\n \\begin{array}{l l l}\n \\eta_u=0.358 & \\eta=0.821 & \\zeta=0.59 \n \\end{array}\n\\ee\n\\item{Large d expansion:} The result is \\cite{DG:88}\n\\be\\label{Larged_flat}\n\\eta=\\frac{2}{d} \\rightarrow \\eta(3)=2/3\n\\ee\n\\end{itemize}\nWe regard the results of the numerical simulation as our most accurate \nestimates, since we can estimate the errors. The results obtained from the\nSCSA, which are the best analytical estimate, are in acceptable agreement\nwith simulations.\n\nFinally there are two experimental measurements \nof critical exponents for the flat phase of crystalline membranes.\nThe static structure factor of the red blood cell cytoskeleton \n(see Sect.\\ref{SECT__Intro}) has been measured by small-angle x-ray\nand light scattering, yielding a roughness exponent of\n$\\zeta=0.65(10)$ \\cite{Skel:93}. Freeze-fracture electron microscopy\nand static light scattering of the conformations of graphitic oxide\nsheets (Sect.\\ref{SECT__Intro}) revealed flat sheets with a fractal\ndimension $d_H=2.15(6)$. Both these values are in good agreement with\nthe best analytic and numerical predictions, but the errors are still too \nlarge to discriminate between different analytic calculations. \n\nThe Poisson ratio of a crystalline membrane (measuring the transverse \nelongation due to a longitudinal stress \\cite{Landau7}) is universal and within the\nSCSA approximation, which we regard as the more accurate analytical estimate,\nis given by\n\\be\\label{FL_PR}\n\\sigma(D)=-\\frac{1}{D+1} \\rightarrow \\sigma(2)=-1/3 \\ ,\n\\ee\nThis result has been accurately checked in numerical simulations \n\\cite{FBGT:97}. Rather remarkably, it turns out to be negative. \nMaterials having a negative Poisson ratio are called {\\em auxetic}. \nThis highlights potential applications of crystalline membranes to \nmaterials science since auxetic materials have a wide variety \nof potential applications as gaskets, seals etc.\n\nFinally another critical regime of a flat membrane is achieved by\nsubjecting the membrane to external tension \\cite{GDLP:88,GDLP:89}.\nThis allows a low temperature phase in which the membrane has a domain\nstructure, with distinct domains corresponding to flat phases with \ndifferent bulk orientations. This describes, physically, a {\\em\nbuckled} membrane whose equilibrium shape is no longer planar. \n \n\n\\subsection{Self-avoiding}\\label{Sub_SECT__SA} \n\nSelf-avoidance is a necessary interaction in any realistic description\nof a crystalline membrane. It is introduced in the form of a delta-function\nrepulsion in the full model Eq.(\\ref{LAN_CR_VER}). We have already analyzed the \nphantom case and explored in detail the distinct phases. \nThe question before us now is the effect of self-avoidance on each\nof these phases.\n\nThe first phase we analyze is the flat phase. Since self-intersections\nare unlikely in this phase, it is intuitively clear that self-avoidance should\nbe irrelevant. This may also be seen if one neglects \nthe effects of the in-plane phonons. In the self-avoiding term for the flat phase\nwe have\n\\bea\\label{SA__term__sim}\n&&\\frac{b}{2}\n\\int d^D{\\bf x}\\, d^D{\\bf y} \\delta^d({\\vec r}({\\bf x})-\n{\\vec r}({\\bf y}))\n\\\\\\nonumber\n&=&\\frac{b}{2}\\int d^D{\\bf x}\\, d^D{\\bf y} \n\\delta^{D}(\\zeta({\\bf x}-{\\bf y})+{\\bf u}({\\bf x})-{\\bf u}({\\bf y}) )\n\\delta^{d-D}(h({\\bf x})-h({\\bf y})) \n\\\\\\nonumber\n&\\sim&\\frac{b}{2}\\int d^D{\\bf x}\\, d^D{\\bf y} \n\\delta^{D}(\\zeta({\\bf x}-{\\bf y}))\n\\delta^{d-D}(h({\\bf x})-h({\\bf y}))=0 \\ ,\n\\eea\nas the trivial contribution where the membrane equals itself is \neliminated by regularization. The previous argument receives \nadditional support from numerical simulations in the\nflat phase, where it is found that self-intersections are extremely rare\nin the typical configurations appearing in those simulations. It seems \nclear that self-avoidance is most likely an irrelevant operator, in the \nRG sense, of the FLFP. Nevertheless, it would be very interesting if one\ncould provide a more rigorous analytical proof for this statement.\nA rough argument can be made as follows. \nShortly we will see that the Flory approximation for self-avoiding\nmembranes predicts a fractal dimension $d_H=2.5$. For bulk dimension\n$d$ exceeding 2.5 therefore we expect self-avoidance to be irrelevant.\nA rigorous proof of this sort remains rather elusive, as it involves\nthe incorporation of both self-avoidance and non-linear elasticity, and this remains \na difficult open problem.\n\nThe addition of self-avoidance in the crumpled phase consists of \nadding the self-avoiding interaction to the free energy \nof Eq.(\\ref{LAN_CR_PH_IRR}) \n\\be\\label{CRU__SA}\nF({\\vec r})=\\frac{1}{2}\\int d^D {\\bf x} (\\parp_{\\alpha} {\\vec r}({\\bf x}))^2\n+\\frac{b}{2}\\int d^D{\\bf x}\\, d^D{\\bf y} \\delta^d({\\vec r}({\\bf x})-\n{\\vec r}({\\bf y})) \\ ,\n\\ee\nwhich becomes the natural generalization of the Edwards' model for polymers to \n$D$-dimensional objects. Standard power counting shows that the \nGFP of the crumpled phase is infra-red unstable \nto the self-avoiding perturbation for\n\\be\\label{SA__eps}\n\\vap(D,d)\\equiv 2D-d\\frac{2-D}{2} > 0 \\ ,\n\\ee\nwhich implies that self-avoidance is a relevant perturbation for $D=2$-objects\nat any embedding dimension $d$. The previous remarks make it apparent that it\nis possible to perform an $\\vap$-expansion of the model \n\\cite{Dup:87,AL:87,KN3:87,KN1:88}.\nIn Appendix~\\ref{APP__SA__CP}, we present the calculation of \nthe $\\beta$-function at lowest order in $\\vap$ using \nthe MOPE (Multi-local-operator-product-expansion) formalism \\cite{DDG:94,DDG:97}.\nThe MOPE formalism has the advantage that it is more easily \ngeneralizable to higher orders in $\\vap$, and enables concrete proofs showing\nthat the expansion may be carried out to all orders. At lowest order,\nthe result for the $\\beta$-function is\n\\bea\\label{SA__CR__B}\n\\beta_b(b_R)&=&-\\vap b_R+\\frac{(2-D)^{-1+\\frac{d}{2}}}{(4\\pi)^{\\frac{d}{2}}}\n\\left(\\frac{2 \\pi^{\\frac{D}{2}}}{\\Gamma(D/2)}\\right)^{2+\\frac{d}{2}}\\left[\n\\frac{\\Gamma(\\frac{D}{2-D})^2}{\\Gamma(\\frac{2D}{2-D})}+\n\\frac{d}{2}\\frac{(2-D)^2}{2D}\\right]\\frac{b^2_R}{2}\n\\nonumber\\\\\n&\\equiv&-\\vap b_R+a_1 b_R^2 \\ .\n\\eea\nThe infra-red stable FP is given at lowest order in $\\vap$ by\n$b^{\\ast}_R=\\frac{\\vap}{a_1}$, which clearly shows that the GFP of the \ncrumpled phase is infra-red unstable in the presence of self-avoidance.\n\nThe preceding results are shown in Fig.\\ref{fig__CR_SA} and may be\nsummarized as\n\\begin{itemize}\n\\item{The flat phase of self-avoiding crystalline membranes is exactly the \nsame as the flat phase of phantom crystalline tethered membranes.}\n\\item{The crumpled phase of crystalline membranes is destabilized by the \npresence of any amount of self-avoidance.}\n\\end{itemize}\n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{Add__SA.eps}}\n \\caption{The addition of Self-avoidance at the Crumpled and Flat \n phases.}\n \\label{fig__CR_SA}\n \\end{figure}\n\nThe next issue to elucidate is whether this new SAFP describes a \ncrumpled self-avoiding phase or a flat one and to give a more quantitative\ndescription of the critical exponents describing the universality class.\nSupposing that the SAFP is, in fact, flat we must understand its\nrelation to the FLFP describing the physics of the flat phase \nand the putative phase transitions between these two.\n\n\\subsubsection{The nature of the SAFP}\n\nLet us study in more detail the model described in Eq.(\\ref{CRU__SA}). The\nkey issue is whether this model still admits a crumpled phase, and if\nso to determine the associated size exponent. \nOn general grounds we expect that there is a critical dimension\n$d_c$, below which there is no crumpled phase (see Fig.\\ref{fig__SA_scenario}).\n\n \\begin{figure}[htb]\n \\epsfxsize=3 in \\centerline{\\epsfbox{Gen__nu.eps}}\n \\caption{The size exponent as a function of $d$. \n\t There is a critical dimension $d_c$ below which the\n crumpled phase disappears.}\n \\label{fig__SA_scenario}\n \\end{figure}\n\nAn estimate for the critical dimension may be obtained from a Flory\napproximation in which minimizes the free energy obtained by replacing \nboth the elastic and self-avoiding terms with the radius of gyration \nraised to the power of the appropriate scaling dimensions. Within \nthe Flory treatment a $D$-dimensional membrane is in a crumpled phase, \nwith a size exponent given by\n\\be\\label{Flory_est}\n\\nu=(D+2)/(d+2) \\ .\n\\ee\nFrom this it follows that $d_c=D$ (see Fig.\\ref{fig__SA_scenario}).\nThe Flory approximation, though very accurate for polymers ($D=1$), remains \nan uncontrolled approximation. \n\nIn contrast the $\\vap$-expansion provides a systematic determination\nof the critical exponents. For the case of membranes, however, some extrapolation\nis required, as the upper critical dimension is infinite.\nThis was done in \\cite{Hwa:90}, where it is shown that reconsidering \nthe $\\vap$-expansion as a double expansion in $\\vap$ and $D$, critical \nquantities may be extrapolated for $D=2$-dimensional objects. At lowest\norder in $\\vap$, the membrane is in a crumpled phase. The enormous task \nof calculating the next correction ($\\vap^2$) was successfully carried \nout in \\cite{DaWi:96}, employing more elaborate extrapolation methods than \nthose in \\cite{Hwa:90}. Within this calculation, the $d=3$ membrane is \nstill in a crumpled phase, but with a size exponent now closer to 1. It \ncannot be ruled out that the $\\vap$-expansion, successfully carried out to \nall orders could give a flat phase $\\nu=1$. In fact, \nthe authors in \\cite{DaWi:96,WiDa:97} present some arguments in favor of a \nscenario of this type, with a critical dimension $d_c\\sim 4$.\n\nOther approaches have been developed with different results. A Gaussian \napproximation was developed in \\cite{Gou:91,LeD:92}. The size exponent of \na self-avoiding membrane within this approach is\n\\be\\label{Gauss_est}\n\\nu=4/d ,\n\\ee\nand since one has $\\nu > 1$ for $d \\leq 4$, one may conclude that the\nmembrane is flat for $d \\leq d_c=4$. Since we cannot determine the accuracy of \nthe Gaussian approximation this estimate must be viewed largely as interesting \nspeculation. Slightly more elaborate arguments of this type \\cite{GuPa:92} \nyield an estimated critical dimension $d_c=3$.\n\n\\vskip 0.5cm\n\\noindent {\\bf Numerical simulations}\n\\vskip 0.5cm\n\nWe have seen that numerical simulations provide good support for\nanalytic results in the case of phantom membranes. \nWhen self-avoidance is included, numerical simulations become\ninvaluable, since analytic results are harder to come by.\nIt is for this reason that we discuss them in greater detail than in \nprevious sections.\n\nA possible discretization of membranes with excluded volume\neffects consists of a network of $N$ particles arranged in a triangular\narray. Nearest neighbors interact with a potential\n\\be\\label{tether_pot}\nV_{NN}({\\vec r})=\\left\\{ \\begin{array}{c c} \n 0 & \\mbox{for $|{\\vec r}|< b$ } \\\\\n\t \\infty & \\mbox{for $|{\\vec r}| > b $}\n\t \\end{array} \\right.\n\t\t\t \\ ,\n\\ee\nalthough some authors prefer a smoothened version, with the same \ngeneral features. The quantity $b$ is of the order of a few lattice \nspacings. This is a lattice version of the elastic term in\nEq.(\\ref{CRU__SA}). The discretization of the self-avoidance \nis introduced as a repulsive hard sphere potential, now acting \nbetween any two atoms in the membrane, instead \nof only nearest neighbors. A hard sphere repulsive potential is, for example,\n\\be\\label{exc_potential}\nV_{Exc}({\\vec r})=\\left\\{ \\begin{array}{c c} \n \\infty & \\mbox{for $|{\\vec r}|< \\sigma $} \\\\\n\t 0 & \\mbox{ for $|{\\vec r}| > \\sigma $ }\n\t \\end{array} \\right.\n\t\t\t \\ ,\n\\ee\nwhere $\\sigma$ is the range of the potential, and $\\sigma < b$. Again, some\nsmoothened versions, continuous at $|{\\vec r}|=\\sigma$, have also been\nconsidered. This model may be pictured as springs, defined by the \nnearest-neighbor potential Eq.(\\ref{tether_pot}), with excluded volume \neffects enforced by balls of radius $\\sigma$\n(Eq.(\\ref{exc_potential})). This model represents a lattice discretization of\nEq.(\\ref{CRU__SA}).\n \nEarly simulations of this type of model \\cite{KKN:86,KKN:87} provided a first estimate \nof the size exponent at $d=3$ fully compatible with the Flory estimate\nEq.(\\ref{Flory_est}). The lattices examined were not very large,\nhowever, and subsequent simulations with larger volumes \\cite{PB:88,ARP:89} found that \nthe $d=3$ membrane is actually flat. This result is even more\nremarkable if one recalls that there is no explicit bending \nrigidity.\n\n \\begin{figure}[htb]\n \\epsfxsize=3 in \\centerline{\\epsfbox{extr__curv.eps}}\n \\caption{Visualization of bending rigidity generated by a hard sphere potential.\n Normals ${\\vec n}_1$ and ${\\vec n}_2$ cannot be anti-parallel.}\n \\label{fig__SA__ext}\n \\end{figure}\n\nThe flat phase was a very surprising result, in some conflict with \nthe insight provided from the analytical estimates discussed in the previous\nsubsection. An explanation for it came from the observation \\cite{AN1:90}\nthat excluded volume effects induce bending rigidity, as depicted in\nFig.\\ref{fig__SA__ext}. The reason is that the excluded volume effects\ngenerate a non-zero expectation value for the bending rigidity, since the\nnormals can be parallel, but not anti-parallel\n(see~\\ref{fig__SA__ext}). \nThis induced bending rigidity was estimated and found to be big enough \nto drive the self-avoiding membrane well within the flat phase of \nthe phantom one. This means that this particular discretization of the\nmodel renders any potential SAFP inaccessible and the physics is\ndescribed by the FLFP. In \\cite{AN2:90} the structure function of the\nself-avoiding model is numerically computed and found to compare well \nwith the analytical structure function for the flat phase of phantom \ncrystalline membranes, including comparable roughness exponents.\n\nThe natural question then to ask is whether it is possible to reduce\nthe bending rigidity sufficiently to produce a crumpled self-avoiding\nphase. Subsequent studies addressed this issue in various ways.\nThe most natural way is obviously to reduce the range of the potential \nsufficiently that the induced bending rigidity is within the crumpled\nphase. This is the approach followed in \\cite{BLLP:89}. \nThe flat phase was found to persist to very small \nvalues of $\\sigma$, with eventual signs of a crumpled phase. \nThis crumpled phase may essentially be due to the elimination at\nself-avoidance at sufficiently small $\\sigma$. A more comprehensive\nstudy, in which the same limit is performed this time with an excluded volume potential \nwhich is a function of the internal distance along the lattice \\cite{KK:93},\nconcluded that for large membranes, inclusion of excluded volume effects, no\nmatter how small, leads to flatness. A different approach to weakening the\nflat phase, bond dilution \\cite{GM:90}, found that the flat phase persists \nuntil the percolation critical point. In conclusion the bulk of\naccumulated evidence indicates that flatness is an intrinsic\nconsequence of self-avoidance. If this is indeed correct the\nSAFP coincides with FLFP and this feature is an inherent consequence\nof self-avoidance, rather than an artifact of discretization. \n\nGiven the difficulties of finding a crumpled phase with a repulsive potential, \nsimulations for larger values of the embedding space dimension $d$ have also\nbeen performed \\cite{Grest:91,BP:94}. These simulations show clear evidence that\nthe membrane remains flat for $d=3$ and $4$ and undergoes a crumpling transition\nfor $d \\geq 5$, implying $d_c \\geq 4$.\n\nAn alternative approach to incorporating excluded volume effects corresponds\nto discretize a surface with a triangular lattice and imposing the self-avoidance \nconstraint by preventing the triangular plaquettes from inter-penetrating. \nThis model has the advantage that is extremely flexible, since there \nis no restriction on the bending angle of adjacent plaquettes\n(triangles) and therefore no induced bending rigidity (see \\ref{APP__Dis}).\n\nThe first simulations of the plaquette model \\cite{B:91} found a size exponent\nin agreement with the Flory estimate Eq.~\\ref{Flory_est}. A subsequent \nsimulation \\cite{KG:93} disproved this result, and found a size \nexponent $\\nu=0.87$, higher than the Flory estimate, but below one. \nMore recent results using larger lattices and more sophisticated\nalgorithms seem to agree completely with the results obtained from the\nball and spring models \\cite{BCTT:2000}.\n\nFurther insight into the lack of a crumpled phase for self-avoiding\ncrystalline membranes is found in the study of folding \n\\cite{DiFGu1:94,DiFGu2:94,BDGG:95,BDGG:96,BGGM:97,CGP1:96,CGP2:96}.\nThis corresponds to the limit of infinite elastic constants studied\nby David and Guitter with the further approximation that the space of\nbending angles is discretized. One quickly discovers that the\nreflection symmetries of the allowed\nfolding vertices forbid local folding (crumpling) of surfaces.\nThere is therefore essentially no entropy for crumpling. There is,\nhowever, local unfolding and the resulting statistical mechanical\nmodels are non-trivial. The lack of local folding is the discrete \nequivalent of the long-range curvature-curvature interactions that\nstabilize the flat phase. The dual effect of the integrity of the\nsurface (time-independent connectivity) and self-avoidance is so\npowerful that crumpling seems to be impossible in low embedding dimensions. \n \n\\subsubsection{Attractive potentials}\n\nSelf-avoidance, as introduced in Eq.(\\ref{exc_potential}) is a totally \nrepulsive force among monomers. There is the interesting possibility of\nallowing for attractive potentials also. This was pioneered in \n\\cite{AN1:90} as a way to escape to the induced bending rigidity \nargument (see Fig.\\ref{fig__SA__ext}), since an attractive potential \nwould correspond to a negligible (or rather a negative) bending \nrigidity. Remarkably, in \\cite{AN1:90}, a compact (more crumpled) \nself-avoiding phase was found, with fractal dimension close to 3.\n\nThis was further studied in \\cite{AK:91}, where it was found that with \nan attractive Van der Waals potential, the \ncrystalline membrane underwent a sequence of folding transitions leading to a\ncrumpled phase. In \\cite{LP:92} similar results were found, but instead of\na sequence of folding transitions a crumpled phase was found with an\nadditional compact (more crumpled phase) at even lower temperatures. \nSubsequent work gave some support to this scenario \\cite{GP:94}.\n\nOn the analytical side, the nature of the $\\Theta$-point for membranes\nand its relevance to the issue of attractive interactions \nhas been addressed in \\cite{DaWi:95}.\n\nWe think that the study of a tether with an attractive potential remains\nan open question begging for new insights.\nA thorough understanding of the nature of the compact phases produced by\nattractive interactions would be of great value. \n\n\n\\subsubsection{The properties of the SAFP}\n\n\\begin{figure}[htb]\n\\epsfxsize=3in\n\\centerline{\\epsfbox{SA_pd_con.eps}}\n\\caption{The conjectured phase diagram for self-avoiding crystalline\nmembranes in $d=3$. With any degree of self-avoidance the flows are to\nthe flat phase fixed point of the phantom model (FLFP).}\n\\label{fig__SApd}\n\\end{figure}\n\nThe enormous efforts dedicated to study the SAFP have not resulted in a\ncomplete clarification of the overall scenario since the existing \nanalytical tools do not provide a clear picture. Numerical results\nclearly provide the best insight. For the physically relevant case \n$d=3$, the most plausible situation is that there is no crumpled phase \nand that the flat phase is identical to the flat phase of the phantom\nmodel. For example, the roughness exponents\n$\\zeta_{SA}$ from numerical simulations of self-avoidance at $d=3$ using\nball-and-spring models \\cite{Grest:91} and the roughness exponent at the FLFP, \nEq.(\\ref{Sim__exponents}), compare extremely well\n\\be\\label{COMP_SA_FL}\n\\zeta_{SA}=0.64(4) \\ , \\ \\zeta=0.64(2) \\ ,\n\\ee\nSo the numerical evidence allows us to conjecture that the SAFP is exactly\nthe same as the FLFP, and that the crumpled self-avoiding phase is\nabsent in the presence of purely repulsive potentials (see Fig.\\ref{fig__SApd}). \nThis identification of fixed points enhances the significance of the\nFLFP treated earlier. \nIt would be very helpful if analytical tools were developed to \nfurther substantiate this statement.\n\n\\section{Anisotropic Membranes}\\label{SECT__POLYMEM_ANI}\n\nAn anisotropic membrane is a crystalline membrane having the property \nthat the elastic or the bending rigidity properties in one \ndistinguished direction are different from those in the $D-1$\nremaining directions. As for the isotropic case we keep the \ndiscussion general and describe the membrane\nby a $d$-dimensional ${\\vec r}({\\bf x}_{\\perp},y)$, where now the\n$D$ dimensional coordinates are split into $D-1 \\ \\ {\\bf x}_{\\perp}$\ncoordinates and the orthogonal distinguished direction $y$.\n\nThe construction of the Landau free energy follows the same steps\nas in the isotropic case. Imposing translational invariance,\n$O(d)$ rotations in the embedding space and $O(D-1)$\nrotations in internal space, the equivalent of Eq.(\\ref{LAN_CR_VER})\nis now\n\\bea\\label{LGW}\nF(\\vec r({\\bf x}))&=& \\frac{1}{2} \\int d^{D-1}{\\bf x}_{\\perp} dy \\left[\n\\kappa_{\\perp}(\\partial_{\\perp}^2 \\vec r)^2 + \\kappa_y (\\pary^2 \\vec\nr)^2 \\right.\n\\nonumber\\\\\n&& + \\kappa_{\\perp y} \\pary^2 \\vec r \\cdot \\parp_{\\perp}^2 \\vec r +\nt_{\\perp}(\\parp_{\\alpha}^{\\perp} \\vec r)^2 + t_y(\\pary \\vec r)^2\n\\nonumber\\\\\n&& + \\frac{u_{\\perp \\perp}}{2}(\\parp_\\alpha^{\\perp} \\vec r \\cdot\n\\parp_{\\beta}^{\\perp} \\vec r)^2 + \\frac{u_{yy}}{2}(\\pary \\vec r \\cdot\n\\pary \\vec r)^2\n\\nonumber\\\\\n&& + u_{\\perp y} (\\parp_{\\alpha}^{\\perp} \\vec r \\cdot \\pary \\vec r)^2\n+ \\frac{v_{\\perp \\perp}}{2}(\\parp_{\\alpha}^{\\perp} \\vec r \\cdot\n\\parp_{\\alpha}^{\\perp} \\vec r)^2\n\\nonumber\\\\\n&& \\left. + v_{\\perp y}(\\parp_{\\alpha}^{\\perp} \\vec r)^2\n(\\pary \\vec r)^2 \\right]\n\\nonumber\\\\\n&&+ \\frac{b}{2} \\int d^D {\\bf x} \\int d^D {\\bf x}^\\prime \\delta^d\n(\\vec r({\\bf x}) - \\vec r({\\bf x}^\\prime)).\n\\eea\nThis model has eleven parameters, representing distinct physical \ninteractions:\n\\begin{itemize}\n\\item{ $ \\kappa_{\\perp},\\kappa_y,\\kappa_{\\perp y} $ \n\\underline{bending rigidity}:} the anisotropic versions of \nthe isotropic bending rigidity splits into three distinct terms.\n\n\\item{ $t_{\\perp},t_y, u_{\\perp \\perp}, u_{yy}, v_{\\perp \\perp}, v_{\\perp y}$\n\\underline{elastic constants}:} there are six quantities describing the \nmicroscopic elastic properties of the anisotropic membrane.\n\n\\item{ $b$, \\underline{self-avoidance coupling}:} This particular\nterm is identical to its isotropic counterpart. \n\\end{itemize}\n\nFollowing the same steps as in the isotropic case, we split\n\\be\\label{mf_variable_any}\n{\\vec r}({\\bf x})=(\\zeta_{\\perp} {\\bf x}_{\\perp}+{\\bf u}_{\\perp}({\\bf x}),\n\\zeta_y y +u_y({\\bf x}), h({\\bf x})) \\ ,\n\\ee\nwith ${\\bf u}_{\\perp}$ being the $D-1$-dimensional phonon in-plane modes, \n$u_y$ the in-plane phonon mode in the distinguished direction $y$ and\n$h$ the $d-D$ out-of-plane fluctuations. If $\\zeta_{\\perp}=\\zeta_y=0$,\nthe membrane is in a crumpled phase and if both\n$\\zeta_{\\perp} \\ne 0$ and $\\zeta_y \\neq 0$ the membrane is in a flat \nphase very similar to the isotropic case (how similar will be\ndiscussed shortly). There is, however, the possibility that \n$\\zeta_{\\perp}=0$ and $\\zeta_y \\ne 0$ or $\\zeta_{\\perp} \\ne 0$ and\n$\\zeta_y = 0$. This describes a completely new phase, in which the membrane\nis crumpled in some internal directions but flat in the remaining ones.\nA phase of this type is called a {\\em tubular phase} and does not\nappear when studying isotropic membranes. \nIn Fig.\\ref{fig__CRTUFL} we show an intuitive visualization of a\ntubular phase along with the corresponding flat and crumpled phases \nof anisotropic membranes.\n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{CRTUFL.eps}}\n \\caption{Examples of a) the tubular phase b) the crumpled phase and\n c) the flat phase of an anisotropic phantom crystalline membrane\n taken from the simulations of \\cite{BFT:97}}\n \\label{fig__CRTUFL}\n \\end{figure}\n\nWe will start by studying the phantom case. We show, using both\nanalytical and numerical arguments, that the phase diagram contains\na crumpled, tubular and flat phase. The crumpled and flat phases are \nequivalent to the isotropic ones, so anisotropy turns out to be an\nirrelevant interaction in those phases. The new physics is contained\nin the tubular phase, which we describe in detail, both \nwith and without self-avoidance.\n\n\\subsection{Phantom}\\label{SECT__phase_ani}\n\n\\subsubsection{The Phase diagram}\\label{SubSECT__tub}\n\nWe first describe the mean field theory phase diagram and then \nthe effect of fluctuations. \nThere are two situations depending on the particular \nvalues of the function $\\Delta$, which depends on the \nelastic constants $u_{\\perp \\perp},v_{\\perp y},u_{yy}$ and $v_{\\perp \\perp}$.\nSince the derivation is rather technical, we refer to Appendix~\\ref{MF__Appen} \nfor the details.\n\n\\begin{itemize}\n{\\item Case A} ($ \\Delta > 0$): the mean field solution displays all possible phases.\nWhen $t_y > 0$ and $t_{\\perp}>0$ the model is in a crumpled phase. \nLowering the temperature, one of the $t$ couplings becomes \nnegative, and we reach a tubular phase (either $\\perp$ or $y$-tubule).\nA further reduction of the temperature eventually leads to a flat phase.\n\n{\\item Case B} ($ \\Delta < 0$): in this case the flat phase disappears from the mean\nfield solution. Lowering further the temperature leads to a continuous \ntransition from the crumpled phase to a tubular phase. Tubular phases\nare the low temperature stable phases in this regime.\n\\end{itemize}\nThis mean field result is summarized in Fig.\\ref{fig__MF_an}. \n\n \\begin{figure}[htb]\n \\epsfxsize=5 in \\centerline{\\epsfbox{meanf_ani.eps}}\n \\caption{The phase diagram for anisotropic phantom membranes} \n \\label{fig__MF_an}\n \\end{figure}\n\nBeyond mean field theory, the Ginsburg criterion applied to this model\ntells us that the phase diagram should be stable for physical membranes\n$D=2$ at any embedding dimension $d$, so the mean field scenario should \ngive the right qualitative picture for the full model.\n\nNumerical simulations have spectacularly confirmed this result. We have \nalready shown in Fig.\\ref{fig__CRTUFL} the results from the numerical \nsimulation in \\cite{BFT:97}, where it was shown that changing the\ntemperature generates a sequence of transitions crumpled-to-tubular \nand tubular-to-flat, in total agreement with case A) in the mean field result\nillustrated in Fig.\\ref{fig__MF_an}. \n\nWe now turn to a more detailed study of both the crumpled and\nflat anisotropic phases. Since we have already studied crumpled and \nflat phases we just outline how those are modified when anisotropy\nis introduced.\n\n\\subsubsection{The Crumpled Anisotropic Phase}\\label{SubSECT__cr_ani}\n\nIn this phase $t_y>0$ and $t_{\\perp} >0$, and \nthe free energy Eq.(\\ref{LGW}) reduces for $D \\ge 2$ to \n\\be\\label{crum_ani_fe}\nF(\\vec r({\\bf x}))=\\frac{1}{2} \\int d^{D-1}{\\bf x}_{\\perp} dy \\left[\nt_{\\perp}(\\parp_{\\alpha}^{\\perp} \\vec r)^2 + t_y(\\pary \\vec r)^2\n\\right]+\\mbox{Irrelevant} \\ .\n\\ee\nBy redefining the $y$ direction as $y^{\\prime}=\\frac{t_{\\perp}}{t_y} y$\nthis reduces to Eq.(\\ref{LAN_CR_PH_IRR}), with $t\\equiv t_{\\perp}$. We have\nproved that anisotropy is totally irrelevant in this particular phase.\n\n\\subsubsection{The Flat Phase}\\label{SubSECT__Fl_ani}\n\nThis phase becomes equivalent to the isotropic case as well. Intuitively,\nthis may be obtained from the fact that if the membrane is flat, the\nintrinsic anisotropies are only apparent at short-distances, and therefore\nby analyzing the RG flow at larger and larger distances the membrane\nshould become isotropic. This argument may be made slightly more \nprecise \\cite{Toner:88}.\n\n\\subsection{The Tubular Phase}\\label{SECT__tubular}\n\nWe now turn to the study of the novel tubular phase, both in the\nphantom case and with self-avoidance. Since the physically relevant\ncase for membranes is $D=2$ the $y$-tubular and $\\perp$-tubular \nphase are the same. So we concentrate on the properties of the\n$y$-tubular phase.\n\nThe key critical exponents characterizing the tubular phase are\nthe size (or Flory) exponent $\\nu$, giving the scaling of the tubular\ndiameter $R_g$ with the extended ($L_y$) and transverse ($L_{\\perp}$)\nsizes of the membrane, and the roughness exponent $\\zeta$ associated with \nthe growth of height fluctuations $h_{rms}$ (see Fig.\\ref{fig__tubdef}):\n\\bea\n\\label{nuzeta}\nR_g(L_{\\perp},L_y) & \\propto & L_{\\perp}^{\\nu} S_R(L_y/L_{\\perp}^z)\\\\\n\\nonumber\nh_{rms}(L_{\\perp},L_y) & \\propto & L_y^{\\zeta} S_h(L_y/L_{\\perp}^z)\n\\eea\nHere $S_R$ and $S_h$ are scaling functions \\cite{RT:95,RT:98}\nand $z$ is the anisotropy exponent.\n \n\\begin{figure}[htb]\n\\epsfxsize=3in \\centerline{\\epsfbox{tubdef.eps}}\n\\caption{A schematic illustration of a tubular configuration\nindicating the radius of gyration $R_g$ and the height \nfluctuations $h_{rms}$.}\n\\label{fig__tubdef}\n\\end{figure}\n \nThe general free energy described in Eq.(\\ref{LGW}) may be simplified\nconsiderably in a $y$-tubular phase. The analysis required is \ninvolved and we refer the interested reader to \\cite{BG:97,BT:99}.\nWe just quote the final result. It is\n\\bea\\label{free_EG}\nF(u,\\vec h)&=&\\frac{1}{2}\\int d^{D-1}{\\bf x}_{\\perp} dy \\left[\n\\kappa (\\pary^2 \\vec h)^2+t(\\parp_{\\alpha} \\vec h)^2 \\right.\n\\nonumber\\\\\n&&+\ng_{\\perp}(\\parp_{\\alpha} u+\\partial_{\\alpha} \\vec h \\pary \\vec h )^2\n\\nonumber\\\\\n&&+\\left.\ng_y(\\pary u+\\frac{1}{2}(\\pary \\vec h)^2)^2\n\\right] \n\\nonumber\\\\\n&&+\n\\frac{b}{2}\\int dy d^{D-1}{\\bf x}_{\\perp}d^{D-1}{\\bf x}_{\\perp}^\\prime\n\\delta^{d-1}(\\vec{h}({\\bf x}_{\\perp},y)-\\vec{h}({\\bf x}_{\\perp}^\\prime,y)) \n\\ .\n\\eea\nComparing with Eq.(\\ref{LGW}), this free energy\ndoes represent a simplification as the number of couplings has been\nreduced from eleven to five. Furthermore, the coupling $g_{\\perp}$ is irrelevant \nby standard power counting. The most natural assumption is to \nset it to zero. In that case the phase diagram one obtains is \nshown in Fig.\\ref{fig__BG}. Without self-avoidance $b=0$,\nthe Gaussian Fixed Point (GFP) is unstable and the infra-red stable\nFP is the tubular phase FP (TPFP). Any amount of self-avoidance,\nhowever, leads to a new FP, the Self-avoiding Tubular FP (SAFP),\nwhich describes the large distance properties of self-avoiding tubules.\n \n \\begin{figure}[htb]\n \\epsfxsize=4in \\centerline{\\epsfbox{BG.eps}}\n \\caption{The phase diagram for self-avoiding anisotropic membranes\n with the Gaussian fixed point (GFP), the tubular phase fixed point\n (TPFP) and the self-avoidance fixed point (SAFP).}\n \\label{fig__BG}\n \\end{figure}\n\nWe just mention, though, that other authors advocate a different\nscenario \\cite{RT:98}. \nFor sufficiently small embedding dimensions $d$, including the\nphysical $d=3$ case, these authors suggest the existence of a new bending rigidity renormalized\nFP (BRFP), which is the infra-red FP describing the actual properties \nof self-avoiding tubules (see Fig.~\\ref{fig__RT}). \n\n\\begin{figure}[htb]\n\\epsfxsize=4in \\centerline{\\epsfbox{RT.eps}}\n\\caption{The phase diagram for self-avoiding anisotropic membranes\nwith the Gaussian fixed point (GFP), the tubular phase fixed point\n(TPFP), the self-avoidance fixed point (SAFP) and the bending rigidity\nfixed point (BRFP).}\n\\label{fig__RT}\n\\end{figure}\n\nHere we follow the arguments presented in \\cite{BT:99} and consider \nthe model defined by Eq.(\\ref{free_EG}) with the $g_{\\perp}$-term as the model describing the large\ndistance properties of tubules. One can prove then than there\nare some general scaling relations among the critical exponents.\nAll three exponents may be expressed in terms of a single exponent \n\\bea\\label{Ani_scaling}\n\\zeta&=&\\frac{3}{2}+\\frac{1-D}{2z} \\nonumber\\\\\n&\\nu&=\\zeta z \\ .\n\\eea\n\nRemarkably, the phantom case as described by Eq.(\\ref{free_EG}) can be \nsolved exactly. The result for the size exponent is \n\\be\\label{Phantom_ani_size}\n\\nu_{Phantom}(D)=\\frac{5-2D}{4} \\ , \\nu_{Phantom}(2)= \\frac{1}{4}\n\\ee\nwith the remaining exponents following from the scaling\nrelations Eq.(\\ref{Ani_scaling}).\n\n \\begin{table}[htb]\n \\centerline{\n \\begin{tabular}{|c||l|l|l|l|l|}\n \\multicolumn{1}{c}{$d$} &\n \\multicolumn{1}{c}{$\\nu$} & \\multicolumn{1}{c}{$\\nu_F$ } &\n \\multicolumn{1}{c}{$\\nu_V$} & \\multicolumn{1}{c}{$\\nu_{Flory}$} \\\\\\hline\n 8 & $0.333(5) $ & $0.34(1)$ & $0.34(1)$ & $0.333$ \\\\\\hline\n 7 & $0.374(8) $ & $0.39(2)$ & $0.39(2)$ & $0.375$ \\\\\\hline\n 6 & $0.42(1) $ & $0.44(2)$ & $0.44(4)$ & $0.429$ \\\\\\hline\n 5 & $0.47(1) $ & $0.51(3)$ & $0.51(5)$ & $0.500$ \\\\\\hline\n 4 & $0.54(2) $ & $0.60(4)$ & $0.60(6)$ & $0.600$ \\\\\\hline\n 3 & $0.62(2) $ & $0.71(6)$ & $0.70(9)$ & $0.750$ \\\\\\hline\n \\end{tabular}}\n \\caption{Different extrapolations for the size exponent at different \n embedding dimensions $d$ \\cite{BT:99}. The first column gives the corrections\n to the mean field result, the second corrections to the Flory estimate and\n the third corresponds to corrections to the Gaussian approximation. The \n last column quotes the Flory estimate for comparison.}\n \\label{tab__EXP_comp}\n \\end{table}\n\nThe self-avoiding case may be treated with techniques similar to those\nin isotropic case. The size exponent may be estimated within a \nFlory approach. The result is\n\\be\\label{Flory_ani}\n\\nu_{Flory}=\\frac{D+1}{d+1} \\ .\n\\ee\nThe Flory estimate is an uncontrolled approximation. Fortunately,\na $\\vap$-expansion, adapting the MOPE technique described for the\nself-avoiding isotropic case to the case of tubules, is also\npossible \\cite{BG:97,BT:99}. The $\\beta$-functions are computed and provide evidence for\nthe phase diagram shown in Fig.\\ref{fig__BG}. Using rather involved\nextrapolation techniques, it is possible to obtain estimates\nfor the size exponent, which are shown \nin Table~\\ref{tab__EXP_comp}. The rest of the exponents may be \ncomputed from the scaling relations.\n\n\\section{Defects in membranes: The Crystalline-Fluid transition and\nFluid membranes}\\label{SECT__Defects}\n\nA flat crystal melts into a liquid when the temperature is increased.\nThis transition may be driven by the sequential liberation\nof defects, as predicted by the KTNHY theory. The KTNHY theory is \nschematically shown in Fig.\\ref{fig__KTNHY}. With increasing\ntemperature, a crystal melts first to an intermediate hexatic \nphase via a continuous transition, and finally goes to a conventional \nisotropic fluid phase via another continuous transition. \n\n \\begin{figure}[htb]\n \\epsfxsize= 4 in \\centerline{\\epsfbox{KTNHY.eps}}\n \\caption{Two stage melting according to KTNHY theory.}\n \\label{fig__KTNHY}\n \\end{figure}\n\t \nWe will not review here either the KTNHY theory or the \nexperimental evidence in its favor {--} \\cite{Nelson:83}. We just\nwant to emphasize here that the KTNHY theory is in general agreement\nwith existing experiments, although there are two main points \nworth keeping in mind when studying the more difficult case of fluctuating\ngeometries. 1) The experimental evidence for the existence of the \nhexatic phase is not completely settled in those transitions which \nare continuous. 2) Some 2D crystals (like Xenon absorbed on graphite)\nmelt to a fluid phase via a first order transition without any \nintermediate hexatic phase. \n\nThe straight-forward translation of the previous results to the\ntethered membrane would suggest a similar scenario. There would then be\na crystalline to hexatic transition and a hexatic to fluid transition, \nas schematically depicted in Fig.\\ref{fig__KTNHY}. Although the \nprevious scenario is plausible, there are no solid experimental\nor theoretical results that establish it. From the theoretical \npoint of view, for example, an important open problem is how to \ngeneralize the RG equations of the KTNHY theory to the case \nof fluctuating geometry.\nThe situation looks even more uncertain experimentally, especially \nconsidering the elusive nature of the hexatic phase even in the case \nof flat monolayers.\n\nIn this review we will assume the general validity \nof the KTNHY scenario and we describe models of hexatic membranes, as well\nas fluid membranes. The study of the KTNHY theory in fluctuating\ngeometries is a fascinating and challenging problem that deserves\nconsiderable effort. In this context,\nlet us mention recent calculations of defects on\nfrozen topographies \\cite{BNT:99}, which show that even in the more simplified\ncase when the geometry is frozen, defects proliferate in an attempt to\nscreen out Gaussian curvature, even at zero temperature, and organize \nthemselves in rather surprising and unexpected structures. These results\nhint at a rich set of possibilities for the more general case of\nfluctuating geometries.\n\n\\subsection{Topological Defects}\n\nA crystal may have different distortions from its ground state. \nThermal fluctuations are the simplest. Thermal fluctuations are\nsmall displacements from the ground state, and therefore one \nmay bring back the system to its original positions by local\nmoves without affecting the rest of the lattice. There are more \nsubtle lattice distortions though, where the lattice cannot be\ntaken to its ground state by local moves. These are the topological\ndefects. There are different possible topological defects that\nmay occur on a lattice, but we just need to consider dislocations\nand disclinations. Let us review the most salient features.\n\n \\begin{figure}[htb]\n \\epsfxsize= 2 in \\centerline{\\epsfbox{mem_burg.eps}}\n \\caption{Example of a dislocation showing the breaking of the \n translational holonomy measured by the Burgers vector $\\vec{b}$.} \n \\label{fig__disl}\n \\end{figure}\n \n\\begin{itemize}\n\\item{\\underline{Dislocation}}: represents the breaking of the \ntranslational holonomy. A path that would naturally close in a \nperfect lattice fails to close by a vector ${\\vec b}$, the Burgers\nvector, as illustrated in Fig.\\ref{fig__disl}. In a flat monolayer, the energy is \n\\be\\label{DISl__EN}\nE=\\frac{K_0 {\\vec b}^2}{8 \\pi} \\ln(\\frac{R}{a}) \\ ,\n\\ee\nwhere $K_0$ is the Young Modulus. It diverges {\\em logarithmically}\nwith system size.\n\\item{\\underline{Disclination}}: represents the breaking of the\nrotational holonomy. The bond angle around the point defect is a multiple\nof the natural bond angle in the ground state ($\\frac{\\pi}{3}$ in\na triangular lattice), as illustrated in Fig.\\ref{fig__discl} for a $+$ and $-$\ndisclination. The energy for a disclination in a flat monolayer is\ngiven by\n\\be\\label{Discl__EN}\nE = \\frac{K_0 s^2}{32 \\pi} R^2 \\ .\n\\ee\nNote the {\\em quadratic} divergence of the energy with system size $R$.\n\\end{itemize}\n\n\n \\begin{figure}[htb]\n \\centerline{\\epsfxsize= 2 in \\epsfbox{mem2_vec.eps}\n \\epsfxsize= 2 in \\epsfbox{mem3_vec.eps}}\n \\caption{Example of a minus disclination (left figure) and a plus \n disclination (right figure). The orientational holonomy is broken by \n $\\pm\\frac{\\pi}{3}$ respectively.}\n \\label{fig__discl}\n \\end{figure}\n\nInspection of Fig.\\ref{fig__disl} shows that a dislocation\nmay be regarded as a tightly bound +,- disclination pair. \n\n\\subsubsection{Topological Defects in fluctuating geometries}\n\nThe problem of understanding topological defects when the geometry \nis allowed to fluctuate was addressed in \\cite{SN:88} (see \\cite{DRNles} \nfor a review). The important new feature is that the energy of a\ndisclination defect may be lowered considerably if the membrane buckles\nout-of-the-plane. That is, the membrane trades elastic energy for \nbending rigidity. The energy for a buckled free disclination is given by\n\\be\\label{new__discl_ENER}\nE = f(\\kappa,K_0,q_i) \\ln(\\frac{R}{a}) \\ ,\n\\ee\nwhere $f$ is some complicated function that may be evaluated numerically\nfor given values of the parameters. It depends explicitly on $q_i$, which\nimplies that the energies for positive and negative disclinations may be\ndifferent, unlike the situation in flat space. The extraordinary reduction in\nenergy from $R^2$ to $\\ln R$ is possible because the buckled membrane\ncreates positive Gaussian curvature for the\nplus-disclination and negative curvature for the negative-disclination. \nThis is a very important physical feature of defects on curved surfaces.\nThe defects attempt to screen out like-sign curvature, and analogously,\nlike-sign defects may force the surface to create like-sign curvature\nin order to minimize the energy.\n\nThe reduction in energy for a dislocation defect is even more remarkable, \nsince the energy of a dislocation becomes a constant, independent of \nthe system size, provided the system is larger than a critical radius \n$R_c$. Again, by allowing the possibility of out-of-plane buckling, a \nspectacular reduction in energy is achieved (from $R^2$ to $\\ln R$).\n\nThe study of other topological defects, e.g. vacancies, interstitials,\nand grain boundaries, may be carried out along the same lines. Since\nwe are not going to make use of it, we refer the reader to the excellent review\nin \\cite{DRNles}. \n\n\\subsubsection{Melting and the hexatic phase}\n\nThe celebrated Kosterlitz-Thouless argument shows that defects will\nnecessarily drive a 2D crystal to melt. The entropy of a \ndislocation grows logarithmically with the system size, so for \nsufficiently high temperature, entropy will dominate over the dislocation\nenergy (Eq.(\\ref{DISl__EN})) and the crystal will necessarily melt.\nIf the same Kosterlitz-Thouless is applied now to \na tethered membrane, the entropy is still growing logarithmically\nwith the system size, while the energy becomes independent of the system\nsize, as explained in the previous subsection, so any finite temperature \ndrive the crystal to melt, and the low temperature phase of a tethered \nmembrane will necessarily be a fluid phase, either hexatic if the KTNHY melting\ncan be applied, or a conventional fluid if a first order transition takes\nplace, or even some other more perverse possibility. This problem has\nalso been investigated in numerical simulations \\cite{GK:97}, which provide some\nconcrete evidence in favor of the KTNHY scenario, although the issue \nis far from being settled.\n\nIt is apparent from these arguments that a hexatic membrane \nis a very interesting and possibly experimentally relevant \nmembrane to understand. \n\n\\subsection{The Hexatic membrane}\n\nThe hexatic membrane is a fluid membrane that, in contrast to a\nconventional fluid, preserves the\norientational order of the original lattice (six-fold (hexatic) for\na triangular lattice). The mathematical\ndescription of a fluid membrane is very different from those with \ncrystalline order. Since the description cannot depend on internal\ndegrees of freedom, the free energy must be invariant under \nreparametrizations of the internal coordinates (that is, should depend\nonly on geometrical quantities, or in more mathematical terminology,\nmust be diffeomorphism invariant). The corresponding free energy was proposed\nby Helfrich \\cite{Helf:73} and it is given by\n\\be\\label{extr_curv}\n\\frac{{\\cal H}_{H}}{T}=\\mu \\int \\sqrt{g} +\n\\frac{\\kappa}{2}\\int d {\\bf x} \\sqrt{g} {\\vec H}^2 \\ ,\n\\ee\nwhere $\\mu$ is the bare string tension, $\\kappa$ the bending rigidity,\n$g$ is the determinant of the metric of the surface \n\\be\\label{metric__srf}\ng_{\\mu \\nu}({\\bf x})=\\partial_{\\mu} {\\vec r}({\\bf x}) \n\\partial_{\\nu} {\\vec r}({\\bf x})\n\\ee\nand ${\\vec H}$ is the mean Gaussian curvature of the surface.\nFor a good description of the differential geometry relevant to the\nstudy of membranes we refer to \\cite{D:92}\nA hexatic membrane has an additional degree of \nfreedom, the bond angle, which is introduced as a field on the \nsurface $\\theta$.\nThe hexatic free energy \\cite{NP:90} is\nobtained from adjoining to the fluid case of Eq.(\\ref{extr_curv}), the\nadditional energy of the bond angle\n\\be\\label{hexatic_energy}\n{\\cal H}_{hex}/T=\\frac{K_A}{2}\\int d {\\bf x} \\sqrt{g}\ng^{\\mu \\nu}(\\partial_{\\mu} \\theta+\\Omega_{sing}-\\Omega^L_{\\mu})\n(\\partial_{\\nu} \\theta+\\Omega_{sing}-\\Omega^L_{\\nu})\n\\ee\nwhere $K_A$ is called the hexatic stiffness, and\n$\\Omega_{\\mu}$ is the connection two form of the metric, which\nmay be related to the Gaussian curvature of the surface by\n\\be\\label{metric__con}\nK({\\bf x})=\\frac{1}{\\sqrt{g}}\\epsilon^{\\mu \\nu}\\partial_{\\mu} \n\\Omega_{\\nu}\n\\ee\nand $\\Omega_{sing}$ is similarly related to the topological defect\ndensity \n\\bea\\label{hexatic_energy_2}\ns({\\bf x})&=&\\frac{1}{\\sqrt{g}}\\epsilon^{\\mu \\nu}\\partial_{\\mu} \n\\Omega_{sing \\nu}\n\\nonumber\\\\\ns({\\bf x})&=&\\frac{\\pi}{3}\\frac{1}{\\sqrt{g}}\\sum_{i=1}^N q_i \n\\delta({\\bf x},{\\bf x}_i) \\ .\n\\eea\nFrom general theorems on differential geometry one has the relation\n\\be\\label{Diff_geom}\n\\int \\sqrt{g} s({\\bf x})=4 \\pi \\rightarrow \\sum_{i=1}^i q_i = 2 \\chi \\ ,\n\\ee\nwhere $\\chi$ is the Euler characteristic of the surface.\n\nTherefore the total free energy for the hexatic membrane is given by\n\\bea\\label{Hex__Mem}\n{\\cal H}/T&=&\\mu \\int \\sqrt{g}+\n\\frac{\\kappa}{2}\\int d {\\bf x} \\sqrt{g} {\\vec H}^2+ \n\\\\\\nonumber\n&+& \\frac{K_A}{2}\\int d {\\bf x} \\sqrt{g}\ng^{\\mu \\nu}(\\partial_{\\mu} \\theta+\\Omega_{sing}-\\Omega^L_{\\mu})\n(\\partial_{\\nu} \\theta+\\Omega_{sing}-\\Omega^L_{\\nu}) \\ .\n\\eea\nThe partition function is therefore\n\\bea\\label{new_part_func}\n{\\cal Z}(\\beta)&=&\\sum_{N_{+},N_{-}} \\frac{\\delta_{N_+-N_{-},2 \\chi}}\n{N_{+}!N_{-}!} y^{N_++N_-} \\times\n\\\\\\nonumber\n&& \\int D[{\\vec r}] D[{\\theta}]\n\\int \\prod_{\\mu=1}^{N_+} d {\\bf\nx}^+_{\\mu}\\sqrt{g} \\prod_{\\nu=1}^{N_-} d {\\bf x}_{\\nu}^{-} \\sqrt{g}\ne^{-{\\cal H}({\\vec r}({\\bf x}),\\theta({\\bf x}))/T} \\ ,\n\\eea\nwhere $y$ is the fugacity of the disclination density.\nThe partition function includes a discrete sum over allowed topological\ndefects, those satisfying the topological constraint Eq.(\\ref{Diff_geom}), and\na path integral over embeddings ${\\vec r}$ and bond angles $\\theta$.\nThe previous model remains quite intractable since the sum over defects\ninteraction is very difficult to deal with.\n\n \\begin{figure}[htb]\n \\epsfxsize= 4 in \\centerline{\\epsfbox{hexatic.eps}}\n \\caption{Phase diagram according to \\cite{GuKa:90}. The hexatic \n membrane interpolates between a crystalline membrane and a fluid one.\n For small rigidity and large hexatic stiffness the RG flows towards \n a fluid crumpled phase. For small hexatic stiffness and large \n $\\kappa$ it flows towards a tethered like phase, whereas for \n both large $\\kappa$ and $K_A$ the flow is to the crinkled phases\n described by the non-trivial FP of Eq.(\\ref{beta__function}).}\n \\label{fig__hexatic}\n \\end{figure}\n\nFortunately, the limit of very low fugacity $y \\rightarrow 0$ is \nanalytically tractable as was shown in the beautiful paper \\cite{DGP:87}. \nThe RG functions can be computed within a combined large $d$ and large\nbending rigidity expansion. The $\\beta$ functions in that limit is given \nby\n\\bea\\label{beta__function}\n\\beta(\\alpha)&=&\\frac{1}{4 \\pi K_A}\\left( -D \\alpha^2+\\frac{3}{4}\\alpha^3+\n{\\cal O}(1/K_A^2)\\right) \\ ,\n\\nonumber\\\\\n\\beta(K_A)&=&0\n\\eea\nwhere $\\alpha=1/\\kappa$. The physics of hexatic membranes in the limit \nof very low fugacity is very rich and show a line of fixed points \nparametrized by the hexatic stiffness $K_A$. The normal-normal correlation \nfunction, for example, reads \\cite{DGP:87}\n\\be\\label{Hex__norm_norm}\n\\langle {\\vec n}({\\bf r}) {\\vec n}({\\bf 0}) \\rangle \\sim \n|{\\bf x}|^{-\\eta} \\ ,\n\\ee\nwith $\\eta=\\frac{2}{3 \\pi} d (d-2)\\frac{k_B T}{K_A}$. The FPs of \nEq.(\\ref{beta__function}) describe a new {\\em crinkled} phase, \nmore rigid than a crumpled phase but more crumpled than a flat one. \nThe Hausdorff dimension at the crinkled phase is given by \\cite{DGP:87} \n\\be\\label{crink__phase}\nd_H=2+\\frac{d(d-2)}{3 \\pi} \\frac{k_B T}{K_A} \\ .\n\\ee\nFrom the RG point of view, the properties of these crinkled phases\nare really interesting, since they involve a line of Fixed Points which\nare inequivalent in the sense that the associated critical exponents depend continuously \non $K_A$, a situation reminiscent of the $XY$-model. In \\cite{GuKa:90}\nthe phase diagram is discussed, and the authors propose the scenario\ndepicted in Fig.\\ref{fig__hexatic}. How these scenarios are modified when\nthe fugacity is considered is not well established and we refer the reader to the \noriginal papers \\cite{PaLu2:96,PaLu3:96,GuKa:90}. \n\nThe shape fluctuations of hexatic vesicles, for large defect core\nenergies, have also been investigated \\cite{ML:91,Evans:95}. \n\n\\section{The Fluid Phase}\\label{SECT__Fluid}\n\nThe study of fluid membranes is a broad subject, currently under intense \nexperimental and theoretical work. The Hamiltonian is given by\n\\be\\label{Fluid_phase}\n{\\cal H}/T=\\mu \\int \\sqrt{g}+\n\\frac{\\kappa}{2}\\int d {\\bf x} \\sqrt{g} {\\vec H}^2+ \n\\frac{\\hat{\\kappa}}{2}\\int \\sqrt{g} K \\ ,\n\\ee\nThis corresponds to the Helfrich hamiltonian together with a term that\nallows for topology changing interactions.\nFor fixed topology it is well-known\n\\cite{PL:85,Poly:86,Forster:86,Klein1:86} \nthat the one loop beta function\nfor the inverse-bending rigidity has a fixed point only at $\\kappa=0$,\nwhich corresponds to the bending rigidity being irrelevant at large\nlength scales. The RG flow of the bending rigidity is given by\n\\be\\label{FluidRG}\n\\kappa(l) = \\kappa_o - \\frac{3T}{4\\pi} {\\rm ln} (l/a) \\ , \n\\ee\nwhere a is a microscopic cutoff length.\nThe fluid membrane is therefore crumpled, for\narbitrary microscopic bending rigidity $\\kappa_0$, at length scales beyond\na persistence length which grows exponentially with $\\kappa_0$. \nFor a fluid membrane out-of-plane fluctuations cost no elastic energy\n(the membrane flows internally to accommodate the deformation)\nand the bending rigidity is therefore softened by thermal undulations\nat all length scales, rather than stiffening at long length scales as\nin the crystalline membrane.\n \nSo far we have assumed an infinite membrane,\nwhich is not always a realistic assumption. A thickness may be \ntaken into account via a spontaneous\nextrinsic curvature ${\\vec H}_0$. The model described by \nEq.~\\ref{Fluid_phase} gets replaced then by\n\\be\\label{Fluid_phase_2}\n{\\cal H}/T=\\mu \\int \\sqrt{g}+\n\\frac{\\kappa}{2}\\int d {\\bf x} \\sqrt{g} ({\\vec H- \\vec H_0})^2+ \n\\frac{\\hat{\\kappa}}{2}\\int \\sqrt{g} K \\ .\n\\ee\nFurther effects of a finite membrane size for spherical topology have\nbeen discussed in \\cite{MM:94,Morse:94,MM:95}.\nThe phase diagram of fluid membranes when topology change is allowed \nis fascinating and not completely understood. A complete description \nof these phases goes beyond the scope of this review {--} we refer the \nreader to \\cite{GK1:97,GK2:97,GS:94,Safran} and references therein.\n\n\\section{Conclusions}\\label{SECT__CONC}\n\nIn this review we have described the distinct universality classes of membranes \nwith particular emphasis on crystalline membranes. In each case we \ndiscussed and summarized the key models describing the interactions of\nthe relevant large distance degrees of freedom (at the micron\nscale). The body of the review emphasizes qualitative and descriptive \naspects of the physics with technical details presented in extensive\nappendices. We hope that the concreteness of these calculations\ngives a complete picture of how to extract relevant physical information \nfrom these membrane models.\n\nWe have also shown that the phase diagram of the phantom crystalline \nmembrane class is theoretically very well understood both by analytical and \nnumerical treatments. To complete the picture it would be \nextremely valuable to find experimental realizations for this particular \nsystem. An exciting possibility is a system of cross-linked DNA\nchains together with restriction enzymes that catalyze cutting and\nrejoining \\cite{BEN}. \nThe difficult chemistry involved in these experiments is not yet \nunder control, but we hope that these technical problems will be \novercome in the near future.\n\nThere are several experimental realizations of self-avoiding polymerized\nmembranes discussed in the text. \nThe experimental results compare very well with the \ntheoretical estimates from numerical simulations. As a future theoretical\nchallenge, analytical tools need to be sharpened since they fail to provide a clear\nand unified picture of the phase diagram. On the experimental side, there\nare promising experimental realizations of tethered membranes which will \nallow more precise results than those presently available. Among them \nthere is the possibility of very well controlled synthesis of DNA networks \nto form physical realizations of tethered membranes.\n\nThe case of anisotropic polymerized membranes has also been described in\nsome detail. The phase diagram contains a new tubular phase which \nmay be realized in nature. There is some controversy about the precise\nphase diagram of the model, but definite predictions for the critical\nexponents and other quantities exist. Anisotropic membranes are also\nexperimentally relevant. They may be created in the laboratory \nby polymerizing a fluid membrane in the presence of \nan external electric field. \n\nProbably the most challenging problem, both theoretically and experimental, \nis a complete study of the role of defects in polymerized membranes. There\nare a large number of unanswered questions, which include \nthe existence of hexatic phases, the properties \nof defects on curved surfaces and its relevance to the possible\nexistence of more complex phases. \nThis problem is now under intense experimental investigation.\nIn this context, let us mention very recent experiments on Langmuir\nfilms in a presumed hexatic phase \\cite{Fischer}. The coalescence of \nair bubbles with the film exhibit several puzzling features which are\nstrongly related to the curvature of the bubble.\n\nCrystalline membranes also provide important insight into the \nfluid case, since any crystalline membrane eventually becomes fluid at\nhigh temperature. The physics of fluid membranes is a complex and\nfascinating subject in itself which goes beyond the scope of this\nreview. We highlighted some relevant experimental realizations and gave a quick\noverview of the existing theoretical models. Due to its relevance in\nmany physical and biological systems and its potential\napplications in material science, the experimental and theoretical \nunderstanding of fluid membranes is, and will continue to be, one of the \nmost active areas in soft condensed matter physics.\n\nWe have not been able in this review, simply for lack of time, to\naddress the important topic of the role of disorder.\nWe hope to cover this in a separate article. \n\nWe hope that this review will be useful for physicists trying to get a \nthorough understanding of the fascinating field of membranes. \nWe think it is a subject with significant prospects for new and exciting\ndevelopments. \n\n\\bigskip\n\\noindent{Note}: The interested reader may also find additional\nmaterial in a forthcoming review by Wiese \\cite{Wiese:00}, \nof which we have seen only the table of contents. \n\n\\bigskip\n\\centerline{\\bf Acknowledgements}\n\\medskip\n\nThis work was supported by the U.S. Department of Energy under \ncontract No. DE-FG02-85ER40237.\nWe would like to thank Dan Branton and Cyrus Safinya for providing us\nimages from their laboratories and Paula Herrera-Sikl\\'ody for \nassistance with the figures. MJB would like to thank Riccardo\nCapovilla, Chris Stephens, Denjoe O'Connor and the other\norganizers of RG2000 for the opportunity to attend a wonderful meeting\nin Taxco, Mexico. \n\n\\newpage\n\n\\appendix\n\n\\section{Useful integrals in dimensional regularization}\\label{APP__SA__Int}\n\nIn performing the $\\vap$-expansion, we will be considering integrals\nof the form\n\\be\\label{int__form}\nI_{\\alpha_1,\\cdots,\\alpha_n}(a,b)({\\vec p})=\\int d^D \\hat{q}\n\\frac{q_{\\alpha_1} \\cdots q_{\\alpha_n}}{(\\vec{p}+\\vec{q})^{2a}\\vec{q}^{2b}}\n\\ ,\n\\ee\nThese integrals may be computed exactly for general \n$D,a,b$ and $\\alpha=1,\\cdots,N$. The result will be published elsewhere.\nWe will content ourselves by quoting what we need, the poles in $\\vap$,\nfor the integrals that appear in the diagrammatic calculations. We just\nquote the results\n\\be\\label{dim_reg_int}\nI_{\\alpha_1 \\alpha_2}(2,2)=-\\frac{1}{8 \\pi^2 p^4}p_{\\alpha_1}p_{\\alpha_2} \n\\frac{1}{\\vap}\n\\ee\n\\be\nI_{\\alpha_1 \\alpha_2 \\alpha_3}(2,2)=\\frac{1}{8 \\pi^2 p^4}\np_{\\alpha_1}p_{\\alpha_2}p_{\\alpha_3} \\frac{1}{\\vap}\n\\ee\n\\bea\n&& I_{\\alpha_1 \\alpha_2 \\alpha_3 \\alpha_4}(2,2)=-\\frac{1}{8 \\pi^2 p^4}\n\\left(p_{\\alpha_1}p_{\\alpha_2}p_{\\alpha_3}p_{\\alpha_4}-\\right.\n\\nonumber\\\\ &&\\left. \\frac{p^4}{24}(\n\\delta_{\\alpha_1 \\alpha_2}\\delta_{\\alpha_3 \\alpha_4}+\n\\delta_{\\alpha_1 \\alpha_3}\\delta_{\\alpha_2 \\alpha_4}+ \n\\delta_{\\alpha_1 \\alpha_4}\\delta_{\\alpha_2 \\alpha_3})\\right)\\frac{1}{\\vap}\n\\eea\n\\bea\nI_{\\alpha_1 \\alpha_2 \\alpha_3}(2,1)&=&-\\frac{1}{8 \\pi^2 p^2}\n(\\frac{p^2}{6}(p_{\\alpha_1}\\delta_{\\alpha_2 \\alpha_3}\n+p_{\\alpha_2}\\delta_{\\alpha_1 \\alpha_3}+\np_{\\alpha_3}\\delta_{\\alpha_2 \\alpha_1})\n\\nonumber\\\\ &&\n-p_{\\alpha_1}p_{\\alpha_2}p_{\\alpha_3}) \\frac{1}{\\vap}\n\\eea\n\n\\section{Some practical identities for RG quantities}\\label{APP_RGstuff}\n\nThe beta functions defined in Eq.~\\ref{define_set_RG} may be \nre-expressed as\n\\be\\label{APP_RG_bet}\n\\left(\\begin{array}{cc} \n\\beta_u(u_R,v_R) \\\\ \\beta_v(u_R,v_R) \n\\end{array}\\right)=-\\vap \\left(\\begin{array}{cc} \n\\frac{\\parp \\ln u}{\\parp u_R} & \\frac{\\parp \\ln u}{\\parp v_R} \\\\\n\\frac{\\parp \\ln v}{\\parp u_R} & \\frac{\\parp \\ln v}{\\parp v_R} \n\\end{array}\\right)^{-1} \n\\left(\\begin{array}{cc} \n1 \\\\ 1 \\end{array}\\right)\n\\ee\nThe previous expression may be further simplified noticing\n\\be\\label{APP_REP}\nA=\\left(\\begin{array}{cc} \n\\frac{\\parp \\ln u}{\\parp u_R} & \\frac{\\parp \\ln u}{\\parp v_R} \\\\\n\\frac{\\parp \\ln v}{\\parp u_R} & \\frac{\\parp \\ln v}{\\parp v_R} \n\\end{array}\\right)=\\left(\\begin{array}{cc} \n\\frac{1}{u_R} & 0 \\\\ 0 & \\frac{1}{v_R} \n\\end{array}\\right)+D \\ ,\n\\ee\nso that \n\\bea\\label{APP_RG_invA}\nA^{-1}&=&\\left(1+\\left(\\begin{array}{cc} \nu_R & 0 \\\\ 0 & v_R \n\\end{array} \\right) D \\right)^{-1}\n\\left(\\begin{array}{cc} \nu_R & 0 \\\\ 0 & v_R \n\\end{array}\\right)\n\\\\\\nonumber\n&=&\n\\left(\\begin{array}{cc} u_R & 0 \\\\ 0 & v_R \\end{array}\\right)-\n\\left(\\begin{array}{cc} u_R & 0 \\\\ 0 & v_R \\end{array}\\right)D\n\\left(\\begin{array}{cc} u_R & 0 \\\\ 0 & v_R \\end{array}\\right)+\n\\cdots\n\\eea\nwhere the last result follows from Taylor-expanding. These formulas\neasily allow to compute the corresponding $\\beta$-functions. If \n\\bea\\label{APP_RG_COU}\nu&=&M^{\\vap}\\left[ u_R+\\frac{1}{\\vap}(a_{11} u^2_R+a_{12} u_R v_R+\na_{13} v^2_R) \\right] \\\\\\nonumber\nv&=&M^{\\vap}\\left[ v_R+\\frac{1}{\\vap}(a_{21} u^2_R+a_{22} u_R v_R+\na_{23} v^2_R) \\right] \\ ,\n\\eea\nfrom Eq.~\\ref{APP_RG_invA} and Eq.~\\ref{APP_RG_bet} we easily derive\nthe leading two orders in the couplings\n\\bea\\label{APP_RG_beta}\n\\beta_u(u_R,v_R)&=&-\\vap u_R+a_{11} u^2_R+a_{12} u_R v_R+\na_{13} v^2_R \\\\\\nonumber\n\\beta_v(u_R,v_R)&=&-\\vap v_R+a_{21} u^2_R+a_{22} u_R v_R+\na_{23} v^2_R \\ .\n\\eea\n\nThe formula for $\\gamma$ in Eq.~\\ref{define_set_RG} may also be\ngiven a more practical expression. It is given by \n\\be\\label{APP_prac_exp}\n\\gamma=(\\beta_u \\frac{\\parp}{\\parp u_R}+\n\\beta_v \\frac{\\parp}{\\parp v_R}) \\ln Z_{\\phi} \\ , \n\\ee\n\nThose are the formulas we need in the calculations we present in this\nreview.\n\n\\section{Discretized Model for tethered membranes}\\label{APP__Dis}\n\nIn this appendix we present appropriate discretized models for \nnumerical simulation of tethered membranes. The surface is discretized by\na triangular lattice defined by its vertices $\\{{\\vec r}\\}_{a=1,\\cdots}$,\nwith a corresponding discretized version of the Landau elastic term\nEq.~\\ref{LAN_FL_PH} given by \\cite{SN:88}\n\\be\\label{discr__ver__flat}\nF_{s}=\\frac{\\beta}{2}\\sum_{\\langle a b \\rangle} ( |{\\vec r}_a-{\\vec r}_b|-1)^2\n\\ ,\n\\ee\nwhere $\\langle a,b \\rangle$ are nearest-neighbor vertices.\nIf we write ${\\vec r}_a={\\bf x}_a+{\\bf u}_a$ with ${\\bf x}_a$ defining the \nvertices of a perfectly regular triangular lattice and ${\\bf u}$ the \nsmall perturbations around it, one gets\n\\be\\label{discr__tensor}\n|{\\vec r}_a-{\\vec r}_b|=1+u_{\\alpha\\beta} x^{\\alpha} x^{\\beta} +\\cdots \n\\ ,\n\\ee\nwith $u_{\\alpha \\beta}$ being a discretized strain tensor and we \nwe have neglected higher order terms.\nPlugging the previous expression into Eq.~\\ref{discr__ver__flat} and\npassing from the discrete to the continuum language we obtain\n\\be\\label{fin__dis_ver}\nF_{s}=\\frac{\\sqrt{3}}{8}\\beta \\int d^2 {\\bf x} (2 u^2_{\\alpha \\beta}+\nu^2_{\\alpha \\alpha})\n\\ee\nwhich is the elastic part of the free energy Eq.~\\ref{LAN_FL_PH} with\n$\\lambda=\\mu=\\frac{\\sqrt{3}}{4} \\beta$. \n\nThe bending rigidity term is written in the continuum as\n\\be\\label{bending__rig}\nS_{ext}=\\int d^2 {\\bf x} \\sqrt{g} K^{\\mu}_{\\alpha \\beta}\nK_{\\mu}^{\\alpha \\beta}=\\int d^2 {\\bf u} \\sqrt{g} g^{\\alpha \\beta}\n\\nabla_{\\alpha} {\\vec n} \\nabla_{\\beta} {\\vec n}\n\\ee\nwhere ${\\vec n}$ is the normal to the surface and $\\nabla$ is the \ncovariant derivative (see \\cite{D:89} for a detailed description of\nthese geometrical quantities). We discretize the\nnormals form the the previous equation by\n\\be\\label{bend__rig_dis}\n\\int d^2 {\\bf u} \\sqrt{g} g^{\\alpha \\beta}\n\\nabla_{\\alpha} {\\vec n} \\nabla_{\\beta} {\\vec n}\n\\rightarrow\n\\sum_{\\langle a b \\rangle} ({\\vec n}_a - {\\vec n}_b)^2=\n2 \\sum_{\\langle a b \\rangle} (1-{\\vec n}_a {\\vec n}_b)\n\\ee\n\nThe two terms Eq.~\\ref{discr__ver__flat} and Eq.~\\ref{bend__rig_dis}\nprovide a suitable discretized model for a tethered membranes. \nHowever, in actual simulations, the even more simplified discretization\n\\be\\label{sim__disc}\nF=\\sum_{\\langle a, b \\rangle} ({\\vec r}_a-{\\vec r}_b)^2+\n\\kappa \\sum_{\\langle i, j \\rangle}(1-{\\vec n}_i {\\vec n}_j)\n\\ee\nis preferred since it is simpler and describes the same universality \nclass (see \\cite{BCFTA:96} for a discussion).\nAnisotropy may be introduced in this model by ascribing distinct \nbending rigidities to bending across links in different intrinsic\ndirections \\cite{BFT:97}.\n\nSelf-avoidance can be introduced in this model by imposing that the \ntriangles that define the discretized surface cannot\nself-intersect. There are other possible discretizations of self-avoidance\nthat we discus in sect.~\\ref{Sub_SECT__SA}.\n\nIn order to numerically simulate the model Eq.~\\ref{sim__disc} different\nalgorithms have been used. A detailed comparison of the \nperformance of each algorithm may be found in \\cite{TF:98}.\n\n\\section{The crumpling Transition}\\label{APP__CT__FP}\n\nThe Free energy is given by Eq.~\\ref{LAN_CRTR_PH}\n\\be\\label{APP_CRTR_PH}\nF({\\vec r})=\\int d^D{\\bf x} \\left[ \n\\frac{1}{2}(\\parp_{\\alpha}^2 {\\vec r})^2+\nu(\\parp_{\\alpha} {\\vec r} \\parp_{\\beta} {\\vec r} -\n\\frac{\\delta_{\\alpha \\beta}}{D}(\\parp_{\\alpha \\vec r})^2 )^2+\nv(\\parp_{\\alpha} {\\vec r} \\parp^{\\alpha} {\\vec r})^2 \\right] \\ ,\n\\ee\nwhere the dependence on $\\kappa$ is trivially scaled out. \nThe Feynman rules for the model are given in Fig.\\ref{fig__Feyn_CT}.\n\n \\begin{figure}[htb]\n \\epsfxsize=3 in \\centerline{\\epsfbox{feyn_crumpled.eps}}\n \\caption{Feynman rules for the model at the crumpling transition}\n \\label{fig__Feyn_CT}\n \\end{figure}\n\nWe need three renormalized constants, namely $Z$, $Z_u$ and $Z_v$ in order\nto renormalize the theory. We define the renormalized quantities by\n\\bea\\label{App_CR__RG}\n{\\vec r}&=&Z^{-1/2} {\\vec r} \n\\\\\\nonumber\nu_R=M^{-\\vap}Z^2 Z_{u}^{-1} u & \\ , &\nv_R=M^{-\\vap}Z^2 Z_{v}^{-1} v \\ .\n\\eea\nThen Eq.~\\ref{APP_CRTR_PH} becomes\n\\bea\\label{REN_CRTR_APP}\nF({\\vec r})&=&\\int d^D{\\bf x} \\left[ \n\\frac{Z}{2}(\\parp_{\\alpha}^2 {\\vec r_R})^2+\nM^{\\vap} Z_{u} u(\\parp_{\\alpha} {\\vec r_R} \\parp_{\\beta} {\\vec r_R} -\n\\frac{\\delta_{\\alpha \\beta}}{D}(\\parp_{\\alpha} \\vec r_R)^2 )^2\n\\right. +\n\\nonumber \\\\ &+& \\left. M^{\\vap} Z_{v} v(\\parp_{\\alpha} {\\vec r_R} \\parp^{\\alpha} \n{\\vec r_R})^2 \\right] \\ ,\n\\eea\n\nIn order to compute the renormalized couplings, one must compute all\nrelevant diagrams at one loop. Those are depicted in \nfig.~\\ref{fig__one_loop_CT}. Within dimensional regularization, \ndiagrams (1a) and (1b) are zero, which in turns imply that the renormalized\nconstant is $Z=1$ at leading order in $\\vap$, similarly as in linear\n$\\sigma$ models.\n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{cr__alldia.eps}}\n \\caption{Diagrams to consider at one loop}\n \\label{fig__one_loop_CT}\n \\end{figure}\n\nUsing the integrals in dimensional regularization (see\nSect.~\\ref{APP__SA__Int}) Diagram (2a) \ngives the result\n\\be\\label{CR__TR__RES2A}\n\\frac{d}{8 \\pi^2} \\frac{1}{\\vap} \\delta^{i_1 i_2} \\delta^{j_1 j_2}\n\\left\\{ \\frac{u^2}{24}( \\vec p_1\\cdot\\vec p_3 \\vec p_2\\cdot\\vec p_4+\n\\vec p_1\\cdot\\vec p_4 \\vec p_2\\cdot\\vec p_3)+\n(v^2-\\frac{u^2}{48})\\vec p_1\\cdot\\vec p_2 \\vec p_3\\cdot\\vec p_4\n\\right\\}\n\\ee\n\nAnd diagram (2b) and (2c) may be computed at once, since the \nresult of (2c) is just (2b) after interchanging \n$\\vec p_3 \\leftrightarrow \\vec p_4$, so the total result \n(2a)+(2b) is\n\n\\bea\\label{CR__TR__RES2BC}\n&& \\frac{1}{16 \\pi^2} \\frac{1}{\\vap} \\delta^{i_1 i_2} \\delta^{j_1 j_2}\n\\left\\{ (\\frac{61}{96}u^2+\\frac{7}{12} uv+\\frac{v^2}{6})\n( \\vec p_1\\cdot\\vec p_3 \\vec p_2\\cdot\\vec p_4\\right.+\n\\vec p_1\\cdot\\vec p_4 \\vec p_2\\cdot\\vec p_3)\\nonumber\\\\\n&&\\left. +(\\frac{v^2}{6}+\\frac{u v}{12}+\\frac{u^2}{96})\n\\vec p_1\\cdot\\vec p_2 \\vec p_3\\cdot\\vec p_4 \\right\\}\n\\eea\n\nAnd the result for (2d) and (2e) is just identical, so the total \nresult (2d)+(2e) is \n\n\\bea\\label{CR__TR__RES2DE}\n&& \\frac{1}{16 \\pi^2} \\frac{1}{\\vap} \\delta^{i_1 i_2} \\delta^{j_1 j_2}\n\\left\\{ (\\frac{u^2}{24}+\\frac{1}{6} uv)\n( \\vec p_1\\cdot\\vec p_3 \\vec p_2\\cdot\\vec p_4\\right.+\n\\vec p_1\\cdot\\vec p_4 \\vec p_2\\cdot\\vec p_3)\\nonumber\\\\\n&&\\left. +(v^2+\\frac{13}{6}u v-\\frac{u^2}{48})\n\\vec p_1\\cdot\\vec p_2 \\vec p_3\\cdot\\vec p_4 \\right\\}\n\\eea\nAdding up all these contributions taking into account \nthe different combinatorial factors (4 the first contribution, 8 the\nlast two ones) and recalling $Z=1$, we get\n\\bea\\label{CR__rencoup}\nu&=& M^{\\vap}\\left[ u_R+\\frac{1}{\\vap}\\frac{1}{8 \\pi^2}\\left( \n(\\frac{d}{3}+\\frac{65}{12})u^2_R+6u_R v_R+\\frac{4}{3} v^2_r\\right)\\right]\n\\nonumber\\\\\nv&=&M^{\\vap}\\left[ v_R+\\frac{1}{\\vap}\\frac{1}{8 \\pi^2}\\left( \n\\frac{21}{16} u_R +\\frac{21}{2} u_R v_R+(4d+5)v^2_R\\right)\\right] \\ .\n\\eea\nThe resultant $\\beta$-functions are then readily obtained by applying\nEq.(\\ref{APP_RG_beta}). \n\n\\section{The Flat Phase}\\label{APP__FP__FP}\n\nThe free energy is given in Eq.~\\ref{LAN_FL_PH},and it is given \nby\n\\be\\label{LAN_FL_PH_APP}\nF({\\bf u},h)=\\int d^D {\\bf x} \\left[\n\\frac{\\hat{\\kappa}}{2} (\\parp_{\\alpha} \\parp_{\\beta} h )^2+\n\\mu u_{\\alpha \\beta} u^{\\alpha \\beta} + \\frac{\\lambda}{2} (u^{\\alpha}_{\\alpha})^2\n\\right] \\, .\n\\ee\nThe Feynman rules are shown in fig.~\\ref{fig__Feyn_FLA}, it is apparent that \nthe in-plane phonons couple different from the out-of-plane, which play the\nrole of Goldstone bosons.\n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{feyn__flat.eps}}\n \\caption{Feynman rules for the model in the flat phase}\n \\label{fig__Feyn_FLA}\n \\end{figure}\n\nWe apply standard field theory techniques to obtain the RG-quantities. \nUsing the Ward identities, the theory can be renormalized using \nonly three renormalization constants $Z$, $Z_{\\mu}$ and $Z_{\\lambda}$,\ncorresponding to the wave function and the two coupling renormalizations. \nRenormalized quantities read\n\\bea\\label{App_def__RG}\nh_R=Z^{-1/2} h & \\ , & {\\bf u}=Z^{-1} {\\bf u}\n\\\\\\nonumber\n\\mu_R=M^{-\\vap}Z^2 Z_{\\mu}^{-1} \\mu & \\ , &\n\\lambda_R=M^{-\\vap}Z^2 Z_{\\lambda}^{-1} \\lambda \\ ,\n\\eea\nThen Eq.~\\ref{LAN_FL_PH_APP} becomes\n\\be\\label{REN_FL_PH_APP}\nF({\\bf u},h)=\\int d^D {\\bf x} \\left[\nZ (\\parp_{\\alpha} \\parp_{\\beta} h_R )^2+\n2 M^{\\vap} Z_{\\mu} \\mu_R u_{R \\alpha \\beta} u^{\\alpha \\beta}_R + \nM^{\\vap} Z_{\\lambda} \\lambda (u^{\\alpha}_{R \\alpha})^2\n\\right] \\, .\n\\ee\nWe now compute the renormalized quantities from the leading divergences\nappearing in the Feynman diagrams. The diagrams to consider \nare given in Fig.\\ref{fig__one_loop_FP}. These can \nbe computed using the integrals given in Sect.\\ref{APP__SA__Int}.\n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{flat__alldia.eps}}\n \\caption{Diagrams to consider at one loop}\n \\label{fig__one_loop_FP}\n \\end{figure}\n\nThe result of diagram (1a) is given by\n\\be\\label{Dia__1A}\n\\frac{1}{\\vap}\\frac{d_c}{6 \\pi^2}\\left[ \n\\mu^2(\\delta_{\\alpha \\beta}-\\frac{p_{\\alpha} p_{\\beta}}{p^2})+\n3(\\mu^2+2\\mu\\lambda+2\\lambda^2)\\frac{p_{\\alpha}p_{\\beta}}{p^2}\\right] p^2\n \\ .\n\\ee\nDiagrams (2b) and (2c) are identically zero, so (2a) is the only \nadditional diagram to be computed. The result is\n\\be\\label{Dia__2A}\n-\\frac{1}{\\vap}\\frac{\\delta^{i j}}{8 \\pi^2} \n\\frac{\\mu(\\mu+\\lambda)}{2 \\mu+\\lambda} 10 (p^2)^2\n\\ee\nfrom Eq.~\\ref{Dia__2A} and the definitions in Eq.~\\ref{App_def__RG}\n\\be\\label{zeta_plain}\nZ=1-\\frac{1}{\\vap}\\frac{10}{8 \\pi^2}\\frac{\\mu_R(\\mu_R+\\lambda_R)}\n{2\\mu_R+\\lambda_R} \\ .\n\\ee\nUsing the previous result in the diagrams (1a) whose result is in \nEq.~\\ref{Dia__1A} we obtain \n\\bea\\label{zetamula}\nZ_{\\mu}&=&1+\\frac{1}{\\vap}\\frac{d_c}{24 \\pi^2} \\mu_R\n\\\\\\nonumber\nZ_{\\lambda}&=&1+\\frac{1}{\\vap}\\frac{d_c}{24 \\pi^2}(\\mu^2_R+6\\mu_R\\lambda_R\n+6\\lambda_R^2)/\\lambda_R\n\\eea\nand we deduce the renormalized couplings\n\\bea\\label{FL__rencoup}\n\\mu&=& M^{\\vap}\\left[ \\mu_R+\\frac{1}{\\vap}\\left( \n\\frac{10}{4 \\pi^2}\\frac{(\\mu_R+\\lambda_R)}{2\\mu_R+\\lambda_R}+\n\\frac{d_c}{24\\pi^2}\\right)\\mu^2_R \\right]\n\\\\\\nonumber\n\\lambda&=&M^{\\vap}\\left[ \\lambda_R+\\frac{1}{\\vap}\\left( \n\\frac{10}{4 \\pi^2}\\frac{(\\mu_R+\\lambda_R)}{2\\mu_R+\\lambda_R}\\mu_R\\lambda_R+\n\\frac{d_c}{24 \\pi^2}(\\mu^2_R+6\\mu_R\\lambda_R+6\\lambda^2_R)\\right)\\right]\n\\eea\nfrom which the $\\beta$-functions trivially follow with the aid of \nEq.(\\ref{APP_RG_beta}).\n\n\\section{The Self-avoiding phase}\\label{APP__SA__CP}\n\nThe model has been introduced in Eq.~\\ref{CRU__SA} and is given by\n\\be\\label{APP_CRU__SA}\nF({\\vec r})=\\frac{1}{2}\\int d^D {\\bf x} (\\parp_{\\alpha} {\\vec r}({\\bf x}))^2\n+\\frac{b}{2}\\int d^D{\\bf x}\\, d^D{\\bf y} \\delta^d({\\vec r}({\\bf x})-\n{\\vec r}({\\bf y})) \\ ,\n\\ee\nWe follow the usual strategy of defining the renormalized quantities by\n\\bea\\label{App_SAdef__RG}\n {\\vec r}&=&Z^{1/2} {\\vec r}_R\n\\\\\\nonumber\n b&=&M^{\\vap}Z_b Z^{d/2} b_R \\ ,\n\\eea\nand the renormalized Free energy by \n\\be\\label{APP_REN__SA}\nF({\\vec r})=\\frac{Z}{2}\\int d^D {\\bf x} (\\parp_{\\alpha} {\\vec r}_R({\\bf x}))^2\n+M^{\\vap} Z_b \n\\frac{b_R}{2}\\int d^D{\\bf x}\\, d^D{\\bf y} \\delta^d({\\vec r}_R({\\bf x})-\n{\\vec r}_R({\\bf y})) \\ .\n\\ee\nThe $\\delta$-function being non-local adds some technical\ndifficulties to the calculation of the renormalized constants $Z$ and $Z_b$. \nThere are different approaches available but we will follow the \nMOPE (Multilocal-Operator-Product-Expansion), which we will just explain\nin a very simplified version. A rigorous description of the method \nmay be found in the literature. \n\nThe idea is to expand the $\\delta$-function term in Eq.~\\ref{APP_REN__SA}\n\\be\\label{APP_SA_exp}\ne^{-F({\\vec r})}=e^\n{-\\frac{Z}{2}\\int d^D {\\bf x} (\\parp_{\\alpha} {\\vec r}_R({\\bf x}))^2}\n\\times \\sum_{n=0}^{\\infty} \\left(-M^{\\vap} Z_b \n\\frac{b_R}{2}\\int d^D{\\bf x}\\, d^D{\\bf y} \\delta^d({\\vec r}_R({\\bf x})-\n{\\vec r}_R({\\bf y})) \\right)^n \\ ,\n\\ee\nwith this trick, the delta-function term may be treated\nas expectation values of a Gaussian free theory. This observation alone\nallows to isolate the poles in $\\vap$. We write the identity\n\\be\\label{APP__Norm_ord}\ne^{i {\\vec k}({\\vec r}({\\bf x}_1)-{\\vec r}({\\bf x}_2))}=\n:e^{i {\\vec k}({\\vec r}({\\bf x}_1)-{\\vec r}({\\bf x}_2))}:e^{k^2 G(x_1-x_2)}\n\\ ,\n\\ee\nwhere $G(x)$ is the two point correlator \n\\be\\label{APP_two_poin}\n-G({\\bf x})=-\\langle {\\vec r}({\\bf x}) {\\vec r}(0) \\rangle =\n\\frac{|{\\bf x}|^{2-D}}{(2-D)S_D} \\ ,\n\\ee\nwith $S_D$ being the volume of the $D$-dimensional sphere.\nThe symbol $::$ stands for normal ordering. A normal ordered operator is\nnon-singular at short-distances, so it may be Taylor-expanded\n\\bea\\label{APP__Norm__Tayl}\ne^{i {\\vec k}({\\vec r}({\\bf x}_1)-{\\vec r}({\\bf x}_2))}&=&\n(1+i({\\bf x_1-x_2})^{\\alpha} ({\\vec k} \\parp_{\\alpha} {\\vec r})\n\\\\\\nonumber\n&& -\\frac{1}{2}({\\bf x_1-x_2})^{\\alpha}({\\bf x_1-x_2})^{\\beta} \n({\\vec k} \\parp_{\\alpha}{\\vec r}) ({\\vec k} \\parp_{\\beta} {\\vec r})\n+\\cdots) e^{k^2 G(x_1-x_2)} \\ .\n\\eea\nTo isolate the poles in $\\vap$ we do not need to consider higher order\nterms as it will become clear.\nIf we now integrate over ${\\vec k}$, we get\n\\bea\\label{APP_MOPE}\n\\delta^d({\\vec r}({\\bf x}_1)-\\vec r({\\bf x}_2))&=&\n\\frac{1}{(4 \\pi)^{d/2}(-G({\\bf x}_1-{\\bf x}_2))^{d/2}} 1\n\\\\\\nonumber\n&&-\n\\frac{1}{4}\\frac{({\\bf x}_1-{\\bf x}_2)^{\\alpha}({\\bf x}_1-{\\bf x}_2)^{\\beta}}\n{(4 \\pi)^{d/2}(-G({\\bf x}_1-{\\bf x}_2))^{d/2+1}} \n \\parp_{\\beta} {\\vec r}({\\bf x}) \\parp_{\\alpha} {\\vec r}({\\bf x})+\\cdots\n\\\\\\nonumber\n&\\equiv& C^1({\\bf x}_1-{\\bf x}_2) 1 + C^{\\alpha \\beta}({\\bf x}_1-{\\bf x}_2) \n \\parp_{\\beta} {\\vec r}({\\bf x}) \\parp_{\\alpha} {\\vec r}({\\bf x})+\\cdots\n\\eea\nwhere we omit higher dimensional operators in ${\\vec r}$, which are \nirrelevant by power counting, so since the theory is renormalizable they\ncannot have simple poles in $\\vap$. Additionally, we have defined\n${\\bf x}=\\frac{{\\bf x}_1+{\\bf x}_2}{2}$. \nOne recognizes in Eq.~\\ref{APP_MOPE} Wilson's Operator product expansion,\napplied to the non-local delta-function operator.\n\nFollowing the same technique of splitting the operator into a normal\nordered part and a singular part at short-distances, it just takes a\nlittle more effort to derive the OPE for the product of two delta\nfunctions, the result is\n\\be\\label{APP_twodel}\n\\delta^d({\\vec r}({\\bf x}_1)-\\vec r({\\bf y}_1))\n\\delta^d({\\vec r}({\\bf x}_2)-\\vec r({\\bf y}_2))=\nC({\\bf x}_1-{\\bf x}_2,{\\bf y}_1-{\\bf y}_2) \n\\delta^d({\\vec r}({\\bf x})-\\vec r({\\bf y}))+\\cdots\n\\ee\nwith $C({\\bf x}_1-{\\bf x}_2,{\\bf y}_1-{\\bf y}_2)= \\frac{1}{(4 \\pi)^{d/2}\n(-G({\\bf x}_1-{\\bf x}_2)-G({\\bf y}_1-{\\bf y}_2))^{d/2}}$. The terms\nomitted are again higher dimensional by power counting so they.\nThe OPE Eq.~\\ref{APP_MOPE} and Eq.~\\ref{APP_twodel} is all we need to \ncompute the renormalization constants at lowest order in $\\vap$,\nbut the calculation may be pursued to higher orders in $\\vap$. In order\nto do that, one must identify where poles in $\\vap$ arise. In the \nprevious example poles in $\\vap$ appear whenever the internal coordinates\n(${\\bf x}_1$ and ${\\bf x}_2$ in Eq.~\\ref{APP_MOPE},\n${\\bf x}_1$,${\\bf x}_2$,${\\bf y}_3$ and ${\\bf y}_2$ in \nEq.~\\ref{APP_twodel}) are pairwise made to coincide. This is diagrammatically\nshown in fig.~\\ref{fig__MOPE}. It is possible to show, that higher poles\nappear in the same way, if more $\\delta$-product terms are considered.\n\n \\begin{figure}[htb]\n \\epsfxsize=4 in \\centerline{\\epsfbox{SA__alldia.eps}}\n \\caption{Diagrammatic expansion to isolate the poles in $\\vap$ within \n the MOPE formalism at lowest non-trivial order. Solid lines represent\n $\\delta$-function terms and dashed lines indicate that points inside\n are taken arbitrarily close. Higher orders contributions arise in the\n same way.}\n \\label{fig__MOPE}\n \\end{figure}\n\nLet us consider the first delta-function term corresponding to $n=1$\nin the sum Eq.~\\ref{APP_SA_exp}. Using Eq.~\\ref{APP_MOPE} we have\n\\bea\\label{APP_comZ}\n&&-\\frac{b_r M^{\\vap}}{2} Z_{b} \\int d^D{\\bf x}\\, d^D{\\bf y} \n\\delta^d({\\vec r}_R({\\bf x})-{\\vec r}_R({\\bf y}))\n\\\\\\nonumber\n&=&-\\frac{b_r M^{\\vap}}{2} Z_{b} \\int d^D{\\bf x}\\, d^D{\\bf y} \n(C^1({\\bf x}-{\\bf y})+ C^{\\alpha \\beta}({\\bf x}-{\\bf y}) \n\\parp_{\\beta} {\\vec r}_R({\\bf x}) \\parp_{\\alpha} {\\vec r}_R({\\bf x})+\n\\cdots\n\\\\\\nonumber\n&=&-\\frac{b_r M^{\\vap}}{2} Z_{b}\\int d^D{\\bf x} \n\\parp_{\\alpha} {\\vec r}_R({\\bf x}) \\parp^{\\alpha} {\\vec r}_R({\\bf x})\n\\int d^D{\\bf y} \\frac{\\delta_{\\alpha \\beta} C^{\\alpha \\beta}}{D}+\\cdots\n\\eea\nThe first term just provides a renormalization of the identity\noperator, which we can neglect. From\n\\be\\label{APP_int_in_MOPE}\n\\int_{|{\\bf y}|> 1/M} d^D{\\bf y} \\frac{\\delta_{\\alpha \\beta} \nC^{\\alpha \\beta}}{D}({\\bf y})=\n-\\frac{1}{4D}\\frac{M^{-\\vap}}{\\vap} (4 \\pi)^{-d/2}(2-D)^{1+d/2}\n\\left(\\frac{2 \\pi^{D/2}}{\\Gamma(D/2)}\\right)^{2+d/2} \\ ,\n\\ee\nand we can absorb the pole by $Z$ if we define\n\\be\\label{APP_Z_factor}\nZ=1+\\frac{b_R}{\\vap}\\frac{(4 \\pi)^{-d/2}}{4D}(2-D)^{1+d/2}\n\\left(\\frac{2 \\pi^{D/2}}{\\Gamma(D/2)}\\right)^{2+d/2} \\ ,\n\\ee\n\nFrom the short distance behavior in the sum Eq.~\\ref{APP_SA_exp}\ncorresponding to $n=2$ we get\n\\bea\\label{APP_comZb}\n&&-\\frac{b_r^2 M^{2\\vap}}{8} \\int d^D{\\bf x}_1 \\, d^D{\\bf y}_1 \nd^D{\\bf x}_2 \\, d^D{\\bf y}_2\n\\delta^d({\\vec r}_R({\\bf x}_1)-{\\vec r}_R({\\bf y}_1))\n\\delta^d({\\vec r}_R({\\bf x}_1)-{\\vec r}_R({\\bf y}_2))\n\\nonumber\\\\\n&&-\\frac{b_r^2 M^{2\\vap}}{8} \\int d^D{\\bf x} \\, d^D{\\bf y} \n\\delta^d({\\vec r}_R({\\bf x})-{\\vec r}_R({\\bf y}))\n\\int d^D{\\bf z} \\, d^D{\\bf w} C(z,w) \\ ,\n\\eea\nwhere, in order to isolate the pole we can perform the following tricks\n\\bea\\label{APP_last__int}\n&& \\int d^D{\\bf z} \\, d^D{\\bf w} C(z,w)\n\\\\\\nonumber\n&=& (4 \\pi)^{-d/2}S_D^{d/2}(2-D)^{d/2} \n\\left(\\frac{2 \\pi^{D/2}}{\\Gamma(D/2)}\\right)^2\n\\int_0^{M^{-1}}dz \\int_0^{M^{-1}}d w \\frac{z^{D-1}w^{D-1}}\n{(z^{2-D}+w^{2-D})^{d/2}}\n\\\\\\nonumber\n&=& (4 \\pi)^{-d/2}S_D^{d/2}(2-D)^{d/2} \n\\left(\\frac{2 \\pi^{D/2}}{\\Gamma(D/2)}\\right)^2\\frac{M^{-\\vap}}{(2-D)^2}\n\\int_0^{1}\\int_0^{1}d x d y \\frac{x^{\\frac{D}{2-D}} y^{\\frac{D}{2-D}}}\n{(x+y)^{-d/2}}\n\\\\\\nonumber\n&=& (4 \\pi)^{-d/2}S_D^{d/2}(2-D)^{d/2} \n\\left(\\frac{2 \\pi^{D/2}}{\\Gamma(D/2)}\\right)^2\\frac{M^{-\\vap}}{(2-D)^2}\n\\int_{x^2+y^2 \\le 1} d x d y \\frac{x^{\\frac{D}{2-D}} y^{\\frac{D}{2-D}}}\n{(x+y)^{-d/2}}\n\\\\\\nonumber\n&=& (4 \\pi)^{-d/2}S_D^{d/2}(2-D)^{d/2} \n\\left(\\frac{2 \\pi^{D/2}}{\\Gamma(D/2)}\\right)^2\\frac{M^{-\\vap}}{(2-D)^3}\n\\frac{\\Gamma(\\frac{D}{2-D})^2}{\\Gamma(\\frac{2D}{2-D})}\\frac{1}{\\vap}\n\\eea\nsince changing the boundary of integration from a square to a circle\ndoes not affect the residue of the pole.\nWe finally have\n\\be\\label{APP_Zb_final}\nZ_b=1+\\frac{b_R}{\\vap}\\frac{1}{2}(2-D)^{-1+d/2} \n\\frac{\\Gamma(\\frac{D}{2-D})^2}{\\Gamma(\\frac{2D}{2-D})}\n\\left(\\frac{2 \\pi^{D/2}}{\\Gamma(D/2)}\\right)^{2+d/2}\n(4 \\pi)^{-d/2} \\ ,\n\\ee\nand the $\\beta$-function follows from the definitions \nEq.(\\ref{App_SAdef__RG}) together with Eq.(\\ref{APP_Z_factor}) \nand Eq.(\\ref{APP_RG_beta}). \n\n\\section{The mean field solution of the anisotropic case}\\label{MF__Appen}\n\nThe free energy has been introduced in Eq.~\\ref{LGW}. Let us first \nshow the constraints on the couplings so that the Free energy\nis bounded from below. \n\\begin{itemize}\n\\item{$u_{yy} > 0$:} This follows trivially.\n\\item{$ u^{\\prime}_{\\perp \\perp} \\equiv v_{\\perp \\perp}+\n\\frac{u_{\\perp \\perp}}{D-1} > 0$ :} \nDefine $A_{\\alpha}^i=\\parp_{\\alpha} r^i({\\bf x})$ then from \nEq.~\\ref{LGW} we\nget\n\\bea\\label{ineq_ani_APP}\n&&\\frac{u_{\\perp \\perp}}{2} Tr (A A^{T})^2+\\frac{v_{\\perp \\perp}}{2}\n(Tr A A^{T})^2\\ge(\\frac{u_{\\perp \\perp}}{D-1}+v_{\\perp \\perp})/2 \n(Tr A A^{T})^2\n\\nonumber \\\\\n&=&\\frac{u^{\\prime}_{\\perp \\perp}}{2}(Tr A A^{T})^2 \\ ,\n\\eea\nwhich implies $u_{\\perp \\perp}^{\\prime} > 0$.\n\\item{$ v_{\\perp y} > -(u^{\\prime}_{\\perp \\perp} u_{yy})^{1/2}$ :}\ndefining ${\\vec b}=\\partial_y {\\vec r}({\\bf x})$, It is derived from\n\\be\\label{further_cons_any}\n\\frac{u^{\\prime}_{\\perp \\perp}}{2}(Tr(A^TA))^2+\\frac{u_{yy}}{2} (b^2)^2+\nv_{\\perp y} b^T b Tr A A^T > 0 \\ .\n\\ee\n\\end{itemize}\n\nIntroducing the variables \n\\be\\label{def__pot_an}\nA=\\left(\\begin{array}{cc}\n v_{\\perp \\perp}+\\frac{u_{\\perp \\perp}}{D-1} & v_{\\perp y} \\\\\n\t v_{\\perp y} & u_{yy}\n\t \\end{array} \\right) \\ , \\ \\\nb=(t_{\\perp},t_y) \n\\ee\nand $w=((D-1)\\zeta^2_{\\perp},\\zeta^2_y)$, the mean field effective \npotential may be written as\n\\be\\label{eff_pot_an}\nV(w)=\\frac{1}{2}L_{\\perp}^{D-1} L_y \\left[ w \\cdot b +\n\\frac{1}{2} w \\cdot A \\cdot w \\right] \\ .\n\\ee\nIn this form, it is easy to find the four minima of Eq.~\\ref{eff_pot_an},\nthose are\n\\begin{enumerate}\n\\item{Crumpled phase:} \n\\be\\label{Flat_crumpled_an}\n\\begin{array}{l}\n\\zeta^2_{\\perp}=0 \\\\\n\\zeta^2_{y}=0\n\\end{array} \\ \\ \\ V_{min}=0\n\\ee\n\\item{Flat phase:}\n\\be\\label{Flat_par_an}\n\\begin{array}{l}\n\\zeta^2_{\\perp}=-\\frac{u_{yy} t_{\\perp}-v_{\\perp} t_y}\n{\\Delta (D-1)} \\\\\n\\zeta^2_{y}=\\frac{-v_{\\perp y} t_{\\perp}+u_{\\perp \\perp}^{\\prime}t_y}{\\Delta} \n\\end{array} \\ \\ \\ \\ V_{min}=-\\frac{L_{\\perp}^{D-1}L_y}{4 \\Delta}\n\\left[ u_{\\perp \\perp}^{\\prime} t^2_y+u_{yy} t^2_{\\perp}-2 v_{\\perp y} \nt_{\\perp} t_y \\right]\n\\ee\n\\item{${\\perp}$-Tubule:}\n\\be\\label{Perptub_par_an}\n\\begin{array}{l}\n\\zeta^2_{\\perp}=\\frac{-t_{\\perp}}{u_{\\perp \\perp}^{\\prime}} \n\\\\ \\zeta^2_{y}=0\n\\end{array} \\ \\ \\ \\ V_{min}=-\\frac{L_{\\perp}^{D-1}L_y}{4}\n\\frac{t^2_{\\perp}}{u_{\\perp \\perp}^{\\prime}}\n\\ee\n\\item{$y$-Tubule:}\n\\be\\label{ytub_par_an}\n\\begin{array}{l}\n\\zeta^2_{\\perp}=0 \n\\\\ \\zeta^2_{y}=-\\frac{t_y}{u_{yy}} \n\\end{array} \\ \\ \\ \\ V_{min}=-\\frac{L_{\\perp}^{D-1}L_y}{4}\\frac{t^2_{y}}{u_yy}\n\\ee\n\\end{enumerate}\n\nThe regions in which each of the four minima prevails depend on the\nsign of $\\Delta$. \n\\begin{itemize}\n\\item{$\\Delta > 0$:} \n Let us see under which conditions the flat phase may occur. \n We must satisfy the equations \n \\bea\\label{App_flat_ani}\n u_{yy}t_{\\perp} & < & v_{\\perp y} t_{y} \n \\nonumber\\\\\n u^{\\prime}_{\\perp \\perp} t_y & < & v_{\\perp y} t_{\\perp y} \n \\eea\n If $v_{\\perp y} > 0$\n this inequalities can only be satisfied if both\n $t_{\\perp}$ and $t_y0$ have the same sign. If they\n are positive, Eq.~\\ref{App_flat_ani} imply\n $\\Delta t_{\\perp} < 0$ or $\\Delta t_y < 0$,\n which by the assumption $\\Delta > 0$ cannot be satisfied.\n The flat phase exists for $t_y < 0$ and $t_{\\perp} < 0$ and \n satisfying Eq.~\\ref{App_flat_ani}. If $t_y > 0$ and $t_{\\perp} > 0$\n then the flat phase or the tubular cannot exist (see\n Eq.~\\ref{Perptub_par_an} and Eq.~\\ref{ytub_par_an}) so those\n are the conditions for the crumpled phase. Any other case is a \n tubular phase, either ${\\perp}$-tubule or $y$-tubule, depending on\n which of the inequalities Eq.~\\ref{App_flat_ani} is not satisfied.\n If $v_{\\perp y} < 0$ it easily checked from Eq.~\\ref{Flat_par_an} that\n the flat phase exists as well and the same analysis apply.\n\n\\item{$\\Delta < 0$:}\n From inequality Eq.~\\ref{further_cons_any} we have $v_{\\perp y} > 0$.\n The inequalities are now\n \\bea\\label{App_flat__ani_2}\n t_{y} & < & \\frac{u_{yy}}{v_{\\perp}} t_{\\perp} \n \\nonumber\\\\\n t_y & > & \\frac{v_{\\perp y}}{u_{\\perp \\perp}^{\\prime} t_{\\perp y} }\n \\eea\n Now, in order to have a solution for both inequalities we must have \n $\\frac{u_{yy}}{v_{\\perp}} > \\frac{v_{\\perp y}}{u_{\\perp \\perp}^{\\prime}}$\n which requires $\\Delta > 0$. This proves that the flat phase cannot\n exist. There is then a crumpled phase for $t_{y}>0$ and $t_{\\perp}> 0$\n and tubular phase when either one of this two conditions are not\n satisfied.\n\\end{itemize}\n\n\\newpage \n\n\\bibliography{melt,examples,reviews,thesis}\n\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n"
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"name": "memxxx.bbl",
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Safran.\n\\newblock {\\em Statistical Thermodynamics of Surfaces, Interfaces, and\n Membranes}.\n\\newblock Addison Wesley, Reading, Massachusetts, 1994.\n\n\\bibitem{Safinya1}\nJ.O. Radler, I.~Koltover, T.~Salditt, and C.R. Safinya.\n\\newblock Structure of {DNA}-{Cationic} {Liposome} {Complexes}: {DNA}\n {Intercalation} in {Multilamellar} {Membranes} in {Distinct} {Interhelical}\n {Packing} {Regimes}.\n\\newblock {\\em Science}, 275:810, (1997).\n\n\\bibitem{Salditt}\nT.~Salditt, I.~Koltover, J.O. Radler, and C.R. Safinya.\n\\newblock Two-{Dimensional} {Smectic} {Ordering} of {Linear} {DNA} {Chains} in\n {Self}-{Assembled} {DNA}-{Cationic} {Liposome} {Mixtures}.\n\\newblock {\\em Phys. Rev. Lett.}, 79:2582, (1997).\n\n\\bibitem{Safinya2}\nI.~Koltover, T.~Salditt, J.O. Radler, and C.R. 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USA}, 93:12943, (1996).\n\n\\bibitem{WiKo:74}\nK.~Wilson and J.~Kogut.\n\\newblock The {Renormalization} {Group} and the $\\epsilon$ {Expansion}.\n\\newblock {\\em Phys. Rep.}, 12:75, 1974.\n\n\\bibitem{Amit:84}\nD.~J. Amit.\n\\newblock {\\em Field {Theory}, the {Renormalization} {Group}, and {Critical}\n {Phenomena}}.\n\\newblock World Scientific, 1984.\n\n\\bibitem{NP:87}\nD.R. Nelson and L.~Peliti.\n\\newblock {\\em J. Phys. (France)}, 48:1085, 1987.\n\n\\bibitem{Jer1}\nD.R. Nelson, T.~Piran, and S.~Weinberg, editors.\n\\newblock {\\em Statistical Mechanics of Membranes and Surfaces}, volume~5 of\n {\\em Jerusalem Winter School for Theoretical Physics}.\n\\newblock World Scientific, Singapore, 1989.\n\n\\bibitem{PKN:88}\nM.~Paczuski, M.~Kardar, and D.R. Nelson.\n\\newblock Landau {Theory} of the {Crumpling} {Transition}.\n\\newblock {\\em Phys. Rev. Lett.}, 60:2638, 1988.\n\n\\bibitem{BCFTA:96}\nM.~J. Bowick, S.~M. 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Kroll.\n\\newblock Fluctuations of polymerized, fluid and hexatic membranes: continuum\n models and simulations.\n\\newblock {\\em Current Opinion in Colloid \\& Interface Science}, 2:373, (1997).\n\n\\bibitem{DG:88}\nF.~David and E.~Guitter.\n\\newblock Crumpling transition in elastic membranes: Renormalization group\n treatment.\n\\newblock {\\em Europhys. Lett.}, 5:709, 1988.\n\n\\bibitem{LDR:92}\nP.~{Le~Doussal} and L.~Radzihovsky.\n\\newblock Self consistent theory of polymerized membranes.\n\\newblock {\\em Phys. Rev. Lett}, 69:1209, 1992.\n\n\\bibitem{ET1:96}\nD.~Espriu and A.~Travesset.\n\\newblock {\\em Nucl. Phys. B}, 468:514, 1996.\n\n\\bibitem{AL:88}\nJ.A. Aronovitz and T.C. Lubensky.\n\\newblock Fluctuations of solid membranes.\n\\newblock {\\em Phys. Rev. Lett.}, 60:2634, 1988.\n\n\\bibitem{Landau7}\nL.~D. Landau and E.~M. Lifshitz.\n\\newblock {\\em Theory of Elasticity}, volume~7 of {\\em Course of Theoretical\n Physics}.\n\\newblock Pergamon Press, Oxford, UK, 1986.\n\n\\bibitem{FBGT:97}\nM.~Falcioni, M.~J. Bowick, E.~Guitter, and G.~Thorleifsson.\n\\newblock The {Poisson} ratio of crystalline surfaces.\n\\newblock {\\em Europhys. Lett.}, 38:67, 1997.\n\\newblock arXiv:cond-mat/9610007.\n\n\\bibitem{GDLP:88}\nE.~Guitter, F.~David, S.~Leibler, and L.~Peliti.\n\\newblock Crumpling and buckling transitions in polymerized membranes.\n\\newblock {\\em Phys. Rev. Lett.}, 61:2949, 1988.\n\n\\bibitem{GDLP:89}\nE.~Guitter, F.~David, S.~Leibler, and L.~Peliti.\n\\newblock Thermodynamical behavior of polymerized membranes.\n\\newblock {\\em J. Phys. (France)}, 50:1787, 1989.\n\n\\bibitem{Dup:87}\nB.~Duplantier.\n\\newblock {\\em Phys. Rev. Lett.}, 58:2733, 1987.\n\n\\bibitem{AL:87}\nJ.A. Aronovitz and T.C. Lubensky.\n\\newblock {\\em Europhys. Lett.}, 4:395, 1987.\n\n\\bibitem{KN3:87}\nM.~Kardar and D.~R. Nelson.\n\\newblock $\\epsilon$ expansion for cumpled manifolds.\n\\newblock {\\em Phys. Rev. Lett.}, 58:1289--1292, 1987.\n\n\\bibitem{KN1:88}\nM.~Kardar and D.~R. Nelson.\n\\newblock {\\em Phys. Rev. A}, 38:966, 1988.\n\n\\bibitem{DDG:94}\nF.~David, B.~Duplantier, and E.~Guitter.\n\\newblock {\\em Phys. Rev. Lett.}, 72:311, 1994.\n\n\\bibitem{DDG:97}\nF.~David, B.~Duplantier, and E.~Guitter.\n\\newblock arXiv:cond-mat/9702136, 1997.\n\n\\bibitem{Hwa:90}\nT.~Hwa.\n\\newblock {\\em Phys. Rev. A}, 41:1751, 1990.\n\n\\bibitem{DaWi:96}\nF.~David and K.~J. Wiese.\n\\newblock Scaling of {Selfavoiding} {Tethered} {Membranes}: 2-{Loop}\n {Renormalization} {Group} {Results}.\n\\newblock {\\em Phys. Rev. Lett.}, 76:4564, 1996.\n\\newblock arXiv:cond-mat/960215.\n\n\\bibitem{WiDa:97}\nK.~J. Wiese and F.~David.\n\\newblock New {Renormalization} {Group} {Results} for {Scaling} of\n {Self}-{Avoiding} {Membranes}.\n\\newblock {\\em Nucl. Phys. B}, 487:529, 1997.\n\\newblock arXiv:cond-mat/9608022.\n\n\\bibitem{Gou:91}\nM.~Goulian.\n\\newblock {\\em J. Phys. II}, 1:1327, 1991.\n\n\\bibitem{LeD:92}\nP.~Le Doussal.\n\\newblock {\\em J. Phys. A}, 25:L469, 1992.\n\n\\bibitem{PB:88}\nM.~Plischke and D.~Boal.\n\\newblock Absence of a crumpling transition in strongly self-avoiding tethered\n membranes.\n\\newblock {\\em Phys. Rev. A}, 38:4943, 1988.\n\n\\bibitem{ARP:89}\nF.F. Abraham, W.~Rudge, and M.~Plishke.\n\\newblock Molecular dynamics of tethered membranes.\n\\newblock {\\em Phys. Rev. Lett.}, 62:1757, 1989.\n\n\\bibitem{AN1:90}\nF.F. Abraham and D.R. Nelson.\n\\newblock Fluctuations in the flat and collapsed phase of polymerized surfaces.\n\\newblock {\\em J. Phys. (France)}, 51:2653, 1990.\n\n\\bibitem{AN2:90}\nF.F. Abraham and D.R. Nelson.\n\\newblock Diffraction from polymerized membranes.\n\\newblock {\\em Science}, 249:393, 1990.\n\n\\bibitem{BLLP:89}\nD.~Boal, E.~Levinson, D.~Liu, and M.~Plischke.\n\\newblock Anisotropic scaling of tethered self-avoiding membranes.\n\\newblock {\\em Phys. Rev. A}, 40:3292, 1989.\n\n\\bibitem{KK:93}\nY.~Kantor and K.~Kremer.\n\\newblock Excluded volume interactions in tethered membranes.\n\\newblock {\\em Phys. Rev. E}, 48:2494--2497, 1993.\n\n\\bibitem{GM:90}\nG.~S. Grest and M.~Murat.\n\\newblock Molecular dynamics simulations of the structure of\n gelation/percolation clusters and random tethered membranes.\n\\newblock {\\em J. Phys. (France)}, 51:1415, 1990.\n\n\\bibitem{Grest:91}\nG.~Grest.\n\\newblock {\\em J. Phys. I (France)}, 1:1695, 1991.\n\n\\bibitem{BP:94}\nS.~J. Barsky and M.~Plischke.\n\\newblock {\\em Phys. Rev. E}, 50:3911, 1994.\n\n\\bibitem{B:91}\n{A. Baumg\\\"artner}.\n\\newblock Does a polymerized membrane crumple?\n\\newblock {\\em J. Phys. I (France)}, 1:1549, 1991.\n\n\\bibitem{KG:93}\nD.M. Kroll and G.~Gompper.\n\\newblock Floppy tethered networks.\n\\newblock {\\em J. Phys. (France)}, 3:1131, 1993.\n\n\\bibitem{BCTT:2000}\nM.~Bowick, A.~Cacciuto, G.~Thorleifsson, and A.~Travesset.\n\\newblock In preparation, 2000.\n\n\\bibitem{DiFGu1:94}\nP.~Di Francesco and E.~Guitter.\n\\newblock {\\em Europhys. Lett.}, 26:455, 1994.\n\n\\bibitem{DiFGu2:94}\nP.~Di Francesco and E.~Guitter.\n\\newblock {\\em Phys. Rev. E.}, 50:4418, 1994.\n\n\\bibitem{BDGG:95}\nM.~Bowick, P.~Di Francesco, O.~Golinelli, and E.~Guitter.\n\\newblock {\\em Nucl. Phys. B}, 450[FS]:463, 1995.\n\\newblock arXiv:cond-mat/9502063.\n\n\\bibitem{BDGG:96}\nM.~Bowick, P.~Di Francesco, O.~Golinelli, and E.~Guitter.\n\\newblock Discrete folding.\n\\newblock 1996.\n\\newblock Talk given by M. Bowick at the 4th Chia meeting on {\\em Condensed\n Matter and High-Energy Physics}: arXiv:cond-mat/9610215.\n\n\\bibitem{BGGM:97}\nM.~Bowick, O.~Golinelli, E.~Guitter, and S.~Mori.\n\\newblock {\\em Nucl. Phys. B}, 495[FS]:583, 1997.\n\\newblock arXiv:cond-mat/9611105.\n\n\\bibitem{CGP1:96}\nE.~Cirillo, G.~Gonnella, and A.~Pelizzola.\n\\newblock {\\em Phys. Rev. E}, 53:1479, 1996.\n\n\\bibitem{CGP2:96}\nE.~Cirillo, G.~Gonnella, and A.~Pelizzola.\n\\newblock {\\em Phys. Rev. E}, 53:3253, 1996.\n\n\\bibitem{AK:91}\nF.~F. Abraham and M.~Kardar.\n\\newblock Folding and unbinding transitions in tethered membranes.\n\\newblock {\\em Science}, 252:419, 1991.\n\n\\bibitem{LP:92}\nD.~Liu and M.~Plischke.\n\\newblock {\\em Phys. Rev. A}, 45:7139, 1992.\n\n\\bibitem{GP:94}\nG.~S. Grest and I.~B. Petsche.\n\\newblock {\\em Phys. Rev. E}, 50:1737, 1994.\n\n\\bibitem{DaWi:95}\nF.~David and K.~J. Wiese.\n\\newblock {\\em Nucl. Phys. B}, 450:495, 1995.\n\\newblock arXiv:cond-mat/9503126.\n\n\\bibitem{BFT:97}\nM.~Bowick, M.~Falcioni, and G.~Thorleifsson.\n\\newblock {\\em Phys. Rev. Lett.}, 79:885, 1997.\n\n\\bibitem{Toner:88}\nJ.~Toner.\n\\newblock {\\em Phys. Rev. Lett.}, 62:905, 1988.\n\n\\bibitem{RT:95}\nL.~Radzihovsky and J.~Toner.\n\\newblock A {New} {Phase} of {Tethered} {Membranes}: {Tubules}.\n\\newblock {\\em Phys. Rev. Lett.}, 75:4752, 1995.\n\n\\bibitem{RT:98}\nL.~Radzihovsky and J.~Toner.\n\\newblock Elasticity, {Shape} {Fluctuations} and {Phase} {Transitions} in the\n {New} {Tubule} {Phase} of {Anisotropic} {Tethered} {Membranes}.\n\\newblock {\\em Phys. Rev. E}, 57:1832, 1998.\n\n\\bibitem{BG:97}\nM.~Bowick and E.~Guitter.\n\\newblock {\\em Phys. Rev. E}, 56:7023, 1997.\n\\newblock arXiv:cond-mat/9705045.\n\n\\bibitem{BT:99}\nM.~Bowick and A.~Travesset.\n\\newblock {\\em Phys. Rev. E}, 59:5659, 1999.\n\\newblock arXiv:cond-mat/9808214.\n\n\\bibitem{Nelson:83}\nDavid~R. Nelson.\n\\newblock {\\em Defect-mediated {Phase} {Transitions}}.\n\\newblock Academic Press, 1983.\n\\newblock Vol. 7 of {Phase} {Transitions} and {Critical} {Phenomena}, Edited by\n C. Domb and J.~L. Lebowitz.\n\n\\bibitem{BNT:99}\nM.~Bowick, D.~R. Nelson, and A.~Travesset.\n\\newblock Interacting {Topological} {Defects} on {Frozen} {Topographies}.\n\\newblock 1999.\n\\newblock arXiv:cond-mat/9911379.\n\n\\bibitem{SN:88}\nH.~S. Seung and D.~R. Nelson.\n\\newblock Defects in flexible membranes with crystalline order.\n\\newblock {\\em Phys. Rev. A}, 38:1005, 1988.\n\n\\bibitem{DRNles}\nD.~R. Nelson.\n\\newblock Defects in superfluids, superconductors and membranes.\n\\newblock In David et~al. \\cite{Les}.\n\\newblock arXiv:arXiv.org/lh94.\n\n\\bibitem{GK:97}\nG.~Gompper and D.M. Kroll.\n\\newblock Freezing flexible vesicles.\n\\newblock {\\em Phys. Rev. Lett}, 78:2859, 1997.\n\n\\bibitem{Helf:73}\nW.~Helfrich.\n\\newblock {\\em Z. Naturforsch}, 28 C:693, 1973.\n\n\\bibitem{D:92}\nF.~David.\n\\newblock Introduction to the statistical mechanics of random surfaces and\n membranes.\n\\newblock In D.J. Gross, T.~Piran, and S.~Weinberg, editors, {\\em Two\n Dimensional Quantum Gravity and Random Surfaces}, volume~8 of {\\em Jerusalem\n Winter School for Theoretical Physics}. World Scientific, Singapore, 1992.\n\n\\bibitem{NP:90}\nD.R. Nelson and L.~Peliti.\n\\newblock {\\em J. Phys. (France)}, 51:2653, 1990.\n\n\\bibitem{GuKa:90}\nE.~Guitter and M.~Kardar.\n\\newblock {\\em Europhys. Lett.}, 13:441, 1990.\n\n\\bibitem{DGP:87}\nF.~David, E.~Guitter, and L.~Peliti.\n\\newblock {\\em J. de Physique}, 48:1085, 1987.\n\n\\bibitem{PaLu2:96}\nJ.M. Park and T.C. Lubensky.\n\\newblock Topological {Defects} on {Fluctuating} {Surfaces}: {General}\n {Properties} and the {Kosterlitz}-{Thouless} {Transition}.\n\\newblock {\\em Phys. Rev. E}, 53:2648, 1996.\n\\newblock arXiv:cond-mat/9512108.\n\n\\bibitem{PaLu3:96}\nJ.M. Park and T.C. Lubensky.\n\\newblock {\\em Phys. Rev. E}, 53:2665, 1996.\n\\newblock arXiv:cond-mat/9512109.\n\n\\bibitem{ML:91}\nF.~C. MacKintosh and T.C. Lubensky.\n\\newblock {\\em Phys. Rev. Lett.}, 67:1169, 1991.\n\n\\bibitem{Evans:95}\nR.M.L. Evans.\n\\newblock {\\em J. de Physique II (France)}, 5:507, 1995.\n\\newblock arXiv:cond-mat/9410010.\n\n\\bibitem{PL:85}\nL.~Peiliti and S.~Leibler.\n\\newblock {\\em Phys. Rev. Lett.}, 54:1690, 1985.\n\n\\bibitem{Poly:86}\nA.~M. Polyakov.\n\\newblock Fine structure of strings.\n\\newblock {\\em Nucl. Phys. B}, 268:406, 1986.\n\n\\bibitem{Forster:86}\nD.~F\\\"orster.\n\\newblock {\\em Phys. Lett. A}, 114:115, 1986.\n\n\\bibitem{Klein1:86}\nH.~Kleinert.\n\\newblock {\\em Phys. Lett. A}, 114:263, 1986.\n\n\\bibitem{MM:94}\nD.~Morse and S.~Milner.\n\\newblock {\\em Europhys. Lett.}, 26:565, 1994.\n\n\\bibitem{Morse:94}\nD.~Morse.\n\\newblock {\\em Phys. Rev. E}, 50:R2423, 1994.\n\n\\bibitem{MM:95}\nD.~Morse and S.~Milner.\n\\newblock {\\em Phys. Rev. E}, 52:5918, 1995.\n\n\\bibitem{BEN}\nD.~Bensimon.\n\\newblock private communication.\n\n\\bibitem{Fischer}\nZ.~Khattari, P.~Steffen, and Th.~M. Fischer.\n\\newblock Viscoplastic properties of hexatic monolayers.\n\\newblock http://gfux005.mpikg-golm.mpg.de/.\n\n\\bibitem{Wiese:00}\nK.~J. Wiese.\n\\newblock {\\em Polymerized Membranes, a Review}.\n\\newblock Academic Press.\n\\newblock Vol. 19 of {Phase} {Transitions} and {Critical} {Phenomena}, Edited\n by C. Domb and J.~L. Lebowitz.\n\n\\bibitem{D:89}\nF.~David.\n\\newblock Geometry and {Field} {Theory} of {Random} {Surfaces} and {Membranes}.\n\\newblock In Nelson et~al. \\cite{Jer1}.\n\n\\bibitem{TF:98}\nG.~Thorleifsson and M.~Falcioni.\n\\newblock Improved algorithms for simulating crystalline membranes.\n\\newblock {\\em Computer Phys. Comm.}, 109:161, 1998.\n\\newblock arXiv:hep-lat/9709026.\n\n\\bibitem{Les}\nF.~David, P.~Ginsparg, and J.~Zinn-Justin, editors.\n\\newblock {\\em Fluctuating Geometries in Statistical Mechanics and Field\n Theory}, Les Houches Summer school, Amsterdam, 1996. North-Holland.\n\\newblock arXiv:arXiv.org/lh94.\n\n\\end{thebibliography}\n"
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cond-mat0002039
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Effect of phase fluctuations on INS and NMR experiments in the pseudo-gap regime of the underdoped cuprates
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"author": "Harry Westfahl Jr.$^1$ and Dirk K. Morr$^{1,2}$"
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We present a theory for inelastic neutron scattering (INS) and nuclear magnetic resonance (NMR) experiments in the pseudo-gap regime of the underdoped high-$T_c$ cuprates. We show that superconducting phase fluctuations greatly affect the temperature and frequency dependence of the spin-susceptibility, $\chi''$, probed by both experimental techniques. This result explains the appearance of a {resonance peak}, observed in INS experiments, below a temperature $T_0 > T_c$. In the same temperature regime, we find that the $^{63}$Cu spin-lattice relaxation rate, $1/T_1$, measured in NMR experiments, is suppressed. Our results are in qualitative agreement with the available experimental data.
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"name": "fluct.tex",
"string": "\\documentstyle[twocolumn,aps,epsfig,floats]{revtex}\n\\begin{document}\n\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname %\n@twocolumnfalse\\endcsname\n\n\n\\title{Effect of phase fluctuations on INS and NMR experiments in the pseudo-gap \nregime of the underdoped\ncuprates}\n\\author{Harry Westfahl Jr.$^1$ and Dirk K. Morr$^{1,2}$ }\n\\address{$^1$ University of Illinois at Urbana-Champaign, Loomis Laboratory of\nPhysics, 1110 W. Green St., Urbana, IL 61801 \\\\ $^2$ Theoretical\nDivision, Los Alamos National Laboratory, Los Alamos, NM 87545 }\n\\date{\\today}\n\\maketitle\n\\draft\n\n\n\\begin{abstract}\nWe present a theory for inelastic neutron scattering (INS) and\nnuclear magnetic resonance (NMR) experiments in the pseudo-gap\nregime of the underdoped high-$T_c$ cuprates. We show that\nsuperconducting phase fluctuations greatly affect the temperature\nand frequency dependence of the spin-susceptibility, $\\chi''$,\nprobed by both experimental techniques. This result explains the appearance\nof a {\\it resonance peak}, observed in INS experiments, below a\ntemperature $T_0 > T_c$. In the same temperature\nregime, we find that the $^{63}$Cu spin-lattice relaxation rate,\n$1/T_1$, measured in NMR experiments, is suppressed. Our results\nare in qualitative agreement with the available experimental data.\n\\end{abstract}\n\\pacs{PACS numbers:74.25.-q,74.25.Ha,74.25.Jb,74.40.+k}\n\n]\n\nOver the last few years intensive research has focused on the\norigin of the pseudo-gap region in the underdoped high-$T_c$\ncuprates \\cite{Eme95,Fra98,Kwo98,Chu98,Ges97,Zha97,Lee97}. This\npart of the phase diagram, below a characteristic temperature $T_{*}>T_{c}$,\nis characterized by a suppression of the low-frequency quasi-particle spectral\ndensity, as observed by angle-resolved photo-emission (ARPES) \\cite{arpes} and\nscanning tunneling spectroscopy (STS) experiments \\cite{sts}. For the same\ncompounds, inelastic neutron scattering (INS) experiments have revealed a\nsharp magnetic mode, the {\\it resonance peak}, below $T_*$, in\ncontrast to the optimally doped cuprates, where it only appears\nbelow $T_c$ \\cite{Dai}. Moreover, nuclear magnetic resonance (NMR)\nexperiments find a strong decrease of the $^{63}$Cu spin-lattice\nrelaxation rate, $ 1/T_{1} $, below $T_{*}$ \\cite{Tak91,Ish98}.\n\nThese experimental observations put tight restrictions on the\nproposed theoretical scenarios for the pseudogap ascribing it to\nspin-charge separation \\cite{Lee97}, SO(5) symmetry \\cite{Zha97},\ncondensation of performed pairs \\cite{Ges97} and spin-fluctuations\n\\cite{Chu98}. Emery and Kivelson (EK) \\cite{Eme95} proposed that,\ndue to the small superfluid density of the underdoped cuprates,\nthermal fluctuations in the phase of the superconducting (SC)\norder parameter destroy the long-range phase coherence in the\npseudo-gap regime, while preserving a finite local amplitude of\nthe order parameter, $|\\Delta(r)|$. In this communication, we\nargue that the presence of phase fluctuations provides an\nexplanation for the results of INS and NMR experiments discussed\nabove. We show that these fluctuations greatly affect the\ntemperature and frequency dependence of the spin-susceptibility,\n$\\chi''$, probed by both experimental techniques. Support for the\nexistence of phase fluctuations comes from recent high frequency\ntransport experiments by Corson {\\it et al.} \\cite{Cor99}. They\ndemonstrated that the SC transition in underdoped\nBi$_2$Sr$_2$CaCu$_2$O$_{8-\\delta}$ (Bi-2212) is of the\nKosterlitz-Thouless (KT) type, where at $T_c=74K$ the unbinding of\nthermally excited vortex-anti-vortex pairs destroys the long-range\nphase coherence. However, they also concluded that $|\\Delta(r)|$\nvanishes at a temperature $T_0 \\sim 100K$, while the onset\ntemperature, $T_*$, for the pseudo-gap regime is much higher. For\nthis reason we will focus our analysis on the region $T_c<T<T_0$.\n\nThe starting point for our calculations is the mean-field BCS\nHamiltonian in which the phase, $\\theta(r)$, of the\nsuperconducting order parameter, $\\Delta(r)=|\\Delta(r)| \\,\ne^{i\\theta(r)}$, varies on the scale of the phase coherence\nlength, $\\xi_{\\theta}$. Such a nonuniform phase $\\theta({\\bf r})$\nis treated via a gauge transformation \\cite{Fra98,Kwo98}\n\\begin{equation}\n\\Psi^{\\dagger}=e^{i\\theta(r) /2}c^{\\dagger }\n\\label{GT}\n\\end{equation}\nwhere $c^{\\dagger }$ is the creation operator of the\noriginal electrons. This transformation induces a coupling of the\n$\\Psi$-fermions to a local superfluid flow\n$\\bf{v}_{s}(\\bf{r})=\\nabla \\theta(\\bf{r})/2m$,\n(we set $\\hbar=1$) whose thermodynamic properties are determined by the 2D-XY\nHamiltonian \\begin{equation}\n{{\\cal H}_{XY}\\over k_BT}={K_{0}(T)\\over 2} \\,\n\\int d^2\\bf{r}\\, |\\nabla \\theta (\\bf{r})|^2 \\ ,\n\\label{Hxy}\n\\end{equation}\nwhere $K_{0}(T)=n_{s}(T)/(4mk_B T)$ is the ``bare'' phase\nstiffness and $n_{s}(T)$ is the 2D superfluid density per CuO$_2$\nlayer which for a d-wave superconductor is given by $n_{s}(T) =\nn_{s}(0)(1-T/T_0)$, where $T_0$ is the BCS mean field temperature\n\\cite{Lee97}. In order to compute $\\chi''$ in the presence of\nphase fluctuations we first compute it for a given configuration\nof $\\bf{v}_{s}(\\bf{r})$ and subsequently perform a thermodynamic\naverage over the ensemble specified by Eq.(\\ref{Hxy}). Our\napproach is similar to the one adopted by Franz and Millis (FM)\n\\cite{Fra98} who computed the single particle Green's function,\n$G(k,\\omega)$, in the pseudo-gap regime. They showed that the\nquantity which determines the ensemble average is the correlator\n$W = m^2 v_F^2 \\langle {\\bf v}^2_{s} \\rangle/2$, whose temperature\ndependence they extracted from fits to ARPES and STS experiments.\nIn the following we show that $W(T)$ can also be obtained from the\nexperiments by Corson {\\it et al.}~\\cite{Cor99}\n\n Assuming that the superfluid velocity is\npurely due to transverse phase fluctuations, we have\n\\begin{equation}\nW(T)=\\pi^2 v_F^2 \\int\n\\frac{d^{2}\\bf{q}}{4\\pi^{2}}\\frac{G(\\bf{q})}{q^{2}} \\ , \\label{WM}\n\\end{equation}\nwhere $ G(\\bf{q}) =\\left\\langle n_{\\bf{q}}n_{-\\bf{q}}\\right\n\\rangle$ is the vortex density correlator. In the limit of large\nvortex density, this correlator is given by \\cite{Minnhagen87} $G(\n{\\bf q})^{-1} \\sim 4\\pi^2 K_{0}(\\xi ^{2}_{\\theta }+q^{-2})$.\nEvaluation of the integral in Eq.(\\ref{WM}) with wave-vector\ncutoff $\\Lambda = 2 \\pi \\xi^{-1} _{GL}$ yields\n\\begin{equation}\nW\\left( T\\right) \\simeq { \\pi^3 \\Delta^2_{0} \\over 8 K_0(T) }\n\\left( { \\xi _{GL} \\over 2 \\pi \\xi _{\\theta}} \\right)^2 \\ln \\left[\n1+\\left({2 \\pi\\xi _{\\theta } \\over \\xi _{GL}} \\right)^{2}\\right] \\ .\n\\label{Wfinal}\n\\end{equation}\nHere, the phase coherence length is given by\n$\\xi_\\theta^{-2}(T)=4\\pi^2 K_0(T)n_F(T)$, where $n_F(T)$ is the\ndensity of free vortices, and we used the BCS result $\\xi\n_{GL}=v_F/(\\pi\\Delta_0)$. Corson {\\it et al.}~found in their analysis that\n$n_F(T)=(2A/\\pi\\xi^{2}_{GL})\\exp(-8 C K_0(T))$ with $A$ and $C$ being\nconstants of order $O(1)$. Assuming that the functional form of $n_F(T)$ (and\nthus that of $W(T)$) remains the same for all underdoped Bi-2212 compounds,\nwith $A$ and $C$ being the only doping dependent parameters, we present $W(T)$\nfrom Eq.(\\ref{Wfinal}) in the inset of Fig.~\\ref{res_peak}, together with FM's\nfits \\cite{Fra98} to STS experiments in Bi-2212 ($T_c = 83K$) \\cite{sts}. With\n$A=0.1$ and $C=0.6$, we find good quantitative agreement of our theoretical\nresults with those of FM up to $T_0 \\approx 150$ K. At this temperature, the\nabove approximation, as well as the analysis by FM, presumably break down\nsince $\\sqrt{W(T)}$ becomes of the order of the maximum\nsuperconducting gap.\n\nWe now turn to the appearance of the resonance peak in the\npseudogap region. Morr and Pines (MP) \\cite{Morr98} recently\nargued that the resonance peak in the superconducting state arises\nfrom a spin-wave mode whose dispersion is given by\n\\begin{equation}\n\\omega_q^2=\\Delta_{sw}^2+c_{sw}^2|\\bf{q}-\\bf{Q}|^2 \\ ,\n\\label{omq}\n\\end{equation}\nwhere $\\Delta_{sw}$ is the spin-wave gap, $c_{sw}$ is the\nspin-wave velocity and $\\bf{Q}$ is the position of the magnetic\npeak in momentum space. Starting from a spin-fermion model\n\\cite{sfmodel}, MP showed that this mode is strongly damped in the\nnormal state, but becomes only weakly damped in the\nsuperconducting state, if $\\Delta_{sw}$ is less than the gap,\n$\\omega_c$, for particle-hole excitations with total momentum\n$\\bf{Q}$. These excitations connect points on the Fermi surface\n(FS) in the vicinity of $(0,\\pi)$ and $(\\pi,0)$ (``hot spots\"),\nand thus for a d-wave gap $ \\Delta_{\\bf{k}}=\\Delta _{0}\\, \\,\n(\\cos(k_{x})-\\cos (k_{y}))/2 \\), $\\omega_c \\approx 2 \\Delta_{0}$.\n\nMP computed $\\chi$ using the Dyson-equation\n\\begin{equation}\n\\label{chifull} \\chi^{-1} =\\chi_0^{-1} - \\Pi \\ ,\n\\end{equation}\nwhere $\\chi_0$ is the ``bare\" susceptibility and $\\Pi$ is the\nbosonic self-energy given by the irreducible particle-hole bubble.\nFor $\\chi_0$, MP made the experimentally motivated ansatz\n\\begin{equation}\n\\label{chi0} \\chi_0^{-1}= { \\omega_q^2-\\omega^2 \\over \\alpha \\,\nc^{2}_{sw} } \\ ,\n\\end{equation}\nwhere $\\omega_q$ is given in Eq.(\\ref{omq}). In the\nsuperconducting state, one obtains for $\\Pi$ to lowest order in\nthe spin-fermion coupling $g$\n\\begin{eqnarray}\n\\Pi({\\bf q}, i \\omega_n) &=& - g^2 \\, T \\sum_{{\\bf k},m} \\\n\\Big\\{ G({\\bf k}, i\\Omega_m) G({\\bf k+q}, i\\Omega_m+i\\omega_n) \\nonumber \\\\\n& & + F({\\bf k}, i\\Omega_m) F({\\bf k+q}, i\\Omega_m+i\\omega_n) \\Big\\} \\ ,\n\\label{Pi}\n\\end{eqnarray}\nwith $G$ and $F$ being the normal and anomalous Green's functions.\nSince, within the spin-fermion model, $\\chi_0$ is obtained by\nintegrating out the high-energy fermionic degrees of freedom, it\nis largely unaffected by the onset of superconductivity or the\npseudo-gap. Moreover, MP argued that due to fermionic self-energy\ncorrections, Re$\\, \\Pi$ in the SC state only leads to an\nirrelevant renormalization of $\\Delta_{sw}$ and $c_{sw}$. Since\nthe same argument also holds within our scenario for the\npseudo-gap region, we neglect Re$\\, \\Pi$ in the following.\n\nOn the other hand, Im$\\, \\Pi$ which determines the damping of the\nspin excitations, changes dramatically in the SC state due to the\nopening of a gap in the fermionic dispersion. Consequently, we\nexpect phase fluctuations to strongly affect Im$\\, \\Pi$. Moreover,\nin each polarization bubble present in the RPA expansion of\nEq.(\\ref{chifull}), the electron-hole pairs probe different parts\nof the sample and thus independent configurations of thermally\nexcited super-currents. It then follows that the susceptibility,\n$\\chi_{pf}$, in the presence of phase fluctuations is obtained\nfrom Eq.(\\ref{chifull}) by using Im$\\, \\Pi_{pf}$ averaged over the\nthermodynamic ensemble determined by Eq.(\\ref{Hxy}).\n\nBefore we discuss the effect of phase fluctuations on Im$\\, \\Pi$,\nwe shortly review its form in the normal and SC state. Extending\nEq.(\\ref{Pi}) to the normal state, we obtain Im$\\, \\Pi({\\bf\nQ})= 4g^2\\omega/ (\\pi v^{2}_{F})$ \\cite{Chu97}, where $v_F$ is the Fermi \nvelocity\nat the hot spots. In contrast, in the SC state, in the limit of $\nT\\ll \\omega_c $, we find to order $O(T/\\omega_c)$\n\\begin{eqnarray}\n{\\rm Im}\\, \\Pi ({\\bf Q}, \\omega) &=& {4 g^2 \\omega_c \\over \\pi\nv_F^2}\\,E(\\sqrt{{1 - \\bar \\omega}^2})\\, \\theta \\left( {\\bar\n\\omega}-1 \\right) \\nonumber \\\\ &\\sim & {g^2 \\omega_c \\over v_F^2}\n( {\\bar \\omega} + 1) \\theta \\left( {\\bar \\omega}-1 \\right) \\ ,\n\\label{PiBCS}\n\\end{eqnarray}\nwhere $\\theta(x)$ is the Heavyside step function, $E(x)$ is the\ncomplete Elliptic integral of the first kind and ${\\bar\n\\omega}=\\omega/\\omega_c$. Thus, Im$\\, \\Pi$ vanishes for\nfrequencies below $\\omega_c$. In Fig.~\\ref{ImPi} we present the\nfrequency dependence of Im$\\, \\Pi$ in the normal and SC state.\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode \\epsfxsize=7.5cm \\epsffile{ImPi.ai}\n\\end{center}\n\\caption{${\\rm Im}\\, \\Pi$ in the normal (black line) and SC state\n(dashed line), and $ {\\rm Im}\\, \\Pi_{pf} ( \\omega)$ in the\npseudo-gap region for {\\bf (a)} ${\\bar W}=0.015$ (dotted line),\nand {\\bf (b)} ${\\bar W}=0.1$ (dashed-dotted line). }\n\\label{ImPi}\n\\end{figure}\n\nWe now consider the effect of phase fluctuations on Im$\\, \\Pi$.\nNote that $\\Pi$, Eq.(\\ref{Pi}), and thus $\\chi$, Eq.(\\ref{chifull}),\nare invariant under the gauge transformation,\nEq.(\\ref{GT}), in contrast to $G(k,\\omega)$, considered by FM.\nThus $G,F$ can be straightforwardly calculated using the\n$\\Psi$-fermions. In the limit $k_F\\xi_\\theta\\gg 1$, where $k_F$ is the Fermi\nmomentum at the hot-spots, the interaction of the $\\Psi$-fermions with the\nsuperfluid flow leads to a Doppler shift in the $\\Psi$-excitation spectrum\n\\cite{Fra98,Kwo98} given by\n\\begin{equation}\n\\label{qp_disp} E^{\\pm }_{\\bf{k}}=\\sqrt{\\epsilon\n_{\\bf{k}}^{2}+|\\Delta _{\\bf{k}}|^{2}}\\pm D_{\\bf{k}} \\\n\\end{equation}\nwhere $\\epsilon _{\\bf{k}}$ is the fermionic dispersion in the\nnormal state, \\(D_{\\bf{k}}=m \\bf{v}_F(\\bf{k}) \\cdot \\bf{v}_{s} \\)\nis the induced Doppler-shift and $\\bf{v}_F(\\bf{k})$ is the Fermi\nvelocity. In the limit $T \\ll D_{\\bf{k}} \\ll \\Delta_0$, Im$\\, \\Pi$\nfor a given superfluid velocity is obtained from Eq.(\\ref{PiBCS})\nvia the frequency shift\n\\begin{equation}\n\\label{om}\n\\omega \\rightarrow \\omega +\\left( D_x +D_y \\right) \\ .\n\\end{equation}\nSimilar to the case of the fermionic spectral function \\cite{Fra98},\nthe thermodynamic average of Im$\\, \\Pi$ over the ensemble specified\nby Eq.(\\ref{Hxy}) is obtained by convoluting Im$\\, \\Pi$ with a\nGaussian distribution of Doppler shifts of the form\n\\begin{equation}\n\\label{prob} P(D_{\\alpha })={ 1 \\over \\sqrt{2\\pi W} }\n\\exp\\left( -\\frac{D_{\\alpha }^2}{2W}\\right) \\, \\, ,\n\\end{equation}\nwhere \\(\\alpha =x,y \\). In the limit $\\sqrt{W} \\ll T \\ll\n\\omega_c$, we can perform this convolution analytically and\nobtain\n\\begin{eqnarray}\n{\\rm Im}\\, \\Pi_{pf} (\\omega) & =& {g^2 \\omega_c \\over 2 v_F^2}\n \\Bigg\\{ \\left( 1+{\\bar\\omega}\\right)\n\\left[ 1+\\Phi \\left( \\frac{ {\\bar\\omega} -1}{\\sqrt{ {\\overline\nW}(T)}})\\right)\\right] \\nonumber \\\\ & + & \\sqrt{ {\\overline\nW}\\left( T\\right)/\\pi } \\ \\exp\\left(-{ ({\\bar \\omega} -1)^2 \\over\n {\\overline W}(T) } \\right)\\Bigg\\} \\ ,\n\\label{ImPi_ave}\n\\end{eqnarray}\nwhere \\( \\Phi (x) \\) is the error function. It follows from\nFig.~\\ref{ImPi}, in which we present the spin-damping for two\ndifferent values of ${\\overline W}=W/\\Delta_0^2$, that the effect of phase\nfluctuations on Im$\\, \\Pi$ is two-fold. First, they lead to a non-zero value of\nIm$\\, \\Pi_{pf} ( \\omega)$ for $\\omega<\\omega_c$, in contrast to\nthe form of Im$\\, \\Pi$ in the superconducting state where the\nspin-damping at $T=0$ vanishes below $\\omega_c$. Second, the\nspin-damping below $\\omega_c$ increases with increasing $W$ while\nat the same time, the sharp step in Im$\\, \\Pi$ is smoothed out.\nNote that in the pseudo-gap region, $T\\ll \\omega_c$, and\nconsequently the temperature dependence of $ {\\rm Im}\\, \\Pi_{pf} (\n\\omega) $ is determined by that of $W(T)$.\n\nFinally, inserting ${\\rm Im}\\, \\Pi_{pf} ( \\omega) $ into\nEq.(\\ref{chifull}), we obtain $\\chi_{pf}''({\\bf Q}, \\omega)$ in\nthe pseudo-gap region. In Fig.~\\ref{res_peak} we present our\ntheoretical results for the frequency and temperature dependence\nof the resonance peak in Bi-2212 ($T_c=83K$), using the $W(T)$\nshown on the inset.\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{chi_W_STS2.ai}\n\\end{center}\n\\caption{The resonance peak in the pseudo-gap region for ${g^2\n\\alpha \\xi^{2} \\over 2 v_F^2}=1/(70 meV)$,$\\Delta_{sw}=35 meV$,\n$\\omega_c=65 meV$ and $W(T)$ from the inset. Inset: $W(T)$ from\nRef.~\\protect\\cite{Fra98} (points) and from Eq.(\\ref{Wfinal}) with\nA=0.1 and C=0.6 (solid line). } \\label{res_peak}\n\\end{figure}\nWe find that, as the temperature is increased above $T_c$, the\nresonance peak becomes broader, while its intensity diminishes.\nSince $W(T)$ is a monotonically increasing function of\ntemperature, it follows from Fig.~\\ref{ImPi} that the spin damping\nfor $\\omega \\approx \\Delta_{sw}<\\omega_c$ also increases with\ntemperature, giving rise to the behavior of the peak\nintensity/width shown in Fig.~\\ref{res_peak}. Unfortunately, no\nexperimental data for the temperature dependence of the resonance peaks in\nthe pseudo-gap region of underdoped Bi-2212 are currently available. However,\nour results are in qualitative agreement with the experimental data on\nunderdoped YBa$_2$Cu$_3$O$_{6+x}$ \\cite{Dai}.\n\nWe now turn to the second experimental probe of $\\chi''$, the \\(\n^{63} \\)Cu spin-lattice relaxation rate, \\( 1/T_{1} \\). For an\napplied field parallel to the \\( c- \\)axis, \\( 1/T_{1} \\) is given\nby\n\\begin{equation} \\label{T1T} \\frac{1}{T_{1}T}={k_{B} \\over 2}\n(\\gamma _{n}\\gamma _{e})^{2}\\frac{1}{N}\\sum _{\\bf{q}}F_{c}(\\bf{q})\\lim\n_{\\omega \\rightarrow 0}\\frac{\\chi'' (\\bf{q},\\omega )}{\\omega }\\, \\, ,\n\\end{equation} where \\begin{equation}\n\\label{form} F_{c}({\\bf q})=\\left[ A_{ab}+2B\\left(\ncos(q_{x})+cos(q_{y})\\right) \\right] ^{2}\\, \\, ,\n\\end{equation}\n and \\( A_{ab} \\) and \\( B \\) are the on-site and transferred\nhyperfine coupling constants, respectively. The spin-lattice\nrelaxation rate in the mixed state, i.e., in the presence of a\nsuperflow, was recently considered by Morr and Wortis (MW)\n\\cite{Morr99}. Using the low-frequency limit of\nEqs.(\\ref{chifull}) and (\\ref{Pi}), they found that the\ntemperature dependence of $1/T_{1}$ is determined by the set\n$\\{D_n/T\\}$, where $D_n$ is the Doppler-shift at the\n$n$th node (see Eq.(\\ref{qp_disp})). In\nthe limit, $|D_n/T| \\gg 1$, they obtained\n\\begin{equation}\n{1 \\over T_1 T} = { {\\cal C} \\over N} \\sum_{i,j} {\\cal F}({\\bf\nq}_{i,j}) |D_i| \\, |D_j| \\ ,\n \\label{T1Tf1}\n\\end{equation}\nwhere ${\\cal C}= (k_B/\\pi)(\\alpha g \\gamma_n\n\\gamma_e)^2/ (4 v_F v_\\Delta)^2$, $v_\\Delta=|\\partial\n\\Delta_{\\bf k} /\\partial {\\bf k}|$ at the nodes, and\n\\begin{equation}\n{\\cal F}({\\bf q}_{i,j})={F_c({\\bf q}_{i,j}) \\over (\\xi^{-2} +\n|{\\bf q}_{i,j}-{\\bf Q}|^2)^2} \\ . \\label{calF}\n\\end{equation}\nHere, ${\\bf q}_{i,j}$ is the wave-vector connecting the nodes $i$\nand $j$, and $\\xi$ is the magnetic correlation length. In the\nlimit $T \\ll \\sqrt{W(T)}$, the convolution of Eq.(\\ref{T1Tf1})\nwith the Gaussian distribution of Eq.(\\ref{prob}) can be performed\nanalytically, and we obtain\n%\\begin{eqnarray}\n% \\left( { 1 \\over T_1 T }\\right)_{pf} &=& 4 {\\cal C} W(T)\n% \\\n%\\Big[ {\\cal F}({\\bf q}_{1,1})+ {\\cal F}({\\bf q}_{1,3}) \\nonumber\n%\\\\\n%& & \\qquad + { 8 \\over \\pi} {\\cal F}({\\bf q}_{1,2}) \\Big] .\n%\\label{T1T_ave}\n%\\end{eqnarray}\n\\begin{equation}\n\\left( { 1 \\over T_1 T }\\right)_{pf} = \\beta \\, W(T)\n\\label{T1T_ave}\n\\end{equation}\nwhere $\\beta=4 {\\cal C}({\\cal F}(0)+ {\\cal F}({\\bf q}_{1,3}) +\n{8 \\over \\pi} {\\cal F}({\\bf q}_{1,2}))$. The constant $\\beta$ can be\nexperimentally obtained \\cite{comm1} by fitting $(T_{1}T)^{-1}$ at $T<T_c$\nwith the d-wave BCS expression $(T_{1}T)^{-1}=\\beta{\\pi^2 \\over 3} T^2$.\nNote that the relaxation rate in Eq.(\\ref{T1T_ave}) directly\nreflects the strength of the classical phase fluctuations. In Fig.~\\ref{nmr}\nwe present our theoretical results for $(T_{1}T)^{-1}_{pf}$,\nEq.(\\ref{T1T_ave}), together with the experimental data by Ishida {\\it et\nal.}~\\cite{Ish98} on underdoped Bi-2212 ($T_c=79K$).\n\\begin{figure} [t]\n\\begin{center}\n\\leavevmode\n\\epsfxsize=7.5cm\n\\epsffile{T1BSCCO.ai}\n\\end{center}\n\\caption{$ 1/T_{1}T$ in the pseudo-gap region of underdoped Bi-2212 ($Tc=79K$). \nSolid\nline: theoretical fits with A=0.05 and C=0.3. Filled squares:\nexperimental data taken from Ref.~\\protect\\cite{Ish98}. We assumed a\nconstant background factor $\\gamma=0.4 K^{-1}s^{-1}$} \\label{nmr}\n\\end{figure}\nUsing $W(T)$, Eq.(\\ref{Wfinal}), with A=0.05 and C=0.3, we find good\nagreement of our theoretical results with the experimental data\nbetween $T_c$ and $T_0 \\approx 130$ K.\n%Moreover, in the inset of\n%Fig.~\\ref{nmr} we present $(T_{1}T)^{-1}_{pf}$ for a $T_c=79K$\n%Bi-2212 sample, calculated with $W(T)$ from the inset of\n%Fig.(\\ref{res_peak})\nNote that the external magnetic\nfield applied in NMR experiments increases the density of free\nvortices by ${\\bar n}_{F}(B) \\sim B/ \\phi_{0}$ \\cite{Minnhagen87}. As a\nresult, the phase coherence length (and in turn $W$) acquires a\nmagnetic field dependence, $\\xi ^{-2}_{\\theta}=4\\pi^2 K_{0}(T)\n\\left(n_{F}(T)+{\\bar n}_{F}(B) \\right)$. For $T \\geq T_c$, and the\nparameter set used above, we find that a magnetic field of $B=10T$\nincreases the vortex density by ${\\bar n}_{F}(B)/ n_{F}(T_c) \\sim\n0.1$; its contribution to $W$ can thus be neglected. Thus the relaxation rate \nabove $T_c$ should be independent of magnetic field for typical values of $B$, \nwhich is consistent with recent experiments by Gorny {\\it et al.} \\cite{Gor99}.\n\nIn the above scenario, we neglected the effect of longitudinal\nphase fluctuations which arise from spin-wave like excitations.\nThis is justified since their excitation spectrum is very\nlikely gapped by the Anderson-Higgs mechanism \\cite{Fra98}, and\nthey are, consequently, irrelevant for the low-frequency\nthermodynamic properties of the underdoped cuprates. It was recently proposed\nin Ref.\\cite{Carl99} that longitudinal phase fluctuations are responsible for\nthe linear temperature dependence of the superfluid density at $T\\ll T_c$. FM\npointed out that longitudinal phase fluctuations at $T \\ll T_c$\nlead to a $W_{long} \\sim T$. In this case it follows from Eq.(\\ref{T1T_ave})\nthat, for $^{63}$Cu and $^{17}$O, $1/T_1T \\sim T$ at $T\\ll T_c$, in contrast\nto the experimentally observed $1/T_1T \\sim T^2$ \\cite{Mar94}. This result \nsuggests\nthat longitudinal phase fluctuations are absent in the superconducting state.\n\nWe assumed above, following the argument applied to STS and ARPES\nexperiments \\cite{Fra98}, that transverse phase fluctuations are\nstatic on the time-scale of INS and NMR experiments which allowed\nus to neglect the quantum dynamical nature of the vortices. While\nthis assumption likely holds for ``fast\" probes like INS, ARPES\nand STS where the quasi-particles are coupled to phase fluctuations\nfor short times, it might be less justified for the much\n``slower'' NMR experiments. In this light, the agreement of our\ntheoretical NMR results with the experimental data, Fig.~\\ref{nmr}\nis remarkable. However, the effects of the vortex quantum dynamics\non various experimental probes is still an open question which\nrequires further study.\n\nIn summary we propose a scenario for INS and NMR experiments in\nthe pseudogap region of the underdoped cuprates. We argue that\nphase fluctuations of the superconducting order parameter\ndrastically affect the frequency dependence of the spin\nsusceptibility and can thus qualitatively account for the\ntemperature dependence of the resonance peak. Moreover, we show\nthat the spin-lattice relaxation rate, $1/T_1T$, is a direct probe\nfor the strength of the phase fluctuations, as reflected in\n$W(T)$. Finally, we showed that $W(T)$ obtained from high\nfrequency transport measurements is in good qualitative agreement\nwith that extracted from STS experiments.\n\nIt is our pleasure to thank A. H. Castro Neto, A.V. Chubukov, \nA. J. Leggett, A. J.\nMillis, D. Pines, R. Ramazashvili, J. Schmalian, R. Stern and M.\nTurlakov for valuable discussions. This work has been supported by\n\\emph{Funda\\c c\\~ ao de Amparo \\`a Pesquisa do Estado de S\\~ ao\nPaulo} (FAPESP), the Center of Advanced Studies (CAS) of the\nUniversity of Illinois (H.W.) and in part by the Science and\nTechnology Center for Superconductivity through NSF-grant\nDMR91-20000 and by DOE at Los Alamos (D.K.M).\n\n\\begin{thebibliography}{99}\n\\bibitem{Eme95}V.J. Emery and S.A. Kivelson, Nature {\\bf 374}, 434 (1995);\n\\textit{ibid.}, Phys. Rev. Lett. \\textbf{74}, 3253 (1995).\n\\bibitem{Fra98}M. Franz and A.J. Millis, Phys. Rev. B, {\\bf 58}, 14572 (1998).\n\\bibitem{Kwo98}H.-J. Kwon and A.T. Dorsey, Phys. Rev. B, {\\bf 59}, 6438\n(1999). \\bibitem{Chu98}A.V. Chubukov and J. Schmalian, Phys. Rev. B\n\\textbf{57}, R11085 (1998).\n\\bibitem{Ges97}V.G. Geshkenbein, L.B. Ioffe, and A.I. Larkin, Phys. Rev. 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{
"name": "cond-mat0002039.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{Eme95}V.J. Emery and S.A. Kivelson, Nature {\\bf 374}, 434 (1995);\n\\textit{ibid.}, Phys. Rev. Lett. \\textbf{74}, 3253 (1995).\n\\bibitem{Fra98}M. Franz and A.J. Millis, Phys. Rev. B, {\\bf 58}, 14572 (1998).\n\\bibitem{Kwo98}H.-J. Kwon and A.T. Dorsey, Phys. Rev. B, {\\bf 59}, 6438\n(1999). \\bibitem{Chu98}A.V. Chubukov and J. Schmalian, Phys. Rev. B\n\\textbf{57}, R11085 (1998).\n\\bibitem{Ges97}V.G. Geshkenbein, L.B. Ioffe, and A.I. Larkin, Phys. Rev. B\n\\textbf{55}, 3173 (1997).\n\\bibitem{Zha97}S.C. Zhang, Science \\textbf{275}, 1089 (1997).\n\\bibitem{Lee97}P.A. Lee and X.-G. Wen, Phys. Rev. Lett. \\textbf{78} 4111 (1997).\n\\bibitem{arpes} H. Ding {\\it et al.}, Phys. Rev. Lett., {\\bf 78}, 2628\n(1997).\n\\bibitem{sts} C. H. Renner {\\it et al.}, Phys. Rev. Lett, {\\bf 80}, 149\n(1998).\n\\bibitem{Dai}P. Dai {\\it et al.}, Science {\\bf 284}, 1344 (1999); H.F. Fong\n{\\it et al.}, preprint, cond-mat/9910041.\n\\bibitem{Tak91}M. Takigawa, Phys. Rev. B \\textbf{43}, 247 (1991).\n\\bibitem{Ish98}K. Ishida {\\it et al.}, Phys. Rev. B {\\bf 58}, R5960 (1998).\n\\bibitem{Cor99} J. Corson, {\\it et al.}, Nature {\\bf 398}, 221 (1999).\n\\bibitem{Minnhagen87}P. Minnhagen, Rev. Mod. Phys. \\textbf{59}, 1001\n(1987).\n\\bibitem{Morr98}D.K. Morr and D. Pines, Phys. Rev. Lett. \\textbf{81}, 1086\n(1998).\n\\bibitem{sfmodel} P. Monthoux and D. Pines, Phys. Rev B {\\bf 47}, 6069 (1993).\n\\bibitem{Chu97} A.V. Chubukov, Europhys. Lett. {\\bf 44}, 655 (1997).\n\\bibitem{Morr99}D.K. Morr and R. Wortis, Phys.~Rev.~B {\\bf 61}, R882 (2000).\n\\bibitem{comm1} We assume that $\\, \\Delta_{sw}$ and $c_{sw}$\nonly exhibit a weak temperature dependence in the\npseudo-gap region \\protect\\cite{Bourges}.\n\\bibitem{Bourges} P. Bourges in {\\it High Temperature\nSuperconductivity}, edited by S.E. Barnes {\\it et al.} 207-212\n(CP483 American Institute of Physics, Amsterdam, 1999) .\n\\bibitem{Gor99} K. Gorny {\\it et al.}, Phys. Rev. Lett. {\\bf 82}, 177 (1999).\n\\bibitem{Carl99} E. W. Carlson, {\\it et al.}, Phys. Rev. Lett, {\\bf 83}, 612\n(1999).\n\\bibitem{Mar94} J.A. Martindale {\\it et al.}, Phys. Rev. B {\\bf\n47}, 9155 (1993).\n\\end{thebibliography}"
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cond-mat0002040
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Finite Size Effects on Spin Glass Dynamics
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[
{
"author": "Y.G. Joh"
}
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The recent identification of a time and temperature dependent spin glass correlation length, $\xi({t_w},T)$, has consequences for samples of finite size. Qualitative arguments are given on this basis for departures from $t/{t_w}$ scaling for the time decay of the thermoremanent magnetization, ${M_{TRM}}(t,{t_w},T)$, where $t$ is the measurement time after a ``waiting time'' $t_w$, and for the imaginary part of the ac susceptibility, ${\chi^{{\prime}{\prime}}}(\omega,t)$. Consistency is obtained for a more rapid decay of $M_{TRM}(t,{t_w},T)$ with increasing $t_w$ when plotted as a function of $t/{t_w}$, for the deviation of the characteristic time for ${M_{TRM}}(t,{t_w},T)$ from a linear dependence upon $H^2$ at larger values of $H$, and for the deviation of the decay of ${\chi^{{\prime}{\prime}}}(\omega,t)$ from $\omega t$ scaling upon a change in magnetic field at large values of $\omega t$. These departures from scaling can, in principle, be used to extract the particle size distribution for a given spin glass sample.
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[
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"name": "LengthScale74.TEX",
"string": "\\documentstyle[aps,prb,epsf]{revtex}\n\\begin{document}\n\\draft\n\\wideabs{\n\\title{Finite Size Effects on Spin Glass Dynamics}\n\\author{Y.G. Joh}\n\\address{National High Magnetic Field Laboratory - Los Alamos National Laboratory, NM 87545}\n\\author{R. Orbach, G.G. Wood}\n\\address{Department of Physics, University of California, Riverside, California 92521-0101}\n\\author{J. Hammann, and E. Vincent}\n\\address{Service de Physique de l'Etat Condens\\'e, Commissariat \\`a l'Energie Atomique,\nSaclay, 91191 Gif sur Yvette, France}\n\\date{Submitted February 5, 2000}\n\\maketitle\n\\begin{abstract}\nThe recent identification of a time and temperature dependent spin glass correlation length, $\\xi({t_w},T)$, has consequences for samples of finite size. Qualitative arguments are given on this basis for departures from $t/{t_w}$ scaling for the time decay of the thermoremanent magnetization, ${M_{TRM}}(t,{t_w},T)$, where $t$ is the measurement time after a ``waiting time'' $t_w$, and for the imaginary part of the ac susceptibility, ${\\chi^{{\\prime}{\\prime}}}(\\omega,t)$. Consistency is obtained for a more rapid decay of $M_{TRM}(t,{t_w},T)$ with increasing $t_w$ when plotted as a function of $t/{t_w}$, for the deviation of the characteristic time for ${M_{TRM}}(t,{t_w},T)$ from a linear dependence upon $H^2$ at larger values of $H$, and for the deviation of the decay of ${\\chi^{{\\prime}{\\prime}}}(\\omega,t)$ from $\\omega t$ scaling upon a change in magnetic field at large values of $\\omega t$. These departures from scaling can, in principle, be used to extract the particle size distribution for a given spin glass sample.\n\\end{abstract}\n\\pacs{PACS numbers: 75.50.Lk, 75.10.Nr, 75.40.Gb}\n%Do not delete this line\n}\n\\narrowtext\n\n\\section{Introduction}\n\nSlow spin-glass dynamics have been investigated through examination of the time dependence of the irreversible magnetization,\\cite{Lundgren} and the out-of-phase dynamical magnetic susceptibility.\\cite{Refregier} Both are related, but typically explore different time domains. A recent paper\\cite{Wood} has shown how the Parisi order parameter,\\cite{Parisi} $x(q)$, can be extracted from high precision measurements of the decay of the thermoremanent magnetization, ${M_{TRM}}(t,{t_w},T)$, where $t$ is the measurement time after waiting a time $t_w$ at the measurement temperature $T(<{T_g})$. The time scale of the experiments restricts the examination to a small region of overlap space, but the shape and magnitude of the results are in quantitative accord with mean field expressions.\\cite{Mezard}\n\nThe barrier model of Nemoto\\cite{Nemoto} and Vertechi and Virasoro\\cite{Vertechi} has been adapted to the metastable states observed in experiment.\\cite{Lederman} Dynamics are ascribed to barrier hopping within ultrametric manifolds of constant magnetization $M$.\\cite{Mezard2} Measurements by Chu {\\it et al.},\\cite{Kenning} along with other observations of a similar nature,\\cite{Bouchaud} suggest that a change in magnetic field {\\it reduces} the barriers in the initially occupied magnetization manifold. This reduction can be conceptually thought of as diffusion between states of constant $M$ through intermediate states of lower Zeeman energy.\\cite{Bouchaud} We designate the reduction in each of the barrier heights by a Zeeman energy, $E_z$, proportional to the number of participating spins, $N_s$, which lie within a coherence length $\\xi({t_w},T)$ from one another. This length scale has been extracted from measurements on both insulating and metallic spin glasses.\\cite{Wood2} As shown in Ref. 12, and exhibited below in this paper as a consequence of further measurements, the spin glass correlation length appears {\\it universal}, having the value $\\xi({t_w},T)=({t_w}/{\\tau_0})^{\\alpha T/{T_g}}$, with $\\alpha=0.153$, for all spin glasses measured so far ($Cu:Mn, Ag:Mn$, and the thiospinel $Cd{Cr_{1.7}}{In_{0.3}}{S_4})$. The length $\\xi({t_w},T)$ is in units of the typical spin-spin spatial separation.\n\nThis paper examines the consequences of a finite length scale associated with spin glass order. In particular, we address the question of what happens when $\\xi({t_w},T)$ exceeds a physical length $r$ associated with finite sample size ({\\it e.g.} defects in an insulating structure, or crystallites in a polycrystalline sample). Typical laboratory waiting times and measurement temperatures result in $\\xi({t_w},T)\\approx 10-100~nm$, approximating the coarseness of powdered or polycrystalline samples. Such materials do not possess a single grain size $r$, but rather a distribution of grains sizes, $P(r)$. This paper will show that it may be possible to extract $P(r)$ from time dependent magnetic measurements.\n\nA consequence of $\\xi({t_w},T)\\approx~r$ is that conventional scaling relationships may be violated. This paper will associate the lack of scaling with $t/{t_w}$ found for the time decay of ${M_{TRM}}(t,{t_w},T)$ by Alba {\\it et al.}\\cite{{Alba},{Ocio}} with $\\xi({t_w},T)\\approx~r$. Other observations which we believe can be associated with\n$\\xi({t_w},T)\\approx~r$ are the deviation of $E_z$ from a proportionality to $H^2$,\\cite{{Wood2},{Vincent2}} and the deviation of ${\\chi^{{\\prime}{\\prime}}}(\\omega,t)$ from $\\omega t$ scaling upon a change in magnetic field for large $\\omega t$.\\cite{Bouchaud}\n\nThis paper will attempt to make plausible the relationship between finite size and lack of scaling in the same spirit as in Bouchaud {\\it et al.}\\cite{Hammann} Sec. II describes the model which underlies our analysis. Sec. III outlines the experimental observations which require a departure from scaling. The relationships between finite size effects and the lack of scaling are developed in Sec. IV. Key experiments and numerical simulations required to fully justify these relationships are discussed in Sec. V. Sec. VI presents our conclusions.\n\n\\section{Description of the Model}\n\nThe model we invoke to quantify our analysis is based on an extension of the ``pure states'' ultrametric geometry found\\cite{Mezard3} for the mean field solution\\cite{Parisi} of the Sherrington-Kirkpatrick infinite range model.\\cite{Sherrington} Previous experiments\\cite{Lederman} have shown that the infinite barriers which separate pure states derive from finite barriers separating metastable states, diverging at a characteristic temperature $T^*$ as the temperature is lowered. Within the time and temperature scale probed experimentally,\\cite{Lederman} the value of $T^*$ depends only on the height of the barrier at a given temperature, and a universal form for $d\\Delta/dT~vs~T$ was derived from experiment.\n\nDynamics are extracted from the assumption that the metastable states possess the same ultrametric geometry as pure states, the barriers separating the metastable states increasing linearly as the Hamming Distance $D$ between states,\\cite{Nemoto,Vertechi} $\\Delta(D)\\propto D$, and activated dynamics.\n\nThe time decay of the thermoremanent magnetization follows from assuming that a temperature quench in constant magnetic field isolates the spin glass states into specific points within phase space. This is represented by a probability density $P(D)$, with $D$ the Hamming distance defined by $D={1\\over 2}({q_{EA}}-q)$, where $q_{EA}$ is the Edwards-Anderson order parameter,\\cite{Edwards} and $q$ the overlap between the states separated by $D$. Immediately after quench, $P(D)=\\delta(D)$. Keeping the magnetic field constant, the system diffuses from $D=0$ to states with $D\\neq 0$ as a function of the ``waiting time'', $t_w$, according to activated dynamics. As stated earlier, the ultrametric tree upon which this diffusion develops is one of constant magnetization $M_{fc}$ associated with the field cooled state.\\cite{Mezard2} The amplitude of the delta function diminishes with increasing $t_w$, with the associated occupancy of states at finite $D$ increasing with $t_w$. Activated dynamics in phase space leads to a maximum barrier surmounted in the time $t_w$ of magnitude $\\Delta({t_w},T)={k_B}T\\ell n({t_w}/{\\tau_0})$. Detailed balance, with experimental confirmation,\\cite{Lederman,Kenning} leads to equilibrium occupation of the metastable states.\n\nExperiments and analysis by Vincent {\\it et al.}\\cite{Bouchaud} establish that, after waiting a time $t_w$, cutting the magnetic field to zero diminishes each barrier height by an amount to which we shall refer as $E_z$, the {\\it Zeeman energy}. This reduction was interpreted\\cite{Lederman} as the origin of the so-called ``reversible'' change in the magnetization. Their model assumes that all of the states occupied for $D<{D_{E_z}}$ [including those at $P(0)$] immediately empty to the zero magnetization manifold $M=0$. If $\\Delta(D)$ is known, then $D_{E_z}$ is the value of $D$ at which ${E_z}=\\Delta(D)$. Field cycling experiments show\\cite{Lederman} that $D$ is ``respected,'' that is, the population of the states within $0\\leq D<{D_{E_z}}$ in the $M$ manifold rapidly transitions to the states $0\\leq D<{D_{E_z}}$ in the $M=0$ manifold. At the measurement time $t$, where the ``clock'' starts when the magnetic field is cut to zero, {\\it i.e.} at $t_w$, the population of the states remaining behind for $D>{D_{E_z}}$ in the $M$ manifold have a total magnetization ${M_{TRM}}(t,{t_w},T)$. They decay to the $M=0$ manifold by diffusing from $D>{D_{E_z}}$ to the ``sink'' created by the magnetic field change $D<{D_{E_z}}$. The characteristic response time is set by the time it takes for the population in the states which have surmounted the highest barrier, $D_{\\Delta({t_w},T)}$, to diffuse to the Hamming distance at the edge of the sink, $D_{E_z}$. By supposition, this is equivalent to surmounting a barrier of characteristic height $\\Delta({t_w},T) - {E_z}$.\n\nFor very small $E_z$, this diffusion process yields\\cite{Nordblad} a peak in the spin glass relaxation rate\n$S(t)=d[-{M_{TRM}}(t,{t_w})/H]/d\\ell n~t$ at $t\\approx{t_w}$. For finite $E_z$, the peak in $S(t)$ is shifted to shorter times ${t_w^{e\\!f\\!f}}$, and was first noted for experiments upon the insulating thiospinel $\\rm {Cd{Cr_{1.7}}{In_{0.3}}{S_4}}$ by Vincent {\\it et al.}\\cite{Bouchaud} and the amorphous system $\\rm {{({Fe_x}{Ni_{1-x}})_{75}}{P_{16}}{B_6}{Al_3}}$ by Djurberg {\\it et al.}.\\cite{Djurberg} The effective characteristic time immediately follows from activated dynamics,\n$$\\Delta({t_w},T)-{E_z}={k_B}T(\\ell n~{t_w^{e\\!f\\!f}}-\\ell n~{\\tau_0})~~.\\eqno(1)$$\n\nThe final piece of the puzzle is the magnitude of $E_z$. The overall magnetization before the magnetic field is cut to zero, $M_{fc}$, is essentially constant at the measurement temperature during the waiting time $t_w$. One can think of $M_{fc}$ as arising from the population of the states in the $M$ manifold, each of which has the same magnetization. If we define the susceptibility {\\it per spin} in the field cooled state as $\\chi_{fc}$, then the magnetization per occupant of each state is just ${\\chi_{fc}}H$. The Zeeman energy, $E_z$, is the amount by which the barriers in the $M$ manifold are reduced. Bouchaud\\cite{Bouchaud} ascribes this ``reduction'' to diffusion {\\it out of the constant $M$ plane}, caused by a ``tilting'' of the overall energy surface.\n\nFor the barriers $\\Delta(D)$ to be uniformly reduced, there must be a coherence associated with the ``hopping'' process. That is, there must be a certain number of spins, $N_s$, which are rigidly locked together, and which participate in the hopping process as a coherent whole. But these spins possess a net magnetization ${N_s}{\\chi_{fc}}H$, or a Zeeman energy ${E_z}={N_s}{\\chi_{fc}}{H^2}$.\n\nHence, use of Eq. (1) in experiments as a function of $H$ generates {\\it absolute values} for ${E_z}$, and thence for $N_s$, because $\\chi_{fc}$ is known. It was this analysis which enabled Joh {\\it et al.},\\cite{Wood2} using $N_s\\propto{\\xi(t,T)^3}$, to find $\\xi({t_w},T)$, and to compare with Monte Carlo calculations\\cite{Parisi2} by extrapolating to time scales ten orders of magnitude shorter than laboratory times.\n\nMeasurements of ${log_{10}}{t_w^{e\\!f\\!f}}~vs~{H^2}$, from Eq. (1), should yield a straight line, the slope of which can be used to obtain an absolute value for ${N_s}(t,T)$, and thence $\\xi(t,T)$. As will be shown in the next Section, the $H^2$ dependence of ${log_{10}}{t_w^{e\\!f\\!f}}$ is found only at the small end of the magnetic field range. At larger magnetic fields, the dependence on $H$ veers away from quadratic to more like linear. Further, dynamics derived from this model should scale as $\\displaystyle{t\\over {t_w}}$, whereas the data quoted by Joh {\\it et al.},\\cite{Wood2} Bouchaud {\\it et al.},\\cite{Hammann} and Vincent {\\it et al.},\\cite{Bouchaud} do not. Sec. III (following) displays the results of experiments which exhibit these deviations, forming the basis for our subsequent analysis in Sec. IV where we account for these deviations on the basis of finite size effects, along the same lines previously proposed by Bouchaud {\\it et al}.\\cite{Hammann}\n\n\\section{Experimental Evidence for Lack of Scaling}\n\nThe previous Section showed that Eq. (1) could be used to obtain a quantitative value for the Zeeman energy ${E_z}={N_s}{\\chi_{fc}}{H^2}$. This requires ${E_z}$ to scale as $H^2$. Fig. 1 reproduces Fig. 2 of Ref. 12.\n\n\n\\begin{figure}\n\\epsfysize=2.3in \\epsfbox{fig1.eps}\n\\caption{A plot of\n${log_{10}}{t_w^{e\\!f\\!f}}$ [equivalently ${E_z}$ from Eq. (1)]\n{\\it vs} $H^2$ for $Cu:Mn~6~at.\\%$ ($T/{T_g}=0.83,~{t_w}=480~{\\rm\nsec})$ and the thiospinel, $Cd{Cr_{1.7}}{In_{0.3}}{S_4}$,\n($T/{T_g}=0.72,~{t_w}=3410~{\\rm sec})$ at fixed $t_w$ and $T$ from\nRef. 12. The dependence is linear in $H^2$ for magnetic fields\nless than 250 and 45 $G$, respectively, then ``breaks away'' to a\nslower dependence. The scale for $H^2$ in the two plots has been\nadjusted to the distance from the de Almeida-Thouless line at the\nrespective temperatures.} \\label{fig1}\n\\end{figure}\n\nThis figure exhibits a quadrative dependence for ${log_{10}}{t_w^{e\\!f\\!f}}$ on $H$ in the very small magnetic field change limit, ``breaking away'' to a slower dependence at a slightly larger field, $H_{break}$. We shall associate $H_{break}$ with the smallest crystallite size. Alternatively, a linear dependence on $H$ can be fitted to the data on the thiospinel\\cite{{Bouchaud},{Vincent2}} at over a range from small to moderate $H$, with a deviation at very small $H$.\n\nTaking the small $H$ region slope from Fig. 1, together with additional data recently obtained by the authors (thiospinel, measured at UC Riverside; and $Ag:Mn 2.6~at.\\%$, measured at SACLAY), allows one to plot $N_s~vs~(T/{T_g})\\ell n({t_w}/{\\tau_0})$ for three different physical systems, over a wide range of reduced temperatures, $T_r$, and waiting times $t_w$. The results are exhibited in Fig. 2. The solid line drawn through the points, setting $\\xi({t_w},T)={{N_s}^{1\\over 3}}$, is the relationship quoted in the Introduction,\n\n$$\\xi({t_w},T)=({t_w}/{\\tau_0})^{0.153T/{T_g}}~~,\\eqno(2)$$\nwhere the unit of length is the typical spin-spin spatial separation.\n\\begin{figure}\n\\vspace{1.0cm} \\epsfysize=2.6in \\epsfbox{fig2.eps} \\caption{A plot\nof $N_s$ on a log scale {\\it vs} $(T/{T_g})ln({t_w}/{\\tau_0})$ for\n$Cu:Mn~6~at.\\%$ (solid circles); thiospinel measured at SACLAY\n(open triangles); thiospinel measured at UC Riverside (solid\ninverted triangles); and $Ag:Mn 2.6~at.\\%$ (open circles).}\n\\label{fig2}\n\\end{figure}\n\n\nIn a similar fashion, the decay of the thermoremanent magnetization ${M_{TRM}}(t,{t_w},T)$ does not obey simple $\\displaystyle{t\\over {t_w}}$ scaling. Fig. 3 reproduces Fig. 1b of Ref. 23, a plot of ${\\displaystyle{{M_{TRM}}(t,{t_w},T)\\over {M_{fc}}}}~vs~{\\displaystyle{t\\over {t_w}}}$, but with an expanded scale. The failure to scale with $t/{t_w}$ is seen clearly.\n\nThe lack of scaling arises from two sources. The first is emptying of the delta function at $D=0$ with increasing $t_w$, appropriate to the barrier model; or, equivalently, from the presence of stationary dynamics in the model of Vincent {\\it et al.}\\cite{Vincent3}. The second contribution arises from transitions between barriers, or, equivalently from the non-stationary dynamics.\n\nIn order to display the part of the decay of $\\displaystyle{M_{TRM}(t,{t_w},T)\\over {M_{fc}}}$ associated with the dynamics of barrier hopping, or, concomitantly, the non-stationary part of the magnetization decay, the data of Fig. 3 are replotted in Fig. 4 with the estimated stationary contribution of Vincent {\\it et al.}\\cite{Vincent3} subtracted from the full measured value.\n\nThe behavior exhibited in Figs. 3 and 4 was ascribed to an ``ergodic'' time, $t_{erg}$ by Bouchaud {\\it et al.}\\cite{{Bouchaud},{Hammann}}, associated with the occupation of the deepest trap in a ``grain'' with finite numbers of states. We shall argue that a finite $t_{erg}$ is also responsible for the behavior exhibited in Fig. 1.\n\n\n\\begin{figure}\n\\epsfysize=2.5in \\epsfbox{fig3.eps}\n\\caption{The decay of\n$\\displaystyle{{M_{TRM}}(t,{t_w},T)\\over {M_{fc}}}$ as a function\nof ${log_{10}}(t/{t_w})$ for $Ag:Mn~2.6~at.\\%$ for ${t_w}=$ 300,\n1000, 3000, 10000, and 30000 seconds from Refs. 14 and 23. The\nlack of scaling with $t/{t_w}$ is evident at large $t/{t_w}$.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n\\epsfysize=2.3in \\epsfbox{fig4.eps} \\caption{Same as Fig. 3, but\nwith the estimated stationary contribution subtracted from the\nfull measured value. The lack of scaling with $t/{t_w}$ continues\nto be evident at large $t/{t_w}$, and it is clearer that\n$\\displaystyle{{M_{TRM}}(t,{t_w},T)\\over {M_{fc}}}$ for longer\nwaiting times decays faster than for shorter waiting times.}\n\\label{fig4}\n\\end{figure}\nIn addition to the lack of $t/{t_w}$ scaling for the ${M_{TRM}}(t,{t_w},T)$, there is also a departure from scaling for the time dependent magnetic susceptibility.\\cite{Bouchaud}\n\nReproducing Fig. 2 of Ref. 11 in Fig. 5 for the thiospinel, it is seen that there is an $\\omega$ dependence for the magnitude of the jump in ${\\chi^{\\prime\\prime}}(\\omega,t)$ when plotted as a function of $\\omega t$, for $\\omega t$ large (of the order of 1,000 or larger), in violation of scaling. However, for small $\\omega t$, scaling is obeyed. These features will be shown to be consistent with a distribution of crystallites of finite size in the next Section.\n\n\\begin{figure}\n\\vspace{1.5cm} \\epsfysize=2.4in \\epsfbox{fig5.eps}\n \\caption{Comparison of the\neffect on $\\chi^{\\prime\\prime}$ of a dc field variation in two\nexperiments at frequencies $\\omega=0.1~{\\rm and}~1~Hz$ from\nVincent {\\it et al.} in Ref. 11. The field variation is applied at\na time $t_1$ (= 350 and 35 $min$, respectively) such that the\nproduct $\\omega \\cdot {t_1}$ is kept constant. The curves are\nplotted as a function of $\\omega\\cdot t$, and have been vertically\nshifted in order to superpose both relaxations before the field\nvariation. At constant $\\omega \\cdot t$, the effect of the\nperturbation is seen to be stronger for the lowest frequency\n(longest $t_1$).} \\label{fig5}\n\\end{figure}\n\nIn summary, three phenomena show departures from the predictions of the model outlined in Sec. II: (a) the ``break away'' from the $H^2$ dependence of ${log_{10}}{t_w^{e\\!f\\!f}}$; (b) the lack of scaling with $t/{t_w}$ of the time dependence of the thermoremanent magnetization ${M_{TRM}}(t,{t_w},T)$; and (c) the lack of scaling with $\\omega t$ of ${\\chi^{\\prime\\prime}}(\\omega,t)$ at large $\\omega t$. These separate, but related, observations are consistent with a particle size distribution $P(r)$ in the sample, such that the spin glass correlation length $\\xi({t_w},T)$ becomes comparable with the size of a component particle $r$.\n\n\n\\section{Finite size effects and the lack of scaling}\n\nWe examine in this Section the dynamics of a small spin glass particle of radius $r$, at a waiting time and temperature where $\\xi({t_w},T)$ is comparable to $r$. Aging ceases in the particle when this occurs, first noted by Bouchaud et al.\\cite{{Bouchaud},{Hammann}}\n\nThe barrier model of Sec. II has the following consequences. For increasing $t_w$, occupied states are separated by barriers which increase in height according to,\\cite{Wood2}\n$$\\Delta({t_w},T)=6.04{k_B}{T_g}\\ell n~\\xi({t_w},T)~~.\\eqno(3)$$\nImmediately after the time $t_w$ when the magnetic field is cut to zero, Eq. (3) specifies the {\\it maximum} barrier height surmounted by the system. Should $\\xi({t_w},T)\\approx r$, there would be no more barriers to surmount! The occupation of all states would be at equilibrium, and aging in the sense of the barrier model ceases. Of course, any physical system will have a distribution of particle sizes, $P(r)$, so that this cessation will be ``smeared'' in time. The purpose of this Section is to explore the impact of $\\xi({t_w},T)\\approx r$ for each of the three experimental departures from scaling presented in Sec. III.\\hfill\\break\n\\par\n\\noindent\n{\\bf (a) ``Break away'' from $H^2$ dependence of ${log_{10}}{t_w^{e\\!f\\!f}}$}\\hfill\\break\n\\par\nAs introduced in Sec. II, a susceptibility, $\\chi_{fc}$, can be associated with each spin in the field cooled state, resulting in ${E_z}={N_s}{\\chi_{fc}}H^2$. In this way, the slope of the plot of $E_z$ vs $H^2$ generates $N_s$, the number of spins locked together in a coherent state. As shown in Sec. III, the actual data do appear to scale as $H^2$ for very small magnetic field changes, but this scaling breaks down at slightly larger fields, ${H_{break}}\\simeq 170~G$ for $Cu:Mn~6at.\\%$ and ${H_{break}}\\simeq 45~G$ for the thiospinel, $Cd{Cr_{1.7}}{In_{0.3}}{S_4}$, with each $H_{break}$ equivalent to about 15\\% of the respective de Almeida - Thouless critical field.\\cite{Almeida} An alternative linear dependence of $E_z$ on $H$ does describe the data over the field range beginning with $H_{break}$ and extending to the largest magnetic field change,\\cite{Wood2,Vincent2} but fails at very small field change. At the time of the publication of Ref. 12, we wrote that ``We do not have a satisfactory explanation for this change in slope.''\n\nWe believe that the departure of the plot of $E_z$ from proportionality to $H^2$ can be understood within the barrier model of Sec. II, modified to include finite size effects.\n\nConsider a spin glass particle of radius $r$. The number of correlated spins would be proportional to $r^3$ should $r<\\xi({t_w},T)$, but proportional to ${\\xi^3}({t_w},T)$ should $r>\\xi({t_w},T)$. Thus, $E_z$ would be less for the smaller particles ($r<\\xi({t_w},T)$) than for the larger particles ($r>\\xi({t_w},T)$) at the same value of $H^2$. Further, the largest barrier overcome on a time scale $t_w$ would be $\\Delta(r)=6.04{k_B}{T_g}\\ell n\\,r$ for $r<\\xi({t_w},T)$ and $\\Delta({t_w},T)=6.04{k_B}{T_g}\\ell n\\,\\xi({t_w},T)$ for $r>\\xi({t_w},T)$. The effective waiting time, $t_w^{e\\!f\\!f}$, depends upon the difference $\\Delta - {E_z}$ from Eq. (1). For small magnetic field changes, this means that $t_w^{e\\!f\\!f}$ is larger for the larger particles and smaller for the smaller particles [$\\ell n\\,\\xi({t_w},T)>\\ell n\\,r$].\n\nAs the magnetic field change increases, $E_z$ increases more rapidly for the infinite [meaning $\\xi({t_w},T)<r$] size particles than for the smaller [meaning $\\xi({t_w},T)>r$] size particles. This means that the peak in $S(t)=d\\Bigl[\\displaystyle{{-M_{TRM}}(t,{t_w},T)\\over H}\\Bigr]\\Bigl/d\\ell n\\,t$ shifts to shorter times {\\it more rapidly} with $H$ for the infinite particles than for the smaller particles. At some value of magnetic field change $H$, $t_w^{e\\!f\\!f}$ for the infinite and smaller particles will become equal. This yields an increase in apparent width for $S(t)$, as observed in many experiments.\\cite{Wood2} For yet larger $H$, the weight of all the smaller particles dominates, slowing the shift of the peak of $S(t)$ with increasing $H$, leading to a less rapid decrease of ${Log_{10}}{t_w^{eff}}$ with increasing $H$.\n\nWe believe this to be the origin of the ``break'' in the slope of ${Log_{10}}{t_w^{eff}}$ versus $H^2$, exhibited in Fig. 1 of Sec. III, and therefore an effect of finite size. A quantitative fit will require knowledge of the particle size distribution $P(r)$. For now, this finite size effect can explain the behavior of the magnetic field dependence of the characteristic response time\\cite{Bouchaud,Wood2,Vincent2}\nwithin the barrier model of Sec. II.\\hfill\\break\n\\par\n\\noindent\n{\\bf (b) Lack of $\\displaystyle {t\\over {t_w}}$ scaling for ${M_{TRM}}(t,{t_w},T)$}\\hfill\\break\n\\par\nThe barrier model of Sec. II to describe spin glass dynamics\\cite{Lederman} predicts scaling as a function of $\\displaystyle {t\\over {t_w}}$. Measurements of Ocio {\\it et al.}\\cite{Ocio,Vincent3}, exhibited in Figs. 3 and 4 of Sec. III show that this is not the case in the long time domain $t\\geq {t_w}$. That is, Fig. 4 corrects Fig. 3 for the stationary contribution.\\cite{Bouchaud} Departure from scaling as $\\displaystyle {t\\over {t_w}}$ is only truly present for $t\\geq {t_w}$, as can be seen in Fig. 4. As is clearly seen in Fig. 4, the barrier hopping or the non-stationary dynamics results in the relaxation of older systems being ``faster'' when plotted versus $\\displaystyle{t\\over {t_w}}$ (although, of course, when plotted versus $t$, the older the system, the slower the relaxation). Bouchaud {\\it et al.}\\cite{Hammann} recalled that all of the data of Alba {\\it et al.}\\cite{Alba,Ocio} could be rescaled with the response times in the spin glass scaling as $({t_w}+t)^\\mu$, with $\\mu<1$. They noted that, from a more fundamental point of view, the scaling variable should be written as $\\displaystyle{t\\over {{{\\tau^*}^{(1-\\mu)}}{t_w^\\mu}}}$, with $\\tau^*$ a characteristic time scale. Their manuscript ascribed a physical meaning to $\\tau^*$, relating it to the {\\it finite number of available metastable states} in real samples made of `grains' of finite size. In their language, ``a finite size system will eventually find the `deepest trap' in its phase space, which corresponds to the equilibrium state.\\cite{Bouchaud} This will take a long, but finite time $t_{erg}$; when $t_w$ exceeds this `ergodic' time, $t_{erg}$, aging is `interrupted' because the phase space has been faithfully probed. Beyond this time scale, conventional stationary dynamics resume.''\n\nThe precise origin of the ergodic time scale was ``...not easy to discuss since we do not know precisely what these `subsystems' are.'' Bouchaud {\\it et al.}\\cite{Hammann} suggested it could be magnetically disconnected regions (such as grains), the size of which was determined by sample preparation, and thus temperature independent; or it could be that the phase space of $3d$ spin-glasses is broken into mutually inaccessible regions (``true'' ergodicity breaking). We argue below that finite size effects can indeed account for this behavior. Our approach is the same as that of Bouchaud {\\it et al.}\\cite{{Bouchaud},{Hammann}}, but our arguments will be based on the barrier model of Sec. II.\n\nThe reasoning follows from the time dependence of the spin glass correlation length. From the data in Fig. 2, and Eq. (2), $\\xi({t_w},T)={(\\displaystyle{{t_w}\\over {\\tau_0}})}^{0.153T/{T_g}}$, where $\\tau_0$ is of the order of an exchange time ($\\approx 10^{-12}~sec$). This means that, for a given waiting time $t_w$, the correlation length $\\xi({t_w},T)$ can be larger than a particle with size $r$. All the available metastable states in that particle would be occupied in thermal equilibrium, and no further aging would take place. The largest barrier in that particle has magnitude\\cite{Wood2} $\\Delta(r)=6.04{k_B}{T_g}\\ell n\\,r$, less than the largest barrier in the particles for which $\\xi({t_w},T)<r$. For these particles [large on the length scale of $\\xi({t_w},T)$],\n$\\Delta({t_w},T)={k_B}T\\ell\\,n\\Bigl({\\displaystyle{{t_w}\\over {\\tau_0}}}\\Bigr)$.\nThe characteristic time for decay of ${M_{TRM}}(t,{t_w},T)$ is proportional to $\\Delta-{E_z}$.\\cite{Kenning,Bouchaud}\nTherefore, for small magnetic field changes, the characteristic decay time of the small particles, proportional to $\\Delta(r)-{r^3}{\\chi_{fc}}{H^2}$, is less than the decay time of the larger particles, proportional to $\\Delta({t_w},T)-{\\xi^3}({t_w},T){\\chi_{fc}}{H^2}$. Averaged over all particles, small and large, the characteristic decay time for ${M_{TRM}}(t,{t_w},T)$ will be less than $t_w$, the magnitude of the difference depending upon what fraction of the sample contains particles of size $r<\\xi({t_w},T)$, {\\it i.e.} the particle size distribution. As $t_w$ increases, more of the particle sizes $r$ will be less than $\\xi({t_w},T)$, thereby {\\it shortening} the characteristic time for ${M_{TRM}}(t,{t_w},T)$ decay. The characteristic time will shorten as $t_w$ increases, leading to a faster decay of $M(t,{t_w},T)$ with increasing $t_w$ when plotted as a function of $\\displaystyle {t\\over {t_w}}$. This is exactly the effect posited by Bouchaud {\\it et al.}\\cite{Hammann}. In addition, the Zeeman energy $E_z$ for the larger particles is proportional to ${\\xi^3}({t_w},T)={N_s}$. Increasing $t_w$ will increase $\\xi({t_w},T)$, causing $E_z$ to increase with increasing $t_w$. This further diminishes $\\Delta({t_w},T)-{\\xi^3}({t_w},T){\\chi_{fc}}{H^2}$ for the larger particles, adding to the reduction of the characteristic time for decay of ${M_{TRM}}(t,{t_w},T)$ with increasing $t_w$. This additional contribution has also been noted by Bouchaud {\\it et al}.\\cite{Hammann}\\hfill\\break\n\\par\n\\noindent\n{\\bf (c) Lack of scaling for ${\\chi^{\\prime\\prime}}(\\omega,t)$ at large $\\omega t$}\\hfill\\break\n\\par\nThe jump in ${\\chi^{\\prime\\prime}}(\\omega,t)$ with the application of a magnetic field is an important test for any dynamical model. Fig. 5 displays the {\\it frequency} dependence of the change in ${\\chi^{\\prime\\prime}}(\\omega,t)$ when plotted against $\\omega \\cdot t$. It is seen that the jump is larger, the smaller $\\omega$. The origin of this effect within the trap model\\cite{Bouchaud} was associated with an increase of coupling to the magnetic field, the deeper the trap.\nWithin the barrier model,\\cite{Joh} this effect was associated with an increase of coupling to the magnetic field, the higher the barrier. The relationship of the jump in ${\\chi^{\\prime\\prime}}(\\omega,t)$ to these non-uniform magnetic field couplings arises naturally from the time dependence of the spin-glass correlation length [Eq. (2)], $\\xi(t,T)$. The smaller $\\omega$, the larger $t$, when ${\\chi^{\\prime\\prime}}(\\omega,t)$ is plotted as a function of $\\omega \\cdot t$. But larger $t$ means larger $\\xi(t,T)$, thence larger ${N_s}~[~\\propto {\\xi^3}(t,T)]$, thence larger ${E_z}~(~\\propto {N_s}{\\chi_{fc}}{H^2})$. This increase in $E_z$ with increasing $t$ maps directly onto the non-uniform magnetic field coupling of the trap model\\cite{Bouchaud} and the barrier model.\\cite{Joh}\n\nThere is, in addition to the non-uniform coupling to the magnetic field, an additional effect arising from the presence of crystallites with radius $r<\\xi(t,T)$. We shall show below that the magnitude of the jump in ${\\chi^{\\prime\\prime}}(\\omega,t)$ will also depend upon $\\omega$ ({\\it i.e.} violate scaling) when the spin glass correlation length becomes of the order of, or larger than, the size of a spin glass particle. Conversely, the dependence of the change in ${\\chi^{\\prime\\prime}}(\\omega,t)$ upon $\\omega$ over the full frequency regime could be used to generate the particle size distribution.\n\nThe effective waiting time, $t_w^{e\\!f\\!f}$, after a magnetic field change, is given by Eq. (1), allowing one to write,\n$${t_w^{e\\!f\\!f}}={t_w}exp\\Bigl(-{{E_z}\\over {k_B}T}\\Bigr)~~.\\eqno(4)$$\nThis scaling was first established by Vincent {\\it et al.}\\cite{Bouchaud} and by Chu {\\it et al.}\\cite{Kenning}\n\nThis scaling can be incorporated into the {\\it ac} susceptibility. Before a {\\it dc} magnetic field change, the relaxation of ${\\chi^{\\prime\\prime}}(\\omega,t)$ is well accounted for by a power law,\\cite{Bouchaud}\n$${\\chi^{\\prime\\prime}}(\\omega,t)-{\\chi_{eq}^{\\prime\\prime}}\\propto {(\\omega t)^{-b}}~~,\\eqno(5)$$\nwith $b>0$, and $\\chi_{eq}^{\\prime\\prime}$ the equilbrium susceptibility ${\\chi^{\\prime\\prime}}(\\omega,t\\rightarrow\\infty)$.\n\nUsing the relationship Eq. (4), the change in ${\\chi^{\\prime\\prime}}(\\omega,t)$ upon a change in magnetic field at time $t_1$ can be written as,\n$$\\Delta {\\chi^{\\prime\\prime}}(\\omega,{t_1})={\\chi^{\\prime\\prime}}(\\omega,{t_1^{e\\!f\\!f}})-{\\chi^{\\prime\\prime}}(\\omega,{t_1})~~.\\eqno(6)$$\nHere, $t_1^{e\\!f\\!f}$ is the effective waiting time upon a magnetic field change, defined through Eq. (4) with ${t_w}\\rightarrow{t_1}$.\n\nWith reference to Fig. 5, the lower the frequency $\\omega$, the greater the time $t_1$ (because the abcissa is the scaling variable $\\omega t$). But the greater the time $t_1$ before the magnetic field is changed, the larger the spin glass correlation length $\\xi({t_1},T)$, and therefore the more the likelihood that $\\xi({t_1},T)>r$, the size of the spin glass particle. But if $\\xi({t_1},T)>r$, the effective response time, $t_1^{e\\!f\\!f}$, will be less than that for an ``infinite'' sample [$\\xi({t_1},T)<r$], and ${\\chi^{\\prime\\prime}}(\\omega,{t_1^{e\\!f\\!f}})$ will be larger, increasing the size of the jump from Eq. (6). However, ${\\chi^{\\prime\\prime}}(\\omega,{t_1})$ will also be slightly larger because it does not decay beyond ${\\chi^{\\prime\\prime}}(\\omega,{t_{erg}})$, where $t_{erg}$ is the time at which $\\xi({t_1},T)=r$, decreasing the size of the jump from Eq. (6).\n\nThe increase in the first term in Eq. (6) turns out to be larger than the increase in the\nsecond because finite size is involved in the argument of an exponential (through the Zeeman energy) in the first, while finite size enters only as an argument of a weak power law in the second (see the Appendix for details). This results in a larger magnitude of the jump in ${\\chi^{\\prime\\prime}}(\\omega,{t_1})$, the larger $t_1$, or, equivalently, the smaller $\\omega$, precisely what is seen experimentally in Fig. 5.\n\nThese arguments are qualitative, based upon Eqs. (4) and (5). A quantitative evaluation of Eq. (6) is made in the Appendix, fully supporting the conclusions of this subsection.\n\nFinally, for very short times, ${\\chi^{\\prime\\prime}}(\\omega,t)$ is seen to scale with $\\omega\\cdot t$ in Fig. 5. Very short times means that $\\xi(t,T)<r$ for all particles. As we have already noted, under these conditions the barrier model calls for scaling with $\\omega\\cdot t$, again in agreement with experiment.\n\n\\section{Key Experiments and Numerical Simulations}\n\nThe arguments given above, especially in Sec. IV, are qualitative in nature, but fully capable of quantitative application. There are two approaches which we feel would be most relevant. The first, experimental, is one of careful magnetic field and waiting time variations upon a variety of samples. The second would be to make use of the model of Sec. II to numerically simulate particular realizations of spin glass materials.\\hfill\\break\n\\par\n\\noindent\n{\\bf (a) Experimental determination of P(r)}\\hfill\\break\n\\par\nPreliminary examination of a number of spin glass samples, using SEM techniques,\\cite{Krassimir} suggest that many are made up of a powder-like array of small crystallites, embedding much larger apparently single crystal pieces. Of course, this division may not be general, and may only apply to those materials with which we have been working. Nevertheless, the analysis of Sec. IV, parts (a) and (b), can in this instance generate a measure of the bounds on the size distribution of the powder-like array component.\n\nThe analysis of Sec. IV, part (a) generates the upper end, $r_{max}$ of the small crystallite length scale distribution $P(r)$. The experiments are at fixed waiting time, with measurements as a function of the change in magnetic field, $H$.\nAt very small magnetic field changes, only the volume occupied by ${\\xi^3}({t_w},T)$ contributes to $E_z$, shifting the peak of $S(t)$ by reducing the energy difference $\\Delta-{E_z}$. As the magnetic field change increases, there will come a point when the energy difference $\\Delta-{E_z}$ becomes comparable to $\\Delta (r)-{{r_{max}}^3}{\\chi_{fc}}{H^2}$ for the {\\it largest} of the small crystallites. We have argued in Sec. IV, part (a), that this causes the ``break'' in the plot of ${log_{10}}{t_w^{e\\!f\\!f}}~vs~H^2$. Thus, the ``break'' field, $H_{break}$, generates a determination of $r_{max}$.\n\nThe other extreme of $P(r)$, $r_{min}$, can be extracted from the departure from scaling, exhibited in Figs. 3 and 4. Here, the magnetic field change is fixed, and the experiments are a function of increasing waiting time, $t_w$. From these two figures, the longer $t_w$ the more rapid the relaxation of ${M_{TRM}}(t,{t_w},T)$ as a function of the reduced time variable, ${\\displaystyle {t\\over {t_w}}}$. For very small $t_w$, $\\xi({t_w},T)$ is less than the ``minimum'' size of the powder-like array of small crystallites. The departure from scaling occurs first when $t_w$ increases to a point where $\\xi({t_w},T)={r_{min}}$, or equivalently $\\Delta({t_w},T)=\\Delta({r_{min}})$. For longer $t_w$, the crystallite with dimension $r_{min}$ is at equilibrium, and for that part of the sample, aging is over. Thus, the waiting time at which departure from scaling is first seen is a direct measure of $r_{min}$, the lower extreme of $P(r)$.\n\nCareful (tedious!) measurements beginning from either domain, $r_{max}$ or $r_{min}$, can of course be used to generate all of $P(r)$. At the very least, these two approaches will given a measure of the width of the $P(r)$ distribution.\\hfill\\break\n\\par\n\\noindent\n{\\bf (b) Numerical Simulations}\\hfill\\break\n\\par\nAn alternate, and certainly complementary approach, is to simulate the spin glass sample by selected choices for $P(r)$. Previous simulations,\\cite{Wood} using the barrier model of Sec. II, were able to duplicate the waiting time dependence of the response function $S(t)$. Any attempt to fit to a particle size distribution $P(r)$ will require the ability to simulate ${log_{10}}{t_w^{e\\!f\\!f}}~vs~{H^2}$ and ${M_{TRM}}(t,{t_w},T)~vs~t$. This procedure will not be unlike that of neutron or X-ray diffraction, where scattering from a specific model is measured against experiment. It is the usual ``inverse'' problem where small iterations from a hypothesized model are used to fit experiment. Previous success at fitting $S(t)$ suggests that a similar procedure, using the data of Sec. III and the analysis of Sec. IV, will be successful. Having the limits $r_{max}$ and $r_{min}$ on $P(r)$ in hand, as discussed in part (a) of this Section, will greatly aid such an analysis.\n\n\\section{Conclusion}\n\nThis paper discusses spin glass dynamics for crystallites or amorphous particles of finite size. Departures from scaling arise when the spin glass correlation length becomes of the order of or larger than particle sizes. Qualitative arguments are given for the associated existence of a ``break field,'' $H_{break}$ away from a linear plot of ${log_{10}}{t_w^{e\\!f\\!f}}~vs~{H^2}$; the departure from $\\displaystyle{t\\over {t_w}}$ scaling of ${M_{TRM}}(t,{t_w},T)$; and the frequency dependence of the magnitude of the jump in ${\\chi^{\\prime\\prime}}(\\omega,t)~vs~\\omega\\cdot t$. A guide to future experiments and numerical simulations, leading to the extraction of the particle size distribution, $P(r)$, are given with specific attention to what can be learned from experimental protocols. SEM measurements\\cite{Krassimir} of the particle size distribution will lead to explicit experimental consequences, setting the stage for a consistency check on the entire model. The authors find it remarkable that the behavior of magnetization measurements in the time domain could so directly depend upon the physical size parameters of the sample particulates.\n\nThe authors have benefited from extensive discussions with Dr. J.-P. Bouchaud, experimental data supplied by Dr. M. Ocio, and from the financial support of the Japan Ministry of Education (Monbusho) and the U.S. National Science Foundation, Grant DMR 96 23195.\n\n\n\\begin{references}\n\\bibitem{Lundgren}\nZero-field cooled magnetization: L. Lundgren, P. Svedlindh, P. Nordblad, and O. Beckman, Phys. Rev. Lett. {\\bf 51}, 911 (1983); field-cooled magnetization: R.V. Chamberlin, Phys. Rev. B{\\bf 30}, 867 (1984).\n\\bibitem{Refregier}\nPh. R\\'efr\\'egier, M. Ocio, J. Hammann, and E. Vincent, J. Appl. Phys. {\\bf 63}, 8 (1988).\n\\bibitem{Wood}\nY.G. Joh, R. Orbach, and J. Hammann, Phys. Rev. Lett. {\\bf 77}, 4648 (1996); Y.G. Joh, R. Orbach, G.G. Wood, J. Hammann, and E. Vincent, Phys. Rev., submitted for publication (1999).\n\\bibitem{Parisi}\nG. Parisi, Phys. Lett. {\\bf 73A}, 203 (1979); Phys. Rev. Lett. {\\bf 43}, 1574 (1979); J. Phys. A {\\bf 13}, L115 (1980).\n\\bibitem{Mezard}\nM. M\\'ezard, G. Parisi, and M.A. Virasoro, {\\it Spin Glass Theory and Beyond} (World Scientific, Singapore, 1987).\n\\bibitem{Nemoto}\nN. Nemoto, J. Phys. A {\\bf 21}, L287 (1988).\n\\bibitem{Vertechi}\nD. Vertechi and M.A. Virasoro, J. Phys. (Paris) {\\bf 50}, 2325 (1989).\n\\bibitem{Lederman}\nM. Lederman, R. Orbach, J. Hammann, M. Ocio, and E. Vincent, Phys. Rev. B {\\bf 44}, 7403 (1991); J. Hammann, M. Lederman, M. Ocio, R. Orbach, and E. Vincent, Physica (Amsterdam) {\\bf 185A}, 278 (1992); G.G. Kenning, Y.G. Joh, D. Chu, and R. Orbach, Phys. Rev. B {\\bf 52}, 3479 (1995).\n\\bibitem{Mezard2}\nM. M\\'ezard and M.A. Virasoro, J. Phys. (Paris) {\\bf 46}, 1293 (1985).\n\\bibitem{Kenning}\nD. Chu, G.G. Kenning, and R. Orbach, Phil. Mag. B{\\bf 71}, 479 (1995); D. Chu, thesis, University of California, Los Angeles, 1994, unpublished.\n\\bibitem{Bouchaud}\nJ.-P. Bouchaud, J. Phys. (France) I {\\bf 2}, 1705 (1992); J.-P. Bouchaud and D.S. Dean, J. Phys. I (France) {\\bf 5}, 265 (1995); E. Vincent, J.-P. Bouchaud, D.S. Dean, and J. Hammann, Phys. Rev. B {\\bf 52}, 1050 (1995).\n\\bibitem{Wood2}\nY.G. Joh, R. Orbach, G.G. Wood, J. Hammann, and E. Vincent, Phys. Rev. Lett. {\\bf 82}, 438 (1999).\n\\bibitem{Alba}\nM. Alba, M. Ocio, J. Hammann, Europhys. Lett. {\\bf 2}, 45 (1986); {\\it ibid} J. Phys. Lett. {\\bf 46}, L-1101 (1985); M. Alba, J. Hammann, M. Ocio, and Ph. Refregier, J. Appl. Phys. {\\bf 61}, 3683(1987); and E. Vincent, J. Hammann, and M. Ocio, {\\it Recent Progress in Random Magnets}, D.H. Ryan Ed. (World Scientific Pub. Co. Pte. Ltd, Singapore, 1992).\n\\bibitem{Ocio}\nM. Ocio, private communication.\n\\bibitem{Vincent2}\nE. Vincent, in conjunction with V. Villar and V. Dupuis (unpublished).\n\\bibitem{Hammann}\nJ.P. Bouchaud, E. Vincent, and J. Hammann, J. Phys. I (France) {\\bf 4}, 139 (1994).\n\\bibitem{Mezard3}\nM. M\\'ezard, G. Parisi, N. Sourlas, G. Toulouse, and M.A. Virasoro, J. Phys. (Paris) {\\bf 45}, 843 (1984).\n\\bibitem{Sherrington}\nD. Sherrington and S. Kirkpatrick, Phys. Rev. B{\\bf 17}, 4384 (1978).\n\\bibitem{Edwards}\nS.F. Edwards and P.W. Anderson, J. Phys. F. {\\bf 5}, 965 (1975).\n\\bibitem{Nordblad}\nP. Nordblad, P. Svedlindh, J. Ferre, and M. Ayadi, J. Magn. Magn. Mater. {\\bf 59}, 250 (1986); M. Ocio, M. Alba, and J. Hammann, J. Phys. (Paris) Lett. {\\bf 46}, L1101 (1985); P. Granberg, L. Sandlund, P. Nordblad, P. Svedlindh, and L. Lundgren, Phys. Rev. B {\\bf 38}, 7097 (1988).\n\\bibitem{Djurberg}\nC. Djurberg, J. Mattsson, and P. Nordblad, Europhys. Lett. {\\bf 29}, 163 (1995).\n\\bibitem{Parisi2}\nE. Marinari, G. Parisi, J. Ruiz-Lorenzo, and F. Ritort, Phys. Rev. Lett. {\\bf 76}, 843 (1996); J. Kisker, L. Santen, M. Schreckenberg, and H. Rieger, Phys. Rev. B {\\bf 53}, 6418 (1996); P. Sibani, C. Sh\\'on, P. Salamon, and J.-O. Andersson, Europhys. Lett. {\\bf 22}, 479 (1993); P. Sibani and J.-O. Andersson, Physica (Amsterdam) {\\bf 206A}, 1 (1994).\n\\bibitem{Vincent3}\nE. Vincent, J. Hammann, M. Ocio, J.-P. Bouchaud, and L.F. Cugliandolo, Proceedings, Complex Behaviour of Glassy Systems, 14. Sitges Conference, Ed. by M. Rubi and C. Perez-Vicente (Springer, Berlin, 1997), pp. 184-219.\n\\bibitem{Almeida}\nJ.R. de Almeida and D.J. Thouless, J. Phys. A {\\bf 11}, 983 (1978).\n\\bibitem{Joh}\nY.G. Joh and R. Orbach, Phil. Mag. {\\bf B 77}, 221 (1998).\n\\bibitem{Krassimir}\nWe are grateful to Mr. Krassimir Bozhilov for performing scanning electron microscope measurements on both the Cu:Mn and thiospinel samples.\n\\end{references}\n\\par\n\\centerline{\\bf Appendix}\n\\vskip .1cm\nSec. IV, subsection (c) gives a qualitative argument for the frequency dependence of the jump ${\\chi^{\\prime\\prime}}(\\omega,t)~vs~\\omega t$ upon a change in magnetic field as a consequence of finite spin glass particle size. This Appendix develops quantitative expressions for these quantities.\n\nTo derive the dependence of the jump in ${\\chi^{\\prime\\prime}}(\\omega,t)$ upon change in magnetic field on particle size, consider two cases: I. $r<\\xi(t,T)$ and II. $r>\\xi(t,T)$, where $r$ is the radius of the spin glass particle, and $\\xi(t,T)$ the spin glass correlation length displayed in Eq. (2). We shall assume that the time dependence of ${\\chi^{\\prime\\prime}}$\nis given by\n$${\\chi^{\\prime\\prime}}={\\chi_{eq}^{\\prime\\prime}}(\\omega)+A{(\\omega t)^{-b}}~~,\\eqno(A1)$$\nwhere ${\\chi_{eq}^{\\prime\\prime}}(\\omega)$ may be different for cases I and II, but will cancel when we consider only the {\\it change} in ${\\chi^{\\prime\\prime}}(\\omega,t)$ upon a change in magnetic field. That is, we assume that the equilibrium ($t\\rightarrow\\infty$) value of ${\\chi^{\\prime\\prime}}(\\omega)$ is magnetic field independent. The exponent $b\\approx 0.20$ from experiment.\\cite{Bouchaud}\\hfill\\break\n\\par\n\\noindent\nI. ${\\rm particle~size}~r<\\xi(t,T)$\\hfill\\break\n\\par\nConsider the effect of a jump in magnetic field at a time $t_1$, and for this subsection, assume that the particle size $r<\\xi({t_1},T)$. Before the jump in magnetic field,\n$${\\chi_{I,before}^{\\prime\\prime}}(\\omega\\cdot{t_1})={\\chi_{eq_I}^{\\prime\\prime}}(\\omega)+A{(\\omega\\cdot {t_{erg}})^{-b}}~~,\\eqno(A2)$$\nwhere we have used $\\xi({t_1},T)=({t_1}/{\\tau_0})^a$, with $a=0.153T/{T_g}$ from Eq. (2), and where $t_{erg}={\\tau_0}{r^{1/a}}$ is defined in Sec. IV, subsection (b).\n\nThe effective time, $t_I^{e\\!f\\!f}$ after a jump in magnetic field, is given by\n$${t_I^{e\\!f\\!f}}={t_{erg}}exp{\\Bigl(}-{{E_z}\\over {k_B}T}\\Bigr)~~.\\eqno(A3)$$\nThis scaling was first established by Vincent {\\it et al.}\\cite{Bouchaud} and by Chu {\\it et al.}, giving\\cite{Kenning}\n$${t_I^{e\\!f\\!f}}={\\tau_0}{r^{1/a}}exp{\\Bigl(-}{{{r^3}{\\chi_{fc}}{H^2}}\\over {{k_B}T}}{\\Bigr)}~~.\\eqno(A4)$$\nThus, after the jump in magnetic field,\n$${\\chi_{I,after}^{\\prime\\prime}}(\\omega\\cdot{t_1})={\\chi_{eq_I}^{\\prime\\prime}}(\\omega)+A{(\\omega\\cdot{t_I^{e\\!f\\!f}})^{-b}}~~.\\eqno(A5)$$\nSubtracting Eq. (A2) from Eq. (A5) gives the jump in ${\\chi^{\\prime\\prime}}(\\omega,{t_1})$ upon a jump in magnetic field:\n$$\\Delta{\\chi_I^{\\prime\\prime}}(\\omega,{t_1})=A{{\\Bigl(}\\omega{\\tau_0}{r^{1/a}}{\\Bigr)}^{-b}}{\\Bigl[}exp\\Bigl({b{r^3}{\\chi_{fc}}\n{H^2}\\over {{k_B}T}}\\Bigr)-1{\\Bigr]}~~.\\eqno(A6)$$\n\\par\n\\noindent\nII. $\\xi(t,T)<{\\rm particle~size}~r$\\hfill\\break\n\\par\n\\noindent\nConsider the effect of a jump in magnetic field at a time $t_1$, and for this subsection, assume that $\\xi({t_1},T)$ is smaller than the particle size. Before the jump in magnetic field,\n$${\\chi_{II,before}^{\\prime\\prime}}(\\omega\\cdot{t_1})={\\chi_{eq_{II}}^{\\prime\\prime}}(\\omega)+A{(\\omega\\cdot {t_1})^{-b}}~~.\\eqno(A7)$$\nAfter the jump in magnetic field,\n$${\\chi_{II,after}^{\\prime\\prime}}(\\omega\\cdot{t_1})={\\chi_{eq_{II}}^{\\prime\\prime}}(\\omega)+A{(\\omega\\cdot{t_{II}^{e\\!f\\!f}})^{-b}}~~,\\eqno(A8)$$\nwhere,\n$${t_{II}^{e\\!f\\!f}}={t_1}exp\\Bigl(-{{\\xi^3}{\\chi_{fc}}{H^2}\\over {k_B}T}\\Bigr)~~.\\eqno(A9)$$\nThe jump in ${\\chi_{II}^{\\prime\\prime}}(\\omega\\cdot{t_1})$ is then given by subtracting Eq. (A7) from Eq. (A8):\\hfill\\break\n$$\\Delta{\\chi_{II}^{\\prime\\prime}}(\\omega\\cdot{t_1})=A{\\Bigl(\\omega{\\tau_0}{\\alpha^{1/a}}{r^{1/a}}\\Bigr)^{-b}}$$\n$$\\times \\Bigl[exp\\Bigl({b{\\alpha^3}{r^3}{\\chi_{fc}}{H^2}\\over {k_B}T}\\Bigr)-1\\Bigr]~~.\\eqno(A10)$$\nFor convenience, $\\xi=\\alpha r,~\\alpha>1$; $\\alpha$ is a function of ${t_1},T$; and ${t_1}={\\tau_0}(\\alpha r)^{1/a}$.\n\nExperimentally, from Fig. 5, the magnitude of the jump in ${\\chi^{\\prime\\prime}}(\\omega,{t_1})$ is larger, the smaller $\\omega$. But the lower the frequency $\\omega$, the greater the time $t_1$ (because the abscissa is the scaling variable $\\omega t$). And the greater the time $t_1$ before the magnetic field is changed, the larger the spin glass correlation length $\\xi ({t_1},T)$, and therefore the more the likelihood that $\\xi ({t_1},T)>r$, the size of the spin glass particle.\n\nThis means that the jump in ${\\chi^{\\prime\\prime}}(\\omega,{t_1})$ for case I should exceed the jump in ${\\chi^{\\prime\\prime}}(\\omega,{t_1})$ for case II, or more simply, that Eq. (A6) should exceed Eq. (A10), more for smaller $\\omega$, or equivalently, larger $t_1$.\n\nIt is a somewhat tedious algebraic exercise, but one can show that this is indeed the case. Thus, finite size effects can generate the $\\omega$ dependence of the jump in ${\\chi^{\\prime\\prime}}(\\omega,{t_1})$. This is a consequence of different $E_z$ values [through Eq. (A3)] as a consequence of differing particle sizes. This feature\\cite{Hammann} adds to the non-uniform magnetic field couplings (larger, the larger the trap depth) introduced in the trap model\\cite{Bouchaud} or (larger, the larger the barrier height) introduced in the barrier model,\\cite{Joh} independent of possible finite size effects, arising from the time dependence of $\\xi(t,T)$ and hence of $E_z$.\n\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002040.extracted_bib",
"string": "\\bibitem{Lundgren}\nZero-field cooled magnetization: L. Lundgren, P. Svedlindh, P. Nordblad, and O. Beckman, Phys. Rev. Lett. {\\bf 51}, 911 (1983); field-cooled magnetization: R.V. Chamberlin, Phys. Rev. B{\\bf 30}, 867 (1984).\n\n\\bibitem{Refregier}\nPh. R\\'efr\\'egier, M. Ocio, J. Hammann, and E. Vincent, J. Appl. Phys. {\\bf 63}, 8 (1988).\n\n\\bibitem{Wood}\nY.G. Joh, R. Orbach, and J. Hammann, Phys. Rev. Lett. {\\bf 77}, 4648 (1996); Y.G. Joh, R. Orbach, G.G. Wood, J. Hammann, and E. Vincent, Phys. Rev., submitted for publication (1999).\n\n\\bibitem{Parisi}\nG. Parisi, Phys. Lett. {\\bf 73A}, 203 (1979); Phys. Rev. Lett. {\\bf 43}, 1574 (1979); J. Phys. A {\\bf 13}, L115 (1980).\n\n\\bibitem{Mezard}\nM. M\\'ezard, G. Parisi, and M.A. Virasoro, {\\it Spin Glass Theory and Beyond} (World Scientific, Singapore, 1987).\n\n\\bibitem{Nemoto}\nN. Nemoto, J. Phys. A {\\bf 21}, L287 (1988).\n\n\\bibitem{Vertechi}\nD. Vertechi and M.A. Virasoro, J. Phys. (Paris) {\\bf 50}, 2325 (1989).\n\n\\bibitem{Lederman}\nM. Lederman, R. Orbach, J. Hammann, M. Ocio, and E. Vincent, Phys. Rev. B {\\bf 44}, 7403 (1991); J. Hammann, M. Lederman, M. Ocio, R. Orbach, and E. Vincent, Physica (Amsterdam) {\\bf 185A}, 278 (1992); G.G. Kenning, Y.G. Joh, D. Chu, and R. Orbach, Phys. Rev. B {\\bf 52}, 3479 (1995).\n\n\\bibitem{Mezard2}\nM. M\\'ezard and M.A. Virasoro, J. Phys. (Paris) {\\bf 46}, 1293 (1985).\n\n\\bibitem{Kenning}\nD. Chu, G.G. Kenning, and R. Orbach, Phil. Mag. B{\\bf 71}, 479 (1995); D. Chu, thesis, University of California, Los Angeles, 1994, unpublished.\n\n\\bibitem{Bouchaud}\nJ.-P. Bouchaud, J. Phys. (France) I {\\bf 2}, 1705 (1992); J.-P. Bouchaud and D.S. Dean, J. Phys. I (France) {\\bf 5}, 265 (1995); E. Vincent, J.-P. Bouchaud, D.S. Dean, and J. Hammann, Phys. Rev. B {\\bf 52}, 1050 (1995).\n\n\\bibitem{Wood2}\nY.G. Joh, R. Orbach, G.G. Wood, J. Hammann, and E. Vincent, Phys. Rev. Lett. {\\bf 82}, 438 (1999).\n\n\\bibitem{Alba}\nM. Alba, M. Ocio, J. Hammann, Europhys. Lett. {\\bf 2}, 45 (1986); {\\it ibid} J. Phys. Lett. {\\bf 46}, L-1101 (1985); M. Alba, J. Hammann, M. Ocio, and Ph. Refregier, J. Appl. Phys. {\\bf 61}, 3683(1987); and E. Vincent, J. Hammann, and M. Ocio, {\\it Recent Progress in Random Magnets}, D.H. Ryan Ed. (World Scientific Pub. Co. Pte. Ltd, Singapore, 1992).\n\n\\bibitem{Ocio}\nM. Ocio, private communication.\n\n\\bibitem{Vincent2}\nE. Vincent, in conjunction with V. Villar and V. Dupuis (unpublished).\n\n\\bibitem{Hammann}\nJ.P. Bouchaud, E. Vincent, and J. Hammann, J. Phys. I (France) {\\bf 4}, 139 (1994).\n\n\\bibitem{Mezard3}\nM. M\\'ezard, G. Parisi, N. Sourlas, G. Toulouse, and M.A. Virasoro, J. Phys. (Paris) {\\bf 45}, 843 (1984).\n\n\\bibitem{Sherrington}\nD. Sherrington and S. Kirkpatrick, Phys. Rev. B{\\bf 17}, 4384 (1978).\n\n\\bibitem{Edwards}\nS.F. Edwards and P.W. Anderson, J. Phys. F. {\\bf 5}, 965 (1975).\n\n\\bibitem{Nordblad}\nP. Nordblad, P. Svedlindh, J. Ferre, and M. Ayadi, J. Magn. Magn. Mater. {\\bf 59}, 250 (1986); M. Ocio, M. Alba, and J. Hammann, J. Phys. (Paris) Lett. {\\bf 46}, L1101 (1985); P. Granberg, L. Sandlund, P. Nordblad, P. Svedlindh, and L. Lundgren, Phys. Rev. B {\\bf 38}, 7097 (1988).\n\n\\bibitem{Djurberg}\nC. Djurberg, J. Mattsson, and P. Nordblad, Europhys. Lett. {\\bf 29}, 163 (1995).\n\n\\bibitem{Parisi2}\nE. Marinari, G. Parisi, J. Ruiz-Lorenzo, and F. Ritort, Phys. Rev. Lett. {\\bf 76}, 843 (1996); J. Kisker, L. Santen, M. Schreckenberg, and H. Rieger, Phys. Rev. B {\\bf 53}, 6418 (1996); P. Sibani, C. Sh\\'on, P. Salamon, and J.-O. Andersson, Europhys. Lett. {\\bf 22}, 479 (1993); P. Sibani and J.-O. Andersson, Physica (Amsterdam) {\\bf 206A}, 1 (1994).\n\n\\bibitem{Vincent3}\nE. Vincent, J. Hammann, M. Ocio, J.-P. Bouchaud, and L.F. Cugliandolo, Proceedings, Complex Behaviour of Glassy Systems, 14. Sitges Conference, Ed. by M. Rubi and C. Perez-Vicente (Springer, Berlin, 1997), pp. 184-219.\n\n\\bibitem{Almeida}\nJ.R. de Almeida and D.J. Thouless, J. Phys. A {\\bf 11}, 983 (1978).\n\n\\bibitem{Joh}\nY.G. Joh and R. Orbach, Phil. Mag. {\\bf B 77}, 221 (1998).\n\n\\bibitem{Krassimir}\nWe are grateful to Mr. Krassimir Bozhilov for performing scanning electron microscope measurements on both the Cu:Mn and thiospinel samples.\n"
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cond-mat0002041
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Dynamics of two-particle granular collisions on a surface
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[
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"author": "Benjamin Painter and R. P. Behringer"
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We experimentally examine the dynamics of two-particle collisions occuring on a surface. We find that in two-particle collisions a standard coefficient of restitution model may not capture crucial dynamics of this system. Instead, for a typical collision, the particles involved slide relative to the substrate for a substantial time following the collision; during this time they experience very high frictional forces. The frictional forces lead to energy losses that are larger than the losses due to particle inelasticity. In addition, momentum can be transfered to the substrate, so that the momentum of the two particles is not necessarily conserved. Finally, we measure the angular momenta of particles immediately following the collision, and find that angular momentum can be lost to the substrate following the collision as well.
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[
{
"name": "2particle.tex",
"string": "\\documentstyle[pre,aps,psfig]{revtex}\n\n\\begin{document}\n \\title{Dynamics of two-particle granular collisions on a surface}\n\\author{ Benjamin Painter and R. P. Behringer}\n\\address{Department of Physics and Center for Nonlinear and Complex\nSystems, Duke University, Durham, North Carolina 27708-0305}\n\\date{\\today}\n\\maketitle\n\n\n\\begin{abstract}\nWe experimentally examine the dynamics of two-particle collisions\noccuring on a surface. We find that in two-particle collisions a\nstandard coefficient of restitution model may not capture crucial\ndynamics of this system. Instead, for a typical collision, the\nparticles involved slide relative to the substrate for a substantial\ntime following the collision; during this time they experience very\nhigh frictional forces. The frictional forces lead to energy losses\nthat are larger than the losses due to particle inelasticity. In\naddition, momentum can be transfered to the substrate, so that the\nmomentum of the two particles is not necessarily conserved. Finally,\nwe measure the angular momenta of particles immediately following the\ncollision, and find that angular momentum can be lost to the substrate\nfollowing the collision as well.\n\\end{abstract}\n\n\\pacs{45.50.Tn, 83.10.Pp, 45.70.-n}\n\n\n\n\\section{Introduction}\nDry granular systems have generated much interest recently in the\nphysics and engineering communities, both for fundamental\nunderstanding and for direct applications\n\\cite{jaeger_96,campbell_90,goldhirsch_93b}.\nThese systems are important both in nature (e.g. avalanches) and in\nindustry (e.g. pharmeceuticals and grain elevators).\n\nParticles in such systems are typically considered to interact only\nthrough interparticle collisions, i.e.~repulsive contact forces.\nExperiments that can yield quantitative data for velocities, collision\nrates and other useful quantities are often performed in two\ndimensions. In order to allow reasonable motion of particles in such\nan experiment, the particles must either be free to roll or they must\nbe levitated, for instance by air flow. Here, we consider particles\nrolling on a smooth flat surface. We note that there are then two\ntypes of friction that the particles experience when in motion. The\nfirst, rolling friction, occurs when the particle is moving without\nsliding on the substrate; its effect is relatively weak, with a\ncoefficient of friction on the order of $10^{-3}$\\cite{kudrolli_97}.\nRolling friction affects individual particles independently of\ncollisions; it tends to damp motion slowly over time. It also affects\nthe mobility of particles on the surface. For example, segregation\noccurs when particles of differing surface properties are shaken on a\nsmooth surface\\cite{tennakoon_99}. The second type of friction\naffecting particles is sliding friction. This occurs when the contact\npoint of the particle and the surface is not instantaneously at rest.\nSliding friction can occur when particles undergoing collisions\nexperience frictional frustration, i.e.~when it is impossible to\nmaintain nonsliding contacts between colliding particles and the\nsubstrate. During a collision the contact force between the particles\nis much greater than the force of gravity, so some sliding on the\nsubstrate will occur. Sliding friction is much more dissipative than\nrolling friction, with a coefficient of friction on the order of\n$10^{-1}$\\cite{kondic_99}. In the experiments described here, the\nsliding interactions with the substrate are the predominant mechanism\nfor energy loss. The sliding of particles following a collision\nleads to an energy dissipation rate that is $\\sim10^2$ times greater\nthan dissipation from rolling friction. The time over which particles\nslide is typically relatively long, $\\sim0.05-0.1{\\rm s}$. Hence, the\neffective time over which a collision influences the dynamics of a\nparticle is much longer than the actual contact time of\n$\\sim10^{-5}{\\rm s}$\\cite{kondic_99}. After a pair of particles has\nstopped sliding, the momentum of their center of mass (in the lab\nframe) need not be the same as the before-collision value. These\nfeatures have tremendous importance on dynamics of systems rolling on\nsurfaces, but have been relatively little explored experimentally.\nHowever, recent related theoretical and numerical work has been\nconducted by Kondic\\cite{kondic_99}.\n\n\nThe purpose of this paper is to examine in detail some important\naspects of the dynamics of two-particle collisions which occur on a\nsurface. We begin by briefly describing the measurement apparatus\nused to follow the particles' motion, and then discuss the surface\neffects.\n\n\n\\section{Apparatus}\n\\label{sec:apparatus}\n\nThe particles used were $2.38 mm$ steel balls, which moved on a flat\naluminum surface. The aluminum was black anodized to improve visual\ncontrast between the steel spheres and the background. The apparatus\nwas illuminated from nearly directly overhead; with this lighting,\neach metal sphere produced a single bright spot near its highest point\ndue to the reflection of the overhead light. In order to track the\ncenters of individual spheres over time, we used high speed video at\nrates of 250 frames per second. We then used particle tracking\ntechniques to follow the particles. We began by finding the positions\nof the centers of all particles within a video frame, identified by\nthe brightest points in the image (the local maxima in the brightness\nfield). Although there were some secondary reflections between\nneighboring balls, these reflections were much less bright than the\nprimary reflections, and they could be eliminated easily. By\nfollowing the positions of individual particles from frame to frame,\nwe obtained trajectories, velocities, and other time-varying\nquantities of interest.\n\n\n\\section{Particle-substrate dynamics}\n\\label{sec:2_particle}\n\n\\subsection{Rolling friction}\n\nThe simplest effect of motion on a substrate is rolling friction, and\nwe consider this effect first. The frictional force from a single\nsphere rolling on a substrate is usually modeled by\n\\begin{equation} \nF_{fr} = \\mu_r F_N, \\label{eqn:rollfrict}\n\\end{equation} \nwhere $F_N$ is the normal force at the sphere-substrate contact and\n$\\mu_r$ is the coefficient of rolling friction.\n\nWe have carried out measurements of the frictional force on a single\nsteel sphere rolling on the aluminum substrate described above. The\nsphere was tracked as described in Section \\ref{sec:apparatus}. We\ndetermined its acceleration by dividing the change in velocities\nbetween two frames by the time between the frames; the resulting\nacceleration versus velocity is shown in Fig.~\\ref{fig:rollfrict}.\nThe solid line in the figure corresponds to a least-squares linear fit\nto the data. We see that the rolling friction for this system is\nvelocity dependent, with higher frictional force at higher velocity.\nThis tends, in principle, to make velocities in rolling granular\nsystems become more uniform. To a reasonable approximation, the\nacceleration due to rolling friction which a particle experiences is\n$a=-C v - D$, where $C=0.135 s^{-1}$ and $D=1.03 cm/s^2$. Using\n$a=\\mu_r g$, with a typical acceleration of $a=-2.5 cm/s^2$, we find\nthat $\\mu_r\\sim 2.5 \\cdot 10^{-3}$, which is comparable to that\nreported by Kudrolli $et~al.$\\cite{kudrolli_97} for steel balls\nrolling on a Delrin surface.\n\n\n\\subsection{Sliding friction}\n\\label{subsec:slidingfriction}\n\nWhile this rolling friction has a dissipative effect over long times,\nthere is another, stronger, mechanism for energy loss to the\nsubstrate: sliding friction. We find that sliding friction with the\nsubstrate immediately after a collision plays a particularly important\nrole in the system dynamics. In order to investigate this effect, we\nconsider the collision of two particles on a substrate. We first\nreview the textbook example of two particles colliding in free space,\nwhich we assume is two-dimensional, and then compare this to\nexperimental observations when the motion occurs on a substrate.\n\nIn the standard case of two inelastic frictionless particles colliding\nin free space, i.e.~with no substrate, the collision is described by\nconservation of momentum and by an energy loss given through the\ncoefficient of restitution, $r$. We introduce the following notation\nto describe this process. The initial momenta of the two particles\nare given by $\\vec{p}_{1i}$ and $\\vec{p}_{2i}$, the final momenta by\n$\\vec{p}_{1f}$ and $\\vec{p}_{2f}$, and $\\vec{p}_i =\n\\vec{p}_{1i}+\\vec{p}_{2i}$. The direction of the vector connecting\nthe centers of mass of the two particles at the time of the collision\nis $\\hat{n}$. The relative velocity of the particles in the $\\hat{n}$\ndirection after colliding is a fraction $r$ of their initial relative\nvelocity, while the relative velocity tangential to $\\hat{n}$ is a\nfraction $s$ of its initial value. Thus,\n\\begin{eqnarray}\n\\vec{p}_{1f} + \\vec{p}_{2f} & = & \\vec{p}_{1i} + \\vec{p}_{2i}, \\\\\np_{1fn} - p_{2fn} & = &-r (p_{1in} - p_{2in}) , \\label{eqn:coeffres}\n\\\\ p_{1ft} - p_{2ft} & = &s (p_{1it} - p_{2it}),\n\\end{eqnarray}\nwhere the subscripts $n$ and $t$ refer to the directions parallel and\nperpendicular to $\\hat{n}$, respectively. We take $s=1$, the\nsimplest case describing an inelastic collision.\n\nThis model is usually used in modeling granular\nsystems\\cite{grossman_97,goldhirsch_93a}. However, it does not\naccurately reflect the dynamics of two rolling particles colliding on\na surface. When two rolling particles collide, there are three\ncontact points: each particle with the substrate, and the particles\nwith each other. In general, these contact points are frictional.\nThis leads to rotational frustration and, after the collision, to\nsliding\\cite{kondic_99}. The following simple argument shows why the\nparticles are likely to slide on the substrate following a collision.\nDuring a collision the frictional force between the particles competes\nwith the frictional forces between the particles and the substrate.\nIf the static friction coefficients at all contacts are comparable,\nthe frictional force will be greatest where the contact forces are\ngreatest. The interparticle contact force is $F_{p-p} \\sim \\Delta\np/\\Delta t$, where $\\Delta p = m \\Delta v$ is the momentum change of a\nparticle and $\\Delta t_{contact} \\approx 10^{-5} {\\rm s}$\n\\cite{kondic_99} is the contact time for a hard-particle collision.\nThe contact force for a particle with the substrate is $F_{p-s} = m\ng$, and the ratio $F_{p-p}/F_{p-s} = \\Delta v/(g \\Delta t_{contact})$.\nThus $g \\Delta t_{contact} \\approx 10^{-2} {\\rm cm/s}$ for hard metal\nspheres defines a crossover velocity, with sliding on the substrate\noccuring for $\\Delta v > g \\Delta t_{contact}$. If sliding has been\ninitiated in the collision, a finite time and distance is required\nafter the spheres separate before dynamic (sliding) friction slows the\nspheres' sliding motion. They will eventually reach a point where\nsliding stops and the particles are simply rolling. During this time,\nboth the direction and the speed of the particles change\nsignificantly, as detailed below.\n\nWe have investigated this effect experimentally by rolling one ball at\nan identical stationary ball, and by tracking their motion before and\nafter the collision. Fig.~\\ref{fig:2tracks} shows a typical set of\ntrajectories in such a two-particle collision. The moving ball (in\nthe lab frame) enters from the left of the image, hitting the\nstationary ball. Since it is difficult to produce a perfectly head-on\ncollision, the incoming ball strikes the stationary ball slightly\noff-center. Immediately after the collision the two balls behave\nalmost as though there were no surface interactions (see inset).\nSomewhat later, the particles begin to show the influence of the\nsubstrate as they change direction and speed.\n\nDuring the time between the collision and the time when the balls\nbegin rolling without sliding, both the direction and the speed of the\nballs change due to sliding. This is shown in\nFig.~\\ref{fig:distapart}, which gives the distance between the two\nballs shown in Fig.~\\ref{fig:2tracks} over time. The collision occurs\nat $t_c \\approx 0.06 sec$. For times $t$ before and well after the\ncollision, the separation $s$ varies nearly linearly with time,\nindicating that the balls roll with nearly constant velocity in these\nperiods. By contrast, during the $\\sim\\,0.06 sec$ immediately\nfollowing the collision, the interparticle separation varies\nnonlinearly in time. This indicates a regime in which the two\nparticles experience dynamic, or sliding, friction with the substrate.\nWe denote the time following the collision at $t = t_c$ and before the\nparticles start rolling without sliding at $t = t_r$ as the\n``relaxation time,'' $\\tau_R = t_r - t_c$. We define $t_c$ as the\ntime at which the particles' centers of mass are closest together, and\n$t_r$ as the point in time after the collision at which a particle\nbegins moving with nearly constant velocity. $\\tau_r$ was typically\n$0.05-0.1 {\\rm s}$ in the systems we studied, which is very large\ncompared to the time the particles are in contact, roughly\n$10^{-5}\\,{\\rm s}$\\cite{kondic_99}. After a period of time equal to\n$\\tau_R$ has elapsed, each particle has nearly constant velocity,\naffected only by rolling friction.\n\n\nFrom collision data we can determine the coefficient of restitution\n$r$, as defined in Eq.~\\ref{eqn:coeffres} by examining the velocities\nimmediately preceeding and after the collision, but before sliding\nfriction has had significant effects. Fig.~\\ref{fig:initcoeff} gives\na histogram of data obtained for a number of measurements of $r$\nobtained this way. To produce these results, we measured velocities\nimmediately before and within $0.01$ seconds after the collision. We\nfind an average value of the coefficient of restitution to be $r =\n0.85 \\pm 0.11$, with no obvious dependence on the velocity of incoming\nparticle. This value is similar to the value of $r$ reported by\nothers for steel-on-steel collisions: $r = 0.93$ in\nref.~\\cite{kudrolli_97} and $r = 0.90$ in ref.~\\cite{luding_94}.\n\nImmediately after the collision, the relative angle of the particles'\nnew directions is also close to what one would expect for an elastic\ncollision between two equal-sized spherical particles with one\ninitially at rest. Fig.~\\ref{fig:anglebetween} shows a typical\nexample. In a collision between two identical spheres of radius $R$,\nwith coefficient of restitution $r$ and impact parameter $b$, the\nangle between the directions of the spheres' motion after the\ncollision is given by\n\\begin{equation}\ncos \\theta_f = {{(1-r)(1-(\\frac{b}{2R})^2)}\\over\n{\\left\\{ \\left[ (1-r)^2(1-(\\frac{b}{2R})^2)+4(\\frac{b}{2R})^2 \\right] \n(1-(\\frac{b}{2R})^2) \\right\\}^{1/2}}}.\n\\end{equation}\nThen as $r \\rightarrow 1$, a perfectly elastic collision, $cos\n\\theta_f = 0$ and $\\theta_f = \\pi/2$, provided that $b/2R \\gg 1-r$. A\ntypical value of the impact parameter in these experiments is $b/2R\n\\sim 0.25$. The collision in Fig.~\\ref{fig:anglebetween} occurs at\ntime $t \\approx 0.06 sec$, indicated by the vertical dotted line; at\nthis time, the relative angle between the particle velcities is near\n$\\pi/2$. As sliding friction begins to affect the particles and they\nare accelerated or decelerated, the angle between the velocities\ndecreases. The direction of the acceleration is discussed in detail\nbelow, in Sec.~\\ref{subsec:momentumloss}.\n\n\nKondic has investigated a model for two particles colliding on a\nsurface that includes both the interaction between the particles via a\ncollision and the interaction of the particles with the substrate\nthrough friction \\cite{kondic_99}. For a system consisting of a\nmoving particle hitting a stationary particle head-on, he predicts\nvelocities and relaxation times of each particle after the collision.\nIf the initial velocity of the moving particle is $v_o$, the final\n(i.e.~purely rolling) velocities of the initially stationary and\ninitially moving particles are $v_{1f}$ and $v_{2f}$ respectively, and\nthe relaxation times of the initially stationary and initially moving\nparticles are $\\tau_1$ and $\\tau_2$ respectively, then:\n\\begin{eqnarray}\nv_{1f} & = & {v_o \\over {2(1+{m R^2 \\over I})}}((1+r){{m R^2}\\over{I}}-2 C) \\equiv F(v_o,r,C) \\label{eqn:kondic1}\\\\ \nv_{2f} & = & {v_o \\over {2(1+{m R^2 \\over\nI})}}(2+(1-r){{m R^2}\\over{I}}-2 C) \\equiv G(v_o,r,C),\\\\\n\\tau_1 & = & {{{{1+r}\\over{2}}+C} \\over {(1+{{m R^2}\\over{I}})\\mu_k g}}v_o \\equiv H(v_o,r,C,\\mu_k),~{\\rm and}\\\\\n\\tau_2 & = &{{{{1+r}\\over{2}}-C}\\over{(1+{{m R^2}\\over{I}})\\mu_k\ng}}v_o \\equiv I(v_o,r,C,\\mu_k) \\label{eqn:kondic4},\n\\end{eqnarray}\nwhere $r$ is the coefficient of restitution of the particles, $\\mu_k$\nis the coefficient of kinetic friction of the particles with the\nsubstrate, $g$ is the acceleration of gravity, and $I$ is the moment\nof inertia of the particles ($I=2 m R^2/5$). $C$ is a measure of the\ntransfer of angular momentum between the particles during the\ncollision, such that immediately after the collision \n\\begin{eqnarray}\n\\omega_1 & = & -C ~\\omega_o,~{\\rm and} \\nonumber \\\\\n\\omega_2 & = & (1-C)~ \\omega_o, \\label{eqn:angulartransfer}\n\\end{eqnarray}\nwhere $\\omega_i$, for $i > 0$, is the angular velocity of each\nparticle and $\\omega_o$ is the angular velocity of the incoming\nparticle before the collision.\n\nWe fitted data for experimentally determined final velocities $v_1$\nand $v_2$ versus $v_o$ and relaxation times $\\tau_1$ and $\\tau_2$\nversus $v_o$ to these predictions by minimizing the squared deviation\nof the model from observed data. Specifically, we minimized\n\\begin{eqnarray}\n\\chi^2 & = {\\sum_i} & \\left [ {(v_{1fi}-F(v_{oi},r,C))^2\n\\over{\\sigma_{v_{1fi}}^2}} +\n{(v_{2fi}-G(v_{oi},r,C))^2 \\over{\\sigma_{v_{2fi}}^2}} + \n{(\\tau_{1i}-H(v_{oi},r,C,\\mu_k))^2 \n\\over{\\sigma_{\\tau_{1i}}^2}} + \\nonumber \\right. \\\\\n & & \\left. {(\\tau_{2i}-I(v_{oi},r,C,\\mu_k))^2 \n\\over{\\sigma_{\\tau_{2i}}^2}} \\right ],\n\\end{eqnarray}\nwith fitting parameters $r$, $\\mu_k$, and $C$. Here $\\sigma_n$\nrepresents the experimental uncertainty in the variable $n$. We found\nthat $r = 0.903 \\pm 0.008$, $\\mu_k = 0.232 \\pm 0.023$, and $C = 0.347\n\\pm 0.008$ in our experiments. Note that this result for the\ncoefficient of restitution $r$ is consistent with, but much more\nprecise than, the value of $r=0.85 \\pm 0.11$ determined from\nFig.~\\ref{fig:initcoeff}. Results of these fits can be seen in\nFig.~\\ref{fig:relaxtime} and Fig.~\\ref{fig:relaxvel}.\n\n\nWe conclude that for a two-particle collision on a substrate, the picture of an\ninstantaneous normal coefficient of friction is inaccurate, and may\nnot be particularly useful. Without surface interactions, the\nrelaxation times are $\\tau = 0$ and the final velocities are $v_1/v_o\n=(1+r)/2$ and $v_2/v_o=(1-r)/2$ (represented by the dotted lines in\nFig.~\\ref{fig:relaxvel}).\n\n\n\\subsection{Energy loss}\n\nFor many-particle systems, important indicators of the properties of\nthe collision are the net energy and momentum losses of the system.\nThus we now turn our attention to them. By the time the particles\nhave reached the point of rolling without sliding, there has been an\nenergy loss much greater than that which would occur in a system\ndescribed only by a standard coefficient of restitution. For two\nparticles undergoing a collision described by a conventional\ncoefficient of restitution, as in Eq.~\\ref{eqn:coeffres}, the\nmaximum fractional energy loss, which occurs in a head-on collision,\nis $(1-r^2)/2$. For steel balls, with $r \\approx 0.903$, $(1- r^2)/2\n\\approx 0.09$. For collisions that are not head-on, the energy losses\nare smaller, as only the component of velocity normal to the collision\ndecreases, assuming the tangential coefficient of restitution equals\n1. Figure \\ref{fig:energyloss} shows the total system energy versus\ntime for the two-particle collision described above. Figure\n\\ref{fig:energyvsangle} shows the fractional energy remaining in the\nsystem at the end of the relaxation time $\\tau_R$ for a series of\ncollisions as a function of the final angle between the velocities.\nFor a head-on collision the system's energy after the collision is on\naverage $\\sim 37\\%$ of the energy before the collision, representing a\nloss of $63\\%$ of system energy as the result of a single collision.\nAlso shown in Fig.~\\ref{fig:energyvsangle} is a prediction based on\nEqs.~\\ref{eqn:kondic1}-\\ref{eqn:kondic4} (solid line) with the\nparameters determined in the fit discussed above. The dashed line\nrepresents energy loss in a system with $r=0.903$ and no surface\ninteractions. The observed energy loss is only weakly dependent on the\ncollision angle for nearly head-on collisions.\n\n\n\n\\subsection{Momentum loss and the direction of sliding frictional forces}\n\\label{subsec:momentumloss}\n\nWe note that the direction of force due to sliding friction is not\nnecessarily parallel to the contact normal, $\\hat{n}$; instead,\nit is in the direction of the relative velocity of the contact between\nthe substrate and the bottom of the particle, which we call the\ncontact velocity $v_{ct}$. Thus,\n\\begin{eqnarray}\n\\vec{v_{ct}} = \\vec{v_{cm}} + \\vec{a} \\times \\vec{\\omega} \\label{eqn:vcdef},\n\\end{eqnarray}\nwhere $\\vec{v}_{cm}$ is the velocity of the center of mass of the\nparticle, $\\vec{a}$ is the vector from the contact point to the center\nof the particle, and $\\vec{\\omega}$ is the particle's angular\nvelocity. A sketch is provided in Fig.~\\ref{fig:veldiagram}. Here,\nwe define the $\\hat{x}$ direction as $\\vec{v_{ct}}/|\\vec{v_{ct}}|$,\nand the $\\hat{y}$ direction as $(\\vec{a} \\times \\hat{x})/|\\vec{a}|$.\nAll momentum loss to the substrate will occur in the $\\hat{x}$\ndirection, as this is the direction of the only force acting on the\nparticle (neglecting rolling friction, which is small compared to\nsliding friction). We experimentally determine the $\\hat{x}$\ndirection by finding the direction in which a particle's velocity\nchanges following a collision. Figures \\ref{fig:pparallel} and\n\\ref{fig:pperp} show the momenta of the individual particles versus\ntime in the collision described above, in the $\\hat{x}$\n(Fig.~\\ref{fig:pparallel}) and $\\hat{y}$ (Fig.~\\ref{fig:pperp})\ndirections. Note that the $\\hat{x}$ and $\\hat{y}$ directions are\nindependently defined for each of the particles, i.e. $\\hat{x}$ for\nthe initially moving particle is different from $\\hat{x}$ for the\ninitially stationary particle. Figure \\ref{fig:pparallel} illustrates\nthe finite time after the collision (the collision time $t_c$ is\nmarked by a vertical dotted line) for which momentum is transferred to\nthe substrate through sliding friction. After this time, the only\nmomentum loss is due to rolling friction. In Fig.~\\ref{fig:pperp} we\nobserve that no momentum is lost in the $\\hat{y}$ direction for either\nparticle after the collision, aside from a slow loss due to rolling\nfriction.\n\n\nWe also examine the net momentum loss in the direction of the initial\nmomentum versus the final angle between the particle velocities\n(Fig.~\\ref{fig:pvsangle}). Note that in the usual case, with no\nsurface interactions, no momentum is lost, so $p/p_0 = 1$ for all\nangles. In contrast, for a head-on collision with surface\ninteractions we see that $\\sim 20\\%$ of $\\vec{p}_i$ is lost. Further,\nwe note that this quantity is weakly dependent on the final angle\nbetween the velocities after the collision for small angles. The\nsolid line in this figure shows the prediction based on\nEqs.~\\ref{eqn:kondic1}-\\ref{eqn:kondic4}, with the parameters\ndetermined above.\n\nThis momentum loss may be important in many-particle systems. For\nexample, inelastic collapse, a condition in which there are an\ninfinite number of collisions in a finite time, occurs in one- and\ntwo-dimensional idealized systems \\cite{mcnamara_92,mcnamara_94}.\nOne-dimensional numerical simulations by Dutt $et~al.$ \\cite{dutt_99}\nshow that if even a very small momentum loss per collision is\nintroduced, inelastic collapse does not occur. This suggests that\ninelastic collapse cannot be observed in experimental granular systems\nwhich interact with a surface.\n\n\n\\section{Angular velocity}\n\nWe would also like to determine the angular velocities of the\nparticles immediately after the collision. These are difficult to\nmeasure directly, but we can derive expressions for them from\nEq.~\\ref{eqn:vcdef}, given the assumption that after sliding stops\neach particle will be rolling without sliding. Then the angular\nvelocity of a particle immediately after the collision, $\\omega_0$, is\n\\begin{equation}\n\\omega_{y0} = {1\\over{a}}(\\nu \\Delta v_{cm} + v_{cmx0}),\n\\end{equation}\nand\n\\begin{equation}\n\\omega_{x0} = - {1\\over{a}} v_{cmy0},\n\\end{equation}\nwhere $\\Delta v_{cm}$ is the change of the center of mass velocity in\nthe $\\hat{x}$ direction due to sliding forces, $v_{cm0}$ is the center\nof mass velocity immediately following the collision, and $\\nu\n\\equiv (1+m a^2/I) = 7/2$. Since we can directly measure the center\nof mass velocity at all times, we can deduce the values of\n$\\omega_{x0}$ and $\\omega_{y0}$.\n\nThese calculations determine the components of $\\vec{\\omega}$ in the\n$\\hat{x}$ and $\\hat{y}$ directions, as defined in\nSec.~\\ref{subsec:momentumloss}. These directions are defined by the\nsliding frictional forces acting on the particles; a more natural\ncoordinate system when examining the effect of the collision itself on\nangular velocities is defined by the $\\hat{n}$ and $\\hat{t}$\ndirections, that is, parallel ($\\hat{n}$) and perpendicular\n($\\hat{t}$) to the vector connecting the centers of mass of the\nparticles at the time of the collision. If surface effects are\nnegligible during the collision, there is no torque in the $\\hat{n}$\ndirection, so we expect that $\\omega_{n}$ for each particle will not\nbe changed by the collision. Indeed, we find that during the\ncollision the mean change of the angular velocity in the $\\hat{n}$\ndirection, averaged over 50 particles and normalized by the angular\nvelocity of the incoming particle in each case, is $d\\omega_n /\n(v_o/a) = 0.01 \\pm 0.02$. In contrast, we expect that during the\ncollision, some angular momentum will be transferred from the moving\nparticle to the stationary particle in the $\\hat{t}$ direction. The\namount of angular velocity transferred can be quantified by\nEq.~\\ref{eqn:angulartransfer}; we find from these calculations that\n$C=0.25 \\pm 0.02$. This is similar to, although slightly smaller\nthan, the value of $C=0.347 \\pm 0.008$ obtained from the fit to\nEqs.~\\ref{eqn:kondic1}-\\ref{eqn:kondic4} above.\n% in Sec.~\\ref{subsec:slidingfriction}.\n\n\n\\section{Conclusion}\n\nIn two-dimensional granular systems, understanding interactions with\nthe substrate is crucial to understanding the system dynamics. As two\nparticles collide, there is rotational frustration between them and\nthe substrate, leading to sliding on the surface. The large contact\nforce between the particles at the time of the collision is much\ngreater than gravity, with $(\\Delta v/\\Delta t_{coll})/{\\rm g} \\sim\n10^3$, guaranteeing that particles will slide on the substrate after\nthe collision. The resulting sliding friction leads to high energy\nlosses and can be modeled simply, as discussed by Kondic\n\\cite{kondic_99}. In fact, we find that up to $63\\%$ of the incoming\nenergy is lost in a single collision between two particles with\ncoefficient of restitution of $0.9$, and most of this is due to\nsliding friction with the substrate. The sliding continues for a\ntime, $\\tau_R$, which is long relative to the collision time.\n$\\tau_R$ is comparable to or longer than the time between collisions\nfor moderately dense, rapidly cooling systems, which means that\nsliding is experimentally important for many-particle systems until\ntypical velocities reach $v \\sim g \\Delta t_{coll}$. Additionally, we\nfind that both momentum and angular momentum are typically lost to the\nsubstrate following a collision.\n\n\n\\bibliography{../../../bibtex/granular}\n\n\\bibliographystyle{unsrt}\n\n\\begin{figure}[t]\n\\center{\\parbox{6in}{\n\\psfig{file=figure1.eps,width=6in,angle=270} \n}\n\\caption{Acceleration due to rolling friction for a single particle\nrolling on a horizontal flat surface. Shown are measured values of\nacceleration vs. velocity, and a best-fit line.}\n\\label{fig:rollfrict}}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\center{\\parbox{6in}{\n\\psfig{file=figure2.eps,height=3in} \n}\n\\caption{Tracks of two particles colliding: A moving particle enters\nfrom the left with $v \\approx 10 cm/s$ and strikes a stationary particle.\nBoth exit toward the right. The circles represent the particles'\npositions every $0.02 sec$. Inset is a detailed view of the particle\ntracks near the collision point, with the length of one particle\ndiameter shown for scale.}\n\\label{fig:2tracks}}\n\\end{figure}\n\n\\begin{figure}[t]\n\\center{\\parbox{6in}{\n\\psfig{file=figure3.ps,width=6in,angle=270}\n}\n\\caption{Separation $s$ between the centers of the particles shown in \nFig.~\\ref{fig:2tracks} vs. time.}\n\\label{fig:distapart}}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\center{\\parbox{6in}{\n\\psfig{file=figure4.eps,width=6in,angle=270} \n}\n\\caption{Histogram for the coefficient of restitution, as determined \nimmediately following a collision, for 28 samples.}\n\\label{fig:initcoeff}}\n\\end{figure}\n\n\\begin{figure}[t]\n\\center{\\parbox{6in}{\n\\psfig{file=figure5.eps,width=6in,angle=270} \n}\n\\caption{Angle between particle velocities. The collision occured at\nt $\\approx 0.06$ sec, as indicated by the vertical dashed line. At\nthis point the particles are moving at nearly $90^\\circ$ to each\nother. The angle between the incoming particle velocity and $\\hat{n}$\nis $5.5^\\circ$. The oscillations are due to experimental noise.}\n\\label{fig:anglebetween}}\n\\end{figure}\n\n\\begin{figure}[t]\n\\center{\\parbox{6in}{\n\\psfig{file=figure6.ps,width=6in,angle=270}\n}\n\\caption{Relaxation time vs. impact velocity. The solid lines are\npredictions based on a fit to\nEqs.~\\ref{eqn:kondic1}-\\ref{eqn:kondic4}. Note that in a system\nwithout surface interactions, $\\tau_r=0$ for all collisions.}\n\\label{fig:relaxtime}}\n\\end{figure}\n\n\\begin{figure}[t]\n\\center{\\parbox{6in}{\n\\psfig{file=figure7.eps,width=6in}\n}\n\\caption{Final velocity vs. initial velocity. The solid lines are\npredictions based on the fit to\nEqs.~\\ref{eqn:kondic1}-\\ref{eqn:kondic4}. Dotted lines represent\ntheoretical final velocities of particles without surface\ninteractions, with $r=0.903$.}\n\\label{fig:relaxvel}}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\center{\\parbox{5in} {\n\\psfig{file=figure8.eps,width=5in} \n}\n\\caption{Total system energy versus time in a single collision.\nThe vertical dotted line marks the approximate time of the collision.\nThe dashed line shows the final energy that would result if the\nfractional energy loss was $(1-r^2)/2$.}\n\\label{fig:energyloss}}\n\\end{figure}\n\n\\begin{figure}[t]\n\\center{\\parbox{6in}{\n\\psfig{file=figure9.eps,width=6in,angle=270} \n}\n\\caption{Energy loss vs. angle between final velocities of two particles.\nThe dashed line represents numerical calculations of two particles\ncolliding without substrate interactions, with $r=0.903$. The solid\nline represents predictions based on\nEqs.~\\ref{eqn:kondic1}-\\ref{eqn:kondic4}, with the fitted parameters\n$r=0.903$, $\\mu_k=0.232$, and $C=0.347$.}\n\\label{fig:energyvsangle}}\n\\end{figure}\n\n\n\\begin{figure}\n\\center{\\parbox{6in}{\n\\psfig{file=figure10.ps,width=6in,angle=270}\n}\n\\caption{Sketch of velocities for a sliding particle.}\n\\label{fig:veldiagram}}\n\\end{figure}\n\n\n\\begin{figure}\n\\center{\\parbox{6in}{\n\\psfig{file=figure11.ps,height=6in,angle=270}\n}\n\\caption{Momentum parallel to $\\hat{x}$ vs. time, for (a) the\ninitially stationary particle and (b) the initially moving particle \n($\\hat{x}$ is different for each particle). The collision occurs at $t\n\\sim 0.06 sec$, as indicated by the vertical dotted lines. Note that\na large part of the momentum is transfered from one particle to the\nother at the time of collision, and that $v_{x2}$ increases after the\ncollision as a result of its spin.}\n\\label{fig:pparallel}}\n\\end{figure}\n\n\\begin{figure}\n\\center{\\parbox{6in}{\n\\psfig{file=figure12.ps,height=6in,angle=270} \n}\n\\caption{Momentum perpendicular to $\\hat{x}$ vs. time, for (a) the\ninitially stationary particle and (b) the initially moving particle.\nThe collision occurs at $t \\sim 0.06 sec$, represented by the vertical\ndotted lines. Very little momentum transfer takes place in this\ndirection after the collision.}\n\\label{fig:pperp}}\n\\end{figure}\n\n\\begin{figure}\n\\center{\\parbox{6in}{\n\\psfig{file=figure13.eps,width=6in,angle=270}\n}\n\\caption{Momentum fraction remaining after relaxation (at time \nt=$t_c+\\tau_r$) in the direction of the initial momentum (in the lab\nframe) versus final angle between velocities. For a head-on\ncollision, approximately 20\\% of the total system momentum is lost.\nThe solid line gives the prediction based on\nEqs.~\\ref{eqn:kondic1}-\\ref{eqn:kondic4}, with the fitted parameters\n$r=0.903$, $\\mu_k=0.232$, and $C=0.347$.}\n\\label{fig:pvsangle}}\n\\end{figure}\n\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n"
}
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[
{
"name": "2particle.bbl",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{jaeger_96}\nHeinrich~M. Jaeger, Sidney~R. Nagel, and Robert~P. Behringer.\n\\newblock Granular solids, liquids, and gases.\n\\newblock {\\em Rev. Mod. Phy.}, 68:1259, 1996.\n\n\\bibitem{campbell_90}\nC.~S. Campbell.\n\\newblock {\\em Annu. Rev. Fluid Mech.}, 22:57, 1990.\n\n\\bibitem{goldhirsch_93b}\nI.~Goldhirsch and G.~Zanetti.\n\\newblock Clustering instability in dissipative gases.\n\\newblock {\\em Phys. Rev. Lett.}, 70:1619, 1993.\n\n\\bibitem{kudrolli_97}\nA.~Kudrolli, M.~Wolpert, and J.~P. Gollub.\n\\newblock Cluster formation due to collisions in granular material.\n\\newblock {\\em Phys. Rev. Lett.}, 78:1383, 1997.\n\n\\bibitem{tennakoon_99}\nS.~Tennakoon and R.~P. Behringer.\n\\newblock (to be published).\n\n\\bibitem{kondic_99}\nLjubinko Kondic.\n\\newblock Dynamics of spherical particles on a surface: Collision-induced\n sliding and other effects.\n\\newblock {\\em Phys. Rev. E}, 60:751, 1999.\n\n\\bibitem{grossman_97}\nE.~L. Grossman, Tong Zhou, and E.~Ben-Naim.\n\\newblock Towards granular hydrodynamics in two dimensions.\n\\newblock {\\em Phys. Rev. E}, 55:4200, 1997.\n\n\\bibitem{goldhirsch_93a}\nI.~Goldhirsch, M.-L. Tan, and G.~Zanetti.\n\\newblock A molecular dynamical study of granular fluids \\uppercase{I}: The\n unforced granular gas in two dimensions.\n\\newblock {\\em J. Sci. Comp.}, 8:1, 1993.\n\n\\bibitem{luding_94}\nS.~Luding, E.~Clement, A.~Blumen, J.~Rajchenbach, and J.~Duran.\n\\newblock Studies of columns of beads under external vibrations.\n\\newblock {\\em Phys. Rev. E}, 49:1634, 1994.\n\n\\bibitem{mcnamara_92}\nSean McNamara and W.~R. Young.\n\\newblock Inelastic collapse and clumping in a one-dimensional granular medium.\n\\newblock {\\em Phys. Fluids A}, 4:496, 1992.\n\n\\bibitem{mcnamara_94}\nSean McNamara and W.~R. Young.\n\\newblock Inelastic collapse in two dimensions.\n\\newblock {\\em Phys. Rev. E}, 50:R28, 1994.\n\n\\bibitem{dutt_99}\nM.~Dutt and R.~P. Behringer.\n\\newblock (to be published).\n\n\\end{thebibliography}\n"
}
] |
cond-mat0002042
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Stability analysis of the $D-$dimensional nonlinear Schr\"odinger equation with trap and two- and three-body interactions
|
[
{
"author": "A. Gammal$^{1}$"
},
{
"author": "T. Frederico$^{2}$"
},
{
"author": "Lauro Tomio$^{1}$"
}
] |
Considering the static solutions of the $D-$dimensional nonlinear Schr\"odinger equation with trap and attractive two-body interactions, the existence of stable solutions is limited to a maximum critical number of particles, when $D\ge 2$. In case $D=2$, we compare the variational approach with the exact numerical calculations. We show that, the addition of a positive three-body interaction allows stable solutions beyond the critical number. In this case, we also introduce a dynamical analysis of the conditions for the collapse. \newline PACS: 03.75.Fi; 47.20.Ky; 02.30.Jr; 31.75.Pf \newline Keywords: Nonlinear Schr\"odinger Equation; trapped two and three-body atomic systems; multidimensional systems
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[
{
"name": "multid.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentstyle[aps,multicol,epsf]{revtex}\n\\begin{document}\n\\title{Stability analysis of the $D-$dimensional nonlinear\nSchr\\\"odinger equation with trap and two- and three-body interactions}\n\\author{A. Gammal$^{1}$, T. Frederico$^{2}$, Lauro Tomio$^{1}$,\nand F.Kh. Abdullaev$^{1}$ \\thanks{On leaving from \nPhysical Technical Institute, Tashkent, Uzbekistan.}} \n\\address{$^{1}$ Instituto de F\\'{\\i}sica Te\\'{o}rica, \nUniversidade Estadual Paulista, \\\\\n01405-900 S\\~{a}o Paulo, Brazil \\\\\n$^{2}$Departamento de F\\'{\\i }sica, Instituto Tecnol\\'{o}gico da\nAeron\\'{a}utica, \\\\\nCentro T\\'{e}cnico Aeroespacial, 12228-900 S\\~{a}o Jos\\'{e} dos Campos, SP,\nBrazil}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nConsidering the static solutions of the $D-$dimensional nonlinear\nSchr\\\"odinger equation with trap and attractive two-body interactions,\nthe existence of stable solutions is limited to a maximum critical number\nof particles, when $D\\ge 2$. In case $D=2$, we compare the variational\napproach with the exact numerical calculations. We show that, the addition\nof a positive three-body interaction allows stable solutions beyond the\ncritical number. In this case, we also introduce a dynamical analysis\nof the conditions for the collapse.\n\\newline\nPACS: 03.75.Fi; 47.20.Ky; 02.30.Jr; 31.75.Pf\n\\newline\nKeywords: Nonlinear Schr\\\"odinger Equation; trapped two and three-body \natomic systems; multidimensional systems \n\\end{abstract}\n\\begin{multicols}{2}\nRecent experiments on Bose Einstein Condensation (BEC)~\\cite{exp} \nhave brought great attention to its theoretical formulation. \nAtomic traps are effectively described by the Ginzburg-Pitaevskii-Gross \n(GPG) formulation of the nonlinear Schr\\\"odinger equation\n(NLSE)~\\cite{gin}, which includes two-body interaction.\nWhen the atoms have negative two-body scattering lengths, a formula\nfor the critical maximum number of atoms was presented in ref.~\\cite{WT}. \nIn ref.~\\cite{ADV,GFT,GFTC}, the formulation was\nextended in order to include the effective potential originated from\nthe three-body interaction. In this case, in three-dimensions, it was\nshown that a kind of first order phase-transition occurs. \nIn this connection, as also considered in the motivations given \nin \\cite{GFT,GFTC}, it is relevant to observe that recently\nit was reported the possibility of altering continuously the two-body \nscattering length, from positive to negative values, by means of an \nexternal magnetic field~\\cite{achange}. Within such perspective, the \ntwo-body binding energy can be close to zero, and one can approach the \nso-called Efimov limit, which corresponds to an increasing number of \nthree-body bound states~\\cite{efimov}. Near this limit, nontrivial \nconsequences can occur in the dynamics of the condensate, such that\none should also consider three-body effects in the effective \nnonlinear potential.\n\nIn the present work, we study the critical number of atoms in\narbitrary $D-$dimensions, using a variational procedure; and also\nby an exact numerical approach in the case of dimension $D=2$.\nThe $D-$dimensional NLSE, with attractive two-body interactions, was\npreviously analyzed in models of plasma and light waves in nonlinear \nmedia~\\cite{zakharov}. The collapse conditions, in this case, were\ninvestigated without~\\cite{weinstein} and with~\\cite{TW} the harmonic\npotential term. In case of $D=$3, it was shown that a\nrepulsive nonlinear three-body interaction term can extend\nconsiderably the critical limit for the existence of stable\nsolutions~\\cite{ADV,GFT,GFTC}. \n\nMotivated by the observed high interest in stable solutions for arbitrary \n$D$, we look for variational solutions in a few significant\ncases ($D=$1,2,4 and 5) not previously considered, when a three-body\ninteraction term, parametrized by $\\lambda_3$, is added to the\neffective non-linear interaction that contains a two-body attractive term.\nOur analysis also shows that, as in case of $D=3$, a kind of\nfirst-order phase-transition can occur when $D\\ge 4$, for certain\ncases of $\\lambda_3\\ge 0$.\nIn the present paper, we have also considered the approach given in\n\\cite{Pit1}, in order to study the stability conditions\nin the case of arbitrary $D$, when the non-linear interaction\ncontains two (attractive) and three-body terms.\n\nIn order to obtain an analytical approach and verify \nthe validity of the variational Ritz method, \nwe consider in detail the case of $D=2$, with and without\nthe three-body term, comparing the variational results with exact\nnumerical calculations for some relevant physical observables. \nIn this case, we also discuss how the method given in \\cite{weinstein}\ncan be extended in order to approach analytically the exact value for\nthe total energy.\n\nBy extending the GPG formalism from three to $D$ dimensions, \nincluding two~\\cite{baym96} and three-body interactions in the \neffective non-linear potential~\\cite{GFT}, we obtain \n\\begin{eqnarray}\ni\\hbar\\frac{d\\psi}{dt} =\n\\left[-\\frac{\\hbar^2}{2m}\\nabla^{2}+\\frac{m\\omega^2 r^2}{2} \n+\\lambda_2|\\psi|^2+\\lambda_3|\\psi|^4\n\\right]\\psi\\label{1} ,\n\\end{eqnarray} \nwhere $\\psi\\equiv\\psi(\\vec{r},t)$ is the wave-function normalized to the\nnumber of atoms $N$, $\\omega$ is the frequency of the trap harmonic\npotential and $m$ is the mass of the atom.\n$\\lambda_2$ and $\\lambda_3$ are, respectively, the\nstrength of the two- and three-body effective interaction, given\nin a $D-$dimensional space. \n$r\\equiv|\\vec{r}|$ is the hyperradius, such that \n$\\vec{r}\\equiv\\sum_{i=1}^{D}r_i\\hat{e}_i$ and \n$\\nabla\\equiv\\sum_{i=1}^{D}\\hat{e}_i\\frac{\\partial}{\\partial r_i}$ \n($\\hat{e}_i$ is the unit vector, with $i=1,2,... D$). \n\nThe stationary solutions for the chemical potential $\\mu$ are given\nby\n\\begin{equation} \ni\\hbar\\frac{d\\psi}{dt} = \\mu\\psi.\n\\label{1.1}\n\\end{equation}\nConsidering the general solution of eq.(\\ref{1}), \n${\\displaystyle i\\hbar\\frac{d\\psi}{dt} = \\frac{\\delta{\\cal\nH}}{\\delta\\psi^\\star}}$,\none can obtain the total energy $E$:\n\\begin{eqnarray}\nE&=&\\int d^D\\vec{r}\\;\\;{\\cal H} ,\\;\\;\\;{\\rm with}\\label{Etot}\\\\\n{\\cal H}&\\equiv&\n\\frac{\\hbar ^{2}}{2m}\\left| \\nabla \\psi\n\\right|^{2}+\\frac{m\\omega ^{2}r^{2}}{2} \\left| \\psi\\right|^{2}\n+\\frac{\\lambda_2}{2}|\\psi|^{4}+\n\\frac{\\lambda_3}{3}|\\psi|^{6}\n. \\nonumber\n\\end{eqnarray} \n\nHere we consider only attractive two-body interaction, which \nis more interesting in the case of trapped atoms. \nFor $D=$3, $\\lambda_2 \\equiv -4\\pi\\hbar^2|a|/m$, where $a$ is\nthe two-body scattering length and $m$ is the mass of the atom. \nIn the case of arbitrary $D$, $\\lambda_2$ has \ndimension of energy times $L^D$, where $L$ is a length scale in such\nspace. However, a convenient redefinition of the wave-function in terms of \ndimensionless variables will absorb this constant, as will be shown. \n\nOur study will be concentrated on the ground state for a spherically\nsymmetric potential.\nWe first consider the case of $\\lambda_3=0$, using a variational\nprocedure, with a trial Gaussian\nwave-function for $\\psi (\\vec{r})$, normalized to $N$, given by\n\\begin{equation}\n\\psi_{var}(\\vec{r})=\\sqrt{N}\\left(\\frac{1}{\\pi\\alpha^{2}}\\frac{m\\omega\n}{\\hbar }\\right)^{{D}/{4}}\n\\exp {\\left[ -\\frac{r^{2}}{2\\alpha^{2}}\\left(\n\\frac{\nm\\omega }{\\hbar }\\right) \\right] }, \\label{varwf}\n\\end{equation}\nwhere $\\alpha $ is a dimensionless variational parameter. \nFrom eq.~(\\ref{Etot}), the corresponding expression for the total\nvariational energy can be expressed as \n\\begin{eqnarray} \nE_{var}&=&\\hbar \\omega \\frac{N}{\\nu}\n{\\cal E}_{var},\\label{EvarND}\\\\\n{\\cal E}_{var}&\\equiv& \\nu\n\\left( \\frac{D}{4\\alpha^2}+\\frac{D\\alpha^2}{4}\\right)-\n\\frac{\\nu^2\\Omega_D}{4(2\\pi)^{D/2}\\alpha^D}+ \n\\frac{G_3}{6\\pi^D}\n\\frac{\\nu^3\\Omega_D^2}{3^{D/2}\\alpha^{2D}}\n,\\nonumber\\\\\n\\end{eqnarray} \nwhere $\\Omega_D$ is the solid angle in $D$ dimensions,\n\\begin{eqnarray}\n&&\\Omega_D\\equiv\\frac{2\\pi^{D/2}}{\\Gamma (D/2)},\n\\;\\;\\;\\;\nG_3\\equiv\\frac{\\lambda_3}{2(\\lambda_2)^2}\n\\hbar \\omega \\;,\n\\nonumber\\\\\n&&{\\rm and} \\;\\;\\;\n\\nu\\equiv - \\frac{N}{\\Omega_D}\\frac{2\\lambda_2}{\\hbar\\omega}\n\\left(\\frac{m\\omega}{\\hbar}\\right)^{D/2}.\n\\end{eqnarray} \nBy using dimensionless variables, \n$\\vec{x}\\equiv \\sqrt{{m\\omega }/{\\hbar }}\\; \\vec{r}$, \nwe redefine the wave-function $\\psi$ as \n\\begin{equation}\n\\phi(\\vec{x})\\equiv\n\\sqrt{\\frac{2|\\lambda_2|}{\\hbar\\omega}} \\psi(\\vec{r}),\n\\label{wf}\n\\end{equation} \nsuch that\n\\begin{equation}\n\\int |\\phi (\\vec{x})|^2 d^{D}\\vec{x} = \nN\\left[\\frac{2|\\lambda_2|}{\\hbar\\omega}\\right]\n\\left(\\frac{m\\omega}{\\hbar }\\right)^{D/2} \n= \\nu\\Omega_D.\n\\label{norm}\n\\end{equation}\nThe dimensionless equation corresponding to \neq.~(\\ref{1}), can be rewritten as \n\\begin{equation}\n\\left[ \\left(- \\sum_1^{D}\n\\frac{d^2}{dx_i^2}+x_i^2\\right)-|\\phi|^{2}\n+G_{3}|\\phi|^{4} -2 \\beta \\right] \\phi= 0 ,\n\\label{schd}\n\\end{equation}\nwhere $\\beta\\equiv\\mu/(\\hbar\\omega)$ is the dimensionless chemical \npotential.\nFrom eqs.~(\\ref{wf}) and (\\ref{varwf}), the trial wave-function can \nbe written as\n\\begin{equation}\n\\phi_{var}(x)\\equiv\n\\sqrt{\\nu\\Omega_D}\\left(\\frac{1}{\\pi\\alpha^2}\\right)^{D/4}\n\\exp\\left(-\\frac{x^2}{2\\alpha^2}\\right),\n\\label{wfv}\n\\end{equation}\n\nThe variational results, obtained by using the above expressions can be\nextended analytically to non-integer values of the dimension $D$. \nMinimization of the energy [eq. (\\ref{EvarND})], with respect to\n$\\alpha^2$, is done numerically by sweeping over $\\alpha^2$ values.\nThe results for the energy and the chemical potential are shown in Fig. 1.\nFor each value of $D$, one can observe a critical\nnumber of atoms, $N_c$, related to the critical parameter $\\nu_c$, only\nwhen $D\\le 2$. This critical limit corresponds to the cusps in the upper\nplot of Fig.1 and is also observed using exact numerical calculation for\n$D=$3.\nIt is also interesting to note that for $D>2$ there are two\nbranches of solutions for ${\\cal E}_{var}$ and $\\beta$,\none stable and the other unstable. \nIn the energy, the lower branch corresponds to stable solutions\n(minima), while the upper one gives unstable solutions (maxima).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIG.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[tbp]\n\\setlength{\\epsfxsize}{0.9\\hsize} \n\\setlength{\\epsfysize}{0.8\\hsize} \n\\centerline{\\epsfbox{ND.eps}} \n\\caption{\nThe variational energy ${\\cal E}$ (upper part) and chemical\npotential $\\beta$ (lower part), as functions of the reduced \nnumber of atoms $\\nu$, for several values of the dimension\n$D$, indicated in each plot. All the quantities are in \ndimensionless units, as defined in the text.\n}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe case with $D=2$ is particularly interesting, as no\nunstable solutions exist and there are stable solutions only for\n$\\nu < 2$, such that $\\nu_c =2$.\nFor $D=2$, the minimization of eq.~(\\ref{EvarND}) in respect to \n$\\alpha^2$ leads to \n\\begin{equation}\n{\\cal E}_{var}=\\nu\\sqrt{1-\\frac{\\nu}{2}}.\n\\end{equation}\nThe behavior of $\\nu$, and the corresponding critical limits, as one\nalters the dimension $D$, has other curious particular results. For\nexample, the critical limit $\\nu_c$ has a minimum for $D=3$ ($\\nu_c^{(D)}\n\\ge \\nu_c^{(3)}$ for all $D$).\n\nIn conclusion of this part of our work, considering arbitrary \n$D$ with $\\lambda_3=0$, there are no stable solutions for eq.~(\\ref{1}),\nif the wave-function $\\phi(x)$, given by eq.~(\\ref{norm}), is \nnormalized to $\\nu > \\nu_c$. Fig. 1 shows that this restriction is \nstrongest for $D=3$: $\\nu_c$ is a minimum when compared with \n$\\nu_c$ for $D\\ne 3$.\nThis is a relevant result, considering that $\\nu$ is directly \nproportional to the number of atoms. \nAlso, it is observed that $\\nu_c$ increases very fast for $D>3$.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIG. 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\begin{figure}[tbp]\n\\setlength{\\epsfxsize}{1.0\\hsize} \n\\setlength{\\epsfysize}{1.0\\hsize} \n\\centerline{\\epsfbox{bec2d.eps}}\n\\caption{For $D=2$, we present the variational and exact numerical\ncalculations of the chemical potential ($\\beta$), total energy (${\\cal\nE}$), mean-square-radius ($\\langle x^2\\rangle$), and central density\n($|\\phi(0)|^2$), as a function of the reduced number of atoms $\\nu$.\nAll the quantities are in dimensionless units (see text).\nThe solid line curves correspond to exact numerical results, while the\ndashed curves are the variational results.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nNext, we also solve equation (\\ref{schd}) exactly \nemploying the shooting and Runge-Kutta methods, and compare the results \nwith the corresponding variational ones. \nIn this case, we consider only the particular \ninteresting case of $D=2$, with $\\lambda_3=0$. The results are shown in \nFig. 2, for \nthe chemical potential $\\beta$, the total energy ${\\cal{E}}$,\nmean-square-radius $\\langle x^2\\rangle$,\nand the central density $|\\phi(0)|^2$. \nIn order to numerically solve eq.~(\\ref{schd}), in the $s-$wave,\nwe first write it in terms of the single variable \n$x\\equiv\\sqrt{(x_1^2+x_2^2)}$ and consider the following \nboundary conditions: \n$\\phi^\\prime(0)=0$ (where $\\prime$ stands for the derivative with \nrespect to $x$)\nand $\\phi (x)$ $\\to C\\exp (-x^{2}/2+[\\beta -{1}]\\ln (x))$ when\n$x\\to \\infty$, where $C$ is a constant to be determined.\nAs observed in Fig.2, the critical limit $\\nu_c=2$ obtained \nanalytically using the variational approach should be compared with\n$\\nu_c=1.862$, obtained by exact numerical calculation. This critical\nlimit was first obtained by Weinstein~\\cite{weinstein}, in a non-linear\napproach with two-body term, without the trapping potential.\nThe coincidence of the value with our exact calculation is due to the \nfact that at the critical limit the mean square radius goes to zero.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIG. 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[tbp]\n\\setlength{\\epsfxsize}{0.9\\hsize} \n\\setlength{\\epsfysize}{0.8\\hsize} \n\\centerline{\\epsfbox{bec2dg3.eps}} \n\\caption{\nExact numerical solutions for $D=$2, of the chemical potential,\n$\\beta$, in dimensionless units, given as function of the reduced\nnumber of atoms $\\nu$, for different values of the three-body\nparameter $G_3$, when the space dimension is $D=$2. \nAs shown, only when $G_3\\le 0$ the number of\natoms is limited to certain critical number.} \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIG. 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[tbp]\n\\setlength{\\epsfxsize}{0.9\\hsize} \n\\setlength{\\epsfxsize}{0.8\\hsize} \n\\centerline{\\epsfbox{var2d.eps}} \n\\vskip 0.5cm\n\\caption{\nThe variational solutions for $D=$2, of the chemical\npotential, corresponding to the exact results given in Fig. 3.} \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe have also compared the results obtained by the variational approach\nwith the exact numerical one, in the case of $D=2$, for several\nvalues\nof the three-body interaction term (positive and negative), as shown in\nFigs. 3 and 4. In Fig. 3 we have the exact numerical approach and \nin Fig. 4 we have the corresponding variational results.\nBy comparing the results we have for $D=2$ (shown in Figs. 3 and 4) with\nthe ones obtained in ref.~\\cite{GFT} for $D=3$, we should observe that no \nfirst order phase-transition exists in two dimensions. As observed in \nrefs.~\\cite{ADV,GFTC}, for $D=3$, a first-order phase-transition\ncan occur in trapped condensed states with negative two-body\nscattering length, when a repulsive three-body (quintic) term is added\nin the Hamiltonian. As shown in Figs. 3 and 4, with $G_3$ positive the\nrange of stability for the number of atoms $N$ can be increased\nindefinitely; with $G_3$ negative this range is reduced.\n\n We can analyze the collapse conditions using ``the virial theorem\"\napproach \\cite{Pit1}. The mean square radius, $\\langle r^2\\rangle$, of a\n$D-$dimensional condensate, is given by \n\\begin{eqnarray}\n&&\\frac{d^2\\langle r^2\\rangle }{dt^2} + 4\\omega^2 \\langle r^2\\rangle = \n\\label{pit}\\\\\n&&\\frac{1}{m}\\left[{4 \\langle H\\rangle } +\n{\\lambda_2}(D-2) \n\\langle |\\psi|^2 \\rangle + \n\\frac{4\\lambda_3}{3}(D-1)\n\\langle |\\psi|^4 \\rangle \\right], \n\\nonumber\\end{eqnarray}\nwhere \n\\begin{equation}\n\\langle {\\cal O}\\rangle\\equiv \\frac{1}{N}\\int d^D\\vec{r}\n\\psi^\\dagger(\\vec{r},t)\n{\\cal O} \\psi(\\vec{r},t)\n\\end{equation}\nand $\\langle H\\rangle = E/N$. When $\\lambda_3 = 0$ we obtain the equation\nderived in \\cite{Pit1}. \n\nWe can also write the eq.~(\\ref{pit}) in dimensionless units, as\nit was done in eqs.~(\\ref{wf}-\\ref{schd}):\n\\begin{eqnarray}\n&&\\frac{d^2\\langle x^2\\rangle }{d\\tau^2} \n+ 4\\langle x^2\\rangle = \\frac{4 {\\cal E} }{\\nu} + 2f(\\tau),\n\\label{pitx}\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nf(\\tau)\\equiv\n\\frac{\\lambda_2}{|\\lambda_2|}\\frac{D-2}{4} \n\\langle |\\phi|^2 \\rangle + \nG_3\\frac{D-1}{3} \n\\langle |\\phi|^4 \\rangle . \n\\label{pitx2}\\end{eqnarray}\nUsing the initial conditions for $\\langle x^2 \\rangle$ and \n$d\\langle x^2 \\rangle/d\\tau$, where, for simplicity, \nwe assume ${d\\langle x^2\\rangle }/{d\\tau} = 0$, \nthe solution of eq.~(\\ref{pitx}) is given by\n\\begin{eqnarray}\n\\langle x^2 \\rangle &=& \\frac{{\\cal E}}{\\nu} + \n\\left[\\left.\\langle x^2\\rangle\\right|_0 -\n\\frac{{\\cal E}}{\\nu}\\right]\\cos(2\\tau) \n\\nonumber\\\\\n&+&\\int_{0}^{\\tau}f(\\tau')\\sin(2(\\tau-\\tau'))d\\tau'.\n\\label{x}\\end{eqnarray} \n\nThe stability regions and the estimates for the collapse time can be\nobtained from the analysis of this solution, like as performed for the\ncase $\\lambda_3 = 0$ in \\cite{Wad2}.\nLet us analyze the dynamics when $D=2$. \nIn this case, $\\lambda_2$ does not appear explicitly in \n$\\langle x^2\\rangle$ and $f(\\tau)$ also does not depend on this \nparameter:\n\\begin{enumerate}\n\\item For a positive $G_3$, negative $\\lambda_2$ and ${\\cal E}>0$ we\nobserve that $\\langle x^2\\rangle$ cannot be zero and the condensate is\nstable. The mean square radius of the condensate oscillates in time around\na finite value. This is confirmed by the numerical simulations\n(see Figs. 3 and 4).\n\\item For a negative $G_3$, positive $\\lambda_2$ an analysis of\nstability like the one performed in ref.~\\cite{Wad2} shows that\n\\newline\n{\\bf a)} \\ When the total energy ${\\cal E}<0$, the condensate is unstable\nand the wavefields collapse in a finite time at any initial conditions;\n\\newline\n{\\bf b)} \\ When ${\\cal E}>0$, as the function $f(\\tau)$ is negative, the \ncontribution of the integral term for $\\tau<\\pi$ is negative. \nThen, we found the collapse condition as \n\\begin{equation}\n\\langle x^2\\rangle|_0 \\ge 2\\frac{\\cal E}{\\nu}.\n\\end{equation} \n\\end{enumerate}\nThe same kind of analysis, for $D>2$, is much involved in the present\napproach, as the sign of the function $f(\\tau)$ is not fixed at opposite\nsigns for the parameters $\\lambda_2$ and $\\lambda_3$.\n\nSome information about the dynamics of the collapse can also be obtained\nby using the techniques based on integral inequalities~\\cite{weinstein,Tur}.\nFor instance, when $D=2$, we can estimate the three-body term \ncontribution in $E$, following the procedure given in \\cite{weinstein}\n\\begin{equation}\n\\int d^2\\vec{r} |\\psi|^6 \\leq C_2 \\left(\\int d^2\\vec{r}\n\\frac{|\\nabla\\psi|^2}{2m} \\right)^2\n\\left(\\int d^2\\vec{r}|\\psi|^2\\right) = C_2 K^2 N,\n\\end{equation}\nwhere $K$ is the kinetic energy and $C_2$ is defined from the \nminimization of the functional\n\\begin{equation}\n{\\cal J} = \\frac{\n\\left(\\int d^2\\vec{r}|\\nabla\\psi|^2\\right)^2\n\\left(\\int d^2\\vec{r}|\\psi|^2 \\right) \n}\n{\\int d^2\\vec{r} |\\psi|^6}.\n\\end{equation}\nCombining with the corresponding estimate for $\\int d^2\\vec{r}|\\psi|^4$, \nwe obtain $E > E(K)$, where\n\\begin{equation}\nE(K) = K + \\frac{\\omega^2N^2}{4K} + \\frac{\\lambda_2}{2} C_1 N K +\n\\frac{\\lambda_3}{3} C_2 K^2 N.\n\\label{EK}\n\\end{equation}\nWhen $\\lambda_3 = 0$ we get the equation derived in \\cite{Pit1}. \nEquation (\\ref{EK}) should be compared with the corresponding \nvariational expression (\\ref{EvarND}), where \nthe kinetic energy is given by $K = N\\hbar\\omega/(2\\alpha^2)$ and \n$\\alpha$ is the width of the cloud. As we see, the expression for the\nenergy (\\ref{EK}) is very similar to the obtained by the variational\napproach. However, (\\ref{EK}) is valid for arbitrary time and\ndescribes the nonstationary dynamics.\nBy using the variational expression (upper limit) for the\nground-state, and the right-hand-side of eq. (\\ref{EK}) (lower limit), \nwe can approach analytically the exact solution for the total energy\n\\begin{equation} \nE(K) < E < E_{var}.\n\\end{equation} \nFor a more deep insight to the problem of stability, we need to obtain\nthe values of the constants $C_1$ and $C_2$.\nThis problem requires a generalization of the method suggested by\nWeinstein in \\cite{weinstein}, to be considered in a future work.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIG. 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[tbp]\n\\setlength{\\epsfxsize}{0.9\\hsize}\n\\setlength{\\epsfysize}{0.8\\hsize} \n\\centerline{\\epsfbox{var1d.eps}} \n\\vskip 0.5cm\n\\caption{\nVariational solutions for the chemical potential\n(in dimensionless units) as functions of $\\nu$, for $D=1$ and\ndifferent values of the three-body parameter $G_3$.} \n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIG. 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[tbp]\n\\setlength{\\epsfxsize}{0.9\\hsize} \n\\setlength{\\epsfysize}{0.8\\hsize} \n\\centerline{\\epsfbox{var4d.eps}} \n\\vskip 0.3cm\n\\caption{\nThe same as in Fig. 5, for $D=4$.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\nWe should observe that exact numerical results, when $G_3=0$, have already\nbeen considered in refs. \\cite{rup95} (for $D=1$ and $D=3$),\nin \\cite{GFT} (for $D=$ 3), and in \\cite{2D} (for $D=$2). \n In \\cite{ADV,GFT,GFTC}, for $D=3$, \nit was also considered the case with $G_3\\ne 0$, and shown a kind\nof first-order phase-transition in the condensate. In the present work,\nwe have extended the variational formalism, in case $G_3\\ne 0$, for an\narbitrary $D-$dimension. In the following Figs. 5, 6 and 7, we present\nour results for the chemical potential as a function of $\\nu$, for a\nset of given values of $G_3$, in case of $D =$ 1, 4 and 5. As one \ncan observe in Fig. 5, even in case of $D=1$ one can reach a critical \nmaximum limit for $\\nu$, when $G_3$ is enough negative.\nFor $D =$ 4 and 5 (Figs. 6 and 7), we observe similar picture of\nfirst-order phase-transition occurring for some specific values of $G_3$.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FIG. 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[tbp]\n\\setlength{\\epsfxsize}{0.9\\hsize} \n\\setlength{\\epsfysize}{0.8\\hsize} \n\\centerline{\\epsfbox{var5d.eps}} \n\\caption{\nThe same as in Fig. 5, for $D=5$.}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn conclusion, in the present work, we first studied the \nstability and the critical number of atoms in arbitrary $D-$dimensions\nusing a variational procedure, for the case we have two-body\n(attractive) and three-body contributions.\nThis part extends a previous analysis done in refs.~\\cite{TW,Wad2}.\nNext, we considered in more detail the case $D=2$.. We \ncompared the variational results with exact numerical calculations\nfor the chemical potential, total energy, mean-square-radius and density. \nFinally, we extended numerically the approach for $D=2$, including an\neffective three-body interaction term.\nWe studied the sensibility of the critical numbers with\nrespect to corrections in the non-linear interaction.\nThe effective interaction considered in the equation contains a\ntrapped harmonic interaction, and two nonlinear terms, proportional to the\ndensity $|\\psi|^2$ (due to first-order two-body interaction) and to \n$|\\psi|^4$ (due to first-order three-body interaction). \nWe also verified, by a variational procedure, that a critical number of\nparticles exists only for $D\\ge 2$, when the nonlinear term of the \nNLSE contains just the cubic term. In case of $D=1$, a critical\nmaximum number of atoms can exist with the addition of a negative\nquintic term (three-body) in the NLSE. In all cases where the number of\natoms is limited, we\nobserved that the addition of a positive $|\\psi|^4$ allows \nstable solutions beyond the critical number.\nWe also introduced an analysis of the collapse conditions, using ``the\nvirial theorem\" approach given in \\cite{Pit1}. \nThe dynamics of the collapse was discussed in terms of the techniques\ndeveloped in \\cite{weinstein}. In particular, we showed how the\nexact energy can be approached in the case of $D=2$ with two and\nthree-body term contributions.\n\n{\\bf Acknowledgments} We are grateful to Jordan M. Gerton for the\nsuggestions and careful reading of the manuscript. \nThis work was partially supported by Funda\\c c\\~ao\nde Amparo \\`a Pesquisa do Estado de S\\~ao Paulo and Conselho Nacional de\nDesenvolvimento Cient\\'\\i fico e Tecnol\\'ogico.\n\n\\begin{references}\n\n\\bibitem{exp} \nM.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, \nE.A. Cornell, Science 269 (1995) 198;\nC.C. Bradley, C.A. Sackett, J.J. Tollet, R.G. Hulet, Phys.Rev. Lett. \n75 (1995) 1687;\nK.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, \nD.S.Durfee, D.M. Kurn, and W.Ketterle, Phys. Rev. Lett. {\\bf 75} 3969 \n(1995); \nD.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, A.C. Moss,\nT.J. Greytak, and D. Kleppner, Phys. Rev. Lett. {\\bf 81}, 3811 (1998).\n\n\\bibitem{gin} V.L. Ginzburg and L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 34\n(1958) 1240 [Sov. Phys. JETP 7 (1958) 858]; \nL.P. Pitaevskii, Sov. Phys. JETP 13 (1961) 451; \nE.P. Gross, J. Math. Phys. 4, 195 (1963).\n\n\\bibitem{WT} M. Wadati and T. Tsurumi, Phys. Lett. A 247 (1998) 287.\n\n\\bibitem{ADV} N. Akhmediev, M.P. Das, and A.V. Vagov, in\n``Condensed Matter Theories'', vol. 12, p. 17-25,\nedited by J.W. Clark and P.V. Panat (Nova Science Publ., New York, 1997).\n\n\\bibitem{GFT}\nA. Gammal, T. Frederico, and L. Tomio, Phys. Rev. E 60 (1999) 2421;\nA. Gammal, T. Frederico, and L. Tomio, in ``Collective Excitations in\nFermi and Bose Systems\", p. 159-168, edited by C. Bertulani, L.F. Canto\nand M. Hussein (World Scientific, Singapore, 1999).\n\n\\bibitem{GFTC}\nA. Gammal, T. Frederico, L. Tomio, and Ph. Chomaz, \n{\\it Liquid-Gas phase transition in Bose-Einstein Condensates with\ntime evolution}, submitted for publication.\n\n\\bibitem{achange} S.Inouye, M.R. Andrews, J. Stenger, H.J. Miesner,\nD.M. Stamper-Kurn, and W. Ketterle, Nature 392 (1998) 151. \n\n\\bibitem{efimov} V. Efimov, Phys. Lett. B 33 (1970) 563.\n\n\\bibitem{zakharov} V.E. Zakharov, Sov. Phys. JETP 35 (1972) 908;\nV.E. Zakharov, V.S. Synakh, Sov. Phys. JETP 41 (1975) 465.\n\n\\bibitem{weinstein} M.I. Weinstein, Commun. Math. Phys. 87 (1983)\n567.\n\n\\bibitem{TW} T. Tsurumi and M. Wadati, J. of Phys. Soc. of Japan\n68 (1999) 1531; L. Berg\\'e, T.J. Alexander, and Y.S. Kivshar,\ne-print: cond-mat/9907408.\n\n\\bibitem{Pit1}\nL.P. Pitaevskii, Phys.Lett. A 221 (1996) 14.\n\n\\bibitem{baym96} G. Baym and C.J. Pethick, Phys. Rev. Lett. 76 \n(1996) 6.\n\n\\bibitem{rup95} M. Edwards and K. Burnett, Phys. Rev. A 51\n(1995) 1382; P.A. Ruprecht, M.J. Holland, K. Burnett, and M. Edwards,\nPhys. Rev. A 51 (1995) 4704.\n\n\\bibitem{Wad2}\nT.Tsurumi, H. Morise, and M. Wadati, e-print: cond-mat/9912470.\n\n\\bibitem{Tur}\nS.K. Turitsyn, Phys.Rev. E 47 (1993) R13.\n\n\\bibitem{2D} S.K. Adhikari, Phys. Lett. A 265 (2000) 91.\n\n\\end{references}\n\\end{multicols}\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002042.extracted_bib",
"string": "\\bibitem{exp} \nM.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, \nE.A. Cornell, Science 269 (1995) 198;\nC.C. Bradley, C.A. Sackett, J.J. Tollet, R.G. Hulet, Phys.Rev. Lett. \n75 (1995) 1687;\nK.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, \nD.S.Durfee, D.M. Kurn, and W.Ketterle, Phys. Rev. Lett. {\\bf 75} 3969 \n(1995); \nD.G. Fried, T.C. Killian, L. Willmann, D. Landhuis, A.C. Moss,\nT.J. Greytak, and D. Kleppner, Phys. Rev. Lett. {\\bf 81}, 3811 (1998).\n\n\n\\bibitem{gin} V.L. Ginzburg and L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 34\n(1958) 1240 [Sov. Phys. JETP 7 (1958) 858]; \nL.P. Pitaevskii, Sov. Phys. JETP 13 (1961) 451; \nE.P. Gross, J. Math. Phys. 4, 195 (1963).\n\n\n\\bibitem{WT} M. Wadati and T. Tsurumi, Phys. Lett. A 247 (1998) 287.\n\n\n\\bibitem{ADV} N. Akhmediev, M.P. Das, and A.V. Vagov, in\n``Condensed Matter Theories'', vol. 12, p. 17-25,\nedited by J.W. Clark and P.V. Panat (Nova Science Publ., New York, 1997).\n\n\n\\bibitem{GFT}\nA. Gammal, T. Frederico, and L. Tomio, Phys. Rev. E 60 (1999) 2421;\nA. Gammal, T. Frederico, and L. Tomio, in ``Collective Excitations in\nFermi and Bose Systems\", p. 159-168, edited by C. Bertulani, L.F. Canto\nand M. Hussein (World Scientific, Singapore, 1999).\n\n\n\\bibitem{GFTC}\nA. Gammal, T. Frederico, L. Tomio, and Ph. Chomaz, \n{\\it Liquid-Gas phase transition in Bose-Einstein Condensates with\ntime evolution}, submitted for publication.\n\n\n\\bibitem{achange} S.Inouye, M.R. Andrews, J. Stenger, H.J. Miesner,\nD.M. Stamper-Kurn, and W. Ketterle, Nature 392 (1998) 151. \n\n\n\\bibitem{efimov} V. Efimov, Phys. Lett. B 33 (1970) 563.\n\n\n\\bibitem{zakharov} V.E. Zakharov, Sov. Phys. JETP 35 (1972) 908;\nV.E. Zakharov, V.S. Synakh, Sov. Phys. JETP 41 (1975) 465.\n\n\n\\bibitem{weinstein} M.I. Weinstein, Commun. Math. Phys. 87 (1983)\n567.\n\n\n\\bibitem{TW} T. Tsurumi and M. Wadati, J. of Phys. Soc. of Japan\n68 (1999) 1531; L. Berg\\'e, T.J. Alexander, and Y.S. Kivshar,\ne-print: cond-mat/9907408.\n\n\n\\bibitem{Pit1}\nL.P. Pitaevskii, Phys.Lett. A 221 (1996) 14.\n\n\n\\bibitem{baym96} G. Baym and C.J. Pethick, Phys. Rev. Lett. 76 \n(1996) 6.\n\n\n\\bibitem{rup95} M. Edwards and K. Burnett, Phys. Rev. A 51\n(1995) 1382; P.A. Ruprecht, M.J. Holland, K. Burnett, and M. Edwards,\nPhys. Rev. A 51 (1995) 4704.\n\n\n\\bibitem{Wad2}\nT.Tsurumi, H. Morise, and M. Wadati, e-print: cond-mat/9912470.\n\n\n\\bibitem{Tur}\nS.K. Turitsyn, Phys.Rev. E 47 (1993) R13.\n\n\n\\bibitem{2D} S.K. Adhikari, Phys. Lett. A 265 (2000) 91.\n\n"
}
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cond-mat0002043
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Probing Pseudogap by Josephson Tunneling
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[
{
"author": "Ivar Martin and Alexander Balatsky"
}
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We propose here an experiment aimed to determine whether there are superconducting pairing fluctuations in the pseudogap regime of the high-$T_c$ materials. In the experimental setup, two samples above $T_c$ are brought into contact at a single point and the differential AC conductivity in the presence of a constant applied bias voltage between the samples, $V$, should be measured. We argue the the pairing fluctuations will produce randomly fluctuating Josephson current with zero mean, however the current-current correlator will have a characteristic frequency given by Josephson frequency $\omega_J = 2 e V /\hbar$. We predict that the differential AC conductivity should have a peak at the Josephson frequency with the width determined by the phase fluctuations time.
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[
{
"name": "jos5.tex",
"string": "\\documentstyle[aps,prl,multicol,fancyheadings,epsfig,graphicx,amsbsy,amssymb,amstex]{revtex}\n\\def\\bbbone{{\\mathchoice {\\rm 1\\mskip-4mu l} {\\rm 1\\mskip-4mu l}\n{\\rm 1\\mskip-4.5mu l} {\\rm 1\\mskip-5mu l}}}\n\\def\\bbbc{{\\mathchoice {\\setbox0=\\hbox{$\\displaystyle\\rm C$}\\hbox{\\hbox\nto0pt{\\kern0.4\\wd0\\vrule height0.9\\ht0\\hss}\\box0}}\n{\\setbox0=\\hbox{$\\textstyle\\rm C$}\\hbox{\\hbox\nto0pt{\\kern0.4\\wd0\\vrule height0.9\\ht0\\hss}\\box0}}\n{\\setbox0=\\hbox{$\\scriptstyle\\rm C$}\\hbox{\\hbox\nto0pt{\\kern0.4\\wd0\\vrule height0.9\\ht0\\hss}\\box0}}\n{\\setbox0=\\hbox{$\\scriptscriptstyle\\rm C$}\\hbox{\\hbox\nto0pt{\\kern0.4\\wd0\\vrule height0.9\\ht0\\hss}\\box0}}}}\n\\pagestyle{fancy}\n\\pagestyle{fancyplain}\n\\footrulewidth 0.4pt\n\\plainheadrulewidth 0.4pt\n\\plainfootrulewidth 0.4pt\n\\lhead{\\large LA-UR-99-XXXX}\n\\chead{ \\today}\n\\rhead{\\sl submitted to PRL}\n\\cfoot{\\sc\\thepage} \n\\lfoot{}\n\\rfoot{}\n\n\\newcommand{\\beq}{\\begin{eqnarray}} \n\\newcommand{\\eeq}{\\end{eqnarray}} \n\\newcommand{\\tp}{{t^\\prime}} \n\\newcommand{\\tap}{{\\tau^\\prime}} \n\\newcommand{\\oJ}{\\omega_J} \n\\newcommand{\\Dp}{\\Delta \\phi} \n\\newcommand{\\nnn}{\\nonumber\\\\} \n\\newcommand{\\bk}{{\\bf k}}\n\\newcommand{\\bp}{{\\bf p}}\n\n\n\n\\begin{document}\n\\title{Probing Pseudogap by Josephson Tunneling}\n\\author{Ivar Martin and Alexander Balatsky}\n\\address{Theoretical Division, Los Alamos National Laboratory, \n\tLos Alamos, NM 87545}\n\n\\date{Received \\today }\n\n\\maketitle\n\n\\begin{abstract}\nWe propose here an experiment aimed to determine whether there are \n superconducting pairing fluctuations in the pseudogap regime of the high-$T_c$ materials.\nIn the experimental setup, two samples above $T_c$ are brought into \ncontact at a single point and the \ndifferential AC conductivity in the presence of a constant applied bias \nvoltage between the samples, $V$, should be measured. \nWe argue the the pairing fluctuations will produce randomly\n fluctuating Josephson current with \nzero mean, however the current-current correlator \n will have a characteristic frequency given\nby Josephson frequency $\\omega_J = 2 e V /\\hbar$. We predict that the \ndifferential AC conductivity should have a peak at the Josephson \nfrequency with the width determined \nby the phase fluctuations time.\n\\end{abstract}\n\\pacs{Pacs Numbers: XXXXXXXXX}\n\n\n\\vspace*{-0.4cm}\n\\begin{multicols}{2}\n\n\\columnseprule 0pt\n\n\\narrowtext\n\\vspace*{-0.5cm}\n\nOne of the long-standing puzzles of the high-temperature superconductivity \nis the nature of the so-called pseudogap regime. The pseudogap \nregime occurs in the wide range of temperatures above superconducting transition\ntemperature in the underdoped cuprate superconductors. \nIt is characterized by the suppressed quasiparticle density of \nstates in the vicinity of the Fermi level. The similarity of the density of states \nin the pseudogap regime and in the superconducting state \nlead many \\cite{kivelson} to believe that that the pseudogap \nitself is of a superconducting origin. In this view, the long range superconducting\norder in the pseudogap regime is destroyed by phase fluctuations. However, locally,\nboth electronic pairing and fluctuating regions of superconductivity should \npersist. \nTherefore, in this picture, the superconducting phase transition is\n believed to be a superconducting phase-ordering transition. Moreover, recent experiments by Orenstein\n and collaborators claim that local superfluid density is present even above \n $T_c$ in Bi2212 materials \\cite{Orenstein}.\n\nWhether or not the pseudogap indeed has a superconducting origin remains to be \nverified experimentally. Such standard experimental test as the vanishing \nresistivity,\nor the Meissner effect are bound to fail. Both these test require spatial and\ntemporal stability of the superconducting phase on the time scale of the experiment. \nOn the other hand, the pseudogap state can at best have superconducting order \nparameter that varies both in space and in time. \nA successful test may be \npossible if the fluctuations could somehow be stabilized by the proximity to a ``real'' \nsuperconductor \\cite{janko}. Another approach is to probe superconductivity locally \nin space on the time scales comparable or shorter than the characteristic \ntime of the phase fluctuations. \n\nIn this letter we propose the first experiment of \nthis type, which is based on the AC Josephson effect. \nThe main point we make is that in the presence of phase fluctuations \nJosephson current $j(t)$ across the tunneling contact is a random time-dependent quantity. It has a dispersion\nthat is the current-current correlator $\\langle j(t)j(t') \\rangle$,\n which is related to the corrections to the conductivity across the junction.\n This current-current correlator ``remembers'' about its Josephson origin and has a scale set by Josephson frequency\n and phase fluctuation time. Therefore, the conductivity of a junction \nwill have a correction\n due to current fluctuations,\n \\beq\\label{main1}\n \\Delta \\sigma \\propto 1/V , \n \\eeq\n where $V$ is the applied voltage across the junction. \n The crucial new aspect of the proposed \napproach is that we will focus on the characteristic time scale\n of frequency fluctuations, assuming that tunneling occurs\nin a small region where the spatial dependence can be ignored. \nWe will focus on the time dynamics of the phase fluctuations \nin our analysis.\n\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width = 3.0 in]{JJ_expt.eps}\n \\caption{Consider two parts of a superconductor brought into a tunneling contact\nat a temperature $T>T_c$, so that the phase of the order parameter is\nno longer stationary. Our approach is to consider the Josephson current (Eq. \n\\protect{\\ref{j1}}) as a randomly fluctuating quantity, with the three major sources of \nrandomness: the amplitude noise, the frequency noise, and the phase noise. \nWe also assume that the contact is small enough for the superconducting \nparameters to be spatially uniform in the vicinity of the contact.}\n \\label{fig1}\n \\end{center}\n\\end{figure}\n\n\nWhen two pieces of a superconductor are joined by a weak link, a superconducting \ncurrent begins to flow through the link in the absence of applied voltage \nbetween the superconductors. The current is related to the difference of the\nphases $\\phi_1$ and $\\phi_2$ of the superconductors,\n\\beq\\label{j1}\nj = j_0 \\sin(\\phi_1 - \\phi_2),\n\\eeq\nwith the parameter $j_0$ related to the coupling strength between the two superconductors. \nFor a superconductor in an equilibrium, evolution of the phase,\n\\beq\n{\\partial\\phi}/{\\partial t} = -2\\mu/\\hbar,\n\\eeq \nis determined\\cite{anderson} by the superconductor chemical potential, $\\mu$.\nTherefore, the phase as a function of time is \n\\mbox{$\\phi(t) = -2\\mu t/\\hbar + \\phi_0$}, \nwhere $\\phi_0$ is the phase at time $t = 0$. For two coupled superconductors \nthe phase difference evolves as\n\\beq\n\\phi_1(t) - \\phi_2(t) = 2(\\mu_2 - \\mu_1) t /\\hbar + \\phi_1(0) - \\phi_2(0).\n\\eeq\nThe difference of the chemical potentials equals the applied voltage, \n$\\mu_2 - \\mu_1 = e V$. Hence, if $V = 0$, both the phase difference and the current \ngiven by Eq.~(\\ref{j1}) remain constant. However, in the presence of a bias, $V$, \nthe phase difference grows linearly with time and the current oscillates according to \n\\beq\nj(t) = j_0 (\\omega_J) \\sin(\\omega_J t + \\Delta \\phi),\n\\eeq\nwith the frequency $\\omega_J = 2 e V/\\hbar$ and the initial phase \n$\\Delta \\phi = \\phi_1(0) - \\phi_2(0)$. The effect of generating an alternating \ncurrent by applying a constant bias to a superconducting tunnel junction is called \nAC Josephson effect. An important feature of this effect is that the frequency of\nthe generated current is only a function of the applied bias voltage and is \nindependent of the microscopic and macroscopic parameters of the system. The scale\nof the frequency is about 0.5 $\\Gamma$Hz per 1 microvolt.\nThe AC Josephson effect is routinely observed in the superconducting regime. It can\nbe observed either directly as the micro-wave emission from the oscillating current, or \nindirectly as ``Shapiro steps'' \\cite{shapiro} in the DC $I-V$ curves measured \nin the presence of the oscillating bias voltage component. \n\nFor AC Josephson effect\nto be observable in the pseudogap regime, the measurement has to be local both in space \nand time. \nSuppose that above the transition temperature, superconductor can be modeled as a \ncollection of superconducting islands of a characteristic size $L$, inside \nwhich phase fluctuates at a rate $\\Lambda$. Then, if the size of a contact between two \nsuperconductors is less than $L$ then the superconducting state is essentially uniform in the\nvicinity of the contact. If the applied bias voltage is such that the \nJosephson frequency is larger than $\\Lambda$ then the Josephson oscillations are \nfaster than the phase fluctuations dynamics, and hence can be approximately modeled as\n\\beq\\label{j2}\nj(t) = j^*(\\omega_J, \\Lambda, L) \\sin(\\omega_J t + \\Delta \\phi (t)),\n\\eeq\nwith the amplitude $j^*$ being renormalized by spatial and temporal fluctuations of the \nsuperconducting phase. In general, the Josephson frequency can also fluctuate around\nits average value due to voltage fluctuations which are particularly important for small\nsamples. However, we assume that these effects can be absorbed into the overall fluctuations\nof the phase, $\\Delta \\phi (t)$.\nIt is important to note that both $\\Lambda$ and $L$ are functions of \ntemperature, with L and $1/\\Lambda$ diverging at the superconducting transition.\nThe parameter $L$ is related to the phase gradient correlation function $W$ \nconsidered by Franz and Millis \\cite{millis}. Relationship between $\\Lambda$ and $L$ is \na subject of an active interest. In the vortex diffusion picture, where phase fluctuations\nare produced by moving vortices, $L$ is a distance \nvortex travels during the time $1/\\Lambda$, namely $L = \\sqrt{D/\\Lambda}$. Here $D$\nis the vortex diffusion constant. This corresponding dynamical critical exponent is \n$z = 2$. Alternatively, if the phase fluctuations are governed by fast ballistic \ndynamics, the relation between the parameters $L$ and $\\Lambda$ should be $L \\Lambda = v^*$,\nwhere $v^*$ is propagation speed for the ballistic modes. This corresponds to $z = 1$.\nUsing different geometries in the \nexperimental setup that we propose below may help to determine the relevant model.\n\n\nA possible experimental setup that can be used to perform the measurement of the \nAC Josephson effect in the pseudogap regime is shown in the figure \\ref{fig1}. \nThe crucial aspect is that the point of contact between the superconductors be as small as \npossible. If the size of the contact becomes larger than $L$ then\nin addition to the temporal phase fluctuations a the point of contact one needs to include \nspatial fluctuations, which can lead to a significant suppression of the effect.\nAnother desirable feature is that the two superconductors be only a few $ab$-planes thick.\nThis is because the size of the superconducting ``islands'' is likely to be more extended \nin the planes, compared to across the planes. Hence, we believe that the effect\nwe propose is more likely to be observed in the geometry of figure \\ref{fig1}, \nalthough c-axis tunneling may also yield similar results. Finally, using very thin samples\nreduces the transition temperature \\cite{1-layerTc}, thereby making the pseudogap regime \naccessible at lower temperatures, where the thermal fluctuations are reduced.\n\nThere are several ways the oscillating super-current in the pseudogap regime \ncan be detected. Here we consider two methods: 1) differential AC conductivity measurements\nin the presence of constant bias voltage, 2) detection of electro-magnetic radiation \ngenerated by the oscillating Josephson current. Although there is no coherent Josephson \ncurrent in the pseudogap regime, the junction is expected to have strong response \nto the perturbations acting at the frequencies near $\\omega_J$. Such super-current is also \nexpected to generate a radiation peak at the frequency $\\omega_J$, with a width of the peak \ngoverned by the phase fluctuations.\n\nTo make our qualitative arguments more formal we have to assign a particular form to the \nphase fluctuations. Here we make an assumption that the the phase difference \nbetween the superconductors, $\\Delta \\phi (t)$ follows a diffusion\nprocess, as shown in Fig.~\\ref{fig:diff}, with a variance \n\\beq\\label{eq:diff}\n\\langle (\\Delta \\phi_t - \\Delta \\phi_{t^\\prime} )^2\\rangle = 2 \\Lambda |t - t^\\prime|.\n\\eeq\nand the initial phase $\\Delta \\phi_0$ distributed uniformly in the interval \n$[0, 2\\pi]$. \nThe factor of 2 appears because for a weak tunnel junction the phases of on the both \nsides of the junction fluctuate independently, each at a rate $\\Lambda$.\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[width = 3.0 in]{JJ_bw.eps}\n \\caption{We assume here that the phase difference, $\\Dp_t$, between the superconductors forming the \nJosephson junction follows a one-dimensional geometrical Brownian walk as a function of time, $t$.\nSuch process is defined by a linearly increasing with time dispersion, Eq. (\\protect{\\ref{eq:diff}}).\n The initial phase, $\\Dp_0$, is\nalso a random quantity evenly distributed in the interval $[0, 2\\pi]$.}\n \\label{fig:diff}\n \\end{center}\n\\end{figure}\n\n\n\nViewing the Josephson current of Eq. (\\ref{j2}) as a random quantity with a zero mean,\n we can characterize it by its dispersion and autocorrelation. The autocorrelation, \naccording to the Kubo formula, determines the correction to the conductivity \ndue to the fluctuating Josephson tunneling, \n\\beq\n\\Delta \\sigma (\\omega) = \\frac {1} {\\omega\\nu}\\int^t _{-\\infty} {e ^{i \\omega(t - \\tp)}\n\t\\langle j (t) j(t^\\prime)\\rangle d\\tp},\n\\eeq\n% Mahan 3.8.8\nwhere brackets correspond to the $\\phi$-averaging, and averaging over time $t$ is implied. \nVolume $\\nu$ is necessary for normalization.\nSubstituting expression for the current from Eq. (\\ref{j2}), we obtain\n\\beq\n\\Delta \\sigma &=& \\frac {j^{*2}} {2\\omega\\nu} \\int^t _{-\\infty} \n\t{e ^{i \\omega(t-\\tp)}\n\t\\langle \\cos(\\oJ (t- \\tp) + \\Dp_t - \\Dp_\\tp) -}\\nnn\n\t&& \\cos(\\oJ (t + \\tp) + \\Dp_t + \\Dp_\\tp) \\rangle d\\tp. \n\\eeq\nSince $\\Dp_t + \\Dp_\\tp = (\\Dp_t - \\Dp_0) + (\\Dp_\\tp - \\Dp_0) + 2\\Dp_0$, after averaging\nover $\\Dp_0$ in the interval $[0, 2\\pi]$,\nthe second cosine disappears. To average over \n$(\\Dp_t - \\Dp_\\tp)$, we invoke the relation \n$\\langle \\exp(i u)\\rangle = \\exp(-\\langle u ^2\\rangle/2)$, valid for any \nnormally distributed variable $u$ with a mean zero. Then after the integration we obtain\n\\beq\n\\Delta \\sigma = \\frac {j^{*2}} {4 \\omega\\nu} \\left[\n\\frac{1}{\\Lambda + i\\omega - i\\oJ} + \\frac {1}{\\Lambda + i\\omega + i\\oJ}\n\\right].\n\\eeq\nAs expected, the the real part of the conductivity, \n\\beq\\label{Res}\n{\\rm Re}(\\Delta \\sigma) = \\frac {j^{*2} \\Lambda} {4 \\omega\\nu}\n\\left[ \\frac{1}{(\\omega - \\oJ)^2 + \\Lambda^2} + \n\t\\frac{1}{(\\omega + \\oJ)^2 + \\Lambda^2}\\right],\n\\eeq\nwhich corresponds to the in-phase response, has two peaks near $\\pm\\omega_J$. The divergence \nas $\\omega \\rightarrow 0$ has no physical meaning, since no superconductivity related \nresponse is expected on the time scales larger than the characteristic phase fluctuation\ntime. This translates into the condition $\\omega \\gtrsim \\oJ$ for validity of Eq. (\\ref{Res}). \nThe imaginary part of the conductivity in the vicinity of $\\oJ$ is about two times smaller\nthan the real part, and hence can be neglected in the total conductivity \n\\mbox{${\\rm Abs}\\ \\sigma= \\sqrt{({\\rm Re}\\ \\sigma)^2 + ({\\rm Im}\\ \\sigma)^2 }$}. \n\n\nTherefore, we predict that if the pseudogap regime is superconducting in origin there should be a peak in \nthe differential AC conductivity at the frequency $\\oJ \\propto V$, with the peak value\nthat scales as shown in Eq. (\\ref{main1}): \n\\beq\\label{main}\n\\Delta \\sigma \\propto j^{*2}/\\Lambda\\oJ \\propto 1/V .\n\\eeq\nThis is the main result of this paper.\nWhile this correction may be small relative to\nthe normal (single-electron) current component, it can be extracted from the background conductivity\ndue to its extremely high sensitivity an applied external magnetic field.\nAs is evident, the magnitude of the correction is inversely proportional to the phase-breaking rate, \n$\\Lambda$, and as a consequence should be more easily observable at temperatures close to the superconducting \ntransition. Consequently, a possible experimental approach is to start arbitrarily close to $T_c$ \nand to measure the microwave radiation from the weakly dephased Josephson current, and or to measure the \ndifferential AC conductivity as proposed above. Then, gradually incrementing the temperature one can probe \nhow the spatial and temporal fluctuations of the order parameter phase grow with the temperature.\n\nIn fact, a similar fluctuational AC Josephson effect can be searched for even in \nthe conventional, ``low-$T_c$,'' \nsuperconductors, in the so-called paraconductivity regime \\cite{AL}. The paraconductivity regime\nis characterized by superconducting order parameter fluctuations above $T_c$, and experimentally \nis associated with the rapidly decreasing (but finite) resistivity in the vicinity of $T_c$. \nUsing the experimental setup we propose here, one could attempt to study the dynamics of \nthe dephasing timescales in the close proximity of $T_c$ in the paraconductivity regime.\nThe difference between the conventional paraconductivity effect and the pseudogap is that the pseudogap \nis believed to extend far beyond the paraconductivity range where the rapid changes in the material \nresistivity occur.\n\n\n\nLet us examine now more closely the assumptions that lead to the expression for the \nJosephson current, Eq.~(\\ref{j2}). Within the standard theory \\cite{mahan}, the \nJosephson current is \n\\beq\\label{jFF}\nj(t) = 2 e {\\rm Im} [e^{-i \\oJ t } \\Phi_{\\rm ret}(eV)],\n\\eeq\nwhere the retarded correlation function $\\Phi_{\\rm ret}(eV)$ can be obtained from the \nMatsubara correlation function \n\\beq\n\\Phi(i\\omega) = 2 \\sum_{\\bk\\bp} T_{\\bk, \\bp} T_{-\\bk, -\\bp} \\int_{\\tau - \\beta}^\\tau {d\\tap \n\tF^\\dagger(\\bk, \\tau, \\tap) F(\\bp, \\tap, \\tau)},\\nonumber\n\\eeq\nvia analytical continuation $i \\omega \\rightarrow eV + i0$. Here $T_{\\bk, \\bp}$ is a matrix \nelement for tunneling from a state $\\bk$ on one side of the\njunction into a state $\\bp$ on the other side of the junction, and \n$F(\\bk, \\tau, \\tap) = \\langle T_\\tau c_{\\bk\\uparrow (\\tau)} c_{-\\bk\\downarrow}(\\tap) \\rangle$\nis an anomalous time-ordered Green functions. \nIn Eq. (\\ref{jFF}), we do not include the regular single electron contribution, proportional \nto $G(\\bk, \\tau, \\tap) G(\\bp, \\tap, \\tau)$. The reason is that it does not carry the \nsuperconducting phase information and, therefore, does not produce the resonant features away from zero \nfrequency.\nIn the absence of phase fluctuations $F$ is \nonly a function of the time difference, \n$F^0(\\bk, \\tau-\\tap) = (n_F(-E_\\bk) e^{-E_{\\bk}|\\tau-\\tap|} - \nn_F(E_\\bk) e^{E_{\\bk}|\\tau - \\tap|})/2E_\\bk$.\nPhase fluctuations can be incorporated phenomenologically into the anomalous Green functions as phase\nfactors\n\\beq\nF(\\bk, \\tau, \\tap) \\rightarrow F(\\bk, \\tau, \\tap) e^{i\\phi(\\tau, \\tap)}.\n\\eeq\nThe form of the function $\\phi(\\tau, \\tap)$ depends on the \nmodel of the phase fluctuations. Here we assume that \n\\beq\\label{phi12}\n\\phi(\\tau, \\tap) = \\phi(\\tau) + \\phi^\\prime(\\tau - \\tap),\n\\eeq\nwhere $\\phi(\\tau)$ and $\\phi^\\prime(\\tau - \\tap)$ are uncorrelated Brownian motions. \nSimilar statistical\nproperties of $F(\\bk,\\tau, \\tap)$ can be obtained \\cite{dirk} from a gauge transformation of the\nelectron operators, $c_t = \\psi_t e^{i \\Theta(t)}$, under which \n$F(\\tau, \\tap) = \\langle T_\\tau \\psi_{\\bk\\uparrow (\\tau)} \\psi_{-\\bk\\downarrow}(\\tap) \\rangle \ne^{i \\Theta(\\tau) + i \\Theta(\\tap)}$. Since $\\Theta(\\tau) + \\Theta(\\tap)= 2 \\Theta(\\tau) - \n(\\Theta(\\tau)-\\Theta(\\tap))$, and under realistic assumptions \n($\\Theta(\\tau) - \\Theta(\\tap)$) is only a function of \n$(\\tau -\\tap)$, this approach yields an expression equivalent to Eq.~(\\ref{phi12}), except for the \ncorrelations induced between the functions $\\phi(\\tau)$ and $\\phi^\\prime(\\tau - \\tap)$. \nIn what follows we assume for simplicity that the correlations are absent. Then doing the average over\nthe Brownian random process $\\phi^{\\prime}$ and integrating over $\\tap$, for the Josephson current \nwe obtain\n\\beq\nj(t) = {\\rm Abs}[j_0(\\oJ + i \\Lambda)] sin (\\oJ t + \\Delta \\phi(t)),\n\\eeq\nwhich is identical to the form of the Josephson current conjectured in Eq. (\\ref{j2}).\nThe function $j_0(z)$ is the analytical continuation of the function $j_0(\\oJ)$ which determines \nthe amplitude of the Josephson current in the absence of the phase fluctuations. In the case of \ns-wave superconductivity with a constant gap $\\Delta$, this function is \n$j_0(eV) = (\\sigma_0\\Delta/e) K(eV/2\\Delta)$, defined in terms of complete\n elliptic integral $K(x)$. In the case of d-wave superconductor\n $j_0(eV)$ is also a nontrivial function of the relative orientation between\n lattices in two crystals. Its specific form is not important for our\n discussion.\n Finally, we should \nmention that in the current-current correlator, both $\\phi$ and $\\phi^\\prime$ averages should be done on \nthe product of currents, while we have done the averaging over $\\phi^\\prime$ independently in $j(t)$ and\n$j(t^\\prime)$. The qualitative results for conductivity, however, remain the same with the two peaks \nat the frequencies $\\pm \\oJ$.\n\nIn conclusion, we propose to test the relevance of the phase fluctuations\n scenario in the\n pseudogap regime of the high-$T_c$ superconductors by investigating fluctuating \n Josephson current at $T>T_c$. We focus on the temporal fluctuations of the phase\n assuming small-contact tunneling to ignore spatial dependence of the phase. We argue\n that although phase fluctuations will\n yield zero mean Josephson current, its autocorrelation function will produce\n finite correction to the conductivity of {\\em normal} current\n across the junction. AC conductivity will exhibit the peak at Josephson\n frequency $\\omega_J = 2eV/\\hbar$ with the width determined by the characteristic \n phase fluctuation rate $\\Lambda$. possible experimental test could be to \n measure the junction AC conductivity $\\sigma(\\omega)$ \n in the presence of constant bias $V$ and determine if it has a peak at $\\omega_J$. \nWe predict specific dependence $\\delta \\sigma(\\omega_J) \\propto V^{-1}$ of the peak.\n Specific experimental set up is shown on Fig. 1.\n\n\n\nWe would like to thank D. Morr, M. Graff, L. Bulayevski, J. Eckstein, and M. Maley for useful \ndiscussions. This work was supported by the DOE.\n\n%\\vspace*{-2.2cm}\n\\begin{thebibliography}{99}\n\\bibitem{kivelson} V. J. Emery and S. A. Kivelson, Nature {\\bf 74}, 434 (1995). For a review see\nT. Timusk and B. Statt, Rep. Prog. Phys. {\\bf 62}, 61 (1999).\n\\bibitem{Orenstein} J. Corson {\\it et al.}, Nature {\\bf 398}, 221 (1999).\n\\bibitem{janko} B. Janko {\\it et al.}, cond-mat/9808215.\n\\bibitem{jos} B. D. Josephson, Phys. Lett. {\\bf 1}, 251, (1962).\n\\bibitem{anderson} P. W. Anderson {\\it et al.}, Phys. Rev. 138, A1157 (1966).\n\\bibitem{shapiro} S. Shapiro, Phys. Rev. Lett. {\\bf 11}, 80 (1963).\n\\bibitem{millis} M. Franz and A. J. Millis, Phys. Rev. B {\\bf 58}, 14572 (1998).\n\\bibitem{1-layerTc} I. Bozovic and J. Eckstein in ``Physical properties of high temperature \n\tsuperconductors V'' D. M. Ginsberg, ed., (World Scientific, Singapore 1996). \n\\bibitem{microwave} A. H. Dayem and C. C. Grimes, Appl. Phys. Lett., {\\bf 9}, 47 (1966).\n\\bibitem{mahan} G. D. Mahan, {\\it Many Particle Physics}, (Plenum, New York 1990), p. 806.\n\\bibitem{dirk} D. Morr, private communication.\n\\bibitem{AL} L. G. Aslamasov and A. I. Larkin, Phys. Lett. A {\\bf 26}, 238 (1968).\n\\end{thebibliography}\n\n\\end{multicols}\n\n\\end{document} \n\n\n%sig1[x_, y_] := 1/x (1/((x+y)^2 + 1) + 1/((x-y)^2 +1));\n%sig2[x_, y_] := 1/x ((x-y)/((x-y)^2 +1) + (x+y)/((x+y)^2 + 1))\n%Plot[sig[x, 3], {x, -10, 10}]\n\n\n\n"
}
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[
{
"name": "cond-mat0002043.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\bibitem{kivelson} V. J. Emery and S. A. Kivelson, Nature {\\bf 74}, 434 (1995). For a review see\nT. Timusk and B. Statt, Rep. Prog. Phys. {\\bf 62}, 61 (1999).\n\\bibitem{Orenstein} J. Corson {\\it et al.}, Nature {\\bf 398}, 221 (1999).\n\\bibitem{janko} B. Janko {\\it et al.}, cond-mat/9808215.\n\\bibitem{jos} B. D. Josephson, Phys. Lett. {\\bf 1}, 251, (1962).\n\\bibitem{anderson} P. W. Anderson {\\it et al.}, Phys. Rev. 138, A1157 (1966).\n\\bibitem{shapiro} S. Shapiro, Phys. Rev. Lett. {\\bf 11}, 80 (1963).\n\\bibitem{millis} M. Franz and A. J. Millis, Phys. Rev. B {\\bf 58}, 14572 (1998).\n\\bibitem{1-layerTc} I. Bozovic and J. Eckstein in ``Physical properties of high temperature \n\tsuperconductors V'' D. M. Ginsberg, ed., (World Scientific, Singapore 1996). \n\\bibitem{microwave} A. H. Dayem and C. C. Grimes, Appl. Phys. Lett., {\\bf 9}, 47 (1966).\n\\bibitem{mahan} G. D. Mahan, {\\it Many Particle Physics}, (Plenum, New York 1990), p. 806.\n\\bibitem{dirk} D. Morr, private communication.\n\\bibitem{AL} L. G. Aslamasov and A. I. Larkin, Phys. Lett. A {\\bf 26}, 238 (1968).\n\\end{thebibliography}"
}
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cond-mat0002044
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Band-theoretical prediction of magnetic anisotropy in uranium monochalcogenides
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[
{
"author": "Tatsuya Shishidou"
}
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Magnetic anisotropy of uranium monochalcogenides, US, USe and UTe, is studied by means of fully-relativistic spin-polarized band structure calculations within the local spin-density approximation. It is found that the size of the magnetic anisotropy is fairly large ($\simeq$10 meV/unit formula), which is comparable with experiment. This strong anisotropy is discussed in view of a pseudo-gap formation, of which crucial ingredients are the exchange splitting of U $5f$ states and their hybridization with chalcogen $p$ states ($f$--$p$ hybridization). An anomalous trend in the anisotropy is found in the series (US$\gg$USe$<$UTe) and interpreted in terms of competition between localization of the U $5f$ states and the $f$--$p$ hybridization. It is the spin-orbit interaction on the chalcogen $p$ states that plays an essential role in enlarging the strength of the $f$--$p$ hybridization in UTe, leading to an anomalous systematic trend in the magnetic anisotropy.
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"name": "UX.tex",
"string": "%\\documentstyle[aps,prb,floats,epsfig]{revtex}\n\\documentstyle[aps,floats,epsfig]{revtex}\n%\\documentstyle[aps,preprint,epsfig]{revtex}\n\\begin{document}\n\\draft\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname\n\\title{Band-theoretical prediction of magnetic anisotropy in uranium\nmonochalcogenides}\n\\author{Tatsuya Shishidou}\n\\address{Japan Science and Technology Corporation, Tokyo 102-0081, Japan}\n\\address{and Department of Quantum Matter, ADSM, Hiroshima University,\nHigashihiroshima 739-8526, Japan}\n\\author{Tamio Oguchi}\n\\address{Department of Quantum Matter, ADSM, Hiroshima University,\nHigashihiroshima 739-8526, Japan}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nMagnetic anisotropy of uranium monochalcogenides, US, USe and UTe, is\nstudied by means of\nfully-relativistic spin-polarized band structure calculations within the\nlocal spin-density \napproximation. \nIt is found that the size of the magnetic anisotropy is fairly large \n($\\simeq$10 meV/unit formula), which is comparable with experiment. \nThis strong anisotropy is discussed in view of a pseudo-gap formation,\nof which crucial ingredients are the exchange splitting of U $5f$ states and their hybridization \nwith chalcogen $p$ states ($f$--$p$ hybridization). \nAn anomalous trend in the anisotropy is found in the series (US$\\gg$USe$<$UTe) \nand interpreted in terms of competition between localization of the U $5f$\nstates and the $f$--$p$ hybridization. \nIt is the spin-orbit interaction on the chalcogen $p$ states that plays an\nessential role \nin enlarging the strength of the $f$--$p$ hybridization in UTe, \nleading to an anomalous systematic trend in the magnetic anisotropy. \n\\end{abstract}\n% insert suggested PACS numbers in braces on next line\n\\pacs{71.20.Gj, 75.30.Gw, 75.50.Cc}\n]\n\nMagnetic moments in solid originate in the spin and orbital components\nof electrons. \nSince electronic states responsible for the magnetism are normally\nlocalized in a particular atomic region, \nthe moments can be regarded as site-selective quantities. \nExperimental techniques such as x-ray magnetic circular dichroism (XMCD)\ncombined with \nthe so-called spin and orbital sum rules \\cite{Thole,Carra} provide such\nseparable information of the spin and orbital magnetic \nmoments of ferro- and ferri-magnets. \nX-ray magnetic scattering \\cite{Blume} can also\ngive us similar information. \n\nUsually $5f$ electrons play major roles in magnetism of uranium compounds. \nSince the spin-orbit interaction (SOI) of the $5f$ electrons is relatively\nlarge, \nthe size of the $5f$ orbital moment is often expected to be greater than\nthe spin counterpart. \nUnlike the $4f$ orbitals in rare-earth element systems, \nthe $5f$ states are more or less extended and may be possibly \naffected by their environmental effects, hybridization and crystal field. \nThus, the magnetic moment in $5f$ systems must be strongly\nmaterial-dependent. Magnetic anisotropy is another fundamental quantity \nin magnetism, which is often even more important for applications. \nHowever, the magnetic anisotropy energy is usually very small to be \nevaluated from first principles \nand furthermore its microscopic origins have never been clearly \nunderstood yet \\cite{Jansen}. \nIt is, therefore, quite interesting to study such issues on the magnetism \nby using a state-of-the-art band-theoretical technique. \n\nUranium monochalcogenides, US, USe and UTe, have NaCl-type cubic crystal\nstructure \nand show a ferromagnetic order at the Curie temperatures, 177K, 160K and\n104K, respectively\\cite{Handbook Actinide}. \nIt is well known that the monochalcogenides show strong magnetic\nanisotropy \\cite{Lander}, \nwhere the [111] ([001]) direction is the easy (hard) axis. \nInterestingly, the saturation magnetic moment depends on the magnetization\naxis \\cite{Tillwick}. \nThe largest moment along the easy axis, the smallest along the hard axis. \nThe total magnetic moment per uranium atom increases from sulfide through\ntelluride with increasing lattice constant. \nAs long as the sulfide is concerned, the $5f$ electrons are considered to\nbe itinerant from photoemission \\cite{Reihl} and other\nexperiments \\cite{Schoenes,Huang,Rudigier}. \n\nIn the present study, mechanism of the magnetic anisotropy in the uranium\nmonochalcogenides is investigated by \nfirst-principles calculations and an anomalous trend in the size of the\nanisotropy in the series is predicted. \nOur method is based on the local spin-density approximation (LSDA) to the\ndensity functional theory. \nOne-electron Kohn-Sham equations are solved self-consistently by using \nan iterative scheme of the\nfull-potential \nlinear augmented plane wave method \\cite{Soler} in a scalar-relativistic\nfashion. \nWe include SOI as the second variation every self-consistent-field step. \nThe improved tetrahedron method proposed by Bl\\\"{o}chl \\cite{Blochl} is used\nfor the Brillouin-zone integration. \nMore details about our methods and calculated results are \npublished elsewhere \\cite{Shishidou2}.\n% fig_band %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[htb]\n\\centerline{\\epsfig{file=band.epsi,width=8cm}} \n%\\centerline{\\epsfxsize=8.0cm\\epsfbox{band.epsi}}\n\\vspace{3mm}\n\\caption{Fully-relativistic spin-polarized band structure of ferromagnetic\nUS with the [111] \nmagnetization. The Fermi energy is chosen as the origin.}\n\\label{fig_band}\n\\end{figure}\n% fig_band %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nFigure \\ref{fig_band} shows calculated fully-relativistic spin-polarized\nband structure of ferromagnetic US with the [111] magnetization. \nShallow core U $6p$ states form $j=1/2$ and $3/2$ bands split by large SOI. \nS $3s$ bands are located just above the U $6p_{3/2}$ bands and have\nrelatively large dispersion. \nS $3p$ bands are situated from 7 to 3 eV below the Fermi energy\n($\\varepsilon_{\\rm F}$). \nDispersive bands appearing just below $\\varepsilon_{\\rm F}$ are made mostly\nof U $d$ states. \nRelatively narrow U $5f$ bands are pinned around $\\varepsilon_{\\rm F}$. \nIt is found that the largest contribution to the magnetic moments comes\nfrom U $5f$ states as expected. \nCalculated $5f$ moments of US are listed in Table \\ref{tab1} for \nthree magnetization axes, [001], [110] and [111]. \n%TABLE1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{table}[htdp]\n\\caption{Calculated $5f$ spin and orbital magnetic moments \nin Bohr magneton of US for different magnetization directions ($\\hat{M}$). \n$\\Delta E$ is a difference in the total energy in meV from the case of the\n[111] magnetization.}\n\\begin{tabular}{ccccc}\n$\\hat{M}$ & spin & orbital & total & $\\Delta E$ \\\\\n\\hline \\\\\n$[001]$\\tablenotemark[1] &1.61 & -2.13 & -0.52 & 14 \\\\\n$[110]$\\tablenotemark[1] &1.60 & -2.24 & -0.64 & 5 \\\\\n$[111]$\\tablenotemark[1] &1.60 & -2.33 & -0.73 & 0 \\\\\n%LAPW & [111] &1.7 & -2.6 & -0.9 & -1.5 & \\\\\n%ASW & [111] & 1.5 & -2.6 & -1.1 & -1.7 & \\\\\n%LMTO & [111] & 2.1 & -3.2 & -1.1 & -1.5 & \\\\\n%\n$[111]$\\tablenotemark[2] & 1.3\\tablenotemark[3] & -3.0\\tablenotemark[3] &\n-1.7 & \\\\\n\\end{tabular}\n\\tablenotetext[1]{Present work.}\n\\tablenotetext[2]{Neutron-scattering experiment (Ref.\\ \\onlinecite{Wedgwood}).}\n\\tablenotetext[3]{From analysis in Ref.\\ \\onlinecite{Severin}.}\n\\label{tab1}\n\\end{table}%\n%TABLE1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% fig_jzdos %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure*}[bht]\n\\centerline{\\epsfig{file=jzdos.eps,width=16cm}} \n\\vspace{3mm}\n\\caption{Calculated $j_z$-projected $5f$ density of states ($j=5/2$) of\nferromagnetic US \nwith the (a) [111] and (b) [001] magnetization directions. \nQuantization axis for specifying $j_z$ is taken as the same direction of\nmagnetization.}\n\\label{fig_jzdos}\n\\end{figure*}\n% fig_jzdos %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe spin moments hardly change with the magnetization axis \nwhile the orbital moments show large dependence of the axis. \nConsequently, the total moments depend strongly on the magnetization axis, \nbeing in qualitatively good agreement with experiment \\cite{Tillwick}. \n%The results for the easy axis is consistent with the previously calculated ones. \n%The discrepancy may be due to the difference in the sphere radius assumed. \nComparing with the experimental data, the spin moment is slightly overestimated \nwhile the orbital moment is underestimated. \nOrigins of the discrepancy in the magnetic moments have been already \ndiscussed in the previous Hartree-Fock type study \\cite{Shishidou}. \nThe total-energy difference shows that the [111] direction is the lowest \nwhile [001] is the highest. \nThe difference of the total energy between the easy and hard axes is nothing but \nthe magnetic anisotropy energy. \nThe size of the magnetic anisotropy is comparable with experimental \ndata \\cite{Lander} and is about $10^3$ times larger than that of \n$3d$ transition metals. \nLet us discuss mechanism of the magnetic anisotropy and its relation to the\nmagnetic moments.\n\nBecause of large SOI of $5f$ electrons, $j=5/2$ and $7/2$ states are \nroughly well separated \nand the occupied states are composed mostly of the $j=5/2$ states. \nThe $j_z$-projected density of states (DOS) in the $j=5/2$ states is plotted in\nFig.~\\ref{fig_jzdos}. \nOne can easily note that a clear pseudo-gap is formed at $\\varepsilon_{\\rm F}$ \nin the [111] magnetization, \nwhile it becomes less apparent in the [001] direction. \nRelative stability of the [111] magnetization must come from the existence \nof the pseudo-gap. \n\nIn the case of [111], the plus and minus components are well separated. \nThis is the exchange splitting of the bands and the pseudo-gap is\nconsidered to be a sort of the exchange gap. \nOnly a small amount of mixing can be seen in the $j_{z} = \\pm 3/2$ states.\nOn the other hand, the pseudo-gap is almost diminished in the case of [001] \nbecause of larger mixing in $j_{z}= \\pm 3/2$ and $j_{z}= \\pm 1/2$. \nBy counting the number of occupied electrons in each partial DOS, \none can get the occupation of each $j_z$ state. \nFor [111] $\\pm j_z$ state are well exchange-polarized \nbut for [001] the occupations in the $j_{z}= \\pm 1/2$ bands are almost\nequal due to the mixing. \n\nIn order to get more intuitive insight into the occupations, \na change-density plot of the $5f$ electrons around the U atom is \ndepicted in Fig.~\\ref{fig_charge}. \n% fig_charge %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure*}[bht]\n\\centerline{\\epsfig{file=charge.epsi,width=12cm}} \n\\vspace{3mm}\n\\caption{Angular dependence of calculated charge density of the $5f$ electrons \naround the U atom with the (a) [111] and (b) [001] magnetization directions. \nThe spherical component has been reduced to $20\\%$ of its value \nto emphasize nonspherical parts.}\n\\label{fig_charge}\n\\end{figure*}\n% fig_charge %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nIn the [111] magnetization, the $5f$ electrons tend to point to \nthe direction of neighboring-U atoms; \nhexagonal brim stretching out in the (111) U plane \nand triangular bell around [111] direction are clearly formed. \nAs a result, the $5f$ electrons have less distribution in the neighboring-S\ndirections \nand they can reduce the hybridization with the S $p$ states and gain the \nexchange splitting. \nThe small mixing found in $j_{z}=\\pm 3/2$ can be understood by the fact\nthat the \n$j_{z}=\\pm 3/2$ orbitals are extended to the S atoms \nwith respect to the polar angle \\cite{quantization}. \nIn the [001] magnetization, on the other hand, \nthe $5f$ electrons again try to point to the U-U bonds, \ntwelve [110] directions. \nIn the $xy$ plane gentle depression is formed in the nearest-neighboring-S\ndirections, \nbecause the $j_{z}=\\pm 5/2$ states which spread in the $xy$ plane \ncan prevent from mixing with the S $p$ states by using their azimuthal\ndegrees of freedom. \nHowever, no depression is seen in the [$00\\pm 1$] directions. \nThe $j_{z}=\\pm 1/2$ states extending to the $z$ direction \nhave no way to refuse the mixing with the S $p$ states \nbecause of less azimuthal degrees of freedom in its $m=0$ component. \nThis hybridization of the $j_{z}=\\pm 1/2$ states with the neighboring S $p$\norbits \ndestroys the pseudo-gap in the corresponding partial band and \nmakes the [001] magnetization unfavorable \\cite{S_111}. \nThis is the most important mechanism of the magnetic anisotropy \nfound in the present study and leads to a very interesting variation \nin the series of the uranium monochalcogenide as we shall discuss below. \n\nA variation in the calculated spin and orbital magnetic moments \nfor USe and UTe as well as for US is shown in Fig.~\\ref{fig_moment}. \n% fig_monent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[htb]\n\\centerline{\\epsfig{file=moment.eps,width=8cm}} \n\\caption{Calculated spin and orbital magnetic moments in $\\mu_{\\rm B}$ of US, USe and UTe \nwith the [111] magnetization. \nFilled rectangles, triangles, and circles represent \nthe spin, orbital, and total $5f$ magnetic moments, respectively.\nOpen circles show experimental $5f$ moments \\protect\\cite{Wedgwood}.}\n%(Ref.\\ \\onlinecite{Wedgwood}).}\n\\label{fig_moment}\n\\end{figure}\n% fig_moment %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nThe spin and orbital moments increase as one goes from sulfide through\ntelluride. \nThis increase of the moments can be interpreted by narrowing of the U $5f$ \nbands due to increase of the lattice constant. \nBased on such theoretical observation on the $5f$-electron nature, \nwe can expect a simple trend in the magnetic anisotropy. \nFrom sulfide to telluride, the lattice constant increases and the $5f$ band\nwidth is reduced. \nAccordingly the magnetic moments are enhanced, approaching to an atomic limit. \nTherefore, the magnetic anisotropy may show a {\\em monotonic decrease\\/} \nfrom sulfide through telluride with increasing free-atom nature. \nBut the reality is not so simple and calculated magnetic anisotropy energy, \nplotted in Fig.~\\ref{fig_aniene}, shows an anomalous behavior: {\\em\ndecrease and increase\\/} with \na minimum appearing at USe. \n% fig_aniene %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}[htb]\n\\centerline{\\epsfig{file=aniene.eps,width=8cm}} \n\\caption{Calculated magnetic anisotropy energy of US, USe and UTe, taken from the \ntotal-energy difference between the [111] and [001] magnetization directions.}\n\\label{fig_aniene}\n\\end{figure}\n% fig_aniene %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nCalling that the strong magnetic anisotropy in the present compounds \noriginates in the hybridization between the U $5f$ and chalcogen $p$ states \n($f$--$p$ hybridization), \ndifferences in the chalcogen $p$ states may be a clue to understand such an \nanomalous feature. \n\nHybridization depends upon spatial extension of the relevant orbitals via\ntransfer integrals. \nSurprisingly, tails of the chalcogen $p$ orbitals are almost the same \nif we consider the nearest neighbor distance between the U and chalcogen atoms. \nThe Te $5p_{3/2}$ orbital has the longest tail of the chalcogens \nbut the difference is not so large. \nIn addition to the orbital extension, the orbital energy is another factor\nto determine the hybridization. \nGenerally, the orbital-energy difference between the U $5f$ and chalcogen\n$p$ states becomes smaller \nwhen going from sulfur through tellurium. \nBecause of the large SOI, the Te $5p_{3/2}$ state raises up and its\norbital-energy difference from the \nU $5f$ decreases substantially. \nTherefore, the $f$--$p$ hybridization \nbecomes very strong in the case of telluride. \nBasically, from sulfide through telluride, \nthe U $5f$ electrons tend to be more localized due to enlargement of the\nlattice constant. \nBut the $f$--$p$ hybridization, which makes the [001] magnetization unfavorable, \nincreases rapidly from selenide to telluride, resulting to large magnetic\nanisotropy in UTe. \nIt should be noted that experimental data about the magnetic anisotropy \nhave not been available for USe and UTe \\cite{Lander2}. \nExperimental efforts to examine our prediction are strongly desired. \n\nIn conclusion, we have carried out fully-relativistic LSDA calculations for\nuranium monochalcogenides, US, USe and UTe. \nThe magnetic anisotropy can be well described by the LSDA calculations. \nThe pseudo-gap formation stabilizes the [111] magnetization. \nWe have emphasized important roles of the $j_{z}=\\pm 1/2$ states, \nwhich make the [001] magnetization unfavorable significantly. \nCalculated magnetic moments show the monotonic increase in the series \nwith increasing lattice constant. \nOn the other hand, the magnetic anisotropy shows the anomalous systematic trend. \nThis can be understood by considering the hybridization of \nthe U $5f$ states with the chalcogen $p$, especially the $p_{3/2}$ components. \nAnyhow, SOI not only on U but also on the chalcogen sites \nis essential to realize such interesting nature in magnetism of the uranium \nchalcogenides. \n\nWe thank Takeo Jo for invaluable discussion. \nThis work was supported in part by Japan Science and Technology Corporation.\n%\n\\begin{references}\n\\bibitem{Thole}B.~T.~Thole, P.~Carra, F.~Sette, and G.~van der Laan, Phys.\\\nRev.\\ Lett.\\ {\\bf 28}, 1943 (1992).\n\\bibitem{Carra}P.~Carra, B.~T.~Thole, M.~Altarelli, and X.~Wang, Phys.\\\nRev.\\ Lett.\\ {\\bf 70}, 694 (1993).\n\\bibitem{Blume}M.~Blume, J.\\ Appl.\\ Phys.\\ {\\bf 57}, 3615 (1985). \n\\bibitem{Jansen}See, for example, H.~J.~F.~Jansen, \nPhysics Today {\\bf 4}, 50 (1995). \n\\bibitem{Handbook Actinide}J.-M.~Fournier and R.~Tro\\'c, in {\\it Handbook\non the Physics and Chemistry of the Actinides}, edited by A.~J.~Freeman and\nG.~H.~Lander (North-Holland, Amsterdam, 1985), Vol.~2, p.~29.\n\\bibitem{Lander}G.~H.~Lander, M.~S.~S.~Brooks, B.~Lebech, P.~J.~Brown,\nO.~Vogt, and K.~Mattenberger, Appl.\\ Phys.\\ Lett.\\ {\\bf 57}, 989 (1990).\n\\bibitem{Tillwick}D.~L.~Tillwick, P.~de V.~Plessis, J.\\ Magn.\\ Magn.\\\nMater.\\ {\\bf 3}, 329 (1976).\n\\bibitem{Reihl}B.~Reihl, J.\\ Less-Common Met.\\ {\\bf 128}, 331 (1987).\n\\bibitem{Schoenes}J.~Schoenes, B.~Frick, and O.~Vogt, Phys.\\ Rev.\\ B {\\bf\n30}, 6578 (1984).\n\\bibitem{Huang}C.~Y.~Huang, R.~J.~Laskowski, C.~E.~Olsen, and J.~L.~Smith, \nJ.\\ Physic {\\bf 40}, 26 (1979).\n\\bibitem{Rudigier}H.~Rudigier, H.~R.~Ott, and O.~Vogt, Phys.\\ Rev.\\ B {\\bf\n32}, 4584 (1985).\n\\bibitem{Soler}J.~M.~Soler and A.~R.~Williams, Phys.\\ Rev.\\ B {\\bf 40},\n1560 (1989).\n\\bibitem{Blochl}P.~E.~Bl\\\"{o}chl, O.~Jepsen, and O.~K.~Andersen, Phys.\\\nRev.\\ B {\\bf 49}, 16 223 (1994).\n\\bibitem{Shishidou2}T.~Shishidou and T.~Oguchi, unpublished. \n\\bibitem{Wedgwood}F.~A.~Wedgwood, J.\\ Phys.\\ C {\\bf 5}, 2427 (1972).\n\\bibitem{Severin}L.~Severin, M.~S.~S.~Brooks, and B.~Johansson, Phys.\\\nRev.\\ Lett.\\ {\\bf 71}, 3214 (1993).\n\\bibitem{Shishidou}T.~Shishidou, T.~Oguchi, and T.~Jo, Phys.\\ Rev.\\ B {\\bf\n59}, 6813 (1999).\n\\bibitem{quantization}Note that $j_z$ is defined by taking the quantization\naxis \nas the same direction as the magnetization. \n\\bibitem{S_111}As for the [111] magnetization \nthere also exist the S atoms in the magnetization direction, \nbut they are the 3rd nearest neighbors and the hybridization effect is much \nweaker.\n\\bibitem{Lander2}G.~H.~Lander: private communication.\n\\end{references}\n%\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002044.extracted_bib",
"string": "\\bibitem{Thole}B.~T.~Thole, P.~Carra, F.~Sette, and G.~van der Laan, Phys.\\\nRev.\\ Lett.\\ {\\bf 28}, 1943 (1992).\n\n\\bibitem{Carra}P.~Carra, B.~T.~Thole, M.~Altarelli, and X.~Wang, Phys.\\\nRev.\\ Lett.\\ {\\bf 70}, 694 (1993).\n\n\\bibitem{Blume}M.~Blume, J.\\ Appl.\\ Phys.\\ {\\bf 57}, 3615 (1985). \n\n\\bibitem{Jansen}See, for example, H.~J.~F.~Jansen, \nPhysics Today {\\bf 4}, 50 (1995). \n\n\\bibitem{Handbook Actinide}J.-M.~Fournier and R.~Tro\\'c, in {\\it Handbook\non the Physics and Chemistry of the Actinides}, edited by A.~J.~Freeman and\nG.~H.~Lander (North-Holland, Amsterdam, 1985), Vol.~2, p.~29.\n\n\\bibitem{Lander}G.~H.~Lander, M.~S.~S.~Brooks, B.~Lebech, P.~J.~Brown,\nO.~Vogt, and K.~Mattenberger, Appl.\\ Phys.\\ Lett.\\ {\\bf 57}, 989 (1990).\n\n\\bibitem{Tillwick}D.~L.~Tillwick, P.~de V.~Plessis, J.\\ Magn.\\ Magn.\\\nMater.\\ {\\bf 3}, 329 (1976).\n\n\\bibitem{Reihl}B.~Reihl, J.\\ Less-Common Met.\\ {\\bf 128}, 331 (1987).\n\n\\bibitem{Schoenes}J.~Schoenes, B.~Frick, and O.~Vogt, Phys.\\ Rev.\\ B {\\bf\n30}, 6578 (1984).\n\n\\bibitem{Huang}C.~Y.~Huang, R.~J.~Laskowski, C.~E.~Olsen, and J.~L.~Smith, \nJ.\\ Physic {\\bf 40}, 26 (1979).\n\n\\bibitem{Rudigier}H.~Rudigier, H.~R.~Ott, and O.~Vogt, Phys.\\ Rev.\\ B {\\bf\n32}, 4584 (1985).\n\n\\bibitem{Soler}J.~M.~Soler and A.~R.~Williams, Phys.\\ Rev.\\ B {\\bf 40},\n1560 (1989).\n\n\\bibitem{Blochl}P.~E.~Bl\\\"{o}chl, O.~Jepsen, and O.~K.~Andersen, Phys.\\\nRev.\\ B {\\bf 49}, 16 223 (1994).\n\n\\bibitem{Shishidou2}T.~Shishidou and T.~Oguchi, unpublished. \n\n\\bibitem{Wedgwood}F.~A.~Wedgwood, J.\\ Phys.\\ C {\\bf 5}, 2427 (1972).\n\n\\bibitem{Severin}L.~Severin, M.~S.~S.~Brooks, and B.~Johansson, Phys.\\\nRev.\\ Lett.\\ {\\bf 71}, 3214 (1993).\n\n\\bibitem{Shishidou}T.~Shishidou, T.~Oguchi, and T.~Jo, Phys.\\ Rev.\\ B {\\bf\n59}, 6813 (1999).\n\n\\bibitem{quantization}Note that $j_z$ is defined by taking the quantization\naxis \nas the same direction as the magnetization. \n\n\\bibitem{S_111}As for the [111] magnetization \nthere also exist the S atoms in the magnetization direction, \nbut they are the 3rd nearest neighbors and the hybridization effect is much \nweaker.\n\n\\bibitem{Lander2}G.~H.~Lander: private communication.\n"
}
] |
cond-mat0002045
|
L\'evy Distribution of Single Molecule Line Shape Cumulants in Low Temperature Glass
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[
{
"author": "E. Barkai$^a$"
},
{
"author": "R. Silbey$^a$ and G. Zumofen$^b$"
},
{
"author": "$^a$ Department of Chemistry and Center for Material Science and Engineering"
},
{
"author": "Cambridge"
},
{
"author": "MA 02139."
},
{
"author": "$^b$ Physical Chemistry Laboratory"
},
{
"author": "ETH-Z"
},
{
"author": "CH-8092 Z$\\ddot{u}$rich."
}
] |
{Abstract} We investigate the distribution of single molecule line shape cumulants, $\kappa_1,\kappa_2,\cdots$, in low temperature glasses based on the sudden jump, standard tunneling model. We find that the cumulants are described by L\'evy stable laws, thus generalized central limit theorem is applicable for this problem.
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[
{
"name": "PRL.tex",
"string": "% use dvips -o name.ps name.dvi\n%\n%\\documentstyle[preprint,eqsecnum,aps]{rrevtex}\n% ^^^^^^^^ this makes bigger letters and double-spacing\n%\\documentstyle[prl,aps,twocolumn]{revtex}\n\\documentstyle[prl,aps,epsf,twocolumn]{revtex}\n%\\documentstyle[12pt]{article}\n\\begin{document}\n\\title{\nL\\'evy Distribution of Single Molecule\nLine Shape Cumulants in Low Temperature Glass\n }\n\\author{E. Barkai$^a$, R. Silbey$^a$ and G. Zumofen$^b$\\\\\n$^a$ Department of Chemistry and Center for Material Science and Engineering,\nMassachusetts Institute of Technology, Cambridge, MA 02139.\\\\\n$^b$ Physical Chemistry Laboratory, Swiss Federal Institute of Technology, ETH-Z, CH-8092 Z$\\ddot{u}$rich. \\\\\n}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\n\n{\\bf Abstract} \n\nWe investigate the distribution of single molecule line shape\n cumulants, $\\kappa_1,\\kappa_2,\\cdots$,\nin low temperature glasses\nbased on the sudden jump, standard tunneling model. \nWe find that the cumulants are described \nby L\\'evy stable laws, thus generalized central limit theorem\nis applicable for this problem.\n\n\\end{abstract}\n\nPacs: 05.40.-a, 05.40.Fb, 61.43.FS, 78.66.Jg \\\\\n\n% 61.43.FS Glass\n\n%\\Huge{Draft Not For Distribution}\n\n% Modern experimental techniques have made it possible to measure\n%line shapes of single molecule (SM) embedded in solids \\cite{MO}. \n%Line shapes of chemically identical SMs in glassy materials \n%vary from one molecule to the other since each\n%molecule is in a unique environment \\cite{Tittel}.\n%By investigating\n%properties of line shapes of SMs in disordered condensed phase \n% one can learn\n%about the dynamical and disorder \n%properties of the host medium \n%\\cite{MO,Tittel,Reilly2,Zumofen,Geva,Pfluegl,Orrit,Barkai1,Barkai2}. \n% Since each line shape is a random function one must consider\n%the statistical properties of the lines. \n%Geva and Skinner \\cite{Geva} have modeled\n%single molecule line shapes in low temperature glass\n% based on the sudden jump approach of Anderson and Kubo \n%\\cite{AK,Reilly2}.\n%The model assumes that low density dynamical defects \n%[e.g. spins, two level systems (TLS)] are \n%distributed randomly and uniformally in the medium. These defects\n%interact with the molecule via {\\em long range interactions} (e.g., the dipole\n%interaction) while the defect-defect interaction is neglected.\n%\n%The line of each SM can\n%be characterized by its cumulants $\\kappa_j$, $(j=1,2,\\cdots)$\n%which will vary from one molecule to the other. In this letter we\n%show\n%that\n%the probability densities $P(\\kappa_1)$, $P(\\kappa_2)$ are\n%symmetrical and one sided L\\'evy stable\n%laws respectively, while\n%higher order cumulants \n%are described \n%by\n%L\\'evy statistics only in the {\\em slow modulation limit}.\n\n Recent experimental advances \\cite{MO} have made it possible to measure\nthe spectral line shape of a single molecule \n(SM) embedded in a condensed phase. \nBecause each molecule is in a unique static and dynamic environment,\nthe line shapes of chemically identical SMs vary from molecule \nto molecule \\cite{Tittel}.\nIn this way, the dynamic properties of the host are encoded\nin the distribution of single molecule spectral line shapes \n\\cite{MO,Tittel,Reilly2,Zumofen,Geva,Pfluegl,Orrit,Barkai1,Barkai2}.\nWe examine the statistical properties of the line shapes\nand show how these are related to the underlying microscopic \ndynamical events occurring in the condensed phase.\n\n We use the\nGeva--Skinner \\cite{Geva}\n model for the SM line shape in a low temperature\nglass \nbased on the sudden jump picture of \nKubo and Anderson \\cite{AK,Reilly}.\nIn this model, a random distribution of low--density (and\nnon--interacting) dynamical defects \n[e.g., spins or two level systems (TLS)] interacts with\nthe molecule via {\\em long range \ninteraction} \n(e.g., dipolar). \nWe show that L\\'evy statistics fully characterizes the\nproperties of the SM spectral line both in the {\\em fast} and {\\em slow\nmodulation limits}, while far from these limits L\\'evy statistics\ndescribes the mean and variance of the line shape. \nWe then compare our\nanalytical results, derived in the slow modulation limit, with results \nobtained from numerical simulation.\nThe good agreement indicates that the slow modulation limit is correct\nfor the parameter set relevant to experiment.\n%Specifically, the spectral line is characterized by its cumulants, \n%Below, we shall show that L\\'evy statistics fully characterizes the\n%properties of the SM spectral line. \n%Specifically, the spectral line is characterized by its cumulants, \n%$\\kappa_j$\n%$(j=1,2,\\cdots)$ that vary from molecule to molecule.\n%We show that the probability densities $P(\\kappa_1)$\n%and $P(\\kappa_2)$ are symmetrical and one sided L\\'evy stable laws,\n%respectively, while higher order cumulants are described by L\\'evy\n%statistics only in the {\\em slow modulation limit}.\n%In the opposite limit of fast modulation we show that the distribution\n%of line widths is a one sided L\\'evy stable law.\n%We then compare our\n%analytical results, derived in the slow modulation limit, with results \n%obtained from numerical simulation.\n%The good agreement indicates that the slow modulation limit is correct,\n%for the parameter set relevant to experiment.\n\n%fast modulation, spectral lines are approximately\n%Lorentzian, then the distribution of line widths can be used\n%to characterize the statistical properties of the lines.\n%For this case the distribution of line width is a one sided L\\'evy\n%stable law. \n\n L\\'evy stable distributions \nserve as\na natural generalization of the normal Gaussian distribution.\nThe importance of the Gaussian in statistical physics\nstems from the central limit theorem. L\\'evy stable laws are used\nwhen analyzing sums of the type $\\sum x_i$, with \n$\\{ x_i \\}$ being independent identically distributed\n random variables\ncharacterized by a diverging variance.\nIn this case the ordinary Gaussian central limit theorem must be replaced\nwith the generalized central limit theorem.\nWith this generalization, \nL\\'evy stable probability densities, $L_{\\gamma,\\eta}(x)$, replace\nthe Gaussian of the standard central\nlimit theorem. \nKhintchine and L\\'evy found\nthat stable characteristic functions,\n$\\hat{L}_{\\gamma,\\eta}(k)$,\n are of the form \\cite{Feller}\n%\n\\begin{equation}\n\\ln\\left[ \\hat{L}_{\\gamma,\\eta} \\left( k \\right) \\right] = i \\mu k \n- z_{\\gamma}|k|^{\\gamma} \n\\left[ 1 - i \\eta { k \\over |k|} \n\\tan\\left( { \\pi \\gamma \\over 2}\\right)\\right]\n\\label{eq1}\n\\end{equation}\n%\nfor $0< \\gamma\\le 2$ (for the case $\\eta \\ne 0$, $\\gamma=1$ see\n\\cite{Feller}). Four parameters are needed for a full description of \na stable law. The constant $\\gamma$ is called the characteristic exponent, \nthe parameter $\\mu$ is a location parameter which is unimportant in\nthe present case, $z_{\\gamma} > 0$ is a scale parameter and $-1\\le \\eta \\le 1$\nis the index of symmetry. When $\\eta=0$ $(\\eta=\\pm 1)$\nthe stable density $L_{\\gamma,\\eta}(x)$ is symmetrical (one sided).\n%Today the term L\\'evy behavior is often used in a wider sense than that\n%implied by the functions in Eq. (\\ref{eq1}) and is used to describe behavior\n%with broad tails. Examples for recent applications of \n%L\\'evy statistics are sub recoil \n%laser cooling \\cite{Cohen} and the characterization of geometrical properties\n%of random fractals \\cite{Hovi}. \nL\\'evy statistics is known to describe\nseveral long range interaction systems in \ndiverse fields such as astronomy \\cite{Feller}, turbulence \\cite{Turbo}\nand spin glass \\cite{Jap}. \n%All these phenomena result from long-range interactions with\n%uniformally distributed defects: gravitational masses, vortexes,\n%magnetic dipoles etc.\nStoneham's theory \\cite{Stoneham}\nof inhomogeneous line broadening in defected crystal, is based on long-range forces\nand parts of it can be\ninterpreted in terms of L\\'evy stable laws \\cite{Barkai2}. \nStoneham's approach \\cite{Stoneham} is inherently static, while the SM\nline shape model \nconsiders both dynamical and static contributions from\nthe defects.\n\n% In \\cite{Geva} the distribution of line widths was investigated\n%based on numerical approach and it was shown that this model\n%can reasonably reproduce single molecule experimental results,\n%as well as hole burning and photon echo measurements \n%in low temperature glass.\n%Based on simulations, we show that the L\\'evy\n%results are reasonable for this model and that indeed a slow modulation\n%limit exists for a relevant choice of parameters.\n%Our finding follows previous\n%numerical work of \n%Pfluegl et al \\cite{Pfluegl} who\n%showed that the \n%distribution of line widths \n%is well approximated by the slow modulation limit.\n%We note that line widths and cumulants are not related\n%by any known formula for the case considered here\n%\\cite{remark1}.\n%Here we use L\\'evy statistics as a test for the validity\n%of the slow modulation limit in low temperature glass.\n%\n\nAn important issue is the slow and fast modulation limits\n\\cite{Pfluegl,Reilly}. \nBriefly, the fast (slow) modulation limit is valid if important\ncontributions to the line shape are from TLSs which satisfy\n$\\nu \\ll K$ $(\\nu \\gg K)$, where \n$\\nu$ is the frequency shift of the SM due to \nSM--TLS\ninteraction\nand $K$ is the \ntransition rate of the flipping TLS\n(see details below).\nIn the fast modulation limit,\nall (or most) lines are Lorentzian \nwith a width \nthat varies from one molecule to the other. For this case, the (L\\'evy)\ndistribution\nof {\\em line widths} fully characterizes the statistical properties\nof the lines. \n%Below we show that in this\n%limit the distribution of line widths is a one sided\n%L\\'evy stable law. \nThe second, more complicated, case corresponds\nto the slow modulation\nlimit. Then the SM line is typically \ncomposed of several peaks\n(splitting) and is not described well by a Lorentzian. \nIf a SM shows splitting, one\ncan investigate the validity of the\nstandard tunneling model of glass \\cite{AHV} in a direct way, \nsince the splitting of\na line is directly associated with SM--TLS interaction \\cite{Orrit}. \n%From an experimental point of view it is difficult, though not \n%impossible to determine whether two distinct peaks\n%arise from two molecules\n%or a split single molecule line. \nAs mentioned, we demonstrate the existence\nof a slow modulation limit in SM--glass system.\n\n%Below, we compare our\n%analytical results, derived in the slow modulation limit, with results \n%obtained from numerical simulation.\n%The good agreement indicates that a slow modulation limit exists\n%for the set of parameters relevant to experiment.\n \n Following \\cite{Geva}\nwe assume a SM coupled to non identical independent TLSs\nat distances ${\\bf r}$ in dimension $d$.\nEach TLS is characterized by its asymmetry variable $A$\nand tunneling element $J$.\nThe energy of the TLS is $E=\\sqrt{A^2 + J^2}$.\nThe TLSs are coupled to phonons\nor other thermal excitations\nsuch that the state of the TLS changes with time. The state of\nthe $n$ th TLS is described by an occupation parameter,\n$\\xi_n(t)$, that is equal $0$ or $1$ if the TLS is in its\nground or excited state respectively. The probability for finding the TLS\nin its upper $\\xi=1$ state, \n$p$, is given \nby the standard Boltzmann form\n$p=1/\\{ 1+ \\exp[ E/(k_b T)]\\}$. \nThe transitions\nbetween the ground and excited state are described by the\nup and down transition rates $K_u,K_d$, which are related to\neach other by the standard detailed balance condition.\n\n The excitation of the $n$ th TLS shifts the SM's\ntransition frequency by $\\nu_n$. Thus, the SM's\ntransition frequency is\n%\n\\begin{equation}\n\\omega(t) = \\omega_0 + \\sum_{n=1}^{N_{\\rm act}} \\xi_n \\left( t \\right) \\nu_n,\n\\label{eq2}\n\\end{equation}\n%\nwhere $N_{\\rm act}$ is the number of active TLSs in the system\n(see details below) and \n$\\omega_0$ is the bare transition frequency that differs\nfrom one molecule to the other depending on the\nlocal static disorder.\nWe consider a wide class of frequency perturbations \n%\n\\begin{equation}\n\\nu = 2 \\pi \\alpha \\Psi\\left( \\Omega\\right) f\\left( A, J \\right)\n{ 1 \\over r^{\\delta}},\n\\label{eq3}\n\\end{equation}\n%\n$\\alpha$ is a coupling constant with units\n$\\left[ \\mbox{Hz}\\ \\mbox{nm}^{\\delta}\\right]$,\n$\\Psi\\left(\\Omega \\right)$ is a dimensionless\nfunction of order unity, $\\Omega$ is a vector of angles\ndetermined by the orientations of the TLS and molecule\n(in some simple cases $\\Omega$ depends on polar angles\nonly),\n$f(A,J)\\ge 0$ is a dimensionless function of the internal\ndegrees of freedom of the fluctuating TLS, \n$\\delta$\nis the interaction exponent. \n The line shape of the SM is given by\nthe complex Laplace transform of the \nrelaxation function\n%\n\\begin{equation}\nI_{SM}\\left( \\omega \\right)={1 \\over \\pi} \\mbox{Re}\n\\left[ \\int_0^\\infty dt e^{i \\omega t} \n%e^{ - {t \\over 2 T_1}}\n\\Pi_{n=1}^{N_{\\rm act}}\\Phi_n\\left( t \\right) \\right]\n\\label{eq4}\n\\end{equation}\n%\nprovided that the natural life time of the SM excited state is long.\nThe relaxation function of a single TLS was evaluated \\cite{Reilly} based\non methods developed in \\cite{AK}\n%\n\\begin{equation}\n\\Phi\\left( t \\right) = e^{-\\left( \\Xi+ i p\\nu\\right) t}\n\\left[\\cosh\\left( \\Omega t \\right) + {\\Xi\\over \\Omega}\n\\sinh\\left( \\Omega t \\right) \\right]\n\\label{eq5}\n\\end{equation}\n%\nwith\n%\n%\\begin{equation}\n$\\Omega = [K^2/ 4 - \\nu^2 / 4 - i \\left( p - 1/2 \n \\right) \\nu K]^{1/2}$,\n%\\label{eq6}\n%end{equation}\n%\n%\\begin{equation}\n$\\Xi = { K \\over 2} - i \\left( p - {1 \\over 2} \\right) \\nu$\n%\\label{eq7}\n%\\end{equation}\n%\nand $K=K_u + K_d$.\nFor a bath of TLSs the line shape, Eq. (\\ref{eq4}), \nis a formidable function of the random TLS\nparameters $(r,\\Omega,A,J)$ as well as the system\nparameters $( \\alpha , T, \\mbox{etc})$.\nIn the fast modulation limit\n$K\\gg |\\nu|$, one finds a simpler behavior: all lines are Lorentzian\nwith half width\n%\n\\begin{equation}\n\\tilde{\\Gamma}= \\sum_n^{N_{act}} p_n(1-p_n)\\nu_n^2/K_n\n\\label{eqGamma}\n\\end{equation}\n%\nwhich varies from one molecule to the other.\nEq. (\\ref{eqGamma}) shows the well known\nphenomena of motional narrowing.\nIn the slow modulation limit $K \\ll |\\nu|$\none finds \n$ \\Phi(t)=1 - p + pe^{- i \\nu t}$ implying that the line shape\nof a molecule coupled to a single TLS \nis composed of two delta peaks, the line shape \nof a molecule coupled to\ntwo TLSs is composed of four delta peaks, etc (splitting).\n\n The spectral line is characterized by its cumulants\n$\\kappa_j$ $(j=1,2,\\cdots)$ that vary from one molecule to the\nother, and we investigate the cumulant probability density\n$P(\\kappa_j)$.\n We have derived the cumulants of the SM line shape, \nand the first four cumulants are\npresented in Table 1 \\cite{remark}. \n%The exact calculation of the cumulants was \n%made possible using symbolic programming without which the\n%calculation becomes cumbersome.\nWe observe that cumulants of order \n$j\\le 2$ are real while generally cumulants of\norder $j>2$ are complex,\nimplying that the moments of the line shape\ndiverge when $j>2$. The summation, $\\sum_n$, in Table $1$\nis over the active TLSs, namely those TLSs which flip\non the time scale of observation $\\tau$ (i.e., $K_n > 1/ \\tau$).\n We consider the slow modulation limit, soon to be justified,\nwhich means that we consider the case $K_n \\ll \\nu_n$.\nTo investigate this limit we set $K_n=0$ in Table $1$,\nthen all the cumulants\nare real and are rewritten as $\\kappa_j=\\sum_n H_{j n} \\nu_n^j$,\nwhere $H_j$ are functions of $p$ only\nand $H_1=p$, $H_2=p(1-p)$, $H_3=p(1-p)(2p-1)$ etc.\nNote that for $\\kappa_1$ and $\\kappa_2$\nno approximation \nis made.\n\n\n\\begin{tabular}{|l|l|}\n\\hline\n%\nj & $\\kappa_j$ \\\\\n%\n$ $ & $$ \\\\\n%\n$1$ &$ \\sum_n p_n \\nu_n $ \\\\\n%\n$ $ & $$ \\\\\n%\n$2$ &$ \\sum_n p_n\\left(1 - p_n\\right) \\nu_n^2 $ \\\\\n%\n$ $ & $$ \\\\\n%\n$3$ &$ \\sum_n p_n\\left(1- p_n\\right)\\left( 2 p_n - 1\\right) \\nu_n^3\n+i \\sum_n p_n(1-p_n)K_n \\nu_n^2 $ \\\\\n%\n$ $ & $$ \\\\\n%\n$4$ &$ \\sum_n p_n\\left(-1+ p_n\\right)\n\\left[ K_n^2 + \\nu_n^2\\left(-1 + 6p_n - 6 p_n^2\\right) \\right]\\nu_n^2 -$\\\\\n$ \\ $ &$\n2 i \\sum_n K_n \\left( - 1 + p_n \\right) p_n \\left( - 1 + 2 p_n\\right) \\nu_n^3\n$ \\\\\n%\n$ $ & $$ \\\\\n%\n\\hline\n\\end{tabular}\n$$\\mbox{Table 1: Cumulants $\\kappa_j$ of the SM line shape} $$\n%\n\nLet \n$\\left\\langle \\cdot \\right\\rangle_{r \\Omega A J}$ denote an averaging\nover\nthe random TLS parameters.\nThe characteristic function\nof the $j$ cumulant can be written in a form\n%\n$$ \\left\\langle \\exp \\left( i k \\kappa_j\\right)\\right \\rangle_{r \\Omega A J} = $$\n\\begin{equation}\n\\exp\\left[ - \\rho_{\\mbox{eff}} \\left\\langle \\int d \\Omega \\int_0^{\\infty}\n{d (r^d) \\over d}\\left( 1 - e^{i k B_j/ r^{\\delta j }}\\right)\\right\\rangle_{AJ}\\right],\n\\label{eq8}\n\\end{equation} \n%\n$ \\rho_{\\mbox{eff}}$ is the density of the active TLS and \n$B_j=(2 \\pi \\alpha)^j \\Psi^j \\left( \\Omega \\right) f^j \\left(A,J\\right)H_j$.\nTo derive Eq. (\\ref{eq8}) we have\nused the assumption of independent\nTLSs uniformly distributed in the system.\n For odd $j$ cumulants we find\n%\n\\begin{equation}\n\\left\\langle \\exp\\left(i k \\kappa_j \\right) \\right\\rangle_{r \\Omega A J}=\n\\hat{L}_{\\gamma,0}( k ),\n\\label{eq9}\n\\end{equation}\n%\nwith characteristic exponent $\\gamma=d/(\\delta j )$ and the scale parameter\n%\n\\begin{equation}\nz_{\\gamma}= \\rho_{\\mbox{eff}} \\left( 2 \\pi \\alpha\\right)^{{d\\over \\delta}}\n\\left\\langle f^{d/\\delta}\\left( A , J \\right) |H_j|^{\\gamma} \\right\\rangle_{AJ}\nc_{\\gamma} \\int d \\Omega |\\Psi^j\\left( \\Omega\\right)|^{\\gamma}\n\\label{eq10}\n\\end{equation}\n%\nwith $c_{\\gamma} = \\cos\\left( \\gamma \\pi/2\\right)\\Gamma(1 - \\gamma)$,\n$c_1=\\pi/2$.\nEq. (\\ref{eq9}) shows that\nodd cumulants are described by symmetrical L\\'evy stable density, i.e.,\n$P(\\kappa_j)=L_{\\gamma,0}(\\kappa_j)$. \nTwo conditions must be satisfied for such\na behavior, $0 <\\gamma < 2$ and \n$\\int d \\Omega \\sin\\left[ \\Psi^j(\\Omega)\\right]=0$. The latter condition\ngives the symmetry condition, $\\eta=0$, which means that negative\nand positive contributions to $\\kappa_j$ are equally probable.\n\n For even cumulants and $0< \\gamma < 1$ we find\n%\n\\begin{equation}\n\\left\\langle \\exp\\left(i k \\kappa_j \\right) \\right\\rangle_{r \\Omega A J}=\n\\hat{L}_{\\gamma,\\eta}(k)\n\\label{eq11}\n\\end{equation}\n%\nwith a scale parameter Eq. (\\ref{eq10})\nand with L\\'evy index of symmetry \n%\n\\begin{equation}\n\\eta= { \\left\\langle f^{j \\gamma} \\left( A, J \\right) |H_j|^{\\gamma} { H_j \\over |H_j|} \\right\\rangle_{AJ} \\over\n\\left\\langle f^{j \\gamma} \\left( A, J \\right) |H_j|^{\\gamma} \\right\\rangle_{AJ}}.\n\\label{eq12}\n\\end{equation}\n%\nEq. (\\ref{eq11}) implies that even cumulants are distributed\naccording to $P(\\kappa_j)=L_{\\gamma,\\eta}(\\kappa_j)$.\nWe note that the asymmetrical L\\'evy functions,\nwith $\\eta\\ne \\pm1,0$, only rarely find\ntheir applications in the literature.\nThe characteristic exponent $\\gamma$ depends only on the general features\nof the model (namely on $d$ and $\\delta$). In contrast the\nL\\'evy index of symmetry $\\eta$ depends on the details\nof the model and on system\nparameters $(T,\\mbox{etc})$. For $j=2$ we\nhave $H_j=|H_j|$ and then $\\eta=1$ so\nthe L\\'evy density is one sided, as is expected since\n$\\kappa_2 > 0$.\n\n As mentioned, in the fast modulation limit, \nthe random line width in Eq. (\\ref{eqGamma}) characterizes the\nstatistical properties of the spectral lines. \nUsing the approach in Eqs. (\\ref{eq8}-\\ref{eq10})\none can show that $P(\\tilde{\\Gamma})=L_{d/(2 \\delta),1}(\\tilde{\\Gamma})$\nwith the scale parameter $z_{d/(2 \\delta)}$ given by Eq. (\\ref{eq10})\nwith $j=2$ and $H_2=p(1-p)/K$.\n \n In what follows we exhibit our results and compare to \nsimulations based on the standard\ntunneling model of low temperature glass \\cite{AHV}. We use system parameters\ngiven by Geva and Skinner \\cite{Geva} to model\nterrylene in polystryrene. \nThe SM-TLS interaction is dipolar,\nhence $\\delta =3$, and we consider spatial dimension $d=3$.\nThe distribution\nof the asymmetry parameter and tunneling element \nis $P(A)P(J)=N^{-1} J^{-1}$ for \n$2.8\\times 10^{-7}{\\rm K} < J <18 {\\rm K}$ and\n$ 0< A < 17{\\rm K}$, $N$ denoting a normalization constant. \nWe use $f(A,J)=A/E$ and define a TLS to be active if\n$K> 1/\\tau$, $\\tau=120$ sec is the time of experimental observation.\nIn this way the averaging $\\left\\langle \\cdots \\right\\rangle_{AJ}$ becomes\n$\\tau$ independent. \nThe rate of the TLS is given by \n$K = c J^2 E \\coth\\left({\\beta E_n/2} \\right)$ \nand $c=3.9 \\times 10^{8} {\\rm K}^{-3}$Hz \nis the TLS phonon coupling constant.\nAdditional system parameters are the coupling constant \n$\\alpha=3.75\\times 10^{11}$ $\\mbox{nm}^{3}$Hz and the\nTLS density $1.15\\times 10^{-2}$ nm$^{-3}$.\nAccording to\nEqs. (\\ref{eq9})-(\\ref{eq12}), only the scale parameter $z_{\\gamma}$\ndepends on \nthe orientation of the TLS and SM, through $\\Psi(\\Omega)$.\nIt is therefore reasonable to assume simple forms for $\\Psi(\\Omega)$.\nWe consider two examples, \nmodel $1$ ($\\mbox{M}1$) for which $\\Psi(\\Omega)$\nis replaced with a two state variable (i.e., a spin model)\n$\\Psi=1$ or $\\Psi=-1$ with equal probabilities of occurrence\nand model $2$ ($\\mbox{M}2$)\n$\\Psi(\\Omega)=\\cos(\\theta)$, with $\\theta$, the standard polar\ncoordinate, distributed uniformly.\nWith these definitions we calculate the symmetry index\n$\\eta$ and the scaling parameter $z_{\\gamma}$ and compare between\nthe theory and numerical simulation.\n\n\n\n%here prediction is distribution of kappa_1 is\n%(1/Pi) z_1/(w^2 + z_1^2) gamma is calculated\n%using Mathematica. We present four cases in a scaling\n%form. 1. T=1.7 and epsilon = +- model z_1=9.07357 GHz\n% 2. T=1.7 and Psi(Omega) = cos(theta) z_1=9.07/2\n% 3. T=1.7 and epsilon = +- model\n% 4. T=1 and Psi(Omega) = cos(theta)\n%all curves collapse on a master curve (scaling)\n%see more details fig2 below and Zu*/1math as well\n%as Zu*/Pr*/cumu.f\n% for the Graphics see Zu*/Pr*R/Graphics Fig1\n\n\\begin{figure}[htb]\n\\epsfxsize=20\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{diskap1.ps} }}\n\\caption {%\\protect\\footnotesize\nScaled probability density of first cumulant\n$P(\\kappa_1)z_1$ versus $\\kappa_1/z_1$.\nSymbols are the simulation results obtained from $4000$\nmolecules for different cases indicated in the figure.\n%The star (dot)\n%represents model 1 (model 2) for the case $T=1.7 {\\bf K}$.\n%while the plus (diamond) represents model 1 (model 2) \n%and $T=1 {\\bf K}$. \nThe theory, plotted as a solid curve, \npredicts a Lorentzian density $P(\\kappa_1)=L_{1,0}(\\kappa_1)$\nwith a scaling parameter $z_1$ which varies from one set of data\nto the other.\n}\n\\label{fig1}\n\\end{figure}\n\n We consider the first two cumulants $\\kappa_1$ and $\\kappa_2$\n(i.e, the line shape mean and variance). \n%We remind the reader\n%that for this case our Eqs. (\\ref{eq9}) and (\\ref{eq11}) are exact\n%within the framework of the model. \nSince $d=\\delta$ we find $P(\\kappa_1)=L_{1,0}(\\kappa_1)$\nwhich is the Lorentzian density,\nand $P(\\kappa_2) = L_{1/2,1}(\\kappa_2)$ which is \nSmirnov's\ndensity. We have considered two temperatures for the two models\nM1 and M2. As shown in Fig. \\ref{fig1} and \\ref{fig2},\na scaling behavior is observed and all data collapse on the\nL\\'evy densities $L_{1,0}(\\kappa_1)$ and $L_{1/2,1}(\\kappa_2)$ respectively.\nIn Fig. \\ref{fig1} and \\ref{fig2} we have rejected TLSs within a sphere\nof radius $r_{\\rm min}=1$ nm,\ndemonstrating that our results are not sensitive to a short cutoff.\nAlso shown in the inset of Fig. \\ref{fig2} is \n$P(\\mbox{Re}[\\kappa_3])$ which is distributed\naccording to $L_{1/3,0}(\\mbox{Re}[\\kappa_3])$ and\na scale parameter $z_{1/3}$ given in \nEq. (\\ref{eq10}).\nThe L\\'evy behavior of $\\kappa_1,\\kappa_2$ and\n$\\mbox{Re}[\\kappa_3]$\nholds generally and is not limited to the slow modulation limit\nsince these random variables do not depend explicitly on the \nrates $K$. \n\n\\begin{figure}[htb]\n\\epsfxsize=20\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{fig2.ps} }}\n\\caption {%\\protect\\footnotesize\nSame as Fig. (\\protect{\\ref{fig1}}) \nfor the second cumulant. We show $P(\\kappa_2)z_{1/2}^2$ versus\n$\\kappa_2/z_{1/2}^2$. The solid curve is Smirnov's density \n$L_{1/2,1}(\\kappa_2)$. \nIn the inset we show\nthe same as Fig. (\\protect{\\ref{fig1}}) for $\\mbox{Re}[\\kappa_3]]$,\nthe solid curve is\nL\\'evy density $L_{1/3,0}(\\mbox{Re}[\\kappa_{1/3}])$.\n}\n\\label{fig2}\n\\end{figure}\n\n\n\n Consider the distribution of $\\mbox{Re}[\\kappa_4]$,\nwhich in the slow modulation limit is distributed according\nto $L_{1/4,\\eta}(\\mbox{Re}[\\kappa_4])$, \nEq. (\\ref{eq11}). The question\nremains if such a slow modulation limit is valid for the standard\ntunneling model parameters we are considering. The slow modulation limit\nis expected to work when $K\\ll |\\nu|$. For large enough $r$\nthis inequality will fail; however,\ndepending on system parameters, we expect that\ncontributions from TLS situated far from the SM\nare negligible. We also note\nthat according to the standard tunneling model the rates\n$K$ are distributed over a broad range, albeit with\nfinite cutoffs that insure that the averaged rate \nis finite.\n% Therefore the existence of a slow modulation limit\n%cannot be generally true. \nTo check if the slow modulation limit\nis compatible with the standard tunneling model approach,\nwe compare \nour slow modulation results \nwith those obtained by simulation \nin Fig. \\ref{fig4}. \nWe also show\nsimulation results in which all rates are set to zero ($K=0$).\nFor model\n$\\mbox{M}1$, we find that the deviation between simulation\nand theory is small so the assumption of slow modulation\nlimit is justified. For model $\\mbox{M} 2$, we see slightly\nlarger deviations between the theory and numerical results,\ndue to\nthe angular dependence of model $\\mbox{M} 2$,\n$\\Psi(\\Omega) = \\cos(\\theta)$, which reduces the typical frequency\nshift $|\\nu|$ compared to model $\\mbox{M}1$. We conclude that\nthe present theory \ncan be used as a criterion for the validity \nof the slow modulation limit. \n\n%\\begin{figure}[htb]\n%\\epsfxsize=20\\baselineskip\n%\\centerline{\\vbox{\n% \\epsffile{disrealkap3.ps} }}\n%\\caption {%\\protect\\footnotesize\n%The same as Fig. (\\protect{\\ref{fig1}}) \n%for $\\mbox{Re}[\\kappa_3]$. The probability density\n%function $P(\\kappa_3)$ is described by the L\\'evy stable \n%density $L_{1/3,0}(\\mbox{Re}[\\kappa_3])$. \n%$8000$ SM were considered and $r_{min}=0$.\n%}\n%\\label{fig3}\n%\\end{figure}\n%\n\\begin{figure}[htb]\n\\epsfxsize=20\\baselineskip\n\\centerline{\\vbox{\n \\epsffile{diskap4.ps} }}\n\\caption {%\\protect\\footnotesize\nSame as Fig. \n(\\protect{\\ref{fig1}}) \nfor $Re[\\kappa_4]$ and for temperature $T=1.7 {\\rm K}$.\nThe symbols are the simulation results obtained for four different\ncases as indicated in the figure and for $12000$\nmolecules.\nThe solid curve is $L_{1/4,\\eta}[\\mbox{Re}(\\kappa_4)]$ with\nan index of asymmetry $\\eta=0.6104$.\nIn the inset we show the power law\ntail of the scaled probability density on a log-log plot.\n}\n\\label{fig4}\n\\end{figure}\n\n Depending on system parameters, L\\'evy statistics\nmay become sensitive to the finite cutoff $r_{\\rm min}$,\nPhysically, the cutoff can be important since the\npower low interaction is not supposed to work well\nfor short distances \\cite{Pfluegl}.\nOur results were derived for $r_{\\rm min}=0$,\nwhile for small though finite $r_{\\rm min}$ \none can find intermittency behavior, i.e.,\nthe ratio $\\langle \\kappa_2^2 \\rangle / \\langle \\kappa_2\\rangle^2$\n(as well as similar dimensionless ratios) is very\nlarge. \nWhen $r_{\\rm min}$ is large one finds a Gaussian behavior.\nThe phenomena of intermittency in the context of a reaction\nof a SM in a random environment was investigated\nin \\cite{Wang}.\nGenerally high order cumulants are more sensitive to finite\ncutoff and for results in Fig (\\ref{fig4}) $r_{\\rm min}=0$\nwas chosen to see the proper decay laws in the\nwing.\n\n% The L\\'evy behavior found in the present study can\n%be partially understood using a\n%simple argument. \n%Consider the cases for which the cumulants can be written as \n%$\\kappa_j=\\sum_n x_n y_n$ where $x_n=1/r_n^{\\delta j}$\n%and $y_n$ are real random variables defined in Table 1.\n%It is easy to show,\n%based on the assumption of uniform distribution of TLSs,\n%that the random variable $x_n$ is distributed \n%according $P(x)\\sim x^{-( 1 + \\gamma)}$. \n%This is why we found a L\\'evy behavior \n%with characteristic exponent $\\gamma$\n%which depends only on the dimensionality and the\n%interaction exponent.\n%The random variables $\\{ y_n \\}$\n%determine the \n%non universal features of the model, i.e., the scale parameter\n%$z_{\\gamma}$ and the index of symmetry $\\eta$ \n%(provided that the distribution of $y$ is not too broad).\n\n% For $j=1$ our theory is a (slight) generalization\n%of Stoneham's \\cite{Stoneham} theory of inhomogeneous \n%line broadening. Briefly, this theory \\cite{Stoneham}\n%treats an ensemble of molecules as though each member of\n%the ensemble is described by a line of zero width.\n%This corresponds to the assumption $\\kappa_j=0$ for $j>1$.\n%On the other hand our theory shows that distribution of $\\kappa_j$\n%is wide and in fact even the average $\\langle \\kappa_j \\rangle$ \n%does not generally exist. It is interesting to note that\n%some parts of the\n%theory of inhomogeneous line\n%broadening can also be interpreted in terms of\n%L\\'evy statistics. For example, in \\cite{Stoneham} it was shown\n%that broadening due to\n%Electric fields, (i.e., $\\delta=2$,$d=3$,$j=1$) is described\n%by a function which we identify as $L_{3/2,0}$, also known as\n%Holtsmark distribution.\n\n\n% The Holtsmark distribution finds its applications in diverse fields\n%like turbulence \\cite{Turbo} and astronomy \\cite{Feller}.\n%Furukawa et al \\cite{Jap}\n%have used M$\\ddot{o}$ssbauer spectroscopy\n%and found that distribution\n%of internal magnetic fields\n%in Au(Fe) spin glass alloys\n%is L\\'evy stable.\n%All these effects result from the long range interaction\n%of uniformally distributed\n% defects, gravitational masses, vortexes, magnetic dipoles etc. \n%Here we went one step further and considered dynamical\n%situation in which a function (i.e., the line shape) is the random\n%quantity of interest. As shown here \n%% the interplay between dynamics (i.e., the rates\n%$K$) and static shifts (i.e., $\\nu$) is a possible cause for \n%deviations from L\\'evy statistics in a fast modulation limit.\n%We believe that approaches similar to ours might find their\n%application in other fields.\n\n% Finally, we note that L\\'evy statistics can be used to analyze\n%other statistical properties of SMs\n%in disordered media.\n%We have shown that, in the slow modulation limit,\n%the averaged line shape is \n%a L\\'evy stable function. \n%Other results based on models similar to ours\n%\\cite{Fleury,Pfluegl,Geva1} can be explained based upon \n%generalized central limit theorem. \n%We shall address these issues in future\n%publication.\n\n To conclude, we showed that the generalized central limit theorem\ncan be used to analyze distribution of cumulants of SM\nline shapes in glass. \nWe note that besides cumulants,\nL\\'evy statistics can be used to analyze other\nstatistical properties of SMs in disordered media \\cite{Barkai2}. \n\n{\\bf Acknowledgment} EB thanks the ETH and Prof. Wild for\ntheir hospitality. This work was supported by the NSF.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{MO} W. E. Moerner and M. Orrit,\n{\\em Science} {\\bf 283} 5408 (1999).\nT. Plakhotnik, E. Donley and U. P. Wild,\n{\\em Annu. Rev. Phys. Chem.} {\\bf 48}, 181 (1997)\nand references therein.\n\n\n\\bibitem{Tittel} L. Fleury, A. Zumbusch, M. Orrit, R. Brown\nand J. Bernard, {\\em J. Lumin.} {\\bf 56} 15\n(1993).\nJ. Tittel, R. Kettner, Th. Basche, C. Brauchle,\nH. Quante, K. Mullen,\n{\\em J. Lumin.} {\\bf 64} 1\n(1995).\nM. Vacha, Y. Liv, H. Nakatsuka, T. Tani,\n{\\em J. Chem. Phys.} {\\bf 106} 8324 (1997).\nB. Kozankiewicz, J. Bernard and M. Orrit, {\\em J. Chem. Phys,\n} {\\bf 101} 9377 (1994).\n\n\\bibitem{Reilly2} P. D. Reilly and J. L. Skinner,\n{\\em Phys. Rev. Lett.} {\\bf 71} 4257 (1993).\n\n\\bibitem{Zumofen}\nG. Zumofen and J. Klafter,\n{\\em Chem. Phys. Lett.} {\\bf 219} 303 (1994).\n\n\\bibitem{Geva} E. Geva and J. L. Skinner,\n{\\em J. Phys. Chem. B} {\\bf 101} 8920 (1997).\nE. Geva and J. L. Skinner,\n{\\em Chem. Phys. Lett.} {\\bf 287} 125 (1998).\n\n\\bibitem{Pfluegl} W. Pfluegl, F. L. H. Brown and R. J. Silbey,\n{\\em J. Chem. Phys.} {\\bf 108} 6876 (1998).\n\n\\bibitem{Orrit} A. M. Boiron, P. Tamarat, B. Lounis, R. Brown and M. Orrit,\n{\\em J. Chem. Phys.} {\\bf 247} 119 (1999).\n\n\\bibitem{Barkai1}\nE. Barkai and R. Silbey,\nChem. Phys. Lett., {\\bf 310} 287 (1999).\n\n\\bibitem{Barkai2}\nE. Barkai and R. Silbey, (submitted).\n\n\\bibitem{AK} P. W. Anderson, {\\em J. Phys. Soc. Jpn.} {\\bf 9}, 316 (1954).\nR. Kubo, {\\em ibid} {\\bf 6} 935 (1954).\n% R. Kubo Adv. Chem. Phys. {\\bf 15} 101 (1969)\n\n% Spectroscopy of a chromophore coupled\n% to a lattice of dynamic two level systems 1. Absorption\n% line shape\n% 959 964\n\\bibitem{Reilly} P. D. Reilly and J. L. Skinner,\n{\\em J. Chem. Phys.} {\\bf 101} (2) 959 (1994).\n\n\\bibitem{Feller} W. Feller,{\\em An introduction to probability Theory and\nIts Applications} Vol. 2 (John Wiley and Sons 1970).\n\n\\bibitem{Turbo} S. G. Llewellyn Smith and S. T. Gille {\\em Phys. Rev. Lett.}\n{\\bf 81} 5249 (1998) \nand references therein.\n\n\n\\bibitem{Jap}\n Y. Furukawa, Y. Nakai and N. Kunitomi,\n{\\em J. of the Phys. Soc. Japan} {\\bf 62} No.1. 306 (1993).\n\n% Theory of single molecule optical line shape\n% distributions in low temperature glasses\n% p 8920 8932\n\n%\\bibitem{Brown} F. L. H. Brown and R. J. Silbey\n%{\\em J. Chem. Phys.} {\\bf 108} 7434 (1998)\n\n\n%\\bibitem{remark1}\n%In fact the line width is not well defined mathematically,\n%and there is no general formula for line width of a line shape\n%given in Eqs. (\\ref{eq4},\\ref{eq5})\n\n%\\bibitem{Cohen} Cohen Tan {\\em Phys. Rev. Lett.} {\\bf 83} 3696 (1999)\n\n%\\bibitem{Hovi} J. P. Hovi, A. Aharony, D. Stauffer and B. B. Mandelbrot, {\n%\\em Phys. Rev. Lett.}, {\\bf 77}, 877, (1996).\n\n\\bibitem{Stoneham} A. M. Stoneham, {\\em Rev. Mod. Phys.} {\\bf 41}\n82 (1969).\n\n\\bibitem{AHV} P. W. Anderson, B. I. Halperin, C. M. Varma, \n{\\em Philos. Mag.} {\\bf 25}, 1 (1971). W. A. Philips,\nJ. Low. Temp. Phys. {\\bf 7}, 351 (1972). \n\n\\bibitem{remark} In Table 1 we use $\\omega_0=0$, for $\\omega_0\\ne 0$\nwe have $\\kappa_1=\\sum_n p_n \\nu_n + \\omega_0$ while the higher order \ncumulants are $\\omega_0$ independent. The distribution of $\\omega_0$ depends\non the static disorder \\cite{Stoneham} while we are considering\nthe dynamic disorder due to the flipping TLSs. \n\n\\bibitem{Wang} J. Wang and P. Wolynes, {\\em Phys. Rev. Lett.} {\\bf 74} \n4317\n(1995).\n \n\n\\end{thebibliography}\n\n\\end{document}\n"
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{
"name": "cond-mat0002045.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{MO} W. E. Moerner and M. Orrit,\n{\\em Science} {\\bf 283} 5408 (1999).\nT. Plakhotnik, E. Donley and U. P. Wild,\n{\\em Annu. Rev. Phys. Chem.} {\\bf 48}, 181 (1997)\nand references therein.\n\n\n\\bibitem{Tittel} L. Fleury, A. Zumbusch, M. Orrit, R. Brown\nand J. Bernard, {\\em J. Lumin.} {\\bf 56} 15\n(1993).\nJ. Tittel, R. Kettner, Th. Basche, C. Brauchle,\nH. Quante, K. Mullen,\n{\\em J. Lumin.} {\\bf 64} 1\n(1995).\nM. Vacha, Y. Liv, H. Nakatsuka, T. Tani,\n{\\em J. Chem. Phys.} {\\bf 106} 8324 (1997).\nB. Kozankiewicz, J. Bernard and M. Orrit, {\\em J. Chem. Phys,\n} {\\bf 101} 9377 (1994).\n\n\\bibitem{Reilly2} P. D. Reilly and J. L. Skinner,\n{\\em Phys. Rev. Lett.} {\\bf 71} 4257 (1993).\n\n\\bibitem{Zumofen}\nG. Zumofen and J. Klafter,\n{\\em Chem. Phys. Lett.} {\\bf 219} 303 (1994).\n\n\\bibitem{Geva} E. Geva and J. L. Skinner,\n{\\em J. Phys. Chem. B} {\\bf 101} 8920 (1997).\nE. Geva and J. L. Skinner,\n{\\em Chem. Phys. Lett.} {\\bf 287} 125 (1998).\n\n\\bibitem{Pfluegl} W. Pfluegl, F. L. H. Brown and R. J. Silbey,\n{\\em J. Chem. Phys.} {\\bf 108} 6876 (1998).\n\n\\bibitem{Orrit} A. M. Boiron, P. Tamarat, B. Lounis, R. Brown and M. Orrit,\n{\\em J. Chem. Phys.} {\\bf 247} 119 (1999).\n\n\\bibitem{Barkai1}\nE. Barkai and R. Silbey,\nChem. Phys. Lett., {\\bf 310} 287 (1999).\n\n\\bibitem{Barkai2}\nE. Barkai and R. Silbey, (submitted).\n\n\\bibitem{AK} P. W. Anderson, {\\em J. Phys. Soc. Jpn.} {\\bf 9}, 316 (1954).\nR. Kubo, {\\em ibid} {\\bf 6} 935 (1954).\n% R. Kubo Adv. Chem. Phys. {\\bf 15} 101 (1969)\n\n% Spectroscopy of a chromophore coupled\n% to a lattice of dynamic two level systems 1. Absorption\n% line shape\n% 959 964\n\\bibitem{Reilly} P. D. Reilly and J. L. Skinner,\n{\\em J. Chem. Phys.} {\\bf 101} (2) 959 (1994).\n\n\\bibitem{Feller} W. Feller,{\\em An introduction to probability Theory and\nIts Applications} Vol. 2 (John Wiley and Sons 1970).\n\n\\bibitem{Turbo} S. G. Llewellyn Smith and S. T. Gille {\\em Phys. Rev. Lett.}\n{\\bf 81} 5249 (1998) \nand references therein.\n\n\n\\bibitem{Jap}\n Y. Furukawa, Y. Nakai and N. Kunitomi,\n{\\em J. of the Phys. Soc. Japan} {\\bf 62} No.1. 306 (1993).\n\n% Theory of single molecule optical line shape\n% distributions in low temperature glasses\n% p 8920 8932\n\n%\\bibitem{Brown} F. L. H. Brown and R. J. Silbey\n%{\\em J. Chem. Phys.} {\\bf 108} 7434 (1998)\n\n\n%\\bibitem{remark1}\n%In fact the line width is not well defined mathematically,\n%and there is no general formula for line width of a line shape\n%given in Eqs. (\\ref{eq4},\\ref{eq5})\n\n%\\bibitem{Cohen} Cohen Tan {\\em Phys. Rev. Lett.} {\\bf 83} 3696 (1999)\n\n%\\bibitem{Hovi} J. P. Hovi, A. Aharony, D. Stauffer and B. B. Mandelbrot, {\n%\\em Phys. Rev. Lett.}, {\\bf 77}, 877, (1996).\n\n\\bibitem{Stoneham} A. M. Stoneham, {\\em Rev. Mod. Phys.} {\\bf 41}\n82 (1969).\n\n\\bibitem{AHV} P. W. Anderson, B. I. Halperin, C. M. Varma, \n{\\em Philos. Mag.} {\\bf 25}, 1 (1971). W. A. Philips,\nJ. Low. Temp. Phys. {\\bf 7}, 351 (1972). \n\n\\bibitem{remark} In Table 1 we use $\\omega_0=0$, for $\\omega_0\\ne 0$\nwe have $\\kappa_1=\\sum_n p_n \\nu_n + \\omega_0$ while the higher order \ncumulants are $\\omega_0$ independent. The distribution of $\\omega_0$ depends\non the static disorder \\cite{Stoneham} while we are considering\nthe dynamic disorder due to the flipping TLSs. \n\n\\bibitem{Wang} J. Wang and P. Wolynes, {\\em Phys. Rev. Lett.} {\\bf 74} \n4317\n(1995).\n \n\n\\end{thebibliography}"
}
] |
cond-mat0002046
|
Mn $K$-edge XANES studies of La$_{1-x}$A$_x$MnO$_3$ systems (A = Ca, Ba, Pb)
|
[
{
"author": "F. Bridges"
},
{
"author": "$^1$ C. H. Booth"
},
{
"author": "$^{2}$ M. Anderson"
},
{
"author": "$^1$ G. H. Kwei"
},
{
"author": "$^2$ J. J. Neumeier$^{3}$"
},
{
"author": "J. Snyder"
},
{
"author": "$^4$ J. Mitchell"
},
{
"author": "$^5$ J. S. Gardner$^{2}$ and E. Brosha$^{2}$"
}
] |
We present Mn $K$-edge XANES data for a number of manganite systems as a function of temperature. The main edge (1s-4p transition) for the Ca substituted samples is very sharp, almost featureless, and shifts uniformly upwards with increasing Ca content. The interpretation of this result is controversial because the lack of structure appears difficult to reconcile with a mixture of Mn$^{+3}$ and Mn$^{+4}$ ions. We propose a possible solution in terms of the extended Mn 4p states and hybridization between the Mn3d and O2p states. A small T-dependent feature is present in the main edge; analysis for the charge ordered (CO) sample suggests a distortion associated with the CO state that increases below T$_{CO}$. The manganite pre-edge structure is quite similar to that for a large number of other Mn compounds, with two or three small peaks that are ascribed to 1s-3d weakly allowed dipole transitions plus possibly a small quadrupole component. The weak dipole transitions are explained as arising from a hybridization of the extended Mn 4p state of the excited atom with an odd symmetry combination of Mn 3d states on adjacent Mn atoms. The first pre-edge peak, A$_1$, has a small shift to higher energy with increasing valence while the next peak, A$_2$, is nearly independent of dopant concentration at 300K. However, for the colossal magnetoresistance (CMR) samples the A$_2$ pre-edge peak shifts to a lower energy below the ferromagnetic (FM) transition temperature, T$_c$, resulting in a decrease in the A$_2$-A$_1$ splitting by $\sim$ 0.4 eV. This indicates a change in the higher energy 3d bands, most likely the minority spin e$_g$ plus some change in covalency. In addition, the amplitudes are temperature dependent for the CMR materials, with the change in A$_1$, A$_2$ correlated with the change in sample magnetization. We discuss these results in terms of some of the theoretical models that have been proposed and other recent XANES studies.
|
[
{
"name": "conclusions.tex",
"string": "\\section{Conclusions}\nWe have addressed several issues related to the Mn valence in the substituted\nLaMnO$_3$ materials. Although discussions of these systems often assume\nisolated Mn$^{+3}$ and Mn$^{+4}$ states, we observe no change ($<$ 0.02 eV)\nin the average edge position through the ferro-magnetic transition for the \nCMR systems (Ca, Ba or Pb doped), and in all cases the total edge shift from \n0 to 300K is $<$ 0.04 eV. Although there is no obvious\nstep or kink in the edge, expected for two well-defined valence states, there\nis a very small shape change that can be observed by taking the difference\nof data files at different temperatures. A dip/peak structure develops as T\ndrops below T$_c$ for the CMR samples; the dip/peak separation is $\\sim$ 2 eV\nand is consistent with the splitting calculated\\cite{Elfimov99} for the\np$_x$ and p$_y$ partial DOS when the \nmanganite structure \nchanges from an undistorted to a distorted (LaMnO$_3$) lattice. For CMR samples,\nsuch changes in the local distortions below T$_c$ were deduced earlier \nfrom EXAFS data\\cite{Booth96,Booth98a,Booth98b,Subias97}. At low T the CMR\nsamples are very well ordered, but as T increases there is a rapid increase\nin the local distortions up to T=T$_c$; above T$_c$ the change in disorder \nchanges slowly. The rapid change just below T$_c$ has been associated with \nthe formation of \npolarons. These distortions, now observed in both the XANES and EXAFS data,\nindicate some change in the local charge distribution. However the small size \nof the effect in the XANES spectra needs to be understood. In part it \ncan be attributed to the extended nature of the broad Mn 4p band which \ntends to make the 1s-4p edge transition an average over several Mn atoms. \nHowever a change in covalency - specifically a transfer of charge between \nMn 3d and O 2p states - might also be associated with this structural\nchange, but produce little change in the edge. Support for this possibility\nis obtained from the pre-edge results, summarized below. \n\nFor the CO sample we observe a similar behavior, but in this case the\nstructure in the difference spectra are inverted relative to that \nfor the CMR sample. This indicates that the local distortions increase in \nthe CO state below T$_{CO}$.\n\nThe pre-edge structure provides additional information about the 3d-bands in\nthese materials. Two or three peaks are observed, labeled A$_1$-A$_3$.\nA$_2$ at 300K is essentially independent of concentration while A$_1$ increases\nslowly with x; A$_3$ is only observed for high Ca concentrations.\nFollowing the work of Elfimov {\\it et al.}\\cite{Elfimov99} we attribute\nthese peaks to a hybridization of Mn 4p on the excited atom with an ungerade\ncombination of 3d states on neighboring Mn atoms, i.e. they are not the result \nof splittings of atomic multiplets on the excited atom as is often assumed.\nSimilar explanations for the pre-edge region have recently been proposed\nfor several other transition metal $K$-edges. Consequently the splittings\nobserved are essentially unaffected by the presence of the core hole and should\nbe a good measure of the splittings of the e$_g$ bands which are influenced\nby the hybridization of the Mn 3d and O 2p states. This interpretation\nof the pre-edge does not depend on small distortions of the crystal and\ntherefore also provides a simple explanation for the large pre-edge features\nobserved in the more ordered CaMnO$_3$ material. \n\nIn the calculations of Elfimov{\\it et al.}\\cite{Elfimov99}, the two \nlowest empty bands are the majority and minority spin e$_g$ bands; the \nminority spin t$_{2g}$ band overlaps the latter but is expected to be more\nweakly coupled. U and J must be reduced slightly to\nfit the experimental splitting (2 eV) of the A$_1$ and A$_2$ peaks; U=4eV \nand J$_H$=0.7eV. This indicates an increase in the covalency. The additional\nsmall decrease in the A$_1$-A$_2$ splitting below T$_c$ may suggest\na further change in covalency or hybridization.\n\nThus the picture that emerges is that there is considerable hybridization\nof the energy states (Mn 4p and 3d, and O 2p), with\nsome hole density in the O bands and possibly only small differences in\nthe charge localized on Mn atoms which have different types of \ne$_g$ orbitals. The possibility of distinct types of orbitals can lead \nto orbital ordering\\cite{Anisimov97}, with displacements of the O atoms\nforming J-T-like Mn-O bond distortions when the hopping charge is \nlocalized for times\nof order the optical phonon periods. As a result, the possibility that part\nof the transport takes place via hole density in the O bands needs to be\nconsidered. Note that slowly hopping holes on the O sites would lead to \ndistorted Mn-O bonds while rapid hopping (faster than phonons) would leave \nthe O atom at an average undistorted position.\n\n%Finally, there are other ways in which the Mn charge could be nearly\n%uniformly spread over all the Mn atoms. In the metallic regime, the\n%charge carriers are moving rapidly enough to average out any variations\n%on individual atoms. Above T$_c$ the polaronic charge carriers move more\n%slowly. However, if the polarons that form are somewhat larger than a\n%unit cell, the charge on the Mn atoms within the polaron may have the\n%same average value while the fraction of the charge associated with the\n%O sites in this region may be more or less localized depending on the\n%situation. Another possibility is that there are clusters present -\n%within each cluster the charge carriers moves rapidly as in the metallic state,\n%resulting in an average charge per Mn atom. In such a model the distortions\n%of the Mn-O would have to be associated with the boundaries between clusters.\n%For the CMR samples, the cluster size would need to change dramatically\n%with T to yield the local structure changes\n%observed in XAFS.\n\n\n\\acknowledgements\nThe authors wish to thank G. Brown, C. Brouder, D. Dessau, T. Geballe, J. \nRehr, G. Sawatzky, and T. Tyson for useful discussions and comments. \nFB thanks K. Terakura for sending some of their unpublished\nresults. The experiments were performed at the Stanford Synchrotron Radiation\nLaboratory, which is operated by the U.S. Department of Energy, Division\nof Chemical Sciences, and by the NIH, Biomedical Resource Technology Program,\nDivision of Research Resources. Some experiments were carried out on\nUC/National Laboratories PRT beam time. The work is supported in part by NSF\ngrant DMR-97-05117.\n\n\n\n"
},
{
"name": "intro.tex",
"string": "\\section{Introduction}\n\n\nThe La$_{1-x}$A$_x$MnO$_{3+\\delta}$ systems exhibit a wide range of different\nphenomena depending on the concentration, x, of the divalent substitutional\natom, A (A = Ca, Ba, Sr, Pb, etc.) and the O concentration.\nThese include ferromagnetism,\nantiferromagnetism, charge ordering, a metal-insulator transition, and\nlarge magnetoresistance\neffects.\\cite{Volger54,Wollan55,Schiffer95,Jonker50,Ramirez96,Chen97}\nFor x roughly in the range 0.2-0.5, these systems have a ferromagnetic\ntransition at a transition temperature, T$_c$, a metal/insulator\ntransition (MI) at T=T$_{MI}$, and a \"colossal'' magnetoresistance (CMR) \nwhich reaches it's maximum at\nT$_{MR}$, with T$_{MI}$ $\\sim$ T$_{MR}$ $\\sim$ T$_c$ in many cases. \nThe substitution\nof a divalent ion for La$^{+3}$ formally changes the average Mn valence to 3+x\n(the Mn valence is +3 in LaMnO$_3$ +4 in CaMnO$_3$), and\nis usually thought to introduce holes\ninto the narrow e$_g$ band of Mn 3d-electrons, which are also hybridized with\nO 2p states. It is the changing occupation of this hybridized band with x that\nleads to many of the observed properties. Excess O in LaMnO$_{3+\\delta}$ \n(actually Mn and La vacancies) or La vacancies\ncan also increase the formal Mn valence, thereby adding carriers to\nthe system. \n\nThe coupling between charge and\nmagnetism has been modeled using the double exchange (DE)\nmechanism\\cite{Zener51,Anderson55,deGennes60} plus strong electron-phonon \ncoupling.\\cite{Millis95,Millis96,Roder96} \nIn the ferromagnetic metallic (FM) state, well below T$_c$,\nthe charge carriers are assumed to be highly delocalized (large polarons)\nand spread out over several unit cells for CMR samples.\n\nIn the paramagnetic (PM) state above T$_c$ for CMR samples, there is a significant\nincrease in the distortions about the Mn atoms compared to the low temperature\ndata.\\cite{Booth96,Booth98a,Booth98b,Subias97,Billinge96} It is\ngenerally assumed that this distortion is a result of the charge carriers\nbecoming localized on the Mn atoms. However, mobile holes could also be \nlocated more on the O atoms as is the case for cuprates.\\cite{Nucker88}\nConsequently, there might be very little change in the charge localized on the \nMn atoms above and below T$_c$. This raises the question as to how the energy \nshift of the absorption edge relates to valence and the local environment in \nthese materials. Is there a mixture of ``ionic-like\" +3 and\n+4 states, an average valence, as in a metal where all Mn atoms are equivalent, or\nsomething in between? \n\nExperimentally\\cite{Booth98b,Subias97}, the Mn $K$-edge absorption for the \nCa substituted manganites is\nsharp with relatively little structure, and shifts almost uniformly with dopant\nconcentration, consistent with an average valence state\nof v = 3+x. The sharpness of the edge is suggestive of a transition into\na state that is uniform throughout the sample, and initially we interpreted\nthis result to mean that all Mn sites have comparable local charge densities.\nThis is difficult to reconcile with the usual assumption of a mixture of\npurely {\\it local} ionic Mn$^{+3}$ and Mn$^{+4}$ sites.\nFor example, below we show explicitly that the observed $K$-edge cannot \nbe modeled as a\nweighted sum of the edges of the end compounds LaMnO$_3$ and CaMnO$_3$ for\nthe charge ordered (CO) material with 65\\% Ca. Note however that\neven with similar charge densities, the d-electron wavefunctions and \nthe local environment need not be identical at each \nsite\\cite{Anisimov97,Mizokawa97}. In addition, Tyson {\\it et al.}\\cite{Tyson00} \nhave investigated the K$_{\\beta}$ emission\nwhich probes Mn 3d states through the 3p-1s decay, and report that\nthese spectra for the substituted manganite materials can be modeled as a\nweighted sum of the end compounds although the shifts with valence are small.\n\n%Our recent XAFS work\\cite{Booth98b} and that of\n%Sub{\\'{i}}as {\\it et al.}\\cite{Subias97} have suggested\n%that the observed Mn $K$-edge for the substituted materials These experimental \n%results are difficult to reconcile with the usual assumption of a mixture of \n%purely {\\it local} ionic Mn$^{+3}$ and Mn$^{+4}$ \n%sites. Here we show explicitly that the observed $K$-edge cannot be modeled as a \n%weighted sum of the edges of the end compounds LaMnO$_3$ and CaMnO$_3$. \n%Tyson {\\it et al.}\\cite{Tyson00} have investigated the K$_{\\beta}$ emission \n%which probes Mn 3d states through the 3p-1s decay, and report that \n%these spectra for the substituted manganite materials can be modeled as a \n%weighted sum of the end compounds. \n%%They also stress the high covalency in these materials. \n%We have investigated the edges of several materials in\n%more detail and address this apparent discrepancy.\n\nThe pre-edge structure for the Mn $K$-edge consists of 2-3 small peaks \nlabeled A$_1$-A$_3$ which have Mn 3d character. These features are observed for\nall the transition metals and are generally ascribed to mixture of\n1s-3d quadrupole and 1s-p dipole transitions (made weakly allowed by a hybridization\nbetween 3d states and p-states). Although the latter are assumed to be\ndominant, the interpretation of the A$_i$ peaks is still controversial\\cite{Joly99}. \nTwo important issues are: 1) How large is the quadrupole\ncontribution and when is it important? and 2) How are the dipole transitions made allowed\nsince in many instances, the local environment has inversion symmetry, and in that\ncase\\cite{Manceau92}, the transition is symmetry forbidden? There have been\na large number of papers in the last five years addressing these issues for \nmany of the transition metals, not all of which are in agreement.\nHowever some questions have been answered. Since quadrupole-allowed pre-edge features have \na strong angular dependence, in contrast to the dipole-allowed transitions, \nmeasurements on single crystals, as a function of angle can separate the two contributions.\nSuch studies have shown that quadrupole transitions contribute to the A$_i$ peaks in \nTi,\\cite{Joly99,Aifa97} V,\\cite{Poumellec98,Sipr99} Ni,\\cite{Heumann97} \nand Fe,\\cite{Heumann97} with the largest contribution at the lowest energies of the \npre-edge. The amplitude can be as large as $\\sim$ 4\\% of the absorption edge height\nfor some systems at optimum orientations; but more generally it is \nof order 1\\%, and could be smaller in powdered samples which are orientational \naverages. The dipole-allowed A$_i$ peaks are often in the 5-15\\% range and often\ndo dominate, but not always. For example, for Ti in rutile\n(TiO$_2$), the small A$_1$ peak appears to be primarily a quadrupole feature.\\cite{Joly99}\n\n\nEarly Mn XANES work\\cite{Manceau92,Belli80} \nassumed that the A$_1$-A$_2$ splitting is produced by the crystal \nfield parameter, often called the 10Dq\nparameter, which splits the t$_{2g}$ and e$_g$ states. These investigations did \nnot consider the possibility of a large on-site Coulomb term, U. Recent work, \nusing the Local Spin Density Approximation (LSDA or sometimes LDA) with and \nwithout U, and including the Hund's rule exchange parameter, J$_H$, find a Coulomb\nsplitting of both the t$_{2g}$ and e$_g$ states, with the e$_g$ states \nfurther split by the Jahn-Teller (J-T)interaction.\\cite{Satpathy96}\n\n\nPickett {\\it et al},\\cite{Pickett96} (LSDA model) suggest that these\nsystems are half metallic, with a gap between the O band and a minority spin \nd-band. They also point out that near 25 \\% Ca, all Mn sites could be \nessentially identical if the Ca were uniformly distributed such that there are \ntwo Ca and six La second neighbors to each Mn. Thus for the concentration range \n20-30\\%, the local environment for each Mn may be very similar.\nAnisimov {\\it et al.} and\nMizokawa {\\it et al.}\\cite{Anisimov97,Mizokawa97} suggest that a large\nfraction of the d-electrons are found on the Mn atoms rather than being\ntransferred to the O atom as in an ionic solid (thereby leaving holes in\nthe O band). These calculations yield nearly the same electron density on\neach Mn atom, for sites associated with formal Mn$^{+3}$ and Mn$^{+4}$ valences.\nOther recent papers\\cite{Dai96a,Zhao96,Zhou97,Kwei97c} have also\nstressed the importance of O, and the question of charge localization on\nthe O atoms\\cite{Ju97} or on the Mn atoms has been considered.\n\nSome promising calculations for considering the pre-edge features are those\nof Elfimov {\\it et al.}\\cite{Elfimov99} These\ncalculations indicate that in addition to U and J$_H$,\nthere are appreciable higher order Coulomb terms that must be included and \nthat strong hybridization occurs between the Mn 4p orbitals and the \nMn 3d states on neighboring Mn atoms. The resulting\nsplitting of the majority and minority e$_g$ spin states results in\na splitting of the Mn pre-edge features.\nWe consider these calculations together with some of the new results\non pre-edges in the discussion section.\n\n\nIn this paper we address the valence question and probe the Mn 3d bands using\nthe near edge structure. Specifically we show there is no Mn $K$-edge shift \n(within 0.04 eV) through T$_c$. We also compare the main edge, which is too \nnarrow to arise from a mixture of ionic Mn$^{+3}$ and Mn$^{+4}$, with the edge\nfor a material, Sr$_3$Mn$_2$O$_{6.55}$, that does have a mixture of these ions. \nIn addition, using a careful subtraction\nmethod, we show that there is indeed a small structure in the main edge that\ncorrelates with T$_c$ for the CMR samples and with T$_{CO}$ for the charge ordered\nmaterial. The structure for CMR material is out of phase with that for the CO \nsample - which suggests that there is a distortion for the CO sample that increases\nat low T. We also note that the pre-edge structure has a temperature\ndependence which again correlates with T$_c$ for CMR samples. The \nsplitting of the pre-edge peaks decreases in the ferromagnetic\nphase which may indicate a change in covalency. Finally our interpretation \nof the XANES differs from earlier work on Mn $K$-edges but is consistent\nwith recent studies of other transition metal atoms.\n\n%consider 1) concentration and temperature dependent shifts of the Mn \n%$K$-edges by fitting to either the edge of the same sample at low\n%temperatures or the edge of the end compounds, 2) the edges for other similar Mn\n%compounds, 3) the tiny change in {\\it shape} of the Mn edge with T, \n%4) the pre-edge region which probes the splitting of the 3d bands, and\n%5) the structural information inferred from the EXAFS data.\n\n%Local structure studies show that there are significant changes in\n%the distortions about the Mn atoms when the temperature is lowered\n%through the FM metal/insulator transition for x in the range 0.2-0.5.\n%The width of the Mn-O pair distribution function in XAFS measurements is\n%large above T$_c$ but small at low temperatures, comparable to that\n%observed in CaMnO$_3$.\n%We have attributed the increased disorder above T$_c$ mostly to the partial\n%localization of small polarons.\\cite{Booth98b} Similar changes in the local\n%structure have also been observed in neutron pair-distribution analysis\n%(PDF).\\cite{Billinge96,Louca97} Our detailed investigation of the near\n%edge features presented here, shows that the position of the main edge\n%changes very little with temperature, except for a slight\n%decrease of the edge position at higher temperatures for every sample\n%studied. Some change would have been expected if\n%the Mn states, particularly the Mn 4p states, were highly localized.\n%In a preliminary report\\cite{Bridges00} we have attributed this result to\n%the broad Mn 4p bands found in the calculations of\n%Elfimov {\\it et al.}\\cite{Elfimov99}\n%We also investigate the pre-edge features which vary considerably with Ca\n%concentration, and more surprisingly, show an unexpected temperature\n%dependence. The splitting of the pre-edge features is decreased in the FM\n%state for the CMR samples.\\cite{Bridges00}\n\nIn Sec II we summarize the samples and experimental setup; some\ndetails were given earlier.\\cite{Booth98b} Then in Sec III, we provide a\nmore extensive discussion of the shift of the Mn $K$-edge as a function of\nconcentration and temperature. Here we also present the pre-edge results.\nWe consider the implications of these results in Sec IV.\n\n\\section{Experimental details}\n\\label{exp_details}\n\nMany samples are used in this study, with the average Mn valence changed in\na variety of ways:- divalent substitutions for La$^{+3}$ and\nchanges in the La$^{+3}$ or O concentrations. \nPowder samples of La$_{1-x}$A$_x$MnO$_3$ were prepared by solid state reaction of\nLa$_2$O$_3$, MnO$_2$, and a dopant compound - CaCO$_3$, PbO, BaO, for various \ndivalent atoms, A. Ca substitutions are $x$=0.0, 0.12, 0.21, 0.25, \n0.3, 0.65 and 1.0, and Ba and Pb are 0.33. Several firings with repeated \ngrindings were carried out using temperatures up to 1400$^\\circ$C, with in some \ncases a final slow cool at 1$^\\circ$C per minute. The dc magnetization was \nmeasured using a commercial SQUID magnetometer. The end compounds, \nCaMnO$_3$ and LaMnO$_{3.006}$,\nshow antiferromagnetic transitions at $\\sim$130 and 125 K, respectively, \nwhile the $x$=0.65 sample showed features consistent with a charge ordered\n(CO) transition at 270 K and an AF transition \nat$\\sim$140 K.\\cite{Ramirez96,Chen97} Similar measurements on the\nsubstituted manganites indicates that they are all orthorhombic.\nThe average Mn valence for several Ca substituted samples was also determined by\ntitration (Sec. \\ref{mainedge}).\nSee Refs. \\onlinecite{Booth98a,Booth98b,Snyder96} for further details.\n\nThe LaMnO$_{3.006}$ sample was prepared by grinding\nstoichiometric amounts of La$_2$O$_3$ (Alfa Aesar Reacton 99.99\\%) and\nMnO$_2$ (Alfa Aesar Puratronic 99.999\\%) in an Al$_2$O$_3$ mortar\nand pestle\nunder acetone until well mixed. The powder sample was formed into a 3/4\"\ndiameter pellet using uniaxially presure (1000lbs), and fired in an\nAl$_2$O$_3$ boat under pure oxygen for 12 hours at 1200-1250$^{\\circ}$C.\nNext the sample was cooled to 800$^{\\circ}$C, re-ground, re-pelletized,\nand refired at 1200-1250$^{\\circ}$C for an additional 24 hours. This process was\nrepeated until a single phase, rhombohedral XRD trace was obtained.\nThe reground powder was placed in an\nAl$_2$O$_3$ boat and post-annealed in UHP Ar at 1000$^{\\circ}$C for 24 hrs. The\noxygen partial pressure was about 60 ppm (determined using an Ametek oxygen\nanalyzer). The sample was then quenched to room temperature. Diffraction,\ntitration and TGA measurements\nindicate this sample is essentially stoichiometric,\n%consistent with Mitchell {\\it et al},\\cite{Mitchell99} \nwith an oxygen content of 3.006.\n\nAdditional LaMnO$_{3+y}$ specimens with various average Mn valences were prepared \nat 1300 $^{\\circ}$C in air, followed by three intermediate regrindings. \nThe original specimen was removed \nfrom the furnace at 1300 $^{\\circ}$C and has a Mn valence of 3.150. \nA piece of this specimen was reacted \novernight at 1000 $^{\\circ}$C and removed from the furnace, producing a \nsample with an average Mn valence of 3.206. A nearly stoichiometric\nspecimen with average Mn valence of 3.063 was prepared at temperatures up\nto 1350 $^{\\circ}$C with 4 intermediate regrindings in flowing helium gas.\nFinally, the nonstoichiometric La$_{0.9}$MnO$_3$ specimen was prepared at \ntemperatures of up\nto 1350 $^{\\circ}$C with three intermediate regrindings. It was slow-cooled\nin air at 1.5 $^{\\circ}$C/min to room temperature and had an average Mn\nvalence of 3.312. For each of these samples the valence was determined\nby titration.\n\n\nA sample that should have isolated Mn$^{+3}$ and Mn$^{+4}$ species is also \nneeded for comparison purposes; such a material is \nSr$_3$Mn$_2$O$_{6.55}$.\\cite{Mitchell98} The two species are \ndue to the oxygen defect structure \nthat puts vacancies into the MnO$_2$ planes to form mixtures of square \npyramids and octahedra. This highly insulating material can then be understood \nfrom chemical reasoning to be Mn$^{+3}$ (square pyramids) and \nMn$^{+4}$ (octahedra). Some further justification for this assignment comes \nfrom the compound Ca$_2$MnO$_{3.5}$, which is all Mn$^{+3}$\nand has only square pyramids with vacancies in the MnO$_2$ planes; it is an\nordered superstructure of the single-layer compound.\\cite{Leonowicz85}\nSr$_3$Mn$_2$O$_{6.55}$ was synthesized by firing a stoichiometric mixture of\nSrCO$_3$ and MnO$_2$ at 1650 $^\\circ$C for 12 hr followed by rapid quenching\ninto dry ice. This procedure is essential to prevent decomposition into\n$\\alpha$-Sr$_2$MnO$_4$ and Sr$_4$Mn$_3$O$_{10}$ on cooling and to prevent\noxidation to Sr$_3$Mn$_2$O$_7$. The oxygen content was measured independently\nby iodometric titration and by thermogravimetric analysis, both techniques\nyielding 6.55(1) oxygen atoms per formula unit.\n\nAll XAFS data were collected at the Stanford Synchrotron Radiation Laboratory\n(SSRL). Most Mn $K$-edge data were collected on beam line 2-3 using Si(220)\ndouble monochromator crystals for all samples. Some data were collected on\nbeam line 4-3 using Si(111) crystals, while most of the Mn $K$-edge\ndata for the Ba substituted sample were collected on beam line 10-2 using\nSi (111). The manganite powders were reground, passed through a\n400-mesh sieve, and brushed onto scotch tape. Layers of tape were stacked\nto obtain absorption lengths $\\mu_{\\rm Mn}t\\sim$1 ($\\mu_{\\rm Mn}$ is the\nMn contribution to the absorption coefficient and $t$ the sample\nthickness) for each sample. Samples were placed in an Oxford LHe flow\ncryostat, and temperatures were regulated to within 0.1 K.\nAll data were collected in transmission mode. A powdered Mn metal sample\nwas used as an energy reference for each scan. The pre-edge absorption\n(absorption from other excitations) was removed by fitting the data to a\nVictoreen formula, and a simple cubic spline (7 knots at \nconstant intervals $\\sim$140 eV in $E$) was used to\nsimulate the embedded-atom absorption, $\\mu_0$, above the edge.\n\nThe edge shifts are reported relative to a Mn powdered metal foil for which\nwe take the position of the first inflection point to be\n6537.4 eV.\\cite{Subias97} For each scan, the position of the\nreference edge was determined by fitting the edge to that of a fiducial scan.\nThis provided a correction to the relative edge position consistent within\n$\\pm$ 0.015 eV - see next section.\n\nIn the pre-edge region there is a remnant of the La L$_{I}$ XAFS that \nmust be considered; the oscillation amplitude is about 0.3 \\% of the Mn\nstep height, just before the pre-edge. However, the La $K$-edge XAFS \nshow that there is a \"beat\" \nin the XAFS from about 8.4-10 \\AA$^{-1}$, which for the La L$_{I}$ XAFS \ncorresponds to the range of the Mn XANES. In this beat region the La L$_{I}$ \nXAFS is reduce by another factor of 4; thus the La oscillations underlying \nthe Mn XANES region has an amplitude of about 0.08\\%, much smaller than the \nchanges we investigate. In addition, this oscillation is slowly\nvarying with energy, and would at most produce a slowly varying background. \nConsequently any remaining La L$_{I}$ XAFS are not a problem for the Mn \nXANES study.\n\n\n"
},
{
"name": "paper.tex",
"string": "% Editor of Physical Review B:\n% Please find below a latex version of the revised paper\n% Mn $K$-edge XANES studies of La$_{1-x}$A$_x$MnO$_3$ systems (A =\n% Ca, Ba, Pb) by F. Bridges, C. H. Booth, M. Anderson,\n% G. H. Kwei, J. J. Neumeier, J. Snyder, J. Mitchell, J. S. Gardner\n% and E. Brosha, which we are resubmitting for publication in PRB. The\n% paper is long and the text is in four files - this file and three\n% others called intro.tex, conclusions.tex and ref.tex. The main paper\n% calls the other sections. These are included with this tar file.\n% The figures are sent as a separate uuencoded tar file.\n%\n% The figure cations have the figures commented out and the positions\n% of the figures are indicated in the text by commented out figures.\n% Please contact me if you have any questions.\n%\n%\tResponse to the referee comments:\n%\n%\tBased on the referee's comments we have shorted the manuscript considerably\n%\tincluding cutting some sections, eliminating some figures and combining \n%\tothers. We have also extensively revised several sections to make them clearer.\n%\tWe thank the referees for their comments.\n% \n%\tHowever, we disagree seriously with some of the comments of the second \n%\treferee who claims our interpretation is not correct. It is not clear \n%\twhether this is a real disagreement or he/she misunderstood what we are \n%\tsaying. We have revised several sections to make them clearer. We have searched \n%\tthe recent literature for edge studies of other transition metal oxides \n%\t(TMO) and found several studies which have a similar interpretation to ours. \n%\tIn the revised version we have compared our edge results to other TMO \n%\t(for Ti, Fe V Ni) rather than give background history on Mn edges. As a \n%\tresult, the paper is strengthened considerably.\n%\n%\tBelow we provide detail responses to each referee.\n%\t\n%\tResponse to referee #1.\n%\t1. We have shortened the introduction and the conclusion (point #7) and \n%\tattempted to improve the clarity. My co-authors agreed with this referee and\n%\tthink the paper reads much better now.\n%\t2. The LaMnO3 sample actually is O3.006. I have now been explicit about the \n%\tO content in each sample. The samples are from different groups - and I have tried\n%\tto be complete in describing each set of samples. The discussion has been \n%\tshortened and clarified.\n%\t3 and 4. Figures have been reduced in size, and some that were in the short paper \n%\thave been removed. I think it helps to have all the data on one page but in \n%\tthe interest of shortening the paper, I have referred to the figures in the recent \n%\tpublication.\n%\t(I'm not sure of the editorial rules because I've seen figures reproduced before)\n%\t5. The importance of old fig 5, now fig 4, is that an earlier work claimed there \n%\tis a small edge shift at Tc relative to the edge position at low and high T.\n%\tFig 4 shows the reproducibility of the data and that there is no feature \n%\tnear Tc (within the error of 0.04 eV). I feel one graph is needed.\n%\t(However, I have eliminated old fig 6)\n%\t6. The referee did not understand the significance of old Fig 12 (now Fig 8) so\n%\twe did not present this figure well in the first version. I hope the referee \n%\tfinds the revised version much clearer. First the extra structure correlates well \n%\twith Tc for the CMR samples and with T_CO for the charge ordered sample. (This \n%\textra structure sits on top of a smooth peak for the end compounds LaMnO3 or CaMnO3). \n%\tThe features in this structure for the CMR samples are out of phase with those\n%\tfor the charge-ordered sample, which suggests that the distortions that produce \n%\tthem have opposite signs for CMR and CO samples; -\n%\tie the distortion associated with the CMR material decreases as T is lowered\n%\twhile that for the CO material increases as T is lowered. This discussion has \n%\tbeen completely rewritten.\n%\t7. Discussed above. \n%\tWe thank the referee for her/his comments and careful reading of the manuscript.\n%\t\n%\n%\n%\tResponse to referee #2.\n%\t1. I agree XANES provide a local probe, but the final state depends on the degree of\n%\tlocalization. Narrow bands have well localized states, while broad energy bands \n%\thave more extended states.\n%\tIn this paper we are only considering the near edge structure so only the Mn 4p\n%\tstates are important. Higher energy p states are obviously present - but because\n%\tthey occur at higher energies, they do not play a role for the near edge absorption.\n%\tFor example, the Ca and La p-states occur at higher energies than \n%\tthe Mn K-edge and thus will not mix much with the Mn 4p states; (the\n%\tO 2p is lower, mostly below the Mn 3d band but partially hybridized with the Mn 3d \n%\tstates). Thus the dominant final states for the Mn K-edge are combinations of the \n%\tMn 4p states. The broad bandwidth for the Mn 4p indicates these states are not \n%\thighly localized - this is a well known effect. \n%\tThe overlap of the Mn 4p states is mainly with 2 neighboring Mn atoms (ie not \n%\tfree-electron-like) - and thus the empty Mn 4p eigenstates are mostly a linear \n%\tcombination over three atomic states. This will tend to partially average out \n%\tany charge variations between sites.\n%\t3. I do not understand point 3 as Mn 4p states dominate at the absorption edge energy\n%\tso mixtures with other p-states are not too important. Whether this would change\n%\tother interpretations would have to be considered for specific systems, but in general\n%\tI do not see how it would completely revise other work om metallic systems.\n%\tMore important is the mixing with Mn 3d states on neighboring atoms - which \n%\trequires some overlap of Mn4p states with neighborin Mn atoms\n%\n%\tOther comments:\n%\t- K-beta results of Tyson etal\n%\tThe K-beta results are a little more consistent with a mixture of +3 and +4 than \n%\tare the K-edges. It is not exact and as I stated, the net shift of the K-beta emission\n%\tis small so it is hard to see how well the \"shape\" agrees. The average edge\n%\tposition basically agrees with the average valence, whether it is the K-edge\n%\tshift or the shift of the K-beta peak. It is the shape that should be different\n%\tif there is a change in localization with T. Tyson's work needs to be mentioned \n%\tbut I have not elaborated on it in the further discussion.\n%\n%\t- Earlier Mn edge studies\n%\tI have eliminated the discussion/comparison with other Mn edge studies. This was \n%\tnot meant to be a comprehensive review and first referee also felt it was too long.\n%\tIf references are not given - or at least some hint given (year, or journal or \n%\tone of the authors) it can be difficult to find the specific references the \n%\treferee is considering. Since I've eliminated most of this section it should no longer\n%\tbe a problem.\n%\n%\t- Comment about \"The position of the absorption edge is bond length dependent- - \"\n%\tIn this paper I was referring to the results of the calculation by Elfimov etal.\n%\tThey show explicitly that the edge shifts for a longer bond length in this material.\n%\tThe substituted materials are quasi-cubic, and highly twinned. Polarized XAFS\n%\twould not see a difference between different axes because there is no average \n%\tmacroscopic anisotropy. We have done several polarized XAFS studies over the last ten\n%\tyears and are very aware that an anisotropic crystal will have different edges for\n%\tdifferent directions - but that is not the case here. Without a hint of the reference\n%\tthe referee thinks is important I'm not sure I will find it. Instead I have referred\n%\tto a chapter by Bianconi in the book edited by Konigsberger and Prins - in that\n%\tchapter Bianconi discusses the shift of edge features for molecules when the bond\n%\tlength changes. 1\n%\tHowever, I strongly disagree with the referee's comment: \n%\t\"I also want to point out that this statement is in controversy with the \n%\taffirmation given by the authors that the position of the absorption K-edge \n%\twould be an average over several Mn atoms.\"\n%\tThe effect we see is very small - about a 2% change. If the high temperature\n%\tphase were really a simple sum of two edges, 2 eV apart, it would give an inflection\n%\tpoint in the edge, similar (but a little smaller for CMR samples) to that shown as \n%\ta simulation in Fig 9. The small magnitude of the observed structure in the edge is \n%\tin part the result of partial averaging because of the overlap of Mn 4p states with \n%\tthe next neighbor Mn atoms.\n%\n%\tThe paper has now been substantially revised - we have addressed most of the referee \n%\tcomments and hope it is now acceptable for publication.\n%\t\n% Sincerely\n% Frank Bridges\n%\n% Corresponding author: Frank Bridges\n% Email: bridges@cats.ucsc.edu\n% Address: Physics Department, Kerr Hall\n% Santa Cruz, CA 95064\n% Phone number: (831) 459 2893\n% FAX number: (831) 459 3043.\n%\n% Article type: paper\n% Experiment or theory: Experiment\n%\n%\n% Suggested principal PACS No.: 75.70.Pa\n% Additional PACS No(s).: 61.10.Ht 71.38.+i 72.20.Jv\n%\n%\n\n\\tolerance1000\n\\documentstyle[aps,preprint,tighten]{revtex}\n%\\documentstyle[aps,prb,twocolumn,tighten]{revtex}\n%\\documentstyle[aps,prl]{revtex}\n\n\\input{psfig}\n\n\\begin{document}\n\n\\draft\n\n\\title{ \\bf\nMn $K$-edge XANES studies of La$_{1-x}$A$_x$MnO$_3$ systems (A = \nCa, Ba, Pb)\n}\n\n\\author{F. Bridges,$^1$ C. H. Booth,$^{2}$ M. Anderson,$^1$ G. H. Kwei,$^2$ \nJ. J. Neumeier$^{3}$, J. Snyder,$^4$ J. Mitchell,$^5$ J. S. Gardner$^{2}$\nand E. Brosha$^{2}$} \n\\address{$^1$Physics Department, University of California, Santa Cruz, CA 95064}\n\\address{$^2$Los Alamos National Laboratory, Los Alamos, NM 87545}\n\\address{$^3$Physics Department, Florida Atlantic University, Boca Raton, FL 33431}\n\\address{$^4$JPL, California Institute of Technology, Pasadena CA 91109-8099}\n\\address{$^5$Argonne National Laboratory, Argonne, IL 60439}\n\n%\\date{draft: \\today}\n\n\\maketitle\n\n\\begin{abstract}\nWe present Mn $K$-edge XANES data for a number of manganite systems as a function\nof temperature. The main edge (1s-4p transition) for the Ca substituted samples \nis very sharp, almost\nfeatureless, and shifts uniformly upwards with increasing Ca content. \nThe interpretation of this result is controversial because\nthe lack of structure appears difficult to reconcile with a \nmixture of Mn$^{+3}$ and Mn$^{+4}$ ions. We propose a possible solution\nin terms of the extended Mn 4p states and hybridization between \nthe Mn3d and O2p states. A small T-dependent feature is present in\nthe main edge; analysis for the charge ordered (CO) sample suggests\na distortion associated with the CO state that increases below T$_{CO}$. The \nmanganite pre-edge structure is quite similar to that for a large number of \nother Mn compounds, with two or three small peaks that are ascribed to 1s-3d\nweakly allowed dipole transitions plus possibly a small quadrupole component.\nThe weak dipole transitions are explained as arising from \na hybridization of the extended Mn 4p state of the excited atom with an \nodd symmetry combination of Mn 3d states on adjacent Mn atoms. The first \npre-edge peak, A$_1$, has a small shift to higher energy with increasing \nvalence while the next peak, A$_2$, is nearly independent of dopant \nconcentration at 300K. However, for the colossal magnetoresistance (CMR) \nsamples the A$_2$ pre-edge peak shifts to a lower energy below the \nferromagnetic (FM) transition temperature, T$_c$, resulting in \na decrease in the A$_2$-A$_1$ splitting by $\\sim$ 0.4 eV. This indicates a\nchange in the higher energy 3d bands, most likely the minority spin e$_g$ plus\nsome change in covalency. In addition, the amplitudes are temperature \ndependent for the CMR materials, with the change in A$_1$, A$_2$ correlated \nwith the change in sample magnetization.\nWe discuss these results in terms of some of \nthe theoretical models that have been proposed and other recent XANES studies.\n\n\\end{abstract}\n\n\\pacs{PACS numbers: 75.70.Pa 61.10.Ht 71.30.+h 71.38.+i}\n\n\n\\narrowtext\n\n\n\\input{intro.tex}\n\n\n\\section{Near edge results}\n\\subsection{Main edge}\n\\label{mainedge}\n\nIn Fig. \\ref{edges} we show the Mn absorption $K$-edge for several concentrations\nof Ca, 33\\% Ba and Pb, La$_{0.35}$Pr$_{0.35}$Ca$_{0.3}$MnO$_3$, \na Sr$_3$Mn$_2$O$_{6.55}$ sample \nthat should have a nearly uniform mixture of ionic Mn$^{+3}$ and Mn$^{+4}$, \nsome O excess samples, and a La deficient sample. \n\n\nFor the Ca substituted samples several\npoints are immediately obvious: (1) To first order the main absorption edges \n(ignoring pre-edge structures for now) have almost\nthe same shape for each dopant concentration and shift nearly rigidly to higher\nenergy as the concentration is increased, (2) the edges for the manganite \nsamples are very sharp, roughly half as wide as the edge for the \nSr$_3$M$_2$nO$_{6.55}$ sample, \n(3) there is no obvious kink or structure in the sharp edges for the \nsubstituted (La,Ca) manganite samples \nthat would indicate a simple mixture of Mn$^{+3}$ and Mn$^{+4}$ ions,\n(4) however, there is a tiny shape change, visible in Fig. \\ref{edges}\nfor samples of different concentration, which shifts the {\\it position} of the \ninflection point on the edge relative to the half height position.\n%Note that this shift of the inflection point could be interpreted as evidence\n%for two distinct Mn sites with slightly different fractional charges.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%/\n%\n%\\begin{figure}\n%\\vspace*{-0.4in}\n%%\\hspace*{-0.4in}\n%\\psfig{file=edges.ps,width=3.2in}\n%%\\vspace{-1.35in}\n%\\caption{The Mn $K$-edges for a number of similar manganite systems, corrected\n%for energy shifts of the monochromator (see text).\n%%The edge energies\n%%are determined by the first inflection point of a Mn-foil reference sample.\n%%We have set this inflection point at 6537.4 eV.\\cite{Subias97}\n%}\n%\\label{edges}\n%\\end{figure}\n\n\nThe data for the La$_{0.35}$Pr$_{0.35}$Ca$_{0.3}$MnO$_3$ sample looks \nvery similar to that for La$_{0.7}$Ca$_{0.3}$MnO$_3$\nindicating that replacing some of the La by Pr does not change the local \nelectronic configuration on the Mn. The O excess and La deficient samples \nshow a similar edge shape to LaMnO$_3$, but the edge shift is considerably \nsmaller than expected based on the Mn valence obtained from TGA. The \nshifts for the O excess data are inconsistent with data from other \ngroups\\cite{Subias98,Maurin99} and are included here to show the \nsharpness of the edge. However, such data suggest that the position of the \nMn $K$-edge is determined by several factors and using the Mn valence and \nO content obtained from TGA may not be sufficient.\n\nIn contrast to the Ca substituted materials, Ba and Pb substitution results in\na significantly broader edge, more comparable to the edges of other Mn \noxides\\cite{Manceau92,Liu97,Yamaguchi98} and the \nSr$_3$Mn$_2$O$_{6.55}$ sample. There is relatively more weight in the lower part\nof the edge.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\n%\\begin{figure}\n%\\hspace*{0.2in}\n%\\psfig{file=edge_fits.ps,width=2.8in}\n%%\\vspace{-1.35in}\n%\\caption{Fits of absorption edges to each other. In a) and b) we show fits of \n%the LaMnO$_3$ edge (solid curve) to the 25\\% and 100\\% Ca samples. In c) and d)\n%we show fits of the 50K data (solid curve) to high temperature data for two\n%different samples. Corrections from the reference Mn foil have not been\n%included in this figure.\n%}\n%\\label{edge_fits}\n%\\end{figure}\n\nIn addition to the shift of the inflection point position on the edge with \nCa concentration, as noted above, the region of steepest slope \nis also quite broad. Consequently using the position of the peak in the \nfirst derivative curve as a measure of the average edge position (as we \nand others have done previously) is only an approximate measure of the average \nedge shift. Using the derivative peak yields a roughly linear shift with \nconcentration.\\cite{Booth98b,Subias97,Croft97} Our data and that of \nSub\\'{i}as {\\it et al.}\\cite{Subias97} have the same edge shift per valence unit,\nwhile the shift reported by Croft {\\it et al.}\\cite{Croft97} is smaller. \nThis may be the result of different O content in the samples.\n\n\n\nTo obtain a better estimate of the average edge shift with concentration\n(at room temperature), we have fit the LaMnO$_3$ edge data (or the CaMnO$_3$ data) \nto that for each of the other samples, over the main part of the edge (above the \npre-edge structure). In this procedure it is \nimportant that when the absorption from other atoms is removed, the data \nbase-line below the pre-edge structure be at zero. Each edge is also\nnormalized using some feature of the data; for the data at different \nconcentrations, we normalized over a range of energies well above the edge,\nwhere the XAFS \noscillations are small. Similarly we fit the corresponding \nreference edges (Mn foil) to a reference scan to obtain a net overall edge shift.\nSeveral examples of these fits are shown in Fig. \\ref{edge_fits}. \nAlthough there is a change in shape between LaMnO$_3$ and CaMnO$_3$, \nthe relative shifts determined with either end compound are nearly\nidentical - less than 0.02 eV difference over the entire concentration range.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\vspace*{-0.4in}\n%\\begin{figure}\n%\\hspace*{0.2in}\n%\\psfig{file=valence.ps,width=2.8in}\n%%\\vspace{-1.35in}\n%\\caption{The shift of the Mn $K$-edge with a) Ca concentration and b)\n%titrated valence for La$_{1-x}$Ca$_x$MnO$_3$. Note we do not have titration\n%data for x=0.30.\n%The relative errors are: edge shift, $\\pm$ 0.02 eV, smaller than symbols; Mn \n%valence $\\pm$ 0.02 units, slightly larger than symbols. In this and several \n%following figures, the lines are a guide to the eye only.}\n%\\label{valence}\n%\\end{figure}\n\n\nIn Fig. \\ref{valence}a we plot the relative shifts obtained \nfrom fits to LaMnO$_3$ at room temperature. The shift with \nx is roughly linear with concentration, with a net shift from 0 to 100\\% Ca of \n$\\sim$3 eV. This is considerably smaller than the value 4.2 obtained from the \nderivative peak\\cite{Booth98b,Subias97} and illustrates the effect of the \nshift of the inflection point relative to the half height. However, over the \nstraight part of the plot from x= 0.3-1.0, the slope is 3.3 eV/valence unit,\nquite close to the 3.5 eV/valence unit obtained by \nRessler {\\it et al.}\\cite{Ressler98} for MnO, Mn$_2$O$_3$, and MnO$_2$. The \npoint at x = 0.12 is anomalous, but titration measurements\ngive about the same Mn valence for the 12 and 21\\% samples, which agrees \nwith the comparable edge shifts. The same data is re-plotted as a function\nof the titrated valence in Fig. \\ref{valence}b; in this\ncase the variation with valence is smoother, but slightly non-linear. The\ndifferent values for the titrated valence, compared to the value expected\nfrom the Ca concentration may indicate that there are slight\nvariations in O content in some samples.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\vspace*{-0.4in}\n%\\begin{figure}\n%\\psfig{file=edges_temp.ps,width=3.1in}\n%%\\vspace{-1.35in}\n%\\caption{The shift of the absorption edge for each sample relative to the low\n%temperature data. In panels a)-c), one of the curves (squares) is shifted\n%downward by 0.1 eV for clarity. Relative errors $\\pm$ 0.015 eV; $\\pm$ 0.03 eV\n%for LaMnO$_{3.03}$.\n%}\n%\\label{edges_temp}\n%\\end{figure}\n\n\nWe have also used a similar analysis to investigate any possible edge shift \nas a function of temperature, by fitting the entire edge of the 50K data for \na given sample to all the higher temperature data files. Two examples are shown in \nFig. \\ref{edge_fits}c,d. This figure shows that changes in the shape of the \nmain edge, above and below T$_c$, are quite small (although measurable). The \nlargest relative change is in the pre-edge peaks, to be discussed later. \nIn Fig. \\ref{edges_temp} we show the shift of the edge position as a \nfunction of temperature up to 320K for several sets of samples. \n(We have similar data out to nearly 500K for the \nBa sample and CaMnO$_3$.) Variations in the fit values for the net\nshift, $\\Delta$E$_o$, for several traces at the same temperature\nare less than $\\pm$ 0.02 eV, and fluctuations about the small average \nshift with T are comparable for a given experimental run. Differences\nbetween experimental set-ups or using Si$<$111$>$ or Si$<$220$>$ monochromators\nare less than 0.1 eV.\nFor T less than 300K, the net shift for each sample is very small \n(less than 0.04 eV), but nearly all appear to have a slight decrease at \nhigh T.\n\n\n\n\\subsection{Pre-edge region}\n\n\nIn Fig. \\ref{pre-edge} we plot the pre-edge region as a \nfunction of temperature on an expanded scale for CaMnO$_3$, a CMR sample \nwith 21\\% Ca, and the 33\\% Ba sample. Data for LaMnO$_3$, the CO sample \nwith 65\\% Ca and another CMR sample have recently been published\\cite{Bridges00}\nin a short paper. For these systems the \nmain features are the lower three peaks labeled A$_1$-A$_3$ (near 6539, \n6541, and 6544 eV) and the B peak. The lower two peaks A$_1$ and A$_2$ are\ncommon to all materials although not resolved for the 33\\% Ba data \ncollected using Si (111) crystals which have a lower energy resolution. \nThe comparison of the two Ba data sets in this figure \nillustrate the importance of using high energy resolution. The A$_{3}$ peak\nis not obviously present in most samples. \n%Unfortunately we do not have high\n%temperature data for Ba with high energy resolution. \nIn Fig. \\ref{pre-edge3}\nwe compare the data for the 30\\% Ca CMR sample with the Sr$_3$Mn$_2$O$_{6.55}$\nsample and also show the pre-edge for the Pb sample, all on a more expanded \nscale.\n\n\nThere are several features to note; all the pre-edge \nfeatures start at very nearly the same position regardless of doping and \nthe amplitude of the pre-edge features labeled A increases with average \nMn valence (Ca \nconcentration) as observed in other Mn compounds\\cite{Manceau92} and in a \nprevious manganite study.\\cite{Croft97} There are, however, small shifts of \nthese features with Ca concentration as shown in Fig. \\ref{pre-edge_shift}. \nThe A$_1$ peak energy increases slightly from LaMnO$_3$ to \nCaMnO$_3$, and the A$_1$-A$_2$ splitting decreases from 2.2 to 1.8 eV. \n(The exception is the 65\\% sample, but here the A$_i$ peaks are poorly resolved.)\nFor the substituted samples, the leading edge of the A$_1$ peak remains \nsteep for all concentrations except the 65 \\% sample. Consequently, the \npre-edge for the intermediate concentrations (CMR samples) cannot be \nmodeled as a simple weighted sum of the end compounds LaMnO$_3$ and CaMnO$_3$.\nNote that the leading edge for the Sr$_3$Mn$_2$O$_{6.55}$ \nsample (See Fig. \\ref{pre-edge3}) is broader, consistent with a mixture of \nMn$^{+3}$ and Mn$^{+4}$ ions, and also has a significant A$_3$ peak. \nThe latter is not present in the data for 30\\% Ca.\n\n\nThe Ba and Pb pre-edges are slightly different; the Ba pre-edge features are \nnot as well resolved even for the higher energy resolution data while \nthe A$_1$ peak is largest for the Pb sample (Compare Fig. \\ref{pre-edge3}\nwith Fig. \\ref{pre-edge}). \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\vspace*{-0.4in}\n%\\begin{figure}\n%\\hspace*{0.2in}\n%\\psfig{file=compd.ps,width=2.7in}\n%%\\vspace{-1.35in}\n%\\caption{The temperature dependence of the pre-edge region for CaMnO$_3$,\n%21\\% Ca, and 33\\% Ba; note different temperature ranges.\n%The CaMnO$_3$ sample shows little change of the pre-edge region below 300K\n%while the A$_1$ and A$_2$ peaks are temperature dependent\n%for the CMR samples. The Ba pre-edge\n%data, collected using Si(220), are much sharper than data collected using\n%Si(111) crystals, and show a splitting of the A peak. Note that the data\n%collected using Si(220) in c) have been shifted down by 0.15 eV for clarity.\n%}\n%\\label{pre-edge}\n%\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nThe most striking feature in Figs. \\ref{pre-edge},\\ref{pre-edge3} is the \nvariation in the \nintensity of the pre-edge peaks and the shift of A$_2$ as T increases through \nT$_c$ for the CMR samples. In contrast, the change for LaMnO$_3$ is small \nup to 300K.\\cite{Bridges00} For the 21\\% Ca sample in Fig. \\ref{pre-edge}\nthe A$_1$ peak decreases in amplitude while the A$_2$ and B-peaks increase \nwith increasing T; the A$_2$ peak is sharpest at 300K and clearly shifts \ndownward below T$_c$ (0.4 - 0.5 eV, depending on the background function \nused). See the solid triangles in Fig. \\ref{pre-edge_shift}. The change \nin the A-peaks for the 33\\% Ba sample (using the high resolution monochromator)\nappear to follow the same trend as observed for the Ca data \n(Fig. \\ref{pre-edge3}) but the A$_2$ peak is not as well resolved. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\begin{figure}\n%\\psfig{file=compd3.ps,width=3.3in}\n%%\\vspace{-1.35in}\n%\\caption{The temperature dependence of the pre-edge region for\n%La$_{0.7}$Ca$_{0.3}$MnO$_3$, Sr$_3$Mn$_2$O$_{6.55}$ and the 33\\% Pb\n%samples up to roughly 300K. The leading edge of the A$_1$\n%peak is broader for Sr$_3$Mn$_2$O$_{6.55}$ than for the 30\\% Ca sample in a).\n%In b) the Pb samples has the largest A$_1$ peak at low T. Note that the\n%scale is expanded compared to the previous figures and the B peak is not shown.\n%}\n%\\label{pre-edge3}\n%\\end{figure}\n\n\nThe largest temperature dependence is observed for the \nCaMnO$_3$ sample above 300K (See Fig. \\ref{pre-edge}a), with the largest \nincrease occurring for the B-peak. Also, the amplitude of the peak at the \ntop of the edge, commonly called the ``white line'', (see Fig \\ref{edges}c \nat 6554 eV for example) decreases slightly at high T.\nThese effects become much larger at only slightly higher temperatures and\nwill be treated in a separate paper. For the CMR samples, we associate \nthe temperature dependent changes in the amplitude of the pre-edge features \nwith changes in charge localization/hybridization.\n\n\n\n\n\\subsection{Difference Spectra}\nMore detailed information can be obtained by examining the change in the shape \nof the XANES region as a function of temperature. The files are first shifted to\ncorrect for any small changes in the energy of the monochromator \nand all spectra are carefully normalized as \ndiscussed earlier. The difference spectra \nare obtained by subtracting the data at 300K from all the \ndata files (at different temperatures) for a given \nsample. This approach was used originally to investigate the pre-edge \nregion for the 21\\% sample,\\cite{Bridges00} but considerable structure \nwas found at energies corresponding to the main edge, for both the CMR \nand CO (x=0.65) samples. Several examples of these difference spectra are \nshown in \nFig. \\ref{diff_spec} for LaMnO$_3$, CaMnO$_3$, and the 21, 30, and 65 \\% \nCa substituted samples. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\begin{figure}\n%\\psfig{file=pre_conc.ps,width=2.9in}\n%\\vspace*{0.1in}\n%\\caption{The positions of the pre-edge features as a function of Ca\n%concentration - open symbols, and solid circles, 300K. The error in peak position is\n%$\\pm$ 0.1 eV. For the CMR samples, the A$_2$ peak is shifted to a\n%lower energy for T $<$ T$_c$ - the filled triangles are for 50K.\n%}\n%\\label{pre-edge_shift}\n%\\end{figure}\n%\n\nIn the pre-edge region, the temperature variation of the A$_1$ and A$_2$ peaks for \nthe 21 and 30\\% Ca (CMR) samples is very clearly visible in Fig. \\ref{diff_spec}c,d;\nit begins at T$_c$, \nwith most of the change occurring over a 60K range just below T$_c$. \nThe temperature-dependent changes of the pre-edge are comparable in both \nsamples, with the magnitude of the change \nof the A$_2$ peak being roughly 50-70\\% that of the A$_1$ peak. \nFor the 65\\% Ca sample,\nchanges of the A$_i$ with T are also observed in the different spectra, but \nthe amplitudes are considerably smaller, and interestingly, the \nphase is inverted - the A$_1$ difference peak decreases instead of increasing. \nFor the LaMnO$_3$ sample (Fig. \\ref{diff_spec}a) there is essentially no \nstructure in the difference \nspectra over the pre-edge energy range, but surprisingly \nthere are small peaks in this range for CaMnO$_3$ (See lower part of \nFig. \\ref{diff_spec}a),\nwith the largest peak in the difference spectra occurring {\\it between}\nA$_2$ and A$_3$ - this suggests\nthat there \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\newpage\n%\\begin{figure*}\n%\\hspace*{0.5in}\n%\\psfig{file=diff_compd.ps,width=5.5in}\n%\\vspace*{-0.2in}\n%\\caption{The difference spectra as a function of temperature for 21, 30 and\n%65\\% Ca, and the end compounds. The main edge is included above each\n%set of traces (multiplied by 0.03 to fit on the graph) to show where\n%the structure is located relative to the edge. Note the inversion of the\n%structure for the CO sample (T$_{CO}$ = 270K) (b), compared to the CMR \n%samples (c) (21\\% Ca, T$_c$ $\\sim$ 210K) and d) (30\\% Ca, T$_c$ $\\sim$ 260K).\n%}\n%\\label{diff_spec}\n%\\end{figure*}\n%\\vspace*{7in}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\noindent are in reality more than three pre-edge peaks.\n\nThere is also well defined structure in the difference spectra over the \nenergy range of the main edge, although it is only a few percent of the \nedge in amplitude. For LaMnO$_3$ there is a broad feature over most of the \nedge region which increases as T is lowered. CaMnO$_3$ has a similar feature\nbut it is larger and narrower \n(Fig. \\ref{diff_spec}a). Both appear to correspond to the temperature \ndependent peak near the top of the edge (the ``white line'' mentioned\nearlier) which is sharpest at low T. For the CMR samples, there is additional \nstructure on top of this broad peak - a dip at 6551 and a peak at 6553 eV \n(2 eV apart). \nAnother dip/peak occurs just above the edge at 6555-6556.5 eV. \nThe CO sample also shows structure over this energy range but again the phase \nis inverted relative to the CMR samples (i.e. a peak/dip at 6552 and6554 eV),\n- this phase inversion thus extends over the entire near-edge region. \n\n\\section{Discussion}\n\n\\subsection{Main edge}\n\n\n\nFor Mn atoms the main $K$-absorption edge represents transitions mainly from \nthe atomic 1s state to the empty Mn 4p band.\nThe XANES results show that this edge is very sharp for the Ca-substituted\nsamples, the O-excess samples and the La deficient sample. The width of\nthe edge (roughly 5-6 eV) is narrower than the edge for most other\nMn compounds and the shift in edge position is $\\sim$ 3 eV for a valence\nchange of +1. No obvious indication of a step or double edge structure is\npresent that would indicate two distinct valence states.\nIf completely localized Mn$^{+3}$ and Mn$^{+4}$ ions were present on time\nscales of 10$^{-14}$ sec, the edge should have a smaller average slope and\ngenerally be broader, as would be expected for a mixture of fine powder\nof LaMnO$_3$ and CaMnO$_3$. To model this explicitly, we compare in\nFig. \\ref{sim_65} the experimental edge for the 65\\% Ca (CO) sample and a\nweighted sum of the +3 and +4 end\ncompounds; clearly the experimental edge is much sharper as noted\npreviously.\\cite{Booth98b,Subias97}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\begin{figure}\n%%\\hspace*{-0.4in}\n%\\psfig{file=sim_65.ps,width=3.0in}\n%%\\vspace{-1.35in}\n%\\caption{A comparison of the data for the 65\\% Ca sample and a simulation\n%obtained from a 35-65\\% weighted sum of the LaMnO$_3$ and CaMnO$_3$ end\n%compounds. The experimental edge is significantly sharper.\n%}\n%\\label{sim_65}\n%\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nIn contrast to the Ca-doped samples, the edge for Sr$_3$Mn$_2$O$_{6.55}$ is\nmuch broader (Fig. \\ref{edges}a), with a width of 10-12 eV. There is also\na change of slope of the main edge for this sample that is consistent with\ntwo valence states, but the shape is more complicated.\nNote that a combination of two edges each $\\sim$5 eV wide, separated by\n$\\sim$3 eV (the\nseparation for a valence change of 1) would yield an\nedge of width $\\sim$ 13eV. Thus the width and structure of the\nSr$_3$Mn$_2$O$_{6.55}$ edge are both consistent with the expectation\nthat two valence states are present in this sample --\nMn$^{+3}$ and Mn$^{+4}$. Similarly our data for Mn$_3$O$_4$, which\nhas a mixture of +2 and +3 valence states, has a very broad edge of\n13-14 eV (not shown).\n\nThe edges for the Ba and Pb samples are also broader; they have\na break in slope and more amplitude in the lower part of the edge that\nmight suggest two valence states. In this regard they are quite\ndifferent from the Ca substituted samples for which the shape of the\nmain edge for the CMR samples does not change much from that\nof LaMnO$_3$. In addition, the net shifts of the edges for the 33\\% Ba,\nand Pb samples are smaller than expected; the additional structure\nnear 6546-48 eV shifts the lower part of the edge down in energy while the\ntop of the edge is close to the position of the 30\\% Ca samples. The net\nresult is a very small overall average edge-shift compared to LaMnO$_3$. \n\nWe have also shown by fitting over most of the edge, that there is \nno significant change in the average edge position for any of \nthe substituted manganite samples near T$_c$. This agrees with\nour earlier result\\cite{Booth98b} in which we averaged data points above\nand below T$_c$. The new analysis also indicates that there is consistently\na slight decrease in edge position at the highest temperatures, that\nis largest for CaMnO$_3$ and Sr$_3$Mn$_2$O$_{6.55}$. The reason for this \ndownward shift is not yet clear but may be related to the temperature\ndependence of the B peak (since an increase in the B-peak intensity effectively\nshifts the lower section of the edge to lower energy).\n\n\nHowever, the lack of any temperature dependence below T$_c$ disagrees with \nthe earlier work of \nSub\\'{i}as {\\it et al.},\\cite{Subias97} who report a 0.1 eV decrease \nin edge position up to T$_c$ for La$_{0.6}$Y$_{0.07}$Ca$_{0.33}$MnO$_3$ \nand then a 0.09 eV increase up to 210K. The small but important change\nin the shape of the main edge in Fig. \\ref{diff_spec} provides a partial\nexplanation for this discrepancy.\nSub\\'{i}as {\\it et al.} assumed no change in edge shape and calculated difference\nspectra for each temperature. Under this assumption, the amplitude of the\npeak in the difference spectra would be proportional to the energy shift of the\nedge. The additional structure observed in the difference spectra indicates \nthere is a shape change rather than an overall edge shift.\n\nThe difference spectra (Fig. \\ref{diff_spec}) also show clearly that the\nadditional structure is dependent on T$_c$. For the CMR samples, the \ndip-peak structure, superimposed on the peak observed for the end compounds,\nbegins to be observable near T$_c$ and grows rapidly in the 60-100K\nrange just below T$_c$. This structure means that compared to the edge at \nlow temperatures (undistorted Mn-O bonds)\nthe edge above T$_c$ (distorted Mn-O bonds)\nhas the upper part of the edge shifted upward in energy\nwhile the lower part is shifted downwards. The separation between \nthe dip-peak structure is about 2 eV (see vertical dotted lines). \n\nThis raises several questions - for the CMR samples is there a mixture of\n+3 and +4 sites as usually assumed? If so why is the edge structure so \nsmall? Can the \nsmall structure observed in the difference spectra be explained in \nsome other way? One aspect that must be included is the very large width \nof the Mn 4p density of states (DOS), roughly 15 eV wide, that is found \nin two quite different \nrecent calculations.\\cite{Elfimov99,Benfatto99} The main edge is due to\ntransitions into this band and the calculated absorption\nedge\\cite{Elfimov99} (broadened by the core-hole lifetime) is very \nsimilar to that observed experimentally.\\cite{Bridges00}\nThe broad width means that the 4p states are\nextended and not localized on one Mn atom. Consequently,\nthe $K$-edge will correspond to a Mn valence partially averaged \nover several Mn atoms, and thus will be less sensitive to variations \nin local charge on different Mn sites.\n\n\nAnother possibility is that the system is more covalent and that there\nare some partial holes in the O 2p band which is hybridized with the Mn 3d\nstates. This is supported by several calculations and by the observation\nof holes in the O 2p band in absorption studies\\cite{Ju97}. Such holes may \nplay an important role in the unusual transport of these materials. In\ncalculations, Anisimov {\\it et al.}\\cite{Anisimov97} and \nMizokawa and Fujimori\\cite{Mizokawa97}\nobtain two types of Mn e$_g$ configurations with almost identical local \ncharge densities. In both calculations\nthere are distortions of the Mn-O bond distances. For the calculation\nof Anisimov {\\it et al.}\\cite{Anisimov97}, one configuration \n%(we label it e$_{g1}$)\nis symmetric in the $ab$ plane with four small equal lobes directed towards\nO while the other \n%(e$_{g2}$) \nhas two large (and two small) lobes, again directed towards O atoms in the\n$ab$ plane. The more symmetric case is\nassociated with a formal Mn$^{+4}$ site and the other state with Mn$^{+3}$\nbut because the charge densities are comparable would not lead to \nsignificantly different edge shifts.\n%These structures are depicted schematically in Fig. \\ref{Mn_dist2} for a\n%50\\% doping level, and if stable, can lead to orbital ordering as \n%observed in neutron scattering\n%experiments;\\cite{Jirak85} they also produce slightly different magnetic\n%moments. \nThe recent calculations of Elfimov {\\it et al.}\\cite{Elfimov99}\nare also relevant. To fit the observed splittings of A$_i$ peaks (2.2 eV for \nLaMnO$_3$ and 1.8 eV for CaMnO$_3$), U and J$_H$ had to be lowered from the \nvalues in the first calculation\\cite{Elfimov99} - to 4 eV and 0.7 eV, \nrespectively,\\cite{Bridges00} which implies higher covalency.\n\nThe remaining question to be answered about the main edge for the CMR samples\nis what is the explanation for the small structure in the difference spectra\nas T is lowered below T$_c$? A possible answer is again found in the calculations\nof Elfimov {\\it et al.}\\cite{Elfimov99}. They find that the position of the \n4p partial DOS is bond-length dependent - it occurs at a lower energy\nwhen the Mn-O bond lengthens (p$_x$ orbitals in their paper) and is at\na higher energy for shorter bond lengths ((p$_y$ and p$_z$).\nSuch a shift is expected; in polarized XAFS experiments on high T$_c$ materials\nwe have observed edge shifts between the c- and a-axes. In addition, studies\nof molecules show that the edge shifts to higher energy when the bond length\nshortens.\\cite{Bianconi88}\nThe separation between the partial DOS for p$_x$ and p$_y$ is about 2 eV\nin Elfinov {\\it etal}'s calculation when they use distortions similar to those \nobserved in LaMnO$_3$; we expect to see some evidence of this splitting \nin the experimental absorption edge, although it is lifetime broadened\nand the 4p states are extended. We propose that the tiny dip-peak\nstructure observed for the CMR materials is the result of the \ndifferent positions for the partial DOS for p$_x$ and p$_y$. The\ndip-peak splitting is also about 2 eV but it is not clear whether this \nis significant or a coincidence.\n\nFor the CO sample there is also structure in the edge but the phase is inverted.\nIf the above explanation for the {\\it dip-peak} structure in the difference spectra\nfor CMR samples is correct then it suggests that the {\\it peak-dip} feature for the\nCO sample is also produced by local distortions - but in this case by a\nlocal distortion that starts at the charge ordering temperature, T$_{CO}$=270K, \nand {\\it increases} as\nT is lowered. Such a model then provides a simple interpretation \nfor the unusual lack of temperature dependence (reported\\cite{Booth98b} but \nnot explained) for \n$\\sigma^2$ for this sample. The surprise is that at least the thermal phonon \nbroadening should have caused some increase in $\\sigma^2$ with T.\nHowever, if there is a distortion associated with the CO state, then \nthere must be an associated broadening contribution $\\sigma_{CO}$, for the \nMn-O pair distribution\nfunction, $\\sigma^2_{CO}$, that is zero above T$_{CO}$ and increases as T \nis lowered below T$_{CO}$. Then the \ntotal variance for the Mn-O bond, $\\sigma^2_{Mn-O}$, will be given by \n\n\\begin{equation}\n\\sigma^2_{Mn-O}(T) = \\sigma^2_{phonon}(T) + \\sigma^2_{CO}(T) + \\sigma^2_{static}\n\\end{equation}\n\n\\noindent where $\\sigma^2_{phonon}(T)$ is the phonon contribution,\nand $\\sigma^2_{static}$ is a static (temperature-independent) contribution\nfrom disorder. $\\sigma^2_{phonon}(T)$ should be comparable to that for\nCaMnO$_3$, since we see the same phonon component for both pure CaMnO$_3$\nand La$_{0.79}$Ca$_{0.21}$MnO$_3$ above T$_c$. To make $\\sigma^2_{total}$\nnearly independent of T means $\\sigma^2_{CO}(T)$ and $\\sigma^2_{phonon}(T)$\nalmost cancel for temperatures below 300K.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 10 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\begin{figure}\n%%\\hspace*{-0.4in}\n%\\psfig{file=D_s2_CO.ps,width=3.2in}\n%%\\vspace{-1.35in}\n%\\caption{\n%A plot of $\\sigma^2_{CO}(T)$ for the Mn-O bond as a function of T (65\\% Ca sample).\n%$\\sigma^2_{CO}(T)$ is extracted from the $\\sigma^2_{Mn-O}$(T) data presented in\n%Ref. \\onlinecite{Booth98b}. T$_{CO}$ $\\sim$ 270K.\n%}\n%\\label{D_s2_CO}\n%\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nWe can extract the CO contribution\nfollowing the method in Ref. \\onlinecite{Booth98b} for calculating $\\Delta \\sigma^2$\n- we fit the two highest T data points (from Ref. \\onlinecite{Booth98b}) to \n$\\sigma^2_{phonon}(T)$ + $\\sigma^2_{static}$\nand then subtract these contributions from the data. In Fig. \\ref{D_s2_CO} we\nplot the result of this analysis for the 65\\% Ca sample. From\nthis figure, the maximum value for $\\sigma^2_{CO}(T)$, is roughly\n10\\% of that associated with polaron formation for the\nCMR samples. Such an increased distortion for charge ordered material makes sense\n- as the sample\nbecomes charge or orbital ordered, there is more room for the longer Mn-O bonds\nto lengthen, while for a random arrangement of orbitals, the series of long\nand short Mn-O bonds are more constrained.\n\n\n\\subsection{Pre-edge region}\n\\label{preedge_region}\n\n\\subsubsection{Background}\n\nThe pre-edge results provide additional information about the nature of the\nelectronic states. For many of the transition elements, 1-3\npre-edge peaks, A$_i$, occur well below the main edge ($\\sim$ 15eV below) and are \nassigned to transitions to empty states with d-like character,\ni.e. these are 1s to 3d$^{(n+1)}$\ntransitions\\cite{Griffith61,Sugano70} where n is the initial number of d electrons \nand n+1 includes the excited electron in the final state, which usually includes \nthe effect of a core hole. The 1s-3d transitions are directly allowed through the very weak\nquadrupole transition\\cite{Manceau92,Joly99,Aifa97,Poumellec98,Sipr99,Heumann97} or allowed \nvia an admixture of 3d and 4p states\\cite{Manceau92,Joly99,Poumellec98,Heumann97,Bridges00}. \nIn the pre-edge region there may also be some hybridization with the O 2p states.\nIf the metal site is centrosymmetric, there is no mixing of 3d and \n4p states on the excited atom and 1s-3d dipole transitions are strictly \nforbidden;\\cite{Manceau92} however local distortions\ncan make such 1s-3d transitions very weakly dipole allowed. Three aspects need to be \nrecognized in considering the Mn pre-edge in the present study.\n\n\\begin{itemize}\n\n\\item {\\it Quadrupole interactions}\\\\\nAlthough the quadrupole interaction is weak it has a clear signature through the \nangular dependence of the absorption process. Based on recent studies on oriented \nsingle crystals\\cite{Joly99,Aifa97,Poumellec98,Sipr99,Heumann97}\n%have shown that this absorption component can be observed in many transition metal $K$-edges\n%but generally it is much smaller than the weakly allowed dipole transitions. From these studies \nwe estimate that quadrupole-allowed peaks will be at most 1\\% in powder samples \nwhich is considerably smaller than the A$_i$ peaks observed for the substituted manganites\nbut perhaps not negligible. A small quadrupole component, as seen for \nexample in FeO,\\cite{Heumann97} may well be present. \n%For these transitions on the absorbing atom, the effect of the core hole must be\n%included.\n\n\\item {\\it Dipole allowed via 3d-4p mixing on the absorbing atom}\\\\\nIf the Mn site lacks inversion symmetry, then in principle there will be mixing of the\nMn 3d and 4p states on the central atom. Consider a system that is nearly cubic\nbut has a small distortion that removes the inversion symmetry - i.e. the metal atom is \nslightly displaced such that the bonds on opposite sides of the metal atom are slightly\ndifferent. Then the mixing parameter is $\\sim$ $\\delta_l/r_o$ and the matrix element will\nbe proportional to $(\\delta_l/r_o)^2$, where $\\delta_l$ is the difference in opposite\nbond lengths and $r_o$ is the average bond length. An example is the V site in V$_2$O$_5$; \nhere the VO$_6$ octahedron is strongly distorted,\\cite{Poumellec98,Sipr99} and along \nthe c-axis, the two \nV-O bond lengths are 1.577 and 2.791 \\AA~respectively. Experimentally there is a large \npre-edge peak for the V $K$-edge that can be modeled assuming 3d-4p mixing on \nthe absorbing atom plus the effect of a corehole. \n%Such calculations \n%suggest that small distortions, at \n%the 1\\% level, would have a very small contribution, comparable to the quadrupole \n%contribution. Thus symmetry breaking by phonons for a cubic system would produce very \n%small peaks. However, if the crystal\n%is already strongly distorted as for V$_2$O$_5$, phonons may make the pre-edge \n%peaks temperature dependent.\n\n\\item {\\it Dipole allowed via 4p mixing with neighboring metal atom 3d states}\\\\\nIn several cases, a significant pre-edge peak is observed in a cubic crystal, which \ncan't be explained by the above 3d-4p mixing on the excited atom. Multi-scattering \ncalculations for such systems often show that large clusters are\nneeded before the pre-edge features are produced - scattering paths are needed which \ninclude many further neighbors, particularly the second neighbor metal atoms. \nAn equivalent result emerges from band theory calculations where the hybridization of \nextended states is important. Here the dipole transition can be made allowed via mixing\nof the 4p state on the central atom with the 3d states on neighboring atoms. Projections\nof the density of states with p character (p-DOS) for such systems show small features \nat the energies of the 3d states; such features are not observed in the p-DOS when the\npre-edge feature is a quadrupole transition. In the limit of multi-scattering \ncalculations with very large clusters, the two approaches (band theory and \nmulti-scattering) should be equivalent.\n\n\\end{itemize}\n\n%The arguments for interactions with the second neighbor metal atoms (third point above) \n%may be more intuitive for the multi-scattering calculations since the neighboring metal \n%atoms need to be included to obtain a pre-edge peak in some nearly cubic systems. \n%The symmetry argument may not be as obvious and it is useful to give an example. \nTo have a mixing of 3d with the 4p states (to make a state of p-character), \none needs a combination\nof 3d states that has odd parity as pointed out by Elfimov {\\it et al.}\\cite{Elfimov99}.\nIt is easy to obtain such a state if a linear combination of\n3d states on two neighboring Mn atoms is used and the p-states are extended enough to\npartially overlap them.\nSpecifically, consider 3 Mn atoms in a line - a central excited atom (0) and\nleft (L) and right (R) atoms - with $\\Psi_{4p}$(0) being the 4p state on the\ncentral atom, and $\\Psi_{3d;x^2-y^2}$(R$_R$) and $\\Psi_{3d;x^2-y^2}$(R$_L$)\nbeing the 3d$_{x^2-y^2}$ states centered on the right and left atoms. Then a state\nwith odd symmetry about the central atom is given by\n\n\\begin{equation}\n\\Psi_{total} = \\alpha \\Psi_{4p}(0) +\n\\frac {\\beta} {\\sqrt{2}}(\\Psi_{3d;x^2-y^2}(R_L) - \\Psi_{3d;x^2-y^2}(R_R))\n\\end{equation}\n\n\\noindent where $\\alpha$ is essentially 1.0 and we ignore the intervening O atom\nvia which the hybridization occurs. The small parameter, $\\beta$, is a measure\nof the hybridization and is strongly dependent on the overlap of the 4p and 3d\nwavefunctions on different Mn atoms, and hence on the distance between them.\n\n\\subsubsection{Application to the substituted manganites}\n\nThe pre-edge for the substituted manganites follows the general trends observed\nfor other Mn systems quite well. Three A-peaks are observed for CaMnO$_3$; A$_2$ is\n larger than the A$_1$ peak, and the A$_2$-A$_1$\nsplitting is smaller (high valence - +4) than for other samples. The LaMnO$_3$ case\nis similar; the A$_2$-A$_1$ splitting is largest (lower valence +3) and the overall\nA peak amplitude is smallest. However, the A$_2$ peak is larger than expected\nfrom the literature for Mn$^{+3}$ states in other compounds\\cite{Manceau92},\npossibly because of increased local distortions in this compound. \n\nHowever, there are difficulties with some of the earlier \ninterpretations\\cite{Belli80,Manceau92} in which the dipole allowed transitions\nare assumed to originate from a 3d-4p mixing on the excited atom. First\nthe A$_i$ peaks appear for both distorted and undistorted systems. Second, the\namplitude (particularly for the relatively undistorted system CaMnO$_3$) is too large\nto be a 1s-3d transition made allowed by a slight breaking of inversion\nsymmetry about the excited Mn atom. Recently, based on the calculations of \nElfimov {\\it et al.}\\cite{Elfimov99}, we have interpreted A$_1$ and A$_2$\nas dipole allowed via a mixing of Mn 4p states with Mn 3d states on neighboring\nmetal atoms\\cite{Bridges00}. The projected p-DOS in the calculations of\nElfimov {\\it et al.}\\cite{Elfimov99}\nshow two features in the pre-edge region, which indicates that dipole-allowed \ntransitions should be present. In addition, the broad Mn 4p band obtained\nin that work and by by Benfatto {\\it et al.}\\cite{Benfatto99}, also implies that \nthe 4p states are indeed extended - a necessary requirement\nfor mixing with the 3d states on the neighboring metal atoms.\nSimilar interpretations have been given recently for other transition metal systems\nthat are cubic or very nearly so; Fe in FeO\\cite{Heumann97} and Ti in \nrutile\\cite{Joly99,Aifa97}.\nA mixing with the 3d states on neighboring Ti atoms was also reported in the layered\ndisulfide TiS$_2$\\cite{Wu97}.\n\nThe calculations of Elfimov {\\it et al.} also show that there is a splitting\nof the unfilled 3d bands - the lowest is the majority e$_g$ band (which may be\npartially filled via doping); the next two are the minority e$_g$ and t$_{2g}$\nbands which partially overlap. The coupling with the t$_{2g}$ is expected to be\nsmaller since these orbitals are of the form d$_{xy}$, which has reduced\noverlap with the Mn 4p in a $\\pi$ bonding configuration. The splitting of\nthese e$_g$ bands depends both on\nJ$_H$ and on the degree of covalency/hybridization. As reported\nrecently,\\cite{Bridges00} adjusting the parameters in this calculation so that\nthe theoretical splitting is close to the 2 eV observed experimentally,\nresulted in U=4 eV and J$_H$=0.7 eV. These lower values also suggest an\nincrease in covalency and hence that the charge is shared between Mn and O.\nConsequently, there is a non-zero density of holes in the O bands, in\nagreement with Ju {\\it et al.}\\cite{Ju97} and these O-holes may play an\nimportant role in the unusual transport of these materials.\nFor the CMR samples, the additional decrease in the A$_1$-A$_2$ splitting \nfor T$<$T$_c$ may suggest a further increase in covalency.\n\nFinally the temperature dependence of the A$_i$ peak amplitudes is still not \nexplained.\nComparing the pre-edges of distorted LaMnO$_3$\\cite{Bridges00} with almost\nundistorted CaMnO$_3$ (See Fig. \\ref{pre-edge}) would suggest that as the \nCMR samples change from distorted above T$_c$ to ordered at low T, the A$_2$\npeak would increase relative to A$_1$. Experimentally the reverse is true.\nHowever, we still suggest that the observed temperature dependence arises\nfrom the change in local structure, based on the fact that the changes\nfor the CMR and CO samples are out of phase for both the pre-edge features\nand the structure in the main edge.\n\n\\subsubsection{Other Models}\nAnother general feature that emerges from our data is that although\nthe main changes occur just below T$_c$, there is also a gradual change\nto the fully ordered state as the sample is cooled well below T$_c$, and\nthe local structure continues to change down to 50K and below. Consequently\nthere may be clusters formed at T$_c$ that grow as T is lowered. We have\ninterpreted our local distortion results earlier in terms of a two component\nmodel.\\cite{Booth98a} Within that model, one of these components (fluids) would\ncorrespond to delocalized states - these could be either delocalized holes\nor delocalized electrons. \nWe also point out that the decreasing distortions observed in EXAFS as T is\ndecreased below T$_c$ and the corresponding increase in resistivity suggest\na changing average mobility of the charge carriers. Within the model we have\nsuggested, the fraction of delocalized carriers would increase as T is\nlowered.\\cite{Booth98b} However, one of these components might also correspond to\nthe Mn atoms in a cluster, the positions of which are\ndominated by small variations in dopant concentration or O vacancies,\npossibly leading to a regime with phase separation. Such inhomogeneities\nlikely play an import role in these materials. In addition,\nJaime {\\it et al.}\\cite{Jaime99} have successfully modeled their resistivity\nand thermoelectric measurements using a two component system of localized and\nitinerant carriers. The recent calculations\nusing the Kondo model\\cite{Yunoki98a,Yunoki98,Dagotto98} also stress phase\nseparation but it is not clear how to compare with their results.\n\n\n\n\n%\\section{Conclusions}\n\\input{conclusions.tex}\n\n\\input{ref.tex}\n%\\bibliography{/exafs/bib/bibli}\n%\\bibliographystyle{/exafs/bib/prsty}\n\n\\newpage\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%/\n\n\\begin{figure}\n%\\vspace*{-0.4in}\n%\\hspace*{-0.4in}\n%\\psfig{file=edges.ps,width=3.2in}\n%\\vspace{-1.35in}\n\\caption{The Mn $K$-edges for a number of similar manganite systems, corrected\nfor energy shifts of the monochromator (see text).\n}\n\\label{edges}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\\begin{figure}\n%\\hspace*{0.2in}\n%\\psfig{file=edge_fits.ps,width=2.8in}\n%\\vspace{-1.35in}\n\\caption{Fits of absorption edges to each other. In a) and b) we show fits of\nthe LaMnO$_3$ edge (solid curve) to the 25\\% and 100\\% Ca samples. In c) and d)\nwe show fits of the 50K data (solid curve) to high temperature data for two\ndifferent samples. Corrections from the reference Mn foil have not been\nincluded in this figure.\n}\n\\label{edge_fits}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\vspace*{-0.4in}\n\\begin{figure}\n%\\hspace*{0.2in}\n%\\psfig{file=valence.ps,width=2.8in}\n%\\vspace{-1.35in}\n\\caption{The shift of the Mn $K$-edge with a) Ca concentration and b)\ntitrated valence for La$_{1-x}$Ca$_x$MnO$_3$. Note we do not have titration\ndata for x=0.30.\nThe relative errors are: edge shift, $\\pm$ 0.02 eV, smaller than symbols; Mn\nvalence $\\pm$ 0.02 units, slightly larger than symbols. In this and several\nfollowing figures, the lines are a guide to the eye only.}\n\\label{valence}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\vspace*{-0.4in}\n\\begin{figure}\n%\\psfig{file=edges_temp.ps,width=3.1in}\n%\\vspace{-1.35in}\n\\caption{The shift of the absorption edge for each sample relative to the low\ntemperature data. In panels a)-c), one of the curves (squares) is shifted\ndownward by 0.1 eV for clarity. Relative errors $\\pm$ 0.015 eV; $\\pm$ 0.03 eV\nfor LaMnO$_{3.03}$.\n}\n\\label{edges_temp}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\vspace*{-0.4in}\n\\begin{figure}\n%\\hspace*{0.2in}\n%\\psfig{file=compd.ps,width=2.7in}\n%\\vspace{-1.35in}\n\\caption{The temperature dependence of the pre-edge region for CaMnO$_3$,\n21\\% Ca, and 33\\% Ba; note different temperature ranges.\nThe CaMnO$_3$ sample shows little change of the pre-edge region below 300K\nwhile the A$_1$ and A$_2$ peaks are temperature dependent\nfor the CMR samples. The Ba pre-edge\ndata, collected using Si(220), are much sharper than data collected using\nSi(111) crystals, and show a splitting of the A peak. Note that the data\ncollected using Si(220) in c) have been shifted down by 0.15 eV for clarity.\n}\n\\label{pre-edge}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n%\\psfig{file=compd3.ps,width=3.3in}\n%\\vspace{-1.35in}\n\\caption{The temperature dependence of the pre-edge region for\nLa$_{0.7}$Ca$_{0.3}$MnO$_3$, Sr$_3$Mn$_2$O$_{6.55}$ and the 33\\% Pb\nsamples up to roughly 300K. The leading edge of the A$_1$\npeak is broader for Sr$_3$Mn$_2$O$_{6.55}$ than for the 30\\% Ca sample in a).\nIn b) the Pb samples has the largest A$_1$ peak at low T. Note that the\nscale is expanded compared to the previous figures and the B peak is not shown.\n}\n\\label{pre-edge3}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n%\\psfig{file=pre_conc.ps,width=2.9in}\n%\\vspace*{0.1in}\n\\caption{The positions of the pre-edge features as a function of Ca\nconcentration - open symbols, and solid circles, 300K. The error in peak position is\n$\\pm$ 0.1 eV. For the CMR samples, the A$_2$ peak is shifted to a\nlower energy for T $<$ T$_c$ - the filled triangles are for 50K.\n}\n\\label{pre-edge_shift}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\begin{figure*}\n%\\hspace*{0.5in}\n%\\psfig{file=diff_compd.ps,width=5.5in}\n%\\vspace*{-0.2in}\n\\caption{The difference spectra as a function of temperature for 21, 30 and\n65\\% Ca, and the end compounds. The main edge is included above each\nset of traces (multiplied by 0.03 to fit on the graph) to show where\nthe structure is located relative to the edge. Note the inversion of the\nstructure for the CO sample (T$_{CO}$ = 270K) (b), compared to the CMR\nsamples (c) (21\\% Ca, T$_c$ $\\sim$ 210K) and d) (30\\% Ca, T$_c$ $\\sim$ 260K).\n}\n\\label{diff_spec}\n\\end{figure*}\n%\\vspace*{7in}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n%\\hspace*{-0.4in}\n%\\psfig{file=sim_65.ps,width=3.0in}\n%\\vspace{-1.35in}\n\\caption{A comparison of the data for the 65\\% Ca sample and a simulation\nobtained from a 35-65\\% weighted sum of the LaMnO$_3$ and CaMnO$_3$ end\ncompounds. The experimental edge is significantly sharper.\n}\n\\label{sim_65}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG 10 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n%\\hspace*{-0.4in}\n%\\psfig{file=D_s2_CO.ps,width=3.2in}\n%\\vspace{-1.35in}\n\\caption{\nA plot of $\\sigma^2_{CO}(T)$ for the Mn-O bond as a function of T (65\\% Ca sample).\n$\\sigma^2_{CO}(T)$ is extracted from the $\\sigma^2_{Mn-O}$(T) data presented in\nRef. {\\protect \\onlinecite{Booth98b}}. T$_{CO}$ $\\sim$ 270K.\n}\n\\label{D_s2_CO}\n\\end{figure}\n\n\n\n\\end{document}\n"
},
{
"name": "ref.tex",
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}
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[
{
"name": "cond-mat0002046.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{Volger54}\nJ. Volger, Physica (Amsterdam) {\\bf 20}, 49 (1954).\n\n\\bibitem{Wollan55}\nE.~O. Wollan and W.~C. Koehler, Phys. Rev. {\\bf 100}, 545 (1955).\n\n\\bibitem{Schiffer95}\nP. Schiffer, A. Ramirez, W. Bao, and {S-W. Cheong}, Phys. Rev. Lett. {\\bf 75},\n 3336 (1995).\n\n\\bibitem{Jonker50}\nG.~H. Jonker and J.~H. {van Santen}, Physica (Amsterdam) {\\bf 16}, 337\n (1950).\n\n\\bibitem{Ramirez96}\nA.~P. Ramirez, P. Schiffer, {S-W. Cheong}, C.~H. Chen, W. Bao, T.~T.~M.\n Palstra, P.~L. Gammel, D.~J. Bishop, and B. Zegarski, Phys. Rev. Lett. {\\bf\n 76}, 3188 (1996).\n\n\\bibitem{Chen97}\nC.~H. Chen, {S-W. Cheong}, and H.~Y. Hwang, J. Appl. Phys. {\\bf 81}, 4326\n (1997).\n\n\\bibitem{Zener51}\nC. Zener, Phys. Rev. {\\bf 82}, 403 (1951).\n\n\\bibitem{Anderson55}\nP.~W. Anderson and H. Hasegawa, Phys. Rev. {\\bf 100}, 675 (1955).\n\n\\bibitem{deGennes60}\nP.~G. de~Gennes, Phys. 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cond-mat0002047
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Phase Transition in a Traffic Model with Passing
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"author": "I. Ispolatov$^1$ and P. L. Krapivsky$^2$"
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\noindent We investigate a traffic model in which cars either move freely with quenched intrinsic velocities or belong to clusters formed behind slower cars. In each cluster, the next-to-leading car is allowed to pass and resume free motion. The model undergoes a phase transition from a disordered phase for the high passing rate to a jammed phase for the low rate. In the disordered phase, the cluster size distribution decays exponentially in the large size limit. In the jammed phase, the distribution of finite clusters is independent on the passing rate, but it accounts only for a fraction of all cars; the ``excessive'' cars form an infinite cluster moving with the smallest velocity. Mean-field equations, describing the model in the framework of Maxwell approximation, correctly predict the existence of phase transition and adequately describe the disordered phase; properties of the jammed phase are studied numerically. \medskip\noindent{PACS numbers: 02.50-r, 05.40.+j, 89.40+k, 05.20.Dd}
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"name": "car.tex",
"string": "\\documentstyle[multicol,epsf,aps]{revtex}\n\\begin{document}\n\\title{Phase Transition in a Traffic Model with Passing}\n\\author{I. Ispolatov$^1$ and P. L. Krapivsky$^2$} \n\\address{$^1$Department of Chemistry, Baker Laboratory, Cornell University,\n Ithaca, NY 14853}\n\\address{$^2$Center for Polymer Studies and \n Department of Physics, Boston University, Boston, MA 02215}\n\\maketitle\n\n\\begin{abstract} \n\\noindent \nWe investigate a traffic model in which cars either move freely with\nquenched intrinsic velocities or belong to clusters formed behind slower\ncars. In each cluster, the next-to-leading car is allowed to pass and\nresume free motion. The model undergoes a phase transition from a\ndisordered phase for the high passing rate to a jammed phase for the low\nrate. In the disordered phase, the cluster size distribution decays\nexponentially in the large size limit. In the jammed phase, the\ndistribution of finite clusters is independent on the passing rate, but\nit accounts only for a fraction of all cars; the ``excessive'' cars form\nan infinite cluster moving with the smallest velocity. Mean-field\nequations, describing the model in the framework of Maxwell\napproximation, correctly predict the existence of phase transition and\nadequately describe the disordered phase; properties of the jammed phase\nare studied numerically.\n\n\\medskip\\noindent{PACS numbers: 02.50-r, 05.40.+j, 89.40+k, 05.20.Dd}\n\\end{abstract}\n\n\\begin{multicols}{2} \n \n\\section {Introduction}\n\nTraffic flows on single-lane roads with no passing exhibit clustering\nsince queues of fast cars accumulate behind slow cars. These clusters\nform and grow even when car density is small. The initial analysis of\ncluster formation was carried out in the earlier days of traffic\ntheory\\cite{newell}, and this subject continued growing ever\nthen\\cite{eps,nag,benjam,krug,evans,wolf,kn,k}. If passing is\nintroduced, the clusters may stop growing after reaching a certain size.\nIndeed, previous work\\cite{ep1,ep2,ep3} indicated that after a transient\nregime a steady state is reached. The models of Refs.\\cite{ep1,ep2,ep3}\nassume that {\\em any} car in a cluster can pass the leading car and the\npassing rate is independent on the location of the car within the\ncluster. This is certainly an oversimplification of the everyday\ntraffic scenarios. The complementary case when only the next-to-leading\ncars can pass is also an idealization, yet it is closer to reality.\nBelow we show that the latter model is also richer phenomenologically as\nit undergoes a dynamical phase transition.\n\nWe first comment on possible theoretical approaches. A mean-field theory\nis the primary candidate, and we believe that it may be very good,\nperhaps even exact, since clustering and passing mix positions and\nvelocities of the cars. The Boltzmann equation approach is an\nappropriate mean-field scheme, and in our earlier work\\cite{ep1,ep2} we\nindeed used it. However, the present model, where only the\nnext-to-leading car is allowed to pass, is significantly more difficult\nthan the model\\cite{ep1,ep2} where passing was possible for all cars.\nIndeed, it appears impossible even to write down closed Boltzmann\nequations for the distribution functions like $P(v,t)$ and $P_m(v,t)$,\nthe density of all clusters moving with velocity $v$, and the density of\nclusters of $m$ cars, respectively. Therefore our theoretical analysis\nis performed in the framework of the Maxwell approach. This scheme\nsimplifies ``collision'' terms by replacing the actual collision rates,\nwhich are proportional to velocity difference of collision partners, by\nconstants. Despite this essentially uncontrolled approximation, the\nMaxwell approximation is very popular in kinetic theory\\cite{rl} and it\nhas already been used in traffic\\cite{ep3}.\n\nThe important feature of our model is {\\em quenched disorder}, which\nmanifests itself in the random assignment of intrinsic velocities. Road\nconditions (construction zones, turns, hills, etc.) present another\nsource of quenched randomness in real driving situations \\cite{other},\nwhich is ignored in our model. Quenched disorder significantly affects\ncharacteristics of many-particle systems, especially in low spatial\ndimensions\\cite{bg}. This general conclusion applies to the present\none-dimensional traffic model as we shall show below.\n\n\\section {Maxwell approximation}\n\nWe now formally define the model. Free cars move with {\\em quenched}\nintrinsic velocities randomly assigned from some distribution $P_0(v)$.\nWhen a car or a cluster encounters a slower one, it assumes its velocity\nand a larger cluster is formed. In every cluster, the next-to-leading\ncar is allowed to pass and resume driving with its intrinsic velocity.\nThe rate of passing is assumed to be a constant. Thus clusters move and\naggregate deterministically, while passing is stochastic. The system is\ninitialized by randomly placing single cars and assigning them\nuncorrelated intrinsic velocities.\n \nWithin the Maxwell approach, the joint size-velocity distribution\nfunction (the density of clusters of size $m$ moving with velocity $v$)\n$P_m(v,t)$ obeys\n\\begin{eqnarray} \n\\label{first} \n{\\partial P_m(v,t)\\over\\partial t}\n&=&\\gamma (1-\\delta_{m,1})[P_{m+1}(v,t)-P_m(v,t)]\\nonumber\\\\ \n&+&\\gamma \\delta_{m,1}[N(v,t)+P_2(v,t)]-c(t)P_m(v,t)\\nonumber \\\\\n&+&\\int_v^{\\infty} dv' \\sum_{i+j=m} P_i(v',t)P_j(v,t).\n\\end{eqnarray}\nHere $\\gamma $ is the passing rate, so terms proportional to $\\gamma $\naccount for escape, while the rest describes clustering. The escape\nterms are the same within Boltzmann and Maxwell approaches, and they are\nactually {\\em exact}. The collision terms are mean-field by nature, and\nthey are different in the Boltzmann and Maxwell approaches. For\ninstance, in the Boltzmann case, the integral term must involve $v'-v$.\nEqs.~(\\ref{first}) also contain $c(t)$, the total cluster density\n\\begin{eqnarray} \n\\label{ct} \nc(t)=\\sum_{j\\geq 1} \\int_0^{\\infty} dv\\,P_j(v,t),\n\\end{eqnarray}\nand $N(v,t)$, the density of clusters in which the next-to-leading car\nhas intrinsic velocity $v$. This $N(v,t)$ causes the major trouble\nsince it cannot be expressed through $P_j(v,t)$. One might try to close\nEqs.~(\\ref{first}) by introducing $F_k(v,v',t)$, the density of clusters\nmoving with the velocity $v'$ whose $k^{\\rm th}$ car has intrinsic\nvelocity $v$. Clearly, $N(v,t)=\\int_0^v dv' F_2(v,v',t)$, and it appears\nthat equations for $F_k(v,v',t)$ are closed. A more careful look,\nhowever, reveals that the governing equation for $F_2(v,v',t)$ includes\nthree-velocity correlators.\n\nThus, at the first sight, the Boltzmann and Maxwell approaches appear to\nbe equally incapable of providing closed equations for the joint\nsize-velocity distribution function. Still, the Maxwell framework has\nan advantage that it does provide a closed description on the level of\nthe cluster size distribution. Indeed, integrating Eqs.~(\\ref{first})\nover velocity and defining $P_m(t)\\equiv \\int_0^\\infty dv\\, P_m(v,t)$,\nwe find that the cluster size distribution $P_m(t)$ obeys\n\\begin{eqnarray} \n\\label{simple} \n{d P_m\\over dt}\n=\\gamma [P_{m+1}-P_m]-c\\,P_m+{1\\over 2}\\sum_{i+j=m} P_iP_j\n\\end{eqnarray}\nfor $m\\geq 2$, and\n\\begin{eqnarray} \n\\label{simple1} \n{d P_1\\over dt}\n=\\gamma [P_2-P_1+c]-c\\,P_1.\n\\end{eqnarray}\nBesides this formal derivation of Eqs.~(\\ref{simple})--(\\ref{simple1})\nby direct integration of Eqs.~(\\ref{first}), it is possible to obtain\nthese equations by enumerating all possible ways in which clusters\nevolve. For instance, consider Eq.~(\\ref{simple1}). Collisions reduce\nthe density of single cars, and the collision rate is clearly equal to\n$c(t)$, as it is velocity-independent in the framework of the Maxwell\napproach. The escape term in Eq.~(\\ref{simple1}) is understood by\nobserving that the rate of return of single cars into the system is\nequal to\n\\begin{eqnarray*} \n\\gamma \\left[2P_2+\\sum_{j\\geq 3} P_j\\right]=\n\\gamma \\left[P_2-P_1+c\\right].\n\\end{eqnarray*}\nHere $P_2(t)$ is singled out since passing transforms it into\ntwo single cars while an escape from larger clusters produces only one\nfreely moving car.\n\nEqs.~(\\ref{simple})--(\\ref{simple1}) are closed. Mathematically similar\nequations were investigated previously in the context of the\naggregation-fragmentation model\\cite{ps,barma}. Therefore, we merely\npresent essential steps of the analysis. Restricting ourselves to the\nsteady state and introducing notations $P_m=\\gamma F_m$,\n$c_\\infty=\\gamma F$, we recast Eqs.~(\\ref{simple})--(\\ref{simple1}) into\n\\begin{equation} \n\\label{F} \nFF_m=F_{m+1}-F_m+\\delta_{m,1}F+{1\\over 2}\\sum_{i+j=m} F_iF_j.\n\\end{equation}\nThese equations should be solved together with the constraints\n$\\sum_{m\\geq 1}P_m=c_\\infty$ and $\\sum_{m\\geq 1} mP_m=1$, i.e.,\n\\begin{equation} \n\\label{sumF} \n\\sum_{m\\geq 1}F_m=F, \\quad\n\\sum_{m\\geq 1}mF_m=\\gamma^{-1}.\n\\end{equation}\nNote that the sum $\\sum_{m\\geq 1}mP_m(t)$ is obviously constant due to car\nconservation. The constant is equal to the initial concentration $c_0$\nas cars were initially unclustered. Here and below we always choose\n$c_0=1$.\n\nAs in Ref.~\\cite{ps}, we introduce the generating function\n\\begin{equation}\n\\label{genF}\n{\\cal F}(z)=\\sum_{m\\geq 1} (z^m-1) F_m.\n\\end{equation}\nThis generating function obeys\n\\begin{equation} \n\\label{Fz} \n{1\\over 2}\\,{\\cal F}^2+{1-z\\over z}\\,{\\cal F}+{(1-z)^2\\over z}\\,F=0,\n\\end{equation}\nwith the solution\n\\begin{equation} \n\\label{Fsol} \n{\\cal F}(z)={z-1\\over z}\\left\\{1-\\sqrt{1-2zF}\\right\\}.\n\\end{equation}\nThe steady state solution (\\ref{Fsol}) exists only when the generating\nfunction is real for all the $0\\leq z\\leq 1$. Hence, we require that\n$2F\\leq 1$. Assuming that this condition is satisfied, we expand the\ngenerating function in the powers of $z$ to obtain the steady state\nconcentrations:\n\\begin{equation}\n\\label{gammaF}\nF_m={(2F)^m\\over 2\\sqrt{\\pi}}\\left\\{{\\Gamma\\left(m-{1\\over 2}\\right)\n\\over \\Gamma(m+1)}-2F\\,\n{\\Gamma\\left(m+{1\\over 2}\\right)\\over \\Gamma(m+2)}\\right\\}.\n\\end{equation}\nThis solution is still incomplete as we have not yet determined $F$. To\nfind $F$ we use the sum rules (\\ref{sumF}). The first sum rule is\nmanifestly obeyed, while the second sum rule yields $\\sum mF_m=d{\\cal\n F}/dz|_{z=1} =1-\\sqrt{1-2F}=\\gamma^{-1}$. Thus, $F={2 \\gamma -1\\over 2 \\gamma ^2}$, \nwhich translates into $c_\\infty=1-1/2\\gamma$.\n\nThe steady state solution (\\ref{gammaF}) exists for sufficiently high\npassing rates, $\\gamma\\geq \\gamma_c=1$. For $\\gamma>1$ and large $m$, \nthe steady state\nsize distribution simplifies to\n\\begin{equation}\n\\label{Pmlarge}\nP_m\\simeq C m^{-3/2}\\left[1-\\left(1-\\gamma^{-1}\\right)^2\\right]^m, \n\\end{equation}\nwith $C=(4\\pi)^{-1/2}\\gamma^{-1}(\\gamma - 1)^2$. Apart from a power-law\nprefactor, the size distribution exhibits an exponential decay, $P_m\\sim\ne^{-m/m^*}$, in the large size limit. The characteristic size diverges,\n$m^*\\sim (\\gamma-1)^{-2}$ as the passing rate approaches the critical\nvalue $\\gamma_c=1$. In the critical case, the size distribution has a\npower-law form\n\\begin{equation}\n\\label{Pmcrit}\nF_m={3\\over 4\\sqrt{\\pi}}\\,{\\Gamma\\left(m-{1\\over 2}\\right)\n\\over \\Gamma(m+2)} \\sim m^{-5/2}.\n\\end{equation}\n\nLet now the passing rate drops below the critical value ($\\gamma<\\gamma\n_c$). Since $F$ cannot grow beyond $F_c=1/2$, it stays constant.\nTherefore, $F_m$ is given by the same Eq.~(\\ref{Pmcrit}) as in the\ncritical case, and the cluster size distribution reads $P_m=\\gamma F_m$.\nThis implies $c_\\infty=\\gamma /2$, i.e., the sum rule $\\sum\nP_m=c_\\infty$ is valid. The second sum rule is formally violated: $\\sum\nmP_m=\\gamma \\ne 1$, i.e. the cluster size distribution (\\ref{Pmcrit})\naccounts only for the fraction of all the cars present in the system.\nThe only possible explanation is the formation of an infinite cluster\nthat contains all the excessive cars. The second sum rule then shows\nthat $1-\\gamma $ of all the cars in the system are in this infinite\ncluster.\n\nThus within the framework of the Maxwell approximation, our\ntraffic model displays a phase transition which separates the disordered\nand jammed phases. The steady state cluster concentration has different\ndependence on the passing rate for these two phases:\n\\begin{equation}\n\\label{cinf}\nc_\\infty=\\cases{1-1/2\\gamma, &$\\gamma>1$;\\cr\n \\gamma/2, &$\\gamma<1$.}\n\\end{equation}\nIn the disordered phase, the size distribution decays exponentially in\nthe large size limit. In the jammed phase, $P_m$ has a power law tail\nand in addition there is an infinite cluster which contains the\nfollowing fraction of cars:\n\\begin{equation}\n\\label{I}\nI=\\cases{0, &$\\gamma>1$;\\cr\n 1-\\gamma , &$\\gamma<1$.}\n\\end{equation}\nThis phase transition is similar to phase transitions in driven\ndiffusive systems {\\em without} passing\\cite{benjam,krug,evans,wolf,k}\nand to phase transitions in aggregation-fragmentation\nmodels\\cite{ps,barma,ziff}. Also, the mechanism of the formation of the\ninfinite cluster has a strong formal analogy to Bose-Einstein\ncondensation \\cite{evans,barma}.\n\nTurning back to the joint size-velocity distribution (\\ref{first}),\nwe note that the\nlack of an {\\em exact} expression for $N(v)$ in terms of $P_m(v)$ does not\nmean the lack of a mean-field relation between these quantities.\nIndeed, the density $N(v)$ of clusters in which the next-to-leading car\nhas intrinsic velocity $v$, can be written as\n\\begin{equation}\n\\label{mv}\nN(v)=\\int_0^v dv'\\,\\sum_{j\\geq 2}P_j(v')\\,\\,\n{C(v)\\over \\int_{v'}^\\infty dv''\\,C(v'')}.\n\\end{equation}\nHere $\\sum_{j\\geq 2}P_j(v')$ is the density of ``true'' clusters (i.e.,\nfreely moving cars are excluded) moving with velocity $v'$. Then,\n$C(v)=P_0(v)-P(v)$ is the density of cars with intrinsic velocity $v$\nwhich are currently slowed down, i.e., they are neither single cars, nor\ncluster leaders. Assuming that the velocities of cars inside clusters\nare perfectly mixed, $C(v)/\\int_{v'}^\\infty dv''\\,C(v'')$ gives the\nprobability density that the next-to-leading car in a true $v'$-cluster\nhas the velocity $v$. The product form of Eq.~(\\ref{mv}) reveals its\nmean-field nature, which is consistent with the spirit of our\ntheoretical approach. One can verify that Eq.~(\\ref{mv}) agrees with\nthe sum rule $\\int dv\\,N(v)=\\sum_{j\\geq 2}P_j$, thus providing a useful\ncheck of self-consistency.\n\nAlthough Eqs.~(\\ref{first}) with $N(v)$ given by (\\ref{mv}) seem very\ncomplex even in the steady-state regime, several conclusions can be\nderived without getting their complete solution. We first\nsimplify Eqs.~(\\ref{first}) by introducing auxiliary functions\n\\begin{equation}\n\\label{Qmv}\nQ_m(v)=\\int_v^\\infty dv'\\,P_m(v').\n\\end{equation}\nBy inserting $P_m=-{dQ_m\\over dv}$ into the Eqs.~(\\ref{first}),\nintegrating resulting equations over $v$, and using the boundary\nconditions $Q_m(v=\\infty)=0$, we find\n\\begin{eqnarray} \n\\label{Qm}\n\\gamma\\left[Q_{m+1}(v)-Q_m(v)\\right]&-&c\\,Q_m(v)+{1\\over 2}\\sum_{i+j=m}\nQ_i(v)Q_j(v)\\nonumber\\\\\n&=&\\delta_{m1}\\,\\gamma\\,q(v),\n\\end{eqnarray}\nwith\n\\begin{eqnarray} \n\\label{qv}\nq(v)=-Q_1(v)-\\int_v^\\infty dv'\\,N(v').\n\\end{eqnarray}\nEqs.~(\\ref{Qm}) are almost identical to the\nEqs.~(\\ref{simple})--(\\ref{simple1}), the velocity is just a parameter.\nConsequently, we anticipate qualitatively similar results, \n$Q_m(v)\\sim m^{-3/2}e^{-m/m^*}$, and\n\\begin{equation}\n\\label{Qmvas}\nP_m(v)\\sim m^{-1/2}e^{-m/m^*}.\n\\end{equation}\nwith the characteristic size $m^*(v,\\gamma)$ dependent on both velocity and\npassing rate. Our more rigorous generating function analysis, performed \nalong the lines described above, confirms the asymptotic form (\\ref{Qmvas}).\n\n\\section{Simulations}\n\nNow let us examine what conclusions obtained within the Maxwell approach\nare relevant for the original model. We first re-derive the condition\nfor the phase transition in the complete velocity-dependent form. Let us\nconsider a system of reference with the origin moving with the slowest\ncar. We assume that the system is sufficiently large for the slowest car\nto have negligible velocity. We compare the total flux of cars\nclustering behind this slowest car, $\\sum mP_m\\langle v\\rangle_m$, to\nthe rate of escape, $\\gamma $. Here $\\langle v\\rangle_m$ is an average\nvelocity of a cluster of size $m$. When the rate of escape becomes less\nthan the rate of accumulation of the cars, the cluster behind the\nslowest car (analog of the ``infinite cluster'' for finite systems)\ngrows to remove the excessive cars from the system. Hence, the phase\ntransition point $\\tilde \\gamma_c$ is defined as\n\\begin{equation}\n\\label{phtr}\n\\sum_{m\\geq 1} mP_m\\langle v\\rangle_m = \\tilde \\gamma_c.\n\\end{equation}\nFor the Maxwell model, where $\\langle v\\rangle_m=1$ for all $m$, \nEq.~(\\ref{phtr}) reduces to $\\sum mP_m=\\gamma_c=1$ as obtained above.\nSince large clusters usually form behind slow cars, $\\langle\nv\\rangle_m$ is a decreasing function of the cluster size $m$. In\nparticular, $\\langle v\\rangle_m$ is always smaller than the average car\nvelocity $\\langle v\\rangle$, implying $\\tilde \\gamma_c<1$. \n\nFor a rough estimate of $\\langle v\\rangle_m$, consider a cluster of $m$\ncars and {\\em assume} that intrinsic velocities of the cars in the\ncluster are independent. The leading car has the minimal velocity, so\nthe size-velocity distribution reads\n\\begin{equation}\n\\label{extrem}\nP_m(v)\\approx mP_0(v)\\left[\\int_v^\\infty dv'\\,P_0(v')\\right]^{m-1}P_m.\n\\end{equation}\nFor concreteness, let us consider intrinsic velocity distributions which\nbehave algebraically near the lower cutoff, $P_0(v)\\sim v^\\mu$ as $v\\to\n0$. Then for large clusters we get\n\\begin{equation}\n\\label{extr}\nP_m(v)\\sim P_m\\,\\exp\\left(-mv^{\\mu+1}\\right).\n\\end{equation}\nThis implies that the average cluster velocity $\\langle v\\rangle_m$\nscales with $m$ according to $\\langle v\\rangle_m \\sim m^{-1/(\\mu+1)}$,\nand hence $\\tilde \\gamma_c \\sim \\sum m^{\\mu/(\\mu+1)}P_m$. We conclude\nthat the phase transition does exist in the original model, although its\nlocation is shifted towards lower passing rate compared to the Maxwell\nmodel prediction. This shift is especially significant for small $\\mu$\n($\\mu>-1$ from the normalization requirement).\n\nTo check the relevance of other predictions of the Maxwell approach, we\nperformed molecular dynamics simulations. We place $N=20000$ single\ncars onto the ring of length $L=N$, so that the average car density is\nequal to one. Initial positions and velocities of cars were assigned\nrandomly. We considered linear $P_0(v)={8\\over 9}\\,v$ ($0<v<3/2$),\nexponential $P_0(v)=e^{-v}$, and $P_0(v)=(2\\pi v)^{-1/2}e^{-v/2}$\nvelocity distributions, which correspond to $\\mu=1, 0, -1/2$ for the\nsmall-$v$ asymptotics. All these three distributions have the average\nvelocity equal to one.\n\nIn Fig.~1, we plot $\\ln [m^{3/2}P_m]$ vs. $m$ for the above three\nvelocity distributions. We take $\\gamma=1$ which, as we concluded\nbefore, lies above the phase transition point $\\tilde\\gamma_c$. We\nexpect the system to be in the disordered phase with $P_m$ being\nexpressed by Eq.~(\\ref{Pmlarge}). For the exponential and $P_0(v)=(2\\pi\nv)^{-1/2} e^{-v/2}$ intrinsic velocity distributions, there is a good\nagreement with the prediction of the Maxwell model (\\ref{Pmlarge}); for\nthe linear velocity distribution, there are some deviations for small\n$m$, but for large $m$ the agreement is satisfactory. The slopes of the\nplots decrease with $\\mu$. Taking into account that at the point of the\nphase transition the slope equals to zero, this qualitatively confirms\nthat $\\tilde\\gamma_c$ gets smaller when $\\mu$ decreases.\n\n\\begin{figure}\n \\centerline{\\epsfxsize=8cm \\epsfbox{fig1.eps}} {{\\small {\\bf Fig.~1}.\n Plot of $\\ln\\left[m^{3/2}P_m\\right]$ vs. cluster size $m$ in the\n high passing rate regime ($\\gamma=1$) for linear ($\\Box $), exponential\n (o), and $P_0(v)=(2\\pi v)^{-1/2}e^{-v/2}$ ($\\nabla$) initial\n velocity distributions.}}\n\\end{figure}\n\nPlots of $P_m$ vs. $m$ for intrinsic velocity distributions\n$P_0(v)=e^{-v}$ and $P_0(v)=(2\\pi v)^{-1/2}e^{-v/2}$, with passing rate\n$\\gamma =0.005$ well below the phase transition point, are shown in\nFig.~2a and Fig.~2b, respectively. The cluster size distribution\nclearly consists of two regions: almost power-law tail for smaller $m$\nand several separate peaks for larger $m$. These peaks correspond to the\nfluctuating size of the infinite cluster, while the power-law tail\ndescribes the regular part of $P_m$. The apparent exponent $\\tau$ of\nthe power-law region $P_m \\sim m^{-\\tau}$ slightly varies for different\npassing rates and $P_0(v)$, though it remains confined between 3/2 and 2. It\nis definitely different form the value 5/2, predicted by the Maxwell\nmodel (\\ref{Pmcrit}). The measured values of $\\tau$ would make the\ntotal amount of cars in the system divergent, $\\sum mP_m \\rightarrow\n\\infty$, so the power-law region ends with an exponential cutoff at\nlarge $m$. \n\nWe now comment on the relationship of our model to earlier work. On the\nmean-field level, our model is similar to the models of\nRefs.\\cite{ps,barma}. On the level of the process, our model reminds an\nasymmetric conserved-mass aggregation model (ASCMAM)\\cite{barma} where\nclusters undergo asymmetric diffusion, aggregation upon contact, and\nchipping (single-particle dissociation). Of course, our model is\ncontinuum while the ASCMAM is the lattice model. More substantial\ndifference between the two models lies in the nature of randomness -- in\nour model intrinsic velocities are quenched random variables, while in\nthe ASCMAM dynamics is the only source of randomness. Nevertheless, the\nphenomenology of the two models appears to be quite similar. In\nparticular, the ASCMAM undergoes a phase transition, and in the jammed\nphase, the cluster size distribution exibits a power law decay with the\nexponent close to 2\\cite{barma}. We should stress that in the jammed\nphase, we have not reached a scale-free critical state which must have\nthe exponent $\\tau\\geq 2$. Maybe quenched randomness does not allow the\nsystem to organize itself into a truly normalizable critical state.\nOther possible explanation relies on large fluctuations in disordered\nsystems, i.e., our system was not large enough to ensure self-averaging.\n\n\\begin{figure}\n \\centerline{\\epsfxsize=8cm \\epsfbox{fig2a.eps}} {{\\small {\\bf\n Fig.~2a}. Plot of the steady state cluster size distribution\n $P_m$ in the low passing rate regime ($\\gamma =0.005$) for the\n exponential initial velocity distribution.}}\n\\end{figure}\n\n\\begin{figure}\n \\centerline{\\epsfxsize=8cm \\epsfbox{fig2b.eps}} {{\\small {\\bf\n Fig.~2b}. Plot of the steady state cluster size distribution\n $P_m$ in the low passing rate regime ($\\gamma =0.005$) for the\n $P_0(v)=(2\\pi v)^{-1/2}e^{-v/2}$ initial velocity distribution.}}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn this paper, we have investigated the model of traffic that involves\nclustering and passing of the next-to-leading car. Despite the fact that\nit is one of the simplest (if not the simplest) possible continuous\nmodel of one-lane traffic with passing, the model has rich kinetic\nbehavior. Depending on the passing rate $\\gamma$ the system organizes\nitself either into disordered phase where density of large clusters is\nexponentially suppressed, or into the jammed phase, where the cluster\nsize distribution becomes independent on $\\gamma$ and the infinite\ncluster is formed. Within the framework of Maxwell approach, which\nplays the role of the mean-field theory in the present context, we have\nshown that the model admits an analytical solution. We have argued that\nthe Maxwell approach correctly predicts the existence of the phase\ntransition and adequately describes the properties of the disordered\nphase which arises when the passing rate is high. For the jammed phase,\nthe Maxwell approach correctly predicts that the system stores excessive\ncars in the infinite cluster and organizes itself into some kind of a\ncritical state. However, the Maxwell approach cannot quantitatively\ndescribe other properties of the jammed phase. It would be interesting\nto design a more accurate theoretical approach which would allow to\nprobe the characteristics of the low passing rate regime analytically.\nSome properties of the jammed state appear similar to the properties of\nthe jammed state of a lattice model of Ref.\\cite{barma} which includes\nan asymmetric lattice diffusion, aggregation, and fragmentation. It\nwould be interesting to gain a deeper understanding of the relationship\nbetween these models, and whether the quenched disorder is the main\nsource of difference.\n\n\\medskip\\noindent We are thankful to E.~Ben-Naim and S.~Redner for\ndiscussions, and to NSF and ARO for support of this work.\n\n\n\n\n\\begin{thebibliography}{99}\n \n\\bibitem{newell} \n G.~F.~Newell, Oper. Res. {\\bf 7}, 589 (1959).\n\\bibitem{eps} \n E.~Ben-Naim, P.~L.~Krapivsky, and S.~Redner, Phys.\\ Rev.\\ \n E {\\bf 50}, 822 (1994). \n\\bibitem{nag} \n T.~Nagatani, Phys.\\ Rev.\\ E {\\bf 51}, 922 (1995). \n\\bibitem{benjam} \n I.~Benjamini, P.~A.~Ferrari,\n and C.~Landim, Stoch.\\ Proc.\\ Appl. {\\bf 61}, 181 (1996).\n\\bibitem{krug} J.~Krug and P.~A.~Ferrari, J.\\ Phys.\\ A {\\bf 29}, L465\n (1996). \n\\bibitem{evans} \n M.~R.~Evans, Europhys.\\ Lett. {\\bf 36}, 13\n (1996); J.\\ Phys.\\ A {\\bf 30}, 5669 (1997). \n\\bibitem{wolf}\n D.~V.~Ktitarev, D.~Chowdhury, and D.~E.~Wolf, J.\\ Phys.\\ A {\\bf 30},\n L221 (1997). \n\\bibitem{kn} \n W.~Knospe, L.~Santen, A.~Schadschneider,\n and M.~Schreckenberg, {\\it cond-mat/9810184}. \n\\bibitem{k} \n For a review, see J.~Krug, in {\\it Traffic and Granular Flow II}, eds.\n D.~Wolf and M.~Schreckenberg (Springer, Singapore, 1998), p.~285.\n\\bibitem{ep1} E.~Ben-Naim and P.~L.~Krapivsky, Phys.\\ Rev.\\ E {\\bf 56},\n 6680 (1997). \n\\bibitem{ep2} \n E.~Ben-Naim and P.~L.~Krapivsky, J.\\ Phys.\\ A {\\bf 31}, 8073 (1998). \n\\bibitem{ep3} \n E.~Ben-Naim and P.~L.~Krapivsky, Phys.\\ Rev.\\ E {\\bf 59}, 88 (1999). \n\\bibitem{rl}\n P.~Resibois and M.~De Leener, {\\it Classical Kinetic Theory of Fluids}\n (Wiley, New York, 1977). \n\\bibitem{other} \n Influence of such spatial disorder, as opposed to particlewise \n disorder included in our model, has been studied by several authors, \n e.g., Z.~Csahok and T.~Vicsek, J.\\ Phys.\\ A {\\bf 27}, L591 (1994); \n O.~J.~O'Loan, M.~R.~Evans, and M.~E.~Cates, Phys.\\ Rev.\\ E {\\bf 58}, \n 1404 (1998); G.~Tripathy and M.~Barma, Phys.\\ Rev.\\ E {\\bf 58}, \n 1911 (1998); J.~Krug, {\\it cond-mat/9912411}.\n\\bibitem{bg} \n For a review of quenched disorder, see J.~P.~Bouchaud and A.~Georges, \n Phys.\\ Rep. {\\bf 195}, 127 (1990). \n\\bibitem{ps} \n P.~L.~Krapivsky and S.~Redner, Phys.\\ Rev.\\ E {\\bf 54}, 3553 (1996). \n\\bibitem{barma} \n S.~N.~Majumdar, S.~Krishnamurthy, and M.~Barma, \n Phys.\\ Rev.\\ Lett. {\\bf 81}, 3691 (1998); {\\it cond-mat/9908443}. \n\\bibitem{ziff} \n R.~D.~Vigil, R.~M.~Ziff, and B.~Lu, Phys.\\ Rev.\\ B {\\bf 38}, 942 (1988).\n \n\n\\end{thebibliography}\n\\end{multicols}\n\\end{document}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002047.extracted_bib",
"string": "\\begin{thebibliography}{99}\n \n\\bibitem{newell} \n G.~F.~Newell, Oper. Res. {\\bf 7}, 589 (1959).\n\\bibitem{eps} \n E.~Ben-Naim, P.~L.~Krapivsky, and S.~Redner, Phys.\\ Rev.\\ \n E {\\bf 50}, 822 (1994). \n\\bibitem{nag} \n T.~Nagatani, Phys.\\ Rev.\\ E {\\bf 51}, 922 (1995). \n\\bibitem{benjam} \n I.~Benjamini, P.~A.~Ferrari,\n and C.~Landim, Stoch.\\ Proc.\\ Appl. {\\bf 61}, 181 (1996).\n\\bibitem{krug} J.~Krug and P.~A.~Ferrari, J.\\ Phys.\\ A {\\bf 29}, L465\n (1996). \n\\bibitem{evans} \n M.~R.~Evans, Europhys.\\ Lett. {\\bf 36}, 13\n (1996); J.\\ Phys.\\ A {\\bf 30}, 5669 (1997). \n\\bibitem{wolf}\n D.~V.~Ktitarev, D.~Chowdhury, and D.~E.~Wolf, J.\\ Phys.\\ A {\\bf 30},\n L221 (1997). \n\\bibitem{kn} \n W.~Knospe, L.~Santen, A.~Schadschneider,\n and M.~Schreckenberg, {\\it cond-mat/9810184}. \n\\bibitem{k} \n For a review, see J.~Krug, in {\\it Traffic and Granular Flow II}, eds.\n D.~Wolf and M.~Schreckenberg (Springer, Singapore, 1998), p.~285.\n\\bibitem{ep1} E.~Ben-Naim and P.~L.~Krapivsky, Phys.\\ Rev.\\ E {\\bf 56},\n 6680 (1997). \n\\bibitem{ep2} \n E.~Ben-Naim and P.~L.~Krapivsky, J.\\ Phys.\\ A {\\bf 31}, 8073 (1998). \n\\bibitem{ep3} \n E.~Ben-Naim and P.~L.~Krapivsky, Phys.\\ Rev.\\ E {\\bf 59}, 88 (1999). \n\\bibitem{rl}\n P.~Resibois and M.~De Leener, {\\it Classical Kinetic Theory of Fluids}\n (Wiley, New York, 1977). \n\\bibitem{other} \n Influence of such spatial disorder, as opposed to particlewise \n disorder included in our model, has been studied by several authors, \n e.g., Z.~Csahok and T.~Vicsek, J.\\ Phys.\\ A {\\bf 27}, L591 (1994); \n O.~J.~O'Loan, M.~R.~Evans, and M.~E.~Cates, Phys.\\ Rev.\\ E {\\bf 58}, \n 1404 (1998); G.~Tripathy and M.~Barma, Phys.\\ Rev.\\ E {\\bf 58}, \n 1911 (1998); J.~Krug, {\\it cond-mat/9912411}.\n\\bibitem{bg} \n For a review of quenched disorder, see J.~P.~Bouchaud and A.~Georges, \n Phys.\\ Rep. {\\bf 195}, 127 (1990). \n\\bibitem{ps} \n P.~L.~Krapivsky and S.~Redner, Phys.\\ Rev.\\ E {\\bf 54}, 3553 (1996). \n\\bibitem{barma} \n S.~N.~Majumdar, S.~Krishnamurthy, and M.~Barma, \n Phys.\\ Rev.\\ Lett. {\\bf 81}, 3691 (1998); {\\it cond-mat/9908443}. \n\\bibitem{ziff} \n R.~D.~Vigil, R.~M.~Ziff, and B.~Lu, Phys.\\ Rev.\\ B {\\bf 38}, 942 (1988).\n \n\n\\end{thebibliography}"
}
] |
cond-mat0002048
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Induced instability for boson-fermion mixed condensate of Alkali atoms due to attractive boson-fermion interaction
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[
{
"author": "T. Miyakawa"
},
{
"author": "T. Suzuki and H. Yabu"
}
] |
Instabilities for boson-fermion mixed condensates of trapped Alkali atoms due to the boson-fermion attractive interaction are studied using a variational method. Three regions are shown for their instabilities according to the boson-fermion interaction strength: stable, meta-stable and unstable ones. The stability condition is obtained analytically from the asymptotic expansion of the variational total energy. The life-time of metastable states is discussed for tunneling decay, and is estimated to be very long. It suggests that, except near the unstable border, meta-stable mixed condensate should be almost-stable against clusterizations. The critical border between meta-stable and unstable phases is calculated numerically and is shown to be consistent with the M{\o}lmer scaling condition.
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[
{
"name": "instcpR3.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentstyle[pra,aps]{revtex}\n\\oddsidemargin=-5mm\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\def\\rvec{\\vec{r}}\n\\def\\xvec{\\vec{x}}\n\\def\\half{{1\\over 2}}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{document}\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\title{Induced instability for boson-fermion mixed condensate of\nAlkali atoms due to attractive boson-fermion interaction}\n%\n\\author{T. Miyakawa, T. Suzuki and H. Yabu}\n%\n\\address{Department of Physics, Tokyo Metropolitan University, \n 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan}\n%\n\\date{\\today}\n%\n\\maketitle\n%\n\\begin{abstract}\nInstabilities for boson-fermion mixed condensates \nof trapped Alkali atoms \ndue to the boson-fermion attractive interaction \nare studied using a variational method. \nThree regions are shown \nfor their instabilities \naccording to the boson-fermion interaction strength: \nstable, meta-stable and unstable ones. \nThe stability condition is obtained \nanalytically from the asymptotic expansion \nof the variational total energy. \nThe life-time of metastable states \nis discussed for tunneling decay, \nand is estimated to be very long. \nIt suggests that, except near the unstable border, \nmeta-stable mixed condensate should be almost-stable\nagainst clusterizations. \nThe critical border between meta-stable and unstable phases \nis calculated numerically \nand is shown to be consistent with the M{\\o}lmer scaling condition. \n\\end{abstract}\n%\n\\pacs{PACS number: 03.75.Fi, 05.30.Fk,67.60.-g}\n%\n%%%%%%%%%%%\n% MAIN BODY\n%%%%%%%%%%%\n%\nDiscoveries of the Bose-Einstein condensates (BEC) \nfor Alkali atoms \\cite{bec,review} \nand the Fermi degeneracy for trapped ${}^{40}$K atoms \\cite{DeMarco}\nencourage the study for \nboson-fermion mixed condensates of trapped atoms. \nOriginally, the boson-fermion condensates have been studied in \nbulk systems such as ${}^3$He-${}^4$He \nand hydrogen-deuteron systems \\cite{hydro}. \nStudies of trapped mixed condensates have progressed recently; \nthe static properties \nof ${}^{39,41}$K-${}^{40}$K condensate \n(with repulsive boson-boson interaction) \\cite{Molmer,Nygaard,Miyakawa}, \ndynamical expansions after the removal \nof trapping potentials \\cite{Amoruso}, \nand the instability changes of ${}^7$Li-${}^6$Li system \nwith attractive boson-boson interaction \\cite{Minniti}. \n\nIn this paper, \nwe study the instabilities and collapses \nof the polarized boson-fermion mixed condensates \ndue to the attractive boson-fermion interaction. \nIn BEC, such instabilities are observed in several systems: \n1) the trapped meta-stable ${}^7$Li BEC \n(observed experimentally \\cite{metaLi})\nequilibrated between the attractive boson-boson interaction \nand the kinetic pressure \ndue to the finite confinement \\cite{Baym}, \n2) two-component uniform BEC (from bosons 1 and 2), \nwhose stability condition is given by \n$g_{11} g_{22} > g_{12}^2$ \n($g_{ij}$ : the coupling constant\nbetween bosons i and j). \nIn the latter case, the stability condition depends only on \nthe interaction strength, \nbut the condition for the case 1) has also \nthe particle number dependence \\cite{Baym}. \n\n%In the boson-fermion mixed system, \n%different particle-statistics play an important role \n%and give a different type of stability conditions. \n%As an illustration, \n%we consider a uniform boson-fermion system \n%(with no spin degeneracy and $T=0$) \n%Using the boson/fermion densities $n_{b,f}$, \n%its free energy becomes\n%\n%\\begin{equation}\n% {\\cal F}[n_b,n_f] =\\frac{g}{2} n^2_b - \\mu_b n_b\n% +\\frac{3}{5} \\frac{\\hbar^2}{2 m_f} \n% (6\\pi^2)^{\\frac{2}{3}} n^{\\frac{5}{3}}_f \n% -\\mu_f n_f+ h n_b n_f, \n%\\label{eQa}\n%%\\end{equation}\n%\n%where $m_{b,f}$ and $\\mu_{b,f}$ are boson/fermion masses \n%and chemical potentials, \n%and, in low-density gas, \n%the boson-boson/boson-fermion coupling constants \n%$g$, $h$ are represented \n%by the s-wave scattering lengths $a_{bb}$ and $a_{bf}$: \n%$g =4\\pi \\hbar^2 a_{bb}/m_b$ and \n%$h =2\\pi \\hbar^2 a_{bf}/m_r$ \n%where $m_r =m_b m_f/(m_b+m_f)$. \n%In eq. (\\ref{eQa}), \n%the fermion-fermion interaction has been neglected \n%because of the Pauli blocking effects. \n%When $g>0$ (repulsive boson-boson interaction), \n%the positive Jacobian determinant\n%$\\frac{\\delta^2 {\\cal F}}{\\delta n^2_b}\n% \\frac{\\delta^2 {\\cal F}}{\\delta n^2_f} > \n% \\left( \\frac{\\delta^2 {\\cal F}}{\\delta n_b \\delta n_f} \\right)^2$\n%gives the stability condition \\cite{Pethick}: \n%\n%\\begin{equation}\n% \\frac{h^2}{g} < \\left(\\frac{h^2}{g}\\right)_c \n% =\\frac{(6\\pi^{2})^{\\frac{2}{3}}}{3} \n% \\frac{\\hbar^2}{m_f} \n% \\frac{1}{n_f^{1}{3}}.\n%\\label{eQbA}\n%\\end{equation}\n% \n%Different from the condition for two-component BEC, \n%the above one has the dependence on fermion density; \n%it shows the effect from particle-statistics. \n\nLet us consider the case of the \ntrapped boson-fermion mixed condensate. \nIn the present paper, \nwe assume $T=0$ and \na spherical harmonic-oscillator \nfor the trapping potential; \nextension to the deformed potential is straightforward. \nUsing the Thomas-Fermi approximation \nfor the fermion degree of freedom \n(appropriate for large $N_f$ condensate), \nthe total energy of the system \nbecomes a functional of \nthe boson order-parameter $\\Phi(\\vec{r})$ \nand the fermion density distribution $n_f(\\vec{r})$: \n%\n\\begin{eqnarray}\n E[\\Phi(\\vec{r}),n_f(\\vec{r})] =\\int{\\,d^3r} \n \\left[ \\frac{\\hbar^2}{2m_b} | \\nabla\\Phi(\\vec{r})|^2\n +\\frac{1}{2} m \\omega^2 \\vec{r}^2 |\\Phi(\\rvec)|^2 \n +\\frac{g}{2} |\\Phi(\\vec{r})|^4 \n \\right] \\nonumber \\\\ \n%\n +\\int{\\,d^3r} \n \\left[ \\frac{3}{5} \\frac{\\hbar^2}{2m} \n (6\\pi^2)^{\\frac{2}{3}} \n n^{\\frac{5}{3}}_f(\\vec{r})\n +\\frac{1}{2} m \\omega^2 \\vec{r}^2 n_f(\\vec{r}) \n +h |\\Phi(\\vec{r})|^2 n_f(\\vec{r}) \n \\right],\n\\label{eQc}\n\\end{eqnarray}\n%\nwhere $\\omega$ and $m$ are \nthe angular frequency for the boson/fermion trapping potential \nand the boson/fermion mass \n($\\omega_b =\\omega_f \\equiv \\omega$ and \nand $m_b =m_f \\equiv m$ \nare assumed for simplicity). \nIn low-density gas, \nthe boson-boson/boson-fermion coupling constants \n$g$, $h$ are represented \nby the s-wave scattering lengths $a_{bb}$ and $a_{bf}$: \n$g =4\\pi \\hbar^2 a_{bb}/m_b$ and \n$h =2\\pi \\hbar^2 a_{bf}/m_r$ \nwhere $m_r =m_b m_f/(m_b+m_f)$. \nIn eq. (\\ref{eQc}), \nthe fermion-fermion interaction has been neglected;\nthe elastic fermion-fermion s-wave scattering for a polarized gas\nis absent because of the Pauli blocking effects \nand the p-wave scattering is suppressed \nbelow 100$\\,\\mu{\\rm K}$ \\cite{DeMarco2}. \n\nEvaluating the total energy \nwith the variational method, \nwe take the Gaussian ansatz \nfor the boson order-parameter: \n%\n\\begin{equation}\n \\Phi(x;R) \\equiv \\xi^{-3/2} \\phi(x;R) \n =\\sqrt{\\left(\\frac{3}{2\\pi}\\right)^{\\frac{3}{2}} \n \\frac{N_b}{(\\xi R)^3} }\n \\exp\\left(\\frac{-3 x^2}{4 R^2} \\right), \n\\label{eQd} \n\\end{equation}\n%\nwhere $x =r/\\xi$ is a radial distance scaled \nby the harmonic-oscillator length $\\xi =(\\hbar/m \\omega)^{1/2}$, \nand $N_b =\\int{d^3x} |\\Phi(x;R)|^2$ is total boson number. \nThe variational parameter $R$ in (\\ref{eQd}) \ncorresponds to the root-mean-square (rms) radius \nof the boson density distribution. \nThis kind of variational functions have also been used \nin the instability investigations of ${}^7$Li-BEC \\cite{Baym} \nand ${}^7$Li-${}^6$Li system \\cite{Minniti}. \nIn the Thomas-Fermi approximation \nwith (\\ref{eQd}), \nthe fermion density distribution $n_f(\\vec{r})$ becomes\n%\n\\begin{equation}\n n_f(x;R) =\\frac{1}{6\\pi^2}\n \\left[ {\\tilde \\mu}_f(R) \n - x^2 \n -{\\tilde h} |\\phi(x;R)|^2 \n \\right]^{\\frac{3}{2}}. \n\\label{eQe}\n\\end{equation}\n%\nwhere \n${\\tilde h} =2h/(\\hbar \\omega \\xi^3)$. \nThis fermion density vanishes at the turning point\n$\\Lambda$ which is determined by \n%\n\\begin{equation}\n F(\\Lambda) \\equiv \n {\\tilde \\mu}_f -\\Lambda^2 \n -\\frac{C}{R^3} e^{-\\frac{3 \\Lambda^2}{2 R^2}} =0, \n\\label{eQf}\n\\end{equation}\n%\nwhere $C =(3/2\\pi)^{3/2} {\\tilde h} N_b$. \nWe take $n_f(x;R) \\equiv 0$ when $x \\geq \\Lambda$. \n\nThe scaled chemical potential \n${\\tilde \\mu}_f =2 \\mu_f/(\\hbar \\omega)$ \nin eq. (\\ref{eQe}) is determined by the normalization condition \n$\\int{d^3x} n_f(x) =N_f$. \nAs a result, the two parameters $\\Lambda$ and ${\\tilde \\mu}_f$ \nbecome functions of $R$. \n\nUsing eqs. (\\ref{eQd}) and (\\ref{eQe}), \nthe total energy $E(R) =E[\\Phi(x;R),n_f(x;R)]$ becomes \n%\n\\begin{eqnarray}\n E(R)/(\\hbar \\omega) \n &=& N_b \\left\\{ \\frac{9}{8} \\frac{1}{R^2} \n +\\frac{1}{2} R^2\n +2^{-5/2} \n \\frac{3}{\\pi} \n {\\tilde g} N_b \\frac{1}{R^3} \\right\\} \n \\nonumber\\\\ \n%\n &+& N_f \\left\\{ I_{ke}(R) + I_{ho}(R) \n +C R^{-3} I_{bf}(R) \\right\\}, \n\\label{eQg}\n\\end{eqnarray}\n%\nwhere ${\\tilde g} =2g/(\\hbar \\omega \\xi^3)$ \nand integrals $I_{ke,ho,bf}(R)$ are defined by \n%\n\\begin{eqnarray}\n I_{ke}(R) &=& \\frac{1}{5 \\pi N_f} \n \\int_0^\\Lambda dx x^2 F(x)^{5/2}, \\nonumber\\\\\n%\n I_{ho}(R) &=& \\frac{1}{3 \\pi N_f} \n \\int_0^\\Lambda dx x^4 F(x)^{3/2}, \\nonumber\\\\\n%\n I_{bf}(R) &=& \\frac{1}{3 \\pi N_f}\n \\int_0^\\Lambda dx x^2 e^{-\\frac{3 x^2}{2 R^2}} F(x)^{3/2}. \n\\label{eQh}\n\\end{eqnarray}\n%\n\nOne of the most important candidates for mixed condensate \nis the ${}^{39,41}$K-${}^{40}$K (boson-fermion) system. \nTheir scattering lengths are not well fixed at present, \nand different values have been reported experimentally \\cite{Cote,Kscat1,Kscat2}. \nIn the present calculations, \nto estimate the qualitative stability behaviors, \nwe take the value for $a_{bb}({}^{41}{\\rm K})$ in ref.\\cite{Cote}, \nand it gives ${\\tilde g} =0.2$ for the boson-boson interactions \nwith $\\omega =450{\\rm\\,Hz}$. \nFor the attractive boson-fermion interaction ${\\tilde h}$, \nseveral negative values are taken: \n$\\alpha \\equiv |{\\tilde h}|/{\\tilde g} =(0 \\sim 3)$. \nIt should be noted that the interaction strength \ncan be shifted using the Feshbach resonance phenomena \\cite{feshbach}, \nwhose applications for K-atom have been discussed in \\cite{Kscat1,Kfesh}. \n\nIn fig.~1, numerical results for $E(R)$ in (\\ref{eQg}) \nare plotted for\n$\\alpha =|{\\tilde h}|/{\\tilde g} =0.1$, $1.0$, \n$2.0$, $2.5$, $3$ (lines a-e) with $N =10^6$, \nwhere three kinds of patterns can be read off: \nstable (a), meta-stable (b,c) and unstable states (d,e). \nIn weak boson-fermion interaction, the system is stable \nagainst the variation of $R$, \nand has an absolute minimum as an equilibrium state. \nFor the intermediate strength, \nit becomes meta-stable with one local minimum (lines b,c), \nand, in strong attractive interactions, \nthe minimum disappears and the system become unstable (lines d,e). \nIt should be noted that similar meta-stable states appear \nin the BEC with attractive interactions (${}^7$Li) \\cite{Baym}, \nbut no stable states can exist in that case. \n\nAs can be found in fig.~1, \nthe stability of mixed condensate can be judged \nfrom the small $R$ behavior of $E(R)$: \npositive divergence in small $R$ suggests\nstable state. \nTo obtain the stability condition, \nwe consider analytically the asymptotic expansion of $E(R)$ \nin $R \\ll 1$. \nFor that purpose, \nthe leading order terms of ${\\tilde \\mu}_f(R)$ and $\\Lambda(R)$\nin small $R$ \nshould be determined from eq. (\\ref{eQf}) \nand the normalization condition: \n$\\int{d^3x} n_f(x;R) =N_f$. \nTo evaluate them, \nwe assume $\\Lambda(R)/R \\ll 1$ when $R \\ll 1$, \nand expand the Gaussian function \nto the order of $(x/R)^2$: \n$\\exp(-3 x^2/2 R^2) \\sim 1 -(3 x^2/2 R^2)$. \nThe consistency of these assumptions will be shown later, \nand can be also checked numerically. \nThese assumptions makes \nthe integral in the normalization condition \nevaluated analytically, and we obtain \n%\n\\begin{eqnarray}\n\\label{mu0}\n \\tilde{\\mu} &\\sim& -\\left( \\frac{3}{2\\pi} \\right)^{\\frac{3}{2}} \n |{\\tilde h}|N_b \\frac{1}{R^3} \n +\\sqrt{\\frac{3}{2}}\n \\left( \\frac{3}{2\\pi} \\right)^{\\frac{3}{4}} \n (48 N_f)^{\\frac{1}{3}} \n ({\\tilde h} N_b)^{\\frac{1}{2}} \n \\frac{1}{R^{\\frac{5}{2}}}, \n\\label{eQi}\\\\\n%\n \\Lambda &\\sim& \\left( \\frac{2}{3} \\right)^{\\frac{1}{4}} \n \\left( \\frac{2\\pi}{3} \\right)^{\\frac{3}{8}} \n \\frac{(48 N_f)^{\\frac{1}{6}}}{\n ({\\tilde h} N_b)^{\\frac{1}{4}}} \n R^{\\frac{5}{4}}. \n\\label{eQj}\n\\end{eqnarray}\n%\nIt should be noted that \neq. (\\ref{eQj}) leads to \n$\\Lambda/R \\propto R^{1/4} \\ll 1$ for $R \\ll 1$, \nwhich shows the consistency of our assumption. \nUsing eqs. (\\ref{eQi}) and (\\ref{eQj}), \nin the leading order in small $R$, \nthe $E(R)$ becomes\n%\n\\begin{eqnarray}\n E(R) /(\\hbar\\omega) \n &\\sim& \\left( \\frac{3}{2\\pi} \\right)^{\\frac{3}{2}}\n \\frac{N_b}{2} \n \\left\\{ \\frac{{\\tilde g} N_b}{2^{\\frac{2}{5}}} \n -|{\\tilde h}| N_f \\right\\} \\frac{1}{R^3} \\nonumber\\\\\n &+&\\frac{1}{4} \\frac{9}{6^{\\frac{1}{6}}} \n \\left( \\frac{3}{2\\pi} \\right)^{\\frac{3}{4}} \n (|{\\tilde h}| N_b)^{\\frac{1}{2}} \n N_f^{\\frac{4}{3}} \\frac{1}{R^{\\frac{5}{2}}} \n +{\\cal O}(R^{-2}), \n\\label{eQk}\n\\end{eqnarray}\n%\nso that, in small $R$ region, \n$E(R)$ is dominated by $R^{-3}$-term, \nand its positivity gives the stability condition: \n%\n\\begin{equation}\n \\alpha \\equiv \\frac{|{\\tilde h}|}{{\\tilde g}}\n < 2^{-5/2} \\frac{N_b}{N_f}. \n\\label{eQkAAA}\n\\end{equation} \n%\nFor the cases of fig.~1 , it becomes $\\alpha < 0.18$, \nwhich shows that only line (a) is for stable state. \nThis condition is also plotted as the dashed line in fig.2 .\n\nFrom the above stability condition, \nlines (b) and (c) in fig.~1 are found to be meta-stable, \nso that, in principle, \nthe equilibrium states (at local minima $R=R_{eq}$) \nshould collapse by the quantum tunneling effects \ninto those of $R=0$ \nthrough the potential barrier \nbetween $R=0$ and $R_{eq}$. \nTo study the collapses of meta-stable states, \nwe estimate their collective tunneling life-time \n$\\tau_{ct}$ applying the Gamow-theory of the nuclear $\\alpha$-decay \nto these states. \nFor the order estimation of the tunneling life-time, \nas an approximation for $E(R)$, \nwe use the simplified linear-plus-harmonic oscillator potential \n$V(R)$: \n%\n\\begin{equation}\n\\label{eQkAAAA}\nV(R) = \\cases{V_1(R) =F R -G, \n &($0 \\le R < R_t$) \\cr\n V_2(R) =\\frac{1}{2} \n M \\Omega^2 (R-R_{eq})^2 \n +V_0. \n &($R_{t} \\le R$) \\cr}\n\\end{equation}\n%\nAs shown in appendix A, \nthe tunneling life time $\\tau_{ct}$ for $V(R)$ becomes \n$\\tau^{-1}_{ct} =D \\exp(-W)$, \nwhere $D$ is a staying probability on barrier surface per unit time \nand $\\exp(-W)$ is a transmission coefficient for the potential barrier. \nThe explicit formulae for $D$ and $\\exp(-W)$ are \n%\n\\begin{eqnarray}\n D &=&\\frac{4\\sqrt{2}\\Omega}{\\sqrt{\\pi}} \n \\sqrt{ \\frac{\\Delta V}{\\hbar\\Omega} \n -\\frac{1}{2}} \n\\label{eQkAAB}\\\\\n%\n W &=&\\frac{4}{3} \n \\sqrt{\\frac{2M}{\\hbar^{2}}} \n \\left( \\Delta{V} -\\frac{\\hbar\\Omega}{2} \\right) \n (R_t -R_E) \n +\\frac{(R_0 -R_t)^2}{a_{HO}^{2}}, \n\\label{eQkAAC}\n\\end{eqnarray}\n%\nwhere $R_E$ is a WKB turning point for the energy $E$ \nand $\\Delta{V} = V_1(R_t) - V_0$ is a barrier height \n(see fig.~3). $a_{HO}=\\sqrt{\\hbar/M\\Omega}$ is harmonic\noscillator length around equilibrium point.\nThe derivation of eqs. (\\ref{eQkAAB},\\ref{eQkAAC}) \nis given in appendix A. \n\nTo apply the above formula for the metastable condensates, \nwe take $M =N m$ and $\\Omega =2 \\omega$. \nThese values have been determined to be the values in\nharmonic oscillator potential around the equilibrium point\nin the non-interacting case.\nFor parameters ($R_0$,$V_0$), ($R_t$,$V_1(R_t)$) and $R_E$, \nwe use the values that can be obtained from \nthe numerically calculated $E(R)$: \nthe equilibrium, maximum, WKB turning points \n(for $E =V_0 +\\hbar\\Omega/2$). \n\nFor the present case of $N =10^6$ and $\\alpha =2.0$\nwe obtain $R_t -R_E \\sim 1$, $R_0 -R_t \\sim 3$ \nin the unit of $\\hbar/m\\omega$, \nand $\\Delta{V} \\sim 40$ in the unit of $N\\hbar\\omega$ \n(see the line (c) in fig.~1). \nUsing these values for eqs. (\\ref{eQkAAB},\\ref{eQkAAC}), \nthe staying probability and the transmission coefficient \nbecome $D \\sim 10^8 \\omega$ and $W \\sim 10^7$. \nAs a result, we obtain \n$\\tau^{-1}_{ct}/\\omega \\sim 10^8 \\times \\exp(-10^7)$, \nand the life-time $\\tau_{ct}$ becomes very longer \nthan, for example, \nthat of clusterization by many-body collisions, \n$\\sim (1 \\sim 10){\\rm\\,sec}$. \nThus, meta-stable states should be really \nalmost stable\nexcept the ones extremely close to the unstable border \n(such as line d). \nThis long lifetime has originated from collectiveness \nof very many particles, \nand is consistent with that for meta-stable BEC \n\\cite{Tunneling}. \nTo shorten the tunneling life-time and \nmake meta-stable states collapse, \nthe parametric excitation with $\\omega =\\omega(t)$ may\ngive an interesting possibility, \nwhich will be discussed in another paper \\cite{MiyaN}. \n\nFor unstable states (e.~g. line e in fig.~1), \nusing eqs. (\\ref{eQi},\\ref{eQj}), \nthe leading $R$-dependence \nof the fermion density at $x=0$ and its rms radius are calculated: \n$n_f(0) \\propto R^{-\\frac{15}{4}}$ \nand $(x_f)_{rms} \\propto R^{\\frac{5}{4}}$, \nwhich are also checked numerically. \nThey show that, \nin the collapses of metastable and unstable states, \nboth boson and fermion distributions become localized \nand compressed, just like gravitational collapses \nof massive stars. \nThose collapses can also be found in \nthe instability by the collective excitations \nevaluated with the sum-rule method \\cite{Miya}, \nwhere the boson-fermion in-phase monopole excitation \nbecomes the zero mode. \n\nFinally, we comment about the critical condition \nfor the unstable regions, \nwhich has originally been obtained by M{\\o}lmer \\cite{Molmer} \nwith the Thomas-Fermi approximations for both fermion and boson \ndistribution in form of the scaling relation: \n$h^2/g \\propto N_f^{-p/(3p+6)}$ \nfor $V \\propto r^p$-type trapping potential\n(the derivation of the scaling law by M{\\o}lmer, \nsee appendix B). \nFor the harmonic oscillator potential ($p=2$), \nit becomes $\\alpha^2 N_f^{1/6} ={\\rm const}/g$. \nIn the present framework of the variational method, \nthe critical $\\alpha$ between the meta-stable \nand unstable states have been evaluated for several $N$ \nand plotted in fig.~2 (filled circles). \nFrom them, \nwe can find that the variational results \nare also consistent with the M{\\o}lmer scaling law: \nthe solid line added in fig.~2 in the unstable region. \n\nIn summary, \nfor stability of boson-fermion \nmixed condensates with boson-boson repulsive and \nboson-fermion attractive interactions, \nthe present variational calculations give\nthe phase diagram with three regions: \nstable, meta-stable and unstable ones. \nThey are clearly shown in fig.~2 \nin the case of $N_f =N_b \\equiv N$ and \n$\\omega_f =\\omega_b \\equiv \\omega$. \nTo estimate borders of stability regions generally, \nwe studied their critical conditions analytically, \nfrom which we can find that \nthe diagram should have qualitatively the similar pattern \nalso in general cases of the mixed condensates. \nFinally, it should be noted that, \nin the boson-fermion collapsing processes \nof unstable and meta-stable states, \nthe increasing boson/fermion densities \nmake the two-body short-range interactions \n(e.g. $p$-wave scattering processes) \nand three-body (or higher) collisions \nmore effective. \nThese interactions, \nwhich are not essential for low-density condensates, \nmay play important roles \nthrough the fermion-paired superfluid formations \nand clusterization into metal states \nby tunneling phenomena \\cite{Tunneling}. \nThey will be discussed in a future publication \\cite{MiyaN}. \n%\n%%%%%%%%%%%\n% Appendix \n%%%%%%%%%%%\n%\n\\appendix\n%\n\\section{Life Time of Collective Tunneling Effect in Metastable Region}\n%\nTo estimate a life time of metastable mixed condensates, \nwe consider the collective tunneling effect \nwith the collective variable $R$ (the boson radius) \nand its effective energy $E(R)$ as a collective potential. \nThe effective energy $E(R)$ is obtained from (\\ref{eQk}), \nand, for the metastable condensates, \na typical shape of it is the line (c) in Fig.~1. \nWe approximate this $E(R)$ \nwith the linear-plus-harmonic-oscillator type potential eq. \n(\\ref{eQkAAAA}) (Fig.~3).\n%\n%\\begin{equation}\n% V(R) = \\cases{V_1(R) =F R -G, \n% &($0 \\le R < R_t$) \\cr\n% V_2(R) =\\frac{1}{2} \n% M \\Omega^2 (R-R_0)^2 \n% +V_0, \n% &($R_{t} \\le R$) \\cr}\n%\\label{aP1}\n%\\end{equation}\n$M$ is inertia mass for the collective variable $R$. \nThe meanings of other parameters \n($F$, $G$, $\\Omega$, $R_{eq}$, $R_t$) \ncan be read off in Fig.~3, \nand they are fixed in order that the potential $V(R)$ reproduce \nthe $E(R)$ numerically obtained by (\\ref{eQk}). \nThis approximation should be enough for the order estimation \nof the life time. \nFor the kinetic term, we take\n%\n\\begin{equation}\n T =-\\frac{\\hbar^2}{2M} \\frac{d^2}{dR^2}. \n\\label{aP2}\n\\end{equation}\n\nIn $V(R)$, the metastable state $\\psi_M(x)$ before tunneling \nis approximately given by the ground-state wave function \nin the harmonic oscillator potential $V_2(R)$:\n%\n\\begin{equation}\n \\psi_M(R) = \\frac{1}{\\sqrt{\\pi^{\\frac{1}{2}} a_{HO}}}\n \\exp{\\left[-\\frac{(R-R_{eq})^2}{a_{HO}^2} \\right]}, \n\\label{aP3}\n\\end{equation}\nwhere $a_{HO}=\\sqrt{\\hbar/M\\Omega}$ is a harmonic oscillator length. \nThis state has the zero-point energy $\\hbar\\Omega/2$\nmeasured from $V_{0}$.\nThe continuum state $\\psi_D(R)$ after the tunneling decay is obtained by \nthe wave function in the linear potential $V_1(R)$, \nand its Schr\\\"odinger equation becomes \n%\n\\begin{equation}\n \\left[ \\frac{\\hbar^2}{2M} \\frac{d^{2}}{dR^2} \n +E -(F R -G) \\right] \\psi_D(R) =0. \n\\label{aP4}\n\\end{equation}\n%\nTo solve the above equation, \nwe use the WKB approximation. \nIt should be noticed that, \nbecause the state energy is lower than the potential maximum \n($E < F R_t -G$), \na turning point exist at $R_E =(E +G)/F$. \nThus, the WKB connection formula should be used at $R_E$ \nfor the continuum state $\\psi_D$: \n%\n\\begin{eqnarray}\n \\psi_D(R) &=&\\frac{A}{2 \\sqrt{K(R)}}\n \\sin{\\left[ \\frac{2}{3} \n \\left( \\frac{2MF}{\\hbar^2} \\right)^{\\frac{1}{2}}\n\t (R_E -R)^{\\frac{3}{2}} \n +\\frac{\\pi}{4} \\right]}, \n%\n \\quad \\hbox{for } 0 \\le R < R_E \n\\label{aP5}\\\\\n%\n \\psi_D(R) &=&\\frac{A}{2 \\sqrt{K(R)}}\n \\exp{\\left[-\\frac{2}{3} \n \\left( \\frac{2MF}{\\hbar^2} \\right)^{\\frac{1}{2}}\n (R-R_{E})^{\\frac{3}{2}} \\right]}, \n%\n \\quad R_E < R\n\\label{aP6}\n\\end{eqnarray}\n%\nwhere\n$A =\\sqrt{ 2M/(\\pi\\hbar^2) }$ and \n$\\hbar K(R) =\\sqrt{2MF |R_E -R|}$. \n\nLet us consider the tunneling decay rate \n$\\Gamma$ from $\\psi_M(R)$ to $\\psi_D(R)$. \nUsing the golden rule of Fermi, \nit becomes\n%\n\\begin{equation}\n \\Gamma =\\frac{2\\pi}{\\hbar} \n \\left| \\int dR \\, \n \\psi_D^*(R) [V(R) -V_2(R)] \n \\psi_M(R) \\right|^2_{E =E_M}, \n\\label{aP7}\n\\end{equation}\n%\nwhere $E_M =V_0 +\\hbar \\Omega/2$ is the energy of $\\psi_M(R)$. \nBecause $V(R) -V_2(R) =0$ for $R_t < R$ and $\\psi_M(0),\n\\psi'_M(0) \\ll 1$, eq. (\\ref{aP7}) becomes \n%\n\\begin{equation}\n \\Gamma =\\frac{2\\pi}{\\hbar} \n \\left( \\frac{\\hbar^2}{2M} \\right)^2\n \\left[ \\frac{d \\psi_D(R)}{dR} \\psi_M(R) \n -\\psi_D(R) \\frac{d \\psi_M(R)}{dR} \n \\right]_{R=R_t}. \n\\label{aP8}\n\\end{equation}\n%\nUsing eqs. (\\ref{aP3}, \\ref{aP5}, \\ref{aP6}), \nthe tunneling rate $\\Gamma$ is shown to have the form \n$\\Gamma =D \\exp(-W)$ : $D$ can be interpreted as \nthe staying probability Per unit time \nand $\\exp(-W)$ is the transition coefficient. \nThe explicit formulae for $D$ and $W$ are\n%\n\\begin{eqnarray}\n D &=&\\frac{4\\sqrt{2}\\Omega}{\\sqrt{\\pi}} \n \\sqrt{ \\frac{\\Delta V}{\\hbar\\Omega} \n -\\frac{1}{2}} \n\\label{aP9}\\\\\n%\n W &=&\\frac{4}{3} \n \\sqrt{\\frac{2M}{\\hbar^{2}}} \n \\left( \\Delta{V} -\\frac{\\hbar\\Omega}{2} \\right) \n (R_t -R_E) \n +\\frac{(R_0 -R_t)^2}{a_{HO}^{2}}, \n\\label{aP10}\n\\end{eqnarray}\n%\nwhere $\\Delta{V} = V_1(R_t) - V_0$ is a barrier height. \nTo represent the parameters $F$ and $G$ in $V_1$ \nby $\\Delta{V}$, \nwe have used \n%\n\\begin{equation}\n V_1(R_t) =F R_t -G, \\quad\n V_1(R_E) =F R_E -G =E =V_0 +\\frac{1}{2} \\hbar \\Omega. \n\\label{aP10A}\n\\end{equation}\n% \nThe tunneling life-time is defined as an inverse of decay rate: \n$\\tau_{ct} = \\Gamma^{-1}$.\n%\n\\section{Critical Condition for Unstable Region}\n%\nIn this appendix, we rederive the M{\\o}lmer scaling law \nwhich gives the critical condition for the unstable region \\cite{Molmer}. \nWe assume the Thomas-Fermi approximations both \nfor the boson and fermion density distributions. \nIn that approximations, the density distributions are given as \nsolutions of the Thomas-Fermi equations: \n%\n\\begin{eqnarray}\n & {\\tilde g} n_b(x) +x^2 +{\\tilde h} n_f(x) ={\\tilde \\mu}_b, \n\\label{ap11}\\\\\n & [6\\pi^2 n_f(x)]^{{2 \\over 3}} +x^2 \n +{\\tilde h} n_b(x) ={\\tilde \\mu}_f, \n\\label{ap12}\n\\end{eqnarray}\n%\nwhere $n_b(x)$ is the boson density distribution \nscaled by the harmonic-oscillator length $\\xi =(\\hbar/m\\omega)^{1/2}$ \n($n_b(x) =|\\Phi(x)|^2 \\xi^3$). \nThe scaled chemical potentials, ${\\tilde \\mu}_{b,f}$, \nand coupling constants, ${\\tilde g}$ and ${\\tilde h}$, \nhave been defined in the main body of this paper.\n\nEliminating $n_b(x)$ in (\\ref{ap12}) by (\\ref{ap11}), \nwe obtain the equation $F[n_f(x)] =G[n_f(x)]$ \nto determine the fermion density $n_f(x)$, \nwhere \n%\n\\begin{equation}\n F[n_f(x)] \\equiv [6\\pi^2 n_f(x)]^{{2 \\over 3}}, \\quad\n%\n G[n_f(x)] \\equiv {\\tilde \\mu}_f \n +{|{\\tilde h}| \\over {\\tilde g}} {\\tilde \\mu}_b \n -\\left(1+{|{\\tilde h}| \\over {\\tilde g}}\\right) x^2 \n +{|{\\tilde h}|^2 \\over {\\tilde g}} n_f(x). \n\\label{ap13}\n\\end{equation}\nWe concentrate on the central density $n_f(0)$. \nIn order to determine $n_{f}(0)$ two conditions\nshould be satisfied: \n%\n\\begin{equation}\n F[n_f(0)] =G[n_f(0)], \\quad\\quad \n {\\delta F \\over \\delta n_f}[n_f(0)] \n \\ge {\\delta G \\over \\delta n_f}[n_f(0)], \n\\label{ap14}\n\\end{equation}\n%\nEvaluating equations in (\\ref{ap14}) with (\\ref{ap13}), \nwe obtain the critical condition for the unstable region:\n%\n\\begin{equation}\n {|{\\tilde h}|^2 \\over {\\tilde g}} \n ={4\\pi^2 \\over \\sqrt{3}} \n \\left( {\\tilde \\mu}_f \n +{|{\\tilde h}| \\over {\\tilde g}} \n {\\tilde \\mu}_b \\right). \n\\label{ap15}\n\\end{equation}\n%\nTo evaluate the right-hand side of (\\ref{ap15}), \nwe should use the relations between ${\\tilde \\mu}_{b,f}$ \nand the boson/fermion particle number $N_{b,f}$, \nwhich are obtained by solving the Thomas-Fermi equations in \n(\\ref{ap12}). \nHere we assume that ${\\tilde \\mu}_b =0$ and \n${\\tilde \\mu}_f =2 (6 N_f)^{1/3}$ \n(${\\tilde \\mu}_f$ for a free fermion system) \nfor (\\ref{ap15}). \nConsequently, we obtain the M{\\o}lmer scaling relation: \n%\n\\begin{equation}\n {|{\\tilde h}|^2 \\over {\\tilde g}} \n ={4\\pi^2 \\over 6 ^{1/6} \\sqrt{6}} N_f^{-1/2} \n \\sim 12.0 N_f^{-1/6}. \n\\label{ap16}\n\\end{equation}\n%\nIt should be noted that the coefficient 12.0 in (\\ref{ap16}) \nare close to the value 13.8 \nwhich is obtained by M{\\o}lmer \\cite{Molmer} with numerically evaluating \nthe Thomas-Fermi equations. \n%\n%%%%%%%%%%%\n% References \n%%%%%%%%%%%\n%\n\\begin{references}\n%\n\\bibitem{bec} \nM.~H. Anderson, J.~R. Ensher, M.~R. Matthews,\nC.~E. Wieman, and E.A. Cornell, Science {\\bf 269}, 198 (1995); \nK.~B. Davis, M.-O. Mewes, M.~R. Andrews, N.~J. van Druten,\nD.~S. Durfee, D.~D. Kurn, and W. Ketterle,\nPhys.~Rev.~Lett. {\\bf 75}, 3969 (1995); \nC.~C. Bradley, C.~A. Sackett, and R.~G. Hulet,\nPhys.~Rev.~Lett. {\\bf 75}, 1687 (1995). \n%\n\\bibitem{review} \nK. Burnett, Contemp.~Phys. {\\bf 37}, 1 (1996); \nM. Lewenstein and L. You, Adv.~Atom.~Mol. Opt.~Phys., {\\bf 36}, 221 (1996); \nA. S. Parkins and H.~D.~F. Walls, Phys.~Reports {\\bf 303}, 1 (1998);\nF. Dalfovo, S. Giorgini, L.~P. Pitaevskii and S. Stringari Rev.~Mod.~Phys.{\\bf 71}, 463 (1999).\n%\n\\bibitem{DeMarco} \nB. DeMarco and D.~S. Jin , Science {\\bf 285}, 1703 (1999). \n%\n\\bibitem{hydro} \nI.~F. Silvera , Physica B {\\bf 109 \\& 110}, 1499 (1982); \nJ. Oliva , Phys.~Rev. B {\\bf 38}, 8811 (1988).\n%\n\\bibitem{Molmer} \nK. M{\\o}lmer, Phys. Rev. Lett. {\\bf 80}, 1804 (1998)\n%\n\\bibitem{Nygaard} \nN. Nygaard and K. M{\\o}lmer, Phys.~Rev. A {\\bf 59}, 2974 (1999).\n%\n\\bibitem{Miyakawa}\nT. Miyakawa, K. Oda, T. Suzuki and H. Yabu, \nJ.~Phys.~Soc.~Jpn. {\\bf 69}, 2997 (2000). \n%\n%\\bibitem{symbf} \n%E. Timmermans and R. C\\^{o}t\\'{e}, Phys.~Rev. ~Lett. {\\bf 80}, 3419 (1998);\n%W. Geist, L. You, and T.~A.~B. Kennedy, Phys.Rev. A {\\bf 59}, 1500 (1999);\n%M.-O. Mewes, G. Ferrari, F. Schreck, A. Sinatra and C. Salomon ; physics/9909007\n%\n\\bibitem{Amoruso} \nM. Amoruso, A. Minguzzi, S. Stringari, M.~P. Tosi\nand L. Vichi, Eur.~Phys.~J. D {\\bf 4}, 261 (1998); \nL. Vichi, M. Inguscio, S. Stringari, G.~M. Tino, J.~Phys. B \n{\\bf 31}, L899 (1998). \n%\n\\bibitem{Wadati}\nT. Tsurumi and M.Wadati, J.~Phys.~Soc.~Jpn. {\\bf 69}, 97 (2000)\n%\n\\bibitem{Minniti} \nM. Amoruso, C. Minniti and M. P. Tosi, Eur.~Phys.~J. D\n{\\bf 8}, 19 (1999). \n%\n\\bibitem{metaLi}\nC.~C. Bradley, C.~A. Sackett and R.~G. Hulet, Phys.~Rev.~Lett. {\\bf 78}, 985 (1997). \nC.~A. Sackett, C.~C. Bradley, M. Welling and R.~G. Hulet,\nAppl. Phys. B: Lasers Opt. {\\bf 65}, 433 (1997). \n%\n\\bibitem{Baym} \nG. Baym and C.~J. Pethick, Phys. Rev. Lett. {\\bf 76}, 6 (1996)\n%\n\\bibitem{Pethick}\nL. Viverit, C.~J. Pethick and H. Smith, cond-mat/9911080\n%\n\\bibitem{DeMarco2}\nB. DeMarco and D.~S. Jin, Phys.~Rev. A {\\bf 58}, 4267 (1998). \n%\n\\bibitem{Cote}\nR. Cot\\'e, A. Dalgarno, H. Wang and W.~C. Stwalley, Phys.~Rev. A \n{\\bf 57}, R4118 (1998). \n%\n\\bibitem{Kscat1} \nJ.~L. Bohn, J.P. Burke,Jr, C.H. Greene, H.Wang\n, P.~L. Gould, W.~C. Stwalley, Phys. Rev. A {\\bf 59}, 3660 (1999). \n%\n\\bibitem{Kscat2}\nB. Demarco, J.~L. Bohn, J.P. Burke,Jr,\nM. Holland and D.~S. Jin, Phys. Rev. Lett. {\\bf 82}, 4208 (1999)\n%\n\\bibitem{feshbach} \nE. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof,\nPhys.~Rev. A {\\bf 46}, R1167 (1992); \nE. Tiesinga, B. J. Verhaar, and H. T. C. Stoof,\nA {\\bf 47}, 4114 (1993). \n %\n\\bibitem{Kfesh} \nJ.~L. Bohn , Phys. Rev. A {\\bf 61}, 053409 (2000). \n%\n\\bibitem{Siemens}\nP.~J. Siemens and A.~S. Jensen, \n{\\it Elements of Nuclei}, p.~270, \n(Addison-Wesley, 1987). \n%\n\\bibitem{Landau}\nL.~D. Landau and E.~M. Lifschitz, \n{\\it Quantum Mechanics}, p.~174, \n(2nd ed., Pergamon Press, 1965)\n%\n\\bibitem{Tunneling}\nH.~T.~C. Stoof, J.~Stat.~Phys. {\\bf 87}, 1353, (1997);\nM. Ueda and A.~J. Leggett, Phys.~Rec.~Lett. {\\bf 80}, 1576 (1998);\nY.~A.~E. Kagan, A.E. Muryshev and G.~V. Shlyapnikov,\nPhys.~Rev.~Lett. {\\bf 81}, 933 (1998);\nA. Sackett, H.~T.~C. Stoof and R. G. Hulet,\nPhys.~Rev.~Lett. {\\bf 80}, 2031 (1998). \n%\n\\bibitem{MiyaN}\nT. Miyakawa et al., in preparation.\n%\n\\bibitem{Miya} \nT. Miyakawa, T. Suzuki and H. Yabu, Phys.~Rev. A {\\bf 62}, No46012 (2000).\n\\end{references}\n\\newpage\n%\n%%%%%%%%%%%%\n% FIGURE CAPTIONS\n%%%%%%%%%%%%\n%\n\\begin{figure}\n\\caption{The total energy variation $E(R)/(N\\hbar\\omega)$ \nof the boson-fermion mixed condensate \nwith the Gaussian boson distribution $\\Phi(x;R)$ of radius $R$: \n$E(R)$ and $\\Phi(x;R)$ are defined in (\\ref{eQg}, \\ref{eQd}). \nThe Thomas-Fermi density function (\\ref{eQe}) is applied for \nthe fermion distribution. \n$N =N_b=N_f =10^6$, \n${\\tilde g} =0.2$, \nand \n$\\alpha ={\\tilde h}/{\\tilde g} =0.1$, $1.0$, $2$, $2.5$, $3$ \nfor lines a-e.} \n\\end{figure}\n\n\\begin{figure}\n\\caption{Stability phase diagram \nof the boson-fermion mixed condensate \nfor $N \\equiv N_b =N_f$ \nand $\\alpha ={\\tilde h}/{\\tilde g}$ \nwhen ${\\tilde g} =0.2$. \nThree regions exist in it: \nstable (S), meta-stable (MS) and unstable (US) ones. \nThe dashed line at $\\alpha =0.18$ corresponds to \nthe border of stable region, \nand the solid line shows the M{\\o}lmer scaling law \nbetween meta-stable and unstable regions. \nThe open circles correspond to the parameters \nof the states in fig.~1, \nand the filled circles show \nnumerically confirmed critical states \nbetween meta-stable and unstable regions. }\n\\end{figure}\n%\n\\begin{figure}\n\\caption{Simplified potential for meta-stable condensates.\nThe $V_{1}(R)$ and $V_{2}(R)$ are the linear and harmonic oscillator \nparts of the potential $V(R)$. \nThe $R_{eq}$, $R_t$, $R_E$ are the equilibrium, maximum \nand the WKB turning points. \n$V_0$ is the equilibrium energy of the potential.}\n\\end{figure}\n\\end{document}"
}
] |
[
{
"name": "cond-mat0002048.extracted_bib",
"string": "\\bibitem{bec} \nM.~H. Anderson, J.~R. Ensher, M.~R. Matthews,\nC.~E. Wieman, and E.A. Cornell, Science {\\bf 269}, 198 (1995); \nK.~B. Davis, M.-O. Mewes, M.~R. Andrews, N.~J. van Druten,\nD.~S. Durfee, D.~D. Kurn, and W. Ketterle,\nPhys.~Rev.~Lett. {\\bf 75}, 3969 (1995); \nC.~C. Bradley, C.~A. Sackett, and R.~G. Hulet,\nPhys.~Rev.~Lett. {\\bf 75}, 1687 (1995). \n%\n\n\\bibitem{review} \nK. Burnett, Contemp.~Phys. {\\bf 37}, 1 (1996); \nM. Lewenstein and L. You, Adv.~Atom.~Mol. Opt.~Phys., {\\bf 36}, 221 (1996); \nA. S. Parkins and H.~D.~F. Walls, Phys.~Reports {\\bf 303}, 1 (1998);\nF. Dalfovo, S. Giorgini, L.~P. Pitaevskii and S. Stringari Rev.~Mod.~Phys.{\\bf 71}, 463 (1999).\n%\n\n\\bibitem{DeMarco} \nB. DeMarco and D.~S. Jin , Science {\\bf 285}, 1703 (1999). \n%\n\n\\bibitem{hydro} \nI.~F. Silvera , Physica B {\\bf 109 \\& 110}, 1499 (1982); \nJ. Oliva , Phys.~Rev. B {\\bf 38}, 8811 (1988).\n%\n\n\\bibitem{Molmer} \nK. M{\\o}lmer, Phys. Rev. Lett. {\\bf 80}, 1804 (1998)\n%\n\n\\bibitem{Nygaard} \nN. Nygaard and K. M{\\o}lmer, Phys.~Rev. A {\\bf 59}, 2974 (1999).\n%\n\n\\bibitem{Miyakawa}\nT. Miyakawa, K. Oda, T. Suzuki and H. Yabu, \nJ.~Phys.~Soc.~Jpn. {\\bf 69}, 2997 (2000). \n%\n%\n\\bibitem{symbf} \n%E. Timmermans and R. C\\^{o}t\\'{e}, Phys.~Rev. ~Lett. {\\bf 80}, 3419 (1998);\n%W. Geist, L. You, and T.~A.~B. Kennedy, Phys.Rev. A {\\bf 59}, 1500 (1999);\n%M.-O. Mewes, G. Ferrari, F. Schreck, A. Sinatra and C. Salomon ; physics/9909007\n%\n\n\\bibitem{Amoruso} \nM. Amoruso, A. Minguzzi, S. Stringari, M.~P. Tosi\nand L. Vichi, Eur.~Phys.~J. D {\\bf 4}, 261 (1998); \nL. Vichi, M. Inguscio, S. Stringari, G.~M. Tino, J.~Phys. B \n{\\bf 31}, L899 (1998). \n%\n\n\\bibitem{Wadati}\nT. Tsurumi and M.Wadati, J.~Phys.~Soc.~Jpn. {\\bf 69}, 97 (2000)\n%\n\n\\bibitem{Minniti} \nM. Amoruso, C. Minniti and M. P. Tosi, Eur.~Phys.~J. D\n{\\bf 8}, 19 (1999). \n%\n\n\\bibitem{metaLi}\nC.~C. Bradley, C.~A. Sackett and R.~G. Hulet, Phys.~Rev.~Lett. {\\bf 78}, 985 (1997). \nC.~A. Sackett, C.~C. Bradley, M. Welling and R.~G. Hulet,\nAppl. Phys. B: Lasers Opt. {\\bf 65}, 433 (1997). \n%\n\n\\bibitem{Baym} \nG. Baym and C.~J. Pethick, Phys. Rev. Lett. {\\bf 76}, 6 (1996)\n%\n\n\\bibitem{Pethick}\nL. Viverit, C.~J. Pethick and H. Smith, cond-mat/9911080\n%\n\n\\bibitem{DeMarco2}\nB. DeMarco and D.~S. Jin, Phys.~Rev. A {\\bf 58}, 4267 (1998). \n%\n\n\\bibitem{Cote}\nR. Cot\\'e, A. Dalgarno, H. Wang and W.~C. Stwalley, Phys.~Rev. A \n{\\bf 57}, R4118 (1998). \n%\n\n\\bibitem{Kscat1} \nJ.~L. Bohn, J.P. Burke,Jr, C.H. Greene, H.Wang\n, P.~L. Gould, W.~C. Stwalley, Phys. Rev. A {\\bf 59}, 3660 (1999). \n%\n\n\\bibitem{Kscat2}\nB. Demarco, J.~L. Bohn, J.P. Burke,Jr,\nM. Holland and D.~S. Jin, Phys. Rev. Lett. {\\bf 82}, 4208 (1999)\n%\n\n\\bibitem{feshbach} \nE. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof,\nPhys.~Rev. A {\\bf 46}, R1167 (1992); \nE. Tiesinga, B. J. Verhaar, and H. T. C. Stoof,\nA {\\bf 47}, 4114 (1993). \n %\n\n\\bibitem{Kfesh} \nJ.~L. Bohn , Phys. Rev. A {\\bf 61}, 053409 (2000). \n%\n\n\\bibitem{Siemens}\nP.~J. Siemens and A.~S. Jensen, \n{\\it Elements of Nuclei}, p.~270, \n(Addison-Wesley, 1987). \n%\n\n\\bibitem{Landau}\nL.~D. Landau and E.~M. Lifschitz, \n{\\it Quantum Mechanics}, p.~174, \n(2nd ed., Pergamon Press, 1965)\n%\n\n\\bibitem{Tunneling}\nH.~T.~C. Stoof, J.~Stat.~Phys. {\\bf 87}, 1353, (1997);\nM. Ueda and A.~J. Leggett, Phys.~Rec.~Lett. {\\bf 80}, 1576 (1998);\nY.~A.~E. Kagan, A.E. Muryshev and G.~V. Shlyapnikov,\nPhys.~Rev.~Lett. {\\bf 81}, 933 (1998);\nA. Sackett, H.~T.~C. Stoof and R. G. Hulet,\nPhys.~Rev.~Lett. {\\bf 80}, 2031 (1998). \n%\n\n\\bibitem{MiyaN}\nT. Miyakawa et al., in preparation.\n%\n\n\\bibitem{Miya} \nT. Miyakawa, T. Suzuki and H. Yabu, Phys.~Rev. A {\\bf 62}, No46012 (2000).\n"
}
] |
cond-mat0002049
|
Second Topological Moment $\langle m^2 \rangle$ of Two Closed Entangled Polymers
|
[
{
"author": "Franco Ferrari$^{(1)}$ Hagen Kleinert$^{(2)}$ and Ignazio Lazzizzera$^{(1)}$"
},
{
"author": "{$^{(1)}$Dipartimento di Fisica, Universit\\`a di Trento, I-38050 Povo, Italy"
},
{
"author": "and INFN, Gruppo Collegato di Trento.}"
},
{
"author": "{$^{(2)}$Institut f\\\"ur Theoretische Physik,"
},
{
"author": "Freie Universit\\\"at Berlin, Arnimallee 14, D-14195 Berlin, Germany.}"
}
] |
We calculate exactly by field theoretical techniques the second topological moment $\langle m^2 \rangle$ of entanglement of two closed polymers $P_1$ and $P_2$. This result is used to estimate approximately the mean square average of the linking number of a polymer $P_1$ in solution with other polymers. %The latter are replaced by %a single very long effective molecule $P_2$.
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[
{
"name": "topolms8.tex",
"string": "\\documentstyle[pra,epsf,aps,twocolumn]{revtex}\n%\\documentstyle[12pt,pra,epsf,aps]{revtex}\n%\\documentstyle[12pt,twocolumn]{revtex}\n\\def\\lfrac#1#2{#1/#2}\n\\def\\sbf#1{{\\footnotesize {\\bf #1}}}\n\\def\\mn#1{\\marginpar{\\scriptsize#1}}\n%\\renewcommand{\\baselinestretch}{2}\n\\begin{document}\n\\sloppy\n\\title{Second Topological Moment $\\langle m^2 \\rangle$\nof Two Closed Entangled Polymers}\n%\n%\n\\author{Franco Ferrari$^{(1)}$\nHagen Kleinert$^{(2)}$\nand Ignazio Lazzizzera$^{(1)}$\\\\\n{$^{(1)}$\\it Dipartimento di Fisica, Universit\\`a di Trento, I-38050 Povo,\nItaly\\\\\nand INFN, Gruppo Collegato di Trento.}\\\\\n{$^{(2)}$\\it Institut f\\\"ur Theoretische Physik,\\\\\nFreie Universit\\\"at Berlin, Arnimallee 14, D-14195 Berlin, Germany.}}\n\\date{February 2000}\n\\maketitle\n%\n\\begin{abstract}\nWe calculate exactly by field theoretical techniques the second topological\nmoment\n$\\langle m^2 \\rangle$ of entanglement\nof two closed polymers $P_1$ and $P_2$.\nThis result is used to estimate approximately the mean square average of the\nlinking number\nof a polymer $P_1$ in solution with other polymers.\n%The latter are replaced by\n%a single very long effective molecule $P_2$.\n\\end{abstract}\n%\n{\\bf 1.}\nConsider two closed polymers $P_1$ and $P_2$\nwhich statistically can be linked with each other any number of\ntimes $m=0,1,2,\\dots~$.\nThe situation is illustrated in Fig.~\\ref{Fig. 1}\nfor $m= 2$.\n\\begin{figure}[tbhp]\n\\input link.tps\n\\caption[]{Closed polymers $P_1, P_2$ with trajectories $C_1, C_2$\nrespectively.}\n\\label{Fig. 1}\\end{figure}\n%\nAn important physical quantity is the probability\ndistribution of the linking number $m$ as a function of\nthe lengths of $P_1$ and $P_2$.\nAs a first step towards finding it\nwe calculate,\nin\nthis note, an exact expression for the second moment of the distribution,\n$\\langle m^2 \\rangle$.%, in the limit of very long polymers.\n\nAn approximate result for this quantity was obtained before\nin Ref.~\\cite{BS} on the basis of a\na mean-field method, considering\nthe density of bond vectors of $P_2$\nas Gaussian random variables. Such methods are usually quite accurate\nwhen a large\nnumber of polymers is involved \\cite{BV,OV}. As an unpleasant feature, however, they\nthey introduce a dependence on the source of\nGaussian noise, and modify the critical behavior of the system, whereas\ntopological interactions are not expected to do that \\cite{MK,ffil}.\nOur note goes therefore an important step beyond this approximation.\nIt treats the two-polymer problem exactly, and contains\nan application to the\ntopological entanglement in diluted solutions.\nThe relevance of the two-polymer systems\nto such systems\nwas emphasized\nin \\cite{BS}. Focusing attention upon a particular molecule,\n$P_1$,\n one may imagine all others to form\n a single long effective molecule $P_2$.\n\n\n\n{\\bf 2.}\nLet\n%\n%\n$G_m ({\\bf x}_1, {\\bf x}_2; L_1, L_2)$ be\nthe configurational probability to\nfind the polymer $P_1$ of length $L_1$\nwith fixed coinciding end points at ${ \\bf x}_1$\nand\nthe polymer $P_2$ of length $L_2$\nwith fixed coinciding end points at ${ \\bf x}_2$,\ntopologically entangled with\na Gaussian linking number $ m$.\n\n\nThe second moment\n $\\langle m^2 \\rangle$ is defined by the ratio of integrals \\cite{tan}\n%\n\\begin{eqnarray}\n \\langle m^2 \\rangle = \\frac{\\int d^3 x_1 d^3 x_2\n \\int^{+\\infty}_{-\\infty } dm ~m^2 G_m\n\\left({\\bf x}_1, {\\bf x}_2; L_1, L_2\\right)}\n {\\int d^3 x_1d^3 x_2 \\int^{+\\infty}_{-\\infty} dm\\, G_m\n \\left( {\\bf x}_1, {\\bf x}_2; L_1, L_2\\right) }\n\\label{1}\\end{eqnarray}\n%\n%performed for either of the two probabilities.\n\n\nThe denominator in (\\ref{1}) plays the role of a partition function:\n\\begin{eqnarray}\nZ\\equiv\n {\\int d^3 x_1 d^3 x_2 \\int^{+\\infty}_{-\\infty} dm\\, G_m\n \\left( {\\bf x}_1, {\\bf x}_2; L_1, L_2\\right) }.\n\\label{1Z}\\end{eqnarray}\n%\n\n\nDue to the translational invariance of the system, the probabilities\ndepend only on the differences between the end point coordinates:\n%\n\\begin{eqnarray}\n G_m \\left( {\\bf x}_1, {\\bf x}_2; L_1, L_2\\right) = G_m \\left({\\bf x}_1\n - {\\bf x}_2;\nL_1, L_2\\right).\n\\label{2}\\end{eqnarray}\n%\nThus, after a shift of variables,\nthe spatial double integrals in (\\ref{1}) can be\nrewritten as\n%\n\\begin{eqnarray}\n%&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\int\\! d^3 x_1 d^3 x_2 G_m ({\\bf x}_1\n \\! - \\!{\\bf x}_2 ; L_1, L_2)\n%\\nonumber \\\\&&\n\\!=\\!\nV\\int\\! d^3 x G_m ({\\bf x} ; L_1, L_2), \\nonumber\n\\label{3}\\end{eqnarray}\n%\nwhere $V$ denotes the total volume of the system.\n%\n\n{\\bf 3.}\nThe most efficient way of describing the statistical\nfluctuations of the\npolymers $P_1$ and $P_2$ is by\ntwo complex polymer fields\n$ \\psi_1^{{ a}_1} ({\\bf x}_1)$ and\n $\\psi_2^{{ a}_2} ({\\bf x}_2)$\nwith $n_1$ and $n_2$ replicas $(a_1=1,\\dots,n_1,~a_2=1,\\dots,n_2)$.\nAt the end we shall take $n_1,n_2\\rightarrow 0$\nto ensure that these fields describe only one polymer each \\cite{pi}.\n\nFor these fields we define an auxiliary\nprobability $G_ \\lambda (\\vec {\\bf x}_1, \\vec{\\bf x}_2 ; \\vec{z} )$\nto find the polymer $P_1$\nwith open ends at ${\\bf x}_1,{\\bf x}_1'$\nand the polymer $P_2$\nwith open ends at ${\\bf x}_2,{\\bf x}_2'$.\nThe double vectors\n$\\vec{{\\bf x}}_1\\equiv ({\\bf x}_1,{\\bf x}_1')$ and\n$\\vec{{\\bf x}}_2\\equiv ({\\bf x}_2,{\\bf x}'_2)$\ncollect initial and final\nendpoints of the two polymers\n $P_1$ and $P_2$.\nHere we follow the\napproach of Edwards \\cite{Ed}, in which one starts\nwith open polymers with fixed ends.\nThe case of closed polymers, where $m$ becomes a true topological number and\nit is thus relevant in the present context,\nis recovered in the limit of coinciding extrema.\nWe notice that in this way one introduces in the configurational probability\nan artificial dependence on the fixed points\n${\\bf x}_1$ and\n${\\bf x}_2$. In physical\nsituations, however, the fluctuations of the polymers are entirely free.\nFor this reason we have averaged in (\\ref{1}) over all possible\nfixed points by means of the integrations in $d^3 {\\bf x}_1d^3 {\\bf x}_2 $.\n\n\n\n\nThe\nauxiliary\nprobability $G_ \\lambda (\\vec {\\bf x}_1, \\vec{\\bf x}_2 ; \\vec{z} )$\nis given\nby a functional integral \\cite{ffil}\n%\n%\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!G_ \\lambda (\\vec{{\\bf x}}_1, \\vec{{\\bf x}}_2 ; \\vec z\n)\n = \\lim _{n_1, n_2 \\rightarrow 0} \\int\n {\\cal D} ({\\rm fields}) \\,\\nonumber \\\\\n & & \\times \\psi_1^{{a}_1} ({\\bf x}_1)\n \\psi_1^{* { a}_1} ({\\bf x}_1')\n\\psi_2^{{ a}_2} ({\\bf x}_2) \\psi_2^{*{ a}_2} ({\\bf x}_2')\n e^{- {\\cal A}},\n\\label{4}\\end{eqnarray}\n%\nwhere\n%\n${\\cal D} (\\mbox{fields})$\nindicates the measure of functional\nintegration and $a_1,a_2$ are now fixed replica indices.\n${\\cal A}={\\cal A}_{\\rm CS}+{\\cal A}_{\\rm pol}$ is the action\ngoverning the fluctuations.\nIt consists of a polymer action\n\\begin{eqnarray}\n {\\cal A}_{\\rm pol} = \\sum _{i=1}^{2} \\int d^3{\\bf x} \\left[ |\\bar {\\bf D}^i\n\\Psi_i|^2 +\n m^2_i |\\Psi_i |^2 \\right].\n\\label{5}\\end{eqnarray}\n%\n%\n and a Chern-Simons action to\ndescribe the linking number $m$\n\\begin{eqnarray}\n {\\cal A}_{\\rm CS}\n =i\n{ \\kappa }\n\\int d^3 x \\,\\varepsilon_{\\mu \\nu \\rho }\n A_1^\\mu \\partial _ \\nu A_2^\\rho ,\n\\label{5CS}\\end{eqnarray}\n%\nIn Eq. (\\ref{5CS}) we have omitted a\ngauge fixing term, which enforces the Lorentz gauge.\nThe effects of self-entanglement and of\nthe so-called {\\em excluded-volume\\/} interactions\nare ignored.\n%\nThe Chern-Simons fields are coupled to the polymer fields by the\n covariant derivatives\n%\n%\\def\\nablab{\\mbox{\\twelvembsy\\symbol{'162}}}\n\\def\\nablab{\\BF \\nabla}\n\\newcommand{\\BF}[1]{\\mbox{\\boldmath $#1$}}\n$\n {\\bf D}^i = {\\nablab} +i \\gamma _i {\\bf A}^i,\n$\n%\nwith the coupling constants $ \\gamma _{1,2}$\ngiven by\n%\n$\n \\gamma _1 = \\kappa,~\n \\gamma _2 = \\lambda .$\n%\nThe square masses of the polymer fields\nare given by\n$ m_i^2 = 2 M z_i$,\n%\nwhere\n $M=3/a$, with $a$ being the length of the polymer links,\nand $z_i$\nthe chemical\npotentials of the polymers,\n measured in units of the temperature.\n%\nThe chemical\npotentials are conjugate variables to the length\nparameters $L_1$ and $L_2$, respectively.\n%\n%\nThe symbols $\\Psi_i$ collect the replicas $\\psi_i^{a_i}$ of the\ntwo polymer fields. Let us note that in the topological Landau-Ginzburg model\n(\\ref{4}) the\nChern-Simons fields do not change the critical behavior of the system,\nas expected.\n\n The parameter $ \\lambda $ is conjugate to the linking number $m$.\nWe can therefore calculate the desired probability\n $G_m (\\vec {\\bf x}_1, \\vec {\\bf x}_2; L_1, L_2)$\nfrom the auxiliary one\n $G_ \\lambda (\\vec {\\bf x}_1, \\vec {\\bf x}_2; \\vec z)$\nby\n Laplace integrals over $\\vec z=(z_1,z_1)$ and an inverse\nFourier transformation\nover $\\lambda$.\n%screens part of the repulsive excluded volume interactions \\cite{ffil}.\n%\n%\n\n%%%%\n{\\bf 4.}\nLet us use the polymer field theory\nto calculate the partition function (\\ref{1Z}).\n%\n It is given by the integral\nover the auxiliary probabilities\n%\n\\begin{eqnarray}\n Z & = &\\int d^3 x_1 d^3 x_2 \\lim_{{\\bf x}_1' \\rightarrow {\\bf x}_1\n \\atop\n {\\bf x}_2 '\\rightarrow {\\bf x}_2}\n \\int^{c + \\infty}_{c - i \\infty} \\frac{Md z_1}{2\\pi i}\n \\frac{Mdz_2}{2\\pi i} e^{z_1 L_1 + z_2 L_2} \\nonumber \\\\\n&&\\times \\int^{+ \\infty}_{-\\infty} d m \\int^{+ \\infty}_{-\\infty} d \\lambda\ne^{-im \\lambda }\n G_ \\lambda \\left( \\vec{\\bf x}_1 ,\\vec{\\bf x}_2 ; \\vec z\\right).\n\\label{15}\\end{eqnarray}\n%\n The integration over $m$ is trivial and gives\n $2 \\pi \\delta ( \\lambda )$, enforcing $ \\lambda =0$, so that\n%\n\\begin{eqnarray}\n Z & = & \\int d^3 x_1 d^3 x_2 \\lim_{{\\bf x}_1 \\rightarrow {\\bf x}'_1\n \\atop {\\bf x}_2' \\rightarrow {\\bf x}_2} \\int^{c + i\\infty}_{c - i \\infty}\n \\frac{Mdz_1}{2\\pi i}\\frac{Mdz_2}{2\\pi i}\ne^{z_1 L_1 + z_2 L_2} \\nonumber \\\\\n & &~~~~~~~~~~~~~\\times~~\n G_{ \\lambda =0} \\left( \\vec{\\bf x}_1 ,\\vec{\\bf x}_2 ; \\vec z \\right).\n\\label{16}\\end{eqnarray}\n%\nTo calculate $\n G_{ \\lambda =0} \\left( \\vec{\\bf x}_1 ,\\vec{\\bf x}_2 ; \\vec z\n \\right)$ we observe\n that the action ${\\cal A}$\n is quadratic\nin $ \\lambda $.\nA trivial calculation gives\n\\begin{eqnarray}\n &&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!G_{ \\lambda =0} \\left( \\vec{\\bf x}_1 ,\n\\vec{\\bf x}_2 ; \\vec z\n \\right)\n = \\int {\\cal D} ({\\rm fields}) e^{-{\\cal A}_0} \\nonumber \\\\\n&& \\times \\psi_1^{{a}_1}({\\bf x}_1)\n \\psi_1^{*{a}_1} ({\\bf x}_1')\\psi_2^{{a}_2} ({\\bf x}_2) \\psi_2^{{a}_2}\n({\\bf x}')\n\\label{22}\\end{eqnarray}\nwhere\n%\n\\begin{eqnarray}\n\\!{\\cal A}_0 &\\equiv &{\\cal A}_{\\rm CS}\n% \\nonumber \\\\&+&\n\\!+\\! \\!\\int d^3 \\!{\\bf x}\\! \\left[ |{\\bf D}_1 \\Psi_1|^2\\! +\\! | \\nablab \\Psi_2\n |^2\\! +\\! \\sum_{i =1} ^2 m_i^2|\\Psi_i|^2 \\right],\n\\label{18}\n\\end{eqnarray}\n%\n%\n{}From Eq.~(\\ref{18}) it is clear that\n$G_{ \\lambda =0} \\left( \\vec{\\bf x}_1 ,\\vec{\\bf x}_2 ; \\vec z\n \\right)$\nis the product of the configurational\nprobabilites of two free polymers.\nIn fact, the fields $\\Psi_2, \\Psi_2^*$ are free, whereas\nthe fields $\\Psi_1, \\Psi^*_1$ are apparently not free because\nof the couplings with the Chern-Simons fields through the covariant\nderivative ${\\bf D}^1$.\nThis is, however, an illusion:\nIntegrating out $A_2^\\mu$ in (\\ref{22}), we find\nthe flatness condition:\n%\n $\\varepsilon^{\\mu \\nu \\rho } \\partial_ \\nu A^i_\\mu= 0.$\n%\n On a flat space with vanishing boundary conditions at infinity this\nimplies $ A_1^\\mu = 0$.\n%%%\nAs a consequence,\nthe functional\nintegral (\\ref{22}) factorizes\n\\begin{eqnarray}\n\\!\\!\\!\n G_{ \\lambda =0} \\left( \\vec{\\bf x}_1 ,\\vec{\\bf x}_2 ; \\vec z \\right)\n = G_0 ({\\bf x}_1 - {\\bf x}_1'; z_1) \\,G_0 ({\\bf x}_2 - {\\bf x}_2'\n ; z_2 ) ,\n\\label{24}\\end{eqnarray}\n%\nwhere $ G_0 ({\\bf x}_i - {\\bf x}_i' ; z_i)$\nare the free correlation functions of the polymer fields\n%\n\\begin{eqnarray}\n G_0 ({\\bf x}_i - {\\bf x}_i'; L_i )\n&=&\\int^{c+i\\infty}_{c-i\\infty} \\frac{Mdz_i}{2\\pi i}e^{z_iL_i} G_0\n ({\\bf x}_i - {\\bf x}_i' ; z_i )\n \\nonumber \\\\\n & &\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n= \\frac{1}{2 } \\left(\\frac{M}{4 \\pi L_i}\\right)^{3/2}\n e^{{- M({\\bf x} _i -{\\bf x}_i')^2}/{2L_i} }.\n\\label{27}\\end{eqnarray}\n%\n%\n%\nThus we obtain for (\\ref{16}) the integral\n%\n\\begin{eqnarray}\n \\!\\!Z\\! =\\! 2 \\pi \\!\\! \\int\\!\\! d^3 x_1 d^3 x_2\n%\\nonumber \\\\ && \\times\n \\lim_{{\\bf x}_1' \\rightarrow {\\bf x}_1 \\atop\n {\\bf x}_2'\\rightarrow {\\bf x}_2 }\n G_0 ({\\bf x}_1 \\!-\\! {\\bf x}_1' ; L_1 )\\, G_0 ({\\bf x}_2 \\!-\\! {\\bf x}_2'\n;L_2),\n \\nonumber \\label{29}\\end{eqnarray}\n%\n%\nyielding the partition function\n\\begin{eqnarray}\nZ =\n \\frac{2 \\pi M^3 V^2}{(4 \\pi )^3} (L_1 L_2)^{-3/2}.\n \\label{30}\\end{eqnarray}\n%\n\nIt is important to realize that in Eq.~(\\ref{15})\n the limits of coinciding end points\n${\\bf x}_i' \\rightarrow {\\bf x}_i$ and\nthe inverse Laplace transformations\n do not commute unless a proper\nrenormalization scheme is chosen to eliminate the divergences\ncaused by the insertion of the composite operators $ |\\psi_i^{a_i}\n({\\bf x})|^2$ and $|\\Psi_i ({\\bf x})|^2$.\n%This can be seen for\n%a single polymer $P$. If we were commuting the limit of coinciding end\n%points with the Laplace\n%transform, we would obtain\n%%\n%\\begin{eqnarray}\n% &&\n%\\!\\!\\int^{c + i\\infty}_{c -i\\infty}\\!\\! \\frac{dz}{2\\pi} {e^{zL}}\n%\\!\\! \\lim_{{\\bf x}' \\rightarrow {\\bf x}} G_0 ({\\bf x}\\! -\\!{\\bf x}'; z)\n%%\\nonumber \\\\& =&\n%\\!=\\!\\int^{c + i\\infty}_{c-i\\infty} \\frac{dz}{2\\pi i}e^{zL}G_0 ({\\bf 0},z)\n%\\label{31}\\nonumber \\end{eqnarray}\n%%\n%with\n%\n%$ G_0 ({\\bf 0};z) = \\langle |\\psi ({\\bf x})|^2\\rangle $\n%being linealy divergent in three dimensions.\n\n{\\bf 5.}\nLet us now turn to the numerator in Eq.~(\\ref{1}). Exploiting\nthe identity $m^2e^{-im\\lambda}=-{\\partial^2}\ne^{-im\\lambda}/{\\partial \\lambda^2}$, and performing two partial integrations\n in $\\lambda$,\nthe same technique\nused above to evaluate the partition function $Z$ yields\n\\begin{eqnarray}\n N & = & \\kappa ^2 \\int d^3 x_1 d^3 x_2 \\lim_{n_1 \\rightarrow 0\n\\atop\n n_2 \\rightarrow 0}\n \\int^{c + i\\infty}_{c - i \\infty} \\frac{Md z_1 }{2\\pi i}\n \\frac{Mdz_2}{2\\pi i} e^{z_1 L_1 + z_2 L_2}\n \\nonumber \\\\\n &\n \\times &\\int {\\cal D}(\\mbox{fields})~ \\exp (-{\\cal A}_0)\n\\vert\\psi_1^{a_1} ({\\bf x}_1)\\vert^2\n \\vert \\psi_2^{a_2} ({\\bf x}_2 ) \\vert^2\n \\nonumber \\\\\n &\n\\times& \\left[ \\left(\\int d^3 x \\,{\\bf A}_1 \\cdot\n \\Psi_1^*\n{\\bf \\nablab} \\Psi_1 \\right)^2\n \\!\\!+ \\frac{1}{2} \\int d^3 x \\, {\\bf A}_1^2\\,\\vert \\Psi_1 \\vert^2\n \\right]\n \\nonumber \\\\\n &\n\\times& \\left[ \\left(\\int d^3 x \\,{\\bf A}_2 \\cdot\n \\Psi_2^*\n{\\bf \\nablab} \\Psi_2 \\right)^2\n\\!\\! + \\frac{1}{2} \\int d^3 x \\, {\\bf A}_2^2\\,\\vert \\Psi_2 \\vert^2\n \\right].\n\\label{39}\\end{eqnarray}\nwhere ${\\cal A}_0$ has been defined in (\\ref{18}).\nIn the above equation we\nhave taken\nthe limits of coinciding endpoints\ninside the\n Laplace integral over $z_1, z_2$.\nThis will be justified later\non the grounds that\nthe potentially dangerous Feynman diagrams containing\nthe insertions of operations like $|\\Psi_i|^2$ vanish\nin the limit $n_1, n_2 \\rightarrow 0$.\nThe functional integral in\nEq.~(\\ref{39})\ncan be calculated exactly\n by diagrammatic\nmethods since\nonly four diagrams shown in Fig.~(\\ref{4dia})\ncontribute.%\n\\begin{figure}[tbhp]\n~\\\\[-2.5mm]{}~\\input 2u4s.tps~\\\\\n\\caption[Four diagrams contributing in Eq.~(\\protect\\ref{39}).\n The lines indicate correlation functions of $\\Psi_i$-fields.\nThe crossed circles with label $i$ denote the insertion of\n$|\\psi_i^{a_i}({\\bf x}_i)|^2$]{Four diagrams\ncontributing in Eq.~(\\protect\\ref{39}).\n The lines indicate correlation functions of $\\Psi_i$-fields.\nThe crossed circles with label $i$ denote the insertion of\n$|\\psi_i^{a_i}({\\bf x}_i)|^2$.\\label{4dia} }\n\\label{@FFF}\\end{figure}%\n%\n%\nOnly the first diagram in Fig.~\\ref{4dia}\nis divergent from the loop integral\nformed by two correlation functions of the vector field.\nThis infinity may be absorbed in the four-$\\Psi$\ninteraction\naccounting for the\nexcluded volume effect which we do not consider at the moment.\nNo divergence arises from the\ninsertion of the composite fields\n$\\vert\\psi_i^{a_i} ({\\bf x}_i)\\vert^2$.\n%%%%%%%%%%%%%%%\n%%%\n\n{\\bf 6.}\n{}In this section we evaluate the first term appearing in the\nright hand side of Eq.~(\\ref{39}):\n%\n\\begin{eqnarray}\n{}~N_1\\!\\! & = & \\frac{ \\kappa ^2}{4}\\lim_{n_1 \\rightarrow 0 \\atop n_2\n\\rightarrow 0}\n \\int^{c + i\\infty }_{c - i \\infty} \\frac{Md z_1}{2\\pi i}\n \\frac{Mdz_2}{2\\pi i} e^{z_1 L_1 + z_2 L_2} \\nonumber \\\\\n &&\n \\int d^3\n{x}_1 d^3 {x}_2\n\\int\n d^3 { x}_1' d^3 { x}_2'\n \\label{40}\\\\\n&& \\bigg\\langle \\vert \\psi_1^{a_1} ({\\bf x}_1)\\vert ^2\n \\vert \\psi_2^{a_2} ({\\bf x}_2) \\vert ^2\n \\left( \\vert \\Psi_1\\vert^2 {\\bf A}_1^2\\right)_{{\\sbf x}_1'}\n\\left(\\vert \\Psi_2\\vert^2{{\\bf A}_2^2}\\right)_{{\\sbf x}_2'}\n \\bigg\\rangle . \\nonumber\n\\end{eqnarray}\n%\nThere is an ultraviolet-divergence which\nmust be\nregularized.\nThis is done by cutting the spatial integrals off\nat\nthe\npersistence length $\\xi$\nover which a polymer is stiff.\nThis contains the stiffness caused by\n the excluded-volume effects. To be rigorous, we define\nthe integral (\\ref{40})\non a lattice with spacing $\\xi$.\n\nReplacing the expectation values by the\nWick contractions corresponding to the first diagram in\nFig.~\\ref{4dia},\n we obtain\n\\begin{eqnarray}\n&&\\!\\!\\!\\!\\!\\!\\!N_1 =\\frac V{4\\pi}\\frac{M^4}{(4\\pi)^6}(L_1L_2)^{-\\frac\n12}\n%\\nonumber \\\\&\\times&\n\\!\\!\\int_0^1\\!\\!ds\\left[(1-s)s\\right]^{-\\frac 3 2}\n \\!\\!\\int\\!\\! d^3xe^{-\\frac {M{\\sbf x}^2}{2s(1-s)}}\\label{ioneint}\\nonumber \\\\\n&\\times&\\!\n\\int_0^1dt\\left[(1-t)t\\right]^{-\\frac 3 2}\n\\!\\int d^3ye^{- \\frac{M{\\sbf y}^2}{2t(1-t)}}\\int\nd^3x_1^{\\prime\\prime}\\frac 1{\\vert{\\bf x}_1^{\\prime\\prime} \\vert^4} .\n \\nonumber\n\\end{eqnarray}\n%\\mn{one more explicit after (\\ref{40}) please}\n%\nThe variables $\\bf x$ and $\\bf y$ have been rescaled with respect to\nthe original ones in order to extract the behavior of $N_1$ in $L_1$\nand $L_2$. As a consequence, the lattices where $\\bf x$ and $\\bf\ny$ are defined\nhave now spacings $ \\xi/{\\sqrt{L_1}}$ and $ \\xi/{\\sqrt{L_2}}$\nrespectively.\nThe ${\\bf x},{\\bf y}$ integrals may be explicitly computed by analytical\nmethods in the\nphysical limit $L_1,L_2>>\\xi$, in which the above spacings become\nsmall.\nThis has a physical explanation.\nIndeed, if the polymer lengths are much larger than the persistence length,\nthe effects due to the finite monomer size become negligible and\ncan be ignored.\n\nFinally, it is possible to approximate the integral in\n${\\bf x}_1^{\\prime\\prime} $ with an integral over a continuous\nvariable $\\rho$ and a cutoff in the ultraviolet region:\n$\n\\int d^3x_1^{\\prime\\prime}\\lfrac{1}{{\\vert{\\bf x}_1^{\\prime\\prime}\n\\vert^4}}\\sim\n4\\pi^2\\int_\\xi^\\infty\\lfrac{d\\rho}{\\rho^2}\\label{appsone}.\n $\nAfter these approximations, we obtain\n\\begin{eqnarray}\nN_1 = {V \\sqrt{ \\pi }} \\frac{M}{(4 \\pi )^3}\n (L_1 L_2)^{-1/2} \\xi^{-1} .\n\\label{41}\\end{eqnarray}\n%\n\n\n\n{\\bf 7.}\nFor the second\ndiagram in Fig.~\\ref{@FFF}\n we have to calculate\n%\n\\begin{eqnarray}\n&&\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\nN_2\n =\n \\kappa ^2 \\lim_{n_1 \\rightarrow 0 \\atop n_2 \\rightarrow 0}\n \\int^{c + i \\infty}_{c - i \\infty} \\frac{Mdz_1}{2\\pi i}\n\\frac{Mdz_2}{2\\pi i}\n e^{z_1 L_1 + z_2 L_2}\n \\nonumber \\\\\n &&\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\times \\int d^3\n{ x}_1 d^3 { x}_2\n\\int\n d^3 {x}_1'\nd^3 {x}_1''\nd^3 {x}_2'\n\\label{42}\n\\nonumber \\\\&&\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\times \\bigg\\langle \\vert \\psi_1^{a_1}\n({\\bf x}_1)\\vert^2\n \\vert \\psi_2^{a_2} ({\\bf x}_2) \\vert^2\n \\left(\n \\,{\\bf A}_1 \\cdot\n \\Psi_1^*\n{\\bf \\nablab} \\Psi_1 \\right)_{\\sbf{x}_1'}\n\\nonumber \\\\\n&& \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\times\n \\left(\n \\,{\\bf A}_1 \\cdot\n \\Psi_1^*\n{\\bf \\nablab} \\Psi_1 \\right)_{\\sbf{x}_1''}\n \\left( {\\bf A}_2^2\\,\\vert \\Psi_2 \\vert^2\\right)_{{\\sbf x}_2'}\n\\bigg\\rangle.\n\\label{@56}\\end{eqnarray}\n%\nThe above amplitude has no ultraviolet divergence, so that no regularization is\nrequired.\nThe Wick contractions pictured in the second Feynman diagrams\nof Fig.~\\ref{4dia}\nlead to the integral\n%\n%\n\\begin{eqnarray}\n N_2 = -4\\sqrt{2} V L_2^{-1/2} L_1^{-1} \\frac{M^3}{\\pi^6} \\int_{0}^{1}\n dt \\int^{t}_{0} dt' C(t,t'),\n\\label{43}\n\\label{@57}\n\\end{eqnarray}\n%\n where $C(t,t')$ is a function independent of $L_1$ and $L_2$:\n%\n%\n\\begin{eqnarray}\n &&C(t,t') = \\left[(1-t) t' (t-t') \\right] ^{-3/2}\n %\\nonumber \\\\& & \\times\n\\int\n d^3 { x} d^3 { y} d^3 { z} e^{-\\frac{ M ({\\sbf y} - {\\sbf x })^2 }{\n 2(1-t)}}\n \\nonumber \\\\ && \\times\n \\left( \\nabla^ \\nu_{\\sbf y} e^{-M {\\sbf y}^2 /2t'}\\right) \\left(\n \\nabla^\\mu_{\\sbf x} e^{-M {\\sbf x}^2 /2 (t-t') }\\right)\nP_{\\mu \\nu }({\\bf x},{\\bf y},{\\bf x}),\n\\label{44}\n\\label{@58} \\nonumber\n\\end{eqnarray}\n%\nwith\n$P_{\\mu \\nu }({\\bf x},,{\\bf y},{\\bf x})\\equiv\n[ \\delta _{\\mu \\nu } {\\bf z} \\cdot\n ({\\bf z} + {\\bf x}) - \\left({ z} + { x}\\right)_\\mu\n { z}_ \\nu ]/\n ({\\vert{\\bf z}\\vert^3 \\vert {\\bf z} + {\\bf x}\\vert^3})$.\nAs in the previous section, the variables $\\bf x, \\bf y, \\bf z$ have been\nrescaled with respect to the original ones in order\nto extract the behavior in $L_1$.\n%\nAgain, if $L_1,L_2>>\\xi$\nthe analytical evaluation of\n$C(t,t')$ becomes possible, leading to %\\mn{an eq for C would be good}\n%\n\\begin{eqnarray}\n N_2 & = & - \\frac{V L_2^{-1/2}L_1^{-1} }{(2\\pi)^6}\n M^{3/2} 4 K,\n\\label{46}\\end{eqnarray}\n%\nwhere $K$ is the constant\n$ \\frac{1}{6} B \\left(\\frac{3}{2},\\frac{1}{2}\\right) +\n \\frac{1}{2} B\\left(\\frac{5}{2}, \\frac{1}{2}\\right)\n- B\n \\left(\\frac{7}{2}, \\frac{1}{2}\\right) + \\frac{1}{3}\n B \\left(\\frac{9}{2}, \\frac{1}{2}\\right) ={19\\pi}/{384} \\approx0.154\n\t\t ,$\n%\nand $B(a,b)$ is the Beta function.\nFor large $L_1 \\rightarrow \\infty $, this diagram gives a negligible\ncontribution with respect to $N_1$.\n\nThe third diagram in Fig.~\\ref{4dia}\ngive the same as the second, except that $L_1$ and $L_2$ are interchanged:\n%\n$\nN_3=N_2|_{L_1\\leftrightarrow L_2}.\n$\n%\n\n\n\n{\\bf 8.}\nThe fourth Feynman diagram in\nFig.~\\ref{@FFF}\nhas no ultraviolet divergence.\n%\n%\\begin{eqnarray}\n% && \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!N_4 = -\\frac{1}{2\\cdot 4^6}\n%\\frac{M^3V}{(2\\pi)^{11}} (L_1L_2)^{-1/2}\n% \\nonumber \\\\\n% &&\\times \\int^{1}_{0} ds\n% \\int^{s}_{0} ds'\n% \\int^{1}_{0} dt\n% \\int^{t}_{0} dt'\n% C (s, s', t, t'),\n%\\label{51}\\end{eqnarray}\n%%\n%where\n%%\n%\\begin{eqnarray}\n% && C(s, s'; t,t') = \\left[ (1-s)s' (s-s') \\right] ^{-3/2}\n%\t\t\t \\left[ (1-t) t' (t-t')\\right] ^{-3/2}\n% \\nonumber \\\\\n%&& \\times \\int \\frac{d^3 p}{ ( 2\\pi )^3 } \\left[\n%\\epsilon_{ \\mu \\lambda \\alpha}\n% \\frac{p^ \\alpha }{{\\bf p}^2 }\\epsilon_{ \\nu \\rho \\beta}\n% \\frac{p^ \\beta }{{\\bf p}^2 } + \\epsilon_{\\mu \\rho \\alpha}\n% \\frac{p^ \\alpha }{{\\bf p}^2 } \\epsilon_{ \\nu \\lambda \\beta}\n% \\frac{p^ \\beta }{{\\bf p}^2 }\\right] \\nonumber \\\\\n%& &\\times \\left[ \\int d^3 { x}' d^3 { y}' e^{-i \\sqrt{L_1} {\\sbf p}\n% ({\\sbf x}' - {\\sbf y}') }e ^{-M{\\sbf x} '{}^2 / 2(1-s)} \\right.\\nonumber\n%\\\\\n%&&\\times \\left.\\left(\\nabla_{{\\sbf y}'}^ \\nu e^{-M{\\sbf y}'{}^2 /2t'}\\right)\n% \\left(\\nabla_{{\\sbf x}'}^{\\mu} e^{-M ({\\sbf x}- {\\sbf y})^2\n%/2(s-s')}\\right)\n% \\right] \\nonumber \\\\\n% & &\\times \\left[ \\int d^3 { u}' d^3 { v}'\n% e^{-i \\sqrt{L_2} {\\sbf p} ({\\sbf u}' -{\\sbf v}' )}\n% e^{-M {\\sbf v}'{}^2 /2 (1-t)} \\right.\\nonumber \\\\\n%&& \\times \\left. \\left( \\nabla^\n%\\rho_{{\\sbf u}'} e^{-M{\\bf u} '{}^2 / 2t'}\\right)\n% \\left(\\nabla^ \\lambda_{{\\sbf v}'}\n%e^{\n%-M ({\\bf u}' -{\\bf v}' )^2 /2(t-t')}\n% \\right)\\right] ,\n%\\label{52}\\end{eqnarray}\n%%\n%and ${\\bf x}', {\\bf y}'$ are scaled variables. To take\n%into account the finite persitence length,\n%they should be defined on a lattice with spacing\n%$\\xi/ \\sqrt{L_1}$, whereas\n%y ${\\bf u}', {\\bf v}'$ have a short-distance cutoff at\n% $\\xi/ \\sqrt{L_2}$.\n%%%%%%%%%%%%%\nAs before, it can be exactly evaluated apart from the lattice integrations.\nHowever, the behavior of the related\nFeynman integral $N_4$\n%performing the\n%space integrations\n%$d^3 x' d^3 {\\bf y}' d^3 {\\bf u}' d^3 {\\bf v}'$,\n%the behavior of $N_4$ as a function of the polymer lengths\ncan be\neasily estimated in the following limits:\n\n1. $L_1 \\gg 1; L_1 \\gg L_2$, where\n%\n$N_4 \\propto L_1^{-1},$\n%\n\n2. $L_2 \\gg 1; L_2 \\gg L_1$ where\n%\n $N_4 \\propto L_2^{-1},$\n\n3. $L_1, L_2 \\gg 1,~ {L_2}/{L_1} = \\mbox{finite} ,$ where\n%\n $N_4 \\propto L_1^{-3/2}$.\\\\\n%\nMoreover, if the lengths of the polymers are considerably larger than\nthe\npersistence length, $N_4$ can be computed in a\nclosed form:\n%\\begin{eqnarray}\n% \\xi/ \\sqrt{L_1} \\sim \\xi/ \\sqrt{L_2} \\sim 0 .\n%\\label{56}\\end{eqnarray}\n%\n% Then the space integrations are easily performed yielding\n%\n\\begin{eqnarray}\n N_4 & \\approx & - \\frac{128V}{\\pi^5} \\frac{M}{\\pi^{3/2}}\n (L_1L_2)^{-1/2}\n \\nonumber \\\\\n& \\times &\\int^{1}_{0} ds \\int^{1}_{0} dt\n (1-s) (1-t) (st)^{1/2} \\nonumber \\\\\n&\\times &\\left[ L_1 t (1-s) + L_2 (1-t)s\\right] ^{-1/2} .\n \\label{57}\\end{eqnarray}\n%\nIt is simple to check that this expression has exactly\nthe above\nbehaviors.\n\n{\\bf 9.}\nCollecting all contributions we obtain the result for the\nsecond topological moment\n%\n$ \\langle m^2\\rangle = \\lfrac{(N_1 + N_2 + N_3 + N_4)}{Z} ,$\n%\nwith $N_1,\\,N_2,\\,N_3,\\,N_4,\\,Z$\ngiven by Eqs.~(\\ref{30}),\n(\\ref{41}),\n(\\ref{46}),\nand\n(\\ref{57}).\nIn all formulas, we have assumed that\n the volume $V$ of the system is much larger\nthan the size of the volume occupied by a single polymer,\ni.e.,\n$V \\gg L_1^3$.\n\nTo discuss the physical content of the above expression for\n$\\langle m^2\\rangle$, we consider a\nnumber $N_p$ of polymers $p_1\\ldots p_{N_p}$ with lengths\n$l_1\\ldots l_{N_p}$ in an uniform solution.\nWe introduce the polymer concentration\n$\\rho= \\lfrac{{\\cal M}}{V}\n $ as the average mass density of the polymers per unit volume,\nwhere ${\\cal M}$ is the total mass of the polymers\n ${\\cal M} = \\sum_{k = 1}^{N_p} m_a \\lfrac{l_k}{a}$ and\n$m_a $ is the mass of a single monomer of length $a$.\nThus ${l_k}/{a}$ is the number of monomers in the\npolymer $p_k$.\nThe polymer $P_1$ is singled out as anyone of the polymers $p_k$, say $p_{\\bar\nk}$,\nof length $L_1=l_{\\bar k}$.\nThe\nremaining ones are replaced by a long effective polymer $P_2$\nof length $L_2= \\sum_{k\\neq \\bar k} l_k$.\n{}From the above relations we may also write\n%\n\\begin{eqnarray}\n L_2 \\approx \\frac{a V \\rho }{m_a}.\n\\label{69}\\end{eqnarray}\n%\nIn this way, the length of the effective molecule\n$P_2$ is expressed in terms of physical parameters.\n%the concentration of polymers, the monomer length,\n%and the mass and volume of the system.\nKeeping only the\nleading terms for $V \\gg 1$,\nwe find for\n the average square number of intersections $\\langle m^2\\rangle_{sol}$\nformed by $P_1$ with\nthe other polymers\nthe approximate result\n%\n\\begin{eqnarray}\n \\langle m^2 \\rangle_{sol} \\approx \\frac{N_1 + N_2}{Z},\n\\label{70}\\end{eqnarray}\n%\nwhich, in turn, has the approximate form\n%\n\\begin{eqnarray}\n\\langle m^2 \\rangle_{sol} = \\frac{a\\rho }{m_a} \\left[\n \\frac{\\xi^{-1} L_1}{2 \\pi ^{1/2} M^2}-\n \\frac{2K L_1^{1/2} }{\\pi^4 M^{3/2}} \\right],\n\\label{71}\\end{eqnarray}\n%\nwith $K$ as defined after Eq.~(\\ref{46}).\nThis is the announced final result. Since the persistence length is of the same\norder of the monomer length $a$ and $M\\sim a^{-1}$,\n $\\langle m^2 \\rangle$ is positive for large $L_1$ as it should.\n\n{\\bf 10.}\nIn conclusion, we have found an exact\nfield theoretic formula\nfor the second topological moment of two polymers.\nOnly the\nfinal\n integrations over the spatial variables in the\nFeynman diagrams of Fig.~\\ref{4dia} were done approximately.\nThese were defined on a lattice related to the finite monomer size.\nOur Chern-Simons-based theory is free of\nthe shortcomings\nof previous mean-field procedures.\nOur formula for $\\langle m^2\\rangle$\nhas been applied to the realistic case of long flexible polymers\nin a solution. When the polymer lengths become large,\nthe Feynman integrals in $N_1,\\ldots,N_4$\ncan be\nevaluated\nanalytically.\nIn this way we have been able to derive\nthe result (\\ref{71}) for the average square number\nof intersections formed by a polymer $P_1$ with all the others.\nThis calculation is\n{\\em exact\\/} in the long-polymer limit.\nThe corrections\nto (\\ref{71}) are\nsuppressed by\nfurther inverse square roots of the polymer lengths.\n\nTo leading order in $L_1$, our result (\\ref{71})\nagrees\n with that of \\cite{BSII}, but our exact subleading\ncorrection go beyond\nthe approximation of\n\\cite{BSII}.\nNote that there is no direct comparison of our result\nwith that of \\cite{tan},\nsince there the polymer $P_2$ was considered as a fixed obstacle\ncausing a dependence on the choice of the\nconfiguration of $P_2$.\n\nFinally, let us emphasize\nthe absence of infrared divergences\nin the topological field theory\n(\\ref{4}) in the limit of vanishing masses $m_1,m_2=0$.\nAs a consequence,\nthe second topological moment does not diverge in the limit of\nlarge $L_1$\nif\n$\\langle m^2 \\rangle$\nis calculated from (\\ref{4})\nfor polymers passing through\ntwo fixed points\n${\\bf x}_1,{\\bf x}_2 $.\nThis indicates a much stronger\nreduction of the\nconfigurational fluctuations by topological\nconstraints than\none might have anticipated.\n\n\\begin{thebibliography}{11}\n\\bibitem{BS}\nM.G.~Brereton,\nS.~Shah, J.~Phys.~{\\bf A}: Math. Gen. {\\bf 13}, (1980), 2751.\n\\bibitem{BV} M. G. Brereton and\nT. A. Vilgis, {\\it Jour. Phys. A: Math. Gen.} {\\bf 28} (1995), 1149.\n\\bibitem{OV} M. Otto and T. A. Vilgis, {\\it Phys. Rev. Lett.} {\\bf 80}\n(1998), 881.\n\\bibitem{MK}\nJ. D. Moroz and\nR. D. Kamien, {\\it Nucl. Phys.} {\\bf B506 [FS]} (1997), 695.\n\\bibitem{ffil} F. Ferrari and I. Lazzizzera,\n{\\it Nucl. Phys.} {\\bf B559} (3), (1999), 673.\n\\bibitem{tan} F. Tanaka, {\\it Prog. Theor. Phys.} {\\bf 68} (1), (1982), 148.\n\\bibitem{pi}\n H. Kleinert, {\\em Path Integrals in Quantum Mechanics\n Statistics and Polymer Physics}, World Scientific,\n Singapore, 1995.\\\\\nhttp://www.physik.fu-berlin.de/\\~{}kleinert/re.html\\#b5\n\\bibitem{Ed} S. F. Edwards, {\\it Proc. Phys. Soc.} {\\bf 91} (1967), 51.\n% \\clearpage\n\\bibitem{BSII} M.G.~Brereton,\nS.~Shah, J.~Phys.~{\\bf A}: Math. Gen. {\\bf 15}, (1982), 985.\n\\end{thebibliography}\n\n\n\n\n\n\\end{document}\n\n\n"
}
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[
{
"name": "cond-mat0002049.extracted_bib",
"string": "\\begin{thebibliography}{11}\n\\bibitem{BS}\nM.G.~Brereton,\nS.~Shah, J.~Phys.~{\\bf A}: Math. Gen. {\\bf 13}, (1980), 2751.\n\\bibitem{BV} M. G. Brereton and\nT. A. Vilgis, {\\it Jour. Phys. A: Math. Gen.} {\\bf 28} (1995), 1149.\n\\bibitem{OV} M. Otto and T. A. Vilgis, {\\it Phys. Rev. Lett.} {\\bf 80}\n(1998), 881.\n\\bibitem{MK}\nJ. D. Moroz and\nR. D. Kamien, {\\it Nucl. Phys.} {\\bf B506 [FS]} (1997), 695.\n\\bibitem{ffil} F. Ferrari and I. Lazzizzera,\n{\\it Nucl. Phys.} {\\bf B559} (3), (1999), 673.\n\\bibitem{tan} F. Tanaka, {\\it Prog. Theor. Phys.} {\\bf 68} (1), (1982), 148.\n\\bibitem{pi}\n H. Kleinert, {\\em Path Integrals in Quantum Mechanics\n Statistics and Polymer Physics}, World Scientific,\n Singapore, 1995.\\\\\nhttp://www.physik.fu-berlin.de/\\~{}kleinert/re.html\\#b5\n\\bibitem{Ed} S. F. Edwards, {\\it Proc. Phys. Soc.} {\\bf 91} (1967), 51.\n% \\clearpage\n\\bibitem{BSII} M.G.~Brereton,\nS.~Shah, J.~Phys.~{\\bf A}: Math. Gen. {\\bf 15}, (1982), 985.\n\\end{thebibliography}"
}
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cond-mat0002050
|
Fermi's golden rule in a mesoscopic metal ring
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[
{
"author": "Peter Kopietz and Axel V\\\"{o}lker"
}
] |
We examine the time-dependent non-equilibrium current in a mesoscopic metal ring threaded by a static magnetic flux $\phi$ that is generated by a time-dependent electric field oscillating with frequency $\omega$. We show that in quadratic order in the field there are three fundamentally different contributions to the current. (a) A time-independent contribution which can be obtained from a thermodynamic derivative. (b) A term increasing linearly in time that can be understood in terms of Fermi's golden rule. The derivation of this term requires a careful treatment of the infinitesimal imaginary parts that are added to the real frequency $\omega$ when the electric field is adiabatically switched on. (c) Finally, there is also a time-dependent current oscillating with frequency $2 \omega$. We suggest an experiment to test our results. \\ { } \\ \noindent Keywords: persistent currents, non-linear response
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[
{
"name": "cond-mat0002050.tex",
"string": "\\tolerance = 10000\n%\\documentstyle[aps,epsf]{revtex}\n%\\documentstyle[twocolumn,aps,epsf,amstex]{revtex}\n%\\documentstyle[preprint,aps,epsf]{revtex}\n\\documentstyle[preprint,tighten,aps,epsf]{revtex}\n%\\documentstyle[12pt]{article}\n%\n%\\renewcommand{\\theequation}{\\arabic{section}.\\arabic{equation}}\n%\\renewcommand{\\thesection}{\\Roman{section}}\n%\n\\begin{document}\n%\n\\draft\n%\n%\n% A B S T R A C T\n%\n%\n\\title{\nFermi's golden rule in a mesoscopic metal ring}\n%\n\\author{\nPeter Kopietz and Axel V\\\"{o}lker}\n%\n\\address{\nInstitut f\\\"{u}r Theoretische Physik der Universit\\\"{a}t G\\\"{o}ttingen,\nBunsenstrasse 9, D-37073 G\\\"{o}ttingen, Germany}\n%\n\\date{June 1, 1999}\n%\n\\maketitle\n%\n\\begin{abstract}\n\nWe examine the time-dependent\nnon-equilibrium current in a\nmesoscopic metal ring threaded by a static\nmagnetic flux $\\phi$ that is generated by \na time-dependent electric field\noscillating with frequency $\\omega$.\nWe show that in quadratic order in \nthe field\nthere are three fundamentally \ndifferent contributions to the current.\n(a) A time-independent contribution which\ncan be obtained from a thermodynamic\nderivative.\n(b) A term increasing linearly in time \nthat can be understood in terms of Fermi's golden rule.\nThe derivation of this term requires a careful treatment\nof the infinitesimal imaginary parts \nthat are added to the real frequency $\\omega$ when \nthe electric field is adiabatically switched on.\n(c) Finally, there is also\na time-dependent current oscillating with\nfrequency $2 \\omega$. \nWe suggest an experiment to test\nour results. \n\\\\\n{ }\n\\\\\n\\noindent\nKeywords: persistent currents, non-linear response\n\n\n\n\n\n\n\n\n\\end{abstract}\n%\n\\pacs{PACS numbers: 73.50.Bk, 72.10.Bg, 72.15.Rn}\n%\n%\n\\narrowtext\n%\n%\n\n\\section{Introduction}\n\\label{sec:intro}\n\nConsider a mesoscopic metal ring threaded by \na time-dependent magnetic flux $\\phi ( t )$ \nthat has a static component $\\phi$ and\na part that oscillates with frequency $\\omega$,\n \\begin{equation}\n \\phi ( t ) =\n\\phi + \\phi_\\omega \\sin ( \\omega t )\n\\; .\n \\label{eq:flux}\n \\end{equation}\nBy Faraday's law of induction, \nthe oscillating part generates\na time-dependent electric field directed along\nof circumference of the ring,\n$E(t)=E_\\omega \\cos(\\omega t)$, with amplitude\n \\begin{equation}\n e L E_\\omega = 2 \\pi \\omega \\frac{\\phi_\\omega }{ \\phi_0 }\n \\; .\n \\label{eq:Faraday}\n \\end{equation}\nHere $L$ is the circumference of the ring, $-e$ is the charge\nof the electron, and $\\phi_0 $ is the flux quantum.\nWe would like to know the induced current\naround the ring. \nIn the limit $\\omega \\rightarrow 0$ this is just the\nusual persistent current\\cite{Hund38,Imry97}.\nBut what happens for frequencies in the range\nbetween $10^{8}$ and $10^{13}$ Hz, which for\nexperimentally relevant rings\\cite{Levy90,Mohanty96} corresponds\nto $\\Delta \\ll \\omega \\ll \\tau^{-1}$?\nHere $\\Delta$ is the average level spacing at the Fermi energy,\nand $\\tau$ is the elastic lifetime.\nWe use units where $\\hbar$ is set equal to unity.\nThis problem has first been studied by\nKravtsov and Yudson\\cite{Kravtsov93} (KY), who found \nthat in quadratic order the time-dependent field induces \n(among other terms that oscillate) a\ntime-independent non-equilibrium current\n$I^{(2)}_0$.\nCalculating the disorder average of this current \nperturbatively, KY found that it has the peculiar\nproperty that for frequencies exceeding the\nThouless energy $E_c = \\hbar {\\cal{D}} / L^2$\n(where ${\\cal{D}}$ is the diffusion coefficient)\nthe average of $I^{(2)}_0$ does not vanish\nexponentially, but \nonly as $\\omega^{-2}$.\nThis is in disagreement with the intuitive expectation that\nthe external frequency $\\omega$ leads to a similar\nexponential suppression of this mesoscopic non-equilibrium\ncurrent\nas a dephasing rate in the case \nof the equilibrium persistent current\\cite{Schmid91,Altshuler91}.\nThe perturbative calculation of KY is based on the assumption of a\ncontinuous energy-spectrum, which means that the level-broadening due to\ndephasing, $1/\\tau_\\varphi$, must exceed the average level spacing at\nthe Fermi energy, $\\Delta$. If we assume that for low temperature $T$ the\ndominant dephasing effect comes from electron-electron interactions, a\nsimple estimate\\cite{Voelker99} shows that \n$1/\\tau_\\varphi ( \\omega ) < \\Delta$ for\n$| \\omega | \\leq E_c$ in the limit $T\\rightarrow 0$. \nHence, for\nfrequencies smaller than the Thouless\nenergy the spectrum is discrete and the perturbative analysis\nbreaks down. \nIn this work we shall\nshow that in this case \nthe term considered by KY\nis not constant, but grows linearly \nin time, a result which can be understood \nsimply\nin terms of Fermi's golden rule of time-dependent\nperturbation theory.\n\n\nIt is important to point out the difference between\nthe current considered here and the\ndirect current due to the\nusual photovoltaic effect.\nIt is well known\\cite{Belinicher80} that\nirradiation of a medium without an inversion\ncenter by an alternating electric field can give rise to\na direct current (photovoltaic effect).\nThe lack of inversion symmetry\ncan be due to impurities and defects in a finite sample.\nFor mesoscopic junctions the photovoltaic direct current\nhas been studied in Ref.\\cite{Falko89}.\nIn this case the average current vanishes, because disorder averaging\nrestores the inversion symmetry. \nIn our case, however, we calculate the direct current\ninduced in a mesoscopic ring threaded by a magnetic flux. \nBecause the magnetic flux breaks the time-reversal symmetry, \nthe direct current considered here has a finite disorder average.\nThus, the physical origin of a mesoscopic non-equilibrium current\ndiscussed in this work is quite different from Ref.\\cite{Falko89}.\n\n\n\n\n\n\\section{\nThe \nquadratic response function: What is wrong with the Green's function approach? \n}\n\n\nWe consider non-interacting disordered electrons\nof mass $m$ on a mesoscopic metal ring\nthreaded by the time-dependent\nmagnetic flux given in Eq.(\\ref{eq:flux}).\nSuppose that we have\ndiagonalized the Hamiltonian \nin the absence of the oscillating\nflux (i.e. for $\\phi_{\\omega} = 0$ in Eq.(\\ref{eq:flux}))\nfor the given realization of the disorder.\nThe time-independent part of the Hamiltonian is\nthen $\\hat{H}_0 = \\sum_\\alpha \\varepsilon_\\alpha \nc^\\dagger_\\alpha c_\\alpha$,\nwhere $\\varepsilon_{\\alpha}$ are the exact \nelectronic eigen-energies \nfor fixed disorder, which are labeled by\nappropriate quantum numbers $\\alpha$.\nThe operators $c^{\\dagger}_{\\alpha}$ create \nelectrons in the corresponding eigenstates\n$| \\alpha \\rangle$.\nIf we now switch on the time-dependent part of the field,\nthe Hamiltonian becomes\n$\\hat H =\n\\hat H_0 + \\hat V(t)$, with\n\\begin{eqnarray}\n \\label{eq:vt}\n \\hat V(t)&=&\\frac{2\\pi}{m\n L}\\delta\\varphi(t)\\sum_{\\alpha,\\beta}\\langle\\alpha|\\hat P_x\n |\\beta\\rangle c^\\dagger_\\alpha c_\\beta \\nonumber \\\\\n&+&\\frac{1}{2m}\\left(\\frac{2\\pi}{L}\\delta\\varphi(t)\\right)^2\n\\sum_\\alpha c^\\dagger_\\alpha c_\\alpha \n\\; .\n\\end{eqnarray}\nHere $\\delta\\varphi(t)=(\\phi_\\omega/\\phi_0)\\sin(\\omega t)$, \nand\n$\\hat P_x=-i d/dx +\n(2\\pi/L)(\\phi/\\phi_0)$ is the $x$-component of the\none particle momentum operator. \nAs usual, the coordinate along the circumference\nis called the $x$-direction, and we impose periodic\nboundary conditions.\nUsing standard non-equilibrium Green's function methods,\nthe contribution to the\nnon-equilibrium current that is quadratic in the\nexternal field is easily obtained\\cite{Fricke95}:\n\\begin{eqnarray}\nI^{(2)}(t) & = & \\frac{(-e)(2\\pi)^2}{(m L)^3} \\int^\\infty_\\infty d\\omega_1\nd\\omega_2 \\delta\\varphi_{\\omega_1}\\delta\\varphi_{\\omega_2} \\nonumber\n\\\\\n&\\times& K^{(2)} (\\omega_1, \\omega_2) e^{-i(\\omega_1+\\omega_2)t}\n\\label{eq:strom}\n\\; ,\n\\end{eqnarray}\nwhere $\\varphi_{\\omega}$ is the Fourier transform\nof the time-dependent part of the\nflux (\\ref{eq:flux}) in units of the flux quantum\n(i.e.\n$\\phi(t) - \\phi = \\phi_0 \n\\int d\\omega' \\delta\\varphi_{\\omega'} e^{-i\\omega' t}$)\nand\nthe response function $K^{(2)} ( \\omega_1 , \\omega_2 )$\nis given by\n\\begin{eqnarray}\n K^{(2)}(\\omega_1,\\omega_2) &=& \\sum_{\\alpha\\beta\\gamma}\n \\frac{P_{\\alpha\\beta\\gamma}}{\\varepsilon_\\gamma-\\varepsilon_\\alpha\n +\\omega_1 +\\omega_2\n +i0} \\nonumber \\\\ \n & & \\hspace{-20mm} \\times \\biggl[\\frac{f(\\varepsilon_\\gamma)-\n f(\\varepsilon_\\beta)}{\\varepsilon_\\gamma-\n \\varepsilon_\\beta +\\omega_2 +i0} \n -\\frac{f(\\varepsilon_\\beta)-\n f(\\varepsilon_\\alpha)}{\\varepsilon_\\beta-\\varepsilon_\\alpha\n +\\omega_1 +i0}\\biggr] \\; ,\n \\label{eq:k2adiabatic}\n \\end {eqnarray}\nwith \n \\begin{equation}\n \\label{Pabc}\n P_{\\alpha\\beta\\gamma}=\\langle\\alpha|\\hat\n P_x|\\beta\\rangle\\langle\\beta|\\hat P_x|\\gamma\\rangle\\langle\\gamma|\\hat\n P_x|\\alpha\\rangle\n \\; .\n \\end{equation}\nHere $f(\\varepsilon_\\alpha)= \n\\langle c^\\dagger_\\alpha c_\\alpha\\rangle $ is the\noccupation number, which in a grand-canonical ensemble\nis the Fermi function. \nKeeping in mind that the time-dependent part\nof the flux (\\ref{eq:flux}) corresponds to\n \\begin{equation}\n \\delta\\varphi_{\\omega'} =\n \\frac{\\phi_\\omega}{2i\\phi_0}[\\delta(\\omega'+\\omega) -\n \\delta(\\omega'-\\omega)]\n \\; ,\n \\end{equation}\nit is clear that in this case\nEq.(\\ref{eq:strom}) contains not only \noscillating terms, but also a time-independent\ncontribution, \n \\begin{equation}\n I^{(2)}_0= A_{\\omega}\n \\bigl[ K^{(2)}(\\omega,-\\omega)\n + K^{(2)}(-\\omega,\\omega)\\bigr]\n \\; ,\n \\label{eq:I2_0}\n \\end{equation}\nwhere\n \\begin{equation}\n A_{\\omega}\n =\n \\frac{(-e)(2\\pi\\phi_\\omega)^2}{4 (L\n m)^3 \\phi_0^2} \n \\label{eq:Aomegadef}\n \\; ,\n \\end{equation}\nand\n \\begin{eqnarray}\n K^{(2)}(\\omega,-\\omega) &=& \\sum_{\\alpha\\beta\\gamma}\n \\frac{P_{\\alpha\\beta\\gamma}}{\\varepsilon_\\gamma-\\varepsilon_\\alpha\n +i0 } \\nonumber \\\\ \n & & \\hspace{-20mm} \\times \\biggl[\\frac{f(\\varepsilon_\\gamma)-\n f(\\varepsilon_\\beta)}{\\varepsilon_\\gamma-\\varepsilon_\\beta\n -\\omega +i0} \n -\\frac{f(\\varepsilon_\\beta)-f(\\varepsilon_\\alpha)}{\n \\varepsilon_\\beta-\\varepsilon_\\alpha\n +\\omega +i0}\\biggr]\n \\; .\n \\label{eq:k20}\n \\end{eqnarray}\nDefining retarded and advanced Green's functions,\n \\begin{equation}\n G^{R}_\\alpha(\\varepsilon)=\\frac{1}{\n \\varepsilon - \\varepsilon_\\alpha + i0}\n \\; \\; , \\; \\; \n G^{A}_\\alpha(\\varepsilon)=\\frac{1}{\n \\varepsilon - \\varepsilon_\\alpha - i0}\n \\; ,\n \\label{eq:green}\n \\end{equation}\nEq.(\\ref{eq:k20}) can also be written as\n \\begin{eqnarray}\n K^{(2)}(\\omega,-\\omega)&=&-\\frac{1}{2\\pi\n i}\\sum_{\\alpha\\beta\\gamma} P_{\\alpha\\beta\\gamma}\n \\nonumber \\\\\n & & \\hspace{-23mm} \\times\n \\Biggl\\{ \n \\int_\\infty^\\infty\n d\\varepsilon\n f(\\varepsilon+\\omega) \n \\bigl[ G^R_\\alpha(\\varepsilon+\\omega)G^R_\\beta(\n \\varepsilon)G^R_\\gamma(\\varepsilon+\\omega)\n \\nonumber\n \\\\\n & & \n \\hspace{7mm} -\n G^A_\\alpha(\\varepsilon+\\omega)G^A_\\beta(\n \\varepsilon)G^A_\\gamma(\\varepsilon+\\omega) \\bigr]\n \\nonumber \\\\\n & & \\hspace{-20mm} -\n \\int_\\infty^\\infty\n d\\varepsilon\n [f(\\varepsilon+\\omega)-f(\\varepsilon)] \\nonumber \\\\\n & \\times & \\bigl[ G^R_\\alpha(\\varepsilon+\\omega)\n G^A_\\beta(\\varepsilon)G^A_\\gamma(\\varepsilon+\\omega)\n \\nonumber \\\\\n & & -\n G^R_\\alpha(\\varepsilon+\\omega)G^R_\\beta(\n \\varepsilon)G^A_\\gamma(\\varepsilon+\\omega) \\bigr]\\Biggr\\} \\; .\n \\label{eq:k20green}\n \\end{eqnarray}\nThe structure of the Green's functions agrees with the one given by\nKY in Ref.\\cite{Kravtsov97}.\nNote, however, that these authors work in a different gauge:\nthey represent \nthe electric field by a scalar potential, so that their\nexpressions contain only a single current vertex.\nThe introduction of Green's function is useful\nfor calculating disorder averages.\nIt is common wisdom that for the calculation of the\ndisorder average\nof Eq.(\\ref{eq:k20green})\nthe terms involving products of only retarded or only advanced\nGreen's functions can be neglected\\cite{common}.\nIn this approximation a perturbative \ncalculation of the disorder average \nof Eq.(\\ref{eq:k20green}) has been given by\nKY\\cite{Kravtsov93}, with the result that the associated time-independent\npart of the non-equilibrium current is proportional\nto $\\omega^{-2}$ for frequencies larger than the Thouless energy.\nAs explained in Sec.\\ref{sec:intro}, for frequencies $\\omega<E_c$ the\nperturbative expansion is not controlled anymore since the energy\nspectrum becomes discrete. In fact, it will turn out, that the physical\nbehavior is completely different in this regime. \n\nTo demonstrate the break down of the diagrammatic perturbation theory\nfor systems with a discrete spectrum, we now show that {\\it{an exact evaluation of the disorder\naverage of\nEq.(\\ref{eq:k20green})}} should actually yield an {\\it{infinite}} result.\nLet us therefore go back\nto the exact spectral representation (\\ref{eq:k20})\nof the response function.\nUsing the formal identity\n \\begin{equation}\n \\label{eq:i0id}\n \\frac{1}{x+i0}=\\wp\\frac{1}{x}-i\\pi\\delta(x)\n \\; ,\n \\end{equation}\nwhere $\\wp$ denotes the Cauchy principal part,\nwe can rewrite Eq.(\\ref{eq:k20}) as\n \\begin{equation}\n \\label{k20b}\n K^{(2)}(\\omega,-\\omega)=\n K^{(2)}_\\wp(\\omega,-\\omega)+\n K^{(2)}_{\\delta\\delta}(\\omega,-\\omega)\n \\; ,\n \\end{equation}\nwith\n \\begin{eqnarray}\n K^{(2)}_\\wp(\\omega,-\\omega)&=& \n 2\\sum_{\\alpha\\beta\\gamma}\n \\frac{ {\\rm Re} P_{\\alpha\\beta\\gamma}}{\n \\varepsilon_\\gamma-\\varepsilon_\\alpha } \n \\wp\\frac{f(\\varepsilon_\\gamma)-f(\n \\varepsilon_\\beta)}{\\varepsilon_\\gamma-\\varepsilon_\\beta\n -\\omega }\n \\; ,\n \\label{eq:K2P}\n \\\\\n K^{(2)}_{\\delta\\delta}(\\omega,-\\omega)&=&\n -2\\pi^2\\sum_{\\alpha\\beta\\gamma} {\\rm Re}\n P_{\\alpha\\beta\\gamma}\n [f(\\varepsilon_\\gamma)-f(\\varepsilon_\\beta)]\\nonumber\\\\\n & \\times & \\delta(\\varepsilon_\\gamma-\\varepsilon_\\alpha)\n \\delta(\\varepsilon_\\gamma-\\varepsilon_\\beta-\\omega)\n \\label{eq:K2deltadelta}\n \\; .\n \\end{eqnarray}\nThe terms with $\\alpha = \\gamma$\nin Eqs. (\\ref{eq:K2P}) and (\\ref{eq:K2deltadelta})\nyield the following contributions,\n \\begin{eqnarray}\n K^{(2)}_{\\wp,{\\rm diag}}(\\omega,-\\omega) &=& \n \\wp\n \\sum_{\\alpha\\beta}\n P_{\\alpha\\beta\\alpha}\\frac{\\partial}{\\partial\\varepsilon_\\alpha} \n \\frac{f(\\varepsilon_\\alpha)-\n f(\\varepsilon_\\beta)}{\\varepsilon_\\alpha-\\varepsilon_\\beta\n -\\omega}\n \\nonumber\n \\\\\n & & \\hspace{-25mm} = \\wp \\sum_{\\alpha\\beta}\n P_{\\alpha\\beta\\alpha} \\Biggl[\n \\frac{ \\frac{\\partial}{\\partial \n \\varepsilon_{\\alpha} } f(\\varepsilon_\\alpha)\n }{\\varepsilon_\\alpha-\\varepsilon_\\beta\n -\\omega} \n - \\frac{f(\\varepsilon_\\alpha)-\n f(\\varepsilon_\\beta)}{(\\varepsilon_\\alpha-\\varepsilon_\\beta\n -\\omega)^2} \\Biggr]\n \\; ,\n \\label{eq:Kp_diag}\n \\end{eqnarray}\n \\begin{eqnarray}\n K^{(2)}_{\\delta\\delta, {\\rm diag}}(\\omega,-\\omega)&=&\n -2\\pi^2\\delta(0)\\sum_{\\alpha\\beta} {\\rm Re}\n P_{\\alpha\\beta\\alpha}[f(\\varepsilon_\\alpha)-\n f(\\varepsilon_\\beta)]\\nonumber\\\\\n & & \\times \\delta(\\varepsilon_\\alpha-\\varepsilon_\\beta-\\omega) \n \\; .\n \\label{eq:k2deltadiag}\n \\end{eqnarray}\nThe right-hand side of Eq.(\\ref{eq:k2deltadiag}) is\nproportional to the infinite\nfactor $\\delta ( 0 )$. Hence,\nthe term\n$K^{(2)}_{\\delta\\delta }(\\omega,-\\omega)$ \nmust also be infinite. Because the singular prefactor\n$\\delta ( 0)$ in Eq.(\\ref{eq:k2deltadiag})\ndoes not depend on the disorder,\nthis singularity survives disorder averaging\\cite{dos}.\nKeeping in mind that Eq.(\\ref{eq:k20green})\nis mathematically equivalent with Eq.(\\ref{eq:k20}),\nwe conclude that\na correct evaluation of the disorder average \n$\\overline{ K^{(2) } ( \\omega , - \\omega )}$\nmust yield an infinite result\\cite{dos}.\nUnfortunately, in an approximate evaluation of\nEq.(\\ref{eq:k20green}) by means of the \nusual diagrammatic methods this $\\delta$-function\nsingularity is\nartificially smoothed out, and one obtains a finite \nresult\\cite{Kravtsov93}. \n\n\n\n\n\n\\section{Adiabatic switching on}\n\\label{sec:adon}\n\nThe infinite term (\\ref{eq:k2deltadiag})\nis clearly unphysical. This term \nis closely related to the\ninfinitesimal imaginary parts\n$i0$ that have been added to the\nreal frequencies in the spectral representation\n(\\ref{eq:k20}) for the response function\n$K^{(2)} ( \\omega , - \\omega)$. As emphasized by KY\\cite{Kravtsov97},\nthe infinitesimal imaginary parts \nare a consequence of the fact that\nthe response function must be causal\nwhen the time-dependent part of the Hamiltonian\nis adiabatically switched on.\nLet us examine the\n''adiabatic switching on'' of the time-dependent\nperturbation more carefully. Following the\nusual recipe\\cite{Baym69},\nwe replace the Hamiltonian\n$\\hat H_0 + \\hat V (t)$ by\n$\\hat H_0 + \\hat V_\\eta (t)$, where $\\hat V_\\eta (t)=\\exp(\\eta t) \\hat\nV(t)$. The limit $\\eta \\rightarrow 0$ is then taken at the end of the\ncalculation of physical quantities.\nFor large enough times $t$ the physical result \nshould be\nindependent of the switching on procedure. \nIndeed, in the appendix\nwe show by explicit calculation\nthat sudden switching on produces the same result\nfor the long-time response as adiabatic switching on.\nHowever, in the latter case one still has to be careful\nto take the limit $\\eta\n\\rightarrow 0$ only after the physical quantity of interest\nhas been calculated. We now show that the \nsingularity in Eq.(\\ref{eq:k2deltadiag}) has been \nartificially generated by taking the limit $\\eta \\rightarrow 0$\nat an intermediate step of the calculation.\n\n\nBy direct expansion of the time evolution operator\nin the interaction representation\nto second order in the time-dependent perturbation, we\nobtain the current for adiabatic switching on with finite $\\eta$\n \\begin{eqnarray}\n I^{(2)}_\\eta (t) & = & \n \\frac{(-e)(2\\pi)^2}{(m L)^3} \\int^\\infty_\\infty d\\omega_1\n d\\omega_2 \\delta\\varphi_{\\omega_1}\\delta\\varphi_{\\omega_2} \\nonumber\n \\\\\n & \\times & K^{(2)}_{\\eta t} (\\omega_1, \\omega_2) e^{-i(\\omega_1+\\omega_2)t}\n \\; ,\n \\label{eq:strometa}\n \\end{eqnarray}\nwith\n \\begin{eqnarray}\n K^{(2)}_{\\eta t} (\\omega_1,\\omega_2) \n & = & e^{2 \\eta t} \\sum_{\\alpha\\beta\\gamma}\n \\frac{P_{\\alpha\\beta\\gamma}}{\n \\varepsilon_\\gamma-\\varepsilon_\\alpha\n +\\omega_1 +\\omega_2\n +2i\\eta} \\nonumber \\\\ \n & & \\hspace{-20mm} \\times \n \\biggl[\\frac{f(\\varepsilon_\\gamma)-\n f(\\varepsilon_\\beta)}{\\varepsilon_\\gamma-\\varepsilon_\\beta\n +\\omega_2 +i\\eta}\n - \\frac{f(\\varepsilon_\\beta)-\n f(\\varepsilon_\\alpha)}{\\varepsilon_\\beta-\\varepsilon_\\alpha\n +\\omega_1 +i\\eta}\\biggr] \\; .\n \\label{eq:k2eta}\n\\end {eqnarray}\nComparing Eq.(\\ref{eq:k2eta}) with\nEq.(\\ref{eq:k2adiabatic}), \nwe see that the former is multiplied by an extra factor of\n$e^{ 2 \\eta t }$. \nIf we directly take the limit $\\eta \\rightarrow 0$,\nthis factor is replaced by unity.\nThis is the limiting procedure adopted in \nthe usual Green's\nfunction approach, where\none takes first the limit\n$\\eta\\rightarrow 0$ in Eq.(\\ref{eq:k2eta}) and then inserts the\nresulting expression into Eq.(\\ref{eq:strometa}). \nIn this case we recover\nEqs.(\\ref{eq:I2_0}) and (\\ref{eq:k20}), which lead to\nthe divergence in Eq.(\\ref{eq:k2deltadiag}). \nWe now show that this unphysical divergence\ndoes not appear if the limit $\\eta \\rightarrow 0$\nis taken {\\it{after the physical current has been calculated}}.\nSubstituting Eq.(\\ref{eq:k2eta}) \ninto Eq.(\\ref{eq:strometa}) we obtain \n \\begin{eqnarray}\n I^{(2)}_\\eta (t) & = & A_{\\omega}\n \\bigl[\n K^{(2)}_{\\eta t}(\\omega,-\\omega) +\n K^{(2)}_{\\eta t}(- \\omega, \\omega)\n \\nonumber\n \\\\\n & & \n + K^{(2)}_{\\eta t}(\\omega,\\omega)e^{-2i \\omega t}\n + K^{(2)}_{\\eta t}(- \\omega, - \\omega)e^{2i \\omega t}\n \\bigr]\n \\; .\n \\label{eq:I2_eta2}\n \\end{eqnarray} \nIn analogy with Eq.(\\ref{k20b}), we express\n$K^{(2)}_{\\eta t}(\\omega,-\\omega)$ in terms of products of real \nand imaginary parts\n \\begin{equation}\n \\label{k2_etab}\n K^{(2)}_{\\eta t} (\\omega,-\\omega)=K^{(2)}_{\\eta\n t, \\wp}(\\omega,-\\omega)+K^{(2)}_{\\eta t, \\delta \\delta}(\\omega,-\\omega)\n \\; ,\n \\end{equation}\nwith \n \\begin{eqnarray}\n K^{(2)}_{\\eta t, \\wp} (\\omega,-\\omega) & = & \n 2 \n e^{2\\eta t} \n \\sum_{\\alpha\\beta\\gamma}\n {\\rm Re}\n P_{\\alpha\\beta\\gamma}[f(\\varepsilon_\\gamma)-\n f(\\varepsilon_\\beta)] \\nonumber \\\\\n & & \\hspace{-20mm} \\times \n \\biggl[\\frac{\\varepsilon_\\gamma-\\varepsilon_\\alpha}{\n (\\varepsilon_\\gamma-\\varepsilon_\\alpha)^2\n +(2\\eta)^2} \\frac{\\varepsilon_\\gamma-\\varepsilon_\\beta\n -\\omega}{(\\varepsilon_\\gamma-\\varepsilon_\\beta -\\omega)^2 +\n \\eta^2}\\biggr] \\; , \n \\label{eq:k2_eta_rr}\n \\end{eqnarray}\n \\begin{eqnarray}\n K^{(2)}_{\\eta t, \\delta \\delta }(\\omega,-\\omega) & = &\n -2 \n e^{2\\eta t} \n \\sum_{\\alpha\\beta\\gamma}\n {\\rm Re}\n P_{\\alpha\\beta\\gamma}[f(\\varepsilon_\\gamma)-f(\\varepsilon_\\beta)]\n \\nonumber\n \\\\\n & & \\hspace{-20mm} \\times\n \\biggl[ \\frac{2\\eta}{(\\varepsilon_\\gamma-\\varepsilon_\\alpha)^2\n +(2\\eta)^2} \n \\frac{\\eta}{(\\varepsilon_\\gamma-\\varepsilon_\\beta -\n \\omega)^2 + \\eta^2}\\biggr]\n \\; .\n \\label{eq:k2_eta_ii}\n \\end {eqnarray} \nFrom Eq.(\\ref{eq:k2_eta_rr}) it is now obvious that\n$K^{(2)}_{\\eta t, {\\wp} }$ does not have any contributions\nfrom the terms \n$\\alpha=\\gamma$. The finite contribution \nin Eq.(\\ref{eq:Kp_diag})\nis thus an artifact of taking the limit $\\eta\\rightarrow\n0$ before calculating any physical quantities. \nLet us now focus on the term\n(\\ref{eq:k2_eta_ii}).\nIf we directly take the limit\n$\\eta \\rightarrow 0$ \nusing\n \\begin{equation}\n \\lim_{\\eta \\rightarrow 0} \\frac{ \\eta}{\\epsilon^2 + \\eta^2}\n = \\pi \\delta ( \\epsilon )\n \\; ,\n \\end{equation}\nwe recover the infinite result (\\ref{eq:k2deltadiag}).\nHowever, the structure of the $\\eta$-dependent part\nof Eq.(\\ref{eq:k2_eta_ii}) is familiar \nfrom the derivation of Fermi's golden rule of elementary\nquantum mechanics.\nAs discussed for example in the classic textbook by Baym\\cite{Baym69},\nterms with this structure should be\ninterpreted as a {\\it{rate}}, \ni.e. as a contribution to the current that grows linearly in time.\nIt is therefore clear that after taking\nthe derivative of Eq.(\\ref{eq:k2_eta_ii}) with respect to\n$t$ we obtain a finite result if we then let\n$\\eta \\rightarrow 0$.\nA simple calculation yields\n \\begin{eqnarray}\n \\lim_{\\eta\\rightarrow 0} \\frac{d}{dt} K^{(2)}_{\\eta\n t, \\delta \\delta }(\\omega,-\\omega) & &\n \\nonumber\n \\\\\n & & \\hspace{-30mm} =\n -2 \\lim_{\\eta\\rightarrow 0}\n \\sum_{\\alpha\\beta}P_{\\alpha\\beta\\alpha}\n \\frac{ [f(\\varepsilon_\\alpha)-\n f(\\varepsilon_\\beta)] \\eta }{(\\varepsilon_\\alpha-\n \\varepsilon_\\beta-\\omega)^2+\\eta^2}\n \\nonumber\\\\\n & & \\hspace{-30mm} = \n -2\\pi \n \\sum_{\\alpha\\beta}P_{\\alpha\\beta\\alpha}\n [f(\\varepsilon_\\alpha)-f(\\varepsilon_\\beta)]\n \\delta(\\varepsilon_\\alpha-\\varepsilon_\\beta-\\omega) \n \\; .\n \\label{eq:k_eta0_ii}\n \\end{eqnarray}\nBecause this expression contains only a single $\\delta$-function,\nafter averaging over disorder it becomes a smooth function\nof $\\omega$.\nWe conclude that to quadratic order in the field\nthe non-equilibrium current\ninduced by the time-dependent\nflux (\\ref{eq:flux}) has the following three contributions,\n \\begin{equation}\n I^{(2)} ( t ) \\equiv \\lim_{\\eta \\rightarrow 0}\n I^{(2)}_{\\eta} ( t ) =\n I^{(2)}_{\\rm th} + t \\frac{d I^{(2)}_{{\\rm kin}}}{dt} + I^{(2)}_{\\rm osc} (t )\n \\; ,\n \\label{eq:Ifinal}\n \\end{equation}\nwhere the time-independent part is given by\n \\begin{eqnarray}\n I^{(2)}_{\\rm th} \n & = & A_{\\omega}\n \\lim_{\\eta \\rightarrow 0}\n \\bigl[\n K^{(2)}_{\\eta t, \\wp }(\\omega,-\\omega) +\n K^{(2)}_{\\eta t, \\wp }(- \\omega, \\omega)\n \\bigr]\n \\nonumber\n \\\\\n & = & 2 A_{\\omega}\n \\sum_{\\alpha\\beta\\gamma, \\alpha \\neq \\gamma }\n \\frac{ {\\rm Re} P_{\\alpha\\beta\\gamma}}{\n \\varepsilon_\\gamma-\\varepsilon_\\alpha } \n \\nonumber\n \\\\\n & \\times & \n \\wp \\left[\n \\frac{f(\\varepsilon_\\gamma)-f(\n \\varepsilon_\\beta)}{\\varepsilon_\\gamma-\\varepsilon_\\beta\n -\\omega }\n + ( \\omega \\rightarrow - \\omega ) \\right]\n \\; .\n \\label{eq:Ith}\n \\end{eqnarray}\nThe coefficient of the term linear in time is\n \\begin{eqnarray}\n \\frac{ d I^{(2)}_{\\rm kin} }{dt}\n & = & A_{\\omega}\n \\lim_{\\eta \\rightarrow 0}\n \\bigl[\n \\frac{d}{dt} K^{(2)}_{\\eta t, \\delta \\delta }(\\omega,-\\omega) +\n \\frac{d}{dt} K^{(2)}_{\\eta t, \\delta \\delta }(- \\omega, \\omega)\n \\bigr]\n \\nonumber\n \\\\\n & = & - 2 \\pi A_{\\omega}\n \\sum_{\\alpha\\beta}P_{\\alpha\\beta\\alpha}\n [f(\\varepsilon_\\alpha)-f(\\varepsilon_\\beta)]\n \\nonumber\n \\\\\n & & \\times\n \\left[ \\delta(\\varepsilon_\\alpha-\\varepsilon_\\beta-\\omega) \n + ( \\omega \\rightarrow - \\omega ) \\right]\n \\; ,\n \\label{eq:Ikin}\n \\end{eqnarray}\nand the oscillating part is \n \\begin{equation}\n I^{(2)}_{\\rm osc} (t) = A_{\\omega}\n \\lim_{\\eta \\rightarrow 0} \\bigl[\n K^{(2)}_{\\eta t}(\\omega,\\omega) e^{- 2 i \\omega t} \n +\n K^{(2)}_{\\eta t}(- \\omega, - \\omega) e^{2 i \\omega t}\n \\bigr]\n \\; . \n \\end{equation}\nThus, \na time-dependent electric field with\nfrequency $\\omega $ induces in quadratic order\nthree fundamentally different currents.\n(a) A time-independent contribution $I^{(2)}_{\\rm th}$;\nas shown in the next section, this contribution\ncan be derived from a thermodynamic calculation.\n(b) A contribution $t d I^{(2)}_{\\rm kin}/ dt$\nwhich increases linearly in time;\nthis term can be understood \nin terms of the usual golden rule\nof time-dependent perturbation theory.\n(c) Finally, there is also a time-dependent contribution $I^{(2)}_{\\rm osc}$ \noscillating with frequency $2 \\omega$. \nWhen this term is averaged over a time-interval\nlarger than $\\omega^{-1}$, its contribution\nto the current is negligibly small.\n\nFrom the above analysis it is clear that the contribution \nthat is proportional to $t$ cannot be calculated \nwithin the usual Green's function machinery, because\nin this approach the limit $\\eta \\rightarrow 0$ is\ntaken at an intermediate step of the calculation, causing\nan unphysical divergence.\nTo further support the correctness \nof the limiting procedure adopted here \nwe show in the appendix that\nEqs.(\\ref{eq:Ifinal}--\\ref{eq:Ikin}) \ncan also be re-derived if the\nperturbation is {\\it{suddenly}} (instead of\nadiabatically) switched on. \n\n\n\n\n\\section{The thermodynamic origin of the time-independent\npart of the current}\n\nThe time-independent part $I^{(2)}_{\\rm th}$ of the\nnon-equilibrium current in Eq.(\\ref{eq:Ifinal}) has\nbeen discussed by us in Ref.\\cite{Kopietz97}.\nThis contribution \ncan be obtained from a thermodynamic calculation.\nIn Ref.\\cite{Kopietz97}\nwe have assumed (without further justification)\nthe existence of such a relation.\nLet us now put this assumption\non a more solid basis.\nWithin the Matsubara (imaginary time) formalism one \ncan directly calculate the imaginary frequency version\nof the response function $K^{(2) } ( \\omega_1 , \\omega_2 )$\ngiven in Eq.(\\ref{eq:k2adiabatic}), i.e.\n \\begin{eqnarray}\n K^{(2)}(i\\omega_1,i\\omega_2) &=& \\sum_{\\alpha\\beta\\gamma}\n \\frac{P_{\\alpha\\beta\\gamma}}{\\varepsilon_\\gamma-\\varepsilon_\\alpha\n +i\\omega_1 +i\\omega_2\n } \\nonumber \\\\ \n &\\times & \\biggl[ \\frac{f(\\varepsilon_\\gamma)-\n f(\\varepsilon_\\beta)}{\\varepsilon_\\gamma-\\varepsilon_\\beta +i\\omega_2} \n -\\frac{f(\\varepsilon_\\beta)-\n f(\\varepsilon_\\alpha)}{\\varepsilon_\\beta-\\varepsilon_\\alpha\n +i\\omega_1}\\biggr] \\; . \n \\label{k2iomega}\n \\end{eqnarray}\nAs pointed out by KY\\cite{Kravtsov97},\nin order to obtain the causal response function, one should\nfirst continue both frequencies to the real axis\nwith positive imaginary part\n($ i \\omega_1 \\rightarrow \\omega_1 + i 0$,\n$ i \\omega_2 \\rightarrow \\omega_2 + i 0$), and \nthen set $\\omega_1 = - \\omega_2$ to obtain the\nconstant part of the physical current.\nOn the other hand, if one performs these steps\nin opposite order\n(i.e. first sets $i \\omega_1 = - i \\omega_2$ and then \ncontinues $i \\omega_1 \\rightarrow \\omega + i 0$) one obtains\nfor the current response function\n \\begin{eqnarray}\n K^{(2)}_{{\\rm th}}(\\omega,-\\omega) & =& {\\rm Re} \n \\sum_{\\alpha\\beta\\gamma}\n \\frac{P_{\\alpha\\beta\\gamma}}{\\varepsilon_\\gamma-\\varepsilon_\\alpha\n } \\nonumber \\\\ \n & & \\hspace{-15mm} \\times \\biggl[\\frac{f(\\varepsilon_\\gamma)-\n f(\\varepsilon_\\beta)}{\\varepsilon_\\gamma-\\varepsilon_\\beta\n -\\omega -i0} \n -\\frac{f(\\varepsilon_\\beta)-f(\\varepsilon_\\alpha)}{\n \\varepsilon_\\beta-\\varepsilon_\\alpha\n +\\omega +i0}\\biggr] \n \\; .\n \\label{eq:k2iomega0}\n \\end{eqnarray}\nComparing this expression with Eqs.(\\ref{k20b}--\\ref{eq:K2deltadelta}),\nit is easy to see that\n \\begin{equation}\n K^{(2)}_{\\rm th} ( \\omega , - \\omega ) =\n K^{(2)}_{\\wp} ( \\omega , - \\omega )\n \\; .\n \\end{equation}\nHence, the time-independent part $I^{(2)}_{\\rm th}$ of the current \ncan indeed be\nobtained from a thermodynamic calculation\\cite{Kopietz97}.\nNote, however, that our analysis of Sec.\\ref{sec:adon} \n(see also the appendix) implies that\nthe terms with $\\alpha = \\gamma$ in Eq.(\\ref{eq:k2iomega0})\nshould be omitted from the sum, i.e. \nthe physical current is given by\n \\begin{equation}\n I_{\\rm th}^{(2)} = A_{\\omega}\n \\left[ \\tilde{K}_{\\rm th}^{(2)} ( \\omega , - \\omega )\n + \\tilde{K}_{\\rm th}^{(2)} ( - \\omega , \\omega )\n \\right]\n \\; .\n \\end{equation}\n where\n \\begin{eqnarray}\n \\tilde{K}^{(2)}_{{\\rm th}}(\\omega,-\\omega) & =& \n {K}^{(2)}_{ \\wp }(\\omega,-\\omega) - \n {K}^{(2)}_{\\wp , {\\rm diag} }(\\omega,-\\omega) \n \\nonumber\n \\\\\n & & \\hspace{-10mm} =\n \\sum_{\\alpha\\beta\\gamma , \\alpha \\neq \\gamma}\n \\frac{P_{\\alpha\\beta\\gamma}}{\\varepsilon_\\gamma-\\varepsilon_\\alpha\n } \\nonumber \\\\ \n & & \\hspace{-10mm} \\times \\wp \\biggl[\\frac{f(\\varepsilon_\\gamma)-\n f(\\varepsilon_\\beta)}{\\varepsilon_\\gamma-\\varepsilon_\\beta\n -\\omega } \n -\\frac{f(\\varepsilon_\\beta)-f(\\varepsilon_\\alpha)}{\n \\varepsilon_\\beta-\\varepsilon_\\alpha\n +\\omega }\\biggr] \n \\; ,\n \\label{eq:k2iomegath}\n \\end{eqnarray}\nsee Eq.(\\ref{eq:Kp_diag}).\nThe direct diagrammatic calculation \nof the disorder average of\n${I^{(2)}_{\\rm th}}$ is difficult, because\nthe restriction $\\alpha \\neq \\gamma$ \nin Eq.(\\ref{eq:k2iomegath}) is not so easy\nto implement.\nIn Ref.\\cite{Kopietz97} the\nfollowing limiting procedure was adopted:\nInstead of directly calculating\n $ \\overline{\\tilde{K}^{(2)}_{{\\rm th}}(\\omega,-\\omega)}$,\nconsider the generalization\nof the imaginary frequency\nresponse function (\\ref{k2iomega}) for\nelectric fields with finite wave-vector ${\\bf{q}}$,\nwhich we denote by\n$\\overline{K^{(2)} ( i \\omega , - i \\omega , {\\bf{q}} )}$.\nThe limit ${\\bf{q}} \\rightarrow 0$ is taken after the\ndisorder averaged current has been calculated.\nAs shown in Ref.\\cite{Kopietz97}, in the diffusive regime\nthe function\n$\\overline{K^{(2)} ( i \\omega , - i \\omega , {\\bf{q}} )}$\nis a smooth function of ${\\bf{q}}$, so that\nthe limit ${\\bf{q}} \\rightarrow 0$ is well defined.\nThe so-defined averaged response function\nvanishes for frequencies exceeding the Thouless energy\nas $\\exp ( - \\sqrt{ | \\omega | / 2 E_c } )$\\cite{Kopietz97}.\nOn the other hand, \nperturbative averaging of the\ncontribution from the (unwanted) diagonal\nterm (\\ref{eq:Kp_diag}) shows that\nthis term vanishes as $ \\omega^{-2}$ for large frequencies.\nThis indicates that the above limiting procedure indeed\neliminates the contribution of the unphysical\ndiagonal term\n(\\ref{eq:Kp_diag})\nto the time-independent part of the non-equilibrium current.\n%At present we cannot give a\n%more rigorous justification for this procedure.\n\n\n\\section{Conclusion}\nIn this work we have shown that\na time-dependent \nflux oscillating with frequency $\\omega$ that pierces the center \nof a\nmesoscopic metal ring generates to quadratic order \nthree fundamentally different contributions to the current:\na constant\nnon-equilibrium current $I^{(2)}_{\\rm th}$, \na current $t d I^{(2)}_{\\rm kin} / dt$ that grows linearly in time, and\na current oscillating with frequency $2 \\omega$.\nAs shown in Ref.\\cite{Kopietz97},\nthe \ndisorder average of the constant term \n$\\overline{I^{(2)}_{\\rm th}}$ \nvanishes \nfor frequencies exceeding the Thouless energy \nas $\\exp ( - \\sqrt{ | \\omega | / 2 E_c } )$.\nThe calculation of the disorder average of \nthe contribution $t d I^{(2)}_{\\rm kin} / dt$ \nremains an open problem. A direct perturbative calculation\nby means of the impurity diagram technique is not straightforward, \nbecause Eq.(\\ref{eq:Ikin}) involves three matrix elements but only\none energy denominator. Therefore this expression cannot be\nsimply written in terms of Green's functions.\n\n\n\nThe main result of this work is the prediction\nof a current\n$t d I^{(2)}_{\\rm kin} / dt$ increasing linearly\nwith time. From the well-known derivation of Fermi's golden rule\\cite{Baym69}\nit is clear that this result is only valid in an intermediate\ntime interval. In particular, the calculation\nof the long-time behavior of the non-equilibrium current\nrequires non-perturbative methods.\n\nOne should keep in mind that our calculation has been\nperformed for non-interacting electrons in a random potential,\nso that our results are valid as long as \nthe spectrum of the system is discrete. We have argued in Sec.\\ref{sec:intro}\nthat at low enough temperatures this should be the case for small\nexternal frequencies, $ | \\omega | <E_c$. \nOn the other hand, \nfor frequencies exceeding $E_c$ the\nspectrum is effectively continuous.\nIn this regime\nthe conventional Green's function methods can be used\nto calculate the direct current, \nso that the results of\nKY\\cite{Kravtsov93} should be valid.\n\n\n\n\nLet us also point out that the linear time-dependence of the current\nis a consequence of the discrete spectrum, and is not related\nto the adiabatic switching on procedure in Eq.(\\ref{eq:k2eta}).\nIn the appendix we show that sudden switching on\nyields the same linear time-dependence of the current.\nIt seems reasonable to expect that for sufficiently short times the\nconstant part $I^{(2)}_{\\rm th}$ of the current is\ndominant\\cite{Kopietz97}. \nWe would like to encourage experimentalists to\nmeasure\nthe non-equilibrium response of mesoscopic metal rings\nto a time-dependent flux in the frequency\nrange $10^8 {\\rm Hz} \\leq \\omega \\leq 10^{13} {\\rm Hz}$.\n\n\n\n\n\\section*{Acknowledgement}\n\nThis work was supported by the\nDeutsche Forschungsgemeinschaft (SFB 345).\nWe thank V. E. Kravtsov for his comments.\n\n\n\\begin{appendix}\n\\section*{Sudden switching on}\n\\label{appendixA}\nTo confirm that the ''switching on procedure'' outlined in \nSec.\\ref{sec:adon} yields the correct physical results, let us\nconsider a\nharmonic perturbation that is\nsuddenly turned on at time $t=0$, \n \\begin{equation}\n \\phi ( t ) =\n \\phi + \\phi_\\omega \\Theta(t)\\sin ( \\omega t )\n \\; ,\n \\label{eq:flux2}\n \\end{equation}\nwhere $\\Theta(t)$ is the step function. \nTo second order in $\\phi_{\\omega}$ the induced current is\n \\begin{eqnarray}\n \\label{eq:stromein}\n I^{(2)}(t)&=&\n 2 A_{\\omega} {\\rm Re}\n \\sum_{\\alpha\\beta\\gamma}\n P_{\\alpha\\beta\\gamma}[f(\\varepsilon_\\beta)-f(\\varepsilon_\\alpha)]\n \\nonumber \\\\\n & & \\hspace{-15mm} \\times\n \\Biggl[ \\frac{ e^{2i\\omega\n t}-e^{i(\\varepsilon_\\gamma-\n \\varepsilon_\\alpha)t}}{(\\varepsilon_\\alpha-\n \\varepsilon_\\gamma+2\\omega)(\\varepsilon_\\alpha-\n \\varepsilon_\\beta +\\omega )} \n \\nonumber\n \\\\\n & & \\hspace{-10mm}\n -\n \\frac{1-e^{i(\\varepsilon_\\gamma-\\varepsilon_\\alpha)t}}{\n (\\varepsilon_\\alpha-\n \\varepsilon_\\gamma)(\\varepsilon_\\alpha-\\varepsilon_\\beta-\\omega)}\n \\nonumber \\\\\n & & \\hspace{-10mm} +\n \\frac{2 \\omega }{ ( \\varepsilon_\\alpha-\\varepsilon_\\beta )^2\n- \\omega^2 }\n \\Biggl[ \\frac{e^{i(\\varepsilon_\\beta-\\varepsilon_\\alpha+\n\\omega)t}-e^{i(\\varepsilon_\\gamma-\\varepsilon_\\alpha)t}}{\n\\varepsilon_\\beta-\\varepsilon_\\gamma\n+\\omega} \\Biggr]\n\\nonumber\n\\\\\n& & \\hspace{-10mm}\n+ (\\omega\\rightarrow -\\omega) \\Biggr]\n\\; .\n\\end{eqnarray}\nThe diagonal term $\\alpha=\\gamma$ is\n \\begin{eqnarray}\n \\label{eq:idiag}\n I^{(2)}_{\\rm diag}(t)&=& 4 A_{\\omega}\n \\sum_{\\alpha\\beta}\n P_{\\alpha\\beta\\alpha}[f(\\varepsilon_\\alpha)-\n f(\\varepsilon_\\beta)] \\nonumber \\\\\n & \\times & \\Biggl[ \\frac{ \\sin^2(\\omega\n t)}{ ( \\varepsilon_\\alpha-\\varepsilon_\\beta )^2 \n - \\omega^2 }\n \\nonumber \\\\\n & & - \\frac{\n \\sin^2\\bigl(\\frac{\\varepsilon_\\beta-\\varepsilon_\\alpha\n +\\omega}{2}t\\bigr) \n + \\sin^2\\bigl(\\frac{\\varepsilon_\\beta-\\varepsilon_\\alpha\n -\\omega}{2}t\\bigr)}{\n ( \\varepsilon_\\alpha-\\varepsilon_\\beta)^2-\n \\omega^2 }\n \\nonumber \\\\\n & & + \n \\Bigl[ \\frac{\\sin \\bigl(\\frac{\\varepsilon_\\beta-\\varepsilon_\\alpha\n +\\omega}{2}t\\bigr)}{\\varepsilon_\\beta-\\varepsilon_\\alpha\n +\\omega} \\Bigr]^2 +\n \\Bigl[ \\frac{\\sin \\bigl(\\frac{\n \\varepsilon_\\beta-\\varepsilon_\\alpha\n -\\omega}{2}t\\bigr)}{\\varepsilon_\\beta-\\varepsilon_\\alpha\n -\\omega} \\Bigr]^2 \\Biggr] \n \\; .\n \\end{eqnarray}\nThe terms in the last line can be interpreted in the same way as is\ndone in Fermi's golden rule\\cite{Baym69} by using the identity \n\\begin{equation}\n\\label{eq:sindelta}\n\\left[\\frac{\\sin\\left(\\frac{\\Delta\\varepsilon}{2}\nt\\right)}{\\Delta\\varepsilon}\n\\right]^2 \\rightarrow \\frac{ \\pi}{2} t \n\\delta(\\Delta\\varepsilon) \\; \\textrm{for} \n\\;\\; t\\rightarrow\\infty\n\\; .\n\\end{equation}\nIt is now easy to see that for large times $I^{(2)}_{\\rm diag}(t)$ \nyields exactly the\nsame linear in time contribution as given in\nEq.(\\ref{eq:Ikin}). \nThe terms with\nno explicit time dependence in Eq.(\\ref{eq:stromein}) can \nbe identified with $I_{\\rm th}^{(2)}$ in Eq.(\\ref{eq:Ith}).\n\n\\end{appendix}\n\n\n\n\n%\n% R E F E R E N C E S\n\\begin{thebibliography}{99}\n%\n\\bibitem{Hund38}\nF. Hund, Ann. Phys. (Leipzig) {\\bf{32}}, 102 (1938);\nM. B\\\"{u}ttiker, Y. Imry, and R. Landauer, Phys. Lett. {\\bf{96 A}}, 365 (1983).\n%\n\\bibitem{Imry97}\nFor a recent review see Y. Imry,\n{\\it{Introduction to Mesoscopic Physics}}, (Oxford University Press,\nNew York, 1997).\n%\n\\bibitem{Levy90}\nL. P. L\\'{e}vy, G. Dolan, J. Dunsmuir, and H. Bouchiat,\nPhys. Rev. Lett. {\\bf{64}}, 2074 (1990). \n%\n\\bibitem{Mohanty96}\nP. Mohanty, E. M. Q. Jariwala, M. B. Ketchen, and R. A. Webb,\nin {\\it{Quantum Coherence and Decoherence}}, edited by\nK. Fujikawa and Y. A. Ono (Elsevier, Amsterdam, 1996).\n%\n\\bibitem{Kravtsov93}\nV. E. Kravtsov and V. I. Yudson, Phys. Rev. Lett. {\\bf{70}}, 210\n(1993).\n%\n\\bibitem{Schmid91}\nA. Schmid, Phys. Rev. Lett. {\\bf{66}}, 80 (1991).\n%\n\\bibitem{Altshuler91}\nA. L. Altshuler, Y. Gefen, and Y. Imry,\nPhys. Rev. Lett. {\\bf{66}}, 88 (1991).\n%\n\\bibitem{Voelker99}\nA. V\\\"{o}lker and P. Kopietz, (unpublished).\n%\n\\bibitem{Belinicher80}\nV. I. Belinicher and B. I. Sturman, Usp. Fiz. Nauk\n{\\bf{130}}, 130 (1980) [Sov. Phys. Usp. {\\bf{23}}, 199 (1980)].\n%\n\\bibitem{Falko89}\nV. I. Fal'ko and D. E. Khmel'nitskii,\nZh. Eksp. Teor. Fiz. {\\bf{95}}, 186 (1989) \n[Sov. Phys. JETP {\\bf{68}}, 186 (1989)];\nV. I. Fal'ko, Europhys. Lett. {\\bf{8}}, 785 (1989).\n%\n\\bibitem{Fricke95}\nJ. Fricke and P. Kopietz, Phys. Rev. B {\\bf{52}}, 2728 (1995).\n%\n\\bibitem{common}\nNote, however, that in a canonical ensemble the chemical\npotential fluctuates, so that disorder averages\ninvolving only retarded or only advanced Green's functions\ncan also generate large contributions \nto response functions, see Ref.\\cite{Fricke95}.\n%\n\\bibitem{Kravtsov97}\nV. E. Kravtsov and V. I. Yudson, preprint cond-mat/9712149 (1997).\n%\n\\bibitem{dos}\nThe same type of singularity occurs if one \ncalculates\nthe disorder average\nof the square of the density of states,\n$\\overline{ \\rho^2 ( \\epsilon ) } =\n\\overline{ \n\\sum_{\\alpha \\beta } \n\\delta ( \\varepsilon - \\varepsilon_{\\alpha} )\n\\delta ( \\varepsilon - \\varepsilon_{\\beta} )}$.\nThe contribution from the terms with $\\alpha = \\beta$\nis proportional to $\\delta ( 0 ) = \\infty$.\nYet, perturbative averaging with the help of the\nimpurity diagram technique produces a finite result.\n%\n\\bibitem{Baym69}\nG. Baym, {\\it{Lectures on Quantum Mechanics}},\n(Benjamin, New York, 1969).\n%\n\\bibitem{Kopietz97}\nP. Kopietz and A. V\\\"olker, \nEur. Phys. J. B {\\bf{3}}, 397 (1998).\n%\n\n\n\n\n\n\n\\end{thebibliography}\n%\n%\n\\end{document}\n"
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{
"name": "cond-mat0002050.extracted_bib",
"string": "\\begin{thebibliography}{99}\n%\n\\bibitem{Hund38}\nF. Hund, Ann. Phys. (Leipzig) {\\bf{32}}, 102 (1938);\nM. B\\\"{u}ttiker, Y. Imry, and R. Landauer, Phys. Lett. {\\bf{96 A}}, 365 (1983).\n%\n\\bibitem{Imry97}\nFor a recent review see Y. Imry,\n{\\it{Introduction to Mesoscopic Physics}}, (Oxford University Press,\nNew York, 1997).\n%\n\\bibitem{Levy90}\nL. P. L\\'{e}vy, G. Dolan, J. Dunsmuir, and H. Bouchiat,\nPhys. Rev. Lett. {\\bf{64}}, 2074 (1990). \n%\n\\bibitem{Mohanty96}\nP. Mohanty, E. M. Q. Jariwala, M. B. Ketchen, and R. A. Webb,\nin {\\it{Quantum Coherence and Decoherence}}, edited by\nK. Fujikawa and Y. A. Ono (Elsevier, Amsterdam, 1996).\n%\n\\bibitem{Kravtsov93}\nV. E. Kravtsov and V. I. Yudson, Phys. Rev. Lett. {\\bf{70}}, 210\n(1993).\n%\n\\bibitem{Schmid91}\nA. Schmid, Phys. Rev. Lett. {\\bf{66}}, 80 (1991).\n%\n\\bibitem{Altshuler91}\nA. L. Altshuler, Y. Gefen, and Y. Imry,\nPhys. Rev. Lett. {\\bf{66}}, 88 (1991).\n%\n\\bibitem{Voelker99}\nA. V\\\"{o}lker and P. Kopietz, (unpublished).\n%\n\\bibitem{Belinicher80}\nV. I. Belinicher and B. I. Sturman, Usp. Fiz. Nauk\n{\\bf{130}}, 130 (1980) [Sov. Phys. Usp. {\\bf{23}}, 199 (1980)].\n%\n\\bibitem{Falko89}\nV. I. Fal'ko and D. E. Khmel'nitskii,\nZh. Eksp. Teor. Fiz. {\\bf{95}}, 186 (1989) \n[Sov. Phys. JETP {\\bf{68}}, 186 (1989)];\nV. I. Fal'ko, Europhys. Lett. {\\bf{8}}, 785 (1989).\n%\n\\bibitem{Fricke95}\nJ. Fricke and P. Kopietz, Phys. Rev. B {\\bf{52}}, 2728 (1995).\n%\n\\bibitem{common}\nNote, however, that in a canonical ensemble the chemical\npotential fluctuates, so that disorder averages\ninvolving only retarded or only advanced Green's functions\ncan also generate large contributions \nto response functions, see Ref.\\cite{Fricke95}.\n%\n\\bibitem{Kravtsov97}\nV. E. Kravtsov and V. I. Yudson, preprint cond-mat/9712149 (1997).\n%\n\\bibitem{dos}\nThe same type of singularity occurs if one \ncalculates\nthe disorder average\nof the square of the density of states,\n$\\overline{ \\rho^2 ( \\epsilon ) } =\n\\overline{ \n\\sum_{\\alpha \\beta } \n\\delta ( \\varepsilon - \\varepsilon_{\\alpha} )\n\\delta ( \\varepsilon - \\varepsilon_{\\beta} )}$.\nThe contribution from the terms with $\\alpha = \\beta$\nis proportional to $\\delta ( 0 ) = \\infty$.\nYet, perturbative averaging with the help of the\nimpurity diagram technique produces a finite result.\n%\n\\bibitem{Baym69}\nG. Baym, {\\it{Lectures on Quantum Mechanics}},\n(Benjamin, New York, 1969).\n%\n\\bibitem{Kopietz97}\nP. Kopietz and A. V\\\"olker, \nEur. Phys. J. B {\\bf{3}}, 397 (1998).\n%\n\n\n\n\n\n\n\\end{thebibliography}"
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cond-mat0002051
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%{PREPRINT v1.4 -- to be submitted in Phys. Rev B (regular)}\\\ \\ Bunching of fluxons by the Cherenkov radiation in Josephson multilayers
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"author": "E.~Goldobin\\cite{gold-mail}"
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A single magnetic fluxon moving at a high velocity in a Josephson multilayer (\eg, high-temperature superconductor such as BSCCO) can emit electromagnetic waves (Cherenkov radiation), which leads to formation of novel stable dynamic states consisting of several bunched fluxons. We find such bunched states in numerical simulation in the simplest cases of two and three coupled junctions. At a given driving current, several different bunched states are stable and move at velocities that are higher than corresponding single-fluxon velocity. These and some of the more complex higher-order bunched states and transitions between them are investigated in detail.
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"name": "CherryBunch02a.tex",
"string": "%%% Alternative title: Cherry-Tree Garden\n%%%\n%%% Date: 13.07.1998 v1.0\n%%% Author: Edward Goldobin\n%%% Reason: created\n%%%\n%%% Date: 26.07.1998 v1.2\n%%% Author: Edward Goldobin and Anton P. Chekhov\n%%% Reason: created\n%%%\n%%% Date: 28.01.2000 v1.4\n%%% Author: Edward Goldobin\n%%% Reason: after Alexey's 6 months reading. \n%%% Had to be a very thorough reading.(Boris)\n\n%stupid macros which are allowed to use by APS\n\\newcommand{\\ie}{{\\it i.e.}}\n\\newcommand{\\eg}{{\\it e.g.}}\n\\newcommand{\\cf}{{\\it c.f.}}\n\\newcommand{\\etc}{{\\it etc.}}\n\\newcommand{\\viz}{{\\it viz.}}\n\\newcommand{\\etal}{{\\it et al.}}\n\\newcommand{\\lhs}{l.h.s.}\n\\newcommand{\\rhs}{r.h.s.}\n\\newcommand{\\BR}[1]{\\linebreak[0]#1\\linebreak[0]}\n\n%\\documentstyle[preprint,prb,aps]{revtex}\n\\documentstyle[prb,aps]{revtex}\n\n\\sloppy\n\\draft\n\n\\begin{document}\n\n\n\\title{\n %{\\bf PREPRINT v1.4 -- to be submitted in Phys. Rev B (regular)}\\\\\\ \\\\\n Bunching of fluxons by the Cherenkov radiation in\n Josephson multilayers\n}\n\n\\author{\n E.~Goldobin\\cite{gold-mail}\n}\n\n\\address{\n Institute of Thin Film and Ion Technology (ISI),\n Research Center J\\\"ulich GmbH (FZJ) \\\\\n D-52425 J\\\"ulich, Germany\n}\n\n\\author{\n B.~A.~Malomed\\cite{Boris-mail}\n}\n\n\\address{\n Department of Interdisciplinary Studies, Faculty of\n Engineering\\\\\n Tel Aviv University, Tel Aviv 69978, Israel\n}\n\n\\author{\n A.~V.~Ustinov\n}\n\n\\address{\n Physikalisches Institut III,\n Uni\\-ver\\-si\\-t\\\"at Er\\-lan\\-gen-N\\\"urn\\-berg,\n D-91054, Erlangen, Germany\n}\n\n\\date{\\today}\n\n\\wideabs{ %REVTeX 3.1\n\n\\maketitle\n\n\\begin{abstract}\n\n A single magnetic fluxon moving at a high velocity in a Josephson\n multilayer (\\eg, high-temperature superconductor such as BSCCO) can\n emit electromagnetic waves (Cherenkov radiation), which leads to\n formation of novel stable dynamic states consisting of several bunched\n fluxons. We find such bunched states in numerical simulation in the\n simplest cases of two and three coupled junctions. At a given driving\n current, several different bunched states are stable and move at\n velocities that are higher than corresponding single-fluxon velocity.\n These and some of the more complex higher-order bunched states and\n transitions between them are investigated in detail.\n\n\\end{abstract}\n\n\\pacs{PACS:\n 74.50.+r, %Josephson effects\n 74.80.Dm, %Superconducting multilayers\n 41.60.Bq, %Cherenkov radiation\n}\n\n} %%End Of \\WideAbs{...}\n\n\n\\section{Introduction}\n\nIn the recent years, a great deal of attention was attracted\nto different kinds of solid state multilayered systems, \\eg,\nartificial Josephson and magnetic multilayers, high-temperature \nsuperconductors (HTS) and perovskites, to name\njust a few. Multilayers are attractive because it is often\npossible to multiply a physical effect achieved in one layer\nby $N$ (and sometimes by $N^2$), where $N$ is the number of\nlayers. This can be exploited for fabrication of novel \nsolid-state devices. In addition, multilayered solid state systems\nshow a variety of new physical phenomena which result from\nthe interaction between individual layers.\n\nIn this article we focus on Josephson multilayer, the\nsimplest example of which is a stack consisting of just two\nlong Josephson junctions (LJJs). The results of our\nconsideration can be applied to intrinsically layered HTS\nmaterials \\cite{Intrinsic}, since the\nJosephson-stack model has proved\nto be appropriate for these structures\n\\cite{Kl-Mue-Kolh:94,Paul:AdvSSP34,pl:pap154}.\n\nIn earlier papers \\cite{image,Cherry1,Cherry2,HexFisherCherry} it\nwas shown that, in some cases, a fluxon (Josephson vortex)\nmoving in one of the layers of the stack may emit\nelectromagnetic (plasma) waves by means of the Cherenkov\nmechanism. The fluxon together with its Cherenkov radiation\nhave a profile of a traveling wave, $\\phi(x-ut)$, having an\noscillating gradually decaying tail. Such a wave profile\ngenerates an effective potential for another fluxon which\ncan be added into the system. If the second fluxon is\ntrapped in one of the minima of this traveling potential, we\ncan get a {\\it bunched state} of two fluxons. In such a state, two\nfluxons can stably move at a small constant distance one from\nanother, which is not possible otherwise.\nFluxons of the same polarity usually repel each other, even\nbeing located in different layers.\n\nSimilar bunched states were already found in a discrete\nJosephson transmission lines\\cite{Ustinov:BunchArray}, as\nwell as in long Josephson junctions with the so-called\n$\\beta$-term due to the surface impedance of the\nsuperconductor\n\\cite{Sakai:BunchBeta,Malomed:BunchBeta,Vernik:BunchBeta}.\nThe dynamics of conventional LJJ is described by the sine-\nGordon equation which does not allow the fluxon to move\nfaster than the Swihart velocity and, therefore, the Cherenkov\nradiation never appears. In both cases mentioned above (the\ndiscrete system or the system with the $\\beta$-term), the\nperturbation of the sine-Gordon equation results in a modified\ndispersion relation for Josephson plasma waves and\nappearance of an oscillating tail. This tail, in turn,\nresults in an attractive interaction between fluxons, \\ie,\nbunching. Nevertheless, the mere presence of an oscillating\ntail is not a sufficient condition for bunching.\n\n%The role of bunching phenomena in physics can not be\n%underestimated. It it enough just to remind that the origin\n%of superconductivity itself lays in the formation of Cooper\n%pairs, \\ie, of a specific bunched state of two electrons. In\n%this case, the electron interaction is mediated by acoustic\n%waves (phonons).\n\nIn this paper, we investigate the problem of fluxon bunching\nin a system of two and three inductively coupled junctions\nwith a primary state $[1|0]$ (one fluxon in the top junction and\nno fluxon in the bottom one) or $[0|1|0]$ (a fluxon only in the\nmiddle junction of a 3-fold stack). We show that bunching is\npossible for some fluxon configurations and specific range\nof parameters of the system. In addition, it is found that\nthe bunched states radiate less than single-fluxon states,\nand therefore can move with a higher velocity. Section\n\\ref{Sec:SimRes} presents the results of numerical\nsimulations, in section \\ref{Sec:Discussion} we discuss the\nobtained results and a feasibility of experimental\nobservation of bunched states. We also derive a simple\nanalytical expression which show the possibility of the\nexistence of bunched states. Section \\ref{Sec:Conclusion}\nconcludes the work.\n\n\n\n\n\\section{Numerical Simulations}\n\\label{Sec:SimRes}\n\nThe system of equations which describes the dynamics of\nJosephson phases $\\phi^{A,B}$ in two coupled LJJ$^A$ and\nLJJ$^B$ is \\cite{SBP,LT21}:\n%\n\\begin{eqnarray}\n \\frac{\\phi^A_{xx}}{1-S^2}\n - \\phi^A_{tt}\n - \\sin{\\phi^A}\n - \\frac{S}{1-S^2} \\phi^B_{xx}\n &=&\\alpha \\phi^A_{t} - \\gamma\n \\ ; \\label{Eq:2:PDEa}\\\\\n \\frac{\\phi^B_{xx}}{1-S^2}\n - \\phi^B_{tt}\n - \\frac{\\sin{\\phi^B}}{J}\n - \\frac{S}{1-S^2} \\phi^A_{xx}\n &=& \\alpha \\phi^B_{t} - \\gamma\n \\ , \\label{Eq:2:PDEb}\n\\end{eqnarray}\n%\nwhere $S$ ($-1<S<0$) is a dimensionless coupling constant, $J=j_c^A/j_c^B$ is the ratio of the critical currents, while $\\alpha$ and $\\gamma=j/j_c^A$ are the damping coefficient and normalized bias current, respectively, that are assumed to be the same in both LJJs. It is also assumed that other parameters of the junctions, such as the effective magnetic thicknesses and capacitances, are the same. As has been shown earlier\\cite{Cherry1,Cherry2}, the Cherenkov radiation in a two-fold stack may take place only if the fluxon is moving in the junction with smaller $j_c$. We suppose in the following that the fluxon moves in LJJ$^A$, that implies $J<1$.\n\nIn the case $N=3$, we impose the symmetry condition $\\phi^{A}\\equiv \\phi^{C}$, which is natural when the fluxon moves in the middle layer, and, thus, we can rewrite equations from Ref.~[\\onlinecite{SBP}] in the form\n%\n\\begin{eqnarray}\n \\frac{\\phi^A_{xx}}{1-2S^2} - \\phi^A_{tt} -\\sin\\phi^A\n -\\frac{S\\phi^B_{xx}}{1-2S^2} &=&\\alpha \\phi^A_t - \\gamma\n \\ ; \\label{Eq:Sym3:AC} \\\\\n \\frac{\\phi^B_{xx}}{1-2S^2}-\\phi^B_{tt}-\\sin\\phi^B\n -\\frac{2S\\phi^A_{xx}}{1-2S^2} &=&\\alpha \\phi^B_t - \\gamma\n \\ . \\label{Eq:Sym3:B}\n\\end{eqnarray}\n%\nNote the factor 2 in the last term on the \\lhs{} of Eq.~(\\ref{Eq:Sym3:B}). In the case of three coupled LJJs, we assume $J=1$, since for more than two coupled junctions the Cherenkov radiation can be obtained for a uniform stack with equal critical currents\\cite{Cherry2}\n\n\n\\subsection{Numerical technique}\n\nThe numerical procedure works as follows. For a given set of the LJJs parameters, we compute the current-voltage characteristic (IVC) of the system, \\ie, $\\bar{V}^{A,B}(\\gamma)$. To calculate the voltages $\\bar{V}^{A,B}$ for fixed values of $\\gamma$, we simulate the dynamics of the phases $\\phi^{A,B}(x,t)$ by solving Eqs.~(\\ref{Eq:2:PDEa}) and (\\ref{Eq:2:PDEb}) for $N=2$ or Eqs.~(\\ref{Eq:Sym3:AC}) and (\\ref{Eq:Sym3:B}) for $N=3$, using the periodic boundary conditions:\n%\n\\begin{eqnarray}\n \\phi^{A,B}(x=L) &=& \\phi^{A,B}(x=0) + 2\\pi N^{A,B}\n ; \\label{Eq:BC:phi}\\\\\n \\phi_x^{A,B}(x=L) &=& \\phi_x^{A,B}(x=0)\n , \\label{Eq:BC:phi'}\n\\end{eqnarray}\n%\nwhere $N^{A,B}$ is the number of fluxons trapped in LJJ$^{A,B}$. In order to simulate a quasi-infinite system, we have chosen annular geometry with the length (circumference) of the junction $L=100$.\n\nTo solve the differential equations, we use an explicit method [expressing $\\phi^{A,B}(t+\\Delta t)$ as a function of $\\phi^{A,B}(t)$ and $\\phi^{A,B}(t-\\Delta t)$], treating $\\phi_{xx}$ with a five-point, $\\phi_{tt}$ with a three-point, and $\\phi_{t}$ with a two-point symmetric finite-difference scheme. The spatial and time steps used for the simulations were $\\delta x = 0.025$, $\\delta t=0.00625$. After the simulation of the phase dynamics for $T=10$ time units, we calculate the average dc voltages $\\bar{V}^{A,B}$ for this time interval as\n%\n\\begin{equation}\n \\bar{V}^{A,B}\n = \\frac{1}{T}\\int_0^T \\phi^{A,B}_t(t) \\:dt\n = \\frac{\\phi^{A,B}(T)-\\phi^{A,B}(0)}{T}\n \\quad . \\label{Eq:V}\n\\end{equation}\n%\nThe dc voltage at point $x$ can be defined as average number of fluxons (the flux) passed through the junction at this point. Since the average fluxon density is not singular in any point of the junction (otherwise the energy will grow infinitely), we conclude that average dc voltage is the same for any point $x$. Therefore, for faster convergence of our averaging procedure, we can additionally average the phases $\\phi^{A,B}$ in (\\ref{Eq:V}) over the length of the stack.\n\nAfter the values of $\\bar{V}^{A,B}$ were found as per Eq. (\\ref{Eq:V}), the evolution of the phases $\\phi^{A,B}(x,t)$ is simulated further during $1.1\\:T$ time units, the dc voltages $\\bar{V}^{A,B}$ are calculated for this new time interval and compared with the previously calculated values. We repeat such iterations further, increasing the time interval by a factor 1.1 until the difference in dc voltages $|\\bar{V}(1.1^{n+1}\\:T)-\\bar{V}(1.1^n\\:T)|$ obtained in two subsequent iterations becomes less than an accuracy $\\delta V=10^{-4}$. The particular factor $1.1$ was found to be quite optimal and to provide for fast convergence, as well as more efficient averaging of low harmonics on each subsequent step. Very small value of this factor, \\eg, $1.01$ (recall that only the values greater than 1 have meaning), may result in a very slow convergence in the case when $\\phi(t)$ contains harmonics with the period $\\ge{}T$. Large values of the factor, \\eg, $\\ge{}2$, would consume a lot of CPU time already during the second or third iteration and, hence, are not good for practical use.\n\nOnce the voltage averaging for current $\\gamma$ is complete, the current $\\gamma$ is increased by a small amount $\\delta\\gamma = 0.005$ to calculate the voltages at the next point of the IVC. We use a distribution of the phases (and their derivatives) achieved in the previous point of the IVC as the initial distribution for the following point.\n\nFurther description of the software used for simulation can be found in Ref.~\\onlinecite{StkJJ}.\n\n\n\\subsection{Two coupled junctions}\n\nFor simulation we chose the following parameters of the system: $S=-0.5$ to be close to the limit of intrinsically layered HTS, $J=0.5$ to let the fluxon accelerate above the ${\\bar c}_{-}$ and develop Cherenkov radiation tail. The velocity ${\\bar c}_{-}$ is the smallest of Swihart velocities of the system. It characterizes the propagation of the out-of-phase mode of Josephson plasma waves. The value of $\\alpha=0.04$ is chosen somewhat higher than, \\eg, in (Nb-Al-AlO$_x$)$_N$-Nb stacks. This choice is dictated by the need to keep the quasi-infinite approximation valid and satisfy the condition $\\alpha L \\gg 1$. Smaller $\\alpha$ requires very large $L$ and, therefore, unaffordably long simulation times. So, we made a compromise and chose the above $\\alpha$ value.\n\nFirst, we simulated the IVC $u(\\gamma)$ in the $[1|0]$ state, by sweeping $\\gamma$ from 0 up to 1 and making snapshots of phase gradients at every point of the IVC. This IVC is shown in Fig.~\\ref{Fig:IVC+Profile[1|0]}(a), and the snapshot of the phase gradient at $\\gamma=0.3$ is presented in Fig.~\\ref{Fig:IVC+Profile[1|0]}(b). As one can see, the Cherenkov radiation tail, which is present for $u>{\\bar c} _{-}$, has a sequence of minima where the second fluxon may be trapped.\n\n\n\\subsubsection{$[1+1|0]$ state}\n\nIn order to create a two-fluxon bunched state and check its stability, we used the following ``solution-engineering'' procedure. By taking a snapshot of the phase profiles $\\phi_{A,B}(x)$ at the bias value $\\gamma_0=0.3$, we constructed an {\\it ansatz} for the bunched solution in the form\n%\n\\begin{equation}\n \\phi_{A,B}^{\\rm new}(x) = \\phi_{A,B}(x) + \\phi_{A,B}\n (x+\\Delta{}x)\n , \\label{Eq:NewTrialFn[2|0]}\n\\end{equation}\n%\nwhere $\\Delta{}x$ is chosen so that the center of the trailing fluxon is placed at one of the minima of the Cherenkov tail. For example, to trap the trailing fluxon in the first, second and third well, we used $\\Delta{}x=0.9$, $\\Delta{}x= 2.4$ and $\\Delta{}x=3.9$, respectively. The phase distribution (and derivatives), constructed in this way, were used as the initial condition for solving Eqs.~(\\ref{Eq:2:PDEa}) and (\\ref{Eq:2:PDEb}) numerically. As the system relaxed to the desired state $[1+1|0]$, we further traced $u(\\gamma)$ curve, varying $\\gamma_0$ down to $0$ and up to $1$.\n\nWe accomplished this procedure for a set of $\\Delta{}x$ values, trying to trap the second fluxon in every well. Fig.~\\ref{Fig:IVC+Profile[1|0]}(c) shows that a stable, tightly bunched state of two fluxons is indeed possible. Actually, all the $[1+1|0]$ states obtained this way have been found to be stable, and we were able to trace their IVCs up and down, starting from the initial value of the bias current $\\gamma=0.3$. For the case when the trailing fluxon is trapped in the first, second and third minima, such IVCs are shown in Fig.~\\ref{Fig:IVC[2|0]}.\n\nThe most interesting feature of these curves is that they correspond to the velocity of the bunched state that is {\\em higher}\\/ than that of the $[1|0]$ state, at the same value of the bias current. Comparing solutions shown in Figs.~\\ref{Fig:IVC+Profile[1|0]}(b) and \\ref{Fig:IVC+Profile[1|0]}(c), we see that the amplitude of the trailing tail is smaller for the bunched state. This circumstance suggests the following explanation to the fact that the observed velocity is higher in the state $[1+1|0]$ than in the single-fluxon one. Because the driving forces acting on two fluxons in the bunched and unbunched states are the same, the difference in their velocities can be attributed only to the difference in the friction forces. The friction force acting on the fluxon in one junction is\n%\n\\begin{equation}\n F_{\\alpha} = \\alpha \\int_{-\\infty}^{+\\infty}\n \\phi_x\\phi_t\\:dx\n \\, , \\label{Eq:Falpha}\n\\end{equation}\n%\nand the same holds for the other junction. By just looking at Fig.~\\ref{Fig:IVC+Profile[1|0]}(b) and (c) it is rather difficult to tell in which case the friction force is larger, but accurate calculations using Eq.~(\\ref{Eq:Falpha}) and profiles from Fig.~\\ref{Fig:IVC+Profile[1|0]}(b) and (c) show that the friction force acting on two fluxons with the tails shown in Fig.~\\ref{Fig:IVC+Profile[1|0]}(b) is somewhat higher than that for Fig.~\\ref{Fig:IVC+Profile[1|0]}(c). This result is not surprising if one recalls that, to create the bunched state, we have shifted the $[1|0]$ state by about half of the tail oscillation period relative to the other single-fluxon state. Due to this, the tails of the two fluxons add up out of phase and partly cancel each other, making the tail's amplitude behind the fluxon in the bunched state lower than that in the $[1|0]$ state.\n\nFrom Fig.~\\ref{Fig:IVC[2|0]} it is seen that every bunched state exists in a certain range of values of the bias current. If the current is decreased below some threshold value, fluxons dissociate and start moving apart, so that the interaction between them becomes exponentially small. When the trailing fluxon sits in a minimum of the Cherenkov tail sufficiently far from the leading fluxon, the IVC corresponding to this bunched state is almost undistinguishable from that of the $[1|0]$ state, as the two fluxons approach the limit when they do not interact. We have found that IVCs for $M>3$, where $M$ is the potential well's number, is indeed almost identical to that of the $[1|0]$ state. In contrast to bunching of fluxons in discrete LJJ\\cite{Ustinov:BunchArray}, the transitions from one bunched state to another with different $M$ do {\\emph not} take place in our system. Thus, we can say that the current range of a bunched state with smaller $M$ ``eclipses'' the bunched states with larger $M$.\n\nThe profiles of solutions found for various values of the bias current are shown in Fig.~\\ref{Fig:Profiles[2|0]}. We notice that at the bottom of the step corresponding to the bunched state the radiation tail is much weaker and fluxons are bunched tighter. This is a direct consequence of the fact that at lower velocities the radiation wavelength and the distance between minima becomes smaller, and so does the distance between the two fluxons. At a low bias current, the radiation wavelength and, hence, width of the potential wells become very small and incommensurable with the fluxon's width. Therefore, the fluxon does not fit into the well and the bunched states virtually disappear.\n\n\n\\subsubsection{$[1|1]$ state}\n\nThe initial condition for this state was constructed in a\nsimilar fashion to the $[1+1|0]$ one, but now using a\ncross-sum of the shifted and unshifted solutions:\n%\n\\begin{equation}\n \\phi_{A,B}^{\\rm new}(x) = \\phi_{A,B}(x) + \\phi_{B,A}\n (x+\\Delta{}x)\n . \\label{Eq:NewTrialFn[1|0+1]}\n\\end{equation}\n%\nIf for the $[1+1|0]$ state, $\\Delta{}x$ was $\\approx(\\lambda-\\frac{1}{2})M$, $M=1,2\\ldots$, then in the $[1|1]$ state we have to take $\\Delta{}x\\approx\\lambda{}M$. We can also take $M=0$ \\, i.e., $\\Delta{}x=0$, which corresponds to the degenerate case of the in-phase $[1|1]$ state. The stability of this state was investigated in detail analytically by Gr{\\o}nbech-Jensen and co-authors\\cite{GrE:Stability}, and is outside the scope of this paper.\n\nOur efforts to create a bound state $[1|1]$ using the phase\nin the form (\\ref{Eq:NewTrialFn[1|0+1]}) with $M=1,2\\ldots$\nhave {\\it not} lead to any stable configuration of bunched fluxons\nwith $\\Delta x \\ne 0$. \n\n% This negative result is not\n% surprising, as in this case the fluxon in one junction\n% could only interact with a very weak ``image\" \n% \\cite{image} induced by the tail\n% belonging to the other junction.\n\n\n\\subsubsection{Higher-order states}\n\nLooking at the phase gradient profiles shown in\nFig.~\\ref{Fig:Profiles[2|0]}, one notes that these profiles\nare qualitatively very similar to the original profile of\nthe soliton with a radiation tail behind it [see\nFig.~\\ref{Fig:IVC+Profile[1|0]}(b)], with the only difference that\nthere are two bunched solitons with a tail. So, we can try\nto construct two pairs of bunched fluxons moving together,\n\\ie, get a $[2+2|0]$ bunched state. As before, the trapping\nof the trailing pair is possible in one of the minima of the\ntail generated by the leading pair. To construct such a\ndouble-bunched state we employ the initial conditions obtained\nusing Eq.~(\\ref{Eq:NewTrialFn[2|0]}) at the bias point\n$\\gamma_0=0.3$, using the steady phase distribution obtained\nfor the $[2|0]$ state at $\\gamma_0=0.3$. The shift $\\Delta{}\nx$ was chosen in such a way that a pair of fluxons fits into\none of the minima of the tail. We note that in this case we\nneeded to vary $\\Delta{}x$ a little bit before we have\nachieved trapping of the trailing pair in a desired well.\n\nSimulations show that the obtained $[2+2|0]$ states are\nstable and demonstrate an even {\\it higher} velocity of the whole\nfour-fluxon aggregate. The corresponding IVCs and\nprofiles are shown in Fig.~\\ref{Fig:IVC[2+2|0]}(a) and (b),\nrespectively. Note that at $\\gamma<0.22$ the bunched state\n$[2+2|0]$ splits first into $[1+1_2+1_3+1_3|0]$ state (the\nsubscripts denote the well's number $M$, counting from the\nprevious fluxon), and at still lower bias current,\n$\\gamma<0.2$, they split into two\nindependent $[1+1_2|0]$ and $[1+1_5|0]$ states. This two\nstates move with slightly different velocities and can\ncollide with each other due to the periodic nature of the\nsystem. As a result of collisions, these states ultimately\nundergo a transformation into two independent $[1+1_5|0]$\nstates. As the bias decreases below $\\approx 0.1$, the\nvelocity $u$ becomes smaller than ${\\bar c}_{-}$ and the\nCherenkov radiation tails disappear. At this point, each of\nthe $[1+1_5|0]$ states smoothly transforms into two\nindependent $[1|0]$ states. The interaction between these\nstates is exponentially small, with a characteristic length\n$\\sim 1$ (or, $\\lambda_J$ in physical units). We note that\nthe interaction between kinks in the region $u>{\\bar c}_{-}\n$, where they have tails, also decreases exponentially, but\nwith a larger characteristic length $\\sim\\alpha^{-1}$.\n\nThe procedure of constructing higher-order bunched states\ncan be performed using {\\em different} states as ``building\nblocks''. In particular, we also tried to form the $[2+1|0]$ bunched\nstate. Note that if two different states are taken as\nbuilding blocks, we need to match their velocities, and,\nhence, the wave lengths of the tail. Thus, we have\nto combine two states at the same velocity, rather than at\nthe same bias current. Since different states have their own\nvelocity ranges, it is not always possible. As an example,\nwe have constructed a $[2+1|0]$ state out of $[2|0]$ state\nat $\\gamma=0.15$ and $[1|0]$ state at $\\gamma=0.45$ using\nan {\\it ansatz} similar to (\\ref{Eq:NewTrialFn[2|0]}). These states\nhave approximately the same velocity $u\\approx0.95$ (see\nto Fig.~\\ref{Fig:IVC[2|0]}). The constructed state was\nsimulated, starting from the points $\\gamma=0.3$ and $\\gamma=\n0.35$, tracing IVC up and down as before. Depending on the\nbias current the system ends up in different states, namely\nin the state $[1+1_1+1_2|0]$ for $\\gamma_0=0.3$, or in the\nstate $[1+1_1+1_1|0]=[3|0]$ for $\\gamma_0=0.35$. The IVCs of\nboth states are shown in Fig.~\\ref{Fig:IVC[2+2|0]}. The\nprofiles of the phase gradients are shown in\nFig.~\\ref{Fig:IVC[2+2|0]}(c).\n\nOur attempts to construct the states with a higher number of bunched fluxons, e.g., $[4+4|0]$, have failed since four fluxons do not fit into one well. We have concluded that such states immediately get converted into one of the lower-order states.\n\n\n\\subsection{Three coupled junctions}\n\nWe have performed numerical simulation of\nEqs.~(\\ref{Eq:Sym3:AC}) and (\\ref{Eq:Sym3:B}), using the\nsame technique as described in the previous section. Our\nintention here is to study the 3-junction case in which the\nfluxon is put in the middle junction ($[0|1|0]$ state). All\nother parameters were the same as in the case of the\ntwo-junction system, except for the ratio of the critical\ncurrents $J$, which was taken equal to one. This simplest\nchoice is made because in a system of $N>2$ coupled {\\em\nidentical} junctions the Cherenkov radiation appears in a\n$[0|\\ldots|0|1|0|\\ldots|0]$ state for $u>{\\bar c}_{-}\n\\approx0.765$ (this pertains to $S=-0.5$).\n\nFig.~\\ref{Fig:IVC[0|2|0]} shows the IVCs of the original\nstate $[0|1|0]$, as well as IVCs of the bunched state\n$[0|1+1|0]$ for $M=1,\\:2,\\:3$. The profiles of the phase\ngradients at points A through D are shown in\nFig.~\\ref{Fig:Profiles[0|2|0]}. Qualitatively, the bunching\nin the 3-fold system takes place in a similar fashion as\nthat in the 2-fold system. Nevertheless, we did not succeed\nin creating a stable fluxon configuration with $M=3$,\nalthough the stable states with other $M$ were obtained. We\nwould like to mention, that when the second fluxon was put\nin the second minimum of the potential to get the state with\n$M=2$, the state with $M=1$ has been finally\nestablished as a result of relaxation. The same behavior was observed\nwhen we put the fluxon initially in the third minimum, the\nsystem ended up in the state $[1+1_2|0]$. For $M\\ge4$, the\nbehavior was as usual. We tried to vary $\\Delta{}x$ smoothly, so\nthat the center of the trailing fluxon would correspond to\ndifferent positions between the second and fourth well, but\nin this case we did not succeed to get $[1+1_2|0]$ state.\n\nFollowing the same way as for two coupled junctions, we\ntried to construct $[0+1|1|0+1]$ states. As in the case $N=\n2$, these states were found unstable for any $M>0$, e.g.,\nthey would split\ninto $[0|1+1_2|0]$ and $[1|-1|1]$.\nThe state $[0|2+2|0]$ was not stable either for $M=1,\\:2,\\:3$ and the\nbias currents $\\gamma_0=0.20$, 0.30, 0.35.\n\nThe state $[0|2+1|0]=[0|3|0]$, constructed by combining the solutions for the $[0|1|0]$ and $[0|2|0]$ states moving with equal velocities was found to be stable when starting at $\\gamma=0.25$ and sweeping bias current up and down. The dependence $u(\\gamma)$ is shown in Fig.~\\ref{Fig:IVC[0|2|0]}. One may note, that for the states $[0|2|0]$ and $[0|3|0]$ the dependence is not smooth. Indeed, for these states the Cherenkov radiation tail is so long ($\\sim L$), that our annular system cannot simulate an infinitely long system, resulting in Cherenkov resonances which inevitably appear in the system with a finite perimeter\\cite{Cherry1,Cherry2}.\n\n\n\\section{Analysis and Discussion}\n\\label{Sec:Discussion}\n\nBecause of the non-linear nature of the bunching problem, it is\nhardly tractable analytically. Therefore, we here present an\napproach in which we analyze the asymptotic behavior of the\nfluxon's front and trailing tails in the linear\napproximation. This technique is similar to that employed\nin Ref.~\\onlinecite{Ustinov:BunchArray}. We assume that, at\ndistances which are large enough in comparison with the\nfluxon's size, the fluxon's profile is exponentially decaying,\n\\begin{equation}\n \\phi(x,t) \\propto \\exp[p(x-ut)]\n \\quad , \\label{Eq:tail}\n\\end{equation}\n%\nwhere $p$ is a complex number which can be found by\nsubstituting this expression into Eqs.~(\\ref{Eq:2:PDEa}) and\n(\\ref{Eq:2:PDEb}). As a result we arrive at an equation\n%\n\\begin{equation}\n \\left|\n \\begin{array}{cc}\n \\frac{p^2}{1-S^2}-p^2u^2-1-\\alpha{}pu & -\\frac{Sp^2}{1-\n S^2}\\\\\n -\\frac{Sp^2}{1-S^2} & \\frac{p^2}{1-S^2}-p^2u^2-\\frac{1}\n {J}-\\alpha{}pu\n \\end{array}\n \\right|=0\n \\quad , \\label{Eq:f(p)=0}\n\\end{equation}\n%\nIn general, this yields a 4-th order algebraic equation which\nalways has 4 roots. If we want to describe a soliton moving\nfrom left to right with a radiation tail behind it, we have\nto find the values $p$ among the four roots which adequately describe\nthe front and rear parts of the soliton. Because the front (right)\npart of the soliton is not oscillating, it is described by\nEq.~(\\ref{Eq:tail}) with real $p<0$. The rear (left) part of the\nsoliton is the oscillating tail, consequently it should be described\nby Eq.~(\\ref{Eq:tail}) with complex $p$ having ${\\rm Re}(p)>0$,\nthe period of oscillations being determined by the imaginary part of\n$p$. Analyzing the 4-th order equation, we conclude that the two \nnecessary types\nof the roots coexist only for $u>{\\bar c}_{-}$,\nwhich is quite an obvious result.\n\nTo analyze the possibility of bunched state formation, we\nconsider two fluxons situated at some distance from each\nother. We propose the following two conditions for the two\nfluxons to form a bunched state:\n%\n\\begin{enumerate}\n\n \\item Since non-oscillating tails result only in repulsion\n between fluxons, while the oscillating tail leads to\n mutual trapping, the condition\n %\n \\begin{equation}\n {\\rm{}Re}(p_l)<|p_r|\n \\quad , \\label{Eq:BunchCond1}\n \\end{equation}\n %\n can be imposed to secure bunching. Here $p_l$ is the root of\n Eq.~(\\ref{Eq:f(p)=0}) which describes the left (oscillating)\n tail of the leading (right) fluxon, and $p_r$ is the root\n of Eq.~(\\ref{Eq:f(p)=0}) which describes the right (non-oscillating) \n tail of the trailing (left) fluxon.\n\n \\item The relativistically contracted fluxon must fit into\n the minimum of the tail, i.e.,\n %\n \\begin{equation}\n \\frac{\\pi}{{\\rm Im}(p)}>\\sqrt{\\frac{u^2}{{\\bar c}_{-}\n ^2}-1}\n \\quad , \\label{Eq:BunchCond2}\n \\end{equation}\n %\n where $\\pi/{\\rm Im}(p)$ is half of the wavelength of the\n tail-forming\n radiation (the well's width), and the expression on\n the \\rhs{} of Eq. (\\ref{Eq:BunchCond2}) approximately\n corresponds to the contraction of the fluxon at the\n trans-Swihart velocities. Although our system is not Lorentz\n invariant, numerical simulations show that the fluxon\n indeed shrinks (not up to zero) when approaching the\n Swihart velocity ${\\bar c}_{-}$ from both sides.\n\n\\end{enumerate}\n%\nFollowing this approach, we have found that the second condition\n(\\ref{Eq:BunchCond2}) is always satisfied. The first\ncondition (\\ref{Eq:BunchCond1}) gives the following result.\nBunching is possible at $u>u_b>{\\bar c}_{-}$. The value of\n$u_b$ can be calculated numerically and for $S=-0.5$, $J=\n0.5$, $\\alpha=0.04$ it is $u_b=0.837$. Looking at\nFig.~\\ref{Fig:IVC[2|0]}, we see that this velocity\ncorresponds to the bias point where the $[1+1_M|0]$ states\ncease to exist. Thus, our crude approximation\nreasonably predicts the velocity range where the\nbunching is possible.\n\n\\section{Conclusion}\n\\label{Sec:Conclusion}\n\nIn this work we have shown by means of numerical simulations\nthat:\n%\n\\begin{itemize}\n\n \\item The emission of the Cherenkov plasma waves by a fluxon moving with high velocity creates an effective potential with many wells, where other fluxons can be trapped. This mechanism leads to bunching between fluxons of the {\\em same} polarity.\n\n \\item We have proved numerically that in the system of two and three coupled junctions the bunched states for the fluxons in the {\\em same} junction such as $[1+1|0]$, $[1+2|0]$, $[2+2|0]$, $[0|1+1|0]$ are stable. The states with fluxons in different junctions like $[1|0+1]$ and $[0+1|1|0+1]$ are numerically found unstable (except for the degenerated case $M=0$, when $[1|1]$ is a simple in- phase state).\n\n \\item Bunched fluxons propagate at a substantially higher velocity than the corresponding free ones at the same bias current, because of lower losses per fluxon.\n\n \\item When decreasing the bias current, transitions between the bunched states with different separations between fluxons were not found. This behavior differs from what is known for the bunched states in a discrete system\\cite{Ustinov:BunchArray}. In addition, a splitting of multi-fluxon states into the states with smaller numbers of bunched fluxons is observed.\n\n\\end{itemize}\n%\n\n\\acknowledgments\n\nThis work was supported by a grant no. G0464-247.07/95 from the German-Israeli Foundation.\n\n\n\\begin{thebibliography}{99}\n% RE to find author's initials:\n% \\([A-Z][a-z]?\\.\\)\\([~:spaces:]\\|\\\\ \\)\\{0,9\\}\n% \\([A-Z][a-z]?\\.\\)\\([~:spaces:]\\|\\\\ \\)\\{0,9\\}\n% \\([A-Z]\\)\n% not very smart!\n\n\\bibitem[*]{gold-mail}\n e-mail: e.goldobin@fz-juelich.de , homepage: http:\\BR{//}www\\BR{.}geocities\\BR{.}com\\BR{/}e\\_goldobin\n\n\\bibitem[\\dag]{Boris-mail}\n e-mail: malomed@eng.tau.ac.il\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Intrinsic}\n R.~Kleiner, F.~Steinmeyer, G.~Kunkel, and P.~M\\\"{u}ller,\n Phys. Rev. Lett. {\\bf 68}, 2394 (1992);\\\\\n %\n R.~Kleiner and P.~M\\\"{u}ller,\n Phys. Rev.~B {\\bf 49}, 1327 (1994).\n\n\\bibitem{Kl-Mue-Kolh:94}\n R.~Kleiner, P.~M\\\"{u}ller, H.~Kohlstedt,\n N.~F.~Pedersen, and S.~Sakai,\n Phys. Rev.~B {\\bf 73}, 3942 (1994).\n\n\\bibitem{Paul:AdvSSP34}\n P.~M\\\"uller,\n In: {\\it Festk{\\\"o}rperprobleme Advances in Solid State\n Physics},\n vol. {\\bf 34}, ed. by Helbig (Vieweg, Braunschweig), p.~1\n (1995).\n\n\\bibitem{pl:pap154}\n A.~V.~Ustinov.\n In: {\\it Physics and Materials Science of Vortex States,\n Flux Pinning and Dynamics}, NATO Science Series E, vol.\n {\\bf 356}, edited by R.~Kossowsky \\etal, Kluwer Acad.\n Publ. (1999), pp.~465--488.\n\n\\bibitem{image}\n Yu.~S.~Kivshar and B.~A.~Malomed, \n Phys. Rev. B {\\bf 37}, 9325 (1988).\n\n\\bibitem{Cherry1}\n E.~Goldobin, A.~Wallraff, N.~Thyssen, and A.~V.~Ustinov,\n Phys. Rev. B {\\bf 57}, 130 (1998).\n\n\\bibitem{Cherry2}\n E.~Goldobin, A.~Wallraff, and A.~V.~Ustinov,\n accepted to J. Low Temp. Phys. (Nov 1999).\n see also cond-mat/9910234\n\n\\bibitem{HexFisherCherry}\n G.~Hechtfischer, R.~Kleiner, A.~V.~Ustinov and\n P.~M\\\"uller,\n Phys. Rev. Lett. {\\bf 79}, 1365 (1997).\n\n\\bibitem{Ustinov:BunchArray}\n A.~V.~Ustinov, B.~A.~Malomed, and S.~Sakai,\n Phys. Rev. B {\\bf 57}, 11691 (1998).\n\n\\bibitem{Sakai:BunchBeta}\n S.~Sakai,\n Phys. Rev. B {\\bf 36}, 812 (1987).\n\n\\bibitem{Malomed:BunchBeta}\n B.~A.~Malomed,\n Phys. Rev. B {\\bf 47}, 1111 (1993).\n\n\\bibitem{Vernik:BunchBeta}\n I.~V.~Vernik, N.~Lazarides, M.~P.~S\\o{}rensen,\n A.~V.~Ustinov, N.~F.~Pedersen, and V.~A.~Oboznov,\n J.~Appl. Phys. {\\bf 79}, 7854 (1996).\n\n\\bibitem{SBP}\n S.~Sakai, P.~Bodin, and N.~F.~Pedersen.\n J.~Appl. Phys. {\\bf 73}, 2411 (1993).\n\n\\bibitem{LT21}\n E.~Goldobin, A.~Golubov, A.~V.~Ustinov,\n Czech. J. Phys. {\\bf 46}, 663 (1996), LT-21 Suppl. S2\n\n\\bibitem{StkJJ}\n E.~Goldobin, available online:\n http:\\BR{//}www\\BR{.}geocities\\BR{.}com\\BR{/}SiliconValley\\BR{/}Heights\\BR{/}7318\\BR{/}StkJJ.htm\n (1999).\n\n\\bibitem{GrE:Stability}\n N.~Gr{\\o}nbech-Jensen, D.~Cai and M.~R.~Samuelsen,\n Phys. Rev. B {\\bf 48}, 16160 (1993).\n\n\\end{thebibliography}\n\n\n\n%\\section{Figure Captions}\n%\\newpage\n\n\\begin{figure}\n \\caption{\n (a) The current-velocity characteristic $u(\\gamma)$ for\n the fluxon moving in the $[1|0]$ state (from left to\n right). (b) The profiles of the phase gradients\n $\\phi^{A,B}_x(x)$ in the state $[1|0]$ at $\\gamma=0.3$,\n corresponding to the bias point A shown in Fig.~(a). The\n Cherenkov tail, present at $u>{\\bar c}_{-} \\approx\n 0.817$, has a set of minima where the second fluxon can\n be trapped. (c) The profiles of $\\phi^{A,B}_x(x)$ in the\n state $[2|0]$ at the same value $\\gamma=0.3$ as (b). Two\n fluxons shown in Fig.~(c) are almost undistinguishable.}\n \\label{Fig:IVC+Profile[1|0]}\n\\end{figure}\n%\n\\begin{figure}\n \\caption{\n Current-velocity characteristics of different bunched\n states $[2|0]$: the second fluxon is trapped in the\n first minimum of the tail (state $[1+1_1|0]$), the second\n minimum (state $[1+1_2|0]$), and the third minimum (state\n $[1+1_3|0]$). The $\\gamma(u)$ curve for the $[1|0]$\n state is shown for comparison. The phase-gradient profiles \n corresponding to the bias points A through D are shown\n in Fig.~\\ref{Fig:Profiles[2|0]}.}\n \\label{Fig:IVC[2|0]}\n\\end{figure}\n%\n%\n\\begin{figure}\n \\caption{\n The profiles of the phase gradients $\\phi^{A,B}_x(x)$ in\n the $[2|0]$ states at the bias points A through D marked in\n Fig.~\\ref{Fig:IVC[2|0]}.\n }\n \\label{Fig:Profiles[2|0]}\n\\end{figure}\n%\n\\begin{figure}\n \\caption{\n (a) Current-velocity characteristics of the bunched\n states $[4|0]$, $[3|0]$, and $[2+1|0]$. Phase profiles\n of $[4|0]$ state and $[3|0]$ state at $\\gamma=0.3$ are\n shown in (b) and (c), respectively.\n }\n \\label{Fig:IVC[2+2|0]}\n\\end{figure}\n%\n%N=3\n%\n\\begin{figure}\n \\caption{\n Current-velocity characteristics of the state $[0|1|0]$,\n bunched state $[0|1+1_M|0]$ for three different cases,\n $M=1,2,3$, and the state $[0|3|0]$. The profiles of\n the Josephson phase gradients at the points A through D are\n shown in Fig.~\\ref{Fig:Profiles[0|2|0]}\n }\n \\label{Fig:IVC[0|2|0]}\n\\end{figure}\n%\n%\n\\begin{figure}\n \\caption{\n The profiles of the Josephson phase gradients\n $\\phi^{A,B}_x(x)$ in $[0|1+1_M|0]$ states at the\n points A through D marked in Fig.~\\ref{Fig:IVC[0|2|0]}.\n }\n \\label{Fig:Profiles[0|2|0]}\n\\end{figure}\n%\n\n\\end{document}\n\n"
}
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[
{
"name": "cond-mat0002051.extracted_bib",
"string": "\\begin{thebibliography}{99}\n% RE to find author's initials:\n% \\([A-Z][a-z]?\\.\\)\\([~:spaces:]\\|\\\\ \\)\\{0,9\\}\n% \\([A-Z][a-z]?\\.\\)\\([~:spaces:]\\|\\\\ \\)\\{0,9\\}\n% \\([A-Z]\\)\n% not very smart!\n\n\\bibitem[*]{gold-mail}\n e-mail: e.goldobin@fz-juelich.de , homepage: http:\\BR{//}www\\BR{.}geocities\\BR{.}com\\BR{/}e\\_goldobin\n\n\\bibitem[\\dag]{Boris-mail}\n e-mail: malomed@eng.tau.ac.il\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Intrinsic}\n R.~Kleiner, F.~Steinmeyer, G.~Kunkel, and P.~M\\\"{u}ller,\n Phys. Rev. Lett. {\\bf 68}, 2394 (1992);\\\\\n %\n R.~Kleiner and P.~M\\\"{u}ller,\n Phys. Rev.~B {\\bf 49}, 1327 (1994).\n\n\\bibitem{Kl-Mue-Kolh:94}\n R.~Kleiner, P.~M\\\"{u}ller, H.~Kohlstedt,\n N.~F.~Pedersen, and S.~Sakai,\n Phys. Rev.~B {\\bf 73}, 3942 (1994).\n\n\\bibitem{Paul:AdvSSP34}\n P.~M\\\"uller,\n In: {\\it Festk{\\\"o}rperprobleme Advances in Solid State\n Physics},\n vol. {\\bf 34}, ed. by Helbig (Vieweg, Braunschweig), p.~1\n (1995).\n\n\\bibitem{pl:pap154}\n A.~V.~Ustinov.\n In: {\\it Physics and Materials Science of Vortex States,\n Flux Pinning and Dynamics}, NATO Science Series E, vol.\n {\\bf 356}, edited by R.~Kossowsky \\etal, Kluwer Acad.\n Publ. (1999), pp.~465--488.\n\n\\bibitem{image}\n Yu.~S.~Kivshar and B.~A.~Malomed, \n Phys. Rev. B {\\bf 37}, 9325 (1988).\n\n\\bibitem{Cherry1}\n E.~Goldobin, A.~Wallraff, N.~Thyssen, and A.~V.~Ustinov,\n Phys. Rev. B {\\bf 57}, 130 (1998).\n\n\\bibitem{Cherry2}\n E.~Goldobin, A.~Wallraff, and A.~V.~Ustinov,\n accepted to J. Low Temp. Phys. (Nov 1999).\n see also cond-mat/9910234\n\n\\bibitem{HexFisherCherry}\n G.~Hechtfischer, R.~Kleiner, A.~V.~Ustinov and\n P.~M\\\"uller,\n Phys. Rev. Lett. {\\bf 79}, 1365 (1997).\n\n\\bibitem{Ustinov:BunchArray}\n A.~V.~Ustinov, B.~A.~Malomed, and S.~Sakai,\n Phys. Rev. B {\\bf 57}, 11691 (1998).\n\n\\bibitem{Sakai:BunchBeta}\n S.~Sakai,\n Phys. Rev. B {\\bf 36}, 812 (1987).\n\n\\bibitem{Malomed:BunchBeta}\n B.~A.~Malomed,\n Phys. Rev. B {\\bf 47}, 1111 (1993).\n\n\\bibitem{Vernik:BunchBeta}\n I.~V.~Vernik, N.~Lazarides, M.~P.~S\\o{}rensen,\n A.~V.~Ustinov, N.~F.~Pedersen, and V.~A.~Oboznov,\n J.~Appl. Phys. {\\bf 79}, 7854 (1996).\n\n\\bibitem{SBP}\n S.~Sakai, P.~Bodin, and N.~F.~Pedersen.\n J.~Appl. Phys. {\\bf 73}, 2411 (1993).\n\n\\bibitem{LT21}\n E.~Goldobin, A.~Golubov, A.~V.~Ustinov,\n Czech. J. Phys. {\\bf 46}, 663 (1996), LT-21 Suppl. S2\n\n\\bibitem{StkJJ}\n E.~Goldobin, available online:\n http:\\BR{//}www\\BR{.}geocities\\BR{.}com\\BR{/}SiliconValley\\BR{/}Heights\\BR{/}7318\\BR{/}StkJJ.htm\n (1999).\n\n\\bibitem{GrE:Stability}\n N.~Gr{\\o}nbech-Jensen, D.~Cai and M.~R.~Samuelsen,\n Phys. Rev. B {\\bf 48}, 16160 (1993).\n\n\\end{thebibliography}"
}
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cond-mat0002052
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Dielectric response of cylindrical nanostructures in a\\ magnetic field
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[
{
"author": "Peter Fulde and Alexander Ovchinnikov"
}
] |
minipage
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[
{
"name": "reprDROCNIAMF.tex",
"string": "\\documentstyle[pre,aps,twocolumn,floats]{revtex}\n\n%\\addtolength{\\textheight}{-1cm}\n\\begin{document}\n\\title{Dielectric response of cylindrical nanostructures in a\\\\ magnetic field}\n\n\\author{Peter Fulde and Alexander Ovchinnikov} \n\n\\address{\nMax-Planck-Institut f\\\"ur Physik komplexer Systeme,\nN\\\"othnitzer Stra\\ss e 38,\n01187 Dresden (Germany)}\n\n\\date{\\today}\n\\maketitle\n\n\\abstract\n\nWe study the magnetic field dependence of the dielectric response of large\ncylindrical molecules such as nanotubes. When a field-induced level crossing\ntakes place, an applied electric field causes a linear instead of the usual\nquadratic Stark effect. This results in a large dielectric response. Explicit\ncalculations are performed for doped nanotubes and a rich structure in the real\npart of the low-frequency dielectric function $\\epsilon'(H)$ is found when a\nmagnetic field is applied along the cylinder axis. It is suggested that studies\nof $\\epsilon'(H, T)$ can serve as a spectroscopic tool for the investigation of\nlarge ring-shaped or cylindrical molecules.\\\\[8ex]\n\n\n\\noindent PACS.~~~~~~~~\\begin{minipage}[t]{15cm}\n77.22.-d Dielectric properties of solids and liquids\\\\\n78.40.Ri Fullerenes and related materials\\\\\n75.20.-g Diamagnetism and paramagnetism\\\\\n73.23.-b Mesoscopic systems\n\\end{minipage}\n\n\\vspace{1.5cm}\n\nDedicated to K. Biedenkopf on the occasion of his 70th birthday.\n\n\\newpage\n\n\\newcommand{\\gsim}{\\mathrel{\\raise.3ex\\hbox{$>$\\kern-.75em\\lower1ex\\hbox{$\\sim$}}}}\n\\newcommand{\\lsim}{\\mathrel{\\raise.3ex\\hbox{$<$\\kern-.75em\\lower1ex\\hbox{$\\sim$}}}}\n\n{\\section{\\bf Introduction}}\n\nDuring the last years considerable progress has been made in precise\nmeasurements of the real part of the low-frequency dielectric function\n$\\epsilon'(\\omega)$. In particular those measurements could be extended to\nultra-low temperatures, i.e., down to a few mK. Ratios of $\\delta\n\\epsilon'/\\epsilon'$ up to $10^{-7}$ were achieved in that temperature\nregime. A rather spectacular success associated with that progress was the\nobservation of a strong magnetic-field dependence of the polarizability of\nmulticomponent glasses in the mK regime \\cite{ref1,ref2}. This development\nsuggests reconsideration of magnetic field effects on ring molecules or related\nstructures like nanotubes. In both cases an applied magnetic field induces a\ndiamagnetic ring current. Due to this current the energy of the ground state\nincreases quadratically with the applied magnetic field. This continues until\none of the excited states which lowers its energy in a field crosses the\nground-state and becomes the new ground state. At the crossing point an applied\nelectric field causes a linear Stark effect, instead of the usual quadratic one\nand hence a divergent electric polarizability. The physical origin of the\ncrossover is easily understood. An excited state carrying a ring current in the\nabsence of a magnetic field becomes a state without a ring current, when a\nsufficiently high magnetic field is applied, because the induced current may\ncancel the original one. When this is the case the energy of that state equals\nthe one of the ground state in the absence of a field. This simple argument\nshows that the ground-state energy is a periodic function of an applied\nmagnetic field. The periodicity is given by the flux enclosed by the ring\ncurrent. When one flux quantum $\\phi_0 = hc/e$ is penetrating the ring the\nground-state energy has returned to its original value. It is well known that a\nhuge field of order $10 ^5~ T$ is needed for a flux quantum $\\phi_0$ to\npenetrate a benzene molecule consisting of a ring of six carbon atoms. Because\nof that large value of the magnetic field, effects of it on ring molecules have\nobtained only little attention in the past \\cite{ref3,ref4}. The purpose of the\npresent paper is to point out that the situation has changed considerably. Not\nonly has there been experimental progress in performing high precision\nmeasurements of $\\epsilon'$ at low temperatures, but also the synthesis of\norganic ring structures has made impressive advances. For example, nanotubes of\nlarge circumference have been produced, lowering the field required for the\nenclosure of a flux unit. In this paper we want to demonstrate that a\nmeasurement of $\\epsilon'(H, T)$ should provide important information on the\nelectronic excitations of ring molecules, in particular on level crossings. As\na first step, we calculate here the dielectric response of a molecule\nconsisting of a square lattice, e.g., of carbon sites bent into a cylindrical\nform and of a nanotube, i.e., a bent honeycomb lattice. Various extensions of\nthe work presented here will follow later. \n\nThe magnetic and electric field are assumed\nto be directed along the cylindrical axis. The electron interactions are\nassumed to be included in effective one-electron parameters like in an extended\nH�ckel theory or in the quasiparticle theory of Landau. In a subsequent\ninvestigation we shall include the electronic interactions more explicitely\nthan done here. This may have profound effects on the results. A rich structure\nin $\\epsilon'(H, T)$ is obtained which should be experimentally observable. It\nis closely related to the low-energy excitations of the systems in an applied\nmagnetic field and in particular to level crossings as the field\nchanges. Although our findings are limited here to the cylindrical structures\ndescribed above, they suggest detailed experimental studies of $\\epsilon'(H,\nT)$ for ring molecules. This seems to us a largely untouched field of research.\n\nIn order to explain the main features of $\\epsilon'(H, T)$ we consider first a\nsingle ring of $N$ sites in a magnetic field along the ring axis. The\nHamiltonian is\n\n\\begin{equation}\n\\label{1} H = t \\sum_{n, \\sigma} (a^+_{(n+1)\\sigma}~ a_{n \\sigma}~ e^\\frac{2\n\\pi i \\phi}{N} + {\\rm h.c.})\n\\end{equation}\n\n\\noindent where $a^+_{n \\sigma}, a_{n \\sigma}$ are electron creation and\nannihilation operators and $\\phi$ is the magnetic flux through the ring in\nunits of the flux quantum $\\phi_0$. The resulting energy eigenvalues are\n\n\\begin{equation}\n\\label{2} \\epsilon (q) = 2t~ {\\rm cos} \\left[ \\frac{2 \\pi}{N} (q +\n\\phi) \\right]~~,~~~~~~~~~~~q = 0, \\pm 1, \\pm 2, ... \n\\end{equation}\n\nThe ground-state energy $E_g$ is periodic in the flux, i.e., $E_g (\\phi + 1) =\nE_g (\\phi)$. More explicitely we write for $E_g$\n\n\\begin{eqnarray}\n\\label{3} E_g (\\phi) & = & \\sum_{\\rm occ} 2t~ \\left[ {\\rm cos} \\left( \\frac{2\n\\pi}{N} q \\right)~ {\\rm cos} \\left( \\frac{2 \\pi}{N} \\phi\n\\right)\\right.\\nonumber \\\\ \n&&\\left. - {\\rm sin}\n\\left( \\frac{2 \\pi}{N} q \\right)~ {\\rm sin} \\left( \\frac{2 \\pi}{N} \\phi\n\\right)\\right]~~. \n\\end{eqnarray}\n \nFor a closed-shell system, i.e., for an electron number $N_e = 4n + 2$, where\n$n$ is an integer we find that\n\n\\begin{equation}\n\\label{4} \\sum_{\\rm occ} {\\rm sin} \\frac{2 \\pi}{N} q = 0~~.\n\\end{equation}\n \n\\noindent In that case the field-dependent contribution to the ground-state\nenergy is \n\n\\begin{eqnarray}\n\\label{5} \\delta E_g (\\phi) & = & \\sum_{\\rm occ} 2t~ {\\rm cos} \\left( \\frac{2\n\\pi}{N} q \\right) \\left( {\\rm cos} \\left( \\frac{2 \\pi}{N} \\phi \\right) -1\n\\right)\\nonumber \\\\ \n& = & -E_g (0) \\left( 1 - {\\rm cos} \\left( \\frac{2 \\pi}{N} \\phi \\right)\n\\right)~~. \n\\end{eqnarray}\n \nFor large $N$ we may expand this expression and obtain\n\n\\begin{equation}\n\\label{6} \\delta E_g (\\phi) = -E_g (0)~ \\frac{2 \\pi^2}{N^2}~ \\phi^2 > 0\n\\end{equation}\n \n\\noindent for $\\phi \\leq \\frac{1}{2}$. When $\\phi = \\frac{1}{2}$ the ground\nstate is twofold degenerate because of a level crossing at that point. For\n$\\phi > \\frac{1}{2}$, the expression (3) is replaced by\n\n\\begin{equation}\n\\label{7} \\delta E_g (\\phi) = -E_g (0)~ \\frac{2 \\pi^2}{N^2}~ (1 -\n\\phi)^2~~,~~~~~~~~~\\phi > \\frac{1}{2}~~.\n\\end{equation}\n \nThe behaviour of $\\delta E_g (\\phi)$ is schematically shown in Fig. 1. The\ncontribution of $\\delta E_g (\\phi)$ to $E_g (\\phi)$ is very small for large\nvalues of $N$ and hardly detectable. This does not hold true though for other\nquantities. For example, when in addition an electric field is applied\nperpendicularly to the ring axis its effect on the ground-state energy is\nstrongly dependent on $\\phi$. For $\\phi = \\frac{1}{2}$ we are dealing with a\nlinear Stark effect instead off a quadratic one when $\\phi \\neq\n\\frac{1}{2}$. Therefore the dielectric constant has a singularity at that\nparticular value of $\\phi$. This simple example sheds light onto the physical\nreason why the dielectric function can be so sensitive to an applied magnetic\nfield. The same feature is found for cylindrical molecules which are subject of\nthis paper. \n\n%z.B. \\cite{ref1,ref2}... functionals were introduced \\cite{ref3,ref4}\n\n\\vspace{1cm}\n\n{\\section{\\bf Magnetic field dependence of the free energy}}\n\nIn order to demonstrate the influence of an applied magnetic field $\\bf{H}$\non the free energy we consider two different systems. One is a model square\nlattice rolled into the form of a cylinder. The other one is a nanotube which\nconsists of a honeycomb lattice rolled into a cylinder in the same way.\n\nWe start with the square lattice forming a cylinder. It consists of $N$ atoms\nalong the perimeter and of $N$ atoms along the cylindrical axis z. The\neigenvalues depend on the flux $\\phi$ though the cylinder and are of the form\n\n\\begin{eqnarray}\n\\label{8} \\epsilon_{pq\\sigma} & = & -{\\rm cos} \\left[ \\frac{2 \\pi}{N}~ (p +\n\\phi)\\right] -t{\\rm cos} \\left[ \\frac{2 \\pi}{M + 1}~q \\right]\\nonumber \\\\\n&& + 2 \\pi^2~\n\\frac{m_{\\rm eff}\\sigma}{m}~ \\frac{\\phi}{N^2}~~. \n\\end{eqnarray} \n\nThe first term corresponds to a transfer integral of size $-\\frac{1}{2}$ along\nthe perimeter and the second to one of magnitude $-\\frac{t}{2}$ along the z\naxis. The parameters $p$ and $q$ take the integer values $p = 1, ..., N$ and\n$q = 1, ..., M$, respectively. The last term is the Zeeman contribution which\nis expressed here in terms of the flux $\\phi$. Since the latter is in units\nof the flux quantum $\\phi_0$, the ratio of the effective mass\n$m_{\\rm eff}$ divided by the electron mass $m$ enters here, with the former\nreferring to an electronic motion perpendicular to the z axis. Furthermore,\n$\\sigma = \\pm 1$. The free energy of the system is of the usual form\n\n\\begin{equation}\n\\label{9} \\beta F = - \\sum_{p q \\sigma}~ {\\rm ln} [1 + {\\rm exp} (-\\beta\n(\\epsilon_{pq\\sigma} - \\mu))] \n\\end{equation}\n\n\\noindent where $\\beta = (k_B T)^{-1}$ and $\\mu$ is the chemical potential. It\nis determined by expressing the number of electrons $N_e$ in terms of it, i.e.,\n\n\\begin{equation}\n\\label{10} N_e = \\sum_{pq\\sigma} \\frac{1}{{\\rm exp}[\\beta(\\epsilon_{pq\\sigma} -\n\\mu)]+ 1}~~. \n\\end{equation}\n\nIn practice we calculate $\\mu$ by first choosing an approximate value $\\mu_0$\nand calculating the corresponding value $N_e^{(1)}$. The correction\n$\\delta\\mu_0$ to $\\mu_0$ can then be determined from\n\n\\begin{equation}\n\\label{11} \\delta \\mu_0 = -\\frac{1}{\\beta} {\\rm ln} \\left( 1 + \\frac{N_e^{(1)} -\nN_e}{a(T)}\\right)~~. \n\\end{equation}\n\nThis expression is more convenient for numerical calculations than its\nlinearized version in $(N_e^{(1)} - N_e)/a(T)$ where the function $a(T)$ is\ngiven by \n\n\\begin{equation}\n\\label{12} a(T) = \\frac{1}{4} \\sum_{pq\\sigma} {\\rm cos}h^{-2}~ \\left[\\frac{\\beta}{2}\n(\\epsilon_{pq\\sigma} - \\mu_0)\\right]~~. \n\\end{equation}\n\nOne can use the corrected potential $\\mu_1 = \\mu_0 + \\delta\\mu_0$ in order to\ncalculate the next correction $\\delta\\mu_1$. We obtain with $\\delta\\mu_1$ the\nchemical potential already with an accuracy of order $N^{-2} M^{-2}$, which\nperfectly serves our purposes. \n\nThe same procedure can be applied to carbon nanotubes. In that case the unit\ncell contains four carbon atoms. Hence the excitation energies form four bands,\ni.e.,\n\n\\begin{eqnarray}\n\\label{13} \\epsilon_{pq\\sigma} (\\phi) & = & \\pm \\left( 1 + u_p \\pm (1 + u_p\nv_q)^\\frac{1}{2} \\right)^\\frac{1}{2}\\nonumber \\\\\n&& +~ 2 \\pi^2 \\frac{m_{\\rm eff}}{m}~ \\frac{\\sigma\\phi}{N^2} \n\\end{eqnarray}\n\n\\noindent with\n\n\\begin{eqnarray}\n\\label{14} u_p & = & 2 \\left(1 + {\\rm cos} \\left[ \\frac{2\\pi}{N} (p +\n\\phi)\\right] \\right)\\nonumber \\\\ \nv_q & = & 2 \\left(1 + {\\rm cos} \\left[ \\frac{2\\pi}{M+1} q\\right] \\right) \n\\end{eqnarray}\n\n\\noindent and $p = 1, ....., N;~~q = 1, ....., M$ \\cite{ref5,ref6}.\n\n\\vspace{1cm} \n\n{\\section{\\bf Induced dipole moment}}\n\nWhen an electric field is applied along the $z$ axis the excitation spectrum of\nthe system can no longer be calculated exactly. Instead, approximations have to\nbe made. We cannot apply linear response theory because of a linear Stark\neffect at level crossings. Since in practise the applied electric field is very\nsmall, the density of electrons changes only slightly along the z axis. This\nenables us to determine the induced density changes by using a quasiclassical\napproximation. Within that scheme the excitation energies depend not only on\n$p, q$ and $\\sigma$ but on the coordinate z as well. We illustrate the\napproximation by considering a chain of $M$ atoms as a simple example. The\nHamiltonians is of the form\n\n\\begin{eqnarray}\n\\label{15} H_{\\rm 1d} & = & - \\sum^{M-1}_{n=1, \\sigma} (a^+_{(n+1)\\sigma}\na_{n\\sigma} + {\\rm h.c.})\\nonumber \\\\\n&& +~ e a_0 F_0~\\sum_{n\\sigma} a^+_{n\\sigma} a_{n\\sigma} \\left( n-\\frac{M+1}{2}\n\\right)~~. \n\\end{eqnarray} \n\nHere $a_0$ is the lattice constant and $F_0$ is an applied electric field along\nthe chain direction.\n\nExact calculations of the induced dipole moment D require an evaluation of\nthe expression\n\n\\begin{eqnarray}\n\\label{16} {\\rm D} & = & \\frac{Sp~ \\hat{d}~ e^{-\\beta(H_{\\rm 1d} - \\mu)}}{Sp~\ne^{-\\beta(H_{\\rm 1d} - \\mu)}}\\nonumber \\\\\n& = & \\sum^M_{k=1,\\sigma} \\frac{(\\hat{d})_{kk}}{1 + e^{\\beta(E_{k\\sigma} -\n\\mu)}}~~. \n\\end{eqnarray}\n\n\\noindent Here $\\hat{d}$ is dipole operator\n\n\\begin{equation}\n\\label{17} \\hat{d} = e a_0 \\sum^M_{n=1,\\sigma} a^+_{n\\sigma} a_{n\\sigma}\n\\left( n - \\frac{M+1}{2} \\right) \n\\end{equation} \n\n\\noindent and $E_{k\\sigma}$ denotes the excitation energies of the chain. In\norder to compute D from (\\ref{16}) we have to diagonalize $H_{\\rm 1d}$ in\norder to find the eigenenergies and eigenfunctions of that Hamiltonian. This\ncan be done if not more than 1000 atoms are involved. Instead of doing that\nwe want to use here a simpler, more effective quasiclassical scheme. In the\nquasiclassical approximation the excitation spectrum is of the form\n\n\\begin{equation}\n\\label{18} \\epsilon_{pm} = - 2 {\\rm cos} \\left[ \\frac{2\\pi}{M+1}~ p\\right] + e\na_0 F_0 \\left( m - \\frac{M+1}{2} \\right) \n\\end{equation} \n\n\\noindent with $p = 1, ....., M$ and $m = 1, ....., M$. The corresponding\nexpression for the induced dipole moment is\n\n\\begin{equation}\n\\label{19} {\\rm D} = \\frac{2e a_0}{M} \\sum_{pm} \\frac{(m -\n(M+1)/2)}{e^{\\beta(\\epsilon_{pm} - \\mu)} + 1}~~. \n\\end{equation} \n\nWe have calculated D for a chain of $M = 201$ atoms by using (\\ref{16}) and\nalternatively (\\ref{18}). The results are compared in Fig. 2 for different\ndensities and temperatures. The deviations caused by the semiclassical\napproximation are less than $1 \\%$ or $\\frac{1}{M}$ in all cases. This\njustifies the use of a quasiclassical approximation when we calculate the\ndielectric response of cylindrical molecules such as nanotubes in an applied\nmagnetic field. \n\n\\vspace{1cm} \n\n{\\section{\\bf Results and discussions}}\n\nIn the following we want to present results for the dielectric response of the\ntwo types of cylindrical molecules described above, i.e., for a square lattice\nrolled into a cylinder and for nanotubes. The induced dipole moment is\ncalculated in close analogy to the one of a ring, although here the electric\nfield $F_0$ is directed along the cylindrical axis. We start with the\nsquare-lattice case. In analogy to (\\ref{19}) the induced dipole is calculated\nfrom\n\n\\begin{equation}\n\\label{20} {\\rm D} (\\phi) = \\frac{e~ a_0}{M} \\sum_{mpq\\sigma}~ \\frac{(m -\n(M+1)/2)}{e^{\\beta [\\tilde{\\epsilon}_{pq\\sigma} (m, \\phi) - \\mu(\\phi)]} + 1}\n\\end{equation} \n\n\\noindent where\n\n\\begin{equation}\n\\label{21} \\tilde{\\epsilon}_{pq\\sigma} (m, \\phi) = \\epsilon_{pq\\sigma} (\\phi) +\ne a_0 F_0~ (m - (M+1)/2) \n\\end{equation} \n\nand $\\epsilon_{pq\\sigma}(\\phi)$ is given by (\\ref{8}). Results for the\nmagnetic-field dependent part ${\\rm D}(\\phi) - {\\rm D}(0)$ are shown in Fig. 3\nfor a cylinder with 100 atoms along the circumference and 1000 atoms along the\naxis, i.e., $N = 100$ and $M = 1000$, respectively. Note that $({\\rm D}(\\phi)\n- {\\rm D}(0))/{\\rm D}(0) = (\\epsilon'(\\phi) - \\epsilon'(0))/ \\epsilon'(0)$\nwhere $\\epsilon'$ is the real part of the dielectric response in the\nlow-frequency limit. The temperature, or more precisely $k_B T$ is $10^{-4}$ in\nunits of the hopping matrix element. The chosen density corresponds to 0,74\nelectrons per site. One notices a rich structure as a function of the applied\nmagnetic field. \n\nA cylinder formed from a square lattice is a hypothetical case. But if one\nassumes a lattice constant $a_0 = 1.40 {\\AA}$ as in the case of an aromatic\ncarbon ring, the field required for the enclosure of a flux unit is of order\n$260 T$. It is derived from the following relation between the flux $\\phi$ (in\nunits of $\\phi_0$) and the applied magnetic field\n\n\\begin{equation}\n\\label{22} \\phi = \\frac{N^2 a^2_0~ eH}{8\\pi^2 \\hbar c}~~. \n\\end{equation} \n\nThe structure in $\\epsilon'(H)$ obtained within that range of fields reflects\nproperties of excited states, in particular crossings of energy levels.\n\nFor nanotubes the calculations are done quite similarly, but here we have to\ntake a sum over all four energy bands. The computational results are shown in\nFig. 4 for a density of $n = 0.89$ $\\pi$-electrons per site. One notices that\nthe rich structure in $\\delta\\epsilon'(H)/\\epsilon'(H)$ in the regime $0<H<40T$\nis of order unity and therefore should be easily detectable. Results for\nother densities look similar, except for $n = 1$ which is special. The reason\nis that a honeycomb or graphite lattice has for $n = 1$ a Fermi surface\nconsisting of a point. Therefore in a a finite system the level spacing close\nto the Fermi energy is particularly large. This leads to small changes in\n$\\delta\\epsilon'(H)/\\epsilon'(0)$ only. The situation changes at high magnetic\nfields. Due to the Zeeman term in the Hamiltonian the spin dependent densities\n$n_\\sigma$ differ more and more from each other, i.e., $n = n_\\uparrow+\nn_\\downarrow$ with $n_\\uparrow (H) \\neq n_\\downarrow (H)$ and the Fermi\nsurface moves away from the special point at half filling. This brings us back\nto the doped case and we obtain again a rich structure in\n$\\delta\\epsilon'(H)/\\epsilon'(0)$ like in Fig. 4. \n\n\\vspace{1cm}\n\n{\\section{\\bf Conclusions}}\n\nThe above calculations show that large molecules of cylindrical or circular\nshape should show detectable magnetic field effects due to the Bohm-Aharonov\neffect. They lead to a strong variation of the dielectric function in the\nlow-frequency limit as function of the applied magnetic field. Those variations\nare predominantly caused by doubly degenerate ground states resulting from\nlevel crossings in the applied field. At a crossing point an applied electric\nfield causes a linear Stark effect instead of a quadratic one when the ground\nstate is nondegenerate. The present investigation requires a number of\nextensions which will be the subject of separate investigations. One concerns\nthe dependence of $\\delta\\epsilon'(H)$ on the directions of the applied\nmagnetic and electric fields. Important is also a proper inclusion of electron\ncorrelations. As pointed out before, the present calculations have been done\nwithin the one-electron approximation. But correlations, in particular when\nthey are strong will clearly result in important modifications of the\ndielectric response \\cite{ref7,ref8}. Finally, we also have to generalize the\nabove theory to the case of mutually interacting molecules. This may become an\nimportant issue when platelike molecules are forming stacks and a magnetic\nfield is applied along the direction of the stack. Although the work presented\nhere needs extensions of the form just described it is fair to state that\nthe results presented here justify efforts towards a systematic investigation\nof the magnetic-field dependent dielectric response of ring- or cylinder\nshaped molecules. We feel that in the future they may develop into a\nspectroscopic tool for studying low-energy excitations of such systems.\n\n\\begin{thebibliography}{99}\n\\vspace{0.5cm}\n\\bibitem{ref1} P. Strehlow, C. Enss, and S. Hunklinger, Phys. Rev. Lett. {\\bf\n80}, 5361 (1998)\n\\bibitem{ref2} P. Strehlow, M. Wohlfahrt, A. G. M. Jansen, R. Haueisen,\nG. Weiss, C. Enss, and S. Hunklinger, preprint \n\\bibitem{ref3} {\\it Electronic Structure Calculations on Fullerenes and their\nDerivatives}, ed. by J. Cioslowski (Oxford University Press, New York, Oxford\n1995), p. 139 \n\\bibitem{ref4} N. Hamada, S. Sawada, and A. Oshiyama, Phys. Rev. Lett. {\\bf\n68}, 1579 (1992) \n\\bibitem{ref5} K. Tanaka, K. Okahara, M. Okada, and T. Yamabe,\nChem. Phys. Lett. {\\bf 191}, 469 (1992)\n\\bibitem{ref6} A. A. Ovchinnikov, Phys. Lett. {\\bf A 195}, 95 (1994)\n\\bibitem{ref7} F. V. Kusmartsev, Phys. Lett. {\\bf A 161}, 433 (1992)\n\\bibitem{ref8} B. Sutherland, and B. S. Shastry, Phys. Rev. Lett. {\\bf 65},\n1833 (1990) \n\\end{thebibliography}\n\n\\newpage\n\n\\noindent \\hspace{4cm} FIGURE CAPTIONS\\\\[-2ex]\n\n\\begin{footnotesize}\n\\newcounter{fig}\n\\begin{list}{Fig. \\arabic{fig}:}{\\usecounter{fig}\n \\setlength{\\labelwidth}{1.6cm} \\setlength{\\leftmargin}{1.8cm}\n \\setlength{\\labelsep}{0.4cm} \\setlength{\\rightmargin}{0cm}\n \\setlength{\\parsep}{0.5ex plus0.2ex minus0.1ex}\n \\setlength{\\itemsep}{0ex plus0.2ex}}\n\\item Schematic plot of $\\delta E_g(\\phi)$ {\\it vs.} $\\phi$ (thick solid\nline). The level crossings are due to parabolas describing different excited\nstates $\\epsilon_i(\\phi)$ shifted by flux units. When levels cross, an applied\nelectric field causes a linear Stark effect and hence induces a divergent\ndielectric response. \n\\item Induced dipole moment as a function of electrons per site for a chain of\n$M = 201$ atoms calculated with the exact quantum-mechanical expression\n(\\ref{16}) (dashed lines) and when a semiclassical approximation (\\ref{19})\n(solid lines) is made. (a) and (b) correspond to temperatures $k_B T = 0.01$\nand $0.05$, respectively (in units of the transfer integral). \n\\item Dielectric response $[\\epsilon'(H) - \\epsilon'(0)]/\\epsilon'(0)$ for a\nmodel square-lattice system with $N=200, M=1000$ in an axial magnetic\nfield. The temperature is $k_BT = 10^{-4}$ (in units of the transfer integral),\nand $a_0 = 1.4 {\\AA}$. The density is 0.74 electrons per site.\n\\item Dielectric response $[\\epsilon'(H) - \\epsilon'(0)]/ \\epsilon'(0)$ for a\nnanotube with $N=100, M=1000$ in an axial magnetic field. The temperature is\n$k_B T= 10^{-4}$ in units of 3 eV and the density is 0.89 electrons per site.\n\\end{list}\n\\end{footnotesize}\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002052.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\\vspace{0.5cm}\n\\bibitem{ref1} P. Strehlow, C. Enss, and S. Hunklinger, Phys. Rev. Lett. {\\bf\n80}, 5361 (1998)\n\\bibitem{ref2} P. Strehlow, M. Wohlfahrt, A. G. M. Jansen, R. Haueisen,\nG. Weiss, C. Enss, and S. Hunklinger, preprint \n\\bibitem{ref3} {\\it Electronic Structure Calculations on Fullerenes and their\nDerivatives}, ed. by J. Cioslowski (Oxford University Press, New York, Oxford\n1995), p. 139 \n\\bibitem{ref4} N. Hamada, S. Sawada, and A. Oshiyama, Phys. Rev. Lett. {\\bf\n68}, 1579 (1992) \n\\bibitem{ref5} K. Tanaka, K. Okahara, M. Okada, and T. Yamabe,\nChem. Phys. Lett. {\\bf 191}, 469 (1992)\n\\bibitem{ref6} A. A. Ovchinnikov, Phys. Lett. {\\bf A 195}, 95 (1994)\n\\bibitem{ref7} F. V. Kusmartsev, Phys. Lett. {\\bf A 161}, 433 (1992)\n\\bibitem{ref8} B. Sutherland, and B. S. Shastry, Phys. Rev. Lett. {\\bf 65},\n1833 (1990) \n\\end{thebibliography}"
}
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cond-mat0002053
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Quantum-Critical Dynamics of the Skyrmion Lattice.
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[
{
"author": "A. G. Green"
}
] |
%Near to filling fraction $\nu=1$, the quantum Hall %ferromagnet contains multiple Skyrmion spin excitations. This multi-Skyrmion %system has a tremendously rich quantum-critical %structure. This is simplified %when Skyrmions are pinned by disorder. We calculate %the nuclear relaxation rate in this case and compare the result with experimen%t. %We discus how such measurements may be used to further probe the quantum-criti%cal structure of the multi-Skyrmion system. %
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[
{
"name": "paper.tex",
"string": "%\\tolerance = 10000\n%\\documentstyle[prb,aps,epsfig]{revtex}\n%\\begin {document}\n%\\bibliographystyle {plain}\n%\\twocolumn[\n%\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname \n%\\title{\\bf Quantum-Critical Dynamics of the Skyrmion Lattice.} \n%\\author{A. G. Green} \n%\\maketitle \n%\\begin{abstract}\n%Near to filling fraction $\\nu=1$, the quantum Hall \n%ferromagnet contains multiple Skyrmion spin excitations. This multi-Skyrmion \n%system has a tremendously rich quantum-critical \n%structure. This is simplified\n%when Skyrmions are pinned by disorder. We calculate \n%the nuclear relaxation rate in this case and compare the result with experimen%t. \n%We discus how such measurements may be used to further probe the quantum-criti%cal structure of the multi-Skyrmion system.\n%\\end{abstract}\n%\\vspace{0.25in}\n%]\n\n\\documentclass{article} \n\\usepackage{epsfig}\n\\title{\\bf Quantum-Critical Dynamics of the Skyrmion Lattice.} \n\\author{A. G. Green} \n\\begin{document}\n\\maketitle \n\\begin{abstract}\nNear to filling fraction $\\nu=1$, the quantum Hall \nferromagnet contains multiple Skyrmion spin excitations. This multi-Skyrmion \nsystem has a tremendously rich quantum-critical \nstructure. This is simplified\nwhen Skyrmions are pinned by disorder. We calculate \nthe nuclear relaxation rate in this case and compare the result with experiment. \nWe discus how such measurements may be used to further probe the quantum-critical \nstructure of the multi-Skyrmion system.\n\\end{abstract}\n\\vspace{0.25in}\n\nAt exact filling of a single Landau level the quantized Hall\nstate forms an almost perfect ferromagnet. This quantum Hall ferromagnet (QHF)\nhas some novel features due to the phenomenology of the underlying quantized \nHall state. \nMagnetic vortices, or Skyrmions, in the QHF carry quantized \nelectrical charge\\cite{Sondhi}. These Skyrmions are stabilised by a chemical \npotential so that the ground state slightly away from filling fraction \n$\\nu=1$ contains a finite density of them. \nThis original proposal of Sondhi {\\it et al.} has been \nconfirmed in a number of experiments\\cite{Sondhi,Experiment}. \n\nThe $T=0$ phase diagram of the multi-Skyrmion system has been \nthoroughly investigated. In the absence of disorder, crystalline \narrangements are expected\\cite{Crystals1,Abolfath1}. \nFor filling\nfractions very close to $\\nu=1$ a triangular lattice is formed with \na transition to a square lattice as the deviation from\n$\\nu=1$ is increased. The statistical mechanics of the possible \nmelting transitions has been considered\\cite{Timm_melt}. \nAt the highest Skyrmion densities, zero point \nfluctuations are expected to give rise to a quantum-melted\nstate\\cite{Quantum_melt}.\n\nDespite this wealth of study, a complete account of the experimental \nobservables has not been achieved. For example, nuclear\nmagnetic resonance provides one of the clearest probes of the spin polarisation\nin the QHF\\cite{Experiment}. Although it is understood in general terms how \nlow-energy \nspin-fluctuations of the Skyrmion system may enhance the relaxation of \nnuclear spins, \nattempts to calculate relaxation rates have been flawed\\cite{Cote}.\nThe fundamental physics, missed in other considerations,\nis the quantum-critical nature\nof fluctuations of the Skyrmion lattice. One immediate consequence of this\nquantum-criticallity is that the limits of temperature/frequency$\\rightarrow0$\nand frequency/temperature$\\rightarrow0$ are very different. Typically, \nexperimental probes are at frequencies much less than temperature and the\nlatter limit is appropriate. This means that zero-temperature calculations \n{\\it cannot} model experiments correctly\\cite{Similar}.\n\nLet us consider these points a little further. In our analysis below, we will\nfind an underlying gappless XY-model governing orientational fluctuations\nof the multi-Skyrmion system. If a spinwave expansion is \nattempted for a gappless magnet at or below its critical dimension, the \noccupation of low-frequency modes is found to diverge. The constraint, \nfixing the \nmagnitude of the local spin, restricts this divergence. Interplay between\ndivergence and constraint gives rise to a finite (temperature dependent) \ncorrelation length, $\\xi(T)$, beyond which correlations of the magnet decay \nexponentially. The dynamics of the critical magnet are very different on \nlength scales greater or less than $\\xi$.\nOn lengthscales less than $\\xi$, the groundstate is ordered (albeit\nin a quantum superposition of all possible orientations due to long\nwavelength spin fluctuations). Fluctuations with wavelength less than $\\xi(T)$ \nmay, therefore, be described by a modified spinwave expansion. \nOn lengthscales greater than $\\xi$, the \ngroundstate is disordered and fluctuations are overdamped. This \n{\\it quantum relaxational dynamics} is a striking feature of quantum-critical \nsystems and leads to interesting universalities\\cite{Read}. \nRegimes of renormalized classical and quantum activated behaviour\nat low-temperature, cross over to universal behaviour in the \nhigh-temperature quantum-critical regime.\n\n The Skyrmion spin-configuration consists of a vortex-like \narrangement of in-plane components of spin with the z-component reversed \nin the centre of the Skyrmion and gradually increasing to match the \nferromagnetic background at infinity. At large distances, the spin \ndistribution \ndecays exponentially to the ferromagnetic background on a length scale \ndetermined by the ratio of spin stiffness to Zeeman energy\\cite{pure_Skyrmion}.\nAn individual \nSkyrmion may be characterized completely by its position ({\\it i.e.} the \npoint at which the spin points in the opposite direction to the ferromagnetic\nbackground) its size ({\\it i.e.} the number of flipped spins) and the \norientation of the in-plane components of spin. The equilibrium size of the Skyrmion \nis determined by a balance between its coulomb and Zeeman energies\\cite{Sondhi}\n(In the presence of a disorder potential, the potential\nenergy of the Skyrmion also enters this balancing act\\cite{Disorder}).\n\nConsider a ferromagnet with a dilute distribution of Skyrmions.\n%({\\it ie } separation greater than $\\sqrt{ \\rho_s/\\bar \\rho g B}$)\nThe normal modes of this system are relatively\neasy to identify. Firstly, ferromagnetic spinwaves propagate in-between\nthe Skyrmions. The spectrum of these is gapped by the Zeeman\nenergy and will be ignored from now on.\nPositional fluctuations, or phonon modes, of the Skyrmions are\ngapless in a pure system, but gapped when\nthe lattice is pinned by disorder. Finally, fluctuations in the in-plane orientation and \nsize must be considered. \nThese two types of fluctuation are intimately connected; rotating a\nSkyrmion changes its size. This follows from the\ncommutation relations of quantum angular momentum \noperators\\cite{Abolfath1,Cote,Nazarov,Piette}.\n\nThe orientation, $\\theta({\\bf x}_i,t)$, of a Skyrmion centred \nat a point ${\\bf x}_i$ is described by the \nfollowing effective action\\cite{Cote,Nazarov}:\n\\begin{equation}\nS=\\frac{1}{2}\n\\int dt \n\\left[\n\\sum_i \nI_i\n\\theta({\\bf x}_i,t) \\partial_t^2\\theta({\\bf x}_i,t)\n- \\sum_{<i,j>} \nJ_{ij}\\cos (\\theta({\\bf x}_i,t)-\\theta({\\bf x}_j,t))\\right].\n\\label{Theta_action}\n\\end{equation}\n%\\begin{eqnarray}\n%S&=&\\frac{1}{2}\n%\\int dt \n%\\left[\n%\\sum_i \n%I_i\n%\\theta({\\bf x}_i,t) \\partial_t^2\\theta({\\bf x}_i,t)\n%\\right.\n%\\nonumber\\\\\n%& &\n%\\left.\n%\\;\\;\\;\\;\\;\\;\\;\n%- \\sum_{<i,j>} \n%J_{ij}\\cos (\\theta({\\bf x}_i,t)-\\theta({\\bf x}_j,t))\\right].\n%\\label{Theta_action}\n%\\end{eqnarray}\n$I_i$ is the moment of inertia of the $i^{th}$ Skyrmion and $J_{ij}$ is the \nstiffness to relative rotations of neighbouring Skyrmions. The first term\nin Eq.(\\ref{Theta_action})\narises due to the change in energy of a Skyrmion when its size fluctuates;\n$\\Delta E=I^{-1} \\delta s^2/8$. Since the z-component of spin and orientation \nare conjugate coordinates, a cross-term \n$i\\delta s({\\bf x}_i,t) \\partial_t \\theta({\\bf x}_i,t)/2$\nappears in their joint effective action. Integrating out \n$\\delta s({\\bf x}_i,t) $ gives\nthe first term in Eq.(\\ref{Theta_action}). \nClearly, $1/4I$ is the second derivative of the Skyrmion energy with \nrespect to its spin. $I$ is related to the Skyrmion size \n(in fact $I=24\\mu_BgB/s$\\cite{Abolfath1}) and, in the absence of disorder, is \nthe same for all Skyrmions.\nCorrelation functions involving \n$\\delta s$ may be calculated using Eq.(\\ref{Theta_action}) by making the \nreplacement \n$\\delta s({\\bf x}_i,t) \\rightarrow 2i I \\partial_t \\theta({\\bf x},t)$-\nthe result of a simple Gaussian integration over $\\delta s$.\n The second term in Eq.(\\ref{Theta_action}) is an effective dipole-interaction of Skyrmions\ndue to the energetics of overlapping Skyrmion tails\\cite{Abolfath1,Piette}.\nIn a square Skyrmion lattice\\cite{hexagonal}, $J_{ij}$ is independent of \nlattice site. \nA continuum limit may be taken where $\\theta_i$ is replaced by a staggered\nfield, $\\theta_i \\rightarrow \\theta_i +\\eta_i \\pi$, with $\\eta_i=0,1$ on \nadjacent sites, and $\\theta_i-\\theta_j$ is replaced by a derivative;\n\\begin{equation}\nS= \\frac{1}{2}\n\\int \\frac{d\\omega d^2k }{(2 \\pi)^3}\n\\theta({\\bf k},\\omega)\n\\left(\n\\delta \\nu \\bar \\rho I \\omega^2\n+\nJ{\\bf k}^2\n\\right)\n\\theta(-{\\bf k},-\\omega).\n\\label{Theta_action2}\n\\end{equation}\nThe frequency integral in this expression is shorthand for a \nMatsubara summation at finite temperature and the momentum integral is over \nthe Brillouin zone\\cite{Brillouin_zone}.\nA factor of the Skyrmion density $\\delta \\nu \\bar \\rho$ has been introduced, \nwhere \n$\\delta \\nu$ is the deviation from filling fraction $\\nu=1$ and\n$\\bar \\rho$ is the electron density.\nThere are a few caveats to the use of Eq.(\\ref{Theta_action2}). We defer\ndiscusion of these until later. \nThis model is perhaps most familiar as an effective theory of the Josephson \njunction array. In this case, $\\theta$ is the phase of the superconducting\norder parameter and its conjugate coordinate is the charge of the \nsuperconducting junction. \n\nIn order to calculate properties of the Skyrmion lattice, we must relate \nfluctuations in the Skyrmion orientation to fluctuations in the orientation\nof local spin. We use a coherent-state representation of the polarization of \nthe local spin, {\\it via} an O(3)-vector field ${\\bf n}({\\bf x},t )$. The \nstatic spin distribution at a point ${\\bf x}$ relative to the centre of a \nsingle Skyrmion is denoted by ${\\bf n}({\\bf x})$\nand its in-plane components by $n_x+in_y=n_r e^{i \\phi_0}$. The in-plane \ncomponents of local spin at a point ${\\bf x}$, in response to rotational\nfluctuations of a Skyrmion centred at ${\\bf x}_i$, are given by \n\\begin{equation}\n%\\hbox{-in-plane Component-}\nn_r e^{i\\theta}({\\bf x},t)=\n(-)^{\\eta_i}\n\\left[ n_r({\\bf x}-{\\bf x}_i)\n+\n2iI \\frac{\\partial n_r({\\bf x}-{\\bf x}_i)}{\\partial s} \\partial_t \\theta({\\bf x}_i,t)\n\\right]\ne^{i\\theta+i\\phi_0}({\\bf x}_i,t).\n\\label{in_plane_component}\n\\end{equation}\nWe have used the conjugate relationship between Skyrmion spin and orientation \nin writing down this expression.\nIn a distribution of many Skyrmions, one must in principle sum the \ncontributions of all Skyrmions to the fluctuation in spin at the point ${\\bf x}$. However, in the dilute limit in which we are performing our explicit \ncalculation, Skyrmions are exponentially localized. The dominant spin \nfluctuations occur near to the centre of Skyrmions and so, to logarithmic\naccuracy, the local fluctuations at a point ${\\bf x}$ are due only to \nthe nearest Skyrmion.\n\n\n\n\nWe will calculate the\nnuclear relaxation rate due to low energy quantum fluctuations of the Skyrmion\nlattice;\n\\begin{equation}\n%\\hbox{-Nuclear-Relaxation-}\n\\frac{1}{T_1}\n=\nT \\gamma \\lim_{\\omega \\rightarrow 0}\n\\int \\frac{d^2k}{(2 \\pi)^2}\n\\frac{\n{\\cal I}m\n\\langle \nS_+({\\bf k}, \\omega) S_-(-{\\bf k}, -\\omega)\n\\rangle\n}{\\omega},\n\\label{nuclear_relaxation}\n\\end{equation}\nwhere $\\gamma$ is the hyperfine coupling constant. Other physical observables,\nsuch as the temperature dependence of magnetization,\n$\n\\langle M \\rangle\n= \n\\int dt\nd^2x \\langle n_z({\\bf x},t) \\rangle,\n$\nmay be calculated similarly.\nOur first task is to replace the expectation of the spin raising and lowering\noperators in Eq.(\\ref{nuclear_relaxation}) by correlators of the Skyrmion\norientation. Substituting from Eq.(\\ref{in_plane_component}) into\nEq.(\\ref{nuclear_relaxation}) and ignoring terms higher order in frequency,\nthe nuclear relaxation rate at a point ${\\bf x}$ is\n\\begin{equation}\n\\frac{1}{T_1}({\\bf x})\n=\nT \\gamma \\lim_{\\omega \\rightarrow 0} n_r^2({\\bf x})\n\\int \\frac{d^2k}{(2 \\pi)^2}\n\\frac{\n{\\cal I}m\n\\langle \ne^{i\\theta}({\\bf k}, \\omega) e^{-i\\theta}(-{\\bf k}, -\\omega)\n\\rangle\n}{\\omega}.\n\\label{nuclear_result}\n\\end{equation}\nThis takes the form of a correlation function of the Skyrmion orientation, \nmultiplied by a profile function characteristic of the Skyrmion groundstate.\nThe average rate is given by integrating this over an area containing a single\nSkyrmion and multiplying by the Skyrmion density $\\delta \\nu \\bar \\rho$. The \nresult is identical to Eq.(\\ref{nuclear_result}) with the replacement \n$n_r^2({\\bf x}) \\rightarrow \\delta \\nu \\overline{n^2_r}$.\n$\\overline{n_r^2}= \\bar \\rho \\int d^2x n_r^2({\\bf x})$ is a number characteristic\nof a single Skyrmion. For a pure Skyrmion spin distribution\\cite{pure_Skyrmion},\n $\\overline{n_r^2}=2s$ \nto logarithmic accuracy in the Skyrmion spin, $s=\\bar \\rho \\int d^2x (1-n_z({\\bf x}))$. \nNotice that radial fluctuations of local spin contribute only to higher order\nin frequency and have been neglected in writing down Eq.(\\ref{nuclear_result}). \n\nThe problem of finding the nuclear relaxation rate has now been reduced to \nevaluating the correlation function in Eq.(\\ref{nuclear_result}) using the\neffective action Eq.(\\ref{Theta_action2}). This is rather tricky. \nThe O(2)-quantum\nrotor, described by Eq.(\\ref{Theta_action2}), is quantum-critical. \nIt is necessary to employ a non-perturbative scheme, such as\n1/N or epsilon expansions, to calculate in the quantum-critical regime of \nthis model.\nHere we use the result of the 1/N expansion of \nChubukov {\\it et al}\\cite{Chubukov}.\n\nAn important feature of the effective action, Eq.(\\ref{Theta_action2}), is \nthat it displays\na zero-temperature phase transition. \nFor $IJ<1$, the Skyrmion moment of inertia is sufficiently small that \nquantum fluctuations destroy long range order even at zero temperatures. \nFor $IJ>1$, the $T=0$\ngroundstate has an infinite correlation length and is \nordered. Notice\nthat arbitrarily small temperatures destroy this long-range order even \nwhen $IJ>1$. This arises due to the interplay between fluctuations \nand constraint and can\nbe seen in a simple mean-field calculation\\cite{Chubukov}. \nIn the O(2) representation of Eq.(\\ref{Theta_action2}),\n$S=\\int dt d^2x \\left[ {\\bf n} (I\\partial_t^2+ J\\partial_{\\bf x}^2 ){\\bf n} \n+\\lambda({\\bf x},t) ({\\bf n}^2-1) \\right]$, where ${\\bf n}({\\bf x},t)$ is \nan O(2)-vector\nfield and $\\lambda({\\bf x},t)$ is an auxiliary field that imposes the \nconstraint\n${\\bf n}^2=1$. Imposing the constraint at mean-field level, \n$\\langle {\\bf n}^2 \\rangle=1$,\ndetermines a temperature dependent gap, $\\lambda(T)$.\nThe spin correlations decay exponentially on a length\nscale $\\xi(T)=\\sqrt{J/\\lambda(T)}$. \nThe results of such a calculation are sketched in Fig.1.\n\\begin{figure}\n\\centering\n\\epsfig{file=Gap.eps,height=1.5in}\n\\caption{T-dependence of the Gap.}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\epsfig{file=Quantum_critical.eps,height=1.5in}\n%\\epsfig{file=Quantum_critical_old.eps,height=1.5in}\n\\caption{Phase diagram of the O(2)-vector model.}\n\\end{figure}\n Above a temperature of about \n$T_{QC}=|2 \\pi J-E_{\\hbox{max}}|$ \n(The cut-off, $E_{\\hbox{max}}=2 \\pi \\sqrt{J/I}$ corresponds to fluctuations \nwith momentum at the Brillouin zone boundary, \n$k_{\\hbox{max}}=\\pi \\sqrt{\\bar \\rho \\delta \\nu}$\\cite{Brillouin_zone}), the\ngap/correlation length develops a universal temperature dependence. In this \nregion, thermal and quantum fluctuations are of similar importance and are very difficult\nto disentangle. This crossover to universal high-temperature behaviour from distinct \nlow-temperature behaviours is a feature of all correlation functions of \nEq.(\\ref{Theta_action2}). It is usual to summarise this behaviour by the phase diagram sketched in\nFig.2. In this figure, $J(T)$ is the renormalized \nstiffness in the ordered phase and $\\lambda(T)$ is the gap in the paramagnetic phase.\n\nThe correlation function required in Eq.(\\ref{nuclear_result}) has been \ncalculated in Ref.\\cite{Chubukov} by means of a 1/N expansion, with the \nresult\n\\begin{eqnarray*}\n& &\n\\;\\;\\;\\;\n\\lim_{\\omega \\rightarrow 0}\\frac{1}{\\omega}\n\\int \\frac{d^2k}{(2 \\pi)^2} \n{\\cal I}m\n\\langle\ne^{i \\theta}({\\bf k}, \\omega) e^{-i\\theta}(-{\\bf k}, -\\omega) \\rangle\n\\\\\n& &\n\\begin{array}{llll}\n= &\n \\frac{1}{T} \\frac{0.015}{\\sqrt{IJ}} \n\\left( \\frac{kT}{\\pi J} \\right)^{\\eta},\n&\n\\mbox{\\scriptsize $T \\gg T_{QC}$} &\n\\\\\n= &\n\\frac{1}{T}\n\\frac{0.085}{\\sqrt{IJ}}\n \\frac{IT^2}{\\lambda(0)} e^{-2 \\sqrt{\\lambda(0)/IT^2}},\n&\n\\mbox{\\scriptsize $T \\ll T_{QC}$,} &\n\\mbox{\\scriptsize $IJ<4$},\n\\\\\n= &\n\\frac{0.18}{T}\n \\left( \\frac{kT}{2\\pi \\lambda(T)} \\right)^{1/2},\n&\n\\mbox{\\scriptsize $T \\ll T_{QC}$,} &\n\\mbox{\\scriptsize $IJ>4$},\n\\end{array}\n\\end{eqnarray*}\nwhere $\\eta$ is a number close to zero. \nSubstituting these results into Eq.(\\ref{nuclear_result}) we obtain\n\\begin{equation}\n\\frac{1}{T_1}=\n\\gamma \\frac{0.03s}{\\sqrt{IJ}} \n\\left( \\frac{kT}{\\pi J} \\right)^{\\eta}\n\\end{equation}\nat high temperature. The full behaviour is sketched in Fig.3. The kink at $T_{KT}$\nis due to the discontinuous change in spin stiffness seen at the Kosterlitz-Thouless transition. This effect is not seen in the 1/N expansion of Ref.\\cite{Chubukov} and must be calculated by some other means.\n\\begin{figure}\n\\centering\n\\epsfig{file=Relaxation.eps,height=1.5in}\n%\\vspace{1.5in}\n\\caption{Temperature of the nuclear relaxation rate, $1/T_1$.}\n\\end{figure}\n\nThe nuclear relaxation rate obtained here is very different from that \nobtained in Ref.\\cite{Cote}. As pointed out in the introduction, this is \ndue to the unphysical limit $T/\\omega \\rightarrow0$ used in Ref.\\cite{Cote}.\nNevertheless, it is instructive to see how the results of Ref.\\cite{Cote}\nrelate to the present formalism. Since $T\\ll \\omega$ in their work, \nC\\^ot\\'e {\\it et al}\nconsider fluctuations on length scales much less than the correlation length.\nThe groundstate is ordered and a spinwave expansion may be used. Long\nwavelength fluctuations lead to a superposition of orientations- rotational\naveraging- the immediate consequence of which is that \n$\\langle e^{i\\theta}({\\bf k},\\omega)e^{-i\\theta}(-{\\bf k},-\\omega)\\rangle=0$.\nReturning to the substitution of Eq.(\\ref{in_plane_component}) into \nEq.(\\ref{nuclear_relaxation}), we must retain terms to next order in \nfrequency. This is a cross-term between radial and transverse \nfluctuations and involves a correlator \n$\\langle \\partial_t \\theta\ne^{i\\theta}({\\bf k}, \\omega) e^{-i\\theta}(-{\\bf k}, -\\omega)\n\\rangle$.\nEvaluating this correlation function {\\it via} a $T=0$ \nspinwave expansion of the effective action, Eq.(\\ref{Theta_action2}), \nreproduces the\nresult of Ref.\\cite{Cote} (up to a numerical factor due to our estimate of the \nSkyrmion profile function, $n_r \\partial n_r/\\partial s$). \nThe zero-temperature phonon contribution may\nbe calculated similarly. \n\n\nUp to now, we have assumed that the Skyrmion lattice is pinned by disorder\n and that phonons may be ignored as a consequence. The situation is rather\nsubtle and a fuller discussion is appropriate at this juncture.\nA quadratic effective action for phonons of the Skyrmion lattice is \nknown\\cite{Crystals1,Abolfath1,Timm_melt}. It is identical to that of an\nelectronic Wigner crystal in a magnetic field with a vanishing effective \nmass\\cite{mass}. Under this effective action, Skyrmions move in small ellipses\nwith a frequency $\\omega_{\\bf k}\\sim |{\\bf k}|^{3/2}/B$ and major axes \norientated transverse to the phonon momentum.\n\nAt finite temperature, the occupation of transverse phonons is infra-red \ndivergent. This divergence is restricted \nby\ninteractions between phonons arising from the non-harmonicity of the \nSkyrmion interaction. Unlike fluctuations in orientation, where the \nspinwave interaction is due to a topological constraint, these phonon \ninteractions are non-universal (at best the universality is hidden in the \ndetails of the groundstate spin distribution and effective Skyrmion interaction\npotential). The resulting physics is very similar to that discussed for the \nrotation mode above; the phonon system is quantum-critical and has \nlow-temperature ordered and quantum-melted\\cite{Quantum_melt} phases and a \nhigh-temperature quantum-critical regime.\n\n\nEven this is not the full story. Although the low energy dispersions of\nphonons and Skyrmion rotations are independent\\cite{Cote}, non-linear interactions exist \nbetween these modes. The lattice stiffness is, in part, due to the dipole interaction of Skyrmions \nand is affected by fluctuations in orientation.\nSimilarly, the dipole interaction between Skyrmions is strongly dependent upon the \nseparation of Skyrmions and is affected by phonons. These non-linearities\noccur on the same footing as the phonon-phonon interactions and \nrotation-rotation interactions. The full quantum-critical structure of the \nmulti-Skyrmion system is tremendously complicated. \n\nThe position taken here in neglecting this wealth of structure is that\nthe Skyrmion lattice is pinned by disorder and the phonon spectrum gapped. \nPhononic fluctuations are suppressed at low temperatures and the associated critical structure occurs \nat higher temperature. The residual effect of phonons is \na slight thermal renormalization of the rotational stiffness.\nThe slight distortion in static positions of Skyrmions, in response to the\ndisorder potential, gives a small random contribution to the stiffness $J$.\nThis randomness produces a small region of Bose-glass\nphase at low temperatures, intervening between the paramagnet and renormalized\nclassical regimes. For weak disorder, this phase only affects the physics very\nclose to the critical point and does not affect our conclusions.\n\nWe now turn to a discussion of the experimental implication of the above \ncalculations. Detailed measurements of $1/T_1$ have been carried out by Bayot \n{\\it et al}\\cite{Bayot}. Above $40mK$, $T_1$ is independent of \ntemperature. This is consistent with the rotational degrees of freedom \nbeing in their quantum-critical regime. Values of $I$ and \n$J$ for this system extracted from TDHFA calculations\\cite{Cote} \n%($I\\sim 2 K^{-1}$, $J\\sim 0.5K$)\nput $IJ$ very close to $1$. The system is close to criticality and \nthe crossover to the quantum-critical regime occurs at correspondingly \nlow temperature.\nAt $40mK$ there is an abrupt step in $1/T_1$(and attendant peak in \nheat capacity). This is consistent with a \nKosterlitz-Thouless transition in the orientational order (notice that\nthe crossover temperature $|2 \\pi J-E_{\\hbox{max}}|$ may be much less\nthan $T_{KT}=2 \\pi J$ and so the behaviour may be quantum-critical\neither side of the transisiton). There are a\nnumber of other candidate transitions\\cite{Crystals1,Timm_melt,Cote},\nhowever, and it is not easy to discriminate between them. Considerations\nalong the lines of those presented here allow some elaboration, but\nthis is necessarily rather speculative and we refrain from its discussion\nat present.\nWe may make some firm predictions for nuclear\nrelaxation measurements below $40mK$. Changing the deviation in filling\nfraction or using tilted filed measurements both change the parameter\n$IJ$ and allow exploration of the phase diagram shown in Fig.1. The divergence\nor otherwise of the nuclear relaxation rate as temperature is reduced to zero\nshould give a clear indication of the quantum-critical structure.\n\nI would like to thank N. R. Cooper, J. R. Chalker, S. M. Girvin and \nN. Read for enlightening discusions, comments and suggestions. \nThis work was supported by Trinity College Cambridge.\n\n\n\n\n\\begin{thebibliography}{99}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Sondhi} S. L. Sondhi, A. Karlhede, S. A. Kivelson and E. H. \nRezayi, Phys. Rev. {\\bf B47}, 16419 (1993).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Experiment}S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West \nand R. Tycko, Phys. Rev. Lett. {\\bf 74} 5112 (1995);\nR. Tycko, S. E. Barrett, G. Dabbagh, L. N. Pfeiffer and K. W. West,\nScience {\\bf 268}, 1460 (1995);\nE. H. Aifer, B. B. Goldberg, D. A. Broido, Phys. Rev. Lett. {\\bf 76} 680\n(1996);\nA. Schmeller, J. P. Eisenstein, L. N. Pfeiffer and K. W. West, Phys. Rev.\nLett. {\\bf 75}, 4290 (1995).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Crystals1}L. Brey, H. A. Fertig, R. C\\^ot\\'e and A. H. MacDonald,\nPhys. Rev. Lett. {\\bf 76}, 2562 (1995);\nA. G. Green, I. I. Kogan and A. M. Tsvelik,\nPhys. Rev. B {\\bf 54} 16838 (1996);\nM. Rao, S. Sengupta and R. Shankar, Phys. Rev. Lett. {\\bf\n79}, 3998, (1997);\n\\bibitem{Abolfath1}M. Abolfath and M. R. Ejtehadi, cond-mat/9807236 (1998);\nPhys. Rev. B{\\bf 58}, 10665 (1998).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Timm_melt} Carsten Timm, S. M. Girvin, H. A. Fertig, Phys. Rev.\nB{\\bf 58}, 10 634 (1998).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Quantum_melt}B. Paredes and J. J. Palacios, cond-mat/9902031 (1999).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Cote}R. C\\^ot\\'e, A. H. MacDonald, Luis Brey, H. A. Fertig, \nS. M. Girvin and H. T. C. Stoof, Phys. Rev. Lett. {\\bf 78}, 4825 (1997); \ncond-mat/9702207 (1997).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Similar} Similar considerations are key in the\ncalculation of finite temperature conductivity of the 2-dimensional \nsupercondutor. \nK. Damle and S. Sachdev, Phys. Rev. B {\\bf 56} 8714 (1997).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Read} N. Read and S. Sachdev, Phys. Rev. Lett. {\\bf 75}, 3509 (1995).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{pure_Skyrmion} The pure Skyrmion spin distribution has the form\n$(n_x+i n_y)({\\bf r})=2(x+iy)e^{i \\theta}r_0/(r^2 + r_0^2)$, \n$n_z({\\bf x})=(r^2-r_0^2)/(r^2+r_0^2)$, out to a radius \n$\\sqrt{\\rho_s/\\bar \\rho gB}$, beyond which the spin distribution decays \nexponentially to the ferromagnetic groundstate. $r_0$ is a parameter \nrelated to the spin of the\nSkyrmion and $\\theta$ its orientation.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Disorder}A. J. Nederveen and Yuli. V. Nazarov, Phys. Rev. Lett. {\\bf 82}, 406 (1999). \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Nazarov}Yuli V. Nazarov and A. V. Khaetski, Phys. Rev. Lett. {\\bf 80},\n576 (1998).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Piette} B. M. A. G. Piette, B. J. Schroers and W. J. Zakrzewshi, Nucl. Phys. B\n{\\bf 439}, 205 (1995).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{hexagonal}In the case of the hexagonal lattice,\nanti-ferromagnetic order is frustrated and a different description is required.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Brillouin_zone} The Skyrmion lattice has an anti-ferromagnetic \nordering of in-plane spins. The unit cell, therefore, contains two Skyrmions\nand the Brillouin zone must be chosen appropriately.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Chubukov}A. V. Chubukov, S. Sachdev and J. Ye, Phys. Rev. B{\\bf 49},\n11919 (1994).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{mass} A curious feature of the nearly filled Landau level is that\nquasi-particles have only a magnus-force term in their effective action and no\nkinetic term.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Bayot}V. Bayot, E. Grivei, S. Melinte, M. B. Santos and M. Shayegan,\nPhys. Rev. Lett. {\\bf 76}, 4585 (1996); V. Bayot, E. Grivei, J.-M. Beuken,\nS. Melinte and M. Shayegan, Phys. Rev. Lett. {\\bf 79}, 1718 (1997).\n%\n%\\bibitem{BKT}V. C. Berezinskii, Zh. Eksp. Teor. Fiz. {\\bf 61}, 1144 (1971)\n%[Sov. Phys. JETP {\\bf 34}, 610 (1972)];\n%J. M. Kosterlitz and D. J. Thouless, J. Phys. C{\\bf 6}, 1181 (1973).\n%\\bibitem{BKTHNY}J. M. Kosterlitz, J. Phys. C{\\bf 7}, 1046 (1974);\n%D. R. Nelson, Phys. Rev. B{\\bf 18},2318 (1978);\n%D. R. Nelson and B. I. Halperin, Phys. Rev. B{\\bf 19}, 2457 (1979);\n%A. P. Young, Phys. Rev. B{\\bf 19}, 1855 (1979).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\bibitem{Abolfath2}M. Abolfath, cond-mat/9712260 (1997);\n%Phys. Rev. B{\\bf 58}, 2013, (1998).\n%\\bibitem{Auerbach}Assa Auerbach, {\\it Interacting Electrons and Quantum Magnet%ism},\n%Springer-Verlag, New York (1995)\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\bibitem{MacDonald}A. H. MacDonald,\n% H. A. Fertig, and L. Brey, Phys. Rev. Lett. {\\bf 76}, 2153, (1996).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\bibitem{Gradsteyn} J. S. Gradsteyn and I. M. Ryzhik, \n%{\\it Tables of integrals, series and products.} Accademic Press (1994).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\bibitem{Anderson} P. W. Anderson, {\\it Lectures on the Many Body Problem}\n%ed. E. R. Caiariello vol. 2 p.127(Academic, New York) (1964)\n%\\bibitem{Simanek} E. Simanek, Phys. Rev. B{\\bf 22}, 459 (1979),\n%Phys. Rev. Lett. {\\bf 45}, 1442 (1980).\n%\\bibitem{Efetov} K. B. Efetov, Zh. Eksp. Teor. Fiz. {\\bf 78}, 17 (1980)\n%[Sov. Phys. JETP {\\bf 51}, 1015 (1980)]\n\n\n\\end{thebibliography}\n\\end{document}\n"
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[
{
"name": "cond-mat0002053.extracted_bib",
"string": "\\begin{thebibliography}{99}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Sondhi} S. L. Sondhi, A. Karlhede, S. A. Kivelson and E. H. \nRezayi, Phys. Rev. {\\bf B47}, 16419 (1993).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Experiment}S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West \nand R. Tycko, Phys. Rev. Lett. {\\bf 74} 5112 (1995);\nR. Tycko, S. E. Barrett, G. Dabbagh, L. N. Pfeiffer and K. W. West,\nScience {\\bf 268}, 1460 (1995);\nE. H. Aifer, B. B. Goldberg, D. A. Broido, Phys. Rev. Lett. {\\bf 76} 680\n(1996);\nA. Schmeller, J. P. Eisenstein, L. N. Pfeiffer and K. W. West, Phys. Rev.\nLett. {\\bf 75}, 4290 (1995).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Crystals1}L. Brey, H. A. Fertig, R. C\\^ot\\'e and A. H. MacDonald,\nPhys. Rev. Lett. {\\bf 76}, 2562 (1995);\nA. G. Green, I. I. Kogan and A. M. Tsvelik,\nPhys. Rev. B {\\bf 54} 16838 (1996);\nM. Rao, S. Sengupta and R. Shankar, Phys. Rev. Lett. {\\bf\n79}, 3998, (1997);\n\\bibitem{Abolfath1}M. Abolfath and M. R. Ejtehadi, cond-mat/9807236 (1998);\nPhys. Rev. B{\\bf 58}, 10665 (1998).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Timm_melt} Carsten Timm, S. M. Girvin, H. A. Fertig, Phys. Rev.\nB{\\bf 58}, 10 634 (1998).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Quantum_melt}B. Paredes and J. J. Palacios, cond-mat/9902031 (1999).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Cote}R. C\\^ot\\'e, A. H. MacDonald, Luis Brey, H. A. Fertig, \nS. M. Girvin and H. T. C. Stoof, Phys. Rev. Lett. {\\bf 78}, 4825 (1997); \ncond-mat/9702207 (1997).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Similar} Similar considerations are key in the\ncalculation of finite temperature conductivity of the 2-dimensional \nsupercondutor. \nK. Damle and S. Sachdev, Phys. Rev. B {\\bf 56} 8714 (1997).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Read} N. Read and S. Sachdev, Phys. Rev. Lett. {\\bf 75}, 3509 (1995).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{pure_Skyrmion} The pure Skyrmion spin distribution has the form\n$(n_x+i n_y)({\\bf r})=2(x+iy)e^{i \\theta}r_0/(r^2 + r_0^2)$, \n$n_z({\\bf x})=(r^2-r_0^2)/(r^2+r_0^2)$, out to a radius \n$\\sqrt{\\rho_s/\\bar \\rho gB}$, beyond which the spin distribution decays \nexponentially to the ferromagnetic groundstate. $r_0$ is a parameter \nrelated to the spin of the\nSkyrmion and $\\theta$ its orientation.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Disorder}A. J. Nederveen and Yuli. V. Nazarov, Phys. Rev. Lett. {\\bf 82}, 406 (1999). \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Nazarov}Yuli V. Nazarov and A. V. Khaetski, Phys. Rev. Lett. {\\bf 80},\n576 (1998).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Piette} B. M. A. G. Piette, B. J. Schroers and W. J. Zakrzewshi, Nucl. Phys. B\n{\\bf 439}, 205 (1995).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{hexagonal}In the case of the hexagonal lattice,\nanti-ferromagnetic order is frustrated and a different description is required.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Brillouin_zone} The Skyrmion lattice has an anti-ferromagnetic \nordering of in-plane spins. The unit cell, therefore, contains two Skyrmions\nand the Brillouin zone must be chosen appropriately.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Chubukov}A. V. Chubukov, S. Sachdev and J. Ye, Phys. Rev. B{\\bf 49},\n11919 (1994).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{mass} A curious feature of the nearly filled Landau level is that\nquasi-particles have only a magnus-force term in their effective action and no\nkinetic term.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Bayot}V. Bayot, E. Grivei, S. Melinte, M. B. Santos and M. Shayegan,\nPhys. Rev. Lett. {\\bf 76}, 4585 (1996); V. Bayot, E. Grivei, J.-M. Beuken,\nS. Melinte and M. Shayegan, Phys. Rev. Lett. {\\bf 79}, 1718 (1997).\n%\n%\\bibitem{BKT}V. C. Berezinskii, Zh. Eksp. Teor. Fiz. {\\bf 61}, 1144 (1971)\n%[Sov. Phys. JETP {\\bf 34}, 610 (1972)];\n%J. M. Kosterlitz and D. J. Thouless, J. Phys. C{\\bf 6}, 1181 (1973).\n%\\bibitem{BKTHNY}J. M. Kosterlitz, J. Phys. C{\\bf 7}, 1046 (1974);\n%D. R. Nelson, Phys. Rev. B{\\bf 18},2318 (1978);\n%D. R. Nelson and B. I. Halperin, Phys. Rev. B{\\bf 19}, 2457 (1979);\n%A. P. Young, Phys. Rev. B{\\bf 19}, 1855 (1979).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\bibitem{Abolfath2}M. Abolfath, cond-mat/9712260 (1997);\n%Phys. Rev. B{\\bf 58}, 2013, (1998).\n%\\bibitem{Auerbach}Assa Auerbach, {\\it Interacting Electrons and Quantum Magnet%ism},\n%Springer-Verlag, New York (1995)\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\bibitem{MacDonald}A. H. MacDonald,\n% H. A. Fertig, and L. Brey, Phys. Rev. Lett. {\\bf 76}, 2153, (1996).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\bibitem{Gradsteyn} J. S. Gradsteyn and I. M. Ryzhik, \n%{\\it Tables of integrals, series and products.} Accademic Press (1994).\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\bibitem{Anderson} P. W. Anderson, {\\it Lectures on the Many Body Problem}\n%ed. E. R. Caiariello vol. 2 p.127(Academic, New York) (1964)\n%\\bibitem{Simanek} E. Simanek, Phys. Rev. B{\\bf 22}, 459 (1979),\n%Phys. Rev. Lett. {\\bf 45}, 1442 (1980).\n%\\bibitem{Efetov} K. B. Efetov, Zh. Eksp. Teor. Fiz. {\\bf 78}, 17 (1980)\n%[Sov. Phys. JETP {\\bf 51}, 1015 (1980)]\n\n\n\\end{thebibliography}"
}
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cond-mat0002054
|
[] |
[
{
"name": "fidelity.tex",
"string": "\\documentstyle[12pt]{article}\n\\begin{document}\n\\centerline {\\Large {\\bf High Reproduction Rate versus Sexual Fidelity}}\n\n\\bigskip\n\\centerline {\\bf A.O. Sousa and S. Moss de Oliveira} \n\n\\bigskip\n\\centerline {\\it Instituto de F\\'{\\i}sica, Universidade Federal \nFluminense} \n\n\\centerline{\\it Av. Litor\\^anea s/n, Boa Viagem, Niter\\'oi 24210-340, RJ, \nBrazil}\n\n\\bigskip\n\\noindent {\\bf Abstract}\n\n\\bigskip\nWe introduce fidelity into the bit-string Penna model for biological \nageing and study the advantage of this fidelity when it produces \na higher survival probability of the offspring due to paternal care. \nWe attribute a {\\it lower reproduction rate} to the faithful males but a {\\it \nhigher \ndeath probability} to the offspring of non-faithful males that abandon \nthe pups to mate other females. The fidelity is considered as a \ngenetic trait which is transmitted to the male offspring (with or without error). \nWe show that {\\it nature may prefer a lower reproduction rate} to warrant \nthe survival of the offspring already born. \n\n\\section{Introduction}\n\nIt is not difficult to find in the animal kingdom species that live and work in \nsexual pairs, but sometimes \nhave extra-pair relations. Biologists believe that these pairs are formed \nin order to better take care of the pups, and that the extra-pair \nrelations have the genetic purpose to maximize the variability \nof their offspring or to produce some fitness benefit for them [1]. \nThe Scandinavian great reed warbler is one of the \nspecies that presents these extra-pair matings. \nHowever, independent of its origin (social or genetic), true monogamy seems to be \nrare in Nature.\n\nThe Penna model for biological ageing [2] is a Monte Carlo simulation technique \nbased on \nthe mutation accumulation hypothesis. It has successfully reproduced many \ndifferent characteristics of living species, as the catastrophic senescence \nof pacific salmon [3], the inheritance of longevity [4] and the self-organization \nof female menopause [5]. The extra-pair relations mentioned above \nhave also been studied through this model [6]. Martins and Penna have obtained \nthat the \noffspring generated by extra-pair relations are genetically stronger and \npresent a higher survival probability than those generated by social relations. \n\nIn this paper we are interested in using the Penna model to study the true \nmonogamy, rarely found in \nNature. One example is the California mouse. In this species a female is \nnot able to sustain one to three pups alone. The pups are born at \nthe coldest time of the year and depend on the parents body heat to survive. \nAccording to the biologist David Gubernick, as cited in Science [1], the situation \nis so dramatic \nthat if the male leaves or is taken away, the female abandons or kills the pups. \nHowever, he also points out that other species of mice that live in the same \nenvironment are promiscuous. That is, the reason for true monogamy is \nstill an open question under study. We have adopted the strategy of considering \nmonogamy as a genetic trait, exclusively related to paternal care. Our assumption \nthat male fidelity is genetically transmitted is analogous to the recent findings \nthat the gene {\\it Mest} regulates maternal care [7,8]. \n\nIn the next section we explain the Penna model and how fidelity is introduced. \nIn section 3 we present our results and in section 4 the conclusions.\n\n\\section{The Sexual Penna model and Fidelity}\n\nWe will now describe the sexual version of the Penna model; \ndetails and applications can be found, for instance, in references [5,9]. \nThe genome of each individual is represented by two bit-strings of 32 bits that are \n{\\it read in parallel}; that is, there are 32 positions to be read, to each \nposition \ncorresponding two bits. One time-step corresponds to read one position of all the \ngenomes. In this way, each individual can live at most for 32 time-steps \n(``years''). Genetic diseases are represented by bits 1. If an individual \nhas two bits 1 (homozygotous) at the third position, for instance, it starts to \nsuffer from a genetic disease at its third year of life. If it is an homozygotous \nposition \nwith two bits zero, no disease appears at that age. If the individual is \nheterozygotous in \nsome position, it will get sick only if that position is a dominant one. The \nnumber of dominant genes and its randomly chosen positions are defined at \nthe beginning of the simulation; they are \nthe same for all individuals and remain constant. When the number of accumulated \ndiseases of any individual reaches a threshold $T$, the individual dies. \n\nThe individuals may also be killed due to a \nlack of space and food, according to the logistic Verhulst factor \n$V=1-N(t)/N_{max}$, where $N(t)$ is the current population size and $N_{max}$ the \ncarrying capacity of the environment. At every time step and for each individual \na random number between zero and 1 is generated and compared with $V$: if this \nnumber is greater than $V$, the individual dies independently of its age or \nnumber of accumulated diseases. \n\nIf a female succeeds in surviving until the minimum reproduction age $R$, \nit generates, with probability $p$, $b$ offspring every year. The female randomly \nchooses a male to mate, the age of which must also be greater or equal to $R$. \nThe offspring genome is constructed from the parents' ones; firstly the strings \nof the mother are randomly crossed, and a female gamete is produced. $M_m$ \ndeleterious mutations are then randomly introduced. The same process occurs with \nthe father's genome (with $M_f$ mutations), and the union of the two remaining \ngametes form the new genome. \nDeleterious mutation means that if the randomly chosen bit of the parent genome \nis equal to 1, it remains 1 in the offspring genome, but if it is equal to zero \nin the parent genome, it is set to 1 in the baby genome. It is well known \n[5,9,10] that due to the dynamics of the model, the bits 1 accumulate, \nafter many generations, at the end part of the genomes, that is, after the \nminimum reproduction age $R$. For this reason ageing appears: the survival \nprobabilities decrease with age. The sex of the baby is randomly chosen,\neach one with probability $50\\%$.\n\nLet's see now how fidelity is introduced. We assume that if a female reproduces \nthis year, she spents the next two following years without reproducing. So we \nconsider two time steps as the {\\it parental care period}. Remembering that in our \nsimulations the female choses the male, if the male is \na faithful one, he will refuse, during this period, to mate any female that \neventually choses him as a partner. The non-faithful male accepts any invitation, \nbut his offspring still under parental care pay the price for the abandonment: they \nhave an extra probability $P_d$ of dying. The male offspring of a faithful father \nwill also be faithful, with probability $P_f$. This means that if the father is \nfaithful and $P_f = 1$, the male offspring will necessarily be faithful. \n$P_f$ is also the probability of a non-faithful male having a non-faithful\noffspring.\n\n\\section{Results} \n\n{\\it We start our simulations with half of the males faithful and half \nnon-faithful}. \nIn Fig.1 we show the final percentages (after many generations) of faithful\nmales as a function of the offspring death probability $P_d$, for the \ncases where the male offspring inherits the father's fidelity state with \nprobability $P_f =~1$ (full line) and with probability $P_f = 0.8$ \n(dashed line). This last case means that the offspring of a faithful \nfather has a $20\\%$ probability of being non-faithful and vice-versa. \nFrom this figure we can see that as the death probability of \nthe abandoned pups increases, the percentage of faithful fathers \nincreases. From the solid curve it is easy to notice that there is a compromise \nbetween the lower reproduction rate of the faithful males and the death \nprobability of \nthe already born offspring abandoned by father: if $P_d < 0.3$, a high \nreproduction rate dominates and after many generations the \nfaithful males disappear from the population. However, for \n$P_d = 1$ the opposite occurs, since there is a strong selection \npressure against the non-faithful males to warrant the survival of \nthe already born offspring. \n\nFrom the dashed curve ($P_f = 0.8$) it can be seen that for $P_d = 0$ a high \npercentage (greater than $20\\%$) of faithful males remains in the final \npopulation. The reason is that for $P_d=0$ there is no selection pressure.\nThere is a probability that non-faithful males, which have a high \nreproduction rate, generate faithful offspring; these offspring \nare introduced into the population and, without any pressure, remain there.\nAt this point ($P_f=0.8$ and $P_d=0$) we have computed which percentages \nof faithful males descend from faithful and non-faithul fathers. We have \nobtained that for the $26.95\\%$ of faithful males that remain in the population, \n$9.28\\%$ of them descend from faithful fathers and $17.67\\%$ from non-faithful \nones. \n\nIn Fig.2a we present the time evolution of the populations for $P_f = 1$, \nand in Fig.2b for $P_f = 0.8$. The inset show the final population sizes \nas a function of $P_d$. From Fig.2a it can be seen that the population sizes \ndecrease until $P_d=0.4$ and then increase for increasing values of $P_d$, \nstabilizing around the same population size of $P_d=0.3$. \nFor $P_f = 0.8$ (Fig.2b) the population sizes decrease until $P_d=0.7$, and then \nstabilize around the same final size for increasing values of $P_d$.\n\nFig.3 shows the survival rates for $P_f = 1$ and $P_d = 0.0$ (circles), 0.5 \n(squares) and 0.9 (triangles). It can be noticed that for $P_d = 0.5$ the \nchild mortality is greater, since $P_d$ is already large and nearly $50\\%$ of \nthe males (see fig.1, solid curve) are not faithful. The results obtained for \n$P_f=0.8$ are similar. \n\nThe survival rate is defined, for a stable population, as the ratio \n$$S(a)=N(a+1)/N(a) \\,\\,\\, ,$$\nwhere $N(a)$ is the number of individuals of age \n$a$. A stable population means that the number of individuals of any given \nage $a$ is constant in time. It is important to emphasize that all \ncurves presented here correspond to already stable situations. To \nobtain each of them we simulated 20 different populations (samples) \nduring 800,000 time steps, and averaged the final results. The \nparameters of the simulations are:\n\n\\bigskip\n\\noindent Initial population = 20,000 individuals (half for each sex);\n\n\\noindent Maximum population size $N_{max} = 200,000$;\n\n\\noindent Limit number of allowed diseases $T = 3$;\n\n\\noindent Minimum reproduction age $R = 10$; \n\n\\noindent Probability to give birth $p = 0.5$; \n\n\\noindent Number of offspring $b = 2$; \n\n\\noindent Number of mutations at birth $M_m = M_f = 1$;\n\n\\noindent Number of dominant positions = 6 (in 32). \n\n\\section{Conclusions}\n\nWe have used the Penna bit-string model for biological ageing to study \nthe problem of true monogamy, rarely found in Nature. In our simulations \na female that gives birth necessarily waits two time steps before giving \nbirth again. We call this period the parental care period. A faithful father \nalso cannot reproduce during this period, but a non-faithful one can accept any \nfemale that randomly choses him to mate, abandoning the pups already born.\nThe abandoned pups have, as a consequence, an extra probability to die. \nIn this way there is a competition between the reproduction rate and \nthe death probability of already born pups. We show that depending on \nthis death probability, nature may prefers a lower reproduction rate to \nwarrant the survival of those babies already born. We consider the paternal \nfidelity an expression of paternal care, and so admit it as a genetic trait \nto be transmitted to the male offspring. \n\n\\bigskip\n\\noindent Acknowledgements: to P.M.C. de Oliveira and D. Stauffer for \nimportant discussions and a critical reading of the manuscript; to CNPq, CAPES \nand FAPERJ for financial support. \n \n\\newpage\n\n\\noindent {\\Large \\bf References}\n\n\\begin{description}\n\n\\item [ 1-] V. Morell, {\\it Science} {\\bf 281}, 1983 (1998).\n\\item [ 2-] T.J.P. Penna, {\\it J.Stat.Phys.} {\\bf 78}, 1629 (1995).\n\\item [ 3-] K.W. Wachter and C.E. Finch, {\\it Betwee Zeus and the Salmon. The \nBiodemography of Longevity}, National Academy Press, Washington D.C.; \nT.J.P. Penna, S. Moss de Oliveira and D. Stauffer, {\\it Phys.Rev.} \n{\\bf E52}, 3309 (1995). \n\\item [ 4-] P.M.C. de Oliveira, S. Moss de Oliveira, A.T. Bernardes \nand D. Stauffer, {\\it Lancet} {\\bf 352}, 911 (1998).\n\\item [ 5-] S. Moss de Oliveira, P.M.C. de Oliveira and D. Stauffer, \n{\\it Evolution, Money, War and Computers}, Teubner, Sttutgart-Leipzig (1999). \n\\item [ 6-] S.G.F. Martins and T.J.P. Penna, {\\it Int.J.Mod.Phys.} {\\bf C9}, 491 \n(1998).\n\\item [ 7-] L. Lefebvre, S. Viville, S. C. Barton, F. \nIshino, E.B. Keverne and M.A. Surani, {\\it Nature Genetics} {\\bf 20}, 163 \n(1998).\n\\item [ 8-] R.S. Bridges, {\\it Nature Genetics} {\\bf 20}, 108 (1998). \n\\item [ 9-] A.T. Bernardes, {\\it Annual Reviews of Computational Physics IV}, \nedited by D. Stauffer, World Scientific, Singapore (1996). \n\\item [10-] J.S. Sa Martins and S. Moss de Oliveira, {\\it Int.J.Mod.Phys.} \n{\\bf C9}, 421 (1998). \n\n\\end{description}\n\n\\newpage\n\\centerline{\\bf Figure Captions}\n\n\\bigskip\n\\noindent Fig.1 - Final percentages of faithful males in the population as a \nfunction of the death probability of the abandoned pups. The solid line \ncorresponds to the cases where the offspring fidelity state is the same of \nthe father. The dashed line corresponds to the cases where the offspring \ninherit the same fidelity of the father with probability $80\\%$. \n\n\\bigskip\n\\noindent Fig.2a - Time evolution of the populations (linear-log scale) for \n$P_f=1$ and different offspring death probabilities $P_d$. The inset shows \nthe final population sizes as a function of $P_d$. For $0.6 \\le P_d \\le 1$ \nthe final sizes are all very close to that for $P_d = 0.3$.\n\n\\bigskip\n\\noindent Fig.2b - The same as Fig.2a for $P_f = 0.8$. \n\n\\bigskip\n\\noindent Fig.3 - Survival rates as a function of age for $P_f = 1$ and \ndifferent values of $P_d$; circles correspond to $P_d = 0.1$, squares \nto 0.5 and triangles to 0.9. A higher child mortality can be noticed \nfor $P_d = 0.5$. \n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"
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cond-mat0002055
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Spin and link overlaps in 3-dimensional spin glasses
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[
{
"author": "F. Krzakala and O.C. Martin"
}
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Excitations of three-dimensional spin glasses are computed numerically. We find that one can flip a finite fraction of an $L~\times~L~\times~L$ lattice with an $O(1)$ energy cost, confirming the mean field picture of a non-trivial spin overlap distribution $P(q)$. These low energy excitations are not domain-wall-like, rather they are topologically non-trivial and they reach out to the boundaries of the lattice. Their surface to volume ratios decrease as $L$ increases and may asymptotically go to zero. If so, link and window overlaps between the ground state and these excited states become ``trivial''.
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[
{
"name": "km.tex",
"string": "\\documentstyle[prl,aps,floats,twocolumn,graphics]{revtex}\n\n\\author{F. Krzakala and O.C. Martin}\n\\title{Spin and link overlaps in 3-dimensional spin glasses}\n\\address{Laboratoire de Physique Th\\'eorique et Mod\\`eles Statistiques,\nb\\^at. 100, Universit\\'e Paris-Sud, F--91405 Orsay, France.}\n\\date{\\today}\n\\draft\n\n\\begin{document}\n\n\\maketitle\n\\begin{abstract}\nExcitations of three-dimensional spin glasses are computed numerically.\nWe find that one can flip a finite fraction of an $L~\\times~L~\\times~L$\nlattice with an $O(1)$ energy cost, confirming the mean field picture\nof a non-trivial spin overlap distribution $P(q)$. These low energy\nexcitations are not domain-wall-like, rather they are topologically\nnon-trivial and they reach out to the boundaries of the lattice.\nTheir surface to volume ratios decrease as $L$ increases and may\nasymptotically go to zero. If so, link and window overlaps between\nthe ground state and these excited states become ``trivial''.\n\\end{abstract}\n\n\\pacs{75.10.Nr, 75.40.Mg, 02.60.Pn}\n\n%\\paragraph*{Introduction ---}\nSpin glasses~\\cite{Young98} are currently at the center of a hot debate.\nOne outstanding question is whether there exists\nmacroscopically different valleys whose\ncontributions simultaneously dominate the partition function.\nAt zero temperature, given the ground state configuration,\nthis leads one to ask whether\nit is possible to flip a finite fraction of the spins and\nreach a state with excess energy $O(1)$. From a mean-field\nperspective~\\cite{MezardParisi87b},\none expects this to be true since it happens in the\nSherrington-Kirkpatrick (SK) model. However it\nis very unnatural in the context of the droplet~\\cite{FisherHuse88}\nor scaling~\\cite{BrayMoore86} approaches where the characteristic\nenergy of an excitation grows with its size.\nRecently it has been suggested that\nthe energy of an excitation may grow with its size $\\ell$ as in\nthe droplet scaling law, $E(\\ell)\\approx \\ell^{\\theta_l}$, but only for\n$\\ell \\ll L$, and that for $\\ell=O(L)$ the energies cross\nover to a different law,\n$E \\approx L^{\\theta_g}$, where $L$ is the size of the\nsystem~\\cite{HoudayerMartin00b}.\nThe first exponent, $\\theta_l$ ($l$ for local), may be given by\ndomain wall estimates, $\\theta_l \\approx 0.2$, while\nthe second exponent, $\\theta_g$ ($g$ for global), could be given by the\nmean field prediction, $\\theta_g = 0$.\nIn this ``mixed'' scenario,\none has coexistence of the droplet model at finite length scales\nand a mean-field behavior (if $\\theta_g = 0$) for\nsystem-size excitations ($\\ell \\approx L$ for which a finite fraction\nof all the spins are flipped).\n\nThe purpose of this article is to provide numerical evidence that such a\nmixed scenario is at work in the three-dimensional\nEdwards-Anderson\n%Edwards-Anderson~\\cite{EdwardsAnderson75}\nspin glass. We have determined ground states\nand excited states for different lattice sizes and have\nanalyzed their geometrical properties. The qualitative picture\nwe reach is that indeed $\\theta_g \\approx 0$.\nSystem-size constant energy excitations are not\nartefacts of trapped domain walls caused by periodic\nboundary conditions, they are intrinsic to\nthis kind of frustrated system. The energy landscape of the\nEdwards-Anderson model then consists\nof many valleys, probably separated by large energy barriers.\nExtrapolating to finite temperature, this picture leads to\na non-trivial equilibrium spin overlap\ndistribution function $P(q)$.\n\nGiven the geometric\nproperties of our excitations, we suggest a new scenario\nfor finite dimensional spin glasses: if\nthe surface to volume ratios of these large\nscale excitations go to zero in the large $L$ limit,\nthen the replica symmetry\nbreaking will be associated with a {\\it trivial} link\noverlap distribution function $P(q_l)$.\n% (tending towards a delta function in the large volume limit).\nWe have coined this scenario TNT for\ntrivial link overlaps yet non-trivial spin\noverlaps. Such a departure from the standard mean field picture\nmight hold in any dimension $d \\ge 3$.\n\n\\paragraph*{The spin glass model ---}\nWe consider an Edwards-Anderson Hamiltonian\non a three-dimensional $L \\times L \\times L$ cubic lattice:\n\\begin{equation}\n\\label{eq_H_EA}\nH_J(\\{S_i\\}) = - \\sum_{<ij>} J_{ij} S_i S_j .\n\\end{equation}\nThe sum is over all nearest neighbor spins of the lattice. The\nquenched couplings $J_{ij}$ are independent random variables,\ntaken from a Gaussian distribution of zero mean\nand unit variance. For the boundaries,\nwe have imposed either periodic or free boundary conditions.\nAlthough in simulations of\nmost systems it is best to use periodic boundary conditions so\nas to minimize finite size corrections,\nthe interpretation of our data is simpler\nfor free boundary conditions. It may also be useful to note\nthat if boundary conditions matter in the infinite\nvolume limit, free boundary conditions are the experimentally\nappropriate ones to use.\n\n\\paragraph*{Extracting excited states ---}\nThe problem of finding the ground state of a spin glass\nis a difficult one. In this study we use a previously\ntested~\\cite{HoudayerMartin99b} algorithmic procedure\nwhich, given enough computational ressources,\ngives the ground state\nwith a very high probability for lattice sizes up to\n$12 \\times 12 \\times 12$. (Since our $J_{ij}$s are continuous,\nthe ground state is unique up to a global spin flip.)\nOur study here is limited to sizes $L \\le 11$;\nthen the rare errors in obtaining the ground states\nare far less important than our statistical errors\nor than the uncertainties in extrapolating our results\nto the $L \\to \\infty$ limit.\n\nOur purpose is to extract low-lying excited states to see whether\nthere are valleys as in the mean field picture or whether the characteristic\nenergies of the lowest-lying large scale excitations grow with $L$ as\nexpected in the droplet/scaling picture. Ideally,\none would like to have a list of all the states whose excess\nenergy is below a given cut-off. However, because there is a\nnon-zero density of states associated with droplets (localized\nexcitations), this is an impossible task for the sizes of\ninterest to us. Thus instead we extract our\nexcitations as follows. Given the\nground state (hereafter called $C_0$), we\nchoose two spins $S_i$ and $S_j$ at random and force their\nrelative orientation to be opposite from what it\nis in the ground state. This constraint can be implemented by replacing\nthe two spins by one new spin giving the orientation of the first spin,\nthe other one being its ``slave''.\nWe then solve for the ground state $C$ of this modified spin glass.\nThe new state $C$ is necessarily distinct from $C_0$ as at least\none spin ($S_i$ or $S_j$) is flipped. That flipped spin may drag along\nwith it some of its surrounding spins, forming a droplet of\ncharacteristic energy $O(1)$. In the droplet picture, this\nis all that happens in the infinite volume limit. However if there\nexist large scale excitations with $O(1)$ energies,\nthen $C$ may be such\nan excitation if its energy is below that of {\\it all} the droplets\ncontaining either $S_i$ or $S_j$.\n\n\\paragraph*{Statistics of cluster sizes ---}\nLet $V$ be the number of sites of the cluster defining\nthe spins that are flipped when going from $C_0$ to $C$\n(by symmetry, $V$ is taken in $\\lbrack 1, L^3/2 \\rbrack$).\nIf $P(V)$ is the probability to have an event of size $V$,\nthe droplet and mean field\npictures lead us to the following parametrization:\n\\begin{equation}\n\\label{eq_P_V}\nP(V) = (1-\\alpha) P_l(V) + \\alpha P_g(V/L^3) .\n\\end{equation}\nHere, $P_l$ and $P_g$ are\nnormalized probability distributions associated\nwith the droplet events ($V$ fixed, $L \\to \\infty$)\nand the global events ($V = O(L^3)$).\nIf large scale excitations\nhave energies $O(L^{\\theta_g})$, the ratio\n$\\alpha / (1 - \\alpha )$ of the two contributions\nshould go as $L^{-\\theta_g}$.\nIn the droplet/scaling picture, the global part\ndecreases as $L^{-\\theta_l}$; that\nis slow since $\\theta_l \\approx 0.2$.\nIn contrast, in the mean field\nscenario, both the $V$ finite and the $V$ growing\nlinearly with $L^3$ contributions converge with non-zero weights,\n$0 < \\alpha < 1$, albeit with $O(L^{-\\theta_l})$ finite\nsize corrections.\n\nGiven that the usable range in $L$ is no more than a\nfactor of two so that $L^{-\\theta_l}$ does not vary much,\nmeasurements of $P(V)$ on their\nown are unlikely to provide stringent tests.\nNevertheless, consider the probability\n$Q(v,v')$ that $V/L^3$ is in the interval $\\lbrack v,v' \\rbrack$.\nUp to finite size corrections,\n$Q(v,v') = \\alpha \\int_{v}^{v'} P_g(x) dx$.\nIn our computations, we\nhave used $5 \\le L \\le 11$, averaging for each $L$ over\n2000 to 10000 randomly generated samples of the\n$J_{ij}$. For each sample, we determined the ground state, and then\nobtained $3$ excitations by choosing successively at random\n$3$ pairs of spins ($S_i$,$S_j$).\nWe find that $Q(v,v')$ decreases slowly with $L$ for both\nperiodic and free boundary conditions, as expected in\nthe droplet and mean field pictures. Because\n$\\theta_l$ is small, when we perform fits of the form\n$Q(v,v') = A + B L^{-\\mu}$, we are not able\nto exclude $A = 0$ nor $A \\ne 0$ with any significant\nconfidence, so a more refined method of analysis is\nnecessary: we will thus consider the geometrical properties\nof the events.\n\nBefore doing so, note that the statistical error on\n$Q(v,v')$ depends on the number of large scale events found\nin the $\\lbrack v, v' \\rbrack$ interval.\nIf the spin $S_i$ or $S_j$ has a small local field,\nthere is a good chance that the corresponding event will\nhave $V=1$, thereby reducing the statistics of the\n``interesting'' events. To amplify our signal of\nlarge $V$ events, we did not consider such spins and focused\nour attention on spins in the top $25$ percentile when\nranked according to their local field. All of our data\nwas obtained with\nthat way of selecting $S_i$ and $S_j$. (Naturally, $P(V)$ and\n$Q(v,v')$ depend on this choice, but the general\nbehavior should be the same for any choice.)\n\n\\paragraph*{Topological features of the clusters ---}\nOur claim that $\\theta_g < \\theta_l$ can be credible\nonly if our large scale excitations\nare different from domain-walls\n(whose energies are believed to\ngrow as $L^{0.2}$). It is thus useful to\nconsider geometrical characterizations of the excitations\ngenerated by our procedure. Figure~\\ref{fig_cuboid}\nshows a typical cluster found for a $12^3$ lattice. It contains 622 spins\nand its (excitation) energy is 0.98 which is $O(1)$.\nThe example displayed is for free boundary conditions which\npermits a better visualization than periodic boundary\nconditions.\n\nThe cluster shown touches\nmany of the 6 faces of the cube, and the same is true\nfor the complement of that cluster.\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.7\\linewidth}{!}{\\includegraphics{fig_cuboid.eps}}\n\\end{center}\n\\caption{Example of excitation found for a $12^3$ lattice with\nfree boundary conditions.}\n\\label{fig_cuboid}\n\\end{figure}\nSuch a cluster has a very non-trivial topology and is thus very far\nfrom being domain-wall like. This motivates the following\nthree-fold classification of the events\nwe obtain when considering free boundary\nconditions. In the first class, a cluster\nand its complement touch all $6$ faces of the cube. In the\nsecond class, a cluster touches at most $3$ faces\nof the cube. The third class consists of all other events.\nFinite size droplets should asymptotically always fall into\nthe second class, albeit with finite size corrections\nof order $L^{-\\theta_l}$.\n\nDoes the first class constitute a non-zero\nfraction of all events?\nAt finite $L$, we find the following fractions: $23.3\\%$ $(L=5)$,\n$23.9\\%$ $(L=6)$, $25.1\\%$ $(L=7)$, $24.4\\%$\n$(L=8)$, $25.0\\%$ $(L=9)$, $25.7\\%$ $(L=10)$,\nand $26.0\\%$ $(L=11)$.\nThe trend of these numbers suggests that the first class does\nindeed encompass a finite fraction\nof all the events when $L \\to \\infty$. We also considered\nthe scaling of cluster sizes with $L$.\nFig.~\\ref{fig_touch} shows $Q(v,1/2)$\nas a function of $v=V/L^3$,\nrestricted to events belonging to the first class.\n(The $v=0$ values are the fractions\nwe just gave above.) The curves for different $L$ show\na small drift, $Q(v,1/2)$ {\\it growing} with $L$.\nWe consider this drift to be a finite size effect and\nthat the correct interpretation of our data is\n$\\theta_g \\approx 0$, in agreement with the mean field picture.\nOur conclusion is then that\nas $L \\to \\infty$, there is a finite probability of having an\n$O(1)$ energy excitation that is non-domain-wall like,\nthe cluster and its complement touching {\\it all} faces of\nthe cube.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.7\\linewidth}{!}{\\includegraphics{fig_Q1class.pstex}}\n\\end{center}\n\\caption{Integrated probability $Q(v,1/2)$ of events in the first\nclass. (From bottom to top, $L=5, 7, 9, 11$.}\n\\label{fig_touch}\n\\end{figure}\n\n\\paragraph*{Surface to volume ratios ---}\nObviously the mean field picture obtained by extrapolating results\nfrom the SK or Viana-Bray spin glasses\ncannot teach us anything about the topology\nof excitations for three-dimensional\nlattices. But mean field may serve\nas a guide for other properties such as the link overlap\n$q_l$ between ground states and excited states. In the SK\nmodel, the {\\it spin} overlap\n$q \\equiv \\sum S_i S_i' / N$ and the {\\it link} overlap\n$q_l\\equiv \\sum (S_i S_j)(S_i' S_j') / (N (N-1)/2)$ satisfy\n$q_l = q^2$, and both $q$ and $q_l$ have non-trivial distributions.\nExtrapolating this to our three dimensional system, the mean field\npicture predicts that the clusters associated with large scale\nexcitations both span the whole system (as we saw with\nfree boundary conditions) and are {\\it space filling}.\nQuantitatively, this implies that their surface grows as the\ntotal volume of the system, {\\it i.e.}, as $L^3$.\n\nTo investigate this question, we have measured the surface\nof our excitations, defined as the number $S$ of links connecting\nthe corresponding cluster to its complement.\n(Then $q_l = 1- 2 S /3 L^3$.)\nIn figure~\\ref{fig_surface} we show the\nmean value of $S / L^3$ as a function of $L$ for\n$v=V/L^3$ belonging to the three intervals\n$\\rbrack 0.20, 0.25 \\rbrack$,\n$\\rbrack 0.30, 0.35 \\rbrack$, and\n$\\rbrack 0.40, 0.45 \\rbrack$.\nThe data shown are for free boundary\nconditions, but the results are very similar\nfor periodic boundary conditions. The most striking feature is that the\ncurves decrease very clearly with\n$L$. For each interval, we have fitted the data to\n$\\langle S \\rangle / L^3 = A + B/L^{\\mu}$ and to a\npolynomial in $1/L$. Of major interest is the value of the constant\nbecause it gives the large $L$ limit of the curves.\n\nTable~\\ref{tab_chi2s} summarizes the quality of the\nfits as given by their $\\chi^2_r$ (chi squared per degree of\nfreedom). In all cases the fits are reasonably good; this\nis not so surprizing because our range of $L$ values is small.\nThe most reliable fits are obtained using a quadratic polynomial in $1/L$,\nthis functional form leading to a smooth and\nmonotone behavior of the parameters\nand to small uncertainties in the parameters.\nFor the large $L$ limits, these fits give\n$A=0.22$, $A=0.27$ and $A=0.30$ for the three intervals.\n(We do not give results for the\nlinear fits which on the contrary are very poor.) The constant\nplus power fits also have good $\\chi^2_r$ but the $A$s obtained were\nsmall and {\\it decreased} with $v$; also they\nhad large uncertainties and seemed to be compatible with $A=0$. Because\nof this, we also performed fits of the form\n$\\langle S \\rangle / L^3 = B/L^{\\mu}$. These are displayed in\nFig.~\\ref{fig_surface} and lead to $\\mu \\approx 0.30$ (the\nexponent varies little from curve to curve),\nagain with reasonable $\\chi^2_r$s. Because of this, we feel we\ncannot conclude from the data that the surface to volume ratios\ntend towards a non-zero asymptote. What can be said is that\nthis asymptote seems to be small, and that it will be difficult to\nbe sure that it is non-zero without going to larger\nvalues of $L$.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.7\\linewidth}{!}{\\includegraphics{fig_surf.pstex}}\n\\end{center}\n\\caption{Mean value of surface to $L^3$ ratios for\n$v=V/L^3$ in intervals around $0.225, 0.325, 0.425$ (bottom to top) using\nfree boundary conditions. Curves are pure power fits.}\n\\label{fig_surface}\n\\end{figure}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccc}\nInterval & $A + B/L^{\\mu}$ & $A + B/L + C/L^2$ & $B/L^{\\mu}$\\\\\n\\hline\n$\\rbrack 0.20,0.25 \\rbrack$ & 0.6 & 0.6 & 2.0\\\\\n$\\rbrack 0.30,0.35 \\rbrack$ & 1.1 & 1.1 & 1.5 \\\\\n$\\rbrack 0.40,0.45 \\rbrack$ & 0.7 & 0.9 & 0.6 \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Chi squared per degree of freedom for the fits to the data\nof Fig.~\\ref{fig_surface}.}\n\\label{tab_chi2s}\n\\end{table}\n\n\n\\paragraph*{Discussion ---}\nFor the three dimensional Edwards-Anderson spin glass model,\nwe have presented numerical evidence\nthat it is possible to flip a finite\nfraction of the whole lattice at an energy cost of $O(1)$, corresponding\nto $\\theta_g \\approx 0$ as predicted by mean field. This\nproperty transpired most clearly through the use of\nfree boundary conditions, allowing one to conclude that\n$\\theta_g \\approx 0$ is not an artefact of trapped domain walls caused by\nperiodic boundary conditions. Extrapolating to finite\ntemperature, we expect the equilibrium $P(q)$ to be non trivial as in the\nmean field picture.\n\nThe other messages of our work concern the nature of\nthese large scale excitations whose energies are $O(1)$. First,\nusing free boundary conditions, we found them to be\ntopologically highly non-trivial:\nwith a finite probability they reach the boundaries\non all 6 faces of the cube. Thus they are not domain-wall-like,\nrather they are sponge-like. Second, our data (both for\nperiodic and free boundary conditions) indicate very clearly that\ntheir surface to volume ratios decrease as $L$ increases. The\nmost important issue here is whether or not these ratios decrease to zero\nin the large $L$ limit. Although our data are\ncompatible with a non-zero limiting value as predicted\nby mean field, the fits were not conclusive\nso further work is necessary.\n\nIf the surface to volume ratios turned out to go to zero, we would\nbe lead to a new scenario that we have coined ``TNT''. In the standard\nmean field picture, the surface to volume ratios cannot go\nto zero; indeed\nin the SK and Viana-Bray spin glass models\nthere are {\\it no} spin clusters with\nsurface to volume ratios going to zero.\nHowever, in finite dimensions, one can have\nsurface to volume ratios going to zero, in which case\n$q_l \\to 1$. This property would then lead to a non-trivial\n$P(q)$ but to a trivial $P(q_l)$.\nThis trivial-non-trivial (TNT)\nscenario does not seem to have been proposed previously.\n\nPerhaps the most dramatic consequence of this new scenario is for window\noverlaps in spin glasses: because in TNT one is asymptotically always\nin the bulk of an excitation,\ncorrelation functions at any finite distance will show\nno effects of replica symmetry breaking. That this may arise in\nfact is supported by work by Palassini and\nYoung\\cite{PalassiniYoung99a}\nwho showed that certain window overlaps seemed to become\ntrivial as $L \\to \\infty$.\n(See also~\\cite{Middleton99} for a similar discussion in\ntwo-dimensions.) These authors\nreferred to this property as evidence for\na ``trivial ground state structure''.\nBut in our picture the {\\it global} (infinite distance)\nstructure is non-trivial, as indicated by\n$\\theta_g = 0$, in sharp contrast to the droplet/scaling\npicture. Also, in very recent work~\\cite{YoungPrivate},\nPalassini and Young have extended their previous investigations\nand have extracted excited states by a quite different\nmethod from ours, and\nthey find that their data is compatible with\nthe TNT scenario.\nNaturally, there is also evidence in {\\it favor}\nof the non-triviality\nof window overlaps~\\cite{MarinariParisi99b}.\nNevertheless, we believe that our mixed scenario is\na worthy candidate to describe the physics of short range\nspin glasses. Furthermore, its plausibility should not\nrestricted to $3$ dimensions, it could hold in\nall dimensions greater than $2$. (Note that in $d=2$, excitations\nare necessarily topologically trivial.) An important indication\nof this was obtained by Palassini and Young whose\ncomputations~\\cite{YoungPrivate} favor the TNT\nscenario over the droplet picture\nin the 4-dimensional Edwards-Anderson model.\n\n%Note: systematic errors due to not obtaining the true ground state\n%are far smaller than statistical errors or uncertainty in the\n%extrapolation to large sizes.\n\n\\paragraph*{Acknowledgements ---}\nWe thank J.-P. Bouchaud and M. M\\'ezard for very stimulating\ndiscussions and for their continuous encouragement, and\nM. Palassini and A.P. Young for letting us know about their\nwork before publication. Finally, we thank\nJ\\'er\\^ome Houdayer; without his superb work on the genetic\nrenormalization approach~\\cite{HoudayerMartin99b}, this numerical\nstudy would not have been possible.\nF.K. acknowledges support from the\nMENRT, and O.C.M. acknowledges support from the Institut Universitaire de\nFrance. The LPTMS is an Unit\\'e de Recherche de\nl'Universit\\'e Paris~XI associ\\'ee au CNRS.\n\n\\bibliographystyle{prsty}\n\\bibliography{../../../Bib/references}\n\n\\end{document}\n\n"
}
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[
{
"name": "km.bbl",
"string": "\\addcontentsline{toc}{chapter}{\\protect\\bibname}\n\\begin{thebibliography}{10}\n\n\\bibitem{Young98}\n{\\em Spin Glasses and Random Fields}, edited by A.~P. Young (World Scientific,\n Singapore, 1998).\n\n\\bibitem{MezardParisi87b}\nM. M{\\'e}zard, G. Parisi, and M.~A. Virasoro, {\\em Spin-Glass Theory and\n Beyond}, Vol.~9 of {\\em Lecture Notes in Physics} (World Scientific,\n Singapore, 1987).\n\n\\bibitem{FisherHuse88}\nD.~S. Fisher and D.~A. Huse, Phys. Rev. B {\\bf 38}, 386 (1988).\n\n\\bibitem{BrayMoore86}\nA.~J. Bray and M.~A. Moore, in {\\em Heidelberg Colloquium on Glassy Dynamics},\n Vol.~275 of {\\em Lecture Notes in Physics}, edited by J.~L. van Hemmen and I.\n Morgenstern ({S}pringer, {B}erlin, 1986), pp.\\ 121--153.\n\n\\bibitem{HoudayerMartin00b}\nJ. Houdayer and O.~C. Martin, Euro. Phys. Lett. {\\bf 49}, 794 (2000).\n\n\\bibitem{HoudayerMartin99b}\nJ. Houdayer and O.~C. Martin, Phys. Rev. Lett. {\\bf 83}, 1030 (1999).\n\n\\bibitem{PalassiniYoung99a}\nM. Palassini and A.~P. Young, Phys. Rev. Lett. {\\bf 83}, 5126 (1999).\n\n\\bibitem{Middleton99}\nA. Middleton, Phys. Rev. Lett. {\\bf 83}, 1672 (1999).\n\n\\bibitem{YoungPrivate}\nM. Palassini and A.~P. Young, private communication.\n\n\\bibitem{MarinariParisi99b}\nE. Marinari {\\it et~al.}, J. Stat. Phys. {\\bf 98}, 973 (2000).\n\n\\end{thebibliography}\n"
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cond-mat0002056
|
Fermi systems with long scattering lengths
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[
{
"author": "Henning Heiselberg"
}
] |
Ground state energies and superfluid gaps are calculated for degenerate Fermi systems interacting via long attractive scattering lengths such as cold atomic gases, neutron and nuclear matter. In the intermediate region of densities, where the interparticle spacing $(\sim 1/k_F)$ is longer than the range of the interaction but shorter than the scattering length, the superfluid gaps and the energy per particle are found to be proportional to the Fermi energy and thus differs from the dilute and high density limits. The attractive potential increase linearly with the spin-isospin or hyperspin statistical factor such that, e.g., symmetric nuclear matter undergoes spinodal decomposition and collapses whereas neutron matter and Fermionic atomic gases with two hyperspin states are mechanically {stable} in the intermediate density region. The regions of spinodal instabilities in the resulting phase diagram are reduced and do not prevent a superfluid transition.
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[
{
"name": "a.tex",
"string": "%%%%%%%%%%%%%%%%%%% MANUSCRIPT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentstyle[aps,prd,twocolumn,epsf]{revtex}\n%\\documentstyle[aps,multicol,psfig]{revtex}\n%\\documentstyle[aps,twocolumn,epsfig]{revtex}\n%\\documentstyle[preprint,aps]{revtex}\n%\\tightenlines\n%\\draft\n\\newcommand \\bea{\\begin{eqnarray}}\n\\newcommand \\eea{\\end{eqnarray}}\n\\newcommand \\ga{\\raisebox{-.5ex}{$\\stackrel{>}{\\sim}$}}\n\\newcommand \\la{\\raisebox{-.5ex}{$\\stackrel{<}{\\sim}$}}\n\\newcommand{\\av}[1]{\\langle{#1}\\rangle}\n\\begin{document}\n%\\wideabs{\n\\title{Fermi systems with long scattering lengths}\n\\author{Henning Heiselberg}\n\\address{NORDITA, Blegdamsvej 17, DK-2100 Copenhagen \\O, Denmark}\n%\\date{Sept. 1999}\n\n\\maketitle\n\n\\begin{abstract}\nGround state energies and superfluid gaps are calculated for\ndegenerate Fermi systems interacting via long attractive scattering\nlengths such as cold atomic gases, neutron and nuclear matter. In the\nintermediate region of densities, where the interparticle spacing\n$(\\sim 1/k_F)$ is longer than the range of the interaction but shorter\nthan the scattering length, the superfluid gaps and the energy per\nparticle are found to be proportional to the Fermi energy \nand thus differs from\nthe dilute and high density limits. The attractive potential increase\nlinearly with the spin-isospin or hyperspin statistical factor such\nthat, e.g., symmetric nuclear matter undergoes spinodal decomposition\nand collapses whereas neutron matter and Fermionic atomic gases with\ntwo hyperspin states are mechanically {\\it stable} in the intermediate\ndensity region. The regions of spinodal instabilities in the \nresulting phase diagram are reduced and do not prevent a superfluid\ntransition.\n\\end{abstract}\n\n\\vspace{1cm}\n\n%PACS numbers: \n\n%Keywords: \n\n\\section{Introduction}\nDilute degenerate Fermi systems with long scattering lengths are of\ninterest for nuclear and neutron star matter (see, e.g.,\n\\cite{Vijay}). Recently, also dilute systems of cold Fermionic atoms\nhave been trapped \\cite{JILA}. The number density is sufficient for\ndegeneracy to be observed and superfluidity is expected at critical\ntemperatures similar to the onset of Bose-Einstein condensates,\n$\\sim10-100$nK. S-wave scattering lengths can be very long,\ne.g.\\footnote{The sign convention of negative scattering length for\nattractive potentials is used. Also $\\hbar=c=1$.} $a=-2160$ Bohr\nradii for triplet $^{6}Li$ and $a\\sim -18.8$~fm for neutron-neutron\ninteractions. These scattering lengths $|a|$ are much longer than\ntypical range of the potentials, respectively $R\\sim1$~fm for strong\ninteractions and $R\\sim 10-100$~\\AA\\ for van der Waals forces.\nGenerally, when $|a|\\gg R$ three density regimes naturally emerges:\nthe low density (or dilute, $k_F^{-1}\\ga |a|$), the high density\n($k_F^{-1}\\la R$), and the {\\it intermediate density region} ($R\\la\nk_F^{-1}\\la |a|$).\n\nThe latter ``novel'' region of densities is the object of study here.\nIt will be shown that previous conjectures based on extrapolations\nfrom the dilute limit fail. Instead it is found that both the energy\nper particle and superfluid gaps scale with the Fermi energy. They\ndepend only on statistical factors but not on the scattering length,\nrange or other details of the interaction. Consequently, phase\ndiagrams are dramatically altered and the stability criteria differ so\nthat two spin systems are actually {\\it stable}.\n\nThe manuscript is organized as follows. In Sec. II the scaling and\npoles of the effective scattering amplitude are studied in a\nhomogeneous many-body system going from dilute to intermediate\ndensities. In Sec. III the ground state energies are calculated at\nintermediate densities and compared to well known results from the\ndilute and high density limits. In Sec. IV extensions to finite\ntemperatures are discussed and a phase diagram is constructed\ndisplaying regions of superfluidity and spinodal instabilities. In\nSec. V the properties of finite systems of Fermions are investigated\nas they are relevant for current experiments with magnetically trapped\ncold atoms. Finally, a summary and conclusion is given.\n\n\n\\section{The effective scattering amplitude and superfluidity}\nConsider a homogeneous many-body system of fermions of mass $m$ at\ndensity: $\\rho=\\nu k_F^3/6\\pi^2$, where $\\nu$ is the statistical factor,\ne.g. $\\nu=4$ for symmetric nuclear matter and $\\nu=2$ for neutron matter\nas well as for the $^{40}K$ atomic gas with two hyperspin states\ncurrently studied at JILA \\cite{JILA}. The scattering lengths and\nFermi momentum $k_F$ are assumed the same for all\nspin-isospin/hyperspin components in the system but interesting\neffects of varying the relative densities of the various components\nwill be discussed at the end. Particles are assumed non-relativistic\nand to interact through attractive two-body contact interactions. The\ndetails of the potential is not important, only its range $\\sim R$ and\nscattering length $a$. We shall be particular interested in cases\nwhere $R\\ll|a|$, which occur when, for example, the two-body\npotential can almost support a bound state or resonance.\n\nAt dilute or intermediate densities the particles interact\nvia short range interactions that appear singular on length scales\nof order the interparticle distance $\\sim k_F^{-1}$.\nSuch systems are best described by resumming the multiple interactions\nin terms of the scattering amplitude. The\nGalitskii's integral equations \\cite{Galitskii} for the effective\ntwo-particle interaction or\nscattering amplitude in the medium is given by the ladder resummation\n\\bea\n \\Gamma({\\bf p,p',P}) &=& \\Gamma_0({\\bf p,p',P})\n + m\\sum_{\\bf k} \\Gamma_0({\\bf p,k,P}) \\nonumber\\\\ \n &\\times&\\left[\\frac{N({\\bf P,k})}{\\kappa^2-k^2} - \\frac{1}{\\kappa^2-k^2}\n \\right] \\Gamma({\\bf k,p',P}) \\, . \\label{Galitskii}\n\\eea\nHere, $\\Gamma_0=4\\pi a/m$ is the s-wave scattering amplitude in\nvacuum; the total energy of the pair in the center of mass is $\\kappa^2/m$;\n${\\bf p,k,p'}$ are the relative momentum of the\npair of interacting particles in the initial, intermediate and final states\nrespectively, and ${\\bf P}$ the total momentum; \n$N({\\bf P,k})=+1$ for particle-particle propagation ($|{\\bf\nP}\\pm{\\bf k}|\\ge k_F$), $N({\\bf P,k})=-1$ for hole-hole propagation ($|{\\bf\nP}\\pm{\\bf k}|\\le k_F$), and zero otherwise. For spin independent\ninteractions the amplitudes contain a factor $(1-\\delta_{\\nu_1\\nu_2})$\nthat takes exchange into account between identical spins $\\nu_1=\\nu_2$.\n\nFor obtaining BCS gaps it is sufficient to study pairs with ${\\bf P}=0$ where\nEq. (\\ref{Galitskii}) is simply\n\\bea\n \\Gamma=\\Gamma_0 + 2m\\sum_{k\\le k_F} \\Gamma_0 \\frac{1}{\\kappa^2-k^2} \\Gamma\n \\,. \\label{Cooper}\n\\eea\nNote the factor of 2 due to particle-particle and hole-hole \npropagation each contributing by the same amount in non-dense systems.\nThe ladder resummation implicit in Eq. (\\ref{Galitskii}) insures that\nonly momenta smaller than Fermi momenta enter.\n$\\Gamma_0$ vary on momentum scales $\\sim 1/R\\gg k_F$ only and\ncan therefore be considered constant at low and intermediate densities.\nEq. (\\ref{Cooper}) is then easily solved for momenta near\nthe Fermi surface\n\\bea\n \\Gamma &=& \\Gamma_0\\left[1-\\frac{2}{\\pi}k_Fa\n (2+\\frac{\\kappa}{k_F}\\ln\\frac{k_F-\\kappa}{k_F+\\kappa})\\right]^{-1}\n \\,. \\label{G3}\n\\eea\nThe in-medium scattering amplitude has a pole due to Cooper pairing when\n\\bea\n \\Delta &\\equiv& \\frac{k_F^2-\\kappa^2}{m} \n = \\frac{(k_F+\\kappa)^2}{m}\\exp(\\frac{\\pi}{2\\kappa a}-2\\frac{k_F}{\\kappa})\n \\nonumber\\\\\n &\\simeq& E_F \\frac{8}{e^2} \\exp(\\frac{\\pi}{2k_Fa}) ,\\quad k_F|a|\\ll 1\\,,\n \\label{gap}\n\\eea\nwhere $E_F=k_F^2/2m$ is the Fermi energy. The critical temperature is\n$T_c=(\\gamma/\\pi)\\Delta$, where $\\gamma=e^C$ and $C=0.577$ is \nEuler's constant.\nEq. (\\ref{gap}) is the BCS gap in the dilute limit which agrees with\ngaps calculated in \\cite{Melo}.\n\nHowever, Gorkov et al. \\cite{Gorkov} showed that spin fluctuations lead to\na higher order correction $\\sim (k_Fa)^2$ in the denominator of \nEq. (\\ref{G3}) that\nis amplified by logarithmic terms $\\sim \\ln(\\Delta)\\sim 1/k_Fa$.\nIt contributes by a (negative) constant in the exponent and leads to a \nreduction of the gap in the dilute limit\nby a factor $(4e)^{1/3}=2.215...$ as compared to Eq. (\\ref{gap})\nfor two spins. Generally for $\\nu$ spins, isospins or hyperspins\nthe gap is \\cite{gap}\n\\bea\n \\Delta = E_F \\frac{8}{e^2} (4e)^{\\nu/3-1} \n \\exp\\left[\\frac{\\pi}{2ak_F}\\right] \\, . \\label{Gorkov}\n\\eea\n\n In the intermediate density region pairing must still occur since\ninteraction are attractive. The validity of the expressions of\nEqs. (\\ref{gap},\\ref{Gorkov}) in this density regime will be discussed\nfurther below. They predict that in the limits $a\\to-\\infty$ and $R\\to\n0$ the gap cannot depend on either $a$, $R$ or other details of the\npotential. For dimensional reasons the gap can therefore only be\nproportional to the Fermi energy.\n\n\n\\section{Ground state energies}\nThe ground state energy is another crucial property of \nthe system. In terms of the on-shell\neffective scattering amplitude it is \\cite{Galitskii,FW}\n\\begin{eqnarray}\n E &=& \\sum_{k_1\\nu_1} \\frac{k_1^2}{2m} + \\frac{1}{2}\n \\sum_{k_1k_2\\nu_1\\nu_2} \\Gamma({\\bf p,p,P})\\, (1-\\delta_{\\nu_1\\nu_2})\n \\nonumber \\\\\n &=& \\frac{3}{5} \\frac{k_F^2}{2m} N\n +\\frac{\\nu(\\nu-1)}{2}\\sum_{k_1k_2} \\Gamma({\\bf p,p,P}) \n \\,. \\label{EV}\n\\end{eqnarray}\nHere,\n$N=V\\rho$ is the number of particles and\nthe summations $\\nu_1,\\nu_2$ include spin and isospin or hyperspin states.\nThe factor $(1-\\delta_{\\nu_1\\nu_2})$ in the amplitude\ndue to exchange has now been written explicitly.\nAntisymmetrization of the wave-function prevents identical particles to\nbe in relative s-states.\nAt low and intermediate densities, $k_FR\\ll 1$, \nthe exchange term is $1/\\nu$ of the direct one for spin independent\ninteractions.\n\nBefore investigating the novel intermediate density region,\na brief review of results at low and high densities is given.\n\n\\subsection{Low densities (dilute): $k_F|a|\\ll 1$}\nAt low densities, $k_F|a|\\ll 1$, gaps are small and have little effect\non the total energy of the system. Expanding the effective scattering\namplitude of Eq. (\\ref{Galitskii})\nin the small quantity $k_Fa$, the energy per particle\nis obtained from Eq. (\\ref{EV}) by summing over momenta of the two\ninteracting particles\n\\bea\n \\frac{E}{N} &=& E_F [\\frac{3}{5}+(\\nu-1)\\frac{2}{3\\pi} k_Fa\n \\nonumber\\\\\n &+&(\\nu-1)\\frac{4(11-2\\ln2)}{35\\pi^2}(k_Fa)^2\n + {\\cal O}((k_Fa)^3) ] \n \\,.\\label{Lenz}\n\\eea\nIt consists of respectively the Fermi kinetic energy, the standard\ndilute pseudo-potential \\cite{Lenz} proportional to the scattering length\nand density, and orders $(k_Fa)^2$ \\cite{HY} and higher \\cite{Efimov}. \n\nThe zero temperature hydrodynamic sound speed squared can\nat low temperatures be expressed as\n\\bea\n s^2 =\\frac{1}{m} \\left(\\frac{\\partial P}{\\partial \\rho}\\right)\n =\\frac{1}{m} \\frac{\\partial}{\\partial\\rho} \\left(\\rho^2\n \\frac{\\partial E/N}{\\partial\\rho}\\right) \\,,\\label{s}\n\\eea\nWith the energy per particle of Eq. (\\ref{Lenz}) \nat low densities, the sound speed can be expanded as\n\\bea\n s^2=\\frac{1}{3} v_F^2 \\left(1+\\frac{2}{\\pi}(\\nu-1)k_F a +...\\right) \n \\,.\\label{sl}\n\\eea\nwhere $v_F=k_F/m$ is the Fermi velocity.\nIt is commonly conjectured from the first two orders that the\nFermion (and Bose) gases undergo spinodal\ninstability when the sound speed squared becomes negative,\nwhich occurs when $k_Fa\\la -\\pi/2(\\nu-1)$.\nHowever, at the same densities the dilute approximation leading\nto Eq. (\\ref{Lenz}) fails and so does the conjecture \nas will be shown below.\n\n\\subsection{High densities, $k_F R\\gg 1$}\nAt high densities, $k_FR\\ga 1$, the particle potentials overlap\nand each particle experience on average the volume integral of the\npotentials.\nThe energy per nucleon consists of the Fermi kinetic energy and the \nHartree-Fock potential, (see, e.g. \\cite{FW} Eq. 40.14):\n\\begin{eqnarray}\n \\frac{E_{HF}}{N} &=& \\frac{3}{5} E_F\n +\\frac{\\rho}{2} \\int d^3r\\,V(r)\\left[ 1\n -\\frac{1}{\\nu}\\left(\\frac{3j_1(rk_F)}{rk_F}\\right)^2 \\right]\n . \\label{Hartree}\n\\end{eqnarray}\nThe latter term is the exchange energy which vanishes at very high\ndensities,\n$k_FR\\sim R/r_0\\gg 1$, leaving the Hartree potential term only.\nAt lower densities, $k_FR\\sim R/r_0\\ll 1$,\nit is identical to the first integral, i.e., the Hartree direct term is\n$\\nu$ times the Fock exchange term as also found in the dilute limit,\nEqs. (\\ref{Ei}-\\ref{Lenz}).\n\nAs shown in \\cite{PPT}, the Hartree potential is considerably less\nattractive than the dilute potential. In fact it vanishes when the range\nof the interaction goes to zero and the scattering length to infinity.\nTake for example a square well potential of range $R$ and depth\n$-V_0$. Long scattering lengths requires $V_0R^2\\to\\pi^2/4m$, and\ntherefore the Hartree potential is $\\propto V_0R^3\\sim R\\to 0$.\nOnly in the Born approximation do the Hartree (\\ref{Hartree})\nand dilute potentials (\\ref{Lenz}) coincide since the\nBorn scattering length is $a_{Born}=(m/4\\pi)\\int d^3r V(r)$.\n\nShort range repulsion complicate the high density limit. In nuclear\nand atomic system the repulsive core is only of slightly shorter range\nthan the attractive force. It makes the liquid strongly correlated\nand the Hartree-Fock approximation fails \\cite{BBG,FW}.\nHow the short range repulsion turn the attraction to repulsion at\nthese even higher densities will, however, not affect the\nintermediate density region.\n\n\\subsection{Intermediate densities, $|a|\\gg k_F^{-1}\\gg R$}\nAt intermediate densities, $|a|\\gg k_F^{-1}\\gg R$, the scattering\nlength expansion in Eq. (\\ref{Lenz}) breaks down. Brueckner, Bethe\nand Goldstone \\cite{BBG} pioneered such studies for nuclear matter and\n$^3He$ where the range of interactions, scattering lengths and\nrepulsive cores all are comparable in magnitude. In our case the\nrange of interaction is small, $k_FR\\ll1$, and therefore all\nparticle-hole diagrams are negligible. Higher order particle-particle\nand hole-hole diagrams do contribute by orders of $\\sim\n\\Gamma_0(mk_F\\Gamma)^n$, It is evident from Eq. (\\ref{gap}) that at\nintermediate densities $\\Gamma$ no longer is proportional to\n$\\Gamma_0$ or the scattering length but instead $\\Gamma\\propto\n(mk_F)^{-1}$. Due to the very restricted phase space such higher\norder terms are usually neglected as in standard Brueckner theory.\n$\\Gamma$ of Eq. (\\ref{Galitskii}) can therefore be considered as a\nresummation of an important class of diagrams.\nThe Cooper instability complicates the calculation of $\\Gamma$. If\nthe gap is small the instability occurs only for pairs near the Fermi\nsystem with opposite momenta and spin and the effect on the total energy\nis small. The momentum dependence of the effective scattering\namplitude also complicates a self-consistent calculation. These\ncomplications can be dealt with by approximating $\\Gamma$ by its momentum\naverage value in Eq. (\\ref{Galitskii}). The momentum integrals\nare then analogous to those in the dilute limit (\\ref{Lenz}), \nand one obtains from Eq. (\\ref{EV}) \n\\bea\n \\frac{E}{N} \\simeq E_F \\left[ \\frac{3}{5} + \\frac{(\\nu-1)\\frac{2}{3\\pi} k_Fa}\n {1-\\frac{6}{35\\pi}(11-2\\ln2)k_Fa} \\right] \\,. \\label{Ei}\n\\eea\nThis expression is valid \nfor dilute systems, where it reproduces Eq. (\\ref{Lenz}),\nand approximately valid within the Galitskii ladder resummation\nat intermediate densities, $R\\ll k_F^{-1}\\ll |a|$, where \nit reduces to\n\\bea\n \\frac{E}{N} = E_F \\left[ \\frac{3}{5} - (\\nu-1)c_1\\right]\n = E_F c_1(\\nu_c-\\nu) \\,, \\label{Ei2}\n\\eea\nwith $c_1=35/9(11-2\\ln2)\\simeq 0.40$ and $\\nu_c=1+3/5c_1\\simeq\n2.5$. Both the attractive and the kinetic part of the energy per\nparticle are proportional to the Fermi energy at these intermediate\nenergies as found for the gaps above.\n\nThe other remarkable feature of Eq. (\\ref{Ei2}) is that the energy per\nparticle changes sign for a critical number of degrees of freedom,\n$\\nu_c\\simeq 2.5$. Fermi systems with more degrees of freedom such as\nsymmetric nuclear matter have negative energy per particle and will\ntherefore implode, undergo spinodal decomposition and fragment\n\\cite{HPR}. Contrarily, systems with $\\nu\\la\\nu_c$ such as neutron matter\nand atomic systems with only two hyperspins have positive energy per\nparticle and will therefore explode, if not contained. This is also\nevident from the sound speed squared which from Eqs. (\\ref{s}) and (\\ref{Ei2})\nbecomes \n\\bea\n s^2=(5/9)c_1(\\nu_c-\\nu)v_F^2 \\,.\n\\eea\n\nCalculations for pure neutron matter and symmetric nuclear matter at\nlow densities by variational Monte Carlo \\cite{Vijay} \nand in neutron matter by Pade' approximants to R-matrix calculations\n\\cite{Baker} independently confirm the above results approximately\nin a limited range of intermediate densities. In the density\nrange $\\rho_0\\ga\\rho\\ga |a|^{-3}\\simeq 10^{-4}\\rho_0$ the energy per\nparticle is positive for neutron matter and negative for symmetric\nnuclear matter \\cite{Vijay}. \nThey scale approximately with $\\rho^{2/3}$ with\ncoefficients compatible with Eq. (\\ref{Ei2}). In symmetric nuclear\nmatter the intermediate density regime is, however, limited since\nprotons also interact through the triplet channel, which has a shorter\nrepulsive scattering length $a_t\\simeq 5.4$~fm, besides the singlet\none, $a_s=-18.8$~fm, relevant for neutron matter. Never-the-less, the\nladder resummation in the Galitskii integral equation\nEqs. (\\ref{Ei},\\ref{Ei2}) are supported by dimensional arguments and\nquantitatively it successfully predicts $\\nu_c$ between that of\nneutron and symmetric nuclear matter. The ladder resummation therefore\nseems to include the most important class of diagrams. However, even\nsmall corrections can be important for the magnitude of the\ngap because they\nappear in the exponent as, e.g., found for induced interaction\n(compare Eq. (\\ref{Gorkov}) with\n(\\ref{gap})). In addition, the superfluidity decrease the energy of the\nsystem by $\\sim\\Delta^2/2E_F$, which can be significant at\nintermediate densities if gaps really are as large as the Fermi\nenergy.\n\nThe interesting feature of the intermediate density region, that\nenergies and gaps are independent of the scattering length, leads\nto the remarkable fact that the system is also insensitive to whether\nthe scattering length goes to plus or minus infinity. In other words,\na many particle system is insensitive to whether the two-body system\nhas a marginally bound state just above or below threshold;\nthe two-body bound state or resonance is dissolved in matter at \nsufficiently high density, $k_F|a|\\ga 1$. \nFor positive scattering lengths a pair condensate of molecules\nmay form at low densities but they dissolve at intermediate densities\nwhen the range of the two-body wavefunction exceeds the\ninterparticle distance.\n\n\\begin{figure}\n\\epsfxsize=8.6truecm \n\\epsfbox{ad.eps}\n\\caption[]{Ground state energy and superfluid gaps for a degenerate\nsystem of fermions interacting via attractive forces with $R\\ll|a|$.\nThe energy per particle $E/N$ (Eq. (\\ref{Ei}), full curves) and the\nBCS gap $\\Delta$ (Eq. (\\ref{Gorkov}), dashed curves) are plotted in\nunits of the Fermi energy as function of density for $\\nu=2$ and\n$\\nu=4$. Dotted curves to the right show qualitatively the gap and\nenergy per particle at high density (see text) as a function of $k_FR$\n(upper axis). }\n\\label{fig1}\n\\end{figure}\n\nIn Fig. 1 the energy per particle is shown as function of density\nextending from the dilute and intermediate densities,\nEqs. (\\ref{Lenz},\\ref{Ei}), to high densities, Eq. (\\ref{Hartree}).\nAt low densities the Fermi kinetic energy dominates but at\nintermediate densities, $R\\la k_F^{-1}\\la |a|$, the attractive\npotential lower the energy by an amount proportional to the\nstatistical factor. The two cases $\\nu=2$ and $\\nu=4$ are seen to\nsaturate at positive and negative energies respectively. In the high\ndensity limit the attractive (Hartree) potential of\nEq. (\\ref{Hartree}) dominates and will lead to collapse of all Fermi\nsystems in the absence of repulsive cores. \n\n Fig. 1 also shows the superfluid gaps of\nEqs. (\\ref{gap},\\ref{Gorkov}) for dilute and intermediate density\nFermi systems. At low densities they decrease exponentially as\n$\\Delta\\sim E_F\\exp(-2/\\pi k_F|a|)$ whereas at intermediate densities\nthe gaps are a finite fraction of the Fermi energy. At high densities\nthe gap generally decreases rapidly with density \\cite{Emery,FW}. For\nexample, for an attractive square well potential of range $R$ and\ndepth $V_0$ with long scattering length (or marginally bound state),\ni.e. $V_0R^2\\simeq \\pi^2/4m$, the gap decrease exponentially as\n$\\Delta\\sim \\exp(-4k_FR/\\pi)$. When $|a|\\gg R$ plateaus appear for\n$(E/N)/E_F$ and $\\Delta/E_F$. Since $E_F$ also decrease with\ndecreasing density, the gap itself is narrowly peaked near $k_F\\simeq\n1/R$ as found in nuclear and neutron matter \\cite{Emery}.\n\nInformation on the density dependence can be obtained independently\nfrom calculations within the Wigner-Seitz cell approximation that has\nrecently been employed for the strongly correlated nuclear liquid\n\\cite{HH}. The periodic boundary condition is a computational\nconvenience which contains the important scale for nucleon-nucleon\ncorrelations given by the interparticle spacing. It naturally gives\nthe correct low density Eq. (\\ref{Lenz}) and high density\nEq. (\\ref{Hartree}) limits. At intermediate densities one obtains\n\\bea\n \\frac{E}{N} &=& E_F \\left[ \\frac{3}{5} \n -c_2 \\frac{\\nu-1}{\\nu^{1/3}} \\right] \\,.\n \\label{eWS}\n\\eea\nFinite crystal momenta complicates the calculation of $c_2$. A lower\n(but reasonable) estimate $c_2\\simeq0.25$ can be calculated. The\npotential energy in Eq. (\\ref{eWS}) is also proportional to the\nkinetic one as found in Eq. (\\ref{Ei2}) and of similar magnitude. The\nscaling with $\\nu^{-1/3}$ arise because energies scale with the square\nof the inverse particle spacing, $r_0^{-2}$, in the Wigner-Seitz cell\napproximation, and $\\rho=\\nu k_F^3/6\\pi^2=(4\\pi r_0^3/3)^{-1}$. The\nenergy per particle can be calculated at all densities and finite\nvalues of $a$ and $R$ and the cross over between the three density\nregimes generally confirm the energy per particle shown in Fig. 1.\n\n\\section{Phase diagram}\n\nConstructing a phase diagram from the low temperature degenerate\nregime to the high temperature classical regime requires a finite\ntemperature generalization. For illustration we shall follow the\nprocedure as in Ref. \\cite{Baym} and employ the high temperature\napproximation for the additional thermal pressure. At high\ntemperatures quantal effects are negligible and the energy per\nparticle is simply given by the classical value $E/N\\simeq\n3T/2$. \nThe isothermal sound speed is within this approximation\n\\bea\n s^2_T = \\frac{T}{m} + s^2_{T=0} \\,,\\label{sT}\n\\eea \nwhere the zero temperature sound speed is given by Eq. (\\ref{s}) with\nenergy per particle from Eqs. (\\ref{Ei},\\ref{Lenz}). The spinodal\ninstability condition, $s_T=0$, determines the line of collapse\n$T(\\rho)$ for long wavelength density fluctuations.\n\n\\begin{figure}\n\\epsfxsize=8.6truecm \n\\epsfbox{phase.eps}\n\\caption[]{Phase diagram at low and intermediate densities $(\\rho=\\nu\nk_F^3/6\\pi$) for a gas of fermions interacting via a long (attractive)\nscattering length $a$. Spinodal lines are shown with full curves and\nthe superfluid transition by dashed curves for various number of spin\nstates $\\nu$ as labeled. The area constrained to the lower right\ncorner are the spinodally unstable and superfluid regions.\nAs systems with two spin states only are stable at intermediate\ndensities the $\\nu=2$ spinodal line is absent. Contrarily, the\n$\\nu=2$ spinodal line based on extrapolating the dilute approximation\nto higher densities (see text) is shown by dash-dotted curves. }\n\\label{figphase}\n\\end{figure}\n\n\nThe resulting phase diagram is shown in Fig. 2 for $\\nu=2,3,4,7,10$\nspin states. The lower ($T\\la T_F$) and right ($k_F|a|\\ga1$) corner\nof the phase diagram is the spinodally unstable region where the\nsystem collapses. The region decreases for fewer spin states and is\nabsent for $\\nu=2$. For comparison the spinodal lines for $\\nu=2$ and $\\nu=3$\nare shown when the dilute approximation of Eq. (\\ref{Lenz}) is\nextrapolated into intermediate densities. Generally, the spinodally\nunstable regions based on the dilute approximation are substantially\noverestimated.\n\nThe regions of superfluidity given by $T_c=(\\gamma/\\pi)\\Delta$ and\nEq. (\\ref{Gorkov}) are also shown in Fig. 2. As for the spinodally\nunstable region it is the lower right corner that is\nsuperfluid. However, superfluidity extends to lower densities and\ntherefore mechanical instability does not prevent the BCS-type pairing\nin the case of fermions. The opposite conclusion was reached for the\npairing transition in Bose-Einstein condensates \\cite{Baym}. As\ncooling becomes increasingly difficult at temperatures below the Fermi\ntemperature we observe that the superfluid transition is readily\nobtained by increasing the density above $k_F|a|\\ga 1$.\n\nThe phase diagram is quantitatively correct at low as well as high\ntemperatures. Around the Fermi energy it gives a qualitative\ndescription only due to the approximate thermal pressure employed.\nFurthermore, at intermediate densities the superfluid gaps become\nlarge exceeding the Fermi energy for large spins, and the\ncorrections to the ground state energies can therefore not be ignored.\n\n\n\\section{Finite systems}\nThe degenerate Fermi gases and Bose-Einstein condensates (BEC) \nproduced so far contain $n\\sim10^3-10^6$ magnetically trapped\nalkali atoms. \nSome of them interact via long scattering lengths such as the \ntriplet $^6Li$ fermions with $a=-2160$ Bohr radii and singlet $^{85}Rb_2$ \nbosons with $|a|\\ga 10^3$ Bohr radii. \nLarge scattering lengths $a\\to\\pm\\infty$ can be taylored by using \nFeshback resonances, i.e. hyperfine\nstates close to threshold further tuned by magnetic fields.\nFermi gases differ from BEC's in several\nrespects. Most importantly, whereas bosons sit at zero momentum states,\nfermions have considerable kinetic energy. Therefore, when\ninteractions are small, a BEC has energy per particle $\\hbar\\omega$ and\nsize $a_{osc}$, where $\\omega=(\\omega_\\perp\\omega_z)^{1/3}$ is the\ngeometric average of the magnetic trap frequencies and\n$a_{osc}=\\hbar/\\sqrt{m\\omega}$ is the oscillator length. A degenerate\ngas of $N$ Fermionic atoms has a larger energy per particle\n$\\sim N^{1/3}\\hbar\\omega$ and\nsize $L\\sim N^{1/6}a_{osc}$. In current experiments with degenerate\nFermi gases and BEC's the densities are low so that the dilute\npotential applies and the energy per particle is approximately\n\\bea\n \\frac{E}{N} \\simeq \\frac{3}{4}\\left(\\frac{6}{\\nu}\\right)^{1/3}\n N^{1/3} \\hbar\\omega \n + \\frac{\\nu-1}{\\nu}\\frac{2\\pi a}{m} \\frac{N}{L^3} \\,,\\label{BEC}\n\\eea\nwhere the average density in the trap has been approximated by\n$\\langle\\rho\\rangle\\simeq N/L^3$ \\cite{Stringari}. For a small number\nof trapped atoms with attractive scattering lengths the system is\nmeta-stable but for a large number of trapped atoms, $N\\ga\n\\nu(a_{osc}/(\\nu-1)a)^6$, the attractive potential overcomes the \nFermi kinetic energy and the degenerate Fermi gas\n becomes unstable and implodes. However, around the same\ndensity $k_F|a|\\ga 1$ and we enter the intermediate density region,\nwhere the potential of (\\ref{Ei}) should be applied instead of the\ndilute potential. The gas is therefore mechanically stable for two\nhyperspins only contrary to conclusions based on the dilute potential\n\\cite{JILA,Houbiers}.\n\nRecent experiment on cold magnetically trapped Fermionic atoms\n\\cite{JILA} observed egeneracy for $^{40}K$ atoms in the two hyperfine\nstates $m_F=9/2,7/2$. Current experimental oscillator lengths\n$a_{osc}\\simeq\\mu m$ are less than one order of magnitude longer than the\natomic scattering length $|a|$ of $^6Li$. It should be possible to\nreach intermediate densities, $k_F|a|\\ga1$, by trapping $N\\ga 10^6$\n$^6Li$ atoms \\cite{Houbiers}. The atomic gases offer the unique\nopportunity to vary the densities as well as the relative amount of\nthe hyperfine states. Varying the composition is a convenient way to\nvary the gaps and attractive potential of Eqs.\n(\\ref{Lenz},\\ref{Ei},\\ref{Hartree}) through $\\nu$ for given density\nand scattering length. In the limit where most atoms are in one of the\nstates, the Fock and Hartree terms almost cancels and effectively\n$\\nu\\to 1_+$.\n\nMore intricate systems of mixtures of fermions and bosons, e.g.\n$^{39,40,41}K$ isotopes can also be studied. If the interaction is\nattractive it will contract the atomic cloud towards higher densities.\nIrrespective of whether the bosons or fermions attract or repel the\ninduced interactions, which are of second order in the fermion-boson\ncoupling, enhance the gap \\cite{gap}.\n\nAn artificial ``gravitational'' or ``Coulomb'' force can be exerted on\nthe atoms by shining laser light on the trapped cloud from many\ndirections \\cite{Odell}. It would add an energy per particle of order\n$\\sim GNm^2/L$ to Eq. (\\ref{BEC}) where $G$ is proportional to the laser\nfield intensity. Such an interaction has several interesting\nconsequences. If $G$ is attractive, it would contract the cloud\ntowards higher densities which would increase gaps (see also\n\\cite{Luciano}) and for sufficiently large $G$ the\nintermediate density region is entered. Depending on\nthe strengths and sign of the scattering amplitude and gravitational\ninteractions the kinetic energy of the atoms will be balanced by the\nmagnetic trap and/or the scattering or gravitational interactions.\nThe resulting phase diagram is much more complex.\n\nIf such a strong attractive laser field is suddenly applied to\nthe gas, the Jeans instability sets in and the gas \ncollapses until balanced again by the kinetic energy.\nSubsequently, the system will ``bounce'' analogous to the initial stages of\na supernova explosion. If, however, intermediate energies\nare reached and the number of spins exceed $\\nu\\ga2.5$, then\nthe collapse will be further accelerated by the attraction between\natoms. The corresponding critical particle number is\n\\bea\n N_c \\simeq (Gm^3a)^{-3/2} \\,,\n\\eea\nat zero temperature. It differs from the standard Chandrasekhar\nmass by a factor $(m|a|)^{3/2}$\nbecause the instability condition is $k_F|a|\\simeq 1$ whereas\nstars go unstable when the particles become relativistic $k_F\\simeq m$.\n\n\n\\section{Summary}\nThe energy per particle and superfluid gaps have been calculated for\nan homogeneous system of fermions interacting via a long attractive\ns-wave scattering length. In the intermediate region of densities, where the\ninterparticle spacing $(\\sim 1/k_F)$ is much longer than the range of\nthe interaction but much shorter than the scattering length or $|a|$,\nthe energy per particle and superfluid gaps are proportional to the\nFermi energy. The energy per particle increases linearly with the\nspin-isospin or hyperspin statistical factor such that, e.g.,\nsymmetric nuclear matter is unstable in the intermediate density\nregions and undergoes spinodal decomposition whereas neutron matter\nand Fermionic atomic gases with few hyperspin states are mechanically\nstable.\n\n A phase diagram of Fermi gases at low and intermediate densities was\nconstructed by including thermal pressures in the high temperature\nclassical approximation. With the proper energy per particle at\nintermediate densities the spinodal region in the phase diagram was\nreduced substantially as compared to conjectures based on\nextrapolations from the dilute limit. Generally, mechanical\ninstability does not prevent a superfluid transition for a wide range\nof densities. This is contrary to Bose gases, where spinodal\ninstabilities exclude pairing transitions \\cite{Baym}.\n\nThe interaction energies of the many-body system were discussed for\nmagnetically trapped cold degenerate gases of Fermi atoms. In such\nsystems both superfluidity and the intermediate density region should\nbe attainable. In these ``novel'' density regions the superfluid gaps\ncan be large and the stability and sensitivity to the statistical\nfactor $\\nu$ can be studied. Adding a gravitationally like force by\nshining laser light on the atomic cloud further increase densities\nwhereby collapse and bounce analogous to the early stages of supernova\nexplosions may be studied.\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Vijay} V. R. Pandharipande, Int. J. of Mod. Phys.\n {\\bf B13}, 543 (1999); H. Heiselberg and M. Hjorth-Jensen, \n Phys. Rep. {\\bf 328}, 237 (2000).\n\\bibitem{JILA} J. Holland, B. deMarco and D.S. Jin, {\\it cond-mat/9911017}; \n Science, {\\bf 285}, 1703 (1999).\n\\bibitem{Galitskii} V. M. Galitskii, JETP {\\bf 34}, 151 (1958).\n\\bibitem{Melo} C.A.R. S\\'a de Melo, M. Randeira, and J.R. Engelbrecht,\n Phys. Rev. Lett. {\\bf 71}, 3202 (1993);\nH.T.C. Stoof et al., Phys. Rev. Lett. {\\bf 76}, 10 (1996);\nG. Papenbrock and G.F. Bertsch, Phys. Rev. {\\bf C59}, 2052 (1999).\n\\bibitem{Gorkov} L.P. Gorkov \\& T.K. Melik-Barkhudarov, \nJETP {\\bf 13}, 1018 (1961).\n\\bibitem{gap} H. Heiselberg, C.J. Pethick, H. Smith and L. Viverit,\ncond-mat/0004360; to appear in {\\it Phys. Rev. Letts.}\n\n\\bibitem{FW} Fetter and Walecka, ``Quantum Theory of Many Particle Physics'',\n McCraw-Hill 1971.\n\\bibitem{Lenz} W. Lenz, Z. Phys. {\\bf 56}, 778 (1929).\n\\bibitem{HY} K. Huang \\& C.N. Yang, Phys. Rev. {\\bf 105}, 767 (1957); \nT.D.Lee \\& C.N. Yang, Phys. Rev. {\\bf 105}, 1119 (1957).\n\\bibitem{Efimov} V.N. Efimov \\& M.Ya. Amus'ya, Sov. Phys. JETP {\\bf 20}, 388 (1965).\n\\bibitem{BBG} K.A. Brueckner, C.A. Levinson, and H.M. Mahmoud,\nPhys. Rev. {\\bf 103}, 1353 (1956); H.A. Bethe and J. Goldstone,\nProc. Roy. Soc. (London), {\\bf A238}, 551 (1957).\n\n\\bibitem{HPR} H.Heiselberg, C.J.Pethick, D.G.Ravenhall,\n Phys. Rev. Letts. {\\bf 61}, 818 (1988).\n\\bibitem{Baker} G.A. Baker, Jr., Phys. Rev. C {\\bf 60}, 054311 (1999).\n\\bibitem{PPT} V.R. Pandharipande, C.J. Pethick and V. Thorsson,\n Phys. Rev. Lett. {\\bf 75}, 4567 (1995).\n\\bibitem{Emery} V.J. Emery \\& A.M. Sessler, Phys. Rev. {\\bf 119}, 248 (1960).\n\\bibitem{HH} J. Carlson, H. Heiselberg, \\& V. R. Pandharipande, nucl-th/9912043.\n\\bibitem{Stringari} L. Vichi, M. Inguscio,\nS. Stringari, G.M. Tino, {\\it cond-mat/9810115}.\n\\bibitem{Houbiers} M. Houbiers et al., Phys. Rev. {\\bf A56}, 4864 (1997).\n\\bibitem{Odell} D. O'Dell, S. Giovanazzi, G. Kurizki, and V.M. Akulin,\n{\\it Phys. Rev. Lett.} {\\bf 84}, 5687 (2000).\n\\bibitem{Luciano} L. Viverit, S. Giorgini, L.P. Pitaevskii, S. Stringari,\n cond-mat/000551.\n\\bibitem{Baym} E.J. Mueller and G. Baym, cond-mat/0005323 \n\\end{thebibliography}\n\n%\\end{multicols}\n\n\\end{document}\n\n"
}
] |
[
{
"name": "cond-mat0002056.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Vijay} V. R. Pandharipande, Int. J. of Mod. Phys.\n {\\bf B13}, 543 (1999); H. Heiselberg and M. Hjorth-Jensen, \n Phys. Rep. {\\bf 328}, 237 (2000).\n\\bibitem{JILA} J. Holland, B. deMarco and D.S. Jin, {\\it cond-mat/9911017}; \n Science, {\\bf 285}, 1703 (1999).\n\\bibitem{Galitskii} V. M. Galitskii, JETP {\\bf 34}, 151 (1958).\n\\bibitem{Melo} C.A.R. S\\'a de Melo, M. Randeira, and J.R. Engelbrecht,\n Phys. Rev. Lett. {\\bf 71}, 3202 (1993);\nH.T.C. Stoof et al., Phys. Rev. Lett. {\\bf 76}, 10 (1996);\nG. Papenbrock and G.F. Bertsch, Phys. Rev. {\\bf C59}, 2052 (1999).\n\\bibitem{Gorkov} L.P. Gorkov \\& T.K. Melik-Barkhudarov, \nJETP {\\bf 13}, 1018 (1961).\n\\bibitem{gap} H. Heiselberg, C.J. Pethick, H. Smith and L. Viverit,\ncond-mat/0004360; to appear in {\\it Phys. Rev. Letts.}\n\n\\bibitem{FW} Fetter and Walecka, ``Quantum Theory of Many Particle Physics'',\n McCraw-Hill 1971.\n\\bibitem{Lenz} W. Lenz, Z. Phys. {\\bf 56}, 778 (1929).\n\\bibitem{HY} K. Huang \\& C.N. Yang, Phys. Rev. {\\bf 105}, 767 (1957); \nT.D.Lee \\& C.N. Yang, Phys. Rev. {\\bf 105}, 1119 (1957).\n\\bibitem{Efimov} V.N. Efimov \\& M.Ya. Amus'ya, Sov. Phys. JETP {\\bf 20}, 388 (1965).\n\\bibitem{BBG} K.A. Brueckner, C.A. Levinson, and H.M. Mahmoud,\nPhys. Rev. {\\bf 103}, 1353 (1956); H.A. Bethe and J. Goldstone,\nProc. Roy. Soc. (London), {\\bf A238}, 551 (1957).\n\n\\bibitem{HPR} H.Heiselberg, C.J.Pethick, D.G.Ravenhall,\n Phys. Rev. Letts. {\\bf 61}, 818 (1988).\n\\bibitem{Baker} G.A. Baker, Jr., Phys. Rev. C {\\bf 60}, 054311 (1999).\n\\bibitem{PPT} V.R. Pandharipande, C.J. Pethick and V. Thorsson,\n Phys. Rev. Lett. {\\bf 75}, 4567 (1995).\n\\bibitem{Emery} V.J. Emery \\& A.M. Sessler, Phys. Rev. {\\bf 119}, 248 (1960).\n\\bibitem{HH} J. Carlson, H. Heiselberg, \\& V. R. Pandharipande, nucl-th/9912043.\n\\bibitem{Stringari} L. Vichi, M. Inguscio,\nS. Stringari, G.M. Tino, {\\it cond-mat/9810115}.\n\\bibitem{Houbiers} M. Houbiers et al., Phys. Rev. {\\bf A56}, 4864 (1997).\n\\bibitem{Odell} D. O'Dell, S. Giovanazzi, G. Kurizki, and V.M. Akulin,\n{\\it Phys. Rev. Lett.} {\\bf 84}, 5687 (2000).\n\\bibitem{Luciano} L. Viverit, S. Giorgini, L.P. Pitaevskii, S. Stringari,\n cond-mat/000551.\n\\bibitem{Baym} E.J. Mueller and G. Baym, cond-mat/0005323 \n\\end{thebibliography}"
}
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cond-mat0002057
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Velocity Autocorrelation and Harmonic Motion in Supercooled Nondiffusing Monatomic Liquids
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[
{
"author": "Eric D.\\ Chisolm"
},
{
"author": "Brad E.\\ Clements"
},
{
"author": "and Duane C.\\ Wallace"
},
{
"author": "Theoretical Division"
},
{
"author": "Los Alamos National Laboratory"
},
{
"author": "Los Alamos"
},
{
"author": "NM~~87545"
}
] |
Studies of the many-body potential surface of liquid sodium have shown that it consists of a great many intersecting nearly harmonic valleys, a large fraction of which have the same frequency spectra. This suggests that a sufficiently supercooled state of this system, remaining in a single valley, would execute nearly harmonic motion. To test this hypothesis, we have compared $\hat{Z}(t)$, the normalized velocity autocorrelation function, calculated from MD simulations to that predicted under the assumption of purely harmonic motion. We find nearly perfect agreement between the two, suggesting that the harmonic approximation captures all essential features of the motion.
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[
{
"name": "paper1.tex",
"string": "\\documentclass[12pt]{article}\n\n%LA-UR-00-120\n\n\\usepackage{graphics}\n\n\\title{Velocity Autocorrelation and Harmonic Motion in Supercooled \n Nondiffusing Monatomic Liquids}\n\\author{Eric D.\\ Chisolm, Brad E.\\ Clements, and Duane C.\\ Wallace \\\\ \n Theoretical Division \\\\ Los Alamos National Laboratory \\\\ Los Alamos, \n NM~~87545}\n\n\\begin{document}\n\n%\\bibliographystyle{unsrt}\n\n\\maketitle\n\n\\begin{abstract}\nStudies of the many-body potential surface of liquid sodium have shown that it \nconsists of a great many intersecting nearly harmonic valleys, a large fraction\nof which have the same frequency spectra. This suggests that a sufficiently \nsupercooled state of this system, remaining in a single valley, would execute \nnearly harmonic motion. To test this hypothesis, we have compared \n$\\hat{Z}(t)$, the normalized velocity autocorrelation function, calculated \nfrom MD simulations to that predicted under the assumption of purely harmonic \nmotion. We find nearly perfect agreement between the two, suggesting that the \nharmonic approximation captures all essential features of the motion.\n\\end{abstract}\n\n\\section{Introduction}\n\\label{intro}\n\nRecent work by Wallace and Clements \\cite{enkin, radang} has uncovered several \nimportant properties of the many-body potential underlying the motion of \nliquid sodium systems. Specifically, it has been shown that (a) the potential \nsurface consists of a large number of intersecting nearly harmonic valleys, (b)\nthese valleys can be classified as symmetric (crystalline, microcrystalline, or\nretaining some nearest-neighbor remnants of crystal symmetry) or random, with \nthe random valleys vastly outnumbering the symmetric ones, (c) the frequency \nspectra of different random valleys are nearly identical (while those of the \nsymmetric valleys vary widely), and (d) below 35 K the system remains in a \nsingle valley throughout the longest molecular dynamics (MD) runs that were \nperformed. Results (a) through (c) verify predictions made by Wallace \nin his theory of liquid dynamics \\cite{liqdyn}, which has been successfully \napplied to account for the high-temperature specific heats of monatomic liquids\n\\cite{specif} and a study of the velocity autocorrelation function \n\\cite{oldvacf}. These four results together suggest that below 35 K the motion\nof the atoms in liquid sodium is purely harmonic to a high degree of \napproximation, again as predicted by Wallace in \\cite{liqdyn}, and we would \nlike to test this hypothesis further. One check is to compare the mean square \ndisplacement from MD with the prediction from purely harmonic motion, which is \ndone in Fig.\\ 12 of \\cite{enkin}, where the two are found to agree closely. \nHowever, it would be more convincing if the theory could be shown to reproduce \nan entire scalar function calculated from MD (instead of just a single number),\nsuch as the normalized velocity autocorrelation function $\\hat{Z}(t)$. That is\nthe aim of this paper. We will show that purely harmonic motion of the atoms \nin a potential valley produces a $\\hat{Z}(t)$ which matches that of MD \ncalculations to within the calculations' accuracy; thus we will conclude that \nthe motion of atoms in a nondiffusing supercooled liquid state is very nearly \nentirely harmonic. For completeness, in Sec.\\ \\ref{theory} we briefly review\nthe calculation of $\\hat{Z}(t)$ assuming harmonic motion, and in Sec.\\ \n\\ref{MD} we compare this result with MD. Finally, in Sec.\\ \\ref{concl} we make\ncontact with work by others in this field, as well as comparing these results \nto Wallace's earlier effort \\cite{oldvacf} mentioned above.\n\n\\section{Harmonic Theory}\n\\label{theory}\n\nIf an $N$-body system is moving in a potential valley, the potential can be \nexpanded about the valley minimum with the resulting Hamiltonian\n\\begin{equation}\nH = {\\sum_{Ki}}' \\, \\frac{p_{Ki}^{2}}{2M} + {\\sum_{Ki,Lj}}' \\, \\Phi_{Ki,Lj} \n u_{Ki} u_{Lj} + \\Phi_{A}\n\\label{Hamiltonian}\n\\end{equation}\nwhere $u_{Ki}$ is the $i$th component of the $K$th particle's displacement \nfrom equilibrium, $p_{Ki}$ is the corresponding momentum, and the anharmonic \nterm $\\Phi_{A}$ contains all of the higher order parts of the expansion. The \nprimed sum indicates that the sum is performed under the constraint that the \ncenter of mass of the system is stationary. (As a result, the system has only \n$3N-3$ independent degrees of freedom.) The matrix $\\Phi_{Ki,Lj}$ is called \nthe dynamical matrix of the system. If the valley is approximately harmonic, \nwe can neglect $\\Phi_{A}$. If coordinates $q_{\\lambda}$ are defined by the\nrelation \n\\begin{equation}\nu_{Ki} = \\sum_{\\lambda} w_{Ki, \\lambda} q_{\\lambda}\n\\label{newcoords}\n\\end{equation}\nwhere the $w_{Ki, \\lambda}$ form a $3N \\times 3N$ orthogonal matrix, \nsatisfying \n\\begin{equation}\n\\sum_{Ki} w_{Ki, \\lambda} w_{Ki, \\lambda'} = \\delta_{\\lambda\\lambda'},\n\\label{orth}\n\\end{equation}\nthen the Hamiltonian in these new coordinates is\n\\begin{equation}\nH = \\sum_{\\lambda} \\frac{p_{\\lambda}^{2}}{2M} + {\\sum_{Ki,Lj}}' \n \\sum_{\\lambda\\lambda'} w_{Ki, \\lambda} \\Phi_{Ki,Lj} w_{Lj, \\lambda'} \n q_{\\lambda} q_{\\lambda'}\n\\end{equation}\nwhere the $p_{\\lambda}$ are the momenta conjugate to the $q_{\\lambda}$. Now \none can always choose the $w_{Ki, \\lambda}$ to diagonalize $\\Phi_{Ki,Lj}$, so \nthat\n\\begin{equation}\n\\sum_{Ki,Lj} w_{Ki, \\lambda} \\Phi_{Ki,Lj} w_{Lj, \\lambda'} = \n M \\omega_{\\lambda}^{2} \\delta_{\\lambda\\lambda'}. \n\\label{diagphi}\n\\end{equation}\n(This equation defines the frequencies $\\omega_{\\lambda}$ in terms of the \neigenvalues of $\\Phi_{Ki,Lj}$.) With this choice, the Hamiltonian becomes\n\\begin{equation}\nH = \\sum_{\\lambda} \\left( \\frac{p_{\\lambda}^{2}}{2M} + \n \\frac{1}{2}M\\omega_{\\lambda}^{2}q_{\\lambda}^{2} \\right).\n\\end{equation}\nThree of the $\\omega_{\\lambda}$ are zero; these modes correspond to uniform \nmotion of the center of mass. Since we have restricted the center of mass \nposition and velocity to zero, these modes are not excited. The classical \nequations of motion for the remaining modes are solved by\n\\begin{equation}\nq_{\\lambda}(t) = a_{\\lambda} \\sin(\\omega_{\\lambda}t + \\alpha_{\\lambda}),\n\\end{equation}\nor, returning to the original coordinates,\n\\begin{equation}\nu_{Ki}(t) = \\sum_{\\lambda} w_{Ki, \\lambda} a_{\\lambda} \\sin(\\omega_{\\lambda}t +\n \\alpha_{\\lambda}),\n\\label{uoft}\n\\end{equation}\nwith the understanding that the sum on $\\lambda$ ranges from $1$ to $3N-3$.\nThe velocities of the particles are\n\\begin{equation}\nv_{Ki}(t) = \\sum_{\\lambda} w_{Ki, \\lambda} \\, \\omega_{\\lambda} a_{\\lambda} \n \\cos(\\omega_{\\lambda}t + \\alpha_{\\lambda}).\n\\label{voft}\n\\end{equation}\nWe compute the $\\langle \\mbox{\\boldmath $v$}(t) \\cdot \\mbox{\\boldmath $v$}(0) \n\\rangle$ in $Z(t)$ by calculating $\\mbox{\\boldmath $v$}_{K}(t) \\cdot \n\\mbox{\\boldmath $v$}_{K}(0)$, summing over $K$ and dividing by $N-1$ (remember \nthat only $3N-3$ coordinates are independent), and averaging over the \namplitudes $a_{\\lambda}$ and phases $\\alpha_{\\lambda}$. Thus\n\\begin{eqnarray}\nZ(t) & = & \\frac{1}{3} \\langle \\mbox{\\boldmath $v$}(t) \\cdot \\mbox{\\boldmath \n $v$}(0) \\rangle \\nonumber \\\\\n& & \\nonumber \\\\\n& = & \\frac{1}{3N-3} \\sum_{Ki} \\sum_{\\lambda\\lambda'} w_{Ki, \\lambda} w_{Ki, \n \\lambda'} \\omega_{\\lambda} \\omega_{\\lambda'} \\langle a_{\\lambda} \n a_{\\lambda'} \\rangle \\langle \\cos(\\omega_{\\lambda}t + \\alpha_{\\lambda}) \n \\cos(\\alpha_{\\lambda'}) \\rangle \\nonumber \\\\\n & & \\nonumber \\\\\n & = & \\frac{1}{3N-3} \\sum_{\\lambda\\lambda'} \\delta_{\\lambda\\lambda'}\n \\omega_{\\lambda} \\omega_{\\lambda'} \\langle a_{\\lambda} a_{\\lambda'} \n \\rangle \\langle \\cos(\\omega_{\\lambda}t + \\alpha_{\\lambda}) \n \\cos(\\alpha_{\\lambda'}) \\rangle \\nonumber \\\\ \n & & \\nonumber \\\\\n & = & \\frac{1}{3N-3} \\sum_{\\lambda} \\omega_{\\lambda}^{2} \\langle \n a_{\\lambda}^{2} \\rangle \\langle \\cos(\\omega_{\\lambda}t + \n \\alpha_{\\lambda}) \\cos(\\alpha_{\\lambda}) \\rangle \\nonumber \\\\\n & & \\nonumber \\\\\n & = & \\frac{1}{6N-6} \\sum_{\\lambda} \\omega_{\\lambda}^{2} \\langle \n a_{\\lambda}^{2} \\rangle \\cos(\\omega_{\\lambda}t).\n\\label{vdotv}\n\\end{eqnarray}\nBy the equipartition theorem, \n\\begin{equation}\n\\left\\langle \\frac{1}{2} M \\omega_{\\lambda}^{2} q_{\\lambda}^{2} \\right\\rangle =\n \\frac{1}{2}kT\n\\end{equation}\nfor any nonzero $\\omega_{\\lambda}$, from which it follows that\n\\begin{equation}\n\\langle a_{\\lambda}^{2} \\rangle = \\frac{2kT}{M\\omega_{\\lambda}^{2}},\n\\label{equipart}\n\\end{equation}\nso\n\\begin{equation}\nZ(t) = \\frac{1}{3N-3} \\frac{kT}{M} \\sum_{\\lambda} \\cos(\\omega_{\\lambda}t).\n\\label{Z}\n\\end{equation}\nNotice that $Z(0) = kT/M$, so $\\hat{Z}(t)$ defined by \n\\begin{equation} Z(t) = Z(0) \\hat{Z}(t) \\end{equation}\nis given in this theory by\n\\begin{equation}\n\\hat{Z}(t) = \\frac{1}{3N-3} \\sum_{\\lambda} \\cos(\\omega_{\\lambda}t).\n\\label{Zhat}\n\\end{equation}\nThis is the result we wish to compare with MD.\n\nTo do so, we need the frequencies $\\omega_{\\lambda}$, which are related to the\neigenvalues of the dynamical matrix $\\Phi_{Ki,Lj}$ as indicated in Eq.\\ \n(\\ref{diagphi}). These were evaluated for five separate random valleys in \n\\cite{enkin} by quenching all the way down to a valley minimum and \ndiagonalizing $\\Phi_{Ki,Lj}$ there; as pointed out in Sec.\\ \\ref{intro}, these \neigenvalues were found to be independent of the specific random valley chosen. \nAll five sets of eigenvalues are shown in Fig.\\ 7 of \\cite{enkin}, and we \npicked one set at random to use in performing the sum in Eq.\\ (\\ref{Zhat}); the\nother sets produce identical graphs of $\\hat{Z}(t)$. \n\nWe can also use the set of eigenvalues to reconstruct the density of \nfrequencies $g(\\omega)$; the results are shown in Fig.\\ \\ref{nvsfreq}. Note \nthat we do not actually integrate over this $g(\\omega)$ to\n\\begin{figure}\n\\includegraphics{gvsomega.ps}\n\\caption{$g(\\omega)$ constructed from one of the five sets of frequencies in \n Fig.\\ 7 of \\cite{enkin}. This same set of frequencies is used to \n calculate $\\hat{Z}(t)$ from Eq.\\ (\\ref{Zhat}).}\n\\label{nvsfreq}\n\\end{figure}\nevaluate $\\hat{Z}(t)$ below; we directly sum over the given set of frequencies \nas indicated in Eq.\\ (\\ref{Zhat}). The Figure is provided only to convey a \nsense of the shape of the frequency distribution. Also note that this \n$g(\\omega)$ is determined from fully mechanical considerations; as a result, it\nis not temperature-dependent as are the frequency spectra used in Instantaneous\nNormal Mode (INM) studies \\cite{RMM, LVS, Stratt, VB, MKS}. We will expand on \nthis point in the Conclusion.\n\n\\section{Comparison with MD}\n\\label{MD}\n\nThe MD setup used to calculate $\\hat{Z}(t)$ to compare with Eq.\\ (\\ref{Zhat}) \nis essentially that described in \\cite{enkin}: $N$ particles interact through \na potential that is known to reproduce accurately a wide variety of \nexperimental properties of metallic sodium (see discussion in \\cite{enkin} for \ndetails). The two significant changes are that we used $N=500$ for all runs \nand that the MD timestep was reduced to $\\delta t = 0.2 t^{*}$, where $t^{*} = \n7.00 \\times 10^{-15}$ s is the natural timescale defined in \\cite{enkin}. (The\nsystem's mean vibrational period $\\tau = 2\\pi/\\omega_{\\rm rms}$, where the rms \nfrequency $\\omega_{\\rm rms}$ is calculated in \\cite{enkin}, is approximately \n$300 \\,\\delta t$.) We cooled the sodium sample to 22.3 K and 6.69 K, and then \nwe ran each at equilibrium to collect velocities $\\mbox{\\boldmath $v$}_{K}(t)$ \nto be used to calculate $Z(t)$ by the formula\n\\begin{equation}\nZ(t) = \\frac{1}{3N} \\sum_{K} \\frac{1}{n+1} \\sum_{t'=0}^{n} \n \\mbox{\\boldmath $v$}_{K}(t+t') \\cdot \\mbox{\\boldmath $v$}_{K}(t').\n\\label{MDZ}\n\\end{equation}\nWe then divided by $Z(0)$ to obtain $\\hat{Z}(t)$. The number $n$ was chosen \nas large as possible without running beyond the data calculated in the MD run. \nWe know that at these temperatures the sodium is nondiffusing for two reasons: \nBoth temperatures are below the 35 K threshold \\cite{enkin}, and $\\hat{Z}(t)$ \nfrom either MD run (shown below in Fig.\\ \\ref{Zvst}) integrates to zero, \nyielding zero diffusion coefficient.\n\nThe formula above may fail to produce reliable values of $\\hat{Z}(t)$ for three\nreasons. First, the number of data points in the time average may be too \nsmall; if the MD simulation is run out to time $t_{\\rm max}$, then for a given \nvalue of $t$ in Eq.\\ (\\ref{MDZ}), the maximum possible value of the upper limit\n$n$ is $t_{\\rm max}-t$. Thus we require $t_{\\rm max} >> t$; we have chosen \n$t_{\\rm max} = 50,000$ timesteps and we have calculated $\\hat{Z}(t)$ only to \n$t = 1000$. To ensure that this value of $t_{\\rm max}$ is large enough, we \nalso performed MD runs out to 200,000 timesteps and calculated $\\hat{Z}(t)$ \nfrom them; the differences from the 50,000 timestep result were of order \n$10^{-3}$. Hence we are confident that 50,000 timesteps is enough if we \ncalculate $\\hat{Z}(t)$ to only 1000 timesteps. Second, it is possible that \nreducing the timestep (thus increasing the accuracy of the simulation) might \nimprove the accuracy of $\\hat{Z}(t)$. To test this, we performed another MD \nrun with $\\delta t$ reduced to $0.05 t^{*}$, keeping the ``real'' time of the \nrun the same; this also produced differences in $\\hat{Z}(t)$ of order \n$10^{-3}$. Thus we are sure that our timestep is small enough. Finally, there\nis the possibility of finite size effects. Since the MD system has periodic \nboundary conditions, an acoustic wave sent out from the system at $t=0$ could \npropagate across the simulation region and return to its point of origin in a \nfinite time, producing spurious correlations that would show up in $\\hat{Z}(t)$\nbut would not be present in a large-$N$ system. To see if this effect is \nrelevant, we estimated the time it would take for an acoustic wave to cross the\nregion, using the numbers from \\cite{enkin}. The speed of sound in sodium at \nits melting temperature is $2.5 \\times 10^{5}$ cm/s, and the volume of the \nregion occupied by one atom is $278 \\, a_{0}^{3}$, so from the fact that there \nare 500 atoms one finds that the time required for an acoustic wave to cross \nthe region is $783 \\, \\delta t$, or about 800 timesteps. (The speed of sound \nin sodium at our lower temperatures varies from that at the melting point by \nroughly 5\\%, so this result is valid to the same accuracy.) In the Figure \nbelow, the MD result for $\\hat{Z}(t)$ begins to show small oscillatory revivals\nat about this time; we conclude that this is a finite size effect, but it does \nnot affect the data before that time. \n\nIn Fig.\\ \\ref{Zvst}, Eq.\\ (\\ref{Zhat}) is plotted on top of the MD data for\n$\\hat{Z}(t)$ of sodium at the two temperatures. Although both temperatures \ncompare exceptionally well to the harmonic theory, the match is visibly poorer \nfor the lower temperature. However, repeated MD runs at the lower temperature \nrevealed that overall variations in $\\hat{Z}(t)$ amount to $10^{-2}$ on\naverage, which is the same order as the differences between theory and MD in \nthis Figure. By 500 timesteps the theory is slightly out of phase with the MD \ndata, and this small difference persists out to more timesteps. \n\\begin{figure}\n\\includegraphics{MDvsmodel.ps}\n\\caption{The theoretical prediction (Eq.\\ (\\ref{Zhat})) for $\\hat{Z}(t)$ of \n sodium moving in a single random valley compared with MD data at $T = \n 6.69$ K and $T = 22.3$ K.}\n\\label{Zvst}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{concl}\n\nThese results show that the motion of a liquid in a single potential valley is \nharmonic to an extremely high approximation; the harmonic prediction for the \nfunction $\\hat{Z}(t)$ matches the calculation from MD very closely. Any \ncontributions due to anharmonicity (which are certainly present) are at most of\nthe same order as the accuracy of the MD calculations.\n\nSome form of harmonic approximation, such as the one used here, has been taken \nup by many workers attempting to understand the dynamics of liquids, and it is \nhelpful to compare their models with our approach. One of the most popular is \nthe theory of Instantaneous Normal Modes (INM), introduced by Rahman, Mandell, \nand McTague \\cite{RMM} and LaViolette and Stillinger \\cite{LVS} and developed \nextensively by Stratt (for example, \\cite{Stratt}). Stratt expands the \nmany-body potential in the neighborhood of an arbitrary point to second order \nin displacements from that point, and he expresses the potential as a quadratic\nsum of normal modes, in which the frequencies may be either real or imaginary. \nHe then replaces the frequencies by their thermal averages over the potential \nsurface, resulting in a temperature-dependent density of frequencies. From \nthis point he calculates the system's motion and considers various time \ncorrelation functions, including $Z(t)$. He observes that his \\mbox{results} \nare accurate to order $t^{4}$ for short times, but his predictions also diverge\nfrom MD results very rapidly, in a time shorter than half of one vibrational \nperiod. The agreement with MD at long times can be improved by omitting the \nimaginary frequencies from the calculation of $Z(t)$, but of course this makes \nthe short time behavior inexact. (The work of Vallauri and Bermejo \\cite{VB} \nfollows Stratt's procedure.) Efforts to improve the long time behavior of the \ncorrelation functions calculated using INM have been made by Madan, Keyes, and \nSeeley \\cite{MKS}, who have attempted to extract from the imaginary part of the\nINM spectrum a damping factor for $Z(t)$ of the general type suggested by \nZwanzig \\cite{Zwan}. Also taking their cue from Zwanzig, Cao and Voth \n\\cite{CV} have followed a slightly different path, replacing the actual \npotential by a set of temperature-dependent effective normal modes which, as \nthey emphasize, bears little resemblance to the mechanical normal modes of a \nsingle many-particle valley. In fact, they state quite explicitly that a \ntheory based on purely mechanical normal modes will have little success in \naccounting for equilibrium or dynamical properties of liquids.\n\nAn obvious difference between our theory and INM is the nature of our \napproximation. In INM, one approximates the potential quadratically at an \narbitrary point, with the result that the motion so predicted is accurate only \nfor very small times; in our theory, we expand the potential only at very \nspecial points where we know the predicted motion will be valid for very long \ntimes. Both theories then face the problem of extending their validity beyond \nthe initial approximation, of course, and we will briefly mention our extension\nin the final paragraph below, but there is one particular reason why we \nstrongly prefer the approach taken here: The other models all replace the true\npotential by a temperature-dependent potential determined by one or another \nthermal averaging process. A temperature-dependent potential does not provide \na true Hamiltonian, and therefore it cannot be used to calculate the quantum or\nclassical motion, i.e., it cannot be used in the Schr\\\"{o}dinger equation or \nNewton's law. (On the dynamical level, temperature is not even a well-defined \nconcept.) Further, the Hamiltonian resulting from a temperature-dependent \npotential cannot be used to do statistical mechanics, except through \nuncontrolled self-consistent procedures. We prefer to build our theory in \nterms of the actual potential, hence in terms of its true Hamiltonian, and to \nfind at least approximate solutions for the Hamiltonian motion, so we can apply\nthe standard procedures of equilibrium and nonequilibrium statistical \nmechanics.\n\nFurther, we would argue that Cao and Voth's skepticism regarding purely \ndynamical approaches is unfounded, given the results here. It is difficult to \ncompare our $\\hat{Z}(t)$ results to those of others, because their MD-simulated\nstates are not always characterized as diffusing or nondiffusing. We are \nfairly confident that Vallauri and Bermejo's Fig.\\ 2b is a comparable state \n(glassy Cs at 20 K), and we believe our fit to MD is slightly better. Madan, \nKeyes, and Seeley's Fig.\\ 3b is an ambiguous case (it is likely that a glass \ntransition has occurred), but there also we are confident that our match with \nMD is better. Hence we would claim that this method shows as much promise as \nthe others currently available, and with the physical potential as opposed to a\nthermal average potential.\n\nIt is also instructive to compare the results of this paper with a model for \n$\\hat{Z}(t)$ previously proposed by Wallace \\cite{oldvacf} in which a single\nparticle oscillates in a three-dimensional harmonic valley, and at each turning\npoint it may with probability $\\mu$ ``transit'' to an adjacent valley. To \napply that model to a nondiffusing case, we set $\\mu = 0$ (indicating no \ntransits), yielding $\\hat{Z}(t) = \\cos(\\omega t)$. Clearly this would not fit \nthe MD data for any $\\omega$, and it is easy to see why: Wallace included only\none frequency in his earlier model, whereas our Eq.\\ (\\ref{Zhat}) contains \ncontributions from many frequencies, all of which are necessary to raise the \nfirst minimum in $\\hat{Z}(t)$ above $-1$ and then damp $\\hat{Z}(t)$ out by \ndephasing. This suggests an alternate path to understanding diffusing states: \nBegin with a mean atom trajectory model that by construction reproduces the \ncorrect result for $\\hat{Z}(t)$ in the nondiffusing regime (Eq.\\ (\\ref{Zhat})),\nand then incorporate Wallace's notion of transits into this model. Our work in\nthis direction, with comparison to MD data for higher-temperature diffusing \nstates of liquid sodium, will be described in a subsequent paper \\cite{chis}.\n\n%\\bibliography{references}\n\n\\begin{thebibliography}{99}\n\n\\bibitem{enkin} D.\\ C.\\ Wallace and B.\\ E.\\ Clements, Phys.\\ Rev.\\ E {\\bf 59}, \n 2942 (1999).\n\n\\bibitem{radang} B.\\ E.\\ Clements and D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 59},\n 2955 (1999).\n\n\\bibitem{liqdyn} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 56}, 4179 (1997). \n\n\\bibitem{specif} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 58}, 1717 (1998). \n\n\\bibitem{oldvacf} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 58}, 538 (1998).\n\n\\bibitem{RMM} A.\\ Rahman, M.\\ J.\\ Mandell, and J.\\ P.\\ McTague, J.\\ Chem.\\ \n Phys.\\ {\\bf 64}, 1564 (1976).\n\n\\bibitem{LVS} R.\\ A.\\ LaViolette and F.\\ H.\\ Stillinger, J.\\ Chem.\\ Phys.\\ {\\bf\n 83}, 4079 (1985). \n\n\\bibitem{Stratt} M.\\ Buchner, B.\\ M.\\ Ladanyi, and R.\\ M.\\ Stratt, J.\\ Chem.\\ \n Phys.\\ {\\bf 97}, 8522 (1992).\n\n\\bibitem{VB} R.\\ Vallauri and F.\\ J.\\ Bermejo, Phys.\\ Rev.\\ E {\\bf 51}, 2654 \n (1995).\n\n\\bibitem{MKS} B.\\ Madan, T.\\ Keyes, and G.\\ Seeley, J.\\ Chem.\\ Phys.\\ {\\bf 94},\n 6762 (1991).\n\n\\bibitem{Zwan} R.\\ Zwanzig, J.\\ Chem.\\ Phys.\\ {\\bf 79}, 4507 (1983).\n\n\\bibitem{CV} J.\\ Cao and G.\\ A.\\ Voth, J.\\ Chem.\\ Phys.\\ {\\bf 103}, 4211 \n (1995).\n\n\\bibitem{chis} E.\\ Chisolm and D.\\ Wallace, submitted to Phys.\\ Rev.\\ E; \n cond-mat/0002058.\n\n\\end{thebibliography}\n\n\\end{document}"
}
] |
[
{
"name": "cond-mat0002057.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{enkin} D.\\ C.\\ Wallace and B.\\ E.\\ Clements, Phys.\\ Rev.\\ E {\\bf 59}, \n 2942 (1999).\n\n\\bibitem{radang} B.\\ E.\\ Clements and D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 59},\n 2955 (1999).\n\n\\bibitem{liqdyn} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 56}, 4179 (1997). \n\n\\bibitem{specif} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 58}, 1717 (1998). \n\n\\bibitem{oldvacf} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 58}, 538 (1998).\n\n\\bibitem{RMM} A.\\ Rahman, M.\\ J.\\ Mandell, and J.\\ P.\\ McTague, J.\\ Chem.\\ \n Phys.\\ {\\bf 64}, 1564 (1976).\n\n\\bibitem{LVS} R.\\ A.\\ LaViolette and F.\\ H.\\ Stillinger, J.\\ Chem.\\ Phys.\\ {\\bf\n 83}, 4079 (1985). \n\n\\bibitem{Stratt} M.\\ Buchner, B.\\ M.\\ Ladanyi, and R.\\ M.\\ Stratt, J.\\ Chem.\\ \n Phys.\\ {\\bf 97}, 8522 (1992).\n\n\\bibitem{VB} R.\\ Vallauri and F.\\ J.\\ Bermejo, Phys.\\ Rev.\\ E {\\bf 51}, 2654 \n (1995).\n\n\\bibitem{MKS} B.\\ Madan, T.\\ Keyes, and G.\\ Seeley, J.\\ Chem.\\ Phys.\\ {\\bf 94},\n 6762 (1991).\n\n\\bibitem{Zwan} R.\\ Zwanzig, J.\\ Chem.\\ Phys.\\ {\\bf 79}, 4507 (1983).\n\n\\bibitem{CV} J.\\ Cao and G.\\ A.\\ Voth, J.\\ Chem.\\ Phys.\\ {\\bf 103}, 4211 \n (1995).\n\n\\bibitem{chis} E.\\ Chisolm and D.\\ Wallace, submitted to Phys.\\ Rev.\\ E; \n cond-mat/0002058.\n\n\\end{thebibliography}"
}
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cond-mat0002058
|
A Mean Atom Trajectory Model for Monatomic Liquids
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[
{
"author": "Eric D.\\ Chisolm and Duane C.\\ Wallace"
},
{
"author": "Theoretical Division"
},
{
"author": "Los Alamos National Laboratory"
},
{
"author": "Los Alamos"
},
{
"author": "NM~~87545"
}
] |
A recent description of the motion of atoms in a classical monatomic system in liquid and supercooled liquid states divides the motion into two parts: oscillations within a given many-particle potential valley, and transit motion which carries the system from one many-particle valley to another. Building on this picture, we construct a model for the trajectory of an average atom in the system. The trajectory consists of oscillations at the normal-mode distribution of frequencies, representing motion within a fluctuating single-particle well, interspersed with position- and velocity-conserving transits to similar adjacent wells. For the supercooled liquid in nondiffusing states, the model gives velocity and displacement autocorrelation functions which exactly match those found in the many-particle harmonic approximation, and which are known to agree almost precisely with molecular dynamics (MD) simulations of liquid Na. At higher temperatures, by allowing transits to proceed at a temperature-dependent rate, the model gives velocity autocorrelation functions which are also in remarkably good agreement with MD simulations of Na at up to three times its melting temperature. Two independent processes in the model relax velocity autocorrelations: (a) dephasing due to the presence of many frequency components, which operates at all temperatures but which produces zero diffusion, and (b) the transit process, which increases with increasing temperature and which produces diffusion. Compared to several treatments of velocity autocorrelations based on instantaneous normal modes, the present model offers an advantage: It provides a single-atom trajectory in real space and time, including transits and valid for arbitrary times, from which all single-atom correlation functions can be calculated, and they are also valid at all times.
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[
{
"name": "paper2.tex",
"string": "\\documentclass[12pt]{article}\n\n%LA-UR-00-637\n\n\\usepackage{graphics}\n\n\\title{A Mean Atom Trajectory Model for Monatomic Liquids}\n\\author{Eric D.\\ Chisolm and Duane C.\\ Wallace \\\\ Theoretical Division \\\\\n Los Alamos National Laboratory \\\\ Los Alamos, NM~~87545}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nA recent description of the motion of atoms in a classical monatomic\nsystem in liquid and supercooled liquid states divides the motion into\ntwo parts: oscillations within a given many-particle potential valley,\nand transit motion which carries the system from one many-particle\nvalley to another. Building on this picture, we construct a model for\nthe trajectory of an average atom in the system. The trajectory\nconsists of oscillations at the normal-mode distribution of\nfrequencies, representing motion within a fluctuating single-particle\nwell, interspersed with position- and velocity-conserving transits to\nsimilar adjacent wells. For the supercooled liquid in nondiffusing\nstates, the model gives velocity and displacement autocorrelation\nfunctions which exactly match those found in the many-particle\nharmonic approximation, and which are known to agree almost precisely\nwith molecular dynamics (MD) simulations of liquid Na. At higher\ntemperatures, by allowing transits to proceed at a\ntemperature-dependent rate, the model gives velocity autocorrelation\nfunctions which are also in remarkably good agreement with MD\nsimulations of Na at up to three times its melting temperature. Two\nindependent processes in the model relax velocity autocorrelations:\n(a) dephasing due to the presence of many frequency components, which\noperates at all temperatures but which produces zero diffusion, and\n(b) the transit process, which increases with increasing temperature\nand which produces diffusion. Compared to several treatments of\nvelocity autocorrelations based on instantaneous normal modes, the\npresent model offers an advantage: It provides a single-atom\ntrajectory in real space and time, including transits and valid for\narbitrary times, from which all single-atom correlation functions can\nbe calculated, and they are also valid at all times.\n\\end{abstract}\n\n\\section{Introduction}\n\\label{intro}\n\nThe motion of atoms in a liquid can be divided into two constituent\n\\mbox{parts}: (a) oscillation in a valley of the liquid's many-body\npotential and (b) transits between many-body valleys. The latter\nprocess is responsible for self-diffusion. As we showed in a previous\npaper \\cite{chis}, which drew upon earlier results of Clements and\nWallace \\cite{enkin, radang}, the former motion can be modeled very\nprecisely under the assumption that the valleys are nearly harmonic,\nwith the majority of valleys (the random valleys) sharing a common\nspectrum of frequencies. Specifically, a purely harmonic model\nprovides an extremely accurate formula for $\\hat{Z}(t)$, the\nnormalized velocity autocorrelation function, in the nondiffusing\nregime. Here we will use this picture and our previous work to\njustify a ``mean atom trajectory'' model, a single-atom model that\napproximates the behavior of an average atom in the liquid and\ncorrectly reproduces its nondiffusing behavior. Then we will\nintroduce a simple intuitive account of the transit process that\nallows us to extend the model to the self-diffusing regime. In Sec.\\\n\\ref{model} we develop this model and explain how it is used to\ncalculate $\\hat{Z}(t)$ for a diffusing liquid. Then we fit the model\nto MD simulations at various temperatures in Sec.\\ \\ref{MD}, and we\ncomment on the quality of the results. In Sec.\\ \\ref{concl} we\ncompare the present mean atom trajectory model with previous work\nbased on Instantaneous Normal Modes (INM), and with an earlier\nindependent atom model \\cite{oldvacf}, and we summarize our\nconclusions.\n\n\\section{The Mean Atom Trajectory Model}\n\\label{model}\n\n\\subsection{General comments}\n\\label{general}\n\nTo form an appropriate basis for our model, we must begin with some\ninitial reasonable approximations about the nature of the valleys and\ntransits in the real liquid. As mentioned in the Introduction,\navailable evidence suggests that the valleys are nearly harmonic, and\nwe will continue to assume that here. Further, we will assume that\ntransits between valleys occur instantaneously and are local in\ncharacter; that is, each transit involves only a few neighboring\natoms. \n\nNow for the nondiffusing supercooled liquid, there are no transits,\nand as we've seen the system's motion is accurately expressed in terms\nof the harmonic normal modes. But it is also legitimate to consider\nthis motion from the point of view of a single atom, and when we do so\nwe see that each atom moves along a complicated trajectory within a\nsingle-particle potential well which fluctuates because of the motion of \nneighboring atoms, but whose center is fixed in space. An\nimportant timescale for this motion is the single-atom mean\nvibrational period $\\tau$, which we take as $2 \\pi / {\\omega_{\\rm\nrms}}$, where ${\\omega_{\\rm rms}}$ is the rms frequency of the set of\nnormal modes. Let us now follow the single-atom description as the\ntemperature is increased. Once the glass transition is passed, the\natom will begin to make transits from one single-particle well to\nanother \\cite{glass}, and even before the melting temperature is\nreached, the transit rate will be on the order of one per mean\nvibrational period. Since each atom has approximately ten neighbors,\nroughly ten transits will occur in its immediate vicinity every\nperiod, changing the set of normal mode eigenvectors each time; thus\nwe conclude that a decomposition of the motion into normal modes will\nnot be useful when the liquid is diffusing, so we have no choice but\nto follow a single-atom description for the liquid state\n\\cite{oldvacf}. That being the case, we shall start from the\nbeginning with a single-atom description, in order to construct a\nunified model for diffusing and nondiffusing motion alike.\n\n\\subsection{Nondiffusing regime}\n\\label{nondiff}\n\nOur starting point will be in the nondiffusing regime, where a normal\nmode analysis is still valid. As shown in \\cite{chis}, the $i$th\ncoordinate of the $K$th atom in an $N$-body harmonic valley can be\nwritten\n\\begin{equation}\nu_{Ki}(t) = \\sum_{\\lambda} w_{Ki, \\lambda} a_{\\lambda}\n \\sin(\\omega_{\\lambda}t + \\alpha_{\\lambda}),\n\\label{uharmoft}\n\\end{equation}\nwhere the $w_{Ki,\\lambda}$ form a $3N \\times 3N$ orthogonal matrix,\nthe $\\omega_{\\lambda}$ are the frequencies of the normal modes, and\nthe $a_{\\lambda}$ are the amplitudes of the modes. Three of the\nmodes have zero frequency and correspond to center of mass motion; here\nwe demand that the center of mass is stationary, so the system has\nonly $3N-3$ independent degrees of freedom, the zero frequency modes\nare absent, and the sum over $\\lambda$ runs from 1 to $3N-3$. To\nmake this an equation for a ``mean'' atom, we first drop the index\n$K$:\n\\begin{equation}\nu_{i}(t) = \\sum_{\\lambda} w_{i \\lambda} a_{\\lambda}\n \\sin(\\omega_{\\lambda}t + \\alpha_{\\lambda}).\n\\label{u1oft}\n\\end{equation}\nNow we must reinterpret the $w_{i \\lambda}$ since they no longer form an \northogonal (or even square) matrix. Let\n\\begin{equation}\n\\mbox{\\boldmath $w$}_{\\lambda} = w_{1 \\lambda} \\hat{\\mbox{\\boldmath $x$}} + \n w_{2 \\lambda} \\hat{\\mbox{\\boldmath $y$}} + \n w_{3 \\lambda} \\hat{\\mbox{\\boldmath $z$}}\n\\label{defw}\n\\end{equation}\nso\n\\begin{equation}\n\\mbox{\\boldmath $u$}(t) = \\sum_{\\lambda} \\mbox{\\boldmath $w$}_{\\lambda} \n a_{\\lambda} \\sin(\\omega_{\\lambda}t + \n \\alpha_{\\lambda}).\n\\label{u2oft}\n\\end{equation}\nWe will ultimately consider situations in which the well center is\nallowed to move, so let \\mbox{\\boldmath $r$}$(t)$ be the atom's\nposition, \\mbox{\\boldmath $R$} be the location of the center of\nthe well, and \\mbox{\\boldmath $u$}$(t)$ be the atom's displacement\nfrom the well center; then\n\\begin{equation}\n\\mbox{\\boldmath $r$}(t) = \\mbox{\\boldmath $R$} + \\sum_{\\lambda} \n \\mbox{\\boldmath $w$}_{\\lambda} a_{\\lambda} \n \\sin(\\omega_{\\lambda}t + \\alpha_{\\lambda})\n\\label{u3oft}\n\\end{equation}\nwith velocity\n\\begin{equation}\n\\mbox{\\boldmath $v$}(t) = \\sum_{\\lambda} \\mbox{\\boldmath\n $w$}_{\\lambda} a_{\\lambda} \\omega_{\\lambda} \\cos(\\omega_{\\lambda}t +\n \\alpha_{\\lambda}).\n\\label{v3oft}\n\\end{equation}\n\nWe now have the basic formula, but we must decide how to assign values\nto the {\\boldmath $w$}$_{\\lambda}$, $a_{\\lambda}$, and\n$\\alpha_{\\lambda}$. Let's do this by calculating $Z(t)$, the velocity\nautocorrelation function, in this model and comparing it to the\nharmonic result derived in \\cite{chis}.\n\\begin{eqnarray}\nZ(t) & = & \\frac{1}{3} \\langle \\mbox{\\boldmath $v$}(t) \\cdot\n \\mbox{\\boldmath $v$}(0) \\rangle \\nonumber \\\\ \n& = & \\frac{1}{3} \\sum_{\\lambda \\lambda'} \\mbox{\\boldmath $w$}_{\\lambda} \n \\cdot \\mbox{\\boldmath $w$}_{\\lambda'} a_{\\lambda} a_{\\lambda'}\n \\omega_{\\lambda} \\omega_{\\lambda'} \\langle \\cos(\\omega_{\\lambda}t +\n \\alpha_{\\lambda}) \\cos(\\alpha_{\\lambda'}) \\rangle\n\\label{1stZ}\n\\end{eqnarray}\nLet's assign the $\\alpha_{\\lambda}$ randomly and average over each \n$\\alpha_{\\lambda}$ {\\em separately}; then if $\\lambda \\neq \\lambda'$,\n\\begin{equation}\n\\langle \\cos(\\omega_{\\lambda}t + \\alpha_{\\lambda}) \\cos(\\alpha_{\\lambda'}) \n\\rangle = \\langle \\cos(\\omega_{\\lambda}t + \\alpha_{\\lambda}) \\rangle \n\\langle \\cos(\\alpha_{\\lambda'}) \\rangle = 0 \n\\end{equation}\nbut if $\\lambda = \\lambda'$, then\n\\begin{equation}\n\\langle \\cos(\\omega_{\\lambda}t + \\alpha_{\\lambda}) \\cos(\\alpha_{\\lambda'}) \n\\rangle = \\langle \\cos(\\omega_{\\lambda}t + \\alpha_{\\lambda}) \n\\cos(\\alpha_{\\lambda}) \\rangle = \\frac{1}{2} \\cos(\\omega_{\\lambda}t).\n\\end{equation}\nThus the $\\lambda \\neq \\lambda'$ terms in Eq.\\ (\\ref{1stZ}) are eliminated and \nwe have\n\\begin{equation}\nZ(t) = \\frac{1}{6} \\sum_{\\lambda} | \\mbox{\\boldmath $w$}_{\\lambda} |^2 \n a_{\\lambda}^2 \\omega_{\\lambda}^2 \\cos(\\omega_{\\lambda}t).\n\\label{2ndZ}\n\\end{equation}\nNote that the formula for $Z(t)$ in the harmonic theory also lacks \noff-diagonal terms \\cite{chis}, but for a different reason: The \northogonality of the matrix $w_{Ki,\\lambda}$ removes them in that case. \nNow let us make the assignment\n\\begin{equation}\n\\mbox{\\boldmath $w$}_{\\lambda} = \\frac{1}{\\sqrt{N-1}} \\hat{\\mbox{\\boldmath \n $w$}}_{\\lambda}\n\\label{defwhat}\n\\end{equation}\nwhere $\\hat{\\mbox{\\boldmath $w$}}_{\\lambda}$ is a randomly chosen unit vector; \nthen\n\\begin{equation}\nZ(t) = \\frac{1}{6N-6} \\sum_{\\lambda} a_{\\lambda}^2 \\omega_{\\lambda}^2 \n \\cos(\\omega_{\\lambda}t).\n\\label{3rdZ}\n\\end{equation}\nThis is the same expression for $Z(t)$ that one derives in the\nharmonic model (see Eq.\\ (10) of \\cite{chis}) for {\\em any}\ndistribution of normal mode amplitudes $a_{\\lambda}$, not just the\nthermal equilibrium distribution. Thus our model with these choices\nof $\\alpha_{\\lambda}$ and \\mbox{\\boldmath $w$}$_{\\lambda}$ correctly\nreproduces $Z(t)$ for any equilibrium or nonequilibrium ensemble in\nthe harmonic theory. To recover the equilibrium result, we make the\nfinal substitution\n\\begin{equation}\na_{\\lambda} = \\sqrt{\\frac{2kT}{M \\omega_{\\lambda}^2}}\n\\end{equation}\nwith the result\n\\begin{equation}\nZ(t) = \\frac{1}{3N-3} \\frac{kT}{M} \\sum_{\\lambda} \\cos(\\omega_{\\lambda}t),\n\\label{4thZ}\n\\end{equation}\nwhich is Eq.\\ (13) from \\cite{chis}. Thus our model assumes that the \nmotion of a mean nondiffusing atom in thermal equilibrium at temperature $T$ \nis given by\n\\begin{equation}\n\\mbox{\\boldmath $r$}(t) = \\mbox{\\boldmath $R$} + \\frac{1}{\\sqrt{N-1}}\n \\sqrt{\\frac{2kT}{M}} \\sum_{\\lambda} \n \\hat{\\mbox{\\boldmath $w$}}_{\\lambda}\n \\omega_{\\lambda}^{-1} \\sin(\\omega_{\\lambda}t + \n \\alpha_{\\lambda})\n\\label{u4oft}\n\\end{equation}\nwhere the phases $\\alpha_{\\lambda}$ and unit vectors\n$\\hat{\\mbox{\\boldmath $w$}}_{\\lambda}$ are randomly chosen. \n\nBy construction this model gets the same result for $Z(t)$, and thus\n$\\langle v^2 \\rangle$, as the harmonic model; does it correctly\nreproduce any other functions? We can check by calculating the analog\nof $Z(t)$ for positions, $\\langle \\mbox{\\boldmath $u$}(t) \\cdot\n\\mbox{\\boldmath $u$}(0) \\rangle$. Using Eq.\\ (\\ref{uharmoft}) for\n$u_{Ki}(t)$ and calculating for the harmonic model as in \\cite{chis},\nand using Eq.\\ (\\ref{u1oft}) for \\mbox{\\boldmath $u$}$(t)$ and\ncalculating as above, one finds in both cases that\n\\begin{equation}\n\\langle \\mbox{\\boldmath $u$}(t) \\cdot \\mbox{\\boldmath $u$}(0) \\rangle = \n\\frac{1}{2N-2} \\sum_{\\lambda} a_{\\lambda}^2 \\cos(\\omega_{\\lambda}t).\n\\end{equation}\nAgain the substitution $a_{\\lambda} = \\sqrt{2kT/M\\omega_{\\lambda}^2}$ \nrecovers the correct thermal equilibrium result. So this model correctly \nreproduces $\\langle \\mbox{\\boldmath $u$}(t) \\cdot \\mbox{\\boldmath $u$}(0) \n\\rangle$ and $\\langle u^2 \\rangle$ as well. Notice that because we have a \nclosed form for \\mbox{\\boldmath $u$}$(t)$, all of the correlation functions \ncalculated so far also have a closed form. Once we introduce transits, this \nwill no longer be the case.\n\nWhile this model compares well with the harmonic model, one might wonder how \nwell it compares with molecular dynamics (MD) results. In Fig.\\ \\ref{vsqvsMD} \na graph of $v^2$ as a function of $t$ for the model in equilibrium at 6.69 K is\ncompared to a randomly chosen particle from an MD run\n\\begin{figure}\n\\includegraphics{vsq_6.69K.ps}\n\\includegraphics{vsq_mu=0.0.ps}\n\\caption{$v^2$ for a randomly chosen atom in an MD simulation of Na at 6.69 K \n (top) is compared with $v^2$ for the mean atom trajectory model in the\n nondiffusing regime (bottom) over 5000 timesteps of duration $\\delta \n t = 1.4 \\times 10^{-15}$ s. Notice that both fluctuations have \n roughly the same magnitude and oscillate at approximately the same \n frequencies.}\n\\label{vsqvsMD}\n\\end{figure}\nof liquid Na also at 6.69 K, a temperature at which it is known the\nsample is nondiffusing (see \\cite{chis} for details). Note that in\nboth graphs $v^2$ has approximately the same amplitude, and the gaps\nbetween peaks are roughly the same size, indicating oscillations at\nfrequencies in the same ranges. Thus not only functions of the motion\nbut the motion itself shows strong qualitative agreement with MD in\nthe nondiffusing regime.\n\nThe MD system with which we will be comparing our model has $N=500$\nparticles, so it has 1497 normal mode frequencies. (Remember that the\nzero frequency modes are removed at the start.) When we introduce\ntransits, we will have to evaluate $Z(t)$ numerically, and for speed\nof computation we would like to use only a representative subset of the normal \nmode frequencies, say 75 instead of all 1497. The original set of\nfrequencies is determined as described in \\cite{chis}. We decided\nwhich subset to use by calculating three moments of the full frequency\ndistribution defined below:\n\\begin{eqnarray}\n\\omega_{2} & = & \\left[ \\frac{5}{3} \\langle \\omega_{\\lambda}^2 \\rangle\n \\right]^{1/2} \\nonumber \\\\\n\\omega_{0} & = & \\exp \\langle \\ln(\\omega_{\\lambda}) \\rangle \\nonumber \\\\\n\\omega_{-2} & = & \\left[ \\frac{1}{3} \\langle \\omega_{\\lambda}^{-2} \\rangle\n \\right]^{-1/2}.\n\\end{eqnarray}\nNote that by this definition $\\omega_{\\rm rms} = \\sqrt{3/5}\\, \\omega_{2}$. \nWe then calculated the same three moments for several sets of 75 frequencies \nevenly spaced throughout the full set, and we chose the set that best fit the \nmoments to use in further calculations. This process is somewhat subjective, \nbecause skewing the sample in favor of lower frequencies improves the accuracy \nof $\\omega_{-2}$ but reduces the accuracy of $\\omega_{2}$, and the opposite is \ntrue if one skews in favor of high frequencies. The table below shows the \nvalues of the moments for the full set and the reduced set of 75 that we \nultimately chose. The frequencies are in units of $\\delta t^{-1}$, where \n$\\delta t = 1.4 \\times 10^{-15}$ s is the timestep of our MD simulations \n\\cite{chis}.\n\\begin{center}\n\\begin{tabular}{ccc}\n & Full & Reduced \\\\ \\hline\n$\\omega_{2}$ & 0.02826 & 0.02824 \\\\\n$\\omega_{0}$ & 0.01807 & 0.01803 \\\\\n$\\omega_{-2}$ & 0.02101 & 0.02111\n\\end{tabular}\n\\end{center}\nTo use the reduced set we must rewrite our formulas for \n\\mbox{\\boldmath $r$}$(t)$ and \\mbox{\\boldmath $v$}$(t)$ \\mbox{slightly}. Let \n$\\Lambda$ be the total number of frequencies; then $\\Lambda = 3N - 3$ for the \nfull set so \\mbox{\\boldmath $r$}$(t)$ and \\mbox{\\boldmath $v$}$(t)$ become\n\\begin{eqnarray}\n\\mbox{\\boldmath $r$}(t) & = & \\mbox{\\boldmath $R$} + \\sqrt{\\frac{3}{\\Lambda}}\n \\sqrt{\\frac{2kT}{M}} \\sum_{\\lambda=1}^{\\Lambda} \n \\hat{\\mbox{\\boldmath $w$}}_{\\lambda} \\,\n \\omega_{\\lambda}^{-1} \\sin(\\omega_{\\lambda}t + \n \\alpha_{\\lambda}) \\nonumber \\\\\n\\mbox{\\boldmath $v$}(t) & = & \\sqrt{\\frac{3}{\\Lambda}} \\sqrt{\\frac{2kT}{M}} \n \\sum_{\\lambda=1}^{\\Lambda} \n \\hat{\\mbox{\\boldmath $w$}}_{\\lambda}\n \\cos(\\omega_{\\lambda}t + \\alpha_{\\lambda}).\n\\label{uoft}\n\\end{eqnarray}\nThis form is also correct for the reduced set of frequencies\n$\\omega_{\\lambda}$, so this is the form we will use. As another check\non the accuracy of our results with only 75 frequencies, we\nrecalculated the original $\\sum \\cos(\\omega_{\\lambda}t)$ expression\nfor $\\hat{Z}(t)$ using both the full and reduced sets; their\ndisagreement is at most 0.01 out to time $1000 \\, \\delta t$. However,\nthe discrepancy grows beyond that time, as the expression using the\nreduced set begins to experience revivals. Hence we will not consider\n$\\hat{Z}(t)$ beyond that point.\n\n\\subsection{Diffusing regime}\n\\label{diff}\n\nTo incorporate diffusion into the mean atom trajectory model, we rely on \nWallace's notion of a single-particle transit \\cite{liqdyn}, a nearly\ninstantaneous transition from one well to another. As discussed in \n\\cite{glass}, we expect transits to be governed not by thermal activation \n(having enough energy to escape a fixed well) but by {\\em correlations} \n(neighbors must be positioned properly for a low-potential path to open between\nwells). We implement this property by having transits occur at a temperature \ndependent {\\em rate} $\\nu(T)$, so in a small time interval $\\Delta t$ the \nprobability of a single transit is $\\nu \\Delta t$.\n\nWe model the transit process itself by assuming it occurs\ninstantaneously in the forward direction; from this we can determine\nthe parameters {\\boldmath $R$}, $\\hat{\\mbox{\\boldmath\n$w$}}_{\\lambda}$, and $\\alpha_{\\lambda}$ appearing in Eq.\\\n(\\ref{uoft}) after the transit in terms of the same quantities before\nthe transit. Since the process is instantaneous, both {\\boldmath\n$r$}$(t)$ and {\\boldmath $v$}$(t)$ are the same before and afterwards.\nLet {\\boldmath $R$}$^{\\rm before}$ and {\\boldmath $R$}$^{\\rm after}$\nbe the well centers before and after the transit and let {\\boldmath\n$u$}$^{\\rm before}$ and {\\boldmath $u$}$^{\\rm after}$ be the\ncorresponding displacements from the well centers. Then\n$\\mbox{\\boldmath $r$}^{\\rm before} = \\mbox{\\boldmath $r$}^{\\rm after}$\nimplies\n\\begin{equation}\n\\mbox{\\boldmath $R$}^{\\rm before} + \\mbox{\\boldmath $u$}^{\\rm before} = \n\\mbox{\\boldmath $R$}^{\\rm after} + \\mbox{\\boldmath $u$}^{\\rm after}. \n\\end{equation}\nTo transit {\\em forward}, we assume the center of the new well lies along the \nline between the old well center and the atom, but it lies on the opposite side\nof the atom from the old well center an equal distance away. This implies\n{\\boldmath $u$}$^{\\rm after} = -\\mbox{\\boldmath $u$}^{\\rm before}$, so \n\\begin{equation}\n\\mbox{\\boldmath $R$}^{\\rm before} + \\mbox{\\boldmath $u$}^{\\rm before} = \n \\mbox{\\boldmath $R$}^{\\rm after} - \\mbox{\\boldmath $u$}^{\\rm before} \n\\end{equation}\nwith the result\n\\begin{equation}\n\\mbox{\\boldmath $R$}^{\\rm after} = \\mbox{\\boldmath $R$}^{\\rm before} + \n 2\\mbox{\\boldmath $u$}^{\\rm before}. \n\\label{Rafttrans}\n\\end{equation}\nThis determines the new well center in terms of the coordinates before the \ntransit. As for the unit vectors $\\hat{\\mbox{\\boldmath $w$}}_{\\lambda}$, since\nthey are randomly generated and play no role in calculating $\\hat{Z}(t)$, we \nhave decided to leave them unchanged by transits. Finally, we have the phases \n$\\alpha_{\\lambda}$. We must use these to implement the relations\n\\begin{equation}\n\\mbox{\\boldmath $u$}^{\\rm after} = -\\mbox{\\boldmath $u$}^{\\rm before}, \\ \\ \n\\mbox{\\boldmath $v$}^{\\rm after} = \\mbox{\\boldmath $v$}^{\\rm before}\n\\end{equation}\nwhich we have assumed above. Since {\\boldmath $u$}$(t)$ is a sum of sines and \n{\\boldmath $v$}$(t)$ a sum of cosines, the simplest way to change the sign of \n{\\boldmath $u$}$(t)$ while preserving that of {\\boldmath $v$}$(t)$ is to \nreverse the signs of the arguments $(\\omega_{\\lambda}t + \\alpha_{\\lambda})$\nin Eq.\\ (\\ref{uoft}). Let the transit occur at time $t_{0}$; then\n\\begin{equation}\n\\omega_{\\lambda}t_{0} + \\alpha_{\\lambda}^{\\rm after} = -(\\omega_{\\lambda}t_{0} \n + \\alpha_{\\lambda}^{\\rm before})\n\\end{equation}\nso\n\\begin{equation}\n\\alpha_{\\lambda}^{\\rm after} = -2\\omega_{\\lambda}t_{0} - \\alpha_{\\lambda}^{\\rm \n before}.\n\\end{equation}\nThus, in this model a transit is implemented at time $t_{0}$ by leaving the \n$\\hat{\\mbox{\\boldmath $w$}}_{\\lambda}$ alone and making the substitutions\n\\begin{eqnarray}\n\\mbox{\\boldmath $R$} & \\rightarrow & \\mbox{\\boldmath $R$} + 2\\mbox{\\boldmath \n $u$}(t_{0}) \\nonumber \\\\\n\\alpha_{\\lambda} & \\rightarrow & -2\\omega_{\\lambda}t_{0} - \\alpha_{\\lambda}.\n\\label{imptrans}\n\\end{eqnarray}\nThis conserves {\\boldmath $r$}, reverses the sign of {\\boldmath $u$}, and \nconserves {\\boldmath $v$}. \n\nNow our mean atom trajectory model consists of nondiffusive motion\nbetween transits as given by Eq.\\ (\\ref{uoft}), with a given\nprobability in each small time interval that {\\boldmath $R$} and the\nphases $\\alpha_{\\lambda}$ will be replaced with new values as\ndetermined in Eq.\\ (\\ref{imptrans}). The addition of transits means\nthat we no longer have closed form expressions for {\\boldmath\n$r$}$(t)$ and {\\boldmath $v$}$(t)$ for all times, so we have no closed\nform expression for $\\hat{Z}(t)$; but this model can be implemented\neasily on a computer in a manner analogous to an MD calculation, and in\nthat way we can calculate autocorrelation functions. We turn to that\ncalculation next.\n\n\\subsection{Evaluating $\\hat{Z}(t)$ in the diffusing regime}\n\\label{evalZ}\n\nTo calculate $\\hat{Z}(t)$, we select a value for the rate $\\nu$,\ngenerate a random set of $\\hat{\\mbox{\\boldmath $w$}}_{\\lambda}$ and\n$\\alpha_{\\lambda}$, and use Eq.\\ (\\ref{uoft}) to calculate {\\boldmath\n$r$}$(t)$ and {\\boldmath $v$}$(t)$ from $t = 0$ to $t=t_{\\rm max}$ in\nincrements of $\\delta t$, where the criterion for choosing $t_{\\rm\nmax}$ is discussed below and $\\delta t$ is the timestep used in our MD\nsimulations (defined in Subsection \\ref{nondiff}). At each timestep,\nwe check to see if a transit occurs, and if so we implement Eq.\\\n(\\ref{imptrans}) and continue with the new {\\boldmath $R$} and\n$\\alpha_{\\lambda}$. We then calculate $\\hat{Z}(t)$ using the formula\n\\begin{equation}\nZ(t) = \\frac{1}{3(t_{\\rm max} - t) + 3} \\sum_{t' = 0}^{t_{\\rm max} - t} \n \\mbox{\\boldmath $v$}(t + t') \\cdot \\mbox{\\boldmath $v$}(t')\n\\end{equation}\nand normalizing. This equation is a modified form of the expression\nused to calculate $Z(t)$ in MD; notice that the average over $t'$ has\nthe same effect as averaging separately over each phase\n$\\alpha_{\\lambda}$ that appears in the velocity vectors. Just as in\nMD, we want to average over a large data set, so we require $t_{\\rm\nmax} >> t$; we have chosen $t_{\\rm max} = 20$ million timesteps and we\ncalculate $\\hat{Z}(t)$ only to $t=1000$ timesteps. We estimate the\ntotal error from using only a subset of all 1497 frequencies and the\nfinite size of the data set to be at most 0.01; in particular, when\n$\\nu = 0$ the calculation converges to the closed form result $\\sum\n\\cos(\\omega_{\\lambda}t)$ to this accuracy.\n\n\\section{Comparison with MD}\n\\label{MD}\n\nThe MD setup with which we compared the predictions of this model is\nthe one described in \\cite{chis}; $N=500$ atoms of Na move under the\ninfluence of a highly realistic pair potential with the timestep\n$\\delta t = 1.4 \\times 10^{-15}$ given in Subsection \\ref{nondiff}.\nWe performed equilibrium runs of the system at 216.3 K, 309.7 K, 425.0\nK, 664.7 K, and 1022.0 K, all temperatures at which the system is\ndiffusing. Since $T_{m} = 371.0$ K for Na at this density, our\nsimulations range from the supercooled regime to nearly three times\nthe melting temperature. We then ran the model for various values of\n$\\nu$, adjusting until the model matched the value of the first\nminimum of $\\hat{Z}(t)$ at each temperature. The values of $\\nu$ that\nwe fit for all temperatures are given below, and the resulting $\\hat{Z}(t)$ \nfor each $\\nu$ is compared to the corresponding MD result in Figs.\\ \n\\ref{216.3Kvsmodel} through \\ref{1022.0Kvsmodel}. Here $\\nu$ is \nexpressed in units of $\\tau^{-1}$ where $\\tau = 2 \\pi / \\omega_{\\rm \nrms}$ is the single-atom mean vibrational period defined in Subsection \n\\ref{general}. \n\\begin{center}\n\\begin{tabular}{rc}\n$T$ (K) & $\\nu \\ (\\tau^{-1})$ \\\\ \\hline\n 216.3 & 0.35018 \\\\\n 309.7 & 0.60276 \\\\\n 425.0 & 0.83985 \\\\\n 664.7 & 1.24858 \\\\\n1022.0 & 1.68774\n\\end{tabular}\n\\end{center}\n\\begin{figure}\n\\includegraphics{216.3Kvsmodel.ps}\n\\caption{The model prediction for $\\hat{Z}(t)$ at $\\nu = 0.35018 \\, \\tau^{-1}$ \n compared with the MD result for supercooled liquid Na at $T = \n 216.3$ K.}\n\\label{216.3Kvsmodel}\n\\end{figure}\n\\begin{figure}\n\\includegraphics{309.7Kvsmodel.ps}\n\\caption{The model prediction for $\\hat{Z}(t)$ at $\\nu = 0.60276 \\, \\tau^{-1}$ \n compared with the MD result for supercooled liquid Na at $T = \n 309.7$ K.}\n\\label{309.7Kvsmodel}\n\\end{figure}\n\\begin{figure}\n\\includegraphics{425.0Kvsmodel.ps}\n\\caption{The model prediction for $\\hat{Z}(t)$ at $\\nu = 0.83985 \\, \\tau^{-1}$ \n compared with the MD result for liquid Na at $T = 425.0$ K.}\n\\label{425.0Kvsmodel}\n\\end{figure}\n\\begin{figure}\n\\includegraphics{664.7Kvsmodel.ps}\n\\caption{The model prediction for $\\hat{Z}(t)$ at $\\nu = 1.24858 \\, \\tau^{-1}$ \n compared with the MD result for liquid Na at $T = 664.7$ K.}\n\\label{664.7Kvsmodel}\n\\end{figure}\n\\begin{figure}\n\\includegraphics{1022.0Kvsmodel.ps}\n\\caption{The model prediction for $\\hat{Z}(t)$ at $\\nu = 1.68774 \\, \\tau^{-1}$ \n compared with the MD result for liquid Na at $T = 1022.0$ K.}\n\\label{1022.0Kvsmodel}\n\\end{figure}\nNotice that in all cases $\\nu$ is of the same order of magnitude as\n$\\tau^{-1}$, indicating roughly one transit per mean vibrational\nperiod, as mentioned in Subsection \\ref{general}, and as predicted in\n\\cite{glass}. \n\nThe most obvious trend exhibited by $\\hat{Z}(t)$ from the five MD runs\nis that its first minimum is rising with increasing $T$; as we mention\nbelow, this is the primary reason for the increasing diffusion\ncoefficient $D$. Note that the model is able to reproduce this most\nimportant feature quite satisfactorily. In fact, all five fits of\nthe model to the MD results capture their essential features, but we\ndo see systematic trends in the discrepancies. First, note that the\nlocation of the first minimum barely changes at all in the model as\n$\\nu$ is raised, but in MD the first minimum moves steadily to earlier\ntimes as the temperature rises. The first minimum occurs at a time\nroughly equal to half of the mean vibrational period ($\\tau = 287 \\,\n\\delta t$ in this system), so the steady drift backward suggests that\nthe MD system is sampling a higher range of frequencies at higher $T$.\nAlso, for the three lowest temperatures the model tends to overshoot\nthe MD result in the vicinity of the first two maxima after the\norigin, and at the highest two temperatures this overshoot is\naccompanied by a positive tail that is slightly higher than the (still\nsomewhat long) tail predicted by MD. These overshoots should clearly\naffect the diffusion coefficient $D$, which is the integral of $Z(t)$.\nTo check this, we calculated the reduced diffusion coefficient\n$\\hat{D}$, the integral of $\\hat{Z}(t)$, which is related to $D$ by\n\\begin{equation} D = \\frac{kT}{M} \\hat{D}. \\end{equation}\nThe results are compared to the values of $\\hat{D}$ calculated from\nthe MD runs in Fig.\\ \\ref{DvsT}. The results from the two\nnondiffusing runs discussed in \\cite{chis} are also included.\n\\begin{figure}\n\\includegraphics{DvsT.ps}\n\\caption{$\\hat{D}$ as a function of $T$ for both the model and MD.}\n\\label{DvsT}\n\\end{figure}\nIn all of the diffusing cases, the model overestimates $\\hat{D}$ by roughly \nthe same amount, which we take to be the effect of the overshoots at the first \ntwo maxima. At the higher temperatures the discrepancy is also higher, \npresumably due to the model's long tail.\n\nIt is interesting to note that this MD system produces results that\nagree very closely with experiment: For example, the MD predicts that\nat $T = 425.0$ K and $\\rho = 0.925$ g/cm$^3$, $D = 6.40 \\times\n10^{-5}$ cm$^2$/s, while experiments by Larsson, Roxbergh, and Lodding\n\\cite{LRL} find that at $T = 425.0$ K and $\\rho = 0.915$ g/cm$^3$, $D\n= 6.020 \\times 10^{-5}$ cm$^2$/s. Hence our agreement with MD results\ngenuinely reflects agreement with properties of real liquid Na.\n\n\\section{Conclusions}\n\\label{concl}\n\nWe have presented a single-atom model of a monatomic liquid that\nprovides a unified account of diffusing and nondiffusing behavior.\nThe nondiffusing motion is modeled as a sum of oscillations at the\nnormal mode frequencies (Eq.\\ (\\ref{uoft})), simulating the trajectory\nof an average atom in a complicated single-body potential well that\nfluctuates due to the motion of its neighbors. Self-diffusion is\naccounted for in terms of instantaneous transits between wells, which\noccur at a temperature-dependent rate $\\nu$. Since this model gives a\nsimple and straightforward account of the motion itself, it can easily\nbe used to calculate any single-atom correlation function one wishes;\nhere we have focussed on the velocity autocorrelation function. It is\ninteresting to note that in this model the velocity correlations\npersist through a transit, instead of being washed out entirely by the\ntransit process; we will return to this point below. The relaxation\nof correlations seen by the decay of $\\hat{Z}(t)$ arises here from two\ndistinct processes: Dephasing as a result of the large number of\nfrequencies in the single-well motion, and transits between wells.\nThe dephasing effect produces relaxation but not diffusion: It causes\n$\\hat{Z}(t)$ to decay \\cite{chis} but its integral remains zero. On\nthe other hand, transits certainly contribute to relaxation (see\n\\cite{oldvacf}, where they provide the only relaxation mechanism), but\nin addition they raise the first minimum of $\\hat{Z}(t)$\nsubstantially, increasing its integral and providing a nonzero $D$.\n\nThe comparison of this model to MD results is generally quite\npositive, particularly for a one-parameter model; the two calculations\nof $\\hat{Z}(t)$ agree strongly over 1000 timesteps. In addition, the\nmatch is encouraging over a very large range of temperatures, from\nessentially 0 K to $3 T_{m}$. The most noticeable discrepancies are\nthe backward drift in the location of the first minimum of\n$\\hat{Z}(t)$, which is present in MD but not the model, and the\ntendency of the model to exaggerate certain characteristics of the MD\nresults (the maxima at intermediate times and the high-$T$ positive\ntail). This latter effect is responsible for the model's overestimate\nof the diffusion coefficient, though we hasten to add that the model\n$\\hat{D}$ is still in satisfactory agreement with MD results,\nespecially for the liquid at $T \\geq T_{m}$ (see Fig.\\ \\ref{DvsT}).\n\nAs in \\cite{chis}, it is useful to compare this model and the\naccompanying results to the work others have done using the formalism\nof Instantaneous Normal Modes (INM) and similar methods. Previously, we\ndiscussed the advantages of our general approach and the superior\nquality of its results when applied to nondiffusing states; here we\nwill consider matters relating explicitly to mechanisms of diffusion.\nAs is noted explicitly by Vallauri and Bermejo \\cite{VB}, the account\nof INM by Stratt (see, for example, \\cite{Stratt}) does not consider\n\\mbox{diffusion} at all; their $Z(t)$ is essentially a sum over\ncosines, and as such it integrates to zero. This is understandable,\nbecause as Stratt et al.\\ repeatedly emphasize, their approximation is\nvalid only for very short times, so they are not attempting to model\neffects with longer timescales. They compare their \\mbox{INM} results with\nMD calculations of $\\hat{Z}(t)$ for states of an LJ system ranging\nfrom a moderately supercooled liquid to well above the melting\ntemperature, and our fits are of roughly the same quality or better in\nall cases. Authors who do attempt to model diffusion usually follow\nthe path suggested initially by Zwanzig \\cite{Zwan}, who thought of\nthe liquid's phase space as divided into \\mbox{``cells''} in which each atom\nspends its time before finding a saddle point in the potential and\njumping from one cell to the next. He imagined as a first\napproximation that the jumps destroy all correlations between the\ncells; since atoms are jumping all the time, he suggested that the net\nresult was to multiply the nondiffusing form of $Z(t)$ by a damping\nfactor $\\exp(-t/\\tau)$ for some timescale $\\tau$ representing the\nlifetime of a stay in a single cell. Notice that Zwanzig provided no\ndynamical model of the jumping process itself. This suggestion has\nbeen developed and transformed extensively by Madan, Keyes, and Seeley\n\\cite{MKS}, who take a general Zwanzig-like functional form for $Z(t)$\nand use a combination of heuristic arguments and constraints on its\nmoments to specify its dependence on a ``hopping rate'' $\\omega_{v}$,\nthe analog of $\\tau^{-1}$ for Zwanzig, which they then extract from\nthe unstable lobe of the INM spectrum. Although we cannot be entirely\nsure, as indicated in \\cite{chis}, we think it most likely that their\nsimulations of LJ Ar at 80 K, 120 K, and 150 K consist of states\ncomparable to our Figs.\\ \\ref{216.3Kvsmodel} and \\ref{309.7Kvsmodel},\nand again we would argue that our fits are somewhat better. Finally,\nCao and Voth \\cite{CV} also approach diffusion by means of a damping\nfactor, and they consider factors of two different types, each of\nwhich contains parameters that can be determined from other calculated\nor experimental quantities. Their matches with MD are actually quite\ngood, but again ours are of at least comparable quality.\n\nHaving claimed that our matches to MD simulations are as good as or\nbetter than all of the others we have surveyed, let us emphasize a\nfundamental difference between our approaches to diffusion: We provide\nan account of the {\\em process} of transiting from well to well, so we\nhave a model of the actual {\\em motion} of a mean atom in space that\nis valid to arbitrary times, and given this model we can {\\em\ncalculate} the effect of transits on correlations. All other\napproaches we know of begin with an account of the motion valid only\nfor very short times, calculate $Z(t)$ from this motion, and then try\nto model the effects of diffusion on $Z(t)$ directly, using parameters\nthat are thought to be characteristic of jumps between wells. In the\ndiffusing regime, no one else we have seen actually calculates\n$\\langle \\mbox{\\boldmath $v$}(t) \\cdot \\mbox{\\boldmath $v$}(0)\n\\rangle$ to find $Z(t)$, as we do. In the process, we find that some\nof the assumptions made by others, in particular Zwanzig's hypothesis\nthat jumps between wells simply erase correlations, are not true.\nThis approach is already yielding insights into the actual motion of\nan atom undergoing a transit.\n\nFinally, we would like to compare the present mean atom trajectory\nmodel with an earlier independent atom model by Wallace\n\\cite{oldvacf}. In developing the independent atom model, two\narguments were made: (a) the high rate of transits in the liquid state\nshows the need to abandon the normal mode description of motion, and\ninstead picture the motion of a single atom among a set of fluctuating\nwells, and (b) the leading approximation to a fluctuating well is its\nsmooth time average well. Accordingly, the independent atom\noscillates with frequency $\\omega$ in a smooth isotropic well, and it\ntransits with probability $\\mu$ to an adjacent identical well at each\nturning point \\cite{oldvacf}. What we have now learned by considering\nlow temperatures, and especially the nondiffusing states of\n\\cite{chis}, is that the MD system exhibits a velocity decorrelation\nprocess which results from the presence of many frequencies in the\nsingle-particle motion, making assumption (b) less reasonable. In the\ncurrent model these many frequencies are retained, and they are\nintended to represent the strong fluctuations in each single-particle\nwell; in this way the current model makes an important improvement\nover the independent atom model, where the well fluctuations were\naveraged out. Beyond this difference, the two models contain similar\nbut not identical treatments of transits, whose rate increases with\nincreasing temperature, and which produce self-diffusion. The less\ndetailed but simpler independent atom model has proven useful in a\ndescription of the glass transition \\cite{glass}, and the\ncorresponding transit parameter has been used to relate shear viscosity\nand self-diffusion in liquid metals \\cite{MT}.\n\nThe next logical step in this work, given the results so far, is to\nextend the mean atom trajectory model and apply it to calculations of\nmore complicated correlation functions of the liquid's motion.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{chis} E.\\ D.\\ Chisolm, B.\\ E.\\ Clements, and D.\\ C.\\ Wallace, \n submitted to Phys.\\ Rev.\\ E; cond-mat/0002057.\n\n\\bibitem{enkin} D.\\ C.\\ Wallace and B.\\ E.\\ Clements, Phys.\\ Rev.\\ E {\\bf 59},\n 2942 (1999).\n\n\\bibitem{radang} B.\\ E.\\ Clements and D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf \n 59}, 2955 (1999).\n\n\\bibitem{oldvacf} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 58}, 538 (1998).\n\n\\bibitem{glass} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 60}, 7049 (1999).\n\n\\bibitem{liqdyn} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 56}, 4179 (1997). \n\n\\bibitem{LRL} S.\\ J.\\ Larsson, C.\\ Roxbergh, and A.\\ Lodding, Phys.\\ Chem.\\ \n Liq.\\ {\\bf 3}, 137 (1972).\n\n\\bibitem{VB} R.\\ Vallauri and F.\\ J.\\ Bermejo, Phys.\\ Rev.\\ E {\\bf 51}, 2654\n (1995).\n\n\\bibitem{Stratt} M.\\ Buchner, B.\\ M.\\ Ladanyi, and R.\\ M.\\ Stratt, J.\\ Chem.\\\n Phys.\\ {\\bf 97}, 8522 (1992).\n\n\\bibitem{Zwan} R.\\ Zwanzig, J.\\ Chem.\\ Phys.\\ {\\bf 79}, 4507 (1983). \n\n\\bibitem{MKS} B.\\ Madan, T.\\ Keyes, and G.\\ Seeley, J.\\ Chem.\\ Phys.\\ {\\bf 94}\n 6762 (1991).\n\n\\bibitem{CV} J.\\ Cao and G.\\ A.\\ Voth, J.\\ Chem.\\ Phys.\\ {\\bf 103}, 4211\n (1995). \n\n\\bibitem{MT} N.\\ H.\\ March and M.\\ P.\\ Tosi, Phys.\\ Rev.\\ E {\\bf 60}, 2402 \n (1999).\n\n\\end{thebibliography}\n\n\\end{document}\n"
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[
{
"name": "cond-mat0002058.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{chis} E.\\ D.\\ Chisolm, B.\\ E.\\ Clements, and D.\\ C.\\ Wallace, \n submitted to Phys.\\ Rev.\\ E; cond-mat/0002057.\n\n\\bibitem{enkin} D.\\ C.\\ Wallace and B.\\ E.\\ Clements, Phys.\\ Rev.\\ E {\\bf 59},\n 2942 (1999).\n\n\\bibitem{radang} B.\\ E.\\ Clements and D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf \n 59}, 2955 (1999).\n\n\\bibitem{oldvacf} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 58}, 538 (1998).\n\n\\bibitem{glass} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 60}, 7049 (1999).\n\n\\bibitem{liqdyn} D.\\ C.\\ Wallace, Phys.\\ Rev.\\ E {\\bf 56}, 4179 (1997). \n\n\\bibitem{LRL} S.\\ J.\\ Larsson, C.\\ Roxbergh, and A.\\ Lodding, Phys.\\ Chem.\\ \n Liq.\\ {\\bf 3}, 137 (1972).\n\n\\bibitem{VB} R.\\ Vallauri and F.\\ J.\\ Bermejo, Phys.\\ Rev.\\ E {\\bf 51}, 2654\n (1995).\n\n\\bibitem{Stratt} M.\\ Buchner, B.\\ M.\\ Ladanyi, and R.\\ M.\\ Stratt, J.\\ Chem.\\\n Phys.\\ {\\bf 97}, 8522 (1992).\n\n\\bibitem{Zwan} R.\\ Zwanzig, J.\\ Chem.\\ Phys.\\ {\\bf 79}, 4507 (1983). \n\n\\bibitem{MKS} B.\\ Madan, T.\\ Keyes, and G.\\ Seeley, J.\\ Chem.\\ Phys.\\ {\\bf 94}\n 6762 (1991).\n\n\\bibitem{CV} J.\\ Cao and G.\\ A.\\ Voth, J.\\ Chem.\\ Phys.\\ {\\bf 103}, 4211\n (1995). \n\n\\bibitem{MT} N.\\ H.\\ March and M.\\ P.\\ Tosi, Phys.\\ Rev.\\ E {\\bf 60}, 2402 \n (1999).\n\n\\end{thebibliography}"
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cond-mat0002059
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Evaluation of the quantitative prediction of a trend reversal on the Japanese stock market in 1999.
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[
{
"author": "Anders Johansen$^1$ and Didier Sornette$^{1,2,3}$"
},
{
"author": "Los Angeles"
},
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"author": "California 90095"
},
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"author": "$^2$ Department of Earth and Space Science"
},
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"author": "$^3$ Laboratoire de Physique de la Mati\\`{e}re Condens\\'{e}e"
},
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"author": "CNRS UMR6622 and Universit\\'{e} de Nice-Sophia Antipolis"
},
{
"author": "B.P. 71"
},
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"author": "Parc Valrose"
},
{
"author": "06108 Nice Cedex 2"
},
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"author": "France"
}
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\noindent In January 1999, the authors published a quantitative prediction that the Nikkei index should recover from its 14 year low in January 1999 and reach $\approx 20500$ a year later. The purpose of the present paper is to evaluate the performance of this specific prediction as well as the underlying model: the forecast, performed at a time when the Nikkei was at its lowest (as we can now judge in hindsight), has correctly captured the change of trend as well as the quantitative evolution of the Nikkei index since its inception. As the change of trend from sluggish to recovery was estimated quite unlikely by many observers at that time, a Bayesian analysis shows that a skeptical (resp. neutral) Bayesian sees her prior belief in our model amplified into a posterior belief $19$ times larger (resp. reach the $95\%$ level). \vspace{1cm} \noindent keywords: Stock market; Log-periodic oscillations; scale invariance; prediction; Gold; Nikkei; Herding behaviour.
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"name": "submprednik.tex",
"string": "\\documentstyle[12pt,epsfig]{article}\n\\setlength{\\topmargin}{-5mm}\n\\setlength{\\evensidemargin}{0cm}\n\\setlength{\\oddsidemargin}{-0.5cm}\n\\setlength{\\textheight}{22cm}\n\\setlength{\\textwidth}{17cm}\n\n\\begin{document}\n\n\\newcommand{\\rp}{\\right)}\n\\newcommand{\\lp}{\\left(}\n\\newcommand \\be {\\begin{equation}}\n\\newcommand \\bea {\\begin{eqnarray}}\n\\newcommand \\ee {\\end{equation}}\n\\newcommand \\eea {\\end{eqnarray}}\n\n\n\\title{Evaluation of the quantitative prediction of a trend\nreversal on the Japanese stock market in 1999.}\n\n\n\\author{Anders Johansen$^1$ and Didier Sornette$^{1,2,3}$ \\\\\n$^1$ Institute of Geophysics and\nPlanetary Physics\\\\ University of California, Los Angeles, California 90095\\\\\n$^2$ Department of Earth and Space Science\\\\\nUniversity of California, Los Angeles, California 90095\\\\\n$^3$ Laboratoire de Physique de la Mati\\`{e}re Condens\\'{e}e\\\\ CNRS UMR6622 and\nUniversit\\'{e} de Nice-Sophia Antipolis\\\\ B.P. 71, Parc\nValrose, 06108 Nice Cedex 2, France}\n\n%\\date{today}\n\n\\maketitle\n\n\\begin{abstract}\n\n\\noindent In January 1999, the authors published a quantitative prediction that\nthe Nikkei index should recover from its 14 year low in January 1999 and reach\n$\\approx 20500$ a year later. The purpose of the present paper is to evaluate\nthe performance of this specific prediction as well as the underlying model:\nthe forecast, performed at a time when the Nikkei was at its lowest (as we\ncan now judge in hindsight), has correctly captured the change of trend as\nwell as the quantitative evolution of the Nikkei index since its inception.\nAs the change of trend from sluggish to recovery\nwas estimated quite unlikely by many observers at that time, a Bayesian\nanalysis shows that a skeptical (resp. neutral) Bayesian\nsees her prior belief in our model amplified into a posterior belief $19$ times\nlarger (resp. reach the $95\\%$ level).\n\\vspace{1cm}\n\n\\noindent keywords: Stock market; Log-periodic oscillations; scale invariance;\nprediction; Gold; Nikkei; Herding behaviour.\n\\end{abstract}\n\\newpage\n\n\\pagenumbering{arabic}\n\nFollowing the general guidelines proposed in \\cite{manisfesto}, the authors\nmade in January 1999 public through the Los Alamos preprint server\n\\cite{lanlsub} a quantitative prediction stating that the Nikkei index should\nrecover from its 14 year low (actually 13232.74 on 5 Jan 1999) and reach\n$\\approx 20500$ a year later corresponding to an increase in the index of\n$\\approx 50\\%$. Furthermore, this prediction was mentioned in a\nwide-circulation journal which appeared in May 1999 \\cite{DSStaupredNi}.\n\nSpecifically, based on a third-order ``Landau'' expansion\n\\be\n\\frac{d F\\lp \\tau \\rp}{d\\log \\tau}=\\alpha F\\lp \\tau \\rp +\\beta\n|F\\lp \\tau \\rp|^2 F\\lp \\tau \\rp + \\gamma |F\\lp \\tau \\rp|^4 F\\lp \\tau\n\\rp\\ldots\n\\label{3expan}\n\\ee\nin terms of $\\tau \\equiv t-tc$, where $t_c =$ 31 Dec. 1989 is the time of\nthe all-time\nhigh of the Nikkei index,\nthe authors arrived at the equation\n$$\n\\log\\lp p(t)\\rp \\approx A' + \\frac{\\tau^\\alpha}{\\sqrt{1+\\left(\\frac{\\tau}\n{\\Delta_t}\\right)^{2\\alpha} + \\left(\\frac{\\tau}{\\Delta_t'}\\right)^{4\\alpha}}}\n$$\n\\be \\label{3feq}\n\\left\\{B'+ C'\\cos\\left[\\omega\\log \\tau +\n\\frac{\\Delta_\\omega}{2\\alpha}\\log\\left(1+\\left(\\frac{\\tau}\n{\\Delta_t}\\right)^{2\\alpha}\\right)+\n\\frac{\\Delta_\\omega'}{4\\alpha}\\log\\left(1+\\left(\\frac{\\tau}\n{\\Delta_t'}\\right)^{4\\alpha}\\right) + \\phi\\right]\\right\\}~,\n\\ee\ndescribing the time-evolution of the Nikkei Index $p(t)$. Equation (\\ref{3feq})\nwas then fitted to the Nikkei index in the time interval from the beginning of\n1990 to the end of 1998, {\\it i.e.}, a total of 9 years. Extending the curve\nbeyond 1998 thus provided us with a quantitative prediction for the future\nevolution of the Index. In figure \\ref{predfig}, we compare the actual and\npredicted evolution of the Nikkei over 1999. We see that not only did the\nNikkei experience a trend reversal as predicted, but it has also followed\nthe quantitative prediction with rather impressive precision, see figure\n\\ref{relerrorfig}. It is important to note that the error between the curve\nand the data has not grown after the last point used in the fit. This tells\nus that the prediction has performed well so far. Furthermore, since the\nrelative error between the fit and the data is within $\\pm 2\\%$ over a\ntime period of 10 years, not only has the prediction performed well, but also\nthe underlying model. This analysis represents the correct quantitative\nevaluation of the performance of the model as well as its predictive power\non the Nikkei Index over a quite impressive time-span of 10 years.\n\nWe wish to stress that the fulfilling of our prediction is even more\nremarkable than the comparison between the curve and the data indicates.\nThis, since it included {\\it a change of trend}: at the time when the\nprediction was issued, the market was declining and showed no tendency to\nincrease. Many economists were at that time very pessimistic and could not\nenvision when Japan and its market would rebounce. For instance, the well-known\neconomist P. Krugman \\cite{Krugman} wrote July 14, 1998 in the Shizuoka Shimbun\nat the time of the banking scandal ``the central problem with Japan right now\nis that there just is not enough demand to go around - that consumers and\ncorporations are saving too much and borrowing too little... So seizing these\nbanks and putting them under more responsible management is, if anything,\ngoing to further reduce spending; it certainly will not in and of itself\nstimulate the economy... But at best this will get the economy back to where\nit was a year or two ago - that is, depressed, but not actually plunging.''\nThen, in the Financial Times, January, 20th, 1999, P. Krugman wrote in an\narticle entitled ``Japan heads for the edge'' the following: ``...the story\nis starting to look like a tragedy. A great economy, which does not deserve\nor need to be in a slump at all, is heading for the edge of the cliff -- and\nits drivers refuse to turn the wheel.'' In a poll of thirty economists\nperformed\nby Reuters (the major news and finance data provider in the world)\nin October 1998 \\cite{poll}, only two economists predicted growth for\nthe fiscal year of 1998-99. For the year 1999-2000 the prediction was a\nmeager 0.1\\% growth. This majority of economists said that\n``a vicious cycle in the economy was unlikely to disappear any\ntime soon as they expected little help from the government's economic stimulus\nmeasures... Economists blamed moribund domestic demand, falling prices,\nweak capital spending\nand problems in the bad-loan laden banking sector for dragging down the\neconomy.''\n\nNevertheless, we predicted a $\\approx 50\\%$ increase of the market in the\nnext 12 months assuming that the Nikkei would stay within the error-bars of\nthe fit. At the time of writing (3rd February 2000), the market is up by\n$\\approx 49.5\\%$ and the error between the prediction and the curve has not\nincreased, see figure \\ref{relerrorfig}. Predictions of trend reversals is\nnoteworthy difficult and unreliable, especially in the linear framework of\nauto-regressive models used in standard economic analyses. The present\nnonlinear framework is well-adapted to the forecasting of change of trends,\nwhich constitutes by far the most difficult challenge posed to forecasters.\nHere, we refer to our prediction of a trend reversal\nwithin the strict confine of equation (\\ref{3feq}): trends are limited\nperiods of times when the oscillatory behavior shown in figure \\ref{predfig}\nis monotonous. A change of trend thus corresponds to crossing a local maximum\nor minimum of the oscillations.\n\nWe report one case. In the standard ``frequentist'' approach to probability\n\\cite{jeffreys}\nand to the establishment of statistical confidence, this bears essentially\nno weight and should be discarded as story telling. We are convinced that\nthe ``frequentist'' approach is unsuitable to assess the quality of such a\nunique\nexperiment as presented here of the prediction of a global financial indicator\nand that the correct framework is Bayesian. Within the Bayesian framework,\nthe probability\nthat the hypothesis is correct given the data can be estimated, whereas this\nis excluded by construction in the standard ``frequentist'' formulation,\nin which\none can only calculate\nthe probability that the null-hypothesis is wrong, not that the alternative\nhypothesis is correct (see also \\cite{Bayes} for recent introductory\ndiscussions).\n\nBayes' theorem states that\n\\begin{equation}\nP(H_i|D) = \\frac{P(D|H_i) \\times P(H_i)}{\\sum_j P(D|H_j)P(D_j)}\\,.\n\\label{eq:bayes2}\n\\end{equation}\nwhere the sum in the denominator runs over all the different\nconflicting hypothesis. In words,\nequation (\\ref{eq:bayes2}) estimates that the probability, that hypothesis\n$H_i$ is\ncorrect given the data $D$, is proportional to the probability $P(D|H_i)$ of\nthe data given the hypothesis $H_i$ multiplied with the prior belief $P(H_i)$\nin the hypothesis $H_i$ divided with the probability of the data. In the\npresent context, we use only the two hypotheses $H_1$ and $H_2$ that our\nprediction of a trend reversal is correct or that it is wrong. For the data,\nwe take the change of trend from bearish to bullish. We now want to estimate\nwhether the fulfillment of our prediction was a ``lucky one''. We quantify\nthe general atmosphere of disbelief that Japan would recover by the value\n$P(D|H_2) = 5\\%$ for the probability that the Nikkei will change trend while\ndisbelieving our model. We assign the classical confidence level of $P(D|H_1)\n= 95\\%$ for the probability that the Nikkei will change trend while believing\nour model.\n\nLet us consider a skeptical Bayesian with prior probability (or belief)\n$P(H_1)\n= 10^{-n}$, $n\\ge 1$ that our model is correct. From (\\ref{eq:bayes2}), we get\n\\be\nP(H_1|D) = {0.95 \\times 10^{-n} \\over 0.95 \\cdot 10^{-n} + 0.05 \\times\n(1-10^{-n})}~ .\n\\ee\nFor $n=1$, we see that her posterior belief in our model has been\namplified compared to her prior belief by a factor $\\approx 7$ corresponding to\n$P(H_1|D) \\approx 70\\%$. For $n=2$, the amplification factor is $\\approx 16$\nand hence $P(H_1|D) \\approx 16\\%$. For large $n$ (very skeptical Bayesian),\nwe see that her posterior belief in our model has been amplified compared\nto her\nprior belief by a factor $0.95/0.05 = 19$.\nAlternatively, consider a neutral Bayesian\nwith prior belief $P(H_1) = 1/2$, {\\it i.e.}, a priori she considers equally\nlikely that our model is correct or wrong. In this case, her prior belief\nis changed\ninto the posterior belief equal to\n\\be\nP(H_1|D) = {0.95 \\cdot {1 \\over 2} \\over 0.95 \\cdot {1 \\over 2} +\n0.05 \\cdot {1 \\over 2}} = 95\\%~.\n\\ee\nThis means that this single case is enough to convince the neutral Bayesian.\n\nWe stress that this specific application of Bayes' theorem only deals with a\nsmall part of the model, {\\it i.e.}, the trend reversal. It does not establish\nthe significance of the quantitative description of {\\em 10 years} of data\n(of which the last one was unknown at the time of the prediction) by\nthe proposed model within a relative error of $\\approx \\pm 2\\%$.\n\nA question that remains is how far into the future will the Japanese stock\nmarket continue to follow equation (\\ref{3feq})? Obviously, the Nikkei Index\nmust ``break away'' at some point in the future even if there are no changes\nin the overall behaviour and the underlying model thus remains valid. The\nreason is that the prediction was made using a third order expansion. This\nmeans that, as the parameter $\\tau = t-t_c$ in equation (\\ref{3feq})\ncontinues to\nincrease, this approximation becomes worse and worse and a fourth order term\nshould be included. Presently, we are not ready to present the derivation\nof such an\nequation. Furthermore, we expect the numerical difficulties involved in\nfitting an even more complex equation than equation (\\ref{3feq}) to be\nconsiderable.\n\nLast, we would like to bring to the attention of the reader that not only can\nbearish markets occasionally be described by the framework underlying equation\n(\\ref{3feq}). In fact, bullish markets exhibits such changes of regimes\neven more frequently, see \\cite{JSL}.\n\n\\vspace{1cm}\n\n\n{\\bf Acknowledgement} The authors wish to thank D. Stauffer for his\nencouragement both with respect to the original work of \\cite{lanlsub} as\nwell as the present re-evaluation.\n\n\\begin{thebibliography}{}\n\n\\bibitem{manisfesto} A. Johansen and D. Sornette,\n{\\it Modeling the Stock Market prior to large crashes}, Eur. Phys. J. B,\n{\\bf 9}, 167-174. Available on http://www.nbi.dk/\\~~johansen/pub.html\n\n\\bibitem{lanlsub} The prediction was made public on the 25 Jan. 1999 by\nposting a preprint on the Los Alamos server, see\nhttp://xxx.lanl.gov/abs/cond-mat/9901268 . The preprint was later published\nas A. Johansen and D. Sornette,\n{\\it Financial ``anti-bubbles'': log-periodicity in Gold and Nikkei collapses},\nInt. J. Mod. Phys. C. {\\bf 10}, 563-575 (1999).\n\n\\bibitem{DSStaupredNi} D. Stauffer, Monte-Carlo-Simulation mikroskopischer\nB\\\"{o}rsenmodelle, Physikalische Bl\\\"{a}tter 55 (1999) 49.\n\n\\bibitem{Krugman} The Official Paul Krugman Web Page:\nhttp://web.mit.edu/krugman/www/\n\n\\bibitem{poll} Reported in Indian Express on the 15 Oct., see\\\\\nhttp://www.indian-express.com/fe/daily/19981016/28955054.html\n\n\\bibitem{jeffreys} H. Jeffreys,\n{\\it Theory of Probability}, ~3rd ed. (Oxford University Press, 1961)).\n\n\\bibitem{Bayes} D. Malakoff,\n{\\it Bayes Offers a 'New' Way to Make Sense of Numbers},\nScience {\\bf 286}, 1460-1464 (1999);\n{\\it A Brief Guide to Bayes Theorem}, ibid {\\bf 286}, 1461 (1999);\nG. D'Agostini, G.,\n{\\it Teaching statistics in the physics curriculum: Unifying and clarifying\nrole of subjective probability}, Am. J. Phys. {\\bf 67}, 1260-1268 (1999).\n\n\\bibitem{JSL} A. Johansen, D. Sornette and O. Ledoit,\n{\\it Predicting Financial Crashes Using Discrete Scale Invariance},\nJ. of Risk, Vol 1 No. 4, pp.5-32 (1999).\nAvailable on http://www.nbi.dk/\\~~johansen/pub.html;\nA. Johansen and D. Sornette,\n{\\it Log-periodic power law bubbles in Latin-American and Asian markets\nand correlated anti-bubbles in Western stock markets: An empirical study}.\nSubm. to J. Empirical Finance. Preprint available on\nhttp://www.nbi.dk/\\~~johansen/pub.html\n\n\n\n\\end{thebibliography}\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=fitnikpredic.eps,width=12cm}\n\\caption{\\protect\\label{predfig}. Logarithm of the Nikkei Index compared to\nequation \\protect\\ref{3feq}. The dots are the data used in the fit of equation\n(\\ref{3feq}) being the ticked line and covers the 9 year period from 31 Dec.\n1989 to 31 Dec. 1998. The solid line is the actual behaviour of the Nikkei\n{\\em after} the last point used in the fit and covers the period 1 Jan. 1999\nto 28 Jan. 2000. The prediction was made public on the 25 Jan. 1999\n\\protect\\cite{lanlsub}. See \\protect\\cite{lanlsub} for details of the fit.}\n\\vspace{5mm}\n\\epsfig{file=errorlognik.eps,width=12cm}\n\\caption{\\protect\\label{relerrorfig} The relative error between the fit with\nequation (\\protect\\ref{3feq}) and the data. The ticked line if the relative\nerror between the fit and the data used in the fit. The solid line is the error\nbetween the prediction and the actual data.}\n\\end{center}\n\\end{figure}\n\n\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"
}
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[
{
"name": "cond-mat0002059.extracted_bib",
"string": "\\begin{thebibliography}{}\n\n\\bibitem{manisfesto} A. Johansen and D. Sornette,\n{\\it Modeling the Stock Market prior to large crashes}, Eur. Phys. J. B,\n{\\bf 9}, 167-174. Available on http://www.nbi.dk/\\~~johansen/pub.html\n\n\\bibitem{lanlsub} The prediction was made public on the 25 Jan. 1999 by\nposting a preprint on the Los Alamos server, see\nhttp://xxx.lanl.gov/abs/cond-mat/9901268 . The preprint was later published\nas A. Johansen and D. Sornette,\n{\\it Financial ``anti-bubbles'': log-periodicity in Gold and Nikkei collapses},\nInt. J. Mod. Phys. C. {\\bf 10}, 563-575 (1999).\n\n\\bibitem{DSStaupredNi} D. Stauffer, Monte-Carlo-Simulation mikroskopischer\nB\\\"{o}rsenmodelle, Physikalische Bl\\\"{a}tter 55 (1999) 49.\n\n\\bibitem{Krugman} The Official Paul Krugman Web Page:\nhttp://web.mit.edu/krugman/www/\n\n\\bibitem{poll} Reported in Indian Express on the 15 Oct., see\\\\\nhttp://www.indian-express.com/fe/daily/19981016/28955054.html\n\n\\bibitem{jeffreys} H. Jeffreys,\n{\\it Theory of Probability}, ~3rd ed. (Oxford University Press, 1961)).\n\n\\bibitem{Bayes} D. Malakoff,\n{\\it Bayes Offers a 'New' Way to Make Sense of Numbers},\nScience {\\bf 286}, 1460-1464 (1999);\n{\\it A Brief Guide to Bayes Theorem}, ibid {\\bf 286}, 1461 (1999);\nG. D'Agostini, G.,\n{\\it Teaching statistics in the physics curriculum: Unifying and clarifying\nrole of subjective probability}, Am. J. Phys. {\\bf 67}, 1260-1268 (1999).\n\n\\bibitem{JSL} A. Johansen, D. Sornette and O. Ledoit,\n{\\it Predicting Financial Crashes Using Discrete Scale Invariance},\nJ. of Risk, Vol 1 No. 4, pp.5-32 (1999).\nAvailable on http://www.nbi.dk/\\~~johansen/pub.html;\nA. Johansen and D. Sornette,\n{\\it Log-periodic power law bubbles in Latin-American and Asian markets\nand correlated anti-bubbles in Western stock markets: An empirical study}.\nSubm. to J. Empirical Finance. Preprint available on\nhttp://www.nbi.dk/\\~~johansen/pub.html\n\n\n\n\\end{thebibliography}"
}
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cond-mat0002060
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{
"name": "cond-mat0002060.tex",
"string": "\\documentstyle[12pt]{article}\n\\headheight 0cm\n\\headsep 0cm\n\\newlength{\\mytopmargin}\n\\newlength{\\myleftmargin}\n\\setlength{\\mytopmargin}{2.8cm}\n\\setlength{\\myleftmargin}{2.5cm}\n\\setlength{\\topmargin}{-1in}\n\\setlength{\\oddsidemargin}{-1in}\n\\addtolength{\\topmargin}{\\mytopmargin}\n\\addtolength{\\oddsidemargin}{\\myleftmargin}\n\\textwidth 16cm\n\\textheight 24cm\n\\setlength{\\parskip}{1ex}\n\n\\def\\zz{\\rlx\\hbox{\\small \\sf Z\\kern-.4em Z}}\n\n\n\\newcommand{\\ml}{\\langle}\n\\newcommand{\\mg}{\\rangle}\n\n\n\\newtheorem{lemma}{Lemma}[section]\n\n\\newtheorem{prop}[lemma]{Proposition}\n\n\n\\setlength{\\parindent}{1.5em}\n\\renewcommand{\\baselinestretch}{1.2}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\begin{document}\n\n\\vspace{1cm}\n\\noindent\n\\begin{center}{ \\large \\bf\nAnalytic properties of the structure function for the \\\\\none-dimensional one-component log-gas\n} \n\\end{center}\n\\vspace{5mm}\n\n\\noindent\n\\begin{center} \n P.J.~Forrester$^*$, B.~Jancovici$^\\#$ and D.S.~McAnally$^*$\\\\\n\n\\it $^*$Department of Mathematics\nand Statistics, University of Melbourne,\\\\ Parkville, Victoria\n3052, Australia\\\\[.1cm]\n$^\\#$Laboratoire de Physique Th\\'eorique, Universit\\'e Paris-Sud,\\\\\n91405 Orsay Cedex, France\\footnote{Laboratoire associ\\'e au Centre\nNational de la Recherche Scientifique - URA D0063}\n\\end{center}\n\n\\begin{center}\nDedicated to R.J.~Baxter on the occasion of his 60$^{\\rm th}$ birthday.\n\\end{center}\n\\vspace{.5cm}\n\n\n\\small\n\\begin{quote}\nThe structure function $S(k;\\beta)$ for the \none-dimensional one-component log-gas is the Fourier\ntransform of the charge-charge, or equivalently the density-density,\ncorrelation function. We show that for $|k| < {\\rm min} \\,(2\\pi \\rho,\n2 \\pi \\rho \\beta)$, $S(k;\\beta)$ is simply related to an analytic\nfunction $f(k;\\beta)$ and this function satisfies the functional equation\n$f(k;\\beta) = f(-2k/\\beta;4/\\beta)$. It is conjectured that the coefficient\nof $k^j$ in the power series expansion of $f(k;\\beta)$ about $k=0$ is of the\nform of a polynomial in $\\beta/2$ of degree $j$ divided by $(\\beta/2)^j$.\nThe bulk of the paper is concerned with calculating these polynomials \nexplicitly up to and including those of degree 9. It is remarked that the\nsmall $k$ expansion of $S(k;\\beta)$ for the two-dimensional one-component\nplasma shares some properties in common with those of the \none-dimensional one-component log-gas, but these\nbreak down at order $k^8$.\n\\end{quote}\n\n\\noindent\n{\\bf Key words} \\quad logarithmic potential; two-dimensional plasma;\nfractional statistics; random matrices; exact solution.\n\\vspace{.5cm}\n\n\\section{Introduction}\n\\setcounter{equation}{0}\nThe one-component log-gas, consisting of $N$ unit charges on a circle of\ncircumference length $L$ interacting via the two-dimensional Coulomb\npotential $\\Phi(\\vec{r},\\vec{r}\\,') = - \\log |\\vec{r} - \\vec{r}\\,'|$,\nis specified by the Boltzmann\nfactor\n\\begin{equation}\\label{1}\nA_{N,\\beta} \\prod_{1 \\le j < k \\le N}\n| e^{2 \\pi i x_k / L} - e^{2 \\pi i x_j / L}|^\\beta, \\qquad 0 \\le x_j \\le L.\n\\end{equation}\nThe constant $A_{N,\\beta}$, which plays no role in the calculation of\ndistribution functions, results from scaling the radius of the circle out\nof the logarithmic potential, and also includes the particle-background\nand background-background interactions (a uniform neutralizing background\nis imposed for thermodynamic stability).\nThe thermodynamic limit\n$N,L \\to \\infty$,$N/L = \\rho$ (fixed) is taken, which gives an infinite\nsystem on a straight line with particle density $\\rho$.\nThis system was first studied because of its relation to\nthe theory of random matrices \\cite{Me91}. The thermodynamic functions\nwere obtained. The pressure $P$ has the simple form\n\\begin{equation}\\label{pressure}\n\\beta P=[1-(\\beta /2)]\\rho\n\\end{equation}\nat any inverse temperature $\\beta$. However, exact (simple) forms for the\ncorrelation functions were obtained by the pioneers only for the special\ntemperatures corresponding to $\\beta=1,2,4$ (See Section 5). More\nrecently, exact expressions for the two-body density were derived for\narbitrary even integer $\\beta$ \\cite{Fo93aa} and then for arbitrary\nrational $\\beta$ \\cite{Ha95}. Unfortunately, these latter exact\nexpressions are complicated multivariable integral representations which\ncannot be easily used as such for actual computations. The purpose of\nthe present paper is to obtain explicit small $k$ expansions for the\nstructure function (the Fourier transform of the two-body density).\n\nThe log-gas is an example of a system interacting via the $d$-dimensional\nCoulomb system (here $d=2$) but confined to a domain of dimension $d-1$.\nIt therefore exhibits universal features --- that is features independent of\nmicroscopic details such as any short range potential between charges \nor the number of\ncharge species --- characteristic of Coulomb systems in this setting\n\\cite{Ja82}. One\nuniversal feature is the existence of an algebraic tail in the leading\nnon-oscillatory term of the large-distance asymptotic expansion of the\ncharge-charge correlation function. For general charged systems in their\nconductive phase, interacting via the two-dimensional Coulomb potential\nin a one-dimensional domain, this is predicted to have the form\n\\cite{FJS83}\n\\begin{equation}\\label{2}\n- {1 \\over \\beta (\\pi r)^2},\n\\end{equation}\nwhere $r$ is the distance. For the\none-component log-gas, (\\ref{2}) can be verified for all $\\beta$\nrational \\cite{FZ96}.\n\nThe verification is possible because the charge-charge correlation function\n(which for a one-component system is the same as the density-density\ncorrelation)\nis known explicitly for $\\beta$ rational \\cite{Ha95} (see (\\ref{Ha})\nbelow). In this work we further analyze the properties of the\nstructure factor $S(k;\\beta)$ (Fourier transform of the charge-charge\ncorrelation) for the one-component log-gas. In particular we are interested\nin the $\\beta$ dependence of the coefficients in the small $k$\nexpansion of $S(k;\\beta)$.\n\nThe large distance behaviour (\\ref{2}) is equivalent to the small $k$\nbehaviour\n\\begin{equation}\\label{3}\nS(k;\\beta) \\sim {|k| \\over \\pi \\beta}.\n\\end{equation}\nFurthermore, by making use of the equivalence of the charge-charge and\ndensity-density correlation in the one-component log-gas,\ntogether with the exact equation of state\nthe second order term in (\\ref{3}) has been predicted for general $\\beta$\n\\cite{FJ97}, giving\n\\begin{equation}\\label{4}\nS(k;\\beta) \\sim {|k| \\over \\pi \\beta} \n+ {(\\beta / 2 - 1) k^2 \\over (\\pi \\beta)^2 \\rho } +\nO(|k|^3).\n\\end{equation}\n\nLet \n\\begin{equation}\\label{4+}\nf(k;\\beta) := {\\pi \\beta \\over |k|} S(k;\\beta), \\quad 0 < k < {\\rm min}\\,\n(2 \\pi\\rho, \\pi \\beta\\rho)\n\\end{equation}\nand define $f$ for $k < 0 $ by analytic continuation\n(we will see below that $f(k;\\beta)$ is analytic for\n$0 \\le |k| < {\\rm min}\\, (2 \\pi \\rho, \\pi \\beta \\rho)$).\nIn Section 2 we use the exact result (\\ref{Ha}) below to derive the\nfunctional equation\n\\begin{equation}\\label{5}\nf(k;\\beta) = f\\Big (-{2 k \\over \\beta}; {4 \\over \\beta} \\Big )\n\\end{equation}\nThe\nsimplest structure consistent with (\\ref{5}) is\n\\begin{equation}\\label{6}\n{\\pi \\beta \\over |k|} S(k;\\beta) =\n1 + \\sum_{j=1}^\\infty p_j(\\beta / 2) \\Big ( {|k| \\over \\pi \\beta \\rho}\n\\Big )^j, \\quad |k| < {\\rm min}\n(2 \\pi \\rho, \\pi \\beta \\rho)\n\\end{equation}\nwhere $p_j(x)$ is a polynomial of degree $j$ which satisfies the functional\nrelation \n\\begin{equation}\\label{6.1}\np_j(1/x) = (-1)^j x^{-j} p_j(x).\n\\end{equation}\nEquivalently, (\\ref{6.1}) can be stated as requiring\n\\begin{eqnarray}\\label{7.1}\np_j(x) & = & \\sum_{l=0}^j a_{j,l} x^l, \\qquad a_{j,l} =\na_{j,j-l} \\quad (j \\: {\\rm even}) \\\\\n\\label{7.2}\np_j(x) & = & \n(x-1)\\sum_{l=0}^{j-1} \\tilde{a}_{j,l} x^l, \\qquad \\tilde{a}_{j,l} =\n\\tilde{a}_{j,j-1-l} \\quad (j \\: {\\rm odd}).\n\\end{eqnarray}\n\nInspection of (\\ref{4}) shows that the conjectured structure (\\ref{6})\nis correct at order $|k|$ and furthermore gives\n\\begin{equation}\\label{6.2}\np_1(x) = (x-1),\n\\end{equation}\nand thus $\\tilde{a}_{1,0} = 1$ in (\\ref{7.2}).\nIn Section 3 we use (\\ref{Ha}) to verify that the structure (\\ref{6})\nis correct at order $k^2$ and we compute $p_2(x)$ explicitly. In Section\n4 we use an exact an exact evaluation of the two-particle\ndistribution function for $\\beta$ even \\cite{Fo93aa} to rederive the\nresult of Section 3, and we also use this formula to\nverify the structure (\\ref{6}) at\norder $k^4$ and to compute $p_4(x)$ explicitly.\n\nAssuming the validity of (\\ref{6}) we see that $p_j(x)$ can be computed\nfrom knowledge of the coefficient of $|k|^j$ in $S(k;\\beta)$, or the\ncoefficient of $|k|^j$ in $\\partial^p S(k;\\beta) / \\partial \\beta^p$\n($p \\le j$), for an appropriate number of distinct values of $\\beta$.\nBecause the functional relation (\\ref{5}) has via (\\ref{7.1}) and (\\ref{7.2})\nbeen made a feature of (\\ref{6}) the values of $\\beta$ cannot be related\nby $\\beta \\mapsto 4/\\beta$. In Section 5 the known exact evaluation of\n$S(k;\\beta)$ to leading order in $\\beta$ is reviewed, as are the \nexact evaluations of\n$S(k;2)$ and $S(k;4)$. Also noted are the exact evaluations of $S(k,1)$\nand $S(k;\\beta)$ to leading order in $1/\\beta$, which according to \n(\\ref{5}) are related to $S(k;4)$ and $S(k;\\beta)$ to leading order\nin $\\beta$ respectively.\nAll of these exact evaluations are in terms of\nelementary functions, and so can be expanded to all orders in $k$. We\nthen present the exact evaluation\nof $\\partial S(k;\\beta) / \\partial \\beta$ to leading\norder in $\\beta$, as well as the exact evaluation of\n$\\partial S(k;\\beta) / \\partial \\beta$ evaluated at $\\beta = 2$ and $\\beta =4$.\nThe details of the latter two calculations are given in separate appendices.\nAgain the final expressions can be expanded to high order in $|k|$.\nUsing this data all polynomials in the expansion (\\ref{6}) \nup to and including the term with $j=9$ can be computed. This expansion\nis written out explicitly in the final section and \nsome special features of the polynomials therein, relating to the sign of the\ncoefficients and the zeros, are noted. \nA physical interpretation of the functional equation, \nbased on an analogy with a quantum many body system, which identifies an\nequivalence between quasi-hole and quasi-particle states contributing to\n$S(k;\\beta)$ for $|k|$ small enough is given.\nWe end with some remarks on the\npossible occurence of a functional equation analogous to (\\ref{5})\nin the two-dimensional one-component plasma.\n\n\\section{The functional equation}\n\\setcounter{equation}{0}\nThe Boltzmann factor (\\ref{1}) also has the physical interpretation as the\nabsolute value squared of the exact ground state wave function, $|0\\mg$\nsay, for the Calogero-Sutherland quantum many body Hamiltonian\n\\begin{equation}\\label{cs}\nH = - \\sum_{j=1}^N{\\partial^2 \\over \\partial x_j^2} +\n\\beta (\\beta / 2 - 1) \\Big ( {\\pi \\over L} \\Big )^2\n\\sum_{1 \\le j < k \\le N}\n{1 \\over \\sin^2 \\pi (x_j - x_k)/L}.\n\\end{equation}\nThis Hamiltonian describes quantum particles on a circle of circumference\nlength $L$ interacting via the inverse square of the distance between the\nparticles. In the thermodynamic limit $N,L \\to \\infty$,\n$N/L = \\rho$ (fixed) the $N$ particle system becomes an infinite system on\na line with particle density $\\rho$. The ground state dynamical \ndensity-density correlation function\n\\begin{equation}\\label{h1}\n\\rho^{\\rm dyn.}(0,x;t) := \\ml 0 | n(0) e^{-i H t} n(x)\ne^{i H t} | 0 \\mg, \\qquad\nn(y) := \\sum_{j=1}^N \\delta(y-x_j)\n\\end{equation}\nin the infinite system has been calculated exactly for all rational $\\beta$\n\\cite{Ha95}. The fact that $(|0\\mg)^2$ is proportional to (\\ref{1}) tells us\nthat at $t=0$ (\\ref{h1}) is equal to \n$$\n\\rho_{(2)}^T(0,x) + \\rho \\delta(x),\n$$\nwhere $\\rho_{(2)}^T$ is the truncated two-body density,\nfor the log-gas system. Thus the exact evaluation of\n$$\nS(k,\\beta) := \\int_{-\\infty}^\\infty \\Big (\n\\rho_{(2)}^T(0,x) + \\rho \\delta(x) \\Big ) e^{i k x} \\, dx\n$$\nfor the log-gas follows from the exact evaluation of (\\ref{h1}) for the\nquantum system. Taking $\\beta$ to be rational and setting\n$$\n\\beta / 2 := p/q =: \\lambda\n$$\nwhere $p$ and $q$ are relatively prime integers, the latter exact result\ngives \\cite{FJ97}\n\\begin{equation}\\label{Ha}\nS(k;\\beta) = \\pi C_{p,q}(\\lambda) \\prod_{i=1}^q\n\\int_0^\\infty dx_i \n \\prod_{j=1}^p \\int_0^1 dy_j Q_{p,q}^2\n F(q,p,\\lambda|\\{x_i,y_j\\})\\,\n \\delta(k - Q_{p,q}),\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{C}\nC_{p,q}(\\lambda ) &:=& {\\lambda^{2p(q-1)} \\Gamma^2(p) \\over 2 \\pi^2 p! q!}\n{\\Gamma^q(\\lambda) \\Gamma^p(1/\\lambda) \\over \n\\prod_{i=1}^q \\Gamma^2(p-\\lambda (i-1)) \\prod_{j=1}^p \\Gamma^2(1-(j-1)/\\lambda)}\n\\nonumber \\\\\nQ_{p,q} & := & 2 \\pi \\rho \\Big ( \\sum_{i=1}^q x_i + \\sum_{j=1}^p y_j \\Big )\n\\nonumber \\\\\nF(q,p,\\lambda|\\{x_i,y_j\\}) & := &\n{\\prod_{i<i'}|x_i - x_{i'}|^{2 \\lambda}\\prod_{j<j'}|y_j - y_{j'}|^{2 /\\lambda}\n\\over \n\\prod_{i=1}^q \\prod_{j=1}^p (x_i + \\lambda y_j)^{2}} \\nonumber \\\\\n&& \\times {1 \\over \\prod_{i=1}^q (x_i(x_i+\\lambda))^{1 - \\lambda}\n\\prod_{j=1}^p(\\lambda y_j(1 - y_j))^{1 - 1/\\lambda}}\n\\end{eqnarray}\n\nIn the domain of integration of (\\ref{Ha}) the integration variables are\nall positive and because of the delta function are restricted to the\nhyperplane\n$$\n\\sum_{i=1}^q x_i + \\sum_{j=1}^p y_j = {|k| \\over 2 \\pi \\rho}.\n$$\nWe see immediately from these constraints that the restriction $y_j < 1$\nin the domain of integration is redundant for \n\\begin{equation}\\label{kl}\n|k| < 2 \\pi \\rho.\n\\end{equation}\nThus assuming (\\ref{kl}) we can extend the integration over $y_j$ to the\nregion $(0,\\infty)$. Doing this and changing variables\n$x_i \\mapsto |k| x_i$ and $y_j \\mapsto |k| y_j$ we see that for $|k|$\nin the region (\\ref{kl})\n\\begin{equation}\\label{d2}\nS(k;\\beta) = \\pi |k| C_{p,q}(\\lambda)\n\\prod_{i=1}^q \\int_0^\\infty dx_i \n \\prod_{j=1}^p \\int_0^\\infty dy_j Q_{p,q}^2\n \\hat{F}(q,p,\\lambda|\\{x_i,y_j\\};k)\\,\n \\delta(1 - Q_{p,q}),\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{d2'}\n\\hat{F}(q,p,\\lambda|\\{x_i,y_j\\};k)& = &\n{1 \\over \\prod_{i=1}^q (x_i(1+kx_i/\\lambda))^{1-\\lambda}\n\\prod_{j=1}^p(y_j(1-ky_j))^{1-1/\\lambda}} \\nonumber \\\\\n&& \\times \n{\\prod_{i<i'}|x_i - x_{i'}|^{2 \\lambda}\\prod_{j<j'}|y_j - y_{j'}|^{2 /\\lambda}\n\\over \n\\prod_{i=1}^q \\prod_{j=1}^p (x_i + \\lambda y_j)^{2}}\n\\end{eqnarray}\nNotice that (\\ref{d2'}) is such that the integral in (\\ref{d2}) is analytic for\n\\begin{equation}\\label{d2a}\n|k| < {\\rm min}\\,(2\\pi \\rho, \\pi \\rho \\beta).\n\\end{equation}\nThus according to the definition (\\ref{4+}) we read off that\n\\begin{equation}\\label{d3}\nf(k;\\beta) = 2 \\pi^2 \\lambda \nC_{p,q}(\\lambda)\n\\prod_{i=1}^q \\int_0^\\infty dx_i \n \\prod_{j=1}^p \\int_0^\\infty dy_j Q_{p,q}^2\n \\hat{F}(q,p,\\lambda|\\{x_i,y_j\\};k)\\,\n \\delta(1 - Q_{p,q}).\n\\end{equation}\n\nThe functional equation (\\ref{5}) is a simple consequence of this exact\nformula. Thus we see that the integral in (\\ref{d3}) is unchanged \nby the mapping\n$\\lambda \\mapsto 1/\\lambda$ (and thus $p \\leftrightarrow q$) followed\nby $k \\mapsto - k/\\lambda$. The precise functional equation (\\ref{5})\nfollows provided we can show that\n$$\nC_{p,q}(\\lambda) = \\lambda^{2pq - 2} C_{q,p}(1/\\lambda),\n$$\nwhich indeed readily follows from the definition of \n$C_{p,q}(\\lambda)$ in (\\ref{C}).\n\n\\section{Expanding $f(k;\\beta)$ in terms of Dotsenko-Fateev type\nintegrals}\n\\setcounter{equation}{0}\nHere we will develop a strategy based on the integral formula\n(\\ref{d3}) to expand $f(k,\\beta)$ at order $k^2$. This relies on our\nability to compute certain generalizations of a limiting case of the\nDotsenko-Fateev integral. This same method has been used in\n\\cite{FZ96,FJ97} to compute the equivalent of $f(k,\\beta)$ and its\nderivative at $k=0$.\n\nWe first expand the integrand in (\\ref{d3}) as a function of $k$.\nAccording to (\\ref{d2'}) we have\n$$\n\\hat{F}(q,p,\\lambda|\\{x_i,y_j\\};k) =\nG(q,p,\\lambda|\\{x_i,y_j\\}) \\Big (\n1 + \\sum_{\\nu=1}^\\infty H_\\nu(q,p,\\lambda|\\{x_i,y_j\\}) k^\\nu \\Big )\n$$\nwhere\n\\begin{eqnarray}\\label{u4}\nG(q,p,\\lambda|\\{x_i,y_j\\}) & = &\n{\\prod_{i<i'}|x_i - x_{i'}|^{2 \\lambda}\\prod_{j<j'}|y_j - y_{j'}|^{2 /\\lambda}\n\\over \n\\prod_{i=1}^q \\prod_{j=1}^p (x_i + \\lambda y_j)^{2}\n\\prod_{i=1}^q x_i^{1-\\lambda} \\prod_{j=1}^p y_j^{1-1/\\lambda}} \\nonumber \\\\\n1 + \\sum_{\\nu = 1}^\\infty\nH_\\nu(q,p,\\lambda|\\{x_i,y_j\\}) k^\\nu & = &\n{1 \\over \\prod_{i=1}^q (1+kx_i/\\lambda)^{1-\\lambda}\n\\prod_{j=1}^p(1-ky_j)^{1-1/\\lambda}}.\n\\end{eqnarray}\nThe coefficients $H_\\nu$ are homogeneous polynomials in $\\{x_i,y_j\\}$ of\ndegree $\\nu$.\n\nLet us now introduce the notation\n\\begin{equation}\\label{u5}\nI_{p,q,\\lambda}[h(\\{x_i,y_j\\})]\n:=\n\\prod_{i=1}^q \\int_0^\\infty dx_i \n\\prod_{j=1}^p \\int_0^\\infty dy_j \\, Q_{p,q}^2\n{G}(q,p,\\lambda|\\{x_i,y_j\\})\\,\n\\delta(1 - Q_{p,q}) h(\\{x_i,y_j\\}).\n\\end{equation}\nBecause of the presence of the delta function the value of $I_{p,q,\\lambda}$\nis unchanged if $Q_{p,q}^2$ is replaced by $Q_{p,q}^n$ for any $n$.\nDoing this and also introducing the usual integral representation of the\ndelta function, we see by a change of variables as detailed in\n\\cite{FJ97} that for $h$ homogeneous of degree $\\nu$\n\\begin{equation}\\label{u6}\nI_{p,q,\\lambda}[h(\\{x_i,y_j\\})] =\n{J_{p,q,\\lambda,n}[h(\\{x_i,y_j\\})] \\over (\\nu + n -1)!} =\n{J_{p,q,\\lambda}[h(\\{x_i,y_j\\})] \\over (\\nu-1)!}\n\\end{equation}\nwhere\n$$\nJ_{p,q,\\lambda,n}[h(\\{x_i,y_j\\})] :=\n\\prod_{i=1}^q \\int_0^\\infty dx_i \n\\prod_{j=1}^p \\int_0^\\infty dy_j \\, Q_{p,q}^n\n{G}(q,p,\\lambda|\\{x_i,y_j\\})\\,\ne^{-Q_{p,q}} h(\\{x_i,y_j\\})\n$$\nand $J_{p,q,\\lambda} := J_{p,q,\\lambda,0}$.\n\nRecalling (\\ref{d3}) and\n(\\ref{u4}) we see that in terms of the notation (\\ref{u5})\n\\begin{equation}\\label{u6f}\nf(k;\\beta) = C_{p,q}(\\lambda) \\Big (\nI_{p,q,\\lambda}[1] + \\sum_{\\nu=1}^\\infty\nI_{p,q,\\lambda}[H_\\nu(q,p,\\lambda|\\{x_i,y_j\\})] k^\\nu \\Big ).\n\\end{equation}\nThe definition of $H_\\nu$ in (\\ref{u4}) shows\n\\begin{equation}\\label{u6b}\nH_2(q,p,\\lambda|\\{x_i,y_j\\}) =\n{(\\lambda - 1)^2 \\over 2 \\lambda^2}\n{Q_{p,q}^2 \\over (2 \\pi \\rho)^2} - {\\lambda - 1 \\over 2 \\lambda^2}\n\\Big ( \\sum_{i=1}^q x_i^2 - \\lambda \\sum_{j=1}^p y_j^2 \\Big ),\n\\end{equation}\nso to compute $f(k,\\beta)$ at order $k^2$ our task is to evaluate\n\\begin{equation}\\label{u7}\nI_{p,q,\\lambda}[Q_{p,q}^2] \\quad {\\rm and} \\quad\nI_{p,q,\\lambda}[ \\sum_{i=1}^q x_i^2 - \\lambda \\sum_{j=1}^p y_j^2].\n\\end{equation}\nNow because of the delta function in (\\ref{u5})\n\\begin{equation}\\label{u7'}\nI_{p,q,\\lambda}[Q_{p,q}^2] = I_{p,q,\\lambda}[1],\n\\end{equation}\nand we know from \\cite{FJ97} that\n\\begin{equation}\\label{u7b}\nC_{p,q}(\\lambda) I_{p,q,\\lambda}[1] = 1.\n\\end{equation}\nThus our remaining task is to compute the second expression in (\\ref{u7})\nor equivalently, using (\\ref{u6}), to compute\n\\begin{equation}\\label{u7a}\nJ_{p,q,\\lambda}[\\sum_{i=1}^q x_i^2 - \\lambda \\sum_{j=1}^p y_j^2]\n= q J_{p,q,\\lambda}[x_i^2] - \\lambda p J_{p,q,\\lambda}[y_j^2]\n\\end{equation}\nwhere the second equality, valid for any $1 \\le i \\le q$ and $1 \\le j \\le p$,\nfollows from the symmetry of the integrand.\n\nFor this purpose we first note formulas for $J_{p,q,\\lambda}[h]$\nin the cases $h=x_i^2$ and $h=y_j^2$. The formulas are\n\\begin{eqnarray}\\label{dd1}\nJ_{p,q,\\lambda}[x_i^2] & = &\n{(2p - \\lambda + 1) \\over 2 \\pi \\rho} J_{p,q,\\lambda}[x_i] -\n{p \\over \\pi \\rho} J_{p,q,\\lambda} \\Big [\n{x_i^2 \\over x_i + \\lambda y_j} \\Big ], \\\\\n\\label{dd2}\nJ_{p,q,\\lambda}[y_j^2] & = &\n{(2q - 1/\\lambda + 1) \\over 2 \\pi \\rho} J_{p,q,\\lambda}[y_j]\n- {\\lambda q \\over \\pi \\rho} J_{p,q,\\lambda} \\Big [\n{x_j^2 \\over x_i + \\lambda y_j} \\Big ].\n\\end{eqnarray}\nThe derivation of (\\ref{dd1}) and (\\ref{dd2}) uses a technique based on\nthe fundamental theorem of calculus. It was first used by Aomoto \n\\cite{Ao87} in the context of the Selberg integral, and has been adapted\nin \\cite{FZ96} to the case of the Dotsenko-Fateev integral.\n\nLet us give the details of the derivation of (\\ref{dd1}) (the derivation\nof (\\ref{dd2}) is similar). From the definition (\\ref{u4}) we see that\n$$\n{\\partial \\over \\partial x_i} G(q,p,\\lambda|\\{x_i,y_j\\})\n= \\Big ( {\\lambda - 1 \\over x_i} - 2 \\sum_{j=1}^p\n{1 \\over x_i + \\lambda y_j} + 2 \\lambda\n\\sum_{i'=1 \\atop i' \\ne i}^q {1 \\over x_i - x_{i'}} \\Big )\nG(q,p,\\lambda|\\{x_i,y_j\\}).\n$$\nThus\n\\begin{eqnarray}\\label{r1}\n0 & = & \n\\prod_{i=1}^q \\int_0^\\infty dx_i \n\\prod_{j=1}^p \\int_0^\\infty dy_j \\, {\\partial \\over \\partial x_i}\n\\Big ( x_i^2 \n{G}(q,p,\\lambda|\\{x_i,y_j\\})\ne^{-Q_{p,q}} \\Big ) \\nonumber \\\\\n& = & (\\lambda + 1) J_{p,q,\\lambda}[x_i] - 2 \\sum_{j=1}^p\nJ_{p,q,\\lambda} \\Big [ {x_i^2 \\over x_i + \\lambda y_j} \\Big ] +\n2 \\lambda \\sum_{i'=1 \\atop i' \\ne i}^q \nJ_{p,q,\\lambda} \\Big [ {x_i^2 \\over x_i - x_{i'}} \\Big ] - 2 \\pi \\rho\nJ_{p,q,\\lambda}[x_i^2] \\nonumber \\\\\n& = & (\\lambda + 1) J_{p,q,\\lambda}[x_i] - 2p\nJ_{p,q,\\lambda} \\Big [ {x_i^2 \\over x_i + \\lambda y_j} \\Big ]\n+2 \\lambda (q-1) \nJ_{p,q,\\lambda} \\Big [ {x_i^2 \\over x_i - x_{i'}} \\Big ] - 2 \\pi \\rho\nJ_{p,q,\\lambda}[x_i^2] \\nonumber \\\\\n\\end{eqnarray}\nwhere the first equality follows from the fundamental theorem of calculus, while\nthe final equality, valid for any $j=1,\\dots,p$ and any \n$i'=1,\\dots,q$, ($i'\\ne i$) follows by the symmetry of the integrand\nwith respect to $\\{x_i\\}$ and $\\{y_j\\}$. The symmetry of the integrand with\nrespect to $\\{x_i\\}$ also gives\n$$\nJ_{p,q,\\lambda} \\Big [ {x_i^2 \\over x_i - x_{i'}} \\Big ]\n= J_{p,q,\\lambda} \\Big [ {x_{i'}^2 \\over x_{i'} - x_{i}} \\Big ]\n$$\nso we have\n$$\nJ_{p,q,\\lambda} \\Big [ {x_i^2 \\over x_i - x_{i'}} \\Big ]\n= {1 \\over 2} \\bigg (\nJ_{p,q,\\lambda} \\Big [ {x_i^2 \\over x_i - x_{i'}} \\Big ] +\nJ_{p,q,\\lambda} \\Big [ {x_{i'}^2 \\over x_{i'} - x_{i}} \\Big ] \\bigg )\n= J_{p,q,\\lambda}[x_i].\n$$\nSubstituting in (\\ref{r1}) implies (\\ref{dd1}).\n\n{}From (\\ref{dd1}) and (\\ref{dd2}) we see that\n\\begin{eqnarray}\\label{u9}\n\\lefteqn{\nq J_{p,q,\\lambda}[x_i^2] - \\lambda p J_{p,q,\\lambda}[y_j^2]} \\nonumber \\\\\n&& = {q(2p - \\lambda + 1) \\over 2 \\pi \\rho}J_{p,q,\\lambda}[x_i]\n- {\\lambda p (2q - 1/\\lambda + 1) \\over 2 \\pi \\rho}\nJ_{p,q,\\lambda}[y_j] - {pq \\over \\pi \\rho}\nJ_{p,q,\\lambda} \\Big [ {x_i^2 - \\lambda^2 y_j^2 \\over x_i + \\lambda y_j}\n\\Big ] \\nonumber \\\\\n&& = {q(-\\lambda +1) \\over 2 \\pi \\rho}\nJ_{p,q,\\lambda}[x_i] - {\\lambda p (-1/\\lambda + 1) \\over 2 \\pi \\rho}\nJ_{p,q,\\lambda}[y_j] \\: = \\:\n- {\\lambda - 1 \\over (2 \\pi \\rho)^2} J_{p,q,\\lambda}[Q_{p,q}]\n\\: = \\: - {\\lambda - 1 \\over (2 \\pi \\rho)^2} J_{p,q,\\lambda}[1]\n\\nonumber \\\\\n\\end{eqnarray}\n\nRecalling (\\ref{u6b}), the results (\\ref{u6}),\n(\\ref{u7'}), (\\ref{u7a}) and (\\ref{u9})\ngive that\n$$\nI_{p,q,\\lambda}[H_2(q,p,\\lambda|\\{x_i\\},\\{y_j\\}] =\n{1 \\over (2 \\pi \\rho)^2} {(\\lambda - 1)^2 \\over \\lambda^2}\nJ_{p,q,\\lambda}[1].\n$$\nUse of (\\ref{u7b}) then gives that the term proportional to $k^2$ in\n(\\ref{u6f}) is equal to\n\\begin{equation}\\label{u6g}\n(\\lambda - 1)^2 \\Big ( {k \\over 2 \\pi \\lambda \\rho} \\Big )^2 =\n(\\beta / 2 - 1)^2 \\Big ( {k \\over \\pi \\beta \\rho} \\Big )^2.\n\\end{equation}\nIt follows from this that the structure (\\ref{6}) is valid at order $k^2$\nwith\n\\begin{equation}\\label{u6h}\np_2(x) = (x-1)^2.\n\\end{equation}\n\n\\section{Large-$x$ expansion of $\\rho_{(2)}^T(0,x)$}\n\\setcounter{equation}{0}\nWe have already remarked that the large-$x$ expansion (\\ref{2}) of the\ncharge-charge correlation, or what is the same thing for the one-component\nlog-gas, the large-$x$ expansion of $\\rho_{(2)}^T(0,x)$, is equivalent to\nthe small-$k$ behaviour (\\ref{3}) of $S(k;\\beta)$. More generally the\nexpansion\n\\begin{equation}\\label{e1}\n\\rho_{(2)}^T(0,x) \\mathop{\\sim}\\limits_{x \\to \\infty}\n\\sum_{n=1}^\\infty {c_n \\over x^{2n}}\n\\end{equation}\nis equivalent to the expansion\n\\begin{equation}\\label{e2}\nS(k;\\beta) \\mathop{\\sim}\\limits_{k \\to 0} \\pi\n\\sum_{n=1}^\\infty {(-1)^n c_n \\over (2n-1)!} |k|^{2n - 1}\n\\end{equation}\nwhere the expansion (\\ref{e2}) contains the terms singular in $k$\n(i.e.~of odd order in $|k|$) only. This follows using the Fourier transform\n$$\n\\int_{-\\infty}^\\infty {e^{ikx} \\over x^{2n}} \\, dx =\n\\pi {(-1)^n |k|^{2n-1} \\over (2n - 1)!}\n$$\nfrom the theory of generalized functions (see e.g.~\\cite{Li58}).\n\n{}From the equivalence between (\\ref{e1}) and (\\ref{e2}) we see the fact,\nfollowing from (\\ref{u6g}), that the term proportional to\n$|k|^3$ in the small $k$ expansion of $S(k;\\beta)$ is equal to\n$$\n\\rho (\\beta / 2 - 1)^2 \\Big ( {|k| \\over \\pi \\beta \\rho} \\Big )^3\n$$\nis equivalent to the statement that the term proportional to\n$1/x^4$ in the large $x$ expansion of $\\rho_{(2)}^T(0,x)$ is equal to\n\\begin{equation}\\label{e3}\n\\rho^2 6 \\beta (\\beta/2 - 1)^2 \\Big ( {1 \\over \\pi \\beta \\rho x}\n\\Big )^4.\n\\end{equation}\nIn this section we will derive (\\ref{e3}) directly. We will also calculate\nthe $O(1/x^6)$ term and so explicitly determine the $O(|k|^5)$ term\nin (\\ref{e2}).\n\nThe starting point for our calculation is an exact $\\beta$-dimensional\nintegral formula for the two-particle distribution\n$\\rho_{(2)}(0,x)$ valid for $\\beta$ even. With\n$$\nS_n(a,b,c) := \\prod_{j=0}^{n-1}{\\Gamma(a+1+jc) \\Gamma(b+1+jc)\\Gamma(1+\n(j+1)c) \\over \\Gamma(a+b+2+(N+j-1)c)\\Gamma(1+c)}\n$$\nthe formula gives that in the thermodynamic limit \\cite{Fo93aa}\n\\begin{eqnarray}\\label{i1}\n\\rho_{(2)}(0,x) & = & \\rho^2 (\\beta / 2)^\\beta\n{((\\beta / 2)!)^3 \\over \\beta ! (3 \\beta / 2)!}\n{e^{-\\pi i \\beta \\rho x} (2 \\pi \\rho x)^\\beta \\over\nS_\\beta(1-2/\\beta,1-2/\\beta,2/\\beta)} \\nonumber \\\\\n&& \\times \\int_{[0,1]} du_1 \\cdots du_\\beta\n\\prod_{j=1}^\\beta e^{2 \\pi i \\rho x u_j} u_j^{-1+2/\\beta}\n(1 - u_j)^{-1+2/\\beta} \\prod_{j < k}|u_k - u_j|^{4/\\beta}.\n\\end{eqnarray}\n\nIn a previous analysis \\cite{Fo93aa} it has been shown that the non-oscillatory\nlarge-$x$ behaviour is determined by the integrand in the vicinity of the\nendpoints 0 and 1, with the requirement that $\\beta/2$ of the integration\nvariables are in the vicinity of the endpoint 0, while the remaining\n$\\beta/2$ integration variables are in the vicinity of the endpoint 1.\nThus we write $u_{\\beta/2 + j} = 1 - v_j$ ($j=1,\\dots, \\beta/2$) \n(this introduces a combinatorial factor $\\beta$ choose $\\beta/2$\nto account for the different ways of so partitioning the integration\nvariables) and then\nexpand the integrand (excluding the exponential\nfactors which involve $x$) in terms of the ``small'' variables $u_j,v_j$\n($j=1,\\dots,\\beta/2$). In particular we must expand\n\\begin{equation}\\label{i2}\n\\prod_{j=1}^{\\beta/2}(1-u_j)^{-1+2/\\beta}(1-v_j)^{-1+2/\\beta}\n\\prod_{l,l'=1}^{\\beta / 2}(1-u_l-v_{l'})^{4/\\beta}.\n\\end{equation}\nThe function (\\ref{i2}) is a symmetric function of the variables\n$\\{u_j\\}$ and $\\{v_j\\}$ separately. Let $\\{q_\\kappa\\}_\\kappa$ be a polynomial\nbasis for symmetric functions with $\\kappa$ denoting a partition (ordered\nset of non-negative integers) of no more than $\\beta/2$ parts, and suppose\nfurthermore that $q_\\kappa$ is homogeneous of order $|\\kappa|\n:= \\kappa_1 + \\cdots + \\kappa_{\\beta / 2}$. Then we\ncan write\n\\begin{eqnarray}\\label{i3}\\lefteqn{\n\\prod_{j=1}^{\\beta/2}(1-u_j)^{-1+2/\\beta}(1-v_j)^{-1+2/\\beta}\n\\prod_{l,l'=1}^{\\beta / 2}(1-u_l-v_{l'})^{4/\\beta}} \\nonumber \\\\&&\n= \\sum_{\\kappa, \\mu} w_{\\kappa, \\mu}\nq_\\kappa(u_1,\\dots,u_{\\beta/2}) q_\\mu(v_1,\\dots,v_{\\beta/2}).\n\\end{eqnarray}\n\nSubstituting (\\ref{i3}) in (\\ref{i1}), then following the procedure of\n\\cite{Fo93aa}, which involves extending the range of integration to\n$u_j \\in (0,\\infty)$, $v_j \\in (0,\\infty)$ and changing variables\n$u_j \\mapsto 2 \\pi i \\rho x u_j$, $v_j \\mapsto -2 \\pi i \\rho x v_j$\nmaking use in the process of the fact that $q_\\kappa$ is homogeneous\nof degree $|\\kappa|$, we obtain the non-oscillatory terms in the\nlarge-$x$ asymptotic expansion of $\\rho_{(2)}(0,x)$. This reads\n\\begin{eqnarray}\\label{i4}\n\\rho_{(2)}(0,x) & \\sim & \\rho^2 \\Big ( {\\beta \\atop \\beta / 2} \\Big )\n(\\beta / 2)^\\beta\n{((\\beta / 2)!)^3 \\over \\beta ! (3 \\beta / 2)!}\n{1\\over\nS_\\beta(1-2/\\beta,1-2/\\beta,2/\\beta)} \\nonumber \\\\ \\times\n&& \\sum_{\\kappa, \\mu}\nw_{\\kappa, \\mu} {K_{\\beta, \\kappa} K_{\\beta, \\mu} \\over\ni^{|\\lambda| - |\\mu|} (2 \\pi \\rho x)^{|\\kappa| + |\\mu|}}\n\\end{eqnarray}\nwhere \n\\begin{equation}\\label{i5}\nK_{\\beta, \\kappa} :=\n\\int_{[0,\\infty)^{\\beta / 2}} du_1 \\cdots du_{\\beta/2} \\,\n\\prod_{l=1}^{\\beta / 2} u_l^{-1 + 2/\\beta} e^{-u_l}\n\\prod_{j < k} |u_k - u_j|^{4/\\beta}\nq_\\kappa(u_1,\\dots, u_{\\beta/2}).\n\\end{equation}\nThe symmetry $w_{\\kappa, \\mu} = w_{\\mu, \\kappa}$ evident from (\\ref{i3})\nimplies terms in (\\ref{i4}) with $|\\kappa| + |\\mu|$ odd cancel. Therefore\nthe sum in (\\ref{i4}) can be restricted to partitions such that\n$|\\kappa| + |\\mu|$ is even, which means the asymptotic expansion only\ncontains inverse even powers of $x$.\n\nTo proceed further we must be able to compute the expansion coefficients\n$w_{\\kappa, \\mu}$ as well as the integrals $K_{\\beta, \\kappa}$. For the\nformer task it is convenient to choose $q_\\kappa$ equal to the monomial\nsymmetric polynomial $m_\\kappa$, which is defined as the symmetrization of\nthe monomial $x_1^{\\kappa_1} \\cdots x_{\\beta/2}^{\\kappa_{\\beta/2}}$ \nnormalized so that the coefficient of \n$x_1^{\\kappa_1} \\cdots x_{\\beta/2}^{\\kappa_{\\beta/2}}$ is unity.\n\nFirst, we have the well known expansion\n\\begin{equation}\\label{g1}\n\\prod_{j=1}^n(1-u_j)^a = \\sum_{\\ell(\\kappa) \\le n} a_\\kappa\nm_\\kappa(u_1,\\dots,u_n)\n\\end{equation}\nwhere\n\\begin{equation}\\label{g2}\na_\\kappa = \\prod_{p=1}^{\\ell(\\kappa)} a_{\\kappa_p}, \\qquad\na_k := {(-a)_k \\over k!}\n\\end{equation}\nwith \n$\\ell(\\kappa)$ denoting the length of $\\kappa$\n(i.e.~number of non-zero parts). We can therefore\nimmediately expand the first product in (\\ref{i3}) in terms of\nmonomial symmetric polynomials.\n\nConsider next the expansion of the double product in (\\ref{i3}). Making use\nof the formulas\n\\begin{eqnarray}\\label{g2a}\n(1-x)^a & = & \\sum_{n=0}^\\infty {(-a)_n \\over n!} x^n \\\\\n\\prod_{j=1}^n \\Big ( \\sum_{k=0}^\\infty a_k t_j^k \\Big ) & = &\n\\sum_{\\ell(\\kappa) \\le n}\na_0^{N-\\ell(\\kappa)} a_\\kappa m_\\kappa(\\{t_j\\}) \\label{g2b}\n\\end{eqnarray}\nwhere $a_\\kappa$ is specified by the first equality in (\\ref{g2}), we see that\n\\begin{equation}\\label{g3}\n\\prod_{j=1}^{\\beta / 2} (1 - u_j - v)^{4/\\beta} =\n\\sum_{\\ell(\\kappa) \\le \\beta / 2} (1 - v)^{2 - |\\kappa|}\nc_\\kappa m_\\kappa(u_1,\\dots, u_{\\beta / 2})\n\\end{equation}\nwhere\n$$\nc_\\kappa = \\prod_{p=1}^{\\ell(\\kappa)} c_{\\kappa_p}, \\qquad\nc_k := {(-4/\\beta)_k \\over k!}.\n$$\nExpanding the factor $(1-v)^{2 - |\\kappa|}$ we can rewrite (\\ref{g3}) as\n$$\n\\prod_{j=1}^{\\beta / 2} (1 - u_j - v)^{4/\\beta} =\n\\sum_{n=0}^\\infty w_n(u_1,\\dots,u_{\\beta/2};\\beta) v^n\n$$\nfor appropriate symmetric functions $w_n$. Replacing $v$ by $v_{j'}$\nand forming the product over $j'$ using (\\ref{g2b}) we obtain\n$$\n\\prod_{j,j'=1}^{\\beta / 2} (1 - u_j - v_{j'})^{4/\\beta} =\n\\sum_{\\ell(\\kappa) \\le \\beta / 2}\nw_0^{\\beta/2 - \\ell(\\kappa)} w_\\kappa m_\\kappa(v_1,\\dots,v_{\\beta/2})\n$$\nwhere $w_\\kappa := \\prod_{p=1}^{\\ell(\\kappa)}\n w_{\\kappa_p}$. The final step is to expand\n$w_0^{\\beta/2 - \\ell(\\kappa)} w_\\kappa$ in terms of $\\{m_\\mu\\}$ and so\nobtain the expansion\n\\begin{equation}\\label{g4}\n\\prod_{j,j'=1}^{\\beta / 2} (1 - u_j - v_{j'})^{4/\\beta} \n= \\sum_{\\mu, \\kappa} t_{\\mu,\\kappa}\nm_\\mu(u_1,\\dots,u_{\\beta/2}) m_\\kappa(v_1,\\dots,v_{\\beta/2}).\n\\end{equation}\nThe practical implementation of this procedure requires the use of computer\nalgebra. We work with arbitrary (positive integer) values of $\\beta/2$.\nFurthermore, we only include terms with $|\\mu| + |\\kappa| \\le 6$ throughout\nsince according to (\\ref{i4}) these terms suffice for the evaluation\nof the coefficients of $1/x^{2n}$, $n \\le 3$.\n\nHaving obtained the coefficients $t_{\\mu,\\kappa}$ in (\\ref{g4}), we\nmultiply the series (\\ref{g4}) with the two series of the form\n(\\ref{g1}) representing the first two products in (\\ref{i2}),\nexpressing the answer in the form of (\\ref{i3}),\nand so determining the coefficients $w_{\\kappa, \\mu}$.\nAgain this step requires computer algebra.\n\nWith $w_{\\kappa, \\mu}$ in (\\ref{i3}) determined, it remains to compute\nthe multiple integral (\\ref{i5}) with $q_\\mu = m_\\mu$. For this task\nwe introduce a further basis of symmetric functions, namely the\nJack polynomials $\\{P_\\kappa^{(\\beta/2)}(u_1,\\dots,u_{\\beta/2})\\}$.\nThe Jack polynomials $P_\\kappa^{(2/\\beta)}(z_1,\\dots,z_N)$ with\n$z_j := e^{2\\pi i x_j/L}$, when muliplied by the ground state wave\nfunction $|0\\mg$, are the eigenfunctions of the Calogero-Sutherland\nSchr\\\"odinger operator (\\ref{cs}) \\cite{Fo94j}. Each polynomial\nis homogeneous of degree $|\\kappa|$ and has the expansion\n\\begin{equation}\\label{g3a}\nP_\\kappa^{(\\alpha)}(z_1,\\dots,z_N) = m_\\kappa +\n\\sum_{\\mu < \\kappa} a_{\\kappa \\mu} m_\\mu\n\\end{equation}\nwhere $<$ is the dominance partial ordering for partitions:\n$\\mu < \\kappa$ if $|\\kappa| = |\\mu|$ with $\\kappa \\ne \\mu$ and\n$\\sum_{i=1}^p \\mu_i \\le \\sum_{i=1}^p \\kappa_i$ for each\n$p=1,\\dots,N$. The coefficients $a_{\\kappa \\mu}$ can be calculated by\nrecurrence \\cite{Ma95}. \n\nThe significance of the Jack polynomial basis is that we have the explicit\nintegral evaluation\n\\begin{equation}\\label{g3b}\n{1 \\over W_{a\\alpha N}} \\prod_{l=1}^N \\int_0^\\infty dt_l \\,\nt_l^a e^{-t_l} P_\\kappa^{(\\alpha)}(t_1,\\dots,t_N)\n\\prod_{j<k}|t_k - t_j|^{2/\\alpha}\n= P_\\kappa^{(\\alpha)}(1^N) [a + (N-1)/\\alpha + 1]_\\kappa^{(\\alpha)},\n\\end{equation}\nwhich is a limiting case of an integration formula due to Macdonald\n\\cite{Ma95}, Kadell \\cite{Ka97g} and Kaneko \\cite{Ka93}. In (\\ref{g3b})\n\\begin{eqnarray*}\nW_{a\\alpha N} & = &\n\\prod_{l=1}^N \\int_0^\\infty dt_l \\, t_l^a e^{-t_l}\n\\prod_{j < k} |t_k - t_j|^{2/\\alpha} \\: = \\:\n\\prod_{j=0}^{N-1} {\\Gamma(1 + (j+1)/\\alpha) \\Gamma(a+1+j/\\alpha) \\over\n\\Gamma(1+1/\\alpha)}, \\\\\n{}[u]_\\kappa^{(\\alpha)} & := &\n\\prod_{j=1}^N {\\Gamma(u-(j-1)/\\alpha + \\kappa_j) \\over\n\\Gamma(u - (j-1)/\\alpha)}\n\\end{eqnarray*}\nand $P_\\kappa^{(\\alpha)}(1^N)$ denotes $P_\\kappa^{(\\alpha)}(x_1,\\dots,\nx_N)$ evaluated at $x_1 = \\cdots = x_N = 1$.\n\nTo make use of (\\ref{g3b}) we must first express the monomial symmetric\npolynomials $m_\\kappa$ in terms of $\\{P_\\mu^{2/\\beta}\\}_{\\mu \\le \\kappa}$,\nwhich can be done using computer algebra\nfrom knowledge of the expansion (\\ref{g3a}). Substituting in (\\ref{g3b})\nallows the integrals $K_{\\kappa \\beta}$ to be computed.\n\nAfter completing this procedure all terms in (\\ref{i4}) for\n$|\\kappa| + |\\mu| \\le 6$ are known explicitly. Performing the sum and\nsimplifying we obtain\n\\begin{equation}\\label{g6}\n\\rho_{(2)}(0,x) \\sim \\rho^2 \\bigg (\n1 - {1 \\over \\beta (\\pi \\rho x)^2} +\n{3 (\\beta - 2)^2 \\over 2 \\beta^3 (\\pi \\rho x)^4} -\n{15 (\\beta - 2)^2 (\\beta^2 - 3\\beta + 4) \\over 2 \\beta^5 (\\pi \\rho x)^6}\n+ \\cdots \\bigg ).\n\\end{equation}\nNote that this agrees with the known form (\\ref{2}) for the term\n$O(1/x^2)$, and the form (\\ref{e3}) for the term $O(1/x^4)$.\nThe term $O(1/x^6)$, due to\nthe equivalence between (\\ref{e1}) and (\\ref{e2}), implies the\nterm $O(|k|^5)$ in the small-$k$ expansion of $S(k,\\beta)$ is equal to\n\\begin{equation}\\label{g6b}\n(\\beta/2-1)^2((\\beta/2)^2 - {3 \\over 2} (\\beta/2) + 1) \\Big ( {|k| \\over \n\\pi \\beta} \\Big )^5\n\\end{equation}\nThis is of the form of the conjecture (\\ref{6}) with\n\\begin{equation}\\label{g6c}\np_4(x) = (x-1)^2 (x^2 - {3 \\over 2} x + 1).\n\\end{equation}\n\n\\section{$S(k;\\beta)$\nfor special $\\beta$}\n\\setcounter{equation}{0}\nLet us assume the validity of (\\ref{6}). The coefficients specifying\nthe polynomials $p_j(x)$ therein can be determined from knowledge of the\ncoefficient of $|k|^{j+1}$ in $S(k;\\beta)$ or\n$\\partial^p S(k;\\beta) / \\partial \\beta^p$ ($p \\le j$) at special values\nof $\\beta$. Now in the context of random matrix theory $S(k;\\beta)$ has\nbeen evaluated in terms of elementary functions for $\\beta = 1,2$ and 4.\nThe results are \\cite{Me91}\n\\begin{eqnarray}\nS(k;1) & = & \\left \\{ \\begin{array}{ll}{|k| \\over \\pi} - {|k| \\over 2 \\pi}\n\\log \\Big ( 1 + {|k| \\over \\pi \\rho} \\Big ), & |k| \\le 2 \\pi \\rho \n\\\\[.1cm]\n2 - {|k| \\over 2 \\pi} \\log \\Big ( \n{1 + |k|/\\pi \\rho \\over -1 + |k|/\\pi \\rho} \\Big ), & |k| \\ge 2 \\pi \\rho \n\\end{array} \\right.\n\\label{er1} \\\\\nS(k;2) & = & \\left \\{ \\begin{array}{ll}{|k| \\over 2\\pi}, & |k| \\le 2 \\pi \\rho \n\\\\[.1cm]\n1, & |k| \\ge 2 \\pi \\rho \\end{array} \\right.\n\\label{er2} \\\\\nS(k;4) & = & \\left \\{ \\begin{array}{ll}{|k| \\over 4 \\pi} - {|k| \\over 8 \\pi}\n\\log \\Big | 1 - {|k| \\over 2 \\pi \\rho} \\Big |, & |k| \\le 4 \\pi \\rho \n\\\\[.1cm]\n1, & |k| \\ge 4 \\pi \\rho \\end{array} \\right.\n \\label{er3}\n\\end{eqnarray}\nRecalling the definition (\\ref{4+}) of $f(k,\\beta)$ we read off\n\\begin{eqnarray}\nf(k;1) & = & 1 - {1 \\over 2} \\log \\Big ( 1 + {k \\over \\pi \\rho} \\Big )\n\\label{er4} \\\\\nf(k;2) & = & 1\n\\label{er5} \\\\\nf(k;4) & = & 1 - {1 \\over 2} \n\\label{er6} \\log \\Big ( 1 - {k \\over 2 \\pi \\rho} \\Big )\n\\end{eqnarray}\n\nThe exact evaluation (\\ref{er5}) implies that for all $j$ $p_j(x)$ contains\na factor of $(x-1)$. In the case of $j$ odd this gives no new information\nsince the factor $(x-1)$ was already deduced as a consequence of\nthe functional equation (\\ref{6.1}). On the other hand, in the case $j$\neven this fact together with the functional equation (\\ref{6.1}) implies\n\\begin{equation}\\label{pf}\np_j(x) = (x-1)^2 \\sum_{l=0}^{j-2}b_{j,l} x^l, \\qquad b_{j,l} = b_{j,j-2-l}\n\\quad (j \\: {\\rm even}).\n\\end{equation}\n\nConsider now the constraints on the coefficients in (\\ref{pf}) and (\\ref{7.2})\nwhich follow from (\\ref{er4}) and (\\ref{er6}). As (\\ref{er4}) and (\\ref{er6})\nare related by the functional equation (\\ref{5}), and this is built\ninto the structures (\\ref{pf}) and (\\ref{7.2}), only one of these exact\nevaluations gives distinct information on $p_j(x)$. For definiteness\nconsider (\\ref{er4}). We see that\n\\begin{equation}\\label{pf1}\n[k^j] f(k;1) = {1 \\over 2} {(-1)^{j} \\over j (\\pi \\rho)^j}, \\quad j \\ge 1\n\\end{equation}\nwhere the notation $[k^j]$ denotes the coefficient of $k^j$.\nRecalling (\\ref{6}), (\\ref{pf}) and (\\ref{7.2}) this implies, for \n$j$ even,\n\\begin{equation}\n{1 \\over j} = {1 \\over 2}\\Big ( (1+2^{-(j-2)})b_{j,0}\n+ (2^{-1} + 2^{-(j-3)})b_{j,1} + \\cdots + (2^{-j/2 +2} + 2^{-j/2})b_{j,j/2-2}\n+ 2^{-j/2 +1} b_{j,j/2-1} \\Big ), \\label{nt1}\n\\end{equation}\nwhile for $j$ odd\n\\begin{eqnarray}\n{1 \\over j} & = & \\Big ( (1+2^{-(j-1)})\\tilde{a}_{j,0}\n+ (2^{-1} + 2^{-(j-2)})\\tilde{a}_{j,1} + \n\\cdots + (2^{-(j-1)/2 +1} + 2^{-(j-1)/2-1})\\tilde{a}_{j,(j-1)/2-1}\n\\nonumber \\\\\n&&\n+ 2^{-(j-1)/2} \\tilde{a}_{j,(j-1)/2} \\Big ). \n\\label{nt2}\n\\end{eqnarray}\nIn the case $j=1$ (\\ref{nt2}) gives $\\tilde{a}_{j,0}=1$ which reclaims\n(\\ref{6.2}), while \nin the case $j=2$ (\\ref{nt1}) gives $b_{j,0}=1$ which reclaims\n(\\ref{u6h}).\n\nThe exact form of $S(k;\\beta)$ in the weak coupling scaling limit\n$\\beta \\to 0$, $k \\to 0$, $k/\\beta$ fixed is also available. Introducing\nthe dimensionless Fourier transforms\n$$\n\\tilde{S}(k;\\beta) := \\rho \\int_{-\\infty}^\\infty\n\\Big ( \\rho_{(2)}^T(0,x) + \\rho \\delta(x) \\Big ) e^{i \\rho x k} \\, dx,\n\\qquad \\tilde{\\Phi}(k) := \\rho \\int_{-\\infty}^\\infty \\Phi(x)\ne^{i \\rho x k} \\, dx\n$$\nwhere \n$\\Phi(x) := - \\log|x|$ is the pair potential of the log-gas (thus the\nintegral in the definition of the $\\tilde{\\Phi}(k)$ is to be interpreted as\na generalized function) we have \\cite{HM90}\n\\begin{equation}\\label{5.30a}\n\\tilde{S}(k;\\beta) \\sim 1 - {\\beta \\tilde{\\Phi}(k) \\over 1 +\n\\beta \\tilde{\\Phi}(k)}.\n\\end{equation}\nSince \n\\begin{equation}\\label{fte}\n\\tilde{\\Phi}(k) = {\\pi \\over |k|},\n\\end{equation}\nand noting $\\tilde{S}(k;\\beta) = S(k\\rho;\\beta)/\\rho$\nwe thus have that in the weak coupling\nscaling limit\n\\begin{equation}\\label{us}\nS(k,\\beta) = \\rho \\Big ( 1 - {1 \\over 1 + |k|/\\pi \\beta \\rho} \\Big ).\n\\end{equation}\nExpanding (\\ref{us}) in the form (\\ref{6}) and recalling (\\ref{pf})\nand (\\ref{7.2}) we deduce\n\\begin{equation}\\label{us1}\n\\tilde{a}_{j,0} = 1 \\qquad {\\rm and} \\qquad b_{j,0}=1\n\\end{equation}\nfor all $j$. Using (\\ref{us1}) in (\\ref{nt1}) and (\\ref{nt2}) gives that\nin the case $j=3$, $\\tilde{a}_{j,1} = - {11 \\over 6}$, and in the case $j=4$,\n$b_{j,1} = - {3 \\over 2}$. The latter result reclaims (\\ref{g6b})\nwhile the former result together with (\\ref{us1}) gives\n\\begin{equation}\\label{p3}\np_3(x) = (x-1)(1 - {11 \\over 6} x + x^2 ).\n\\end{equation}\n\nAn alternative way to derive (\\ref{us1}) is to consider the $\\beta \\to \\infty$\nlow temperature limit. In this limit the system behaves like an harmonic \ncrystal, for which we have available the analytic formula\n\\cite{Fo93}\\footnote{The denominator of the exponent in (3.10) of\n\\cite{Fo93} contains a spurious factor of $\\pi^2$ which is corrected in\n(\\ref{har})}\n\\begin{equation}\\label{har}\n\\rho_{(2)}^{\\rm (har)}(x;0) = \\rho^2 \\sum_{p=-\\infty \\atop p \\ne 0}^\\infty\n\\Big ( {\\beta \\over 4 \\pi f(p)} \\Big )^{1/2}\ne^{-\\beta(p-\\rho x)^2/4f(p)}\n\\end{equation}\nwhere\n$$\nf(p) = {1 \\over \\pi^2} \\int_0^{1/2} {1 - \\cos 2 \\pi p t \\over t - t^2}\n\\, dt.\n$$\nTaking the Fourier transform gives for $|k| < 2 \\pi \\rho$\n\\begin{eqnarray}\nS^{\\rm (har)}(k;\\beta) & = & \\rho \\sum_{p=-\\infty}^\\infty \\Big (\ne^{-k^2 f(p) / \\beta \\rho^2} - 1 \\Big ) e^{ikp/\\rho} \\nonumber \\\\\n& \\mathop{\\sim}\\limits_{\\beta \\to \\infty} & -\n\\rho {k^2 \\over \\beta \\rho^2} \\sum_{p=-\\infty}^\\infty f(p)\ne^{ikp/\\rho} \\: = \\: {|k|/\\pi \\beta \\over 1 - |k|/2\\pi \\rho}.\n\\end{eqnarray}\nThis formula maps to the weak coupling result (\\ref{5.30a}) under the\naction of the functional equation (\\ref{5}) and so implies (\\ref{us1}).\n\n\n\n\\section{Perturbation about $\\beta = 0$}\n\\setcounter{equation}{0}\nThe formula (\\ref{us}) is just the first term in a systematic weak\ncoupling renormalized Mayer series expansion in $\\beta$. In the case of\nthe two-dimensional one-component plasma, low order terms of this\nexpansion have recently been analyzed by Kalinay et al.~\\cite{KMST99}.\nResults from that study can readily be transcribed to the case of the\none-component log-gas.\n\nFormally, the renormalized Mayer series expansion is for the\ndimensionless free energy $\\beta \\bar{F}^{\\rm ex}$ (in \\cite{KMST99}\nour $\\beta \\bar{F}^{\\rm ex}$ is written $-\\beta \\bar{F}^{\\rm ex}$), and\none computes the direct correlation function via the \nfunctional differentiation formula\n\\begin{equation}\\label{o1}\nc(0,x) = - {\\delta^2 (\\beta \\bar{F}^{\\rm ex}) \\over \\delta \\rho_{(1)}(0)\n\\delta \\rho_{(1)}(x)}.\n\\end{equation}\nThe Ornstein-Zernicke relation gives that the dimensionless Fourier transform\nof the direct correlation function, $\\tilde{c}(k,\\beta)$ say, is related to\nthe dimensionless structure function $\\tilde{S}(k;\\beta)$ by\n\\begin{equation}\\label{o2}\n\\tilde{c}(k;\\beta) = 1 - {1 \\over \\tilde{S}(k;\\beta)}\n\\end{equation}\nso expanding $\\tilde{c}(k,\\beta)$ about $\\beta = 0$ with $k/\\beta$ fixed is\nequivalent to expanding $\\tilde{S}(k;\\beta)$ about $\\beta = 0$ with \n$k/\\beta$ fixed.\n\nNow, transcribing the results of \\cite{KMST99} we read off that the\nweak coupling diagrammatic expansion of $c(x_1,x_2)$ starts as\n\\begin{equation}\nc(x_1,x_2) = -\\beta \\Phi(x_1,x_2) + {1 \\over 2!} \\Big (K(x_1,x_2) \\Big )^2 \n+ \\cdots\n\\end{equation}\nwhere\n\\begin{equation}\nK(x_1,x_2) = -\\beta \\pi \\int_{-\\infty}^\\infty\n{e^{i k (x_1 - x_2)} \\over |k| + \\kappa} \\, dk \n\\end{equation}\nwith $\\kappa = \\beta \\pi \\rho$.\nThis implies\n\\begin{equation}\n\\tilde{c}(k;\\beta) = - {\\beta \\pi \\over |k|} + {1 \\over 2} \\rho\n\\int_{-\\infty}^\\infty {dl \\over 2 \\pi} \\, {\\beta \\pi \\over |l| +\n\\kappa} {\\beta \\pi \\over |\\rho k - l| + \\kappa}.\n\\end{equation}\nThe integral is straightforward (consider separately the ranges of $l$ such \nthat $l >0$ ($l <0$) and $\\rho k-l >0$ ($\\rho k - l <0$)). In terms of\n$k' := \\rho k / \\kappa = k/\\pi \\beta$,\n\\begin{equation}\n\\tilde{c}(k;\\beta) =\n- {1 \\over |k'|} + \\beta {1 + |k'| \\over |k'|(2 + |k'|)}\n\\log(1 + |k'|) + O(\\beta^2),\n\\end{equation}\nor equivalently using (\\ref{o2})\n\\begin{equation}\\label{re1}\nS(k;\\beta) = \\rho {|k/\\kappa| \\over 1 + |k/\\kappa|} +\n\\beta \\rho {|k/\\kappa| \\over (1 + |k/\\kappa|)(2+|k/\\kappa|)}\n\\log(1 + |k/\\kappa|) + O(\\beta^2).\n\\end{equation}\nNotice that the leading order term in (\\ref{re1}) reproduces (\\ref{5.30a}).\n\nThe exact result (\\ref{re1}) gives the explicit value of the coefficient\nof $x$ in the polynomial $p_j(x)$. Thus recalling (\\ref{7.2}) and\n(\\ref{pf}) we have\n\\begin{eqnarray}\\label{re2}\n{1 \\over 2}(b_{j,1} - 2) & = & [x^j] {1 \\over (1+x)(2+x)}\n\\log (1+x), \\qquad (j \\quad {\\rm even}) \\nonumber \\\\\n{1 \\over 2} (1 - \\tilde{a}_{j,1}) & = & [x^j] {1 \\over (1+x)(2+x)}\n\\log (1+x), \\qquad (j \\quad {\\rm odd}).\n\\end{eqnarray}\nFurthermore, a simple calculation gives\n\\begin{equation}\\label{re3}\n[x^j] {1 \\over (1+x)(2+x)}\n\\log (1+x) = (-1)^j \\sum_{q=1}^j {1 \\over q}(1 - 2^{q-j})\n\\end{equation}\nso we have for example\n\\begin{equation}\\label{re4}\n\\tilde{a}_{5,1} = - {91 \\over 30}, \\quad\nb_{6,1} = - {31 \\over 15}, \\quad\n\\tilde{a}_{7,1} = - {1607 \\over 420}, \\quad\nb_{8,1} = - {263 \\over 84}, \\quad\n\\tilde{a}_{9,1} = - {791 \\over 180}.\n\\end{equation}\n\nSubstituting $\\tilde{a}_{5,1}$ from (\\ref{re4}) and \n$\\tilde{a}_{5,0}$ from\n(\\ref{us1}) in (\\ref{nt1}) shows $\\tilde{a}_{5,2} = {62 \\over 15}$.\nSimilarly, the value of $b_{6,1}$ above allows us to deduce\nthat $b_{6,2} = {13 \\over 4}$. Thus we have\n\\begin{eqnarray}\\label{re4'}\np_5(x) & = & (x-1)(x^4 - {91 \\over 30} x^3 + {62 \\over 15} x^2\n- {91 \\over 30} x + 1) \\nonumber \\\\\np_6(x) & = & (x-1)^2(x^4 - {37 \\over 15} x^3 + {13 \\over 4} x^2\n- {37 \\over 15} x + 1).\n\\end{eqnarray}\n\nWe remark that according to the conjecture (\\ref{6}), the expansion of\n$S(k,\\beta)$ about $\\beta = 0$ should have the structure\n\\begin{equation}\\label{re5}\nS(k,\\beta) = f_0(k/\\kappa) + \\beta f_1(k/\\kappa) + \\beta^2\nf_2(k/\\kappa) + \\cdots\n\\end{equation}\nwhere\n\\begin{equation}\\label{re6}\nf_j(u) = u^j ( c_{j,0} + c_{j,1} u + \\cdots ).\n\\end{equation}\nConsideration of the analysis of \\cite{KMST99} reveals that the\nstructure (\\ref{re5}) will indeed result from the weak coupling\nexpansion, however the structure (\\ref{re6}) is not immediately evident.\n(Of course the explicit form $f_2$ as revealed by (\\ref{re1}) exhibits\nthis structure.) \n\n\\section{Perturbation about $\\beta = 2$ and $\\beta = 4$}\n\\setcounter{equation}{0}\nA feature of the couplings $\\beta=1,2$ and 4 is that the $n$-particle\ndistribution functions are known for each $n=2,3,\\dots$\n\\cite{Me91}. Introducing the\ndimensionless distribution\n$$\ng(x_1,\\dots,x_n) := \\rho_{(n)}(x_1,\\dots,x_n) / \\rho^n\n$$\nwe can use our knowledge of $g(x_1,\\dots,x_n)$ for $n=2,3$ and 4 at these\nspecific $\\beta$ to expand $g(x_1,x_2)$ about $\\beta = \\beta_0$\nto first order in $\\beta - \\beta_0$. Thus with $\\Phi(x_1,x_2) :=\n- \\log |x_1 - x_2|$ we have \\cite{Ja81}\n\\begin{eqnarray}\\label{ny0}\ng(x_1,x_2;\\beta) & = & g(x_1,x_2) + (\\beta - \\beta_0) \\bigg \\{\n- g(x_1,x_2) \\Phi(x_1,x_2) \\nonumber \\\\\n&& - 2 \\rho \\int_{-\\infty}^\\infty\n\\Big ( g(x_1,x_2,x_3) - g(x_1,x_2) \\Big ) \\Phi(x_1,x_3) \\, dx_3 \\nonumber \\\\\n&& - {1 \\over 2} \\rho^2 \\int_{-\\infty}^\\infty \\Big (\ng(x_1,x_2,x_3,x_4) - g(x_1,x_2) g(x_3,x_4) - g(x_1,x_2,x_3)\n\\nonumber \\\\\n&& - g(x_1,x_2,x_4) + 2 g(x_1,x_2) \\Big ) \\Phi(x_3,x_4) \\, dx_3 dx_4\n\\bigg \\} + O((\\beta - \\beta_0)^2)\n\\end{eqnarray}\nwhere on the right hand side the dimensionless distributions are evaluated \nat $\\beta = \\beta_0$. Here we will compute this first order correction,\nand the corresponding first order correction for $S(k;\\beta)$, in the cases\n$\\beta_0=2$ and $\\beta_0 = 4$ (we do not consider $\\beta_0=1$ because\nof its relation to $\\beta_0=4$ via the functional equation (\\ref{5})). \n\nNow, in the case $\\beta_0 = 2$ we have \n\\begin{equation}\\label{ny1}\ng(x_1,\\dots,x_n) = \\det \\Big [ P_2(x_j,x_k) \\Big ]_{j,k=1,\\dots,n},\n\\qquad\nP_2(x,y) := {\\sin \\pi \\rho (x-y) \\over \\pi \\rho (x-y)}\n\\end{equation}\nwhile in the case $\\beta_0=4$\n\\begin{equation}\\label{ny2}\ng(x_1,\\dots,x_n) = {\\rm qdet} \\Big [ P_4(x_j,x_k) \\Big ]_{j,k=1,\\dots,n}\n\\end{equation}\nwhere\n\\begin{equation}\\label{ny2'}\nP_4(x_j,x_k) = \\left [ \\begin{array}{cc} \\displaystyle\n{\\sin 2 \\pi \\rho x_{jk} \\over 2 \\pi \\rho x_{jk}} & {\\rm Si}\\, \n( 2 \\pi \\rho x_{jk}) \\\\[.2cm]\n\\displaystyle {1 \\over 2 \\pi \\rho} {d \\over d x_{jk}}\n\\Big ( {\\sin 2 \\pi \\rho x_{jk} \\over 2 \\pi \\rho x_{jk}} \\Big ) &\n\\displaystyle\n{\\sin 2 \\pi \\rho x_{jk} \\over 2 \\pi \\rho x_{jk}} \\end{array} \\right ]\n\\end{equation}\nwith $x_{jk} := x_j - x_k$ and Si$(x)$ denoting the complimentary\nsine integral, defined in terms of the sine integral si$(x)$ by\n\\begin{equation}\\label{si}\n{\\rm Si}\\,(x) = \\int_0^x {\\sin t \\over t} \\, dt =\n{\\pi \\over 2} + {\\rm si}\\,(x), \\quad\n {\\rm si}\\,(x) := - \\int_x^\\infty {\\sin t \\over t} \\, dt.\n\\end{equation}\nIn (\\ref{ny2}) qdet denotes quaternion determinant, which can be defined as\n\\begin{equation}\\label{qdet}\n{\\rm qdet} \\Big [ P_4(x_j,x_k) \\Big ]_{j,k=1,\\dots,n} =\n\\sum_{P \\in S_n} (-1)^{n-l} \\prod_1^l\\Big ( P_4(x_a,x_b) P_4(x_b,x_c)\n\\cdots P_4(x_d,x_a) \\Big )^{(0)}\n\\end{equation}\nwhere the superscript $(0)$ denotes the operation ${1 \\over 2} {\\rm Tr}$,\n$P$ is any permutation of the indicies $(1,\\dots,n)$ consisting of $l$\nexclusive cycles of the form $(a \\to b \\to c \\cdots \\to d \\to a)$ and\n$(-1)^{n-l}$ is equal to the parity of $P$. Note that this reproduces the\ndefinition of an ordinary determinant in the case that $P_4$ is a\nmultiple of the identity.\n\nThe task now is to substitute (\\ref{ny1}) in the case $\\beta_0=2$ and\n(\\ref{ny2}) in the case $\\beta_0=4$, and to compute the integrals. Consider\nfirst the case $\\beta_0=2$. After some calculation (see Appendix A) we find\n\\begin{eqnarray}\\label{af0}\ng(0,x;\\beta) & = & 1 - \\Big ( {\\sin \\pi \\rho x \\over \\pi \\rho x} \\Big )^2\n+ (\\beta - 2) \\bigg \\{ {1 \\over 2}\n\\Big ( {\\sin \\pi \\rho x \\over \\pi \\rho x} \\Big )^2\n- {\\sin 2 \\pi \\rho x \\over 2 \\pi \\rho x} + {\\rm ci}\\, (2 \\pi \\rho x) \\nonumber \\\\\n&& + {1 \\over 2 (\\pi \\rho x)^2} \\Big (\n(\\log 2 \\pi \\rho |x| + C) \\cos 2 \\pi \\rho x - {\\rm ci}\\,(2 \\pi \\rho x)\n\\Big ) \\bigg \\} + O((\\beta - 2)^2)\n\\end{eqnarray}\nwhere $C$ denotes Euler's constant while\n\\begin{equation}\\label{ci}\n{\\rm ci}(x) = C + \\log |x| + \\int_0^x {\\cos t - 1 \\over t} \\, dt =\n- \\int_x^\\infty {\\cos t \\over t} \\, dt\n\\end{equation}\ndenotes the cosine integral. From this we can compute (again see Appendix\nA) that up to terms $O((\\beta - 2)^2)$ \n\\begin{equation}\\label{af1}\nS(k;\\beta) = \\left \\{ \\begin{array}{ll}\n {|k| \\over 2 \\pi} + (\\beta - 2) \\rho \\bigg \\{\n{1 \\over 2} \\log \\Big ( 1 - {k^2 \\over (2 \\pi \\rho)^2} \\Big ) +\n{|k| \\over 4 \\pi \\rho} \\log {2 \\pi \\rho + |k| \\over 2 \\pi \\rho - |k|}\n- {|k| \\over 4 \\pi \\rho} \\bigg \\}, & \n|k| < 2 \\pi \\rho, \\\\\n\\rho + (\\beta - 2) \\rho \\bigg \\{\n{1 \\over 2} \\log {|k| + 2 \\pi \\rho \\over |k| - 2 \\pi \\rho}\n+ {|k| \\over 4 \\pi \\rho} \\log \\Big ( 1 - {(2 \\pi \\rho)^2 \\over\nk^2} \\Big )\n- {\\pi \\rho \\over |k|} \\bigg \\}, & \n|k| > 2 \\pi \\rho. \\end{array} \\right.\n\\label{af2}\n\\end{equation}\n\nLet us consider the consequence of (\\ref{af1}) in regards to the expansion\n(\\ref{6}). For $|k| < 2 \\pi \\rho$\nwe observe that all terms but the one proportional to $|k|$\nare even in $k$. This is consistent with $p_j(x)$ having the quadratic\nfactor $(x-1)^2$ for $j$ odd (recall (\\ref{pf})), but only a linear\nfactor for $j$ even (recall (\\ref{7.2})). Moreover, we can use\n(\\ref{af1}) to derive a linear equation for the coefficients\n$\\{\\tilde{a}_j\\}$. First we differentiate (\\ref{af1}) with respect to\n$\\beta$, set $\\beta = 2$ and expand about $k=0$ to obtain\n$$\n{\\partial S(k;\\beta) \\over \\partial \\beta} \\Big |_{\\beta = 2}\n= - {1 \\over 2} {|k| \\over 2 \\pi \\rho}\n+ \\sum_{j=1}^\\infty {1 \\over 2j (2j - 1)}\n\\Big ( {|k| \\over 2 \\pi \\rho} \\Big )^{2j}, \\quad |k| < 2 \\pi \\rho.\n$$\nRecalling (\\ref{6}) and (\\ref{pf}) this in turn implies\n\\begin{equation}\\label{cw}\n{1 \\over 2j(2j-1)} = {1 \\over 2} \\Big ( 2 \\tilde{a}_{2j-1, 0} +\n2 \\tilde{a}_{2j-1,1} + \n\\cdots + 2 \\tilde{a}_{2j-1,j-2} + \\tilde{a}_{2j-1,j-1} \\Big ).\n\\end{equation}\nIn the case $j=4$ we deduce from this equation, (\\ref{us1}), (\\ref{re4})\nand (\\ref{nt2}) that\n\\begin{equation}\\label{p7}\np_7(x) = (x-1) \\Big ( 1 - {1607 \\over 420} x +\n{2011 \\over 280} x^2 - {911 \\over 105} x^3 + {2011 \\over 280} x^4\n- {1607 \\over 420} x^5 + x^6 \\Big ).\n\\end{equation}\n\nConsider now the case $\\beta_0=4$. Due to $P_4$ in (\\ref{ny2})\nbeing a $2 \\times 2$ matrix, the calculation required to compute (\\ref{ny0})\nis more lengthy and tedius than in the case $\\beta_0=2$, although the\nthe common structure of $n$-point distributions means the two cases are\nanalogous. Some details are given in Appendix B. Our final expression\nfor $g(x_1,x_2;\\beta)$ is given by (\\ref{Bg}). We find its Fourier transform\ncan be computed explicitly in terms of elementary functions, together with\nthe dilogarithm\n\\begin{equation}\\label{dilog}\n{\\rm dilog}\\, (x) := \\int_1^x {\\log t \\over 1 - t} \\, dt.\n\\end{equation}\nExplicitly, with $\\rho = 1$ for notational convenience, up to terms\n$O((\\beta - 4)^2)$\n\\begin{equation}\\label{b4}\nS(k,\\beta) = S(k,4) + (\\beta - 4) \\Big ( - {\\pi \\over |k|} +\n\\hat{B}_0 (k) + 2 \\hat{B}_1(k) - 4 \\hat{B}_3(k) + \n2 \\hat{B}_5(k) + \\hat{B}_6(k) - \\hat{B}_7(k) \\Big )\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{B0}\n\\hat{B}_0(k) & = & - {3 \\over 2} + {3|k| \\over 8 \\pi} +\n{|k| \\over 4 \\pi} \\log \\Big ( {4 \\pi + |k| \\over |k|} \\Big )\n+ \\Big ( C + {1 \\over 2} \\log(16 \\pi^2 - k^2) \\Big ) \\Big ( 1 -\n{|k| \\over 4 \\pi} + {|k| \\over 8 \\pi} \\log \\Big | 1 -\n{|k| \\over 2 \\pi} \\Big | \\Big ) \\nonumber \\\\\n&& + {|k| \\over 16 \\pi} \\bigg ( {\\rm dilog} \\, \\Big (\n{|k| \\over 2 \\pi + |k|} \\Big ) -\n{\\rm dilog} \\, \\Big ( {4 \\pi + |k| \\over 2 \\pi + |k|} \\Big ) -\n\\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big |\n\\log \\Big ( {4 \\pi + |k| \\over |k|} \\Big ) + g_1(k) \\bigg ) \\nonumber \\\\\n&& + {2 \\pi - |k| \\over 8 \\pi} \\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big |,\n\\qquad |k| < 4 \\pi,\n\\end{eqnarray}\n\\begin{eqnarray}\n\\hat{B}_0(k) & = &\n{1 \\over 2} \\log \\Big ( {|k| + 4 \\pi \\over |k| - 4 \\pi} \\Big ) \n+ {|k| \\over 8 \\pi} \\log \\Big ( {k^2 - 16 \\pi^2 \\over k^2} \\Big )\n+ {|k| \\over 16 \\pi} \\bigg ( {\\rm dilog} \\,\n\\Big ( {|k| \\over |k| + 2 \\pi} \\Big ) \\nonumber \\\\ &&\n+\n{\\rm dilog} \\,\n\\Big ( {|k| \\over |k| - 2 \\pi} \\Big ) - {\\rm dilog} \\,\n\\Big ( {|k| + 4 \\pi \\over |k| + 2 \\pi} \\Big ) -\n{\\rm dilog} \\,\n\\Big ( {|k| - 4 \\pi \\over |k| - 2 \\pi} \\Big ) \\bigg ), \\qquad |k| > 4 \\pi,\n\\end{eqnarray}\n\\begin{equation}\n\\hat{B}_1(k) = \\left \\{\\begin{array}{ll}\n{\\pi \\over |k|} \\Big ( 1 - {|k| \\over 4 \\pi} + {|k| \\over 8 \\pi}\n\\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big | \\Big ), & |k| < 4 \\pi \\\\[.2cm]\n0, & |k| > 4 \\pi, \\end{array} \\right.\n\\end{equation}\n\\begin{eqnarray}\n\\hat{B}_3(k) & = & - {3 \\over 2} + {3 |k| \\over 8 \\pi} +\nC \\Big ( 1 - {|k| \\over 4 \\pi} + {|k| \\over 8 \\pi} \\log \\Big |\n1 - {|k| \\over 2 \\pi} \\Big | \\Big )\n+ \\Big ( {1 \\over 8} - {3|k| \\over 32 \\pi} \\Big )\n\\log \\Big |\n1 - {|k| \\over 2 \\pi} \\Big | \\nonumber \\\\&&\n+ {|k| \\over 64 \\pi} \\Big ( \\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big |\n\\Big )^2 + {1 \\over 8 \\pi} (4 \\pi - |k|)\n\\log (4 \\pi - |k|) - {1 \\over 8 \\pi} |k| \\log |k| +\n{1 \\over 2} \\log 4 \\pi \\nonumber \\\\\n&& + {|k| \\over 32 \\pi} \\bigg (\n{\\rm dilog} \\, \\Big ( {|k| \\over 2 \\pi + |k|} \\Big ) + {\\pi^2 \\over 12}\n- {\\rm dilog} \\,\\Big ({4 \\pi \\over 2 \\pi + |k|} \\Big )\n- {\\rm dilog} \\, \\Big ( {|k| \\over 2 \\pi} \\Big ) -\n{\\rm dilog} \\, \\Big ({4 \\pi - |k| \\over 2 \\pi} \\Big ) \\nonumber \\\\&&\n+2 \\log(2 \\pi) \\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big |\n+ \\log \\Big (2 \\pi + |k| \\Big ) \\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big |\n+ g_2(k) \\bigg ), \\quad |k| < 4 \\pi, \\nonumber \\\\\n\\hat{B}_3(k) & = & 0, \\qquad |k| > 4 \\pi,\n\\end{eqnarray}\n\\begin{equation}\n\\hat{B}_5(k) = \\left \\{ \\begin{array}{ll}\n\\hat{B}_3(k) - {|k| \\over 128 \\pi} \\Big ( \\log \\Big | \n1 - {|k| \\over 2 \\pi} \\Big |\n\\Big )^2 + {|k| \\over 32 \\pi} g_3(k), & |k| < 4 \\pi \\\\\n0, & |k| > 4 \\pi \\end{array} \\right.\n\\end{equation}\n\\begin{eqnarray}\n\\hat{B}_6(k) & = & - {3 \\over 2} + {3|k| \\over 8 \\pi} - {|k| \\over 16 \\pi}\n\\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big | +\n\\Big ( C + \\log (4 \\pi - |k|) \\Big ) \\Big ( 1 - {|k| \\over 4 \\pi}\n+ {|k| \\over 8 \\pi} \\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big | \\Big )\n\\nonumber \\\\\n&& + {|k| \\over 32 \\pi}\\bigg ( {\\pi^2 \\over 3} -\n{\\rm dilog} \\, \\Big ( {|k| \\over 2 \\pi} \\Big ) -\n\\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big | \\log \\Big ( {|k| \\over 2 \\pi}\n\\Big ) - 2 {\\rm dilog} \\,\\Big ( {4 \\pi - |k| \\over 2 \\pi} \\Big )\n\\nonumber \\\\ &&\n- 2 \\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big |\n\\log \\Big ( {4 \\pi - |k| \\over 2 \\pi} \\Big ) + g_4(k) \\bigg ), \\quad\n|k| < 4 \\pi, \\nonumber \\\\\n\\hat{B}_6(k) & = & 0, \\quad |k| > 4 \\pi,\n\\end{eqnarray}\n\\begin{equation}\\label{B7}\n\\hat{B}_7(k) = \\left \\{ \\begin{array}{ll}\n{\\pi \\over |k|} \\Big ( 1 - {|k| \\over 4 \\pi}\n+ {|k| \\over 8 \\pi} \\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big | \\Big )^2, &\n|k| < 4 \\pi \\\\\n0, & |k| > 4 \\pi, \\end{array} \\right.\n\\end{equation}\nwith\n\\begin{eqnarray}\\label{gg}\ng_1(k) & = & \\left \\{ \\begin{array}{ll}\n{\\rm dilog} \\Big ( {4 \\pi - |k| \\over 2 \\pi - |k|} \\Big ) -\n{\\rm dilog} \\Big ( {2 \\pi \\over 2 \\pi - |k|} \\Big ) - {\\pi^2 \\over 6}\n- \\log \\Big ( 1 - {|k| \\over 2 \\pi} \\Big ) \\log \\Big ( {4 \\pi - |k| \\over \n2 \\pi - |k|} \\Big ), & |k| < 2 \\pi \\\\[.2cm]\n{\\rm dilog} \\Big ( {|k| \\over |k| - 2 \\pi} \\Big ) -\n{\\rm dilog} \\Big ( {2 \\pi \\over |k| - 2 \\pi} \\Big ) - {\\pi^2 \\over 6}\n+ \\log \\Big ( {2 \\pi \\over |k| - 2\\pi} \\Big ) \\log \\Big (\n{|k| \\over |k| - 2 \\pi} \\Big ), & 2 \\pi < |k| < 4 \\pi, \\end{array}\n\\right. \\nonumber \\\\\ng_2(k) & = & \\left \\{ \\begin{array}{ll}\n{\\rm dilog} \\Big ( {4 \\pi - |k| \\over 2 \\pi - |k|} \\Big ) - {\\pi^2 \\over 6}\n+ \\log(2 \\pi - |k|) \\log \\Big (1 - {|k| \\over 2 \\pi} \\Big ),\n& |k| < 2 \\pi \\\\[.2cm]\n- {\\rm dilog} \\Big ( {2 \\pi \\over |k| - 2 \\pi} \\Big ) +\n\\log(4 \\pi - |k|) \\log \\Big ( {|k| \\over 2 \\pi} - 1 \\Big ),\n& 2 \\pi < |k| < 4 \\pi \\end{array} \\right.\n\\nonumber \\\\\ng_3(k) & = & \\left \\{ \\begin{array}{ll}\n{1 \\over 2} \\Big ( {\\pi^2 \\over 6} - 2 {\\rm dilog} \\Big (\n{2 \\pi \\over |k|} \\Big ) -\n\\log \\Big (\n{2 \\pi \\over |k|} \\Big ) \\log \\Big ( {(2 \\pi - |k|)^2 \\over 2 \\pi |k|}\n\\Big ) \\Big ), & |k| < 2 \\pi, \\\\[.2cm]\n{1 \\over 2} \\Big ( \n{\\rm dilog} \\Big ( {|k| - 2 \\pi \\over 2 \\pi} \\Big ) -\n{\\rm dilog} \\Big (\n{2 \\pi \\over |k|} \\Big ) +\n\\log \\Big ( {|k| - 2 \\pi \\over 2 \\pi} \\Big ) \\log \\Big ( {|k| \\over 2 \\pi}\n\\Big ) \\Big ), & 2 \\pi < |k| < 4 \\pi, \\end{array} \\right.\n\\nonumber \\\\\ng_4(k) & = & \\left \\{ \\begin{array}{l}\n- {\\pi^2 \\over 6} - {\\rm dilog} \\Big ( {2 \\pi \\over 2 \\pi - |k|} \\Big )\n+ 2 {\\rm dilog} \\Big ( {4 \\pi - |k| \\over 2 \\pi - |k|} \\Big )\n+ 2 \\log \\Big ( {2 \\pi \\over 2 \\pi - |k|} \\Big )\n\\log \\Big ( {4 \\pi - |k| \\over 2 \\pi - |k|} \\Big ), \\: \\: |k| < 2\\pi\\\\[.2cm]\n{\\rm dilog} \\Big ( {|k| \\over |k| - 2 \\pi} \\Big ) -\n2 {\\rm dilog} \\Big ( {2 \\pi \\over |k| - 2 \\pi} \\Big )\n+ \\log \\Big ( {2 \\pi \\over |k| - 2 \\pi} \\Big )\n\\log \\Big ( {|k| \\over |k| - 2 \\pi} \\Big ), \\: \\: \n2\\pi < |k| < 4 \\pi. \\end{array} \\right. \\nonumber \\\\\n\\end{eqnarray}\n\nThe above formula for $S(k;\\beta)$ in the case $|k| < 2 \\pi$ (recall here\n$\\rho = 1$) can be used to expand $\\partial S(k;\\beta) / \\partial \\beta$\nabout $k=0$. For this task we use computer algebra, which gives the\nresult\n\\begin{eqnarray}\\label{da}\n{\\partial S(k;\\beta) \\over \\partial \\beta} \\bigg |_{\\beta = 4} & = &\n- {|k| \\over 16 \\pi} + {|k|^3 \\over 256 \\pi^3} +\n{5 k^4 \\over 3072 \\pi^4} + {3 |k|^5 \\over 4096 \\pi^5} +\n{27 k^6 \\over 81920 \\pi^6} \\nonumber \\\\\n&& + {37 |k|^7 \\over 245760 \\pi^7} +\n{1273 k^8 \\over 18350080 \\pi^8} +\n{887 |k|^9 \\over 27525120 \\pi^9} + {4423 k^{10} \\over 293601280\n\\pi^{10}} \\nonumber \\\\\n&& + {1949 |k|^{11} \\over 275251200 \\pi^{11}} + \\cdots\n\\end{eqnarray}\nThis allows us to deduce a further equation for $\\{b_{8,j}\n\\}_{j=0,\\dots,4}$ and $\\{\\tilde{a}_{9,j}\\}_{j=0,\\dots,4}$, which in\ncombination with (\\ref{cw}), (\\ref{re4}), (\\ref{us1}), (\\ref{nt1}) and\n(\\ref{nt2}) implies\n\\begin{eqnarray}\np_8(x) & = & (x-1)^2 \\Big ( 1 - {263 \\over 84} x + {1697 \\over 315} x^2\n- {6337 \\over 1008} x^3 + {1697 \\over 315} x^4 - {263 \\over 84} x^5\n+ x^6 \\Big ) \\label{p8}\\\\\np_9(x) & = & (x-1) \\Big ( 1 - {791 \\over 180} x + \n{73603 \\over 7560} x^2 - {7355 \\over 504} x^3 +\n{2231 \\over 135} x^4 - {7355 \\over 504} x^5 +\n{73603 \\over 7560} x^6 - {791 \\over 180} x^7 + x^8 \\Big ) \\nonumber\n\\\\ \\label{p9}\n\\end{eqnarray}\n\n\\section{Conclusion}\n\\setcounter{equation}{0}\n\nCollecting together the evaluations (\\ref{6.2}), (\\ref{u6h}),\n(\\ref{p3}), (\\ref{g6c}), (\\ref{re4'}), (\\ref{p7}), (\\ref{p8})\nand (\\ref{p9}), and substituting in (\\ref{6}) we have that for\n$|k| < {\\rm min}\\,(2\\pi \\rho, \\pi \\beta \\rho)$\n\\begin{eqnarray}\\label{pr}\n\\lefteqn{\n{\\pi \\beta \\over |k|} S(k;\\beta) =} \\nonumber \\\\\n&&\n1 \\nonumber \\\\&&\n+ (x-1)y \\nonumber \\\\&&\n+ (x-1)^2y^2 \\nonumber \\\\&&\n+ (x-1) (x^2 - {11 \\over 6} x + 1) y^3 \\nonumber \\\\&&\n+ (x-1)^2(x^2 - {3 \\over 2} x + 1) y^4 \\nonumber \\\\&&\n+ (x-1) (x^4 - {91 \\over 30} x^3 + {62 \\over 15} x^2 - {91 \\over 30} x + 1)\ny^5 \\nonumber \\\\&&\n+ (x-1)^2(x^4 - {37 \\over 15} x^3 + {13 \\over 4} x^2 - {37 \\over 15} x + 1)\ny^6 \\nonumber \\\\&&\n+ (x-1)(x^6 - {1607 \\over 420} x^5 + {2011 \\over 280} x^4 -\n{911 \\over 105} x^3 + {2011 \\over 280} x^2 - {1607 \\over 420} x + 1) y^7\n\\nonumber \\\\&&\n+ (x-1)^2(x^6 - {263 \\over 84} x^5 + {1697 \\over 315} x^4 - {6337 \\over\n1008} x^3 + {1697 \\over 315} x^2 - {263 \\over 84} x + 1) y^8\n\\nonumber \\\\&&\n+ (x-1) ( x^8 - {791 \\over 180} x^7 + \n{73603 \\over 7560} x^6 - {7355 \\over 504} x^5 +\n{2231 \\over 135} x^4 - {7355 \\over 504} x^3 +\n{73603 \\over 7560} x^2 - {791 \\over 180} x + 1 ) y^9 \\nonumber \\\\&&\n+ O(y^{10})\n\\end{eqnarray}\nwhere $x=\\beta/2$ and $y=|k|/\\pi\\beta \\rho$. With the \ncoefficient of $y^j$ denoted $p_j(x)$ as has been throughout, we recall \nfrom our workings above that $p_0(x)$, $p_1(x)$, $p_2(x)$ and $p_4(x)$ have\nbeen calculated for general values of $\\beta$. In all other cases the\ncalculation has relied on the assumption that the $p_j(x)$ are indeed\npolynomials. On this point we remark that in such cases, excluding\n$j=8$ and 9, we have more data points than is necessary to uniquely\nspecify $p_j(x)$, assuming it is a polynomial, and our extra data points\nare consistent with the explicit forms presented in (\\ref{pr}).\n\nWe remark that the structure exhibited by (\\ref{pr}) is familiar from\nthe study of exactly solvable two-dimensional lattice models\n\\cite{Gu99}. In this field one encounters two-variable generating\nfunctions $G(x,y)$ say with series expansions of the form\n\\begin{equation}\\label{cny}\nG(x,y) = \\sum_{n=0}^\\infty H_n(x) y^n\n\\end{equation}\nin which $H_n(x)$ is a rational function, and furthermore the denominator\npolynomial in $H_n(x)$ only has a small number of (typically no more than\ntwo) distinct zeros. For example, the two-dimensional Ising model\nwith couplings $J_1$ ($J_2$) between bonds in the horizontal (vertical)\ndirection and $x:= \\exp(-4J_1/k_BT)$, $y:= \\exp(-4J_2/k_BT)$ has for its\nspontateous magnetisation the celebrated exact expression (see e.g.~\\cite{Ba82})\n\\begin{equation}\\label{cny1}\nM(x,y) = \\Big ( 1 - {16xy \\over (1-x)^2 (1-y)^2} \\Big )^{1/8}.\n\\end{equation}\nWhen written in the form (\\ref{cny}) one finds\n\\begin{equation}\\label{cny2}\nH_n(x) = {2x P_n(x) \\over (1-x)^n}\n\\end{equation}\nwhere $P_n(x)$ is a polynomial of degree $2n-2$ which satisfies the\nfunctional relation\n\\begin{equation}\\label{cny3}\nP_n(x) = x^{2n-2} P_n(1/x).\n\\end{equation}\nAs emphasized in \\cite{Gu99}, the exact solution (\\ref{cny1}) can be\nuniquely determined by the functional form (\\ref{cny2}), together with\nthe functional (inversion) relation (\\ref{cny3}) and the symmetry\nrelation $M(x,y) = M(y,x)$. For the structure function of the log-gas\nwe have no analogue of the symmetry relation and so cannot\ncharacterize (\\ref{d2}) this way.\n\nOne immediate feature of the polynomials $p_j(x)$ in (\\ref{pr}) is that\nfor $j$ even the polynomial $p_j(-x)$ has all coefficients positive, while\nfor $j$ odd the polynomial $p_j(-x)$ has all coefficients negative.\nAnother general feature of the $p_j(x)$ in (\\ref{pr}), obtained from\nnumerical computation, is that all the zeros lie on the unit circle in\nthe complex $x$-plane.\nThis can be rigorously determined numerically because\nthe symmetry (\\ref{6.1}) implies that if $x_0$ is a zero of\n$p_j(x)$, then so is $1/x_0$, which will be the complex conjugate of\n$x_0$ if and only if $|x_0|=1$.\n\nThe quantum many body interpretation of (\\ref{1}) allows us to give\na physical interpretation to the functional relation\n(\\ref{5}). As the functional relation\nis derived from the integral representation\n(\\ref{d3}), it is appropriate to recall \\cite{Ha95} the physical\ninterpretation of that formula. In (\\ref{d3}), with $\\beta/2 = p/q$,\nthere are $q$ integrals over $x_i \\in (0,\\infty)$ and $p$\nintegrals over $y_j \\in (0,1)$. The variables $x_i$ can be interpreted as\nbeing rapidities of quasi-particle excitations, while the $y_j$ are\nrapidities of quasi-hole excitations. Thus the transformation\n$\\beta \\mapsto 4/\\beta$ is equivalent to interchanging $p$ and $q$\nand thus the quasi-holes and quasi-particles. In (\\ref{d3}) this does not\nlead to an integral of the same functional form as before;\nalthough the functional form of the integrand is conserved, apart\n from a renormalization of $k$, the\ndomain of integration is different for $\\{x_i\\}$ and $\\{y_j\\}$. But\nwith $k$ restricted as in (\\ref{d2a}) both sets of variables can take\nany value in $(0,\\infty)$. The quasi-particles and quasi-holes play an\nidentical role and the functional equation results.\n\nIt is of interest to consider the small $k$ expansion of\n$S(k;\\Gamma)$, $\\Gamma := q^2/k_BT$ ($q$ = charge), for the two-dimensional\none-component plasma. As mentioned earlier, this has recently been the\nobject of study of Kalinay et al.~\\cite{KMST99}. They obtain results which\nimply\n\\begin{equation}\\label{sos}\n{2 \\pi \\Gamma \\over k^2} S(k;\\Gamma) = 1 +\n({\\Gamma \\over 4}-1) {k^2 \\over 2 \\pi \\Gamma \\rho} +\n({\\Gamma \\over 4} - {3 \\over 2})( {\\Gamma \\over 4} - {2 \\over 3})\n\\Big ( {k^2 \\over 2 \\pi \\Gamma \\rho} \\Big )^2 + O(k^6).\n\\end{equation}\nwhere $k := |\\vec{k}|$. The structure of (\\ref{sos}) bears a striking\nresemblence to (\\ref{pr}) with $\\Gamma/4$ corresponding to $x$ and\n$k^2/2\\pi \\Gamma \\rho$ to $y$. In particular with $g(x,y) :=\n(2\\pi \\Gamma / k^2) S(k;\\Gamma)$, the expansion (\\ref{sos}) to the\ngiven order is such that\n\\begin{equation}\\label{sos1}\ng(x,y) = g( {1 \\over x}; - yx).\n\\end{equation}\nFurthermore, writing\n\\begin{equation}\\label{sos2}\ng(x,y) = 1 + \\sum_{l=1}^\\infty u_{l}(x) y^{l}\n\\end{equation}\nwe have $u_1(x) = (x-1)$, $u_2(x) = (x-3/2)(x-2/3)$ so $u_j(x)$ is a\nmonic $j$th degree polynomial for $j \\le 2$. However we can demonstrate that\nthis analogy breaks down for the $l=3$ term in (\\ref{sos2}).\n\nTo demonstrate this fact, suppose instead that the functional equation\n(\\ref{sos2}) was valid at order $l=3$ in (\\ref{sos2}) and $u_3(x)$ is\na monic polynomial. Then $u_3$ must be of the form\n\\begin{equation}\\label{sos3}\nu_3(x) = (x-1)(x^2 + ax + 1).\n\\end{equation}\nFrom the definition of $g(x,y)$ we can check that this is equivalent to\nthe statement that\n\\begin{equation}\\label{sos4}\n{1 \\over \\rho} \\Big ( {\\pi \\Gamma \\rho \\over 2} \\Big )^4\n\\int_{{\\bf R}^2} {r}^8 S({r};\\Gamma) \\, d\\vec{r} =\n(4!)^2(x-1)(x^2 + ax + 1).\n\\end{equation}\nBut as noted in \\cite{KMST99}, it follows from the perturbation expansion\nof \\cite{Ja81} that\n\\begin{eqnarray}\\label{sos5}\n{1 \\over \\rho} \\Big ( {\\pi \\Gamma \\rho \\over 2} \\Big )^4\n\\int_{{\\bf R}^2} {r}^8 S({r};\\Gamma) \\, d\\vec{r} & = &\n- 4! + (\\Gamma - 2)4! \\Big ( \\sum_{k=0}^4\n{2^k - 1 \\over k+1} - 2 \\Big ) + O((\\Gamma - 2)^2) \\nonumber \\\\\n& = & - 4! + (\\Gamma - 2)4! {17 \\over 4} + O((\\Gamma - 2)^2).\n\\end{eqnarray}\nThe term in (\\ref{sos5}) proportional to $\\Gamma - 2$ is incompatible\nwith (\\ref{sos4}) which gives instead\n$$\n(\\Gamma - 2)4! {18 \\over 4}\n$$\nindependent of the value of $a$.\nIndeed in \\cite{KMST99} evidence is presented which indicates\n$u_3(x)$ is an infinite series in $x$, although we have no way of\ndetermining if the functional equation (\\ref{sos1}) also breaks down at\nthis order.\n\n\\section*{Acknowledgements}\nThe work of PJF and DSM was supported by the Australian Research Council.\n\n\\setcounter{equation}{0}\n\\setcounter{section}{1}\n\\renewcommand{\\thesection}{\\Alph{section}}\n\\section*{Appendix A}\nIn this appendix some details of the derivation of (\\ref{af0}) and (\\ref{af1})\nwill be given. To simplify notation we take $\\rho =1$ throughout. The first\nstep is to substitute (\\ref{ny1}) and (\\ref{ny0}) and simplify by\nexpanding out the determinant and cancelling terms where possible. This\nshows that up to terms $ O((\\beta - 2)^2)$\n\\begin{eqnarray}\\label{fi1}\n\\lefteqn{\ng_2(x_1,x_2;\\beta)} \\nonumber \\\\ && = 1 - \\Big ( P_2(x_1,x_2) \\Big )^2 +\n(\\beta - 2) \\bigg \\{ - (1 - (P_2(x_1,x_2))^2) \\Phi(x_1,x_2) \\nonumber \\\\&&\n- 2 \\int_{-\\infty}^\\infty \\Big ( - (P_2(x_2,x_3))^2 -\n(P_2(x_1,x_3))^2 + 2 P_2(x_1,x_2) P_2(x_2,x_3) P_2(x_3,x_1) \\Big )\n\\Phi (x_1,x_3) \\, dx_3 \\nonumber \\\\&&\n- {1 \\over 2} \\int_{-\\infty}^\\infty \\Big (\n4 P_2(x_1,x_3) P_2(x_3,x_4)P_2(x_4,x_1) -\n4 P_2(x_1,x_2) P_2(x_2,x_3) P_2(x_3,x_4) P_2(x_4,x_1) \\nonumber \\\\&&\n- 2P_2(x_1,x_3) P_2(x_3,x_2) P_2(x_2,x_4) P_2(x_4,x_1) +\n2 \\Big ( P_2(x_1,x_3) \\Big )^2 \\Big (P_2(x_2,x_4) \\Big )^2 \\Big )\n\\Phi(x_3,x_4) \\, dx_3 dx_4 \\bigg \\} . \\nonumber \\\\\n\\end{eqnarray}\nThe convolution structure\n$$\n\\int_{-\\infty}^\\infty f(y_1 - x) g(x - y_2) \\, dx\n$$\noften occurs in the above integrals. Such an integral can be transformed\nby introducing the Fourier transforms $\\hat{f}$ ($\\hat{g}$) according\nto the formula\n\\begin{equation}\\label{fi1'}\n\\int_{-\\infty}^\\infty f(y_1 - x) g(x - y_2) \\, dx =\n{1 \\over 2 \\pi} \\int_{-\\infty}^\\infty \\hat{f}(l) \\hat{g}(l)\ne^{-il(y_1 - y_2)} \\, dl.\n\\end{equation}\nMaking use of this formula typically leads to simplifications.\n\nFor example, consider the first integral in (\\ref{fi1}). Starting with\nthe Fourier transform\n\\begin{equation}\\label{fi2}\n\\int_{-\\infty}^\\infty {\\sin^2 \\pi x \\over (\\pi x)^2} e^{ikx} \\, dx\n= \\left \\{ \\begin{array}{ll} 1 - {|k| \\over 2 \\pi}, & |k| < 2 \\pi \\nonumber \n\\\\[.1cm]\n0, & |k| > 2 \\pi \\end{array} \\right.\n\\end{equation}\nand (\\ref{fte}), application of (\\ref{fi1'}) gives\n\\begin{equation}\\label{fi3}\nA_1(x_{12}) := \\int_{-\\infty}^\\infty \\Big ( P_2(x_2,x_3) \\Big )^2\n\\Phi (x_3,x_1) \\, dx_3 =\n\\int_{-2 \\pi}^{2 \\pi} \\Big ( 1 - {|k| \\over 2 \\pi} \\Big )\n{\\pi \\over |k|} \\cos k x_{12} \\, {dk \\over 2 \\pi}.\n\\end{equation}\nThis expression is indeed simpler than the original, but it suffers from\nbeing ill-defined, due to the singularity at the origin. However its\nderivative is well-defined, and can furthermore be evaluated in terms\nof elementary functions giving\n\\begin{equation}\\label{fi4}\n{d \\over dx} A_1(x) = {\\sin 2 \\pi x \\over 2 \\pi x^2} - {1 \\over x}.\n\\end{equation}\nAlso, we have \\cite{GR80}\n\\begin{equation}\\label{fi4'}\nA_1(0) = - \\int_{-\\infty}^\\infty dx \\,\n{\\sin^2 \\pi x \\over (\\pi x)^2} \\log |x| = C + \\log 2 \\pi - 1,\n\\end{equation}\nwhere $C$ denotes Euler's constant. Together (\\ref{fi4}) and (\\ref{fi4'})\nimply\n\\begin{equation}\\label{fi5}\nA_1(x) = - {\\sin 2 \\pi x \\over 2 \\pi x} + {\\rm ci} \\, (2 \\pi x) \n- \\log |x|\n\\end{equation}\nwhere ci$(x)$ denotes the cosine integral (\\ref{ci}). \n\nThe other six integrals in (\\ref{fi1}) yield to similar techniques. We\nfind\n\\begin{eqnarray}\\label{fi7}\nA_2 & := & \\int_{-\\infty}^\\infty \\Big ( P_2(x_1,x_3) \\Big )^2\n\\Phi(x_1,x_3) \\, dx_3 \\: = \\: A_1(0) \\nonumber \\\\\nA_3(x_{12}) & := & \\int_{-\\infty}^\\infty P_2(x_2,x_3) P_2(x_3,x_1)\n\\Phi (x_3,x_1) \\, dx_3 \\: = \\:\n\\int_{-\\pi}^\\pi \\Big ( C + \\log(\\pi + k) \\Big ) \\cos kx_{12} \\, {dk \\over 2 \\pi}\n\\nonumber \\\\\n& = & {1 \\over 2} \\Big ( C + \\log 2 \\pi - \\log |x_{12}| + {\\rm ci} \\, \n(2 \\pi x_{12})\n\\Big ) {\\sin \\pi x_{12} \\over \\pi x_{12}}\n- {1 \\over 2} \\Big ( {\\rm si} \\, (2 \\pi x_{12}) + {\\pi \\over 2} \\Big )\n{\\cos \\pi x_{12} \\over \\pi x_{12}} \\nonumber \\\\\nA_4 & := & \\int_{-\\infty}^\\infty P_2(x_1,x_3) P_2(x_3,x_4) P_2(x_4,x_1)\n\\Phi (x_3,x_4) \\, dx_3 dx_4 \\: = \\: A_3(0) \\: = \\: A_1(0), \\nonumber \\\\\nA_5(x_{12}) & := & \n\\int_{-\\infty}^\\infty P_2(x_2,x_3) P_2(x_3,x_4) P_2(x_4,x_1)\n\\Phi (x_3,x_4) \\, dx_3 dx_4 \\: = \\: A_3(x_{12}), \\nonumber \\\\\nA_6(x_{12}) & := & \n\\int_{-\\infty}^\\infty P_2(x_1,x_3) P_2(x_3,x_2) P_2(x_2,x_4)\nP_2(x_4,x_1) \\Phi(x_3,x_4)\\, dx_3 dx_4 \\nonumber \\\\\n& = & {1 \\over 2 (\\pi x_{12})^2} \\Big ( C + \\log 2 \\pi +\n\\cos 2 \\pi x_{12} \\Big ( \\log |x_{12}| - {\\rm ci} \\, (2 \\pi x_{12}) \\Big )\n- \\sin 2 \\pi x_{12}\n \\Big ( {\\rm si} \\, (2 \\pi x_{12}) + {\\pi \\over 2} \\Big ) \\Big ),\n\\nonumber \\\\\nA_7(x_{12}) & := & \\int_{-\\infty}^\\infty\n\\Big ( P_2(x_1,x_3) \\Big )^2 \\Big ( P_2(x_2,x_4) \\Big )^2\n\\Phi (x_3, x_4) \\, dx_3 dx_4 \\: = \\:\n\\int_{-2 \\pi}^{2 \\pi} {dk \\over 2 \\pi} \\,\n\\Big ( 1 - {|k| \\over 2 \\pi} \\Big )^2 {\\pi \\over |k|} \\cos k x_{12}\n\\nonumber \\\\\n& = & - \\log |x_{12}| - {1 - \\cos 2 \\pi x_{12} \\over (2 \\pi x_{12})^2}\n- {\\sin 2 \\pi x_{12} \\over 2 \\pi x_{12}} + {\\rm ci} \\, (2 \\pi x_{12}),\n\\end{eqnarray}\nwhere si$(x)$ denotes the sine integral defined in (\\ref{si}).\n\nOf the results (\\ref{fi7}), the evaluation of $A_6$ is the most difficult,\nso it is appropriate to give details in that case also. We observe that\n$A_6$ consists of the convolution of $P_2(x_1,x_3) P_2(x_3,x_2)$\nregarded as a function of $x_3$, and $\\Phi(x_3,x_4)$, and\n$P_2(x_4,x_1) P_2(x_2,x_4)$ regarded as a function of $x_4$. It\nsimplifies the calculation to take as the origin in both integrations\nthe centre of the interval between particle 1 and particle 2, which is\nachieved by the change of variables $x_3 \\mapsto x_3 + (x_1 + x_2)/2$,\n$x_4 \\mapsto x_4 + (x_1 + x_2)/2$. Use of (\\ref{fi1'}) then shows\n\\begin{equation}\\label{se1}\nA_6(x_{12}) = \\int_{-\\infty}^\\infty {dk \\over 2 \\pi} \\,\n\\Big ( \\hat{V}(k,x_{12}) \\Big )^2 {\\pi \\over |k|},\n\\end{equation}\n\\begin{equation}\\label{se2}\n \\hat{V}(k,x_{12}) := \\int_{-\\infty}^\\infty {\\sin \\pi x_{13} \\over \\pi x_{13}}\n{\\sin \\pi x_{32} \\over \\pi x_{32}} \\cos k x_3 \\, dx_3 =\n\\left \\{ \\begin{array}{ll} {1 \\over \\pi x_{12}} \\sin \n\\Big ( \\pi - {|k| \\over 2} \\Big ) x_{12}, & |k| < 2 \\pi \\\\[.2cm]\n0, & |k| > 2 \\pi \\end{array} \\right.\n\\end{equation}\nwhere the second equality in (\\ref{se2}) follows after further use of\n(\\ref{fi1'}). Thus\n\\begin{equation}\\label{se3}\nA_6(x) = \\int_{-2\\pi}^{2\\pi} {dk \\over 2 \\pi} \\,\n\\Big ( {\\sin ( \\pi - |k| / 2)x \\over \\pi x}\n\\Big )^2 {\\pi \\over |k|}.\n\\end{equation}\nAs in (\\ref{se3}), this integrand is ill-defined. To proceed further, we write\n$$\nA_6(x) = A_6^{(1)}(x) + A_6^{(2)}(x)\n$$\nwhere\n\\begin{eqnarray*}\nA_6^{(1)}(x) & = & \\int_{-\\infty}^\\infty {dk \\over 2 \\pi} \\,\n\\bigg \\{ \\Big ( {\\sin ( \\pi - |k| / 2)x \\over \\pi x}\n\\Big )^2 - \\Big ( {\\sin \\pi x \\over \\pi x} \\Big )^2 \\bigg \\}\n{\\pi \\over |k|} \\nonumber \\\\\nA_6^{(2)}(x) & = &\n\\Big ( {\\sin \\pi x \\over \\pi x} \\Big )^2 \\int_{-2 \\pi}^{2 \\pi}\n{dk \\over 2 \\pi} \\, {\\pi \\over |k|}.\n\\end{eqnarray*}\nThe integral defining $A_6^{(1)}$ is well defined and can be computed\nby elementary means. The integral defining $A_6^{(2)}$ is singular. It\ncoincides with the singular part of $A_1(0)$ (recall (\\ref{fi3})), and\nso from (\\ref{fi4'}) we have\n$$\nA_6^{(1)}(0) = \\Big ( {\\sin \\pi x \\over \\pi x} \\Big )^2 \n( C + \\log 2 \\pi ).\n$$\n\nCollecting together the above evaluations of $A_1$--$A_7$ and substituting\nas appropriate in (\\ref{fi1}) gives (\\ref{af0}).\n\n The next task is to evaluate\nthe Fourier transform. Now the evaluations of $A_1$ and $A_7$ are given\nas Fourier integrals, so their Fourier transform is immediate:\n\\begin{eqnarray}\\label{a1a7}\n{\\rm FT} \\, A_1(x) & = & {\\pi \\over |k|} - {1 \\over 2}, \\quad |k| < 2\\pi\n \\nonumber \\\\\n{\\rm FT} \\, A_7(x) & = & {\\pi \\over |k|} - 1 + {|k| \\over 4 \\pi}, \n\\quad |k| < 2\\pi,\n\\end{eqnarray}\nwhile for $|k| > 2 \\pi$\n\\begin{equation}\\label{a1a7'}\n{\\rm FT} \\, A_1(x) = {\\rm FT} \\, A_7(x) = 0.\n\\end{equation}\nWe can check that the constants\n$A_2$ and $A_4$ cancel when substituted in (\\ref{fi1}), and so play\nno further part in the calculation.\n\nOf the remaining terms, consider first the first term proportional\nto $\\beta - 2$ in (\\ref{fi1}), $A_0(x)$ say. Making use of (\\ref{fi1'}) we\nsee that\n\\begin{equation}\\label{ii}\n{\\rm FT} \\, A_0(x) = - {\\pi \\over |k|} + \n\\int_{-2\\pi}^{2\\pi} {dl \\over 2 \\pi} \\,\n{\\pi \\over |l-k|} \\Big ( 1 - {|l| \\over 2 \\pi} \\Big ).\n\\end{equation}\nFor $|k| < 2 \\pi$ minor manipulation allows the singular part\n\\begin{equation}\\label{a0a}\n\\int_{-2\\pi}^{2 \\pi} {dl \\over 2 \\pi} \\, {\\pi \\over |l|} = C +\n\\log 2 \\pi\n\\end{equation}\nto be separated, while the remaining convergent integrals are elementary.\nWe thus find that for $|k| < 2 \\pi$\n\\begin{equation}\\label{iia}\n{\\rm FT} \\, A_0(x) = - {\\pi \\over |k|} + \\Big \\{ C + \\log 2 \\pi\n+ {1 \\over 2} \\log \\Big ( 1 - \\Big ( {k \\over 2 \\pi} \\Big )^2 \\Big )\n\\Big \\} \\Big ( 1 - {|k| \\over 2 \\pi} \\Big ) -1 + {|k| \\over 2 \\pi}\n+ {|k| \\over 2 \\pi} \\log \\Big ( {2 \\pi + |k| \\over |k|} \\Big ).\n\\end{equation}\nFor $|k| > 2 \\pi$ the integrals in (\\ref{ii}) are convergent and also\nelementary. In this case we find\n\\begin{equation}\\label{a0b}\n{\\rm FT} \\, A_0(x) = - {\\pi \\over |k|} + \n{1 \\over 2} \\log {|k| +2 \\pi \\over |k| - 2 \\pi} +\n{|k| \\over 4 \\pi} \\log \\Big ( 1 - {4 \\pi^2 \\over k^2} \\Big ).\n\\end{equation}\n\nTo compute the Fourier transform of $A_6$, we begin by making use of\n(\\ref{fi1'}) in (\\ref{se1}) thereby obtaining\n$$\n{\\rm FT} \\, A_6 = \\int_{-\\infty}^\\infty {dl \\over 2 \\pi}\n\\int_{-\\infty}^\\infty {dk_1 \\over 2 \\pi} \\, \\hat{V}(l,k_1)\n\\hat{V}(l,-(k_1-k)) {\\pi \\over |l|}\n$$\nwhere\n$$\n\\hat{V}(l,k) := \\int_{-\\infty}^\\infty dx \\, \\hat{V}(l,x) e^{ikx} =\n\\chi_{|k| < \\pi - |l|/2}\n$$\nwith the equality in the latter formula following from the explicit form\n(\\ref{se2}) of $\\hat{V}(l,x)$ and then computation of the resulting integral,\nand where $\\chi_T = 1$ for $T$ true and $\\chi_T = 0$ otherwise. Thus\n\\begin{eqnarray}\\label{a6}\n{\\rm FT} \\, A_6(x) & = & \\int_{-\\infty}^\\infty {dl \\over 2 \\pi}\n\\int_{-\\infty}^\\infty {dk_1 \\over 2 \\pi} \\,\n\\chi_{|k_1| < \\pi - |l|/2} \\chi_{|k_1 - k| < \\pi - |l|/2} \n{\\pi \\over |l|} \\nonumber \\\\\n& = & \\left \\{ \\begin{array}{ll}\n\\Big ( 1 - {|k| \\over 2 \\pi} \\Big ) ( C + \\log 2 \\pi ) -\n \\Big ( 1 - {|k| \\over 2 \\pi} \\Big ) \\log {2 \\pi \\over 2 \\pi - |k|}\n- {1 \\over 2 \\pi} (2 \\pi - |k|), & |k| < 2 \\pi \\\\\n0, & |k| > 2 \\pi \\end{array} \\right. \\nonumber \\\\\n\\end{eqnarray}\nwhere use has been made of the generalized integral evaluation (\\ref{a0a}).\n\n\nThe final Fourier transform to consider is\n\\begin{eqnarray*}\\lefteqn{{\\rm FT} \\, {\\sin \\pi x \\over \\pi x} A_3(x) =\n{\\rm FT} \\, {\\sin \\pi x \\over \\pi x}\n\\int_{-\\pi}^\\pi {dk_1 \\over 2 \\pi} \\,\n\\Big ( C + {1 \\over 2} \\log (\\pi + k_1) + {1 \\over 2} \\log (\\pi - k_1)\n\\Big ) e^{ik_1x}} \\\\ &&\n= {1 \\over 2 \\pi} \\int_{-\\infty}^\\infty dl \\,\n\\chi_{l \\in [-\\pi,\\pi]} \\chi_{l \\in [-\\pi+k,\\pi+k]}\n\\Big ( C + {1 \\over 2} \\log (\\pi + l-k) + {1 \\over 2} \\log (\\pi - (l-k))\n\\Big )\n\\end{eqnarray*}\nwhere to obtain the equality use has been made of (\\ref{fi1'}).\nEvaluating the integral gives\n\\begin{equation}\\label{a3}\n{\\rm FT} \\, {\\sin \\pi x \\over \\pi x} A_3(x) = \n(C + \\log 2 \\pi) \\Big ( 1 - {|k| \\over 2 \\pi} \\Big ) - {|k| \\over 4 \\pi}\n\\log {|k| \\over 2 \\pi} + {1 \\over 2} \\Big ( 1 - {|k| \\over 2 \\pi} \\Big )\n\\log \\Big ( 1 - {|k| \\over 2 \\pi} \\Big )\n\\end{equation}\nfor $|k| < 2 \\pi$, while for $|k| > 2 \\pi$ \n\\begin{equation}\\label{a3'}\n{\\rm FT} \\, {\\sin \\pi x \\over \\pi x} A_3(x) = 0.\n\\end{equation}\n\nSubstituting the above results as appropriate in the Fourier transform of\n(\\ref{fi1}) gives the result (\\ref{af1}).\n\n\n\n\\setcounter{equation}{0}\n\\setcounter{section}{2}\n\\renewcommand{\\thesection}{\\Alph{section}}\n\\section*{Appendix B}\nIn this appendix we outline some details of the calculation of (\\ref{ny0})\nin the case $\\beta_0=4$ and show how this leads to (\\ref{b4}). Because\n(\\ref{ny1}) and (\\ref{ny2}) formally have the same structure upon expansion\n(recall the definition of qdet (\\ref{qdet})), the formula (\\ref{fi1})\nformally maintains its structure when generalized to the case $\\beta_0=4$.\nThus we have\n\\begin{eqnarray}\\label{gi1}\n\\lefteqn{\ng_2(x_1,x_2;\\beta)} \\nonumber \\\\ && = \n1 - \\Big ( P_4(x_1,x_2) P_4(x_2,x_1) \\Big )^{(0)} +\n(\\beta - 4) \\bigg \\{ - \\Big (1 - \n(P_4(x_1,x_2) P_4(x_2,x_1))^{(0)} \\Big ) \\Phi(x_1,x_2) \\nonumber \\\\&&\n- 2 \\int_{-\\infty}^\\infty \\Big ( - (P_4(x_2,x_3) P_4(x_3,x_2))^{(0)} -\n(P_4(x_1,x_3) P_4(x_3,x_1))^{(0)} \\nonumber \\\\&& + \n2 (P_4(x_1,x_2) P_4(x_2,x_3) P_4(x_3,x_1))^{(0)} \\Big )\n\\Phi (x_1,x_3) \\, dx_3 \\nonumber \\\\&&\n- {1 \\over 2} \\int_{-\\infty}^\\infty \\Big (\n4 (P_4(x_1,x_3) P_4(x_3,x_4)P_4(x_4,x_1))^{(0)} -\n4 (P_4(x_1,x_2) P_4(x_2,x_3) P_4(x_3,x_4) P_4(x_4,x_1))^{(0)} \\nonumber \\\\&&\n- 2(P_4(x_1,x_3) P_4(x_3,x_2) P_4(x_2,x_4) P_4(x_4,x_1))^{(0)} \\nonumber \\\\&&\n+ 2( P_4(x_1,x_3) P_4(x_3,x_1))^{(0)} (P_4(x_2,x_4)P_4(x_4,x_2))^{(0)} \\Big )\n\\Phi(x_3,x_4) \\, dx_3 dx_4 \\bigg \\} +O((\\beta - 4)^2). \n\\end{eqnarray}\nWe treat each of the seven distinct integrals in (\\ref{gi1}) in an\nanalogous way to their counterparts in (\\ref{fi1}), although extra working\nis involved due to $P_4$ being a matrix rather than a scalar.\n\nThe final results are\n\\begin{eqnarray}\\label{B}\nB_1(x_{12}) & := & \\int_{-\\infty}^\\infty\n\\Big ( P_4(x_2,x_3) P_4(x_3,x_2) \\Big )^{(0)} \\Phi(x_1,x_3) \\, dx_3 \n\\nonumber \\\\\n& = & \\int_{-4\\pi}^{4\\pi} {dk \\over 2 \\pi} \\,\n\\Big ( 1 - {|k| \\over 4 \\pi} + {|k| \\over 8 \\pi}\n\\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big | \\Big ) {\\pi \\over |k|}\n\\cos k x_{12} \\nonumber \\\\\n& = & - \\log |x_{12}| - {\\sin 4 \\pi x_{12} \\over 4 \\pi x_{12}}\n+{\\rm ci} \\, (4 \\pi x_{12}) - {\\cos 2 \\pi x_{12} \\over 4 \\pi x_{12}}\n{\\rm Si} \\, (2 \\pi x_{12}) \\nonumber \\\\\nB_2 & := & \\int_{-\\infty}^\\infty\n\\Big ( P_4(x_1,x_3) P_4(x_3,x_1) \\Big )^{(0)} \\Phi(x_1,x_3) \\, dx_3 \n\\: = \\: B_1(0) \\: = \\: C + \\log 4 \\pi - {3 \\over 2} \\nonumber \\\\\nB_3(x_{12}) & := & \\int_{-\\infty}^\\infty\nP_4(x_2,x_3) P_4(x_3,x_1) \\Phi(x_1,x_3) \\, dx_3 \n\\nonumber \\\\\n& = &\n\\left [ \\begin{array}{cc}\n{1 \\over 4} f_1(x_{12}) - {1 \\over 4} f_3(x_{12}) &\n- {1 \\over 4} \\int_0^{x_{12}} (f_1(t) + f_2(t)) \\, dt \\nonumber \\\\\n- {1 \\over 4}f_1'(x_{12}) + {1 \\over 4} f_3'(x_{12}) &\n{1 \\over 4} f_1(x_{12}) + {1 \\over 4} f_2(x_{12}) \\end{array} \\right ]\n\\nonumber \\\\\nB_4 & := & \\int_{-\\infty}^\\infty \n\\Big ( P_4(x_1,x_3) P_4(x_3,x_4) \nP_4(x_4,x_1) \\Big )^{(0)} \\Phi(x_1,x_3) \\, dx_3 dx_4 \n\\: = \\: B_1(0) \\nonumber \\\\\nB_5(x_{12}) & := & \\int_{-\\infty}^\\infty\nP_4(x_2,x_3) P_4(x_3,x_4) P_4(x_4,x_1) \\Phi(x_3,x_4) \\, dx_3 dx_4\n\\nonumber \\\\\n& = &\n\\left [ \\begin{array}{cc}\n{1 \\over 4} f_1(x_{12}) + {1 \\over 8} f_2(x_{12}) \n- {1 \\over 8} f_3(x_{12}) &\n- \\int_0^{x_{12}} ({1 \\over 4}f_1(t) + {1 \\over 8} f_2(t)-\n{1 \\over 8} f_3(t)) \\, dt \\nonumber \\\\\n- ({1 \\over 4}f_1'(x_{12}) + {1 \\over 8} f_2'(x_{12})\n- {1 \\over 8} f_3'(x_{12}))\n &\n{1 \\over 4} f_1(x_{12}) + {1 \\over 8} f_2(x_{12})\n- {1 \\over 8} f_3(x_{12}) \\end{array} \\right ]\n\\nonumber \\\\\nB_6(x_{12}) & := & \\int_{-\\infty}^\\infty \\Big (\nP_4(x_1,x_3) P_4(x_3,x_2) P_4(x_2,x_4)P_4(x_4,x_3)\n\\Big )^{(0)} \\Phi(x_3,x_4) \\, dx_3 dx_4\n\\nonumber \\\\\n&=& \\int_{-4\\pi}^{4\\pi} {1 \\over 2 |k|} \\bigg \\{ (g_1(k,x_{12}))^2\n+ \\cos(kx_{12}/2)g_1(k,x_{12})\\Big (\\frac{|k|}{4\\pi}g_2(k,x_{12})\n-\\frac{ik}{4\\pi}g_3(k,x_{12})\\Big ) \\nonumber \\\\\n&& + \\cos(kx_{12})\\left(\\frac{|k|}{8\\pi}g_2(k,x_{12})\n-\\frac{ik}{8\\pi}g_3(k,x_{12})\\right)^2\n-\\Big (\\frac{\\sin(|k|x_{12}/2)}{4\\pi}g_2(k,x_{12})\n\\nonumber \\\\&& +\\frac{\\cos(kx_{12}/2)}{4\\pi}g_3(k,x_{12})\\Big ) \n\\Big (\\frac{(4\\pi-|k|)\\cos((2\\pi -|k|/2))x_{12})}{4\\pi x_{12}}\n\\nonumber \\\\&&\n-\\frac{\\sin((2\\pi-|k|/2))x_{12})}\n{2\\pi x_{12}^2}\\Big ) \\bigg \\} \\,dk \\nonumber \\\\\nB_7(x_{12}) & := & \\int_{-\\infty}^\\infty \\Big (\nP_4(x_1,x_3) P_4(x_3,x_1) \\Big )^{(0)} \\Big ( P_4(x_2,x_4)P_4(x_4,x_2)\n\\Big )^{(0)} \\Phi(x_3,x_4) \\, dx_3 dx_4\n\\nonumber \\\\\n&=& \n\\int_{-4\\pi}^{4\\pi} \\left(1-\\frac{|k|}{4\\pi}\n+\\frac{|k|}{8\\pi}\\ln\\left|1-\\frac{|k|}{2\\pi}\\right|\\right)^2 \\frac{\\pi}{|k|}\n\\cos kx_{12} \\frac{dk}{2\\pi} \\nonumber \\\\\n& = & -\\log |x| - \\frac{\\sin 4\\pi x}{4\\pi x} + {\\rm ci} \\,(4\\pi x)\n- \\frac{{\\rm Si} \\, (2\\pi x) \\sin 2\\pi x}{8\\pi^2x^2}\n-\\frac{{\\rm Si} (2\\pi x) \\cos 2\\pi x}{4\\pi x} \\nonumber \\\\\n&& + \\int_{-4\\pi}^{4\\pi} |k| \\log \\Big |1-\\frac{|k|}{2\\pi}\\Big |^2 \\cos kx\n\\frac{dk}{128\\pi^2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nf_1(x) & := & \n\\int_{-2\\pi}^{2\\pi} (2C+\\ln(4\\pi^2-k^2)) \\cos kx \\frac{dk}{2\\pi} \\nonumber \\\\\n&=& \\frac{\\sin 2\\pi x}{\\pi x}(C+\\ln 4\\pi-\\ln|x|+{\\rm ci}(4\\pi x))\n- \\frac{\\cos 2\\pi x}{\\pi x} {\\rm Si}(4\\pi x) \\nonumber \\\\\nf_2(x) & := & - {1 \\over \\pi x} {\\rm Si}\\,(2 \\pi x), \\qquad\nf_3(x) \\: := \\: {\\sin 2 \\pi x \\over \\pi x} \\nonumber \\\\\ng_1(k,x) & = & {\\sin((2\\pi - |k|/2)x) \\over 2 \\pi x} \\nonumber \\\\\ng_2(k,x) & = & {\\rm ci}\\,((2\\pi - |k|)x) - {\\rm ci}\\,(2\\pi x) \\nonumber \\\\\ng_3(k,x) & = & {\\rm Si}\\,((2\\pi - |k|)x) + {\\rm Si}\\,(2\\pi x) \n\\end{eqnarray}\n\nWhen substituted in (\\ref{gi1}), the constant terms $B_2$ and $B_4$ cancel,\nand we obtain the formula\n\\begin{eqnarray}\\label{Bg}\ng(x_1,x_2;\\beta) & = & 1 - \\Big (P_4(x_1,x_2)P_4(x_2,x_1) \\Big )^{(0)}\n\\nonumber \\\\\n&& + (\\beta - 4) \\bigg \\{ \\Big ( -1 + (P_4(x_1,x_2)P_4(x_2,x_1))^{(0)}\n \\Phi(x_1,x_2) \\nonumber \\\\\n&& +2 B_1(x_{12}) - 4 \\Big ( P_4(x_{12}) B_3(x_{12}) \\Big )^{(0)} +\n2 \\Big ( P_4(x_{12}) B_5(x_{12}) \\Big )^{(0)} \\nonumber \\\\\n&& + B_6(x_{12}) - B_7(x_{12}) \\bigg \\} +\nO((\\beta - 4)^2).\n\\end{eqnarray}\n\nWe will demonstrate the close analogy with the $\\beta =2$ calculation of\nAppendix A by giving the derivation of the integral formula in\n(\\ref{B}) for $B_1(x_{12})$. As in the derivation of the integral\nformula (\\ref{fi3}) for $A_1(x_{12})$, our strategy is to use the\nconvolution formula (\\ref{fi1'}). However here the Fourier transform of\n$P_4(x_1,x_2) P_4(x_2,x_1)$ is not immediate. What is immediate is the\nFourier transform of $P_4(x_1,x_2)$. Thus from the definition\n(\\ref{ny2'}) we see that\n\\begin{eqnarray*}\n{\\rm FT} \\, P_4(x_1,x_2) & := & \\int_{-\\infty}^\\infty P_4(x_1,x_2)\ne^{ikx_{12}} \\, dx_{12} \\nonumber \\\\\n& = & \\left \\{ \\begin{array}{ll} \\left [ \\begin{array}{cc} 1/2 & i/2k \\\\\n-ik/2 & 1/2 \\end{array} \\right ], & |k| < 2 \\pi \\\\\n\\left [ \\begin{array}{cc} 0 & 0 \\\\ 0 & 0 \n\\end{array} \\right ], & |k| > 2 \\pi. \\end{array} \\right.\n\\end{eqnarray*}\nUse of (\\ref{fi1'}) then shows that for $|k| < 4 \\pi$\n\\begin{eqnarray}\\label{fi2a}\n{\\rm FT} \\, P_4(x_1,x_2)P_4(x_2,x_1) & := & \n\\int_{-2\\pi}^{2\\pi}\n\\left [ \\begin{array}{cc} 1/2 & i/2l \\\\\n-il/2 & 1/2 \\end{array} \\right ]\n\\left [ \\begin{array}{cc} 1/2 & -i/2(k-l) \\\\\ni(k-l)/2 & 1/2 \\end{array} \\right ] \\chi_{|k-l|<2\\pi} \\,\n{dl \\over 2 \\pi} \\nonumber \\\\\n& = &\n\\left [ \\begin{array}{cc}\n1 - {|k| \\over 4 \\pi} + {|k| \\over 8 \\pi} \\log \\Big |\n1 - {|k| \\over 2 \\pi} \\Big | & 0 \\\\\n0 & 1 - {|k| \\over 4 \\pi} + {|k| \\over 8 \\pi} \\log \\Big |\n1 - {|k| \\over 2 \\pi} \\Big | \\end{array} \\right ],\n\\end{eqnarray}\nwhile for $|k| > 4 \\pi$\n\\begin{equation}\\label{fi2b}\n{\\rm FT} \\, P_4(x_1,x_2) P_4(x_2,x_1) = 0.\n\\end{equation}\nThe results (\\ref{fi2a}) and (\\ref{fi2b}) are the analogue of (\\ref{fi2})\nin the working leading to the evaluation of $A_1(x_{12})$. The integral\nformula for $B_1(x_{12})$ in (\\ref{B}) now follows from (\\ref{fi2a}),\n(\\ref{fi2b}) and (\\ref{fte}) upon a further application of (\\ref{fi1'}).\n\nThe Fourier transform of (\\ref{Bg}) can be computed explicitly. The final\nresult has already been stated in (\\ref{b4}). This is obtained through the\nintermediate results\n$$\n{\\rm FT} \\, B_j(x) = \\hat{B}_j(k) \\quad {\\rm for} \\quad j=0,1,3,5,6,7\n$$\nwith\n$$\nB_0(x_{12}) := \\Big ( P_4(x_1,x_2) P_4(x_2,x_1) \\Big )^{(0)}\n\\Phi(x_1,x_2)\n$$\nand the $\\hat{B}_j$ specified by (\\ref{B0})--(\\ref{B7}). We will\nillustrate the working by giving some details of the computation\nof $\\hat{B}_0(k)$ for $|k| < 4 \\pi$.\n\nUsing (\\ref{fi2a}) and (\\ref{fte}) we see from (\\ref{fi1'}) that\n\\begin{eqnarray}\\label{see}\n{\\rm FT} \\, B_0(x_{12}) & = &\n\\int_{-4\\pi}^{4\\pi} \\Big (\n1 - {|l| \\over 4 \\pi} + {|l| \\over 8 \\pi}\n\\log \\Big | 1 - {|l| \\over 2 \\pi} \\Big | \\Big ) {\\pi \\over |k-l|} \\,\n{dl \\over 2 \\pi} \\nonumber \\\\\n& = & \\Big (\n1 - {|k| \\over 4 \\pi} + {|k| \\over 8 \\pi}\n\\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big | \\Big ) \\int_{-4\\pi}^{4\\pi}\n{\\pi \\over |k-l|} \\, {dl \\over 2 \\pi} \\nonumber \\\\\n&& + \\int_{-4\\pi}^{4\\pi} \\Big ( {|k| - |l| \\over 4 \\pi} \\Big )\n{\\pi \\over |k-l|} \\,{dl \\over 2 \\pi} +\n\\int_{-4\\pi}^{8 \\pi} \\Big ( {|l| - |k| \\over 8 \\pi} \\Big )\n\\log \\Big | 1 - {|l| \\over 2 \\pi} \\Big |\n{\\pi \\over |k-l|} \\,{dl \\over 2 \\pi} \\nonumber \\\\\n&& + {|k| \\over 8 \\pi}\n\\int_{-4\\pi}^{4\\pi} \\log \\Big | {2 \\pi - |l| \\over 2 \\pi - |k|} \\Big |\n{\\pi \\over |k-l|} \\,{dl \\over 2 \\pi}\n\\end{eqnarray}\nwhere the second equality, which follows from minor manipulation of the\nfirst integral, is motivated by the desire to separate the singular integral.\nThus in the second equality of (\\ref{see}) only the first integral is\nsingular. It is essentially the same as the first singular integral in\n(\\ref{ii}), and is evaluated as\n\\begin{equation}\\label{nw1}\n\\int_{-4\\pi}^{4\\pi}\n{\\pi \\over |k-l|} \\, {dl \\over 2 \\pi} = C + {1 \\over 2}\n\\log (16 \\pi^2 - k^2), \\qquad |k| \\le 4 \\pi.\n\\end{equation}\n\nThe second integral in the second equality of (\\ref{see}) also appears in\nthe evaluation of (\\ref{ii}). An elementary calculation shows\n\\begin{equation}\\label{nw2}\n\\int_{-4\\pi}^{4\\pi} {|k| - |l| \\over 4 \\pi} {\\pi \\over |k-l|} \\,\n{dl \\over 2 \\pi} = - 1 + {|k| \\over 4 \\pi} + {|k| \\over 4 \\pi}\n\\log \\Big ( {4\\pi + |k| \\over |k|} \\Big ).\n\\end{equation}\n\nTo evaluate the third integral in (\\ref{see}) we suppose without loss of \ngenerality that $k>0$ and write\n\\begin{eqnarray}\\label{nw3}\n\\lefteqn{\\int_{-4\\pi}^{4\\pi} {|l| - |k| \\over 8 \\pi}\n\\log \\Big | 1 - {|l| \\over 2 \\pi} \\Big | {\\pi \\over |k-l|} \\,\n{dl \\over 2 \\pi} = \\int_0^{2\\pi} \\Big ( {1 \\over 16 \\pi} + {k \\over 8 \\pi (l-k)} \\Big )\n\\log \\Big | 1 + {l \\over 2 \\pi} \\Big | \\, dl\n} \\nonumber \\\\ &&\n - \\int_0^k \\log \\Big | 1 - {|l| \\over 2 \\pi} \\Big | \\, {dl \\over 16 \\pi}\n+ \\int_k^{4 \\pi} \\log \\Big | 1 - {|l| \\over 2 \\pi} \\Big | \\, {dl \\over 16 \\pi}.\n\\end{eqnarray}\nThe only non-elementary integral is the second term of the first integral.\nThis can be computed by checking from the definition (\\ref{dilog})\nthat for $-4\\pi < l < 0$\n\\begin{equation}\\label{nw3'}\n{d \\over dl} \\bigg (\n{\\rm dilog} \\, \\Big ( {k-l \\over k + 2\\pi} \\Big ) +\n\\log \\Big | 1 + {l \\over 2 \\pi} \\Big |\n{\\rm log} \\, \\Big ( {k-l \\over k + 2\\pi} \\Big ) \\bigg ) =\n{1 \\over l - k} \\log \\Big | 1 + {l \\over 2 \\pi} \\Big |.\n\\end{equation}\nIn total we therefore have\n\\begin{eqnarray}\\label{nw4}\n\\lefteqn{\n\\int_{-4\\pi}^{4\\pi} {|l| - |k| \\over 8 \\pi}\n\\log \\Big | 1 - {|l| \\over 2 \\pi} \\Big | {\\pi \\over |k-l|} \\,\n{dl \\over 2 \\pi}} \\nonumber \\\\\n&& = - {1 \\over 2} + {|k| \\over 8 \\pi} +\n{|k| \\over 8 \\pi} \\Big ( {\\rm dilog} \\, \\Big ( {|k| \\over 2 \\pi + |k|} \\Big )\n- {\\rm dilog} \\, \\Big ( {4 \\pi + |k| \\over 2 \\pi + |k|} \\Big ) \\Big )\n+ {2 \\pi - |k| \\over 8 \\pi} \\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big |.\n\\end{eqnarray}\n\nTo evaluate the final integral in (\\ref{see}), a similar approach to that\nleading to the evaluation (\\ref{nw4}) is adopted. Minor complications\narise because of the need to modify the formula (\\ref{nw3'}) for\n$l > k$. We find\n\\begin{eqnarray}\\label{nw5}\n\\lefteqn{\n\\int_{-4\\pi}^{4\\pi} {|k| \\over 8 \\pi} \\log \\Big | {2 \\pi - |l| \\over\n2 \\pi - |k|} \\Big | {\\pi \\over |k-l|} \\, {dl \\over 2 \\pi}} \\nonumber \\\\\n&& = {|k| \\over 16 \\pi} \\bigg \\{\n{\\rm dilog} \\, \\Big ( {4 \\pi + |k| \\over 2 \\pi + |k|} \\Big ) -\n{\\rm dilog} \\, \\Big ( {|k| \\over 2 \\pi + |k|} \\Big ) -\n\\log \\Big | 1 - {|k| \\over 2 \\pi} \\Big | \n\\log \\Big ( {4 \\pi + |k| \\over |k|} \\Big ) + g_1(k) \\bigg \\} \\nonumber \\\\\n\\end{eqnarray}\nwhere $g_1$ is defined in (\\ref{gg}). Substituting (\\ref{nw1})--(\\ref{nw5})\nin (\\ref{see}) gives the result \n(\\ref{B0}).\n\n%\\bibliographystyle{plain}\n%\\bibliography{book}\n\n\\begin{thebibliography}{10}\n\n\\bibitem{Ao87}\nK.~Aomoto.\n\\newblock Jacobi polynomials associated with {Selberg's} integral.\n\\newblock {\\em SIAM J. Math. 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Forrester, B.~Jancovici, and E.R. Smith.\n\\newblock The {two-dimensional} {one-component} plasma at $\\Gamma = 2$: the\n semi-periodic strip.\n\\newblock {\\em J. Stat. Phys.}, 31:129--140, 1983.\n\n\\bibitem{FZ96}\nP.J. Forrester and J.A. Zuk.\n\\newblock Applications of the {Dotsenko-Fateev} integral in random-matrix\n models.\n\\newblock {\\em Nucl. Phys. B}, 473:616--630, 1996.\n\n\\bibitem{GR80}\nI.S. Gradshteyn and I.M. Ryzhik.\n\\newblock {\\em Tables of Integrals, Series and Products}.\n\\newblock Academic Press, New York, 4th edition, 1980.\n\n\\bibitem{Gu99}\nA.J. Guttmann.\n\\newblock Indicators of solvability for lattice models.\n\\newblock {\\em Discr. Math.}, to appear.\n\n\\bibitem{Ha95}\nZ.N.C. Ha.\n\\newblock Fractional statistics in one dimension: view from an exactly solvable\n model.\n\\newblock {\\em Nucl. Phys. B}, 435:604--636, 1995.\n\n\\bibitem{HM90}\nJ.P. Hansen and I.R. McDonald.\n\\newblock {\\em Theory of Simple Liquids}.\n\\newblock Academic Press, London, 2nd edition, 1990.\n\n\\bibitem{Ja81}\nB.~Jancovici.\n\\newblock Exact results for the two-dimensional one-component plasma\n\\newblock {\\em Phys. Rev. Lett.}, 46:386--388, 1981.\n\n\\bibitem{Ja82}\nB.~Jancovici.\n\\newblock Classical {Coulomb} systems near a plane wall. {II} \n\\newblock {\\em J. Stat. Phys.}, 29:263--280, 1982.\n\n\\bibitem{Ka97g}\nK.W.J. Kadell.\n\\newblock The {Selberg-Jack} symmetric functions.\n\\newblock {\\em Adv. Math.}, 130:33--102, 1997.\n\n\\bibitem{KMST99}\nP.~Kalinay, P.~Markos, L.~Samaj, and I.~Travenec.\n\\newblock The {sixth-moment} sum rule for the pair correlations of the\n two-dimensional {one-component} plasma: exact results.\n\\newblock {\\em J. Stat. Phys.}, 2000.\n\n\\bibitem{Ka93}\nJ.~Kaneko.\n\\newblock Selberg integrals and hypergeometric functions associated with {Jack}\n polynomials.\n\\newblock {\\em SIAM J. Math Anal.}, 24:1086--1110, 1993.\n\n\\bibitem{Li58}\nJ.~Lighthill.\n\\newblock {\\em Introduction to Fourier Analysis and Generalized Functions}.\n\\newblock CUP, Cambridge, 1958.\n\n\\bibitem{Ma95}\nI.G. Macdonald.\n\\newblock {\\em Hall polynomials and symmetric functions}.\n\\newblock Oxford University Press, Oxford, 2nd edition, 1995.\n\n\n\\bibitem{Me91}\nM.L. Mehta.\n\\newblock {\\em Random Matrices}.\n\\newblock Academic Press, New York, 2nd edition, 1991.\n\n\\end{thebibliography}\n\\end{document}\n\n"
}
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[
{
"name": "cond-mat0002060.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{Ao87}\nK.~Aomoto.\n\\newblock Jacobi polynomials associated with {Selberg's} integral.\n\\newblock {\\em SIAM J. Math. Analysis}, 18:545--549, 1987.\n\n\\bibitem{Ba82}\nR.J. Baxter.\n\\newblock {\\em Exactly Solved Models in Statistical Mechanics}.\n\\newblock Academic Press, London, 1982.\n\n\\bibitem{Fo93aa}\nP.J. Forrester.\n\\newblock Exact integral formulas and asymptotics for the correlations in the\n $1/r^2$ quantum many body system.\n\\newblock {\\em Phys. Lett. A}, 179:127--130, 1993.\n\n\\bibitem{Fo93}\nP.J. Forrester.\n\\newblock Recurrence equations for the computation of correlations in the\n $1/r^2$ quantum many body system.\n\\newblock {\\em J. Stat. Phys.}, 72:39--50, 1993.\n\n\\bibitem{Fo94j}\nP.J. Forrester.\n\\newblock Addendum to {Selberg} correlation integrals and the $1/r^2$ quantum\n many body system.\n\\newblock {\\em Nucl. Phys. B}, 416:377--385, 1994.\n\n\\bibitem{FJ97}\nP.J. Forrester and B.~Jancovici.\n\\newblock Exact and asymptotic formulas for overdamped {Brownian} dynamics.\n\\newblock {\\em Physica A}, 238:405--424, 1997.\n\n\\bibitem{FJS83}\nP.J. Forrester, B.~Jancovici, and E.R. Smith.\n\\newblock The {two-dimensional} {one-component} plasma at $\\Gamma = 2$: the\n semi-periodic strip.\n\\newblock {\\em J. Stat. Phys.}, 31:129--140, 1983.\n\n\\bibitem{FZ96}\nP.J. Forrester and J.A. Zuk.\n\\newblock Applications of the {Dotsenko-Fateev} integral in random-matrix\n models.\n\\newblock {\\em Nucl. Phys. B}, 473:616--630, 1996.\n\n\\bibitem{GR80}\nI.S. Gradshteyn and I.M. Ryzhik.\n\\newblock {\\em Tables of Integrals, Series and Products}.\n\\newblock Academic Press, New York, 4th edition, 1980.\n\n\\bibitem{Gu99}\nA.J. Guttmann.\n\\newblock Indicators of solvability for lattice models.\n\\newblock {\\em Discr. Math.}, to appear.\n\n\\bibitem{Ha95}\nZ.N.C. Ha.\n\\newblock Fractional statistics in one dimension: view from an exactly solvable\n model.\n\\newblock {\\em Nucl. Phys. B}, 435:604--636, 1995.\n\n\\bibitem{HM90}\nJ.P. Hansen and I.R. McDonald.\n\\newblock {\\em Theory of Simple Liquids}.\n\\newblock Academic Press, London, 2nd edition, 1990.\n\n\\bibitem{Ja81}\nB.~Jancovici.\n\\newblock Exact results for the two-dimensional one-component plasma\n\\newblock {\\em Phys. Rev. Lett.}, 46:386--388, 1981.\n\n\\bibitem{Ja82}\nB.~Jancovici.\n\\newblock Classical {Coulomb} systems near a plane wall. {II} \n\\newblock {\\em J. Stat. Phys.}, 29:263--280, 1982.\n\n\\bibitem{Ka97g}\nK.W.J. Kadell.\n\\newblock The {Selberg-Jack} symmetric functions.\n\\newblock {\\em Adv. Math.}, 130:33--102, 1997.\n\n\\bibitem{KMST99}\nP.~Kalinay, P.~Markos, L.~Samaj, and I.~Travenec.\n\\newblock The {sixth-moment} sum rule for the pair correlations of the\n two-dimensional {one-component} plasma: exact results.\n\\newblock {\\em J. Stat. Phys.}, 2000.\n\n\\bibitem{Ka93}\nJ.~Kaneko.\n\\newblock Selberg integrals and hypergeometric functions associated with {Jack}\n polynomials.\n\\newblock {\\em SIAM J. Math Anal.}, 24:1086--1110, 1993.\n\n\\bibitem{Li58}\nJ.~Lighthill.\n\\newblock {\\em Introduction to Fourier Analysis and Generalized Functions}.\n\\newblock CUP, Cambridge, 1958.\n\n\\bibitem{Ma95}\nI.G. Macdonald.\n\\newblock {\\em Hall polynomials and symmetric functions}.\n\\newblock Oxford University Press, Oxford, 2nd edition, 1995.\n\n\n\\bibitem{Me91}\nM.L. Mehta.\n\\newblock {\\em Random Matrices}.\n\\newblock Academic Press, New York, 2nd edition, 1991.\n\n\\end{thebibliography}"
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cond-mat0002061
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Magnetism and superconductivity in underscreened Kondo chains
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"author": "N. Andrei${}^\\dagger$ and E. Orignac${}^{\\dagger,*}$"
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We present a one dimensional model of electrons coupled to localized moments of spin $S\ge 1$ in which magnetism and superconductivity interplay in a nontrivial manner. This model has a non-Fermi liquid ground state of the chiral spin liquid type. A non-conventional odd-frequency pairing is shown to be the dominant instability of the system, together with antiferromagnetism of the local moments. We argue that this model captures the physics of the Kondo-Heisenberg spin $S=1$ chain, in the limit of strong Kondo coupling. Finally, we discuss briefly the effect of interchain coupling.
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"name": "cond-mat0002061.tex",
"string": "\\documentstyle[prl,aps,twocolumn]{revtex}\n\\begin{document}\n\\title{ Magnetism and superconductivity in underscreened Kondo chains}\n\\draft\n\\author{N. Andrei${}^\\dagger$ and E. Orignac${}^{\\dagger,*}$}\n\\address{${}^\\dagger$ Serin Physics Laboratory, Rutgers University,\nP. O. Box 849, Piscataway NJ 08855-0849 USA \\\\\n${}^*$ Laboratoire de Physique Th\\'eorique de l'Ecole Normale\nSup\\'erieure, 24 Rue Lhomond, 75231 Paris Cedex 05}\n\\wideabs{\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nWe present a one dimensional model of electrons coupled to localized \nmoments of spin $S\\ge 1$ in which\n magnetism and superconductivity interplay in a \nnontrivial manner. This model has a non-Fermi liquid ground state of\nthe chiral spin liquid type. A non-conventional\nodd-frequency pairing is shown to be the dominant instability of the\nsystem, together with antiferromagnetism of the local moments. \nWe argue that this model captures the physics of the\nKondo-Heisenberg spin $S=1$ chain, in the limit of strong Kondo\ncoupling. Finally, we discuss briefly the effect of interchain\ncoupling. \n\\end{abstract}\n\\pacs{PACS: 71.27.+a 71.10.Pm 74.20.Mn 75.30.Mb }\n}\nThe interplay of magnetism and superconductivity is a fundamental\nproblem in condensed matter physics. Superconductivity is associated\nwith the pairing of electron states related by time\nreversal. Magnetic states, in which time reversal symmetry is\nlost, should therefore strongly compete with superconductivity. \nThus it came as a surprise\nwhen it was discovered in 1984 by Schlabitz et al.\\cite{schlabitz86}\nthat magnetism and superconductivity actually coexisted in the heavy\nfermion compound $\\mathrm{URu_2Si_2}$. Since then, other heavy\nfermion superconductors were shown to present magnetic moments in\ntheir superconducting phase\\cite{amato_heavy_musr}. All these compounds\ncontain rare earth or actinide ions with very localized $4f$ or $5f$\norbitals, strongly interacting\n with the conduction band. (This is in contrast to such cases\nas the Chevrel phases where\nmagnetism and superconductivity coexist because the magnetic moments\nresponsible for magnetism are only very weakly coupled with the\nelectrons that form the condensate\\cite{fischer_chevrel_phases}.) \n\n\nThe physics of heavy fermion\ncompounds is believed to be described by the Kondo\nLattice model in which conduction electrons interact with the local\nmoments associated with the localized $f$ electrons, and \na large amount of theoretical work was carried out. However,\nalthough the single impurity Kondo problem is well\nunderstood\\cite{kondo_papers}, the\ntheoretical analysis of the Kondo Lattice model has proven extremely\ndifficult. This is because the Kondo effect (the\n quenching of the local moments) competes, in the lattice\nproblem, with the\nRudermann-Kittel-Kasuya-Yosida interaction which orders the local\nmoments. This frustration is believed to be at the origin of the rich\nphase diagram of heavy fermions systems. There exist at present\nvarious theories of the superconducting ground state of the Kondo\nLattice Model. Conventional scenarios involve the pairing of fermionic\nquasiparticles into either a spin triplet ``p-wave'' state or a spin\nsinglet ``d-wave'' state\\cite{varma_vf_review}. A less\nconventional pairing\\cite{coleman_three_body} involves the formation of \na\n spin \nsinglet, isotropic, odd in time superconducting order parameter, known \nas\nodd-frequency pairing\\cite{berizinskii_odd_freq,balatsky_odd_freq}.\n\nIn one dimension, the theoretical situation is more\nfavorable\\cite{tsunetsugu_kondo_1d}, \nsince there are\npowerful methods to deal with strong interactions. \nBosonization \\cite{tsvelikb} and Density Matrix Renormalization\nGroup\\cite{white} have clarified much of the physics\nof the single channel $S=1/2$ Kondo lattice. One finds a \nparamagnetic metallic\nphase for small Kondo\ncoupling and away from half filling. However, conventional \nsuperconducting fluctuations are strongly\nreduced\\cite{shibata_dmrg_kondo_1d}. Adding a direct Heisenberg \ninteraction\nbetween the spins has been shown to cause an enhancement of\nodd-frequency pairing correlations. However, the formation of a spin\ngap precluded the observation of fluctuations towards magnetic\nordering\\cite{zachar_exotic_kondo}. \n\nThe aim of this paper is to discuss a one dimensional\nKondo Lattice model with\n{\\it underscreened moments} in which magnetism can be expected to \ndominate. We\nwill show that in this one dimensional problem strong fluctuations to\ncomposite superconductivity coexist with strong fluctuations towards\nmagnetic ordering. \n\n\n\n\n\n The generalized Kondo lattice Hamiltonian in one\ndimension is:\n\\begin{eqnarray}\nH&=&-t \\sum_{i,\\sigma} (c^\\dagger_{i,\\sigma}c_{i+1,\\sigma} +\nc^\\dagger_{i+1,\\sigma}c_{i,\\sigma}) \\nonumber \\\\ &+& \\lambda_K \\sum_i\n\\vec{S}_i. c^\\dagger_{i,\\alpha} \\vec{\\sigma}_{\\alpha,\\beta} c_{i,\\beta}\n+ \\sum_i f(\\vec{S}_i \\cdot \\vec{S}_{i+1})\n\\end{eqnarray}\nwhere $c_{i,\\sigma}$ annihilates an electron, $\\vec{S}_i$ is a spin-$S$ \noperator, and the function $f$ describes the spin-spin interaction. In \nthe\nfollowing, we will consider the case $\\lambda_K \\ll f,t$.\n\nWe shall be interested in magnetism and underscreening coexisting with \nsuperconductivity. To that purpose choose $S \\ge 1$. Several forms of \nmagnetic interaction need to be considered in this case.\n The simplest possibility is to take a Heisenberg\ncoupling $f(\\vec{S}_i \\cdot \\vec{S}_{i+1})=\\lambda_H\n\\vec{S}_i \\cdot\\vec{S}_{i+1}$, but no interesting magnetic effects \nwould ensue.\n We would\n get at $\\lambda_K=0$ either a spin gap for integer $S$ or an\neffective spin 1/2 chain for half-odd integer $S$\\cite{haldane_gap}. \nTurning on\n$\\lambda_K\\ll t,\\lambda_H$, the case of half-odd integer $S$ would \nreduce\nto exact screening, and a metal with a spin\ngap will form\\cite{sikkema_spingap_kondo_1d}. The case of integer $S$ \nwould lead to a completely trivial\nresult, the local moments being already completely screened by the\nformation of the Haldane gap preventing the Kondo effect and \nleaving the electrons \nessentially free. \n\n\nMagnetic effects, on the other hand, would result\nfrom interactions in the chain, $f$, which lead to low energy dynamics \ndescribed by effective Hamiltonians richer than the $SU(2)_{1}$\nWess--Zumino--Novikov--Witten (WZNW) model\nwhich govern the half-odd integer $S$\nHeisenberg\\cite{tsvelikb} discussed above. We shall discuss systems \nwith fixed points\ndescribed \n by $SU(2)_{2S}$ ($S \\ge 1$).\nAn example is provided by\n the integrable spin-S\nchains \n\\cite{takhtajan_spin_s,babujian_spin_s,affleck_strongcoupl,affleck_log_c\norr},\nwhere $f =P_S=\\sum_{j=1}^S a_j P_j$, $P_j$ is the spin-$j$ projector and\n$a_j=\\sum_{i=1}^j i^{-1}$. Such $SU_{2S}(2)$ models arise naturally \nunder some \ncircumstances: it is known \nin particular that the Heisenberg spin-1 chain \ncan be described as a $SU(2)_2$ WZNW model perturbed by a mass\nterm\\cite{tsvelik_field} of the order of the Haldane gap. The\ngeneralization of this result to arbitrary spin $S$ chains, based on\na perturbed $SU(2)_{2S}$ WZNW model, was obtained in Ref. \n\\onlinecite{cabra_spin_s},\n(and will be further discussed below). Thus, our study\nshould describe the\nUnderscreened Kondo--Heisenberg lattice when the Kondo coupling \nexceeds the Haldane gap. For $S=1$,\nthis corresponds to $0.4\\lambda_H < \\lambda_K <\\lambda_H$.\nWe shall study the model away \nfrom half filling and for all $S$, revealing the appearance of a \ncritical\n point describing a non Fermi-liquid where magnetism and \nsuperconductivity\ninterplay \\cite{note_lehur}.\n\n\n\n\nWe now proceed to determine the low energy\nbehavior of the theory. Consider the Hamiltonian in the continuum limit.\nAccording to the\nstandard prescriptions of non-abelian\nbosonization\\cite{tsvelikb,witten_wz,affleck_houches}, the electrons are\ndescribed by the following Hamiltonian:\n\\begin{eqnarray}\nH_{\\text{el}}&=&H_\\rho + H_\\sigma \\\\\nH_\\rho&=&\\int dx \\frac{v_F}{2\\pi}\\left[ (\\pi \\Pi_\\rho)^2 + (\\partial_x\n\\phi_\\rho)^2 \\right] \\nonumber \\\\\nH_\\sigma&=&\\frac{2\\pi v_F}{3} \\int dx (\\vec{J}_R.\\vec{J}_R + \n\\vec{J}_L.\\vec{J}_L)\n\\end{eqnarray}\nwhere the canonically conjugate fields $\\Pi_\\rho,\\phi_\\rho$ describe\nthe charge excitations, and the non abelian $SU(2)_1$ currents\n$\\vec{J}_{R,L}$\ndescribe the electron spin excitations. \nThe local moments are described in the low energy regime \nby the Hamiltonian:\n\\begin{equation}\nH_{\\text{mom}}=\\frac{2\\pi v_s}{2(1+S)} \\int dx (\\vec{S}_R.\\vec{S}_R + \n\\vec{S}_L.\\vec{S}_L)\n\\end{equation}\nwhere $\\vec{S}_{R,L}$ are $SU(2)_{2S}$ WZNW currents.\nThe Kondo interaction $\\lambda_K$ at\nincommensurate filling becomes:\n\\begin{equation}\n\\lambda_K a \\int dx (\\vec{J}_R + \\vec{J}_L).(\\vec{S}_L+\\vec{S}_R)(x)\n\\end{equation}\nand preserves spin-charge separation. \nCarrying out standard RG calculations we find that\nthis interaction is a combination of terms that are\n(marginally) relevant in the RG sense and purely marginal terms.\n The former drive\nus to a strong coupling fixed point and we need a\nnon-perturbative way to determine it. To do so we may\n neglect the\nmarginal couplings $\\vec{J}_R.\\vec{S}_R$ and $ \\vec{J}_L.\\vec{S}_L$\nas well as the velocity difference between electron and local moments\nspin excitations, and check, subsequently, their relevance at the fixed \npoint.\n This leads us to the following spin Hamiltonian:\n\\begin{eqnarray}\n\\label{eq:chiral_hamiltonian}\nH_{\\text{spin}}&=&H_1+H_2 \\\\\nH_1&=&\\int dx \\left[ \\frac{2\\pi v}{2(1+S)} \\vec{S}_R.\\vec{S}_R + \\frac{2\n\\pi v}{3} \\vec{J}_L.\\vec{J}_L + \\lambda_K a\n\\vec{S}_R.\\vec{J}_L\\right]\\nonumber \\\\ \nH_2&=&\\int dx \\left[ \\frac{2\\pi v}{2(1+S)} \\vec{S}_L.\\vec{S}_L + \\frac{2\n\\pi v}{3} \\vec{J}_R.\\vec{J}_R + \\lambda_K a\n\\vec{S}_L.\\vec{J}_R\\right] \\nonumber\n\\end{eqnarray}\nA similar Hamiltonian was proposed in Refs. \n\\onlinecite{azaria_csl,azaria_composite} to\ndescribe the spin sector of a Hubbard chain coupled to $N$ spin 1/2\nchains (chain cylinder model). The $N=2$ fixed\npoint was analyzed in detail using an exact\nsolution at a Toulouse point\\cite{azaria_csl}, and the correlation \nfunctions of the composite\norder parameters were obtained\\cite{azaria_composite}. \n\nThe fixed point Hamiltonian can be determined by arguments\nof {\\it chiral\nstabilization}\\cite{andrei_chiral_nfl}, since $H_1$ and $H_2$ in Eq. \n(\\ref{eq:chiral_hamiltonian}) are both chirally asymmetric (only their \nsum is chirally\nsymmetric)\\cite{note_andrei}. We find, that\nthe electron spin degrees of\nfreedom are described by the coset CFT - $\\frac{SU(2)_1\\otimes \nSU(2)_{2S-1}}{ SU(2)_{2S}}$, or equivalently, the minimal\nmodel- $M_{2S-1}$,\nof central charge $c=1-\\frac{6}{(2S+1)(2S+2)}$, while the local\nmoments spin\nexcitations are described by the $SU(2)_{2S-1}$ WZNW model:\n\\begin{equation}\nH^*=\\frac{SU(2)_1\\otimes SU(2)_{2S-1}}{ SU(2)_{2S}}+SU(2)_{2S-1}.\n\\end{equation}\nThis fixed point describes an interesting interplay of magnetism and \nsuperconductivity. We shall discuss the structure of the ground state, \nthe\nthermodynamics and then the correlation functions of the model.\n\nThe ground state is a coset singlet, formed between electrons and local\nmoments. A fraction\n$\\frac{6}{(2S+1)(2S+2)}$ of the electron spin degrees of freedom \nis absorbed by the spins, leading to its complete screening. This\n manifests itself in the\n magnetic susceptibility and the specific heat:\n\\begin{eqnarray}\n\\label{eq:susceptibility}\n\\chi&=&\\frac 1 {2\\pi v} (2S-1) \\nonumber \\\\\nC&=&\\frac{\\pi}{3} \\left( 1+ \\frac{3(2S)}{2S+2}+\n1-\\frac{12}{(2S+1)(2S+2)}\\right) T \\nonumber\n\\end{eqnarray}\nwhere in the latter quantity we included also charge contributions.\nThis leads to a Wilson ratio: $\nR_W=\\frac{10S^2+9S-4}{(2S+1)(S+1)(2S-1)}$\n going to zero as $S\\to \\infty$.\n\n \nWe\nnow calculated the long distance behavior of the physical\ncorrelation functions. To do so we\nhave to express the physical operators in terms\n of the operators around the fixed point, which we proceed\nto discuss. The primary operators of $SU(2)_N$ model are\n $\\Phi_N^{(j)}$ ($j\\le \\frac N 2$) and carry spin-$j$. The coset\nprimaries, $\\phi^{j\nj'}_{j^{\\prime \\prime}},\\;(0\\le j\n\\le 1/2$, $0\\le j'\n\\le S-1/2$ and $0\\le j^{\\prime \\prime} \\le S$), are related to the \n$SU(2)_N$ WZNW spin-$j$ \nprimaries via the decomposition:\n\\begin{equation}\n\\Phi_{L,1}^{(j)} \\Phi_{L,2S-1}^{(j')}=\\sum_{|j-j'| \\le j^{\\prime \n\\prime}\\le j+j'} \\phi^{j,j'}_{L,j^{\\prime \\prime}} \n\\Phi_{L,2S}^{(j^{\\prime \\prime})}\n\\end{equation}\n The conformal weight of the coset primary\n is such that the two sides of the equality have\nthe same conformal weight, implying\n that $\\phi^{j\nj'}_{L,j^{\\prime \\prime}}$ has left conformal weight \n$\\left(\\frac{j(j+1)} 3\n +\\frac{j'(j'+1)} {2S+1}- \\frac{j^{\\prime \\prime}(j^{\\prime \\prime}\n +1)}{2S+2}\\right)$. Similar identities hold for the right component. \nNext, note that\nat the fixed point products of left primary operators of $SU(2)_1$ and\nright $SU(2)_{2S}$ WZNW models have to decompose into a sum of products \nof primary\noperators of the left minimal model and right $SU(2)_{2S-1}$ \nmodel. Moreover, the total spin has to be conserved. \nTherefore, one can write the decomposition:\n\\begin{equation}\\label{decomposition_ir}\n\\Phi_{L,1}^{(j)} \\Phi_{R,2S}^{(j^{\\prime \\prime})} \\sim \n\\sum_{|j-j^{\\prime\n\\prime}| \\le j^{\\prime}\\le j+j^{\\prime \\prime}} \\phi^{j\nj'}_{L, j^{\\prime \\prime}} \\Phi_{R,2S-1}^{(j')}\n\\end{equation}\nThis decomposition satisfies the requirements on the indices in \n$\\phi^{j\nj'}_{j^{\\prime \\prime}}$. It is formally similar to a Clebsch-Gordan\ndecomposition. \n\n\nThus, at the fixed point, the spin operator is given by:\n\\begin{equation}\n\\vec{S}_n \\sim a (\\vec{S}^\\prime_R +\\vec{S}^\\prime_L) (na) +\n\\phi^{0,1/2}_{1/2} \\mathrm{Tr}(\\vec{\\sigma}g')(na)\n\\end{equation}\nwhere $\\vec{S}^\\prime_{R,L}$ and $g'$ are respectively the currents\nand the $SU(2)$ matrix field of the $SU(2)_{2S-1}$\nWZNW model, and $ \\phi^{0,1/2}_{1/2}$ is the field of the minimal\nmodel with scaling dimension $\\frac{3}{2(2S+1)(2S+2)}$. \nAs a result, the spin-spin correlation function behaves as:\n\\begin{equation}\n\\langle \\vec{S}_n.\\vec{S}_m\\rangle \\sim \\frac 1\n{(n-m)^2} + \\frac{(-)^{n-m}}{(n-m)^{\\delta_S}}\n\\end{equation}\nwhere $\\delta_S=\\frac {3 (2S+3)}{(2S+1)(2S+2)} \\ll 2$. There is a\nstrong tendency to antiferromagnetism.\n\n\nThe fermion operator $\\psi_L$, is given by:\n\\begin{equation}\n\\psi_L=e^{\\imath (\\theta_\\rho+\\phi_\\rho)/\\sqrt2} \\phi^{1/2,1/2}_{L \\;0} \ng^\\prime_R\n\\end{equation}\nwhere $\\theta_\\rho(x)=\\pi \\int^x \\Pi_\\rho(x') dx'$, \n$\\phi^{1/2,1/2}_{L \\;0}$ is the\nantiholomorphic component of the $\\phi^{1/2,1/2}_{0}$ field of the\nminimal model, and $ g^\\prime_R$ is the holomorphic component of the\n$g'$ field of the $SU(2)_{2S-1}$\nWZNW model. A similar expression holds for $\\psi_R$, in which $L$ and\n$R$ are exchanged and $\\phi_\\rho \\to -\\phi_\\rho$. \nThe resulting fermion Green's function is:\n\\begin{eqnarray}\n\\langle T c_n(t) c_0^\\dagger(0) \\rangle \\sim \\frac {e^{\\imath k_F n\na}}{(na-vt)^{1+\\frac{3}{2(2S+1)}} (na+vt)^{\\frac{3}{2(2S+1)}}} \n\\nonumber \\\\\n+ \\frac {e^{-\\imath k_F n\na}}{(na+vt)^{1+\\frac{3}{2(2S+1)}} (na-vt)^{\\frac{3}{2(2S+1)}}}\n\\end{eqnarray}\n\nConsidering now charge density wave (CDW), spin density\nwave (SDW),\nsinglet superconductor (SS) and triplet superconductor (TS)\n correlations, it is easy to see that they all\ndecay with the same exponent $2+\\frac{6}{2S+1}$. \nThese correlations are thus very strongly suppressed with respect to a\nfree electron gas. Only for $S\\to \\infty$, do we \nrecover the exponents of the free electron gas for the conventional \norder\nparameters. If we specialize to\n$N=2$, we obtain the same exponents as in Ref. \\onlinecite{azaria_csl}. \n\nThis leads us to investigate the presence of composite\norder\\cite{coleman_kondo_odd_pair,zachar_exotic_kondo}: the\n composite Charge Density Wave (c-CDW) order parameter $\nO_{c-CDW}=\\vec{n}(x).\\vec{O}_{SDW}(x)$ and the composite singlet order\nparameter $\nO_{c-S}=\\vec{n}(x).\\vec{O}_{TS}(x)$.\n\nIt is easy to show that the composite Charge Density Wave order\nparameter can be expressed at the fixed point as:\n\\begin{equation}\nO_{c-CDW} \\sim e^{\\imath \\sqrt{2} \\phi_\\rho} \\phi^{1/2,0}_{1/2} \n\\end{equation}\nwhere the field $\\phi^{1/2,0}_{1/2} $ has conformal weight $(\\frac 1 4\n- \\frac 3 {4(2S+2)},\\frac 1 4 - \\frac 3 {4(2S+2)})$. Similarly,\nfor the composite singlet order, one has:\n\\begin{equation}\nO_{c-S} \\sim e^{\\imath \\sqrt{2} \\theta_\\rho} \\phi^{1/2,0}_{1/2}. \n\\end{equation}\nand both correlations decay with the same exponent,\n\\begin{equation}\n\\langle O_{composite}(n)O_{composite}^\\dagger(n') \\rangle \\sim \\frac 1\n{|n-n'|^{2-\\frac{3}{2S+2}}} \n\\end{equation}\nTherefore, for any $S$, composite order parameters are dominant. This\n can be understood in a simple way: electrons are tightly bound to\n local moments, so that composite correlations, that involve coherent\n motion of local moments excitations and electrons, decay more slowly\n than conventional ones.\n For $S\\to \\infty$, conventional and composite\norder parameters become degenerate. This can be understood by\n remarking that in this limit, the local moments become classical and\n acquire a non-zero average so that composite and conventional\n order parameters become identical. \n\nAll our discussion up to now was concerned with the coupling of an\nintegrable spin $S$ chain with fermions. We now turn to the\neffect of the perturbation $\\lambda \\Phi_R^{(1)}\\Phi_L^{(1)}$ that \nrestores the behavior of the Heisenberg\nspin $S$ chain \\cite{cabra_spin_s}. If we assume that this\nterm is small, i. e. that the Kondo coupling is larger than the\nlargest gap induced by the perturbation for zero Kondo coupling,\n we can analyze its effect by determining whether it\nis relevant or irrelevant at the fixed point. \nThe case of the spin 1 chain is special since Kac-Moody selection\nrules prevent the appearance of a primary operator of spin 1 in the\nfixed point theory. Generalizing (\\ref{decomposition_ir}) to include\nalso non primary operators, a simple\ncalculation shows that $\\Phi^{(1)}_R \\to J_R \\xi_L'$, where $J_R$ is\nthe \n$SU(2)_1$ current, and $\\xi_L'$ is a Majorana fermion of the Ising\n($M_1$ minimal) model. This result can also be obtained from the \nToulouse\nlimit solution\\cite{azaria_csl,lecheminant_pc}. \nAs a result, the mass term becomes a term of dimension $3$ at\nthe fixed point and is irrelevant. We therefore expect that the\nunderscreened regime we have found will be present if $\\lambda_K$ \nexceeds\nthe Haldane gap of the isolated spin 1 chain. For a spin $S>1$, the \noperator\n $\\Phi^{(1)}_{R,2S}$ becomes at the fixed point\n$\\phi_{L,1}^{0\\;1}\\Phi^{(1)}_{R,2S-1}$. This operator has dimension\n$\\frac{8}{2S+1}-\\frac 4 {2S+2}<2$ and is therefore \\emph{relevant} at \nthe\nfixed point. It is then likely that for $S>1$\n the models that we have described\nflow when perturbed to the trivial Kondo-Heisenberg fixed point even \nfor a\nstrong Kondo coupling. \n\nWe may also examine what happens when two Kondo chains are coupled \ntogether to form\na {\\it Kondo ladder}. There are several ways to do it. One can \ncouple the local moments of the Kondo chains by a Heisenberg coupling \n$ \\lambda_\\perp \\sum_i\n\\vec{S}_{i,1}.\\vec{S}_{i,2}$. At the fixed point, this gives rise to\nrelevant terms of \nscaling dimension $\\frac{3 (2S+3)}{(2S+1)(2S+2)}$, that induce a spin\ngap. Second, one could consider an exchange interaction between the\nelectrons, $\\lambda_{\\text{ee}} \\sum_i c^\\dagger_{i,\\alpha,1} \n\\vec{\\sigma}_{\\alpha,\\beta} \nc_{i,\\beta,1}. c^\\dagger_{i,\\gamma,2} \\vec{\\sigma}_{\\gamma,\\delta}\nc_{i,\\delta,2}$. It is easily seen that such an interaction only leads\nto irrelevant terms, and does not affect our fixed point. Finally, one\ncould consider interchain hopping of the electrons $t_\\perp \\sum_i\n (c^\\dagger_{i,\\sigma,1} c_{i,\\sigma,2} +\\text{h. c.})$. At the fixed\npoint, this leads to an operator of dimension\n $1+ \\frac{3}{2S+1}$, relevant for $S\\ge 3/2$. \nFor $S=1$, this term is marginal but it generates a relevant\nRKKY interaction between the local moments that destabilizes \n the chiral fixed point.\n\nWe have obtained a physical picture of the one-dimensional\nunderscreened Kondo Lattice. The formation of a chiral non-Fermi\nLiquid results in strong antiferromagnetic fluctuations accompanied\nwith composite pairing. This picture is reminiscent of the situation\nthat obtains in \nthree dimensional heavy fermion systems\\cite{amato_heavy_musr}. It\nwould be very worthwhile \nto try to determine if some analog of a chiral non-Fermi liquid can be\nfound in higher dimensional Kondo Lattice models. A good starting\npoint would be an array of Kondo-underscreened chains coupled by\ninterchain hopping. \n\n\nWe thank P. Azaria, P. Lecheminant, H.-Y. Kee, O. Parcollet and A. \nRosch for\nilluminating discussions. E. O. acknowledges support from NSF under\ngrant DMR 96-14999.\n\\bibliographystyle{prsty}\n\n\n\\begin{thebibliography}{10}\n\n\\bibitem{schlabitz86}\nW. Schlabitz {\\it et~al.}, Z. Phys. B {\\bf 62}, 171 (1986).\n\n\\bibitem{amato_heavy_musr}\nA. Amato, Rev. Mod. Phys. {\\bf 69}, 1119 (1997).\n\n\\bibitem{fischer_chevrel_phases}\nO. Fischer, Appl. Phys. {\\bf 16}, 1 (1978).\n\n\\bibitem{kondo_papers}\nP.~W. Anderson, J. Phys. C {\\bf 3}, 2346 (1970);\n P.~W. Anderson, G. Yuval, and D.~R. Hamann, Phys. Rev. B {\\bf 1}, 4464\n (1970);\nK.~G. Wilson, Rev. Mod. Phys. {\\bf 47}, 773 (1975);\nN. Andrei, K. Furuya, and J.~H. Lowenstein, Rev. Mod. Phys. {\\bf 55}, \n331\n (1983);\nA.~M. Tsvelik and P.~B. Wiegmann, Adv. Phys. {\\bf 32}, 453 (1983);\nI. Affleck, Acta Phys. Polon. B {\\bf 26}, 1869 (1995).\n\n\\bibitem{varma_vf_review}\nC.~M. Varma, Comments Solid State Phys. {\\bf 11}, 221 (1985).\n\n\\bibitem{coleman_three_body}\nP. Coleman, E. Miranda, and A.~M. Tsvelik, Phys. Rev. Lett. {\\bf 74}, \n1653\n (1995).\n\n\\bibitem{berizinskii_odd_freq}\nV.~L. Berizinskii, JETP Lett. {\\bf 20}, 287 (1974).\n\n\\bibitem{balatsky_odd_freq}\nA. Balatsky and E. Abrahams, Phys. Rev. B {\\bf 45}, 13125 (1992).\n\n\\bibitem{tsunetsugu_kondo_1d}\nH. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys. {\\bf 69}, 809 \n(1997).\n\n\\bibitem{tsvelikb}\nA. Tsvelik, {\\em Quantum Field Theory in Condensed Matter Physics} \n(Cambridge\n University Press, Cambridge, 1995).\n\n\\bibitem{white}\nS.~R. White, Phys. Rev. Lett. {\\bf 69}, 2863 (1992).\n\n\\bibitem{shibata_dmrg_kondo_1d}\nN. Shibata and K. Ueda, J. Phys. Condens. Matter {\\bf 11}, R1 (1999).\n\n\\bibitem{zachar_exotic_kondo}\nO. Zachar and A.~M. Tsvelik, cond-mat/9909296 (unpublished).\n\n\\bibitem{haldane_gap}\nF.~D.~M. Haldane, Phys. Rev. Lett. {\\bf 50}, 1153 (1983).\n\n\\bibitem{sikkema_spingap_kondo_1d}\nA.~E. Sikkema, I. Affleck, and S.~R. White, Phys. Rev. Lett. {\\bf 79}, \n929\n (1997).\n\n\\bibitem{takhtajan_spin_s}\nL. Takhtajan, Phys. Lett. A {\\bf 87}, 479 (1982).\n\n\\bibitem{babujian_spin_s}\nJ. Babujian, Phys. Lett. A {\\bf 90}, 479 (1982).\n\n\\bibitem{affleck_strongcoupl}\nI. Affleck and F. Haldane, Phys. Rev. B {\\bf 36}, 5291 (1987).\n\n\\bibitem{affleck_log_corr}\nI. Affleck, D. Gepner, T. Ziman, and H.~J. Schulz, J. Phys. A {\\bf 22}, \n 511\n (1989).\n\n\\bibitem{tsvelik_field}\nA.~M. Tsvelik, Phys. Rev. B {\\bf 42}, 10499 (1990).\n\n\\bibitem{cabra_spin_s}\nD.~C. Cabra, P. Pujol, and C. {von Reichenbach}, Phys. Rev. B {\\bf 58}, \n 65\n (1998).\n\\bibitem{note_lehur}\nThe case $S=1$ at half filling\nhas been investigated by K. Lehur, Phys. Rev. Lett. {\\bf 83}, 848 \n(1999).\n A charge\ngap and gapless spin excitations described by the $SU(2)_1$ WZNW\nmodel were found. \n\n\n\n\n\\bibitem{witten_wz}\nE. Witten, Commun. Math. Phys. {\\bf 92}, 455 (1984).\n\n\\bibitem{affleck_houches}\nI. Affleck, in {\\em Fields, Strings and Critical Phenomena}, edited by \nE.\n Br\\'ezin and J. Zinn-Justin (Elsevier Science Publishers, Amsterdam, \n1988).\n\n\\bibitem{azaria_csl}\nP. Azaria, P. Lecheminant, \nand A.~A. Nersesyan, Phys. Rev. B {\\bf 58}, R8881 (1998); P. Azaria \nand P. Lecheminant, Chirally stabilized critical state in Marginally\n coupled spin and doped systems, 1999, cond-mat/9912406.\n\n\\bibitem{azaria_composite}\nP. Azaria and P. Lecheminant, Composite Pairings in Chirally Stabilized\n Critical Fluids, 2000, cond-mat/0001072.\n\n\\bibitem{andrei_chiral_nfl}\nN. Andrei, M.~R. Douglas, and A. Jerez, Phys. Rev. B {\\bf 58}, 7619 \n(1998).\n\n\\bibitem{note_andrei}\nSame arguments were recently applied to\n the Multichannel Kondo-Heisenberg lattice\nproblem [N. Andrei and E. Orignac, Low energy dynamics of the one \ndimensional\n multichannel Kondo-Heisenberg Lattice, 1999, cond-mat/9912372.]\n\n\\bibitem{coleman_kondo_odd_pair}\nP. Coleman, A. Georges, and A. Tsvelik, J. Phys. Condens. Matter {\\bf \n79}, 345\n (1997).\n\n\\bibitem{lecheminant_pc}\nP. Lecheminant, private communication.\n\n\\end{thebibliography}\n\n\n\n\\end{document}\n\n\n\n"
}
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[
{
"name": "cond-mat0002061.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{schlabitz86}\nW. Schlabitz {\\it et~al.}, Z. Phys. B {\\bf 62}, 171 (1986).\n\n\\bibitem{amato_heavy_musr}\nA. Amato, Rev. Mod. Phys. {\\bf 69}, 1119 (1997).\n\n\\bibitem{fischer_chevrel_phases}\nO. Fischer, Appl. Phys. {\\bf 16}, 1 (1978).\n\n\\bibitem{kondo_papers}\nP.~W. Anderson, J. Phys. C {\\bf 3}, 2346 (1970);\n P.~W. Anderson, G. Yuval, and D.~R. Hamann, Phys. Rev. B {\\bf 1}, 4464\n (1970);\nK.~G. Wilson, Rev. Mod. Phys. {\\bf 47}, 773 (1975);\nN. Andrei, K. Furuya, and J.~H. Lowenstein, Rev. Mod. Phys. {\\bf 55}, \n331\n (1983);\nA.~M. Tsvelik and P.~B. Wiegmann, Adv. Phys. {\\bf 32}, 453 (1983);\nI. Affleck, Acta Phys. Polon. B {\\bf 26}, 1869 (1995).\n\n\\bibitem{varma_vf_review}\nC.~M. Varma, Comments Solid State Phys. {\\bf 11}, 221 (1985).\n\n\\bibitem{coleman_three_body}\nP. Coleman, E. Miranda, and A.~M. Tsvelik, Phys. Rev. Lett. {\\bf 74}, \n1653\n (1995).\n\n\\bibitem{berizinskii_odd_freq}\nV.~L. Berizinskii, JETP Lett. {\\bf 20}, 287 (1974).\n\n\\bibitem{balatsky_odd_freq}\nA. Balatsky and E. Abrahams, Phys. Rev. B {\\bf 45}, 13125 (1992).\n\n\\bibitem{tsunetsugu_kondo_1d}\nH. Tsunetsugu, M. Sigrist, and K. Ueda, Rev. Mod. Phys. {\\bf 69}, 809 \n(1997).\n\n\\bibitem{tsvelikb}\nA. Tsvelik, {\\em Quantum Field Theory in Condensed Matter Physics} \n(Cambridge\n University Press, Cambridge, 1995).\n\n\\bibitem{white}\nS.~R. White, Phys. Rev. Lett. {\\bf 69}, 2863 (1992).\n\n\\bibitem{shibata_dmrg_kondo_1d}\nN. Shibata and K. Ueda, J. Phys. Condens. Matter {\\bf 11}, R1 (1999).\n\n\\bibitem{zachar_exotic_kondo}\nO. Zachar and A.~M. Tsvelik, cond-mat/9909296 (unpublished).\n\n\\bibitem{haldane_gap}\nF.~D.~M. Haldane, Phys. Rev. Lett. {\\bf 50}, 1153 (1983).\n\n\\bibitem{sikkema_spingap_kondo_1d}\nA.~E. Sikkema, I. Affleck, and S.~R. White, Phys. Rev. Lett. {\\bf 79}, \n929\n (1997).\n\n\\bibitem{takhtajan_spin_s}\nL. Takhtajan, Phys. Lett. A {\\bf 87}, 479 (1982).\n\n\\bibitem{babujian_spin_s}\nJ. Babujian, Phys. Lett. A {\\bf 90}, 479 (1982).\n\n\\bibitem{affleck_strongcoupl}\nI. Affleck and F. Haldane, Phys. Rev. B {\\bf 36}, 5291 (1987).\n\n\\bibitem{affleck_log_corr}\nI. Affleck, D. Gepner, T. Ziman, and H.~J. Schulz, J. Phys. A {\\bf 22}, \n 511\n (1989).\n\n\\bibitem{tsvelik_field}\nA.~M. Tsvelik, Phys. Rev. B {\\bf 42}, 10499 (1990).\n\n\\bibitem{cabra_spin_s}\nD.~C. Cabra, P. Pujol, and C. {von Reichenbach}, Phys. Rev. B {\\bf 58}, \n 65\n (1998).\n\\bibitem{note_lehur}\nThe case $S=1$ at half filling\nhas been investigated by K. Lehur, Phys. Rev. Lett. {\\bf 83}, 848 \n(1999).\n A charge\ngap and gapless spin excitations described by the $SU(2)_1$ WZNW\nmodel were found. \n\n\n\n\n\\bibitem{witten_wz}\nE. Witten, Commun. Math. Phys. {\\bf 92}, 455 (1984).\n\n\\bibitem{affleck_houches}\nI. Affleck, in {\\em Fields, Strings and Critical Phenomena}, edited by \nE.\n Br\\'ezin and J. Zinn-Justin (Elsevier Science Publishers, Amsterdam, \n1988).\n\n\\bibitem{azaria_csl}\nP. Azaria, P. Lecheminant, \nand A.~A. Nersesyan, Phys. Rev. B {\\bf 58}, R8881 (1998); P. Azaria \nand P. Lecheminant, Chirally stabilized critical state in Marginally\n coupled spin and doped systems, 1999, cond-mat/9912406.\n\n\\bibitem{azaria_composite}\nP. Azaria and P. Lecheminant, Composite Pairings in Chirally Stabilized\n Critical Fluids, 2000, cond-mat/0001072.\n\n\\bibitem{andrei_chiral_nfl}\nN. Andrei, M.~R. Douglas, and A. Jerez, Phys. Rev. B {\\bf 58}, 7619 \n(1998).\n\n\\bibitem{note_andrei}\nSame arguments were recently applied to\n the Multichannel Kondo-Heisenberg lattice\nproblem [N. Andrei and E. Orignac, Low energy dynamics of the one \ndimensional\n multichannel Kondo-Heisenberg Lattice, 1999, cond-mat/9912372.]\n\n\\bibitem{coleman_kondo_odd_pair}\nP. Coleman, A. Georges, and A. Tsvelik, J. Phys. Condens. Matter {\\bf \n79}, 345\n (1997).\n\n\\bibitem{lecheminant_pc}\nP. Lecheminant, private communication.\n\n\\end{thebibliography}"
}
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cond-mat0002062
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Many-body solitons in a one-dimensional condensate of hard core bosons
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[
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"author": "M. D. Girardeau$^{1,2}$ and E. M. Wright$^2$"
}
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A mapping theorem leading to exact many-body dynamics of impenetrable bosons in one dimension reveals dark and gray soliton-like structures in a toroidal trap which is phase-imprinted. On long time scales revivals appear that are beyond the usual mean-field theory.
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[
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"name": "soliton.tex",
"string": "%soliton.tex : Exact soliton dynamics for 1d hard core bosons\n%\n%\\documentstyle[prl,preprint,aps]{revtex}\n\\documentstyle[prl,twocolumn,aps]{revtex}\n\\RequirePackage{epsfig}\n% A useful Journal macro\n\\def\\Journal#1#2#3#4{{#1} {\\bf #2}, #3 (#4)}\n%\n% Journal names used in this paper\n\\def\\PR{\\em Phys. Rev.}\n\\def\\PRL{\\em Phys. Rev. Lett.}\n\\def\\PRA{{\\em Phys. Rev.} A}\n\\def\\JMP{\\em J. Math. Phys.}\n\\def\\PO{\\em Progress in Optics}\n\\def\\PM{\\em Philos. Mag.}\n\\def\\JPII{{\\em J. Phys. II} (France)}\n\\def\\PF{\\em Phys. Fluids}\n\\def\\Nature{\\em Nature}\n\\def\\JETP{\\em Soviet Phys. JETP}\n\\def\\Science{\\em Science}\n\\def\\JPB{\\em J. Phys. B}\n%\n\\begin{document}\n\\draft\n\\title {Many-body solitons in a one-dimensional condensate\n of hard core bosons}\n\\author{M. D. Girardeau$^{1,2}$ and E. M. Wright$^2$}\n\\address{$^1$Institute of Theoretical Science, University of Oregon, \nEugene, OR 97403\\\\\n\t$^2$Optical Sciences Center and Department of Physics,\nUniversity of Arizona, Tucson, AZ 85721}\n \\date{\\today}\n\\maketitle\n\\begin{abstract}\nA mapping theorem leading to exact many-body dynamics of impenetrable bosons\nin one dimension reveals dark and gray soliton-like structures in a\ntoroidal trap which is phase-imprinted. On long time scales revivals appear\nthat are beyond the usual mean-field theory.\n\\end{abstract}\n\\pacs{03.75.Fi,03.75.-b,05.30.Jp}\n%\nDark and gray solitons are a generic feature\nof the nonlinear Schr\\\"{o}dinger\nequation with repulsive interactions, and several calculations of\ntheir dynamics\nbased on the Gross-Pitaevskii (GP) equation have appeared\n\\cite{Reinhardt,Dum,Scott,Jackson,Burger,Denschlag,Muryshev,Busch},\nas well as experiments demonstrating their existence in atomic BECs\n\\cite{Burger,Denschlag}.\nSince the GP equation is a nonlinear approximation to the more exact \n{\\em linear} many-body Schr\\\"{o}dinger equation, this raises the question of \nhow observed solitonic behavior arises in a theory which is linear at a\nfundamental level. Here this issue will be examined with the aid of\nexact many-body solutions. It has been shown by Olshanii \\cite{Olshanii} that \nat sufficiently low temperatures, densities, and\nlarge positive scattering length, a BEC in a thin atom waveguide\nhas dynamics which approach those of\na one-dimensional (1D) gas of impenetrable point bosons. This is\na model for which the exact many-body energy eigensolutions were\nfound in 1960 using an exact mapping from the Hilbert space of energy\neigenstates of an {\\em ideal} gas of fictitious spinless fermions to that of \nmany-body eigenstates of impenetrable, and therefore \n{\\em strongly interacting}, bosons \\cite{map,map2}. \nThe term ``Bose-Einstein condensation\" is used here in a\ngeneralized sense; it was shown by Lenard \\cite{Lenard} and by Yang and \nYang \\cite{Yang} that for the many-boson ground state of this system, the\noccupation of the lowest single-particle state is of order $\\sqrt{N}$ where \n$N$ is the total number of atoms, in contrast to $N$ for usual BEC.\nNevertheless, since $N\\gg 1$ and the momentum distribution has a sharp peak\nin the neighborhood of zero momentum \\cite{Olshanii}, this system shows strong\ncoherence effects typical of BEC. The response of a BEC of this type to \napplication of a delta-pulsed optical lattice was recently calculated by \nRojo {\\it et al.} \\cite{Rojo}, using the Fermi-Bose mapping theorem \n\\cite{map,map2}, as an exactly calculable model of dynamical optical lattice \nbehavior. They found spatial focussing and periodic self-imaging (Talbot \neffect), which decay as a result of interactions. This decay is absent in the \nGP approximation and therefore serves as a signature of many-body interaction \neffects omitted in GP.\n\nIn this Letter we examine the appearence of dark soliton-like structures\nusing the model of a 1D hard-core Bose gas\nin a toroidal trap, or ring, with cross section so small\nthat motion is essentially circumferencial\n\\cite{JavPaiYoo98,SalParRea99,BusAng99,BenRagSme97,Rok}. The Fermi-Bose\nmapping is employed to generate exact solutions for this problem.\nWe identify\nstationary solutions which reflect some properties of dark solitons\nfrom the GP theory when the ring is pierced at a point by an intense\nblue-detuned laser. We also present dynamical solutions when half of\nan initially homogeneous ring BEC is phase-imprinted via the light-shift\npotential of an applied laser, leading to gray soliton-like\nstructures whose velocity depends on the imposed phase-shift\n\\cite{Burger,Denschlag}. Such structures are apparent for times less than\nthe echo time $\\tau_e=L/c$, with $L$ the ring circumference and $c$ the\nspeed of sound in the BEC. On longer time scales the dynamics becomes\nvery complex showing Talbot recurrences which are beyond\nthe GP theory.\n\n{\\it Time-dependent Fermi-Bose mapping theorem:}\n%\nThe original proof\n\\cite{map,map2} was restricted to energy eigenstates, but the generalization\nto the time-dependent case is almost trivial. \nThe Schr\\\"{o}dinger Hamiltonian is assumed to have the structure\n\\begin{equation}\\label{eq1}\n\\hat{H}=\\sum_{j=1}^{N}-\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x_{j}^{2}}\n+V(x_{1},\\cdots,x_{N};t) ,\n\\end{equation}\nwhere $x_j$ is the one-dimensional position of the $j{\\it th}$ particle\nand $V$ is symmetric (invariant) under permutations of the particles.\nThe two-particle interaction \npotential is assumed to contain a hard core of 1D diameter $a$. This is\nconveniently treated as a constraint on allowed wave functions\n$\\psi(x_{1},\\cdots,x_{N};t)$:\n\\begin{equation}\\label{eq2}\n\\psi=0\\quad\\text{if}\\quad |x_{j}-x_{k}|<a\\quad,\\quad 1\\le j<k\\le N ,\n\\end{equation}\nrather than as an infinite contribution to $V$, which then consists of all\nother (finite) interactions and external potentials. The time-dependent \nversion starts from fermionic solutions $\\psi_{F}(x_{1},\\cdots,x_{N};t)$ of \nthe time-dependent many-body Schr\\\"{o}dinger equation (TDMBSE) \n$\\hat{H}\\psi=i\\hbar\\partial\\psi/\\partial t$ which are antisymmetric under\nall particle pair exchanges $x_{j}\\leftrightarrow x_{k}$, hence all\npermutations. As in the original theorem \\cite{map},\ndefine a ``unit antisymmetric function\"\n\\begin{equation}\\label{eq3}\nA(x_{1},\\cdots,x_{N})=\\prod_{1\\le j<k\\le N}\\text{sgn}(x_{k}-x_{j}) ,\n\\end{equation}\nwhere $\\text{sgn}(x)$ is the algebraic sign of the coordinate difference\n$x=x_{k}-x_{j}$, i.e., it is +1(-1) if $x>0$($x<0$). For given \nantisymmetric $\\psi_F$,\ndefine a bosonic wave function $\\psi_B$ by\n\\begin{equation}\\label{eq4}\n\\psi_{B}(x_{1},\\cdots,x_{N};t)=A(x_{1},\\cdots,x_{N})\\psi_{F}(x_{1},\\cdots,\nx_{N};t)\n\\end{equation}\nwhich defines the Fermi-Bose mapping. $\\psi_B$ satisfies\nthe hard core constraint (2) if $\\psi_F$ does, is totally\nsymmetric (bosonic) under permutations, obeys the same\nboundary conditions as $\\psi_F$, {\\it e.g.} periodic boundary conditions\non a ring, and\n$\\hat{H}\\psi_{B}=i\\hbar\\partial\\psi_{B}/\\partial t$ follows from\n$\\hat{H}\\psi_{F}=i\\hbar\\partial\\psi_{F}/\\partial t$ \\cite{map,map2}.\n\n{\\it Exact solutions for impenetrable point bosons:} The mapping theorem\nleads to explicit expressions for all many-body energy eigenstates and\neigenvalues of a 1D scalar condensate (bosons all of the same spin) \nunder the assumption that the only two-particle interaction is a\nzero-range hard core repulsion, represented by the $a\\rightarrow 0$\nlimit of the hard-core constraint. Such solutions were obtained in \nSec. 3 of the original work \\cite{map} for periodic\nboundary conditions and no external potential. The exact many body\nground state was found to be a pair product of Bijl-Jastrow form: \n$\\psi_{0}=\\text{const.}\\prod_{i>j}|\\sin[\\pi L^{-1}(x_{i}-x_{j})]|$.\nIn spite of the very long range\nof the individual pair correlation factors $|\\sin[\\pi L^{-1}(x_{i}-x_{j})]|$,\nthe pair distribution function $D(x_{ij})$, the integral of $|\\psi_{0}|^2$\nover all but two coordinates, was found to be of short range:\n$D(x_{ij})=1-j_0^2(\\pi\\rho x_{ij})$, with\n$j_{0}(\\xi)=\\sin\\xi/\\xi$, the spherical Bessel function of order zero.\nThe system was\nfound to support propagation of sound with speed $c=\\pi\\hbar\\rho/m$ where\n$\\rho=N/L$, the 1D atom number density. \n\nTo generalize to the time-dependent case, assuming that \nthe many-body potential of Eq. (1) is a sum of one-body \nexternal potentials $V(x_{j},t)$, one generalizes the time-independent \ndeterminantal many-fermion wavefunction \\cite{map} to a determinant \n\\begin{equation}\\label{eq5}\n\\psi_{F}(x_{1},\\cdots,x_{N};t)=C\\det_{i,j=1}^{N}\\phi_{i}(x_{j},t) ,\n\\end{equation}\nof solutions $\\phi_{i}(x,t)$ of the {\\em one-body} TDSE in the external \npotential $V(x,t)$. It then follows that $\\psi_F$ satisfies the TDMBSE, \nand it satisfies the impenetrability constraint (vanishing when any \n$x_{j}=x_{\\ell}$) trivially due to antisymmetry. Then by the mapping theorem \n$\\psi_B$ of Eq.(4) satisfies the same TDMBSE.\n\n{\\it Dark solitons on a ring:} Consider $N$ bosons in a tight toroidal trap,\nand denote their 1D positions measured around the circumference by\n$x_j$. This is equivalent to the exactly-solved model \\cite{map} of\n$N$ impenetrable point bosons in 1D with wave functions satisfying periodic\nboundary conditions with period $L$ equal to the torus circumference, and\nthe fundamental periodicity cell may be chosen as $-L/2<x_{j}<L/2$.\nHowever, the rotationally invariant quantum states of this problem do not\nreveal any dark soliton-like structures. To proceed we therefore\nconsider the case that a blue-detuned laser field pierces the ring\nat $x=0$ by virtue of the associated repulsive dipole force: The light\nsheet then provides a reference position for the null of the dark\nsoliton. Assume that the\nlight sheet is so intense and narrow that it may be replaced by a\nconstraint that the many-body wave function (hence the orbitals\n$\\phi_i$) must vanish whenever any $x_{j}=0$. Then\nthe appropriate orbitals $\\phi_{i}(x)$ are free-particle\nenergy eigenstates vanishing at $x=0$ and periodic with period $L$.\nThe complete orthonormal set of even-parity eigenstates $\\phi_{n}^{(+)}$\nand odd-parity eigenstates $\\phi_{n}^{(-)}$ are\n%Eq. (6)\n\\begin{eqnarray}\n\\phi_{n}^{(+)}(x) & = & \\sqrt{2/L}\\sin[(2n-1)\\pi|x|/L] , \\nonumber\\\\\n\\phi_{n}^{(-)}(x) & = & \\sqrt{2/L}\\sin(2n\\pi x/L) ,\n\\end{eqnarray}\nwith $n$ running from $1$ to $\\infty$. The odd eigenstates are\nthe same as those of free particles with no $x=0$ constraint, since these\nalready vanish at $x=0$. However, the even ones are strongly affected by the\nconstraint, their cusp at $x=0$ being a result of the impenetrable light sheet\nat that point. If one bends a 1D box $-L/2<x<L/2$ with impenetrable \nwalls into a ring, identifying the walls at $\\pm L/2$, then those\nparticle-in-a-box eigenfunctions which are even about the box center become\nidentical with the $\\phi_{n}^{(+)}$, and their cusp results from\nthe nonzero slope of these functions at the walls. The $N$-fermion ground state\nis obtained by inserting the lowest $N$ orbitals (6) into the determinant (5) \n(filled Fermi sea). Assume that $N$ is odd.\nSince $\\phi_{1}^{(+)}$ is lower than $\\phi_{1}^{(-)}$, this Fermi sea \nconsists of the first $(N+1)/2$ of the $\\phi_{n}^{(+)}$\nand the first $(N-1)/2$ of the $\\phi_{n}^{(-)}$. The $N$-boson\nground state is then given by (4). Since $A^{2}=1$, its one-particle \ndensity $\\rho(x)$ is the same as that of the $N$-fermion ground state,\nthe sum of partial densities contributed by all one-particle states in the \nFermi sea. Thus it is the sum of\n%Eq. (7)\n\\begin {equation}\n\\rho^{(+)}(x)\n=\\frac{N+1}{2L}-\\frac{\\sin[2(N+1)\\pi x/L]}{2L\\sin(2\\pi x/L)} ,\n\\end{equation}\nand\n%Eq. (8)\n\\begin{equation}\n\\rho^{(-)}(x)\n=\\frac{N-1}{2L}-\\frac{\\sin[(N-1)\\pi x/L]\\cos[(N-3)\\pi x/L]}\n{L\\sin(2\\pi x/L)}\n\\end{equation}\n%\nIn the thermodynamic limit\n$N\\rightarrow\\infty$, $L\\rightarrow\\infty$, $N/L\\rightarrow\\rho$ for fixed \n$x$, $\\rho^{(\\pm)}$ each contribute half of the total density\n$\\rho(x)$:\n%9\n\\begin{equation}\n\\rho(x)\\sim \\rho[1-j_{0}(2\\pi\\rho x)] .\n\\end{equation}\n%\nSince $j_{0}(0)=1$, $\\rho(x)$ vanishes at $x=0$ and approaches the mean\ndensity $\\rho$ over a healing length $L_h=1/2\\rho$ with damped spatial\noscillations about its limiting value. This differs in detail from the\ndensity $\\rho_\\infty\\tanh^{2}(x/w)$ of a GP dark soliton \\cite{Gin},\nwith $\\rho_\\infty$ the background density and $w$ the corresponding\nhealing length,\nbut has some qualitative similarity. However, it is only the odd\ncomponent $\\rho^{(-)}(x)\\approx\\rho(x)/2$ which has the feature of a dark\nsoliton that the corresponding odd orbitals have a $\\pi$ phase-jump at $x=0$\n(and also at $x=\\pm L/2$ to obey the periodic boundary conditions). But the\nodd and even components can never be separated physically, so the\nodd dark soliton-like component is always accompanied by the even\nnon-soliton component.\n\nNext, suppose that the light-sheet is turned off at $t=0$ by removing\nthe constraint that the wave function vanish at $x=0$.\nThe solution of the TDMBSB for the many-boson system is then given by\n(4) where the Slater determinant (5) is built from the first $(N+1)/2$\nof the $\\phi_{n}^{(+)}(x,t)$ and the first $(N-1)/2$ of the \n$\\phi_{n}^{(-)}(x,t)$, where these time-dependent orbitals are solutions\nof the single-free-particle TDSE which (a) reduce to the orbitals (6) at \n$t=0$, and (b) satisfy periodic boundary conditions with periodicity cell\n$-L/2<x<L/2$. The odd solutions are trivial: Since these never\n``see\" the $x=0$ constraint even for $t<0$, they differ from the odd orbitals\n(6) only by time-dependent phase shifts: \n$\\phi_{n}^{(-)}(x,t)=\\phi_{n}^{(-)}(x)e^{-i\\omega_{n}t}$ with \n$\\omega_{n}=\\hbar k_{n}^{2}/2m$ and $k_{n}=2n\\pi/L$. It\nfollows that $\\rho^{(-)}(x,t)$ is time-independent, and given in the\nthermodynamic limit by\n%10\n\\begin{equation}\n\\rho^{(-)}(x,t)\\sim (\\rho/2)[1-j_{0}(2\\pi\\rho x)] .\n\\end{equation}\n%\nThis further reinforces our view that the odd component of the density\nshares features of a dark soliton. The even-parity orbitals\n$\\phi_{n}^{(+)}(x,t)$ are complicated since the\nremoval of the light sheet constitutes a large, sudden perturbation. Indeed,\nthe periodic even-parity solutions of the free-particle \nSchr\\\"odinger equation are \n$\\chi_{p}^{(+)}(x)=\\sqrt{(2-\\delta_{p0})/L}\\cos(2p\\pi x/L)$\nwith $p=0,1,2,\\cdots$, and these are very different from the solutions\n$\\phi_{n}^{(+)}(x)$ with the $x=0$ constraint [Eq. (6)]. Nevertheless, since\nthe $\\chi_{p}^{(+)}(x)$ are complete for the subspace of even-parity, spatially\nperiodic functions, one can expand the $\\phi_{n}^{(+)}(x,t)$ in terms of the\n$\\chi_{p}^{(+)}(x)$, which evolve with time-dependent phases \n$e^{-i\\omega_{p}t}$ with $\\omega_{p}=\\hbar k_{p}^{2}/2m$ and\n$k_{p}=2p\\pi/L$. One finds \n%11\n\\begin{equation}\n\\phi_{n}^{(+)}(x,t)=\\frac{2(2n-1)}{\\pi}\\sqrt{\\frac{2}{L}}\n\\sum_{p=0}^{\\infty}\\frac{(2-\\delta_{p0})\\cos(k_{p}x)e^{-i\\omega_{p}t}}\n{(2n-1)^{2}-4p^{2}}\n\\end{equation}\n$\\rho^{(+)}(x,t)$ is the sum of absolute squares of the first\n$(N+1)/2$ of the sums (11), generalizing (7). Adding the time-independent \nexpression $\\rho^{(-)}(x,t)$, given in the thermodynamic limit by (10) or \nexactly by (8), one finds the time-dependent total density $\\rho(x,t)$.\nThere are two important time scales: One is the Poincar\\'{e}\nrecurrence time $\\tau_r$. Noting that $\\omega_p$ in (11) is proportional to\n$p^2$, one finds that all terms in the sum are time-periodic with period\n$\\tau_{r}=mL^{2}/\\pi\\hbar$, which is therefore the recurrence time for the\ndensity and in fact all properties of our model \\cite{Rojo}.\nThe other important time is the echo\ntime $\\tau_e$, the time for sound to make one circuit around the torus.\nRecalling\nthat the speed of sound in this system is $c=\\pi\\hbar\\rho/m$ \\cite{map},\none finds $\\tau_{e}=\\tau_{r}/N$. For $t<<\\tau_e$ after the constraint is\nremoved, the initial density develops sound waves that propagate around\nthe ring, and that we examine below in the context of phase-imprinting.\nFor $t>\\tau_e$ the evolution is very complex, but complete recurrences\noccur for times $t=n\\tau_r$ with fractional revivals in between.\n\n{\\it Gray soliton formation by phase-imprinting:} Consider next\na toroidal BEC in its ground state to which a phase-imprinting\nlaser is applied over half the ring at $t=0$. This is\ndescribed by the single-particle Hamiltonian\n%12\n\\begin{equation}\n\\hat{H}=\\sum_{j=1}^{N}\\left[-\\frac{\\hbar^2}{2m}\\frac{\\partial^2}\n{\\partial x_{j}^{2}}-\\hbar\\Delta\\theta\\delta(t)S(x_{j})\\right]\n\\end{equation}\nwhere $S(x)=\\theta(L/4-|x|)$, i.e., unity for $-L/4<x<L/4$ and\nzero elsewhere. This is the technique used in recent experiments\n\\cite{Burger,Denschlag}, here idealized to\na delta-function in time and to sharp spatial edges. Before the pulse the\nmost convenient free-particle orbitals in (5)\nare plane waves $\\phi_{n}(x)=\\sqrt{(1/L)}e^{ik_{n}x}$ where $k_{n}=2n\\pi/L$\nand $n=-n_{F},-n_{F}+1,\\cdots,n_{F}-1,n_{F}$ with $n_{F}=(N-1)/2$. Let\n$\\phi_{n}(x,t)$ be the solution\nof the TDSE with the Hamiltonian (12) reducing to the above\n$\\phi_{n}(x)$ just before the pulse. Then the solutions just\nafter the pulse are $\\phi_{n}(x,0+)=\\phi_{n}(x)e^{iS(x)\\Delta\\theta}$. The\npotential gradients at the pulse edges impart momentum kicks to the particles\nthere which induce both compressional waves propagating at the speed, $c$, \nof sound and density dips (gray solitons) moving at speeds $|v|<c$.\nThe expansion of $\\phi_{n}(x,t)$ in terms of the unperturbed plane waves is\nevaluated as\n%Eq.(13)\n\\begin{eqnarray}\n\\phi_{n}(x,t) & = & \\frac{1}{2}\\left(1+e^{i\\Delta\\theta}\\right)\n-\\frac{1-e^{i\\Delta\\theta}}{\\pi}\n\\sum_{\\ell=-\\infty}^{\\infty}\\nonumber\\\\\n& \\times & \\frac{(-1)^{\\ell}\\phi_{n-2\\ell-1}(x)\ne^{-i\\omega_{n-2\\ell-1}t}}{2\\ell+1}\n\\end{eqnarray}\n%\nand the time-dependent density is the sum of the absolute squares of the\nlowest $N$ of these. Figure \\ref{Fig:one} shows numerical simulations obtained\nusing Eq. (13) for $N=51$, $t/\\tau_e=0.051$, and\n$\\Delta\\theta=\\pi$ (solid line), and $\\Delta\\theta=0.5\\pi$\n(dashed line): due to symmetry we show only half\nof the ring $-L/2<x<0$, the phase-shift being imposed at $x=-L/4$.\nConsidering times short compared to the echo time means that the\ncorresponding results are not very sensitive to the periodic\nboundary conditions, and also therefore apply to a linear geometry.\nThe initial density profile is flat with a value $\\rho_0 L=51$.\nFor both phase-shifts two distinct maxima are seen,\nwhich travel at close to the\nspeed of sound $c$, and two distinct minima, which are analogous to\ngray solitons and travel at velocities $|v|/c<1$. \n%\n\\begin{figure}\n\\epsfxsize 3in\n\\epsfbox{figure1.soliton.eps}\n\\caption{Scaled density $\\rho(x,t)L$ versus scaled position around the\nring $x/L$ for $N=51$, $t/\\tau_e=0.051$, and $\\Delta\\theta=\\pi$\n(solid line), and $\\Delta\\theta=0.5\\pi$ (dashed line).\nDue to symmetry we show only half\nof the ring $-L/2<x<0$, the phase-jump being imposed at $x=-L/4$.}\n\\label{Fig:one}\n\\end{figure}\n%\nIn addition, there are\nalso high wavevector oscillations which radiate at velocities greater than\n$c$. In the case of a phase-shift\n$\\Delta\\theta=\\pi$, the density is symmetric about $x=-L/4$,\nwhereas for a phase-shift other than a multiple of $\\pi$ the evolution is\nnot symmetric, see the dashed line where the global minimum moves to the\nright in reponse to the phase-shift. \n%\n\\begin{figure}\n\\epsfxsize 3in\n\\epsfbox{figure2.soliton.eps}\n\\caption{Dark soliton velocity $|v|/c$ scaled to the speed of sound\n$c$ as a function of phase-shift $\\Delta\\theta/\\pi$ for $N=51$.}\n\\label{Fig:two}\n\\end{figure}\n%\nIn Fig. \\ref{Fig:two} we plot the \ncalculated\nvelocity of the global density minimum relative to the speed of sound\nfor a variety of phase-shifts $\\Delta\\theta$. The basic trend is that\nlarger phase-shift means lower velocity, in qualitative agreement with\nrecent experiments \\cite{Burger,Denschlag}, but there is a sharp velocity\npeak at $\\Delta\\theta\\approx 0.83\\pi$: This peak results from the cross-over\nbetween two local minima in the density. These general features,\nthe generation of gray solitons\nand density waves, agree with those of the GP theory, but here arise out of\nthe exact many-body calculation.\n\nIn conclusion, using our exactly-soluble 1D model we hope to have shown\nthat the dark solitonic features of atomic BECs normally described within\nthe mean-field GP theory arise naturally from consideration of the\nexact {\\it linear} many-body theory for times less than the echo time.\nAn advantage of this approach is\nthat it is number-conserving and does not rely on any symmetry-breaking\napproximation. In addition, long time dynamics such as collapses and\nrevivals are accounted for \\cite{Rojo}. A detailed comparison between\nour results and current experiments is not possible\nas they do not conform to the conditions\nfor a 1D system. However, some estimates are in order to set the\nappropriate time scales: If we consider $^{87}$Rb with a ring of\ncircumference $L=100$ $\\mu$m, and a high transverse trapping frequency\n$\\omega_\\perp=2\\pi\\times 10^5$ Hz, then we are limited to atom\nnumbers $N<300$ \\cite{Olshanii}, so these are small condensates.\nWe then obtain $\\tau_r=4.6$ s, and $\\tau_e=90$ ms for $N=51$.\nFinally, we remark that since our approach relied on the mapping\nbetween the strongly-interacting Bose system and a non-interacting\n``spinless Fermi gas\" model, this suggests that dark and gray solitons\nshould also manifest themselves in the density for the 1D Fermi system.\nAlthough real fermions have spin,\nthe interactions used here to generate solitons were spin-independent.\n\\vspace{0.2cm}\n\n\\noindent\nThis work was supported by the Office of Naval Research Contract\nNo. N00014-99-1-0806.\n%\n\\begin{references}\n%1\n\\bibitem{Reinhardt} W.P. Reinhardt and C.W. Clark, \\Journal{\\JPB}{30}{L785}\n{1997}.\n%\n%2\n\\bibitem{Dum} R. Dum {\\it et al.}, \\Journal{\\PRL}{80}{2972}{1998}.\n%\n%3\n\\bibitem{Scott} T.F. Scott, R.J. Ballagh, and K. Burnett, \\Journal{\\JPB}{31}\n{L329}{1998}.\n%\n%4\n\\bibitem{Jackson} A.D. Jackson, G.M. Kavoulakis, and C.J. Pethick, \n\\Journal{\\PRA}{58}{2417}{1998}.\n%\n%5\n\\bibitem{Burger} S. Burger {\\it et al.}, \\Journal{\\PRL}{83}{5198}{1999}.\n%\n%6\n\\bibitem{Denschlag} J. Denshlag {\\it et al.} \\Journal{\\Science}{287}\n{97}{1999}.\n%\n%7\n\\bibitem{Muryshev} A.E. Muryshev {\\it et al.}, \\Journal{\\PRA}{60}{R2665}{1999}.\n%\n%8\n\\bibitem{Busch} Th. Busch and J.R. Anglin, cond-mat/9809408.\n%\n%9\n\\bibitem{Olshanii} M. Olshanii, \\Journal{\\PRL}{81}{938}{1998}.\n%\n%10\n\\bibitem{map} M. Girardeau, \\Journal{\\JMP}{1}{516}{1960}.\n%\n%11\n\\bibitem{map2} M.D. Girardeau, \\Journal{\\PR}{139}{B500}{1965}. See particularly\nSecs. 2, 3, and 6.\n%\n%12\n\\bibitem{Lenard} A. Lenard, \\Journal{\\JMP}{7}{1268}{1966}.\n%\n%13\n\\bibitem{Yang} C.N. Yang and C.P. Yang, \\Journal{\\JMP}{10}{1115}{1969}.\n%\n%14\n\\bibitem{Rojo} A. G. Rojo, G. L. Cohen, and P. R. Berman, \\Journal{\\PRA}\n{60}{1482}{1999}.\n%\n\\bibitem{JavPaiYoo98} J. Javanainen, S. M. Paik, S. M. Yoo,\n\\Journal{\\PRA}{58}{580}{1998}.\n%\n\\bibitem{SalParRea99} L. Salasnich, A. Parola, and L. Reatto,\n\\Journal{\\PRA}{59}{2990}{1999}.\n%\n\\bibitem{BusAng99} Th. Busch and J. R. Anglin,\n\\Journal{\\PRA}{60}{R2669}{1999}.\n%\n\\bibitem{BenRagSme97} M. Benlaki {\\it et al.}, cond-mat/9711295.\n%\n\\bibitem{Rok} D. S. Rokhsar, cond-mat/9709212.\n%\n%15\n%\\bibitem{WWG} E. M. Wright, D. F. Walls, and J. C. Garrison,\n%\\Journal{\\PRL}{77}{2158}{1996}.\n%\n%16\n%\\bibitem{LY} M. Lewenstein and L. You, \\Journal{\\PRL}{77}{3489}{1996}.\n%\n%17\n%\\bibitem{Andrews1} M.R. Andrews {\\it et al.}, \n\\Journal{\\Science}{275}{637}{1997}.\n%\n%18\n%\\bibitem{Andrews2} M.R. Andrews {\\it et al.}, \\Journal{\\PRL}{79}{553}{1997}.\n%\n%19\n\\bibitem{Gin} V.L. Ginzburg and L.P. Pitaevskii, \\Journal{\\JETP}{34}{858}\n{1958}, Sec. 2; P.O. Fedichev, A.E. Muryshev, and G.V. Shlyapnikov,\n\\Journal{\\PRA}{60}{3220}{1999}.\n%\n\\end{references}\n\\end{document}\n\n\n"
}
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[
{
"name": "cond-mat0002062.extracted_bib",
"string": "\\bibitem{Reinhardt} W.P. Reinhardt and C.W. Clark, \\Journal{\\JPB}{30}{L785}\n{1997}.\n%\n%2\n\n\\bibitem{Dum} R. Dum {\\it et al.}, \\Journal{\\PRL}{80}{2972}{1998}.\n%\n%3\n\n\\bibitem{Scott} T.F. Scott, R.J. Ballagh, and K. Burnett, \\Journal{\\JPB}{31}\n{L329}{1998}.\n%\n%4\n\n\\bibitem{Jackson} A.D. Jackson, G.M. Kavoulakis, and C.J. Pethick, \n\\Journal{\\PRA}{58}{2417}{1998}.\n%\n%5\n\n\\bibitem{Burger} S. Burger {\\it et al.}, \\Journal{\\PRL}{83}{5198}{1999}.\n%\n%6\n\n\\bibitem{Denschlag} J. Denshlag {\\it et al.} \\Journal{\\Science}{287}\n{97}{1999}.\n%\n%7\n\n\\bibitem{Muryshev} A.E. Muryshev {\\it et al.}, \\Journal{\\PRA}{60}{R2665}{1999}.\n%\n%8\n\n\\bibitem{Busch} Th. Busch and J.R. Anglin, cond-mat/9809408.\n%\n%9\n\n\\bibitem{Olshanii} M. Olshanii, \\Journal{\\PRL}{81}{938}{1998}.\n%\n%10\n\n\\bibitem{map} M. Girardeau, \\Journal{\\JMP}{1}{516}{1960}.\n%\n%11\n\n\\bibitem{map2} M.D. Girardeau, \\Journal{\\PR}{139}{B500}{1965}. See particularly\nSecs. 2, 3, and 6.\n%\n%12\n\n\\bibitem{Lenard} A. Lenard, \\Journal{\\JMP}{7}{1268}{1966}.\n%\n%13\n\n\\bibitem{Yang} C.N. Yang and C.P. Yang, \\Journal{\\JMP}{10}{1115}{1969}.\n%\n%14\n\n\\bibitem{Rojo} A. G. Rojo, G. L. Cohen, and P. R. Berman, \\Journal{\\PRA}\n{60}{1482}{1999}.\n%\n\n\\bibitem{JavPaiYoo98} J. Javanainen, S. M. Paik, S. M. Yoo,\n\\Journal{\\PRA}{58}{580}{1998}.\n%\n\n\\bibitem{SalParRea99} L. Salasnich, A. Parola, and L. Reatto,\n\\Journal{\\PRA}{59}{2990}{1999}.\n%\n\n\\bibitem{BusAng99} Th. Busch and J. R. Anglin,\n\\Journal{\\PRA}{60}{R2669}{1999}.\n%\n\n\\bibitem{BenRagSme97} M. Benlaki {\\it et al.}, cond-mat/9711295.\n%\n\n\\bibitem{Rok} D. S. Rokhsar, cond-mat/9709212.\n%\n%15\n%\n\\bibitem{WWG} E. M. Wright, D. F. Walls, and J. C. Garrison,\n%\\Journal{\\PRL}{77}{2158}{1996}.\n%\n%16\n%\n\\bibitem{LY} M. Lewenstein and L. You, \\Journal{\\PRL}{77}{3489}{1996}.\n%\n%17\n%\n\\bibitem{Andrews1} M.R. Andrews {\\it et al.}, \n\\Journal{\\Science}{275}{637}{1997}.\n%\n%18\n%\n\\bibitem{Andrews2} M.R. Andrews {\\it et al.}, \\Journal{\\PRL}{79}{553}{1997}.\n%\n%19\n\n\\bibitem{Gin} V.L. Ginzburg and L.P. Pitaevskii, \\Journal{\\JETP}{34}{858}\n{1958}, Sec. 2; P.O. Fedichev, A.E. Muryshev, and G.V. Shlyapnikov,\n\\Journal{\\PRA}{60}{3220}{1999}.\n%\n"
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cond-mat0002063
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Characterization of One-Dimensional Luttinger Liquids in Terms of Fractional Exclusion Statistics
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"author": "Yong-Shi Wu $^1$ and Yue Yu $^2$ and Huan-Xiong Yang$^{3,2}$"
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We develop a bosonization approach to study the low temperature properties of one-dimensional gas of particles obeying fractional exclusion statistics (FES). It is shown that such ideal gas reproduces the low-energy excitations and asymptotic exponents of a one-component Luttinger liquid (with no internal degrees of freedom). The bosonized effective theory at low energy (or temperature) is identified to a $c=1$ conformal field theory (CFT) with compactified radius determined by the statistics parameter $\lambda$. Moreover, this CFT can be put into a form of the harmonic fluid description for Luttinger liquids, with the Haldane controlling parameter identified with the statistics parameter (of quasi-particle excitations). Thus we propose to use the latter to characterize the fixed points of 1-d Luttinger liquids. Such a characterization is further shown to be valid for generalized ideal gas of particles with mutual statistics in momentum space and for non-ideal gas with Luttinger-type interactions: In either case, the low temperature behavior is controlled by an effective statistics varying in a fixed-point line.
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"name": "cond-mat0002063.tex",
"string": "%cond-mat/0002063\n%Filename: Longp-2.tex\n%Version: Wu, Nov 08, 1999\n%Version: Yu, Sep 11, 1999\n\n%\\documentstyle[aps,preprint,eqsecnum]{revtex}\n\\documentstyle[twocolumn,prl,aps,eqsecnum]{revtex}\n\\begin{document}\n\\draft\n%%%%%%%%%%%%%%%%%%%%%%%\n\\title{Characterization of One-Dimensional Luttinger \nLiquids in Terms of Fractional Exclusion Statistics}\n%%%%%%%%%%%%%%%%%%%%%%%\n\\author{ Yong-Shi Wu $^1$ and Yue Yu $^2$ and Huan-Xiong Yang$^{3,2}$}\n\\address{1. Department of Physics, University of Utah, \nSalt Lake City, UT 84112, U.S.A.}\n\\address{2. Institute of Theoretical Physics, Chinese \nAcademy of Sciences, Beijing 100080, P. R. China}\n\\address{3. Department of Physics, Zhejiang University, \nHangzhou, 310027, P. R. China} \n%%%%%%%%%%%%%%%%%%%%%%%\n%\\date{\\today} \n%%%%%%%%%%%%%%%%%%%%%%%\n%\\receipt{}\n\\maketitle\n\\begin{abstract}\nWe develop a bosonization approach to study the \nlow temperature properties of one-dimensional \ngas of particles obeying fractional exclusion \nstatistics (FES). It is shown that such ideal \ngas reproduces the low-energy excitations and \nasymptotic exponents of a one-component \nLuttinger liquid (with no internal degrees \nof freedom). The bosonized effective theory at \nlow energy (or temperature) is identified to \na $c=1$ conformal field theory (CFT) with \ncompactified radius determined by the statistics \nparameter $\\lambda$. Moreover, this CFT can be \nput into a form of the harmonic fluid description \nfor Luttinger liquids, with the Haldane controlling \nparameter identified with the statistics parameter \n(of quasi-particle excitations). Thus we propose \nto use the latter to characterize the fixed points \nof 1-d Luttinger liquids. Such a characterization \nis further shown to be valid for generalized ideal \ngas of particles with mutual statistics in momentum \nspace and for non-ideal gas with Luttinger-type \ninteractions: In either case, the low temperature \nbehavior is controlled by an effective statistics \nvarying in a fixed-point line.\n\\end{abstract}\n\n%\\pacs{71.27+a,05.30-d,11.10.Kk,67.40Db}\n\n%\\begin{narrowtext}\n\n\\section{Introduction}\n\n\nIt is well-known that the Landau theory of Fermi \nliquids fails to describe most of one-dimensional \n(1-d) interacting many-body systems. To provide\na substitute, Haldane proposed, years ago, the \nconcept of the Luttinger liquid \\cite{Hald1}, \ndefined by a set of low-lying excitations and \ncritical exponents of the asymptotic correlation \nfunctions. Like Fermi liquids, there is a \n(pseudo-)Fermi surface for the quasiparticle-like \nexcitations in Luttinger liquids, so that the \nclassification of low-lying excitations is similar \nto that in Fermi liquids. However, the exponents \nof the asymptotic correlation functions (at low \ntemperature) are distinct from those for Fermi \nliquid theory. For one-component systems (without \ninternal degrees of freedom), the low-energy or \nlow-temperature behavior of a Luttinger liquid \nis controlled by a single parameter, the Haldane \ncontrolling parameter. It controls not only all \nexponents, but also the velocity ratios between \ndifferent types of elementary excitations. The \nFermi liquid theory is a special case of the \nLuttinger liquids with the Haldane parameter \n$\\lambda=1$. \n\nIn recent years, the failure of Landau's theory \nof Fermi liquids to describe several newly \ndiscovered strongly correlated electron systems \nhave revived the interests in the theory of \nLuttinger liquids. Among other questions, \ncompared with Fermi liquids, one would like \nvery much to know the answer to the following \nquestions: \n\\begin{itemize}\n\\item What is the physical meaning of Haldane's \ncontrolling parameter? Or more precisely, how to \nuse physical properties of low-lying excitations \nto characterize the concept of Luttinger liquids?\n\\item Does Haldane's theory of Luttinger liquids\npossess universality in one dimension, just like \nLandau's theory of Fermi liquids in three \ndimensions? Or equivalently, in terms of modern \nlanguage of renormalization group, does the \nLuttinger liquids describe the infrared (or \nlow-energy) fixed points in 1-d systems?\n\\item In what directions could one expect to \ngo for generalizing the concept of Luttinger \nliquids to higher than one dimensions? \n\\end{itemize}\nIn short, a characterization of 1-d Luttinger \nliquids, other than using a bunch of excitations\nand exponents, is in demand for gaining more \ninsights and looking for possible generalization. \n\nTo achieve this, let us recall what motivated \nLandau's concept of Fermi liquids, which is \nknown to describe an infrared fixed point (or \na universality class) of interacting electron \nsystems. The basic idea behind it is based on the \nfollowing {\\it organizing principle} for \ninteracting many-body systems: {\\it At low \ntemperature, the low-lying excited states of an \ninteracting many-body system above a stable ground \nstate can be viewed as consisting of weakly coupled \nelementary excitations.} Here \"weakly coupled\" only \nmeans that the total energy can be written as a sum \nof single-particle (dressed) energies, while the \ndispersion of the dressed energy may well depend on \nthe total particle number, a signal of remnant \ninteractions between the quasiparticles. According \nto Landau, the ground state and the low-lying excited \nstates of a Fermi liquid are approximately, to a good\naccuracy at sufficiently low temperature, described \nby those of an ideal Fermi gas with dressed energy \nfor the quasiparticles.\n\nWe note the significant role played by the ideal \nFermi gas distribution (with dressed energy) in \nthis description of Fermi liquids. Actually it is\nthe ideal Fermi gas that gives a characterization \nto the Fermi liquid fixed point, and a meaning to\nthe universality of the concept of Fermi liquids.\nThis inspires us to try to give a characterization \nof the 1-d Luttinger liquids along a similar line \nof thoughts, namely using a properly generalized \nconcept of {\\it exclusion statistics}, of which \na special case is the usual Fermi statistics.\nBecause the concept of quantum statistics in \nstatistical mechanics is independent of the \ndimensionality of a system, a characterization \nof 1-d infrared fixed points using statistics, if\nsuccessful, would shed light on how to generalize \nto higher dimensional non-Fermi liquids.\n \nFortunately, a generalization as such has been \navailable recently, under the name of fractional \nexclusion statistics (FES). It is based on a new \ncombinatoric rule for the many-body state counting \n\\cite{Hald2,Wu}, which is essentially an abstraction \nand generalization of Yang-Yang's state counting \n\\cite{YangYang,BerWu} in 1-d soluble many-body \nmodels. FES has been shown to be applicable to \nelementary excitations in a number of exactly \nsolvable models for strongly correlated systems \n\\cite{Hald2,Wu,BerWu,NaWil,Ha,Hatsu,WuYu},\nanyons in the lowest Landau level \\cite{Ouvry,Wu}, \nand quasiparticle excitations in the fractional \nquantum Hall effect \\cite{Hald2,Wu,Ha,HKWY}. The \nthermodynamics of the so-called generalized ideal \ngas (GIG) associated with FES have been studied \n\\cite{Wu} in a general framework. \n\nInspired by these results, the thoughts along the \nlines indicated in the above paragraphs have led \ntwo of present authors \\cite{WuYu1} to propose that \nat least {\\it for some strongly correlated systems \nor non-Fermi liquids, their low-energy or \nlow-temperature fixed point may be described by \na GIG associated with FES}, similar to the way \nthat of the Fermi liquid fixed point by the ideal \nFermi gas \\cite{RG}. As a testimony to this \nproposition, a sketchy proof was given in that \nshort letter \\cite{WuYu1} that the low-$T$ critical \nproperties of the 1-d Luttinger liquids are exactly \nreproduced by those of 1D ideal excluson gas(IEG), \nif one identifies the Haldane parameter of the \nformer with the statistics parameter $\\lambda$ of \nthe latter. (We call the particles obeying the \nFES without mutual statistics {\\it exclusons}). \nThrefore {\\it IEG can be used to describe the \nfixed points of the Luttinger liquids}. In this \npaper, we will present our results obtained \nin \\cite{WuYu1} in details, much of which was \nnot published before. \n\nA main tool we use in this study is bosonization\nof the 1-d excluson systems at low $T$, {\\it \\`a \nla} Tomonaga \\cite{Tomonaga} and Mattis and \nLieb \\cite{ML}. To bosonize an IEG system is a \nlittle bit tricky, because at low temperature the \nlinearized dispersion of dressed energy versus\npseudo-momentum has different slope outside and \ninside the pseudo-Fermi sea: There is `refraction' \nat both pseudo-Fermi points. In spite of this, \nwe still manage to construct well-defined density \nfluctuation operators that obey the $U(1)$ current\nalgebras and physically describe free phonons. Then, \nthe Tomonaga-Mattis-Lieb bosonization applies, \nresulting a bosonized effective field theory, in \nagreement with Haldane's harmonic fluid description \nof the Luttinger liquid \\cite{Hald3}. Then the \nasymptotic correlation functions and their \nexponents can be systematically calculated. In \nthis way, the critical properties of IEG reproduce \nthose of the Luttinger liquids. \n\nAn important consequence of our bosonization is \nthat the low energy behavior of IEG is controlled \nby an {\\it orbifold} conformal field theory (CFT) \nwith central charge $c=1$ and compactified radius \n\\cite{CFT} $R=\\sqrt{1/\\lambda}$. This variant of \n$c=1$ CFT is {\\it not} the ordinary $c=1$ CFT \ncompactified on a circle $S^1$, rather it is \ncompactified on an orbifold $S^1/Z_2$, which is \ntopologically an interval \\cite{CFT}. The \ndifferences arise due to different selection rules \nfor vertex operators, that constrain quantum \nnumbers of possible quasiparticle excitations in \nthe system. In the usual literature this difference \nquite often is overlooked. Only within the orbifold \nCFT the IEG with statistics parameter $\\lambda=1$ \nrecovers ideal Fermi gas, as it should be. Also the \ntwo classes of $c=1$ CFT's have different duality \nrelation; only the one in orbifold CFT reproduces \nthe known particle-hole duality in IEG, $\\lambda \n\\leftrightarrow 1/\\lambda$, as given in \n\\cite{BerWu,NaWil}. (For the details and more\nelaboration, see below. We note \\cite{WuYu}\nthat a similar situation happens for the \nCalogero-Sutherland (C-S) model\\cite{Cal,Suth}: \nThe low-energy effective field theory for the \nbosonic and fermionic C-S models belongs to, \nrespectively, the above-mentioned two classes \nof $c=1$ CFT.) \n\nThe fact that the low-$T$ behavior of IEG is \ncontrolled by a {\\it conformally invariant} \ntheory is significant, implying that indeed \nIEG provides a characterization of infrared \nfixed points, having the conformal invariance \nas required by renormalization group. We have \nalso studied the effects of mutual statistics \nbetween different pseudomomenta and of the \nLuttinger-type (density-density) interactions \namong exclusons. In either case, the low-$T$ \nbehavior is controlled by an effective \nstatistics $\\lambda_{eff}$ for excitations \nnear the Fermi points, the same way as \n$\\lambda$ in the case of IEG. In one \ndimension both the momentum-independent \npart of interactions and change in chemical \npotential $\\mu$ are {\\it relevant} \nperturbations \\cite{RG,Schulz}, leading to \na continuous shift in the fixed-point line \nparameterized by $\\lambda$. All these will be \nexplained in details in the present paper.\n\n\nTo make this paper self-contained, we devote \nthe next two sections, Sec. II and Sec. III, \nto reviewing the Luttinger liquid theory and \nthe GIG associated with FES, respectively. In \nSec. IV, we discuss the low-energy behavior \nof the IEG system and achieve its bosonization. \nIn Sec. V, the generalization to the GIG with \nmutual statistics as well as the non-ideal gas \nwith FES are provided. The last section is \ndedicated to conclusions and discussions. \n\n\n\n\\section{Luttinger liquid}\n\nThe Luttinger liquid, which describes a very\nlarge class of one-dimensional interacting \nmany-body systems, is introduced because of \nthe infrared divergence of certain vertices \nin the Fermi liquid description of the 1-d\nsystems. Some pioneering works have been done \nin the Luttinger model before the Luttinger \nliquid concept \\cite{Lutt,ML}. The model has \nbeen exactly solved by using the bosonization \ntechnique \\cite{ML}. Haldane \\cite{Hald1}\nre-solved the model with the following \nimportant observations:\n\n\\noindent{\n(i) Besides a linearized spectrum of non-zero \nmode excitations, i.e., the density fluctuations \n(sound waves), there are two kinds of zero mode \nexcitations, single-particle excitations by \nadding extra particles to the system and \npersistent currents by making Galileo boosts.\n}\n\n\\noindent{\n(ii) There is a fundamental relation among the \nvelocities of these three types of excitations\n}\n\\begin{equation}\nv_s = \\sqrt{v_N \\, v_J}, \n\\label{VeRe}\n\\end{equation} \nwhere $v_s$ is the sound velocity, $v_J$ the \ncurrent velocity and $v_N$ a velocity related \nto the change in particle number. The velocity \nratios define a controlling parameter, \n$e^{-2\\varphi}$, by \n\\begin{equation}\nv_N=v_se^{-2\\varphi},\\hspace{.2in}\nv_J=v_se^{2\\varphi}.\n\\label{VeRePh}\n\\end{equation}\n\n\\noindent{\n(iii) The above defined controlling parameter \nmeasures the essential renormalized coupling \nconstant, and is the unique parameter that \ndetermines the exponents of power-law decay \nin the zero-temperature correlation functions.\n}\n\nBased on these observations, Haldane defined \nthe Luttinger liquids as 1-d systems that have\nsimilar behavior (i)-(iii) at low temperature \njust like the Luttinger model. In this way the \nLuttinger liquids are characterized through \ntheir excitations and the exponents of the \nasymptotic correlation functions. \n\nTo be more precise, recall that the Luttinger \nmodel describes a one-dimensional interacting \nfermion system with the Hamiltonian\n\\begin{equation}\nH=\\int dx |\\nabla\\psi|^2+\\frac{1}{2}\n\\int\\int dx dy V(x-y)\\rho(x)\\rho(y).\n\\label{Hm}\n\\end{equation}\nIn the low energy limit, the Hamiltonian \n(\\ref{Hm}) can be bosonized as\n\\begin{equation}\nH=v_s\\sum_{q} |q| b_q^\\dagger b_q\n+\\frac{1}{2}(\\pi/L)(v_N\\, M^2+v_J\\,J^2),\n\\label{HmB}\n\\end{equation}\nwhere $b_q$ are the standard boson annihilation \noperators, and $M$ and $J$ the operators \ncorresponding to adding extra particle and \nboosting persistent currents, whose eigenvalues \nobey the following selection rule,\n\\begin{equation}\n(-1)^J=(-1)^M.\n\\label{select}\n\\end{equation}\nThe total momentum of the system also has \na bosonized form\n\\begin{equation}\nP=[k_F+(\\pi/)M]J+\\sum_qqb^\\dagger_qb_q,\n\\label{MmB}\n\\end{equation}\nwith $k_F$ being the Fermi momentum. \n\nEqs.(\\ref{VeRe},\\ref{VeRePh},\\ref{HmB}-\\ref{MmB}) \nturned to be universally valid for the description \nof the low-energy properties of gapless \ninteracting one-dimensional spinless fermion \nsystems even for those not exactly soluble with \na conserved current $J$. This universality class \nis named as the Luttinger liquid by Haldane \n\\cite{Hald1}. The Luttinger liquid has a \nmodel-independent representation, namely the \nharmonic fluid description \\cite{Hald3}, which \nis convenient for calculating the correlation \nfunctions. The results of the harmonic fluid\nrepresentation are listed in the Appendix, for \nlater use to be compared with our bosonization \ntheory of the IEG.\n\nThe Haldane theory of Luttinger liquids is \nbased on the significant observation that \nthe low-$T$ behavior of the Luttinger model \nis universal. Naturally arises the question: \nWhy is it so? In this paper we intend to \nanswer this question by pointing out a \nprofound coincidence of the low-$T$ behavior \nof the Luttinger model and that of ideal \nexcluson gas (IEG), i.e., ideal gas of particles \nobeying fractional exclusion statistics: The\nuniversality of the former is due to that of \nthe latter. \n\n\\section{Generalized ideal gas}\n\nIn quantum mechanics, there are two ways to \ndefine the statistics of particles. One is in \nterms of the symmetry of the many-body wave \nfunction under particle exchange. The other \nis based on the state counting. Here we are \ninterested in the latter definition. As is\nwell-known, bosons and fermions have different \ncountings for many-body states, or different \nstatistical weights $W$: The number of \nquantum states of $N$ particles occupying a \ngroup of $G$ states is, for bosons and \nfermions respectively, given by\n\\begin{equation}\nW_{b}= {(G+N-1)! \\over N!~ (G-1)!}~,~~~\n{\\rm{or}}~~ W_{f}= {G! \\over N!~ (G-N)!}~. \n\\label{2}\n\\end{equation}\nA simple interpolation between bosons and \nfermions is given by \\cite{Hald2,Wu}\n\\begin{equation}\n W = {[G+(N-1)(1-\\lambda)]!\n\\over N!~ [G-\\lambda N-(1-\\lambda)]!}~, \n\\label{2'}\n\\end{equation}\nwith $\\lambda=0$ corresponding to bosons \nand $\\lambda=1$ to fermions. The physical \nmeaning of this equation is the following: \nBy assumption, the statistical weight remains \nto be {\\it a single combinatoric number}, so \none can count the states by thinking of the \nparticles {\\it effectively either as bosons \nor as fermions}, with the effective number \nof available single-particle states being \n{\\it linearly dependent on the particle \nnumber}:\n\\begin{equation}\nG_{eff}^{(b)} = G - \\lambda (N-1), \\;\\;\\;\n{\\rm or}\\;\\;\\;\\; G_{eff}^{(f)} \n= G - (1-\\lambda) (N-1). \n\\label{3}\n\\end{equation}\nObviously, for genuine bosons (or fermions), \n$G_{eff}^{(b)}\\;$ (or $G_{eff}^{(f)}\\;$) is \nindependent of the particle number. In all \nother cases, either of the two $G_{eff}$ is \nlinearly dependent on the particle number. \nThis is the defining feature of the FES. The \nstatistics parameter $\\lambda$ tells us, on \nthe average, how many single-particle states \nthat a particle can exclude others to occupy. \nA proper understanding of this has been \ndiscussed in \\cite{WYS}. Thus, the expression \n(\\ref{2'}) for the statistical weight, $W$, \nformulates a generalized Pauli exclusion \nprinciple, as first recognized by Haldane \n\\cite{Hald2}.\n\nIt is easy to generalize this state\ncounting to more than one species,\nlabeled by the index $i$:\n\\begin{equation}\nW = {\\prod}_i ~ { [G_i + N_{i}-1 -\n\\sum_j \\lambda_{ij}(N_j-\\delta_{ij})]!\n\\over (N_i)!~ [G_i - 1- \\sum_j \\lambda_{ij}\n(N_j-\\delta_{ij})]! }~. \n\\label{4}\n\\end{equation}\nHere $G_i$ is the number of states when the \nsystem consists of only a single particle\nof species $i$. By definition, the diagonal \n$\\lambda_{ii}$ is the ``self-exclusion'' \nstatistics of species $i$, while the \nnon-diagonal $\\lambda_{ij}$ (for $i\\neq j$) \nis the mutual-exclusion statistics. Note \nthat $\\lambda_{ij}$, which Haldane \n\\cite{Hald2} called {\\it statistical \ninteractions}, may be {\\it asymmetric} in \n$i$ and $j$. The interpretation is similar \nto that of the one-species case: The number \nof available single-particle states for \nspecies $i$, in the presence of other \nparticles, is again linearly dependent on \nparticle numbers of all species:\n\\begin{eqnarray}\n&&G^{(b)}_{eff,i}= G_i -\\sum_{j}\n\\lambda_{ij} (N_j - \\delta_{ji}),\\nonumber\\\\ \n&& {\\rm or}\\nonumber\\\\\n&&G^{(f)}_{eff,i}= G^{(b)}_{eff,i}\n+ N_i -1. \\label{5}\n\\end{eqnarray}\n\nThe definition (\\ref{2'}) or (\\ref{4}) starts \nwith a postulated form for the statistical \nweight, and thus is more direct and convenient \nfor the purpose of formulating quantum \nstatistical mechanics. One of us \\cite{Wu} \nhas first formulated the quantum statistical \nmechanics by proposing the notion of generalized\nideal gas(GIG): A GIG satisfies the following \ntwo conditions: (i) The total energy (eigenvalue) \nis always of the form of a simple sum, in which \nthe $i$-th term is linear in the particle number \n$N_{i}$:\n\\begin{equation}\n E=\\sum_{i} N_{i} \\varepsilon^0_{i}, \n\\label{7}\n\\end{equation}\nwith $\\varepsilon^0_{i}$ identified as \nthe energy of a particle of species $i$; \n(ii) The state-counting (\\ref{4}) for \nstatistical weight $W$ is applicable. \nWhen there are no statistical interactions\n(i.e., $\\lambda_{ij}=0$ for $i\\neq j$), we\nhave the usual ideal gas, which we call\nas IEG. \n \nWith the assumptions (\\ref{7}) and (\\ref{4}), the \nthermodynamics of a GIG can be worked out by the \nusual techniques in statistical mechanics. \nConsider a grand canonical ensemble at temperature \n$T$ and with chemical potential $\\mu_{i}$ for \nspecies $i$, whose partition function is given by\n\\begin{equation}\nZ= \\sum_{\\{N_{i}\\}} W(\\{N_{i}\\})~\n\\exp \\{\\sum_{i} N_{i} (\\mu_{i} \n-\\varepsilon^0_{i})/T \\}~. \n\\label{8}\n\\end{equation}\nAs usual, we expect that for very large $N_{i}$, \nthe summation has a very sharp peak around the set \nof most-probable (or mean) particle numbers \n$\\{\\bar{N}_{i}\\}$. Using the Stirling formula,\nintroducing the average ``occupation number\nper state'' defined by \n$n_{i} \\equiv \\bar{N}_{i}/ G_{i}$, \nand maximizing\n\\begin{equation}\n{\\partial \\over \\partial n_{i}}\\,\n\\bigl[ \\ln W + \\sum_{i}\nG_{i} n_{i}\\,(\\mu_{i} -\n\\varepsilon^0_{i})/T \\bigr] =0~, \n\\label{9}\n\\end{equation}\none obtains the equations that determine the \nmost-probable distribution of $n_{i}$ \n\\begin{equation}\n\\sum_{j} (\\delta_{ij}w_j +g_{ij}) n_j = 1~, \n\\label{10}\n\\end{equation}\nwith $g_{ij}\\equiv \\lambda_{ij} G_{j}/G_{i}$, \nand $w_i$ being determined by the functional\nequations\n\\begin{equation}\n(1+w_i) \\prod_{j} \\Bigl({w_j\n\\over 1+w_j}\\Bigr)^{\\lambda_{ji}}\n= e^{(\\varepsilon^0_i-\\mu_{i})/T}. \n\\label{11}\n\\end{equation}\n\nThe thermodynamic potential $\\Omega=-T \\ln Z$ \nand the entropy $S$ are then given by\n\\begin{eqnarray}\n\\Omega &\\equiv &- PV = -T \\sum_i G_i\n\\log {1+ n_i - \\sum_j g_{ij} n_j\n\\over 1- \\sum_j g_{ij} n_j}~\\nonumber\\\\\n&=&-T \\sum_i G_i\\ln (1+w_i^{-1}); \\label{12}\n\\end{eqnarray}\n\\begin{eqnarray}\nS&=& \\sum_i G_i\n\\Bigl\\{ n_i {\\varepsilon^0_i - \\mu_{i} \\over T} +\n\\ln {1+ n_i - \\sum_j g_{ij} n_j\n\\over 1- \\sum_j g_{ij} n_j } \\Bigr\\}\\nonumber\\\\\n&=&\\sum_i G_i\\Bigl\\{ n_i {\\varepsilon^0_i \n- \\mu_{i} \\over T} + \\ln (1+w_i^{-1}) \\Bigr\\} . \n\\label{13}\n\\end{eqnarray}\nOther thermodynamic functions follow\nstraightforwardly. As usual, one can \neasily verify that the fluctuations, \n$({\\overline{{N_{i}}^{2}}}-{\\bar{N_{i}}}^{2})/\n{\\bar{N_{i}}}^{2}$, of the occupation numbers\nare negligible, which justifies the validity \nof the above approach.\n\n\\section{Bosonization of 1-d ideal excluson gas} \n\nLet us first consider the simplest case, the \n1-d IEG without internal degrees of freedom. \nWe expect to obtain a continuous interpolation\nbetween the usual ideal Bose and ideal Fermi gas. \nMoreover, we want to show that the low-energy \nbehavior of the IEG reproduces that of the \nLuttinger liquid and, therefore, provides a \nbetter characterization of the infrared fixed \npoints associated with the Luttinger liquid.\n\n\\subsection{Ideal Excluson Gas}\n \nConsider a GIG of $N$ particles on a ring with \nsize $L$. Single-particle states are labeled \nby pseudo-momenta $k_i$. The total energy and \nmomentum are given by \n\\begin{equation}\nE=\\sum k^2_i, \\hspace{.2in}P=\\sum k_i.\n\\end{equation}\nAccording to (\\ref{5}), in the thermodynamic \nlimit the hole density, $\\rho_a (k,T)$, (or \nthe density of available single-particle states) \nis {\\it linearly} dependent on the particle \ndensity, $\\rho(k,T)$. By definition, the \nstatistics interaction matrix is given by\n\\begin{equation}\n\\lambda(k_i,k_j)=\n-\\frac{\\Delta\\rho_a(k_i)}{\\Delta\\rho(k_j)}.\n\\end{equation}\nOr in the thermodynamic limit, one has\n \\begin{equation}\n\\lambda(k,k')= - \\, \\delta\\rho_a(k)/ \n\\delta\\rho(k').\n\\label{stat} \n \\end{equation}\nThe system is called an IEG of statistics \n$\\lambda$ (with {\\it no mutual statistics} \nbetween different momenta), if \n\\begin{equation}\n\\lambda(k_i,k_j)=\\lambda\\, \\delta(k_i-k_j),\n\\end{equation} \nor (\\ref{5}) reads\n\\begin{equation}\n\\rho(k_j)=\\frac{1}{2\\pi}\n+\\frac{1}{L}(1-\\lambda)\\sum_{i\\not=j}\n\\delta(k_j-k_i)\\rho(k_i)\\Delta k,\\label{rhoD}\n\\end{equation}\nwhich, in the thermodynamic limit, can be \nsimply written as \\cite{BerWu}\n\\begin{equation} \n\\rho_a(k,T)+\\lambda\\rho(k,T)=\\rho_{0}(k,T),\n\\label{RHOT}\n\\end{equation} \nwhere $\\rho_0(k)\\equiv 1/2\\pi$ is the bare \ndensity of single-particle states. Thus, \n$\\lambda=1$ corresponds to fermions, and \n$\\lambda=0$ to bosons. The thermodynamic \npotential, now reads, in terms of (\\ref{12})\n \\begin{equation}\n\\Omega=-\\frac{T}{2\\pi}\\int_{-\\infty}^\\infty dk \n\\,\\ln(1+w(k,T)^{-1}),\n\\label{Omega}\n\\end{equation}\nwith the function $w(k,T)\\equiv \n\\rho_{a}(k)/\\rho(k)$ \nsatisfying an algebraic equation,\n\\begin{equation}\nw(k,T)^{\\lambda} [1+w(k,T)]^{1-\\lambda}\n=e^{(k^2-\\mu)/T}. \n\\label{forW}\n\\end{equation}\n\nFirstly, we consider the ground state, in \nwhich the particles are distributed in a \nfinite and origin-symmetric interval in the \npseudo-momentum space. The (pseudo-)Fermi \nmomentum is defined by \n\\begin{equation}\nk_F^2=\\mu\n\\end{equation}\nand its value is fixed by the average \nparticle density \n$\\bar{d}_0=N_0/L$ in the ground state, \n\\begin{equation}\n\\int_{-k_F}^{k_F} dk\\rho(k)=\\bar{d}_0.\n\\end{equation}\nBecause holes are absent in the ground \nstate, the particle density in the ground \nstate is easily obtained from\n(\\ref{RHOT}),\n\\begin{equation}\n\\rho(k)=\\Biggl\\{\\begin{array}{ll}\n\\displaystyle{1\\over2\\pi\\lambda},\n&{\\rm for}~~ |k|<k_{F};\\\\\n0,&{\\rm for}~~ |k|>k_{F}.\n\\end{array}\\label{RHOZ}\n\\end{equation} \nHence, one has\n\\begin{equation}\nk_F=\\pi\\lambda\\bar{d}_0,\\hspace{.2in} \n\\mu=(\\pi\\lambda\\bar{d}_0)^2.\n\\end{equation}\nThen the ground state energy and momentum \nare given by\n\\begin{eqnarray}\n&&\\frac{E_0}{L}=\\int_{-k_F}^{k_F}dk \\rho(k)k^2\n={1\\over 3}\\pi^2\\lambda^2 \n\\bar{d}_{0}^{3},\\nonumber\\\\\n&&P_{0}=\\int_{-k_F}^{k_F}dk \\rho(k)k=0.\n\\end{eqnarray}\n\nNow let us examine possible excitations in an \nIEG. First there are density fluctuations due \nto particle-hole excitations, i.e., sound \nwaves with velocity (see the next subsection)\n\\begin{equation}\nv_{s} =v_{F}\\equiv 2k_{F}.\n\\label{VSF}\n\\end{equation}\nBesides, by adding extra $M$ particles to the \nground state, one can create particle excitations, \nand by Galileo boosts a persistent current. We \nobserve that the velocities of these three \nclasses of elementary excitations in IEG also \nsatisfy the fundamental relation (\\ref{VeRe}). \nIndeed, shifting $N_0$ to $N=N_0+M$, the change \nin the ground state energy is \n\\begin{eqnarray}\n\\delta_M E_0&=&{1\\over 3}\\pi^2\\lambda^2 \n(N/L)^{3}-{1\\over 3}\\pi^2\\lambda^2 (N_0/L)^{3}\n\\nonumber\\\\\n&=&\\pi^2\\bar{d}_0^2 M+\\pi(\\lambda k_F)M^2\n+O(M^3/L^3),\n\\end{eqnarray}\nwhile a persistent current, created by the \nboost of the Fermi sea $k\\to k+\\pi J/L$,\nleads to the energy shift \n\\begin{eqnarray}\n\\delta_JE_0 &=&\\int_{-k_F+\\pi J/L}^{k_F+\\pi J/L}dk \n\\rho(k)k^2-\\int_{-k_F}^{k_F}dk \\rho(k)k^2\\nonumber\\\\\n&=&\\pi(k_F/\\lambda)J^2.\n\\end{eqnarray}\nTherefore the total change in energy,\ndue to charge and current excitations, is\n\\begin{equation} \n\\delta E_0-\\mu M=\n\\frac{\\pi}{2L}v_F (\\lambda M^2\n+\\lambda^{-1} J^2).\n\\label{EMC}\n\\end{equation}\nThe total momentum change due to the current \nexcitations is\n\\begin{equation}\n\\delta P_0=\\sum_k \\frac{\\pi J}{L}\n=\\pi(\\bar{d}_0+\\frac{M}{L})J.\n\\label{MC}\n\\end{equation}\nIf we denote the variation in free energy as \n$\\delta F_0=\\delta E_0-\\mu M$, and identify \n$\\lambda$ as the controlling parameter \n$e^{-2\\varphi}$ in the Luttinger liquid \ntheory, (\\ref{EMC}) just recuperates the \nzero-mode contributions \\cite{comm0} in \n(\\ref{HmB}). Comparing (\\ref{EMC}) with\n(\\ref{HmB}) we identify the velocities\n$v_N$ and $v_J$ to be\n \\begin{equation}\nv_N=v_F\\lambda, \\qquad v_J=v_{F}/\\lambda, \n\\label{Velo}\n\\end{equation}\nThen we see the velocity relation (\\ref{VeRe}), \ni.e., $v_{s} = \\sqrt{v_{N}\\,v_{J}}$, that \nHaldane used to characterize the Luttinger \nliquids, is satisfied in IEG. The selection \nrule (\\ref{select}) also holds for the IEG, \nsince the system should correspond to the \nideal Fermi gas if $\\lambda=1$.\n\nEncouraged by this relationship between the \nIEG and Luttinger liquids, we want to calculate \nthe critical exponents of IEG to see whether\nthey reproduce those of the Luttinger liquids. \nThis motivates to develop a bosonization \nfor the density fluctuations in IEG.\n\n\\subsection{Low Energy Limit and Bosonization} \n\n%To see the non zero modes at low energy and\n%calculate the exponents in IEG, we need to \n%approach. \nFollowing Yang and Yang\\cite{YangYang,Suth}, \nwe introduce the dressed energy \n$\\epsilon (k,T)$ by writing \n \\begin{equation}\nw(k,T)=e^{\\epsilon (k,T)/T}.\n\\label{dressE}\n \\end{equation}\nThe point is that the grand partition function \n$Z_G$, corresponding to the thermodynamic \npotential (\\ref{Omega}), is of the form of that \nfor an ideal system of fermions with a \ncomplicated, $T$-dependent energy dispersion \ngiven by the dressed energy: \n\\begin{equation}\nZ_G=\\prod_k(1+e^{-\\epsilon (k,T)/T}).\n\\label{ptf}\n\\end{equation}\nHowever, this fermion representation is not \nvery useful, because of the implicit \n$T$-dependence of the dressed energy. To \nsimplify, we consider the low-$T$ limit. By \nusing the dressed energy, (\\ref{forW}) reads\n\\begin{equation}\n\\epsilon(k,T)=k^2-\\mu-T(1-\\lambda)\n\\ln(1+e^{-\\epsilon(k,T)/T}).\n\\end{equation}\nBecause there is no singularity in \n$\\epsilon(k,T)$ at $T=0$, the zero \ntemperature dressed energy is given by\n\\begin{equation}\n\\epsilon(k)=\\Biggl\\{ {\\begin{array}{ll}\n (k^2-k^2_F)/\\lambda , & |k|<k_F, \\\\\n \\;\\; k^2-k_F^2,&|k|>k_F.\n \\end{array}}\\label{EXP2}\n\\end{equation}\nDenote\n\\begin{equation}\n\\epsilon(k,T)=\\epsilon(k)\n+\\tilde{\\epsilon}(k,T),\n\\label{EXP1}\n\\end{equation}\nwhere \n\\begin{equation}\n\\tilde{\\epsilon}(k,0)=0.\n\\end{equation}\nIn the low-$T$ limit, one has\n\\begin{equation}\n\\epsilon(k,T)=\\Biggl\\{ \n{\\begin{array}{ll}\n\\frac{k^2-\\mu}{\\lambda}-(\\lambda^{-1}-1)T\n\\ln(1+e^{-|\\epsilon(k)|/T}),&|k|<k_F,\\\\\n(k^2-\\mu)-(1-\\lambda)T\n\\ln(1+e^{-|\\epsilon(k)|/T}),&|k|>k_F,\n\\end{array}}.\n\\label{lowTd}\n\\end{equation}\nHence, \n\\begin{equation}\n\\tilde{\\epsilon}(k)=\\Biggl\\{ \n{\\begin{array}{ll}\n(1-\\lambda^{-1})T\\ln(1+e^{-|\\epsilon(k)|/T}) , \n& |k|<k_F, \\\\\n\\;\\;(\\lambda-1)T\\ln(1+e^{-|\\epsilon(k)|/T}) ,\n&|k|>k_F.\n \\end{array}}\n\\label{EXP3}\n\\end{equation}\nFor low energies, one can consider only the \nexcitations around the Fermi surface,\n\\begin{eqnarray}\\displaystyle\n\\frac{\\Omega(T)}{L}\n&\\approx&-\\frac{T}{2\\pi}\\int_{-\\infty}^{\\infty}\ndk \\ln(1+e^{-\\epsilon(k)/T})\\nonumber\\\\\n&+&\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty dk\n\\frac{ \\tilde{\\epsilon}(k,T)}\n{1+e^{\\epsilon(k)/T}}\\nonumber\\\\\n&\\approx&\\frac{1}{2\\pi}\\int_{-k_F}^{k_F}\ndk \\epsilon(k)-\\frac{T}{\\pi\\lambda}\n\\int_{k_F-\\delta}^{k_F}dk\\ln(1\n+e^{-|\\epsilon(k)|/T})\\nonumber\\\\\n&&-\\frac{T}{\\pi}\\int^{k_F+\\delta}_{k_F}dk\n\\ln(1+e^{-|\\epsilon(k)|/T}), \n\\label{OmegaLT}\n\\end{eqnarray}\nwhere the first term on the right hand side \nof the last equality is recognized as \n$\\Omega(0)/L$. The cut-off $\\delta$ is of \norder $O(T/v_s)$ (actually, a few times of $T/v_s$). \nMathematically, we take the limit of $T\\to 0$ \nfollowed by $\\delta\\to 0$. Using the integral \nformula\n$$\n\\int_0^\\infty dx \\ln(1+e^{-x})=\\frac{\\pi^2}{12},\n$$\nwe have the low-$T$ thermodynamic potential\n\\begin{equation}\n\\frac{\\Omega(T)}{L}-\\frac{\\Omega(0)}{L}\n=-\\frac{\\pi T^2}{6v_s},\n\\end{equation}\nwhich implies that the theory is cut-off \nindependent at low temperature.\nNotice that $F=\\Omega-\\mu N$. Because we \nonly consider the particle-hole excitations \nnear the Fermi surface contribute to thermal \nexcitations, $N(T)-N(0)=0$, which can be \nchecked by an explicit calculation in terms \nof the definition of $\\rho(k,T)$. Thus, we have\n\\begin{equation}\n\\frac{F(T)}{L}-\\frac{F(0)}{L}\n=\\frac{\\Omega(T)}{L}-\\frac{\\Omega(0)}{L}\n=-\\frac{\\pi T^2}{6v_s}.\\label{FREE}\n\\end{equation}\nThis means that the low energy behavior of \nthe IEG is controlled by a $c=1$ CFT. This \nresult can be verified by a finite-size \nscaling in the spatial direction,\n\\begin{equation}\n\\frac{F_L(0)}{L}-\\frac{F(0)}{L}\n=-\\frac{\\pi v_s}{6L^2},\n\\end{equation}\nwhere $F_L(0)$ is the zero temperature free \nenergy for a system with size $L$. (For \ndetails, see \\cite{WuYu}.) \n\nThe above relation agrees with the finite-size \nscaling of a conformally invariant system with \ncentral charge $c=1$. So we want to see whether \nthe low-energy effective theory of the IEG is \nreally a CFT. Let us start with the grand \npartition function (\\ref{ptf}). At low \ntemperature, the solution (\\ref{lowTd}) \nleads to $\\tilde\\epsilon(k,T)=\nO(Te^{-|\\epsilon|/T})$, so one can simply \nreplace $\\epsilon(k,T)$ with $\\epsilon(k)$ \nin the grand partition function:\n\\begin{equation}\nZ_G\\simeq \\prod_k(1+e^{\\beta\\epsilon(k)}).\n\\end{equation}\nNote that the dressed energy with $k$ \noutside the Fermi points $\\pm k_F$ has a \nslope different from that with $k$ inside \n$\\pm k_F$. The former is $\\frac{2\\pi}{L}$ \nand the latter $\\frac{2\\pi\\lambda}{L}$. \nIt is necessary to keep this in mind for \nwriting down the correct ground state wave \nfunctions of the excluson system. Now that \nthe dispersion $\\epsilon(k)$ is $T$-independent, \nthe grand partition function in the low-$T$ \nlimit can be expressed in a fermionic \nrepresentation as\n\\begin{equation}\nZ_G={\\rm Tr}e^{-\\beta H_{\\rm eff}},\n\\end{equation}\nwhere the effective Hamiltonian is given by\n\\begin{equation}\nH_{\\rm eff}=\\sum_k {\\epsilon}(k) \n\\;c_k^\\dagger c_k,\n\\end{equation}\nwhere $c_k^\\dagger$ are fermionic creation \noperators. We also see that $\\epsilon(k_F)=0$,\nwhich can be used to define the Fermi momentum.\n\nPhysically, it is the phonon excitations that \ndominate the low-energy behavior of the system. \nIn the low-$T$ limit, it is enough to consider \nthe density fluctuations only near the Fermi \npoints, $k\\sim \\pm k_F$, where the left- and \nright-moving sectors are separable and \ndecoupled: \n\\begin{equation}\nH_{\\rm eff}=H_{+} + H_{-}.\n\\end{equation}\nBesides this, another important simplification \nfor excitations near Fermi points in the low-$T$ \nlimit is that their energy, $H_{\\pm}$, has a \nlinearized dispersion:\n\\begin{equation}\n\\epsilon_\\pm (k)=\\Biggl\\{ {\\begin{array}{ll}\n\\pm v_F(k\\mp k_F), \\;\\;\\; &|k|>k_F,\\\\ \n\\pm v_F (k\\mp k_F)/\\lambda, &|k|<k_F.\\\\\n\\end{array}}\n\\label{LNH} \n\\end{equation}\nWe note the `refractions' at $k=\\pm k_F$, which \nimplies to create a particle with pseudo-momentum \n$k$ and to create a hole with $k'$ cost different \nenergies, even if $|k-k_F|=|k'-k_F|$. The reason \nfor this is that $k$ is not the actual momentum \ncarried by $c_k^\\dagger$, as we will see soon.\n\nThe key thing for bosonization is to construct \na density fluctuation operator. Taking into \naccount the different slopes for dressed\nenergy inside and outside the Fermi points, \nthe density fluctuation operator at $k\\sim k_F$ \nis constructed as follows:\n\\begin{eqnarray}\n%\\begin{array}{rcl}\n&&\\rho_q^{(+)}=\\displaystyle \\sum_{k>k_F}\n:c^\\dagger_{k+q}c_k: \n+ \\displaystyle \\sum_{k<k_F-\\lambda q}\n:c^\\dagger_{k+\\lambda q}c_k:\n\\nonumber\\\\&&+ \\displaystyle \n\\sum_{k_F-\\lambda q< k < k_F}\n:c^\\dagger_{\\frac{k-k_F}{\\lambda}+k_F+q}c_k: \n%\\end{array}\\\\\n\\label{cdensity}\n\\end{eqnarray}\nfor $q>0$. A similar density operator \n$\\rho_q^{(-)}$ can also be defined at \n$k\\sim -k_F$,\n\\begin{eqnarray}\n%\\begin{array}{rcl}\n&&\\rho_q^{(-)}=\\displaystyle \n\\sum_{k<-k_F}:c^\\dagger_{k-q}c_k: \n+ \\displaystyle \\sum_{k>-k_F+\\lambda q}\n:c^\\dagger_{k-\\lambda q}c_k:\n\\nonumber\\\\ && \n+ \\displaystyle \\sum_{-k_F+\\lambda q> k >- k_F}\n:c^\\dagger_{\\frac{k+k_F}{\\lambda}-k_F-q}c_k: \n%\\end{array}\n\\end{eqnarray} \nTo define the normal ordering we write, \ne.g.,\n\\begin{equation}\nc_k=\\Biggl\\{{\\begin{array}{ll}\nc_k,&k>k_F,\\\\\nd^\\dagger_k,&k<k_F,\\end{array}}\n\\end{equation}\nwhere $d^\\dagger_k$ is understood as a creation \noperator of a hole. Then normal ordering is done\nas usual: putting the annihilation operators to\nthe right of the creation ones. Hence we have, \ne.g.,\n\\begin{eqnarray}\n&&\\rho_q^{(+)}=\\displaystyle \\sum_{k>k_F}\n:c^\\dagger_{k+q}c_k: \n+ \\displaystyle \\sum_{k<k_F-\\lambda q}\n:d_{k+\\lambda q}d^\\dagger_k:\\nonumber\\\\ \n&&+ \\displaystyle \n\\sum_{k_F-\\lambda q< k < k_F}\n:c^\\dagger_{\\frac{k-k_F}{\\lambda}+k_F+q}d^\\dagger_k: \n\\end{eqnarray}\n\nWithin the Tomonaga approximation \\cite{comm1}, \nin which commutators are taken to be their \nground-state expectation value, we obtain \n\\begin{eqnarray}\n&&[\\rho_q^{(\\pm)},\\rho_{q'}^{(\\pm)\\dagger}]\n\\approx \\langle 0|[\\rho_q^{(\\pm)},\n\\rho_q^{(\\pm)\\dagger}]|0\\rangle\\nonumber\\\\&&\n=\\sum_{k_F-\\lambda q< k < k_F}\n\\langle 0|c_{k+\\lambda q} \nc^\\dagger_{k+\\lambda q'}|0\\rangle\\nonumber\\\\\n&&=\\delta_{q, q'}\\sum_{k_F-\\lambda q< k < k_F}\n1= \\frac{ L}{2\\pi} q\\delta_{q,q'}\n\\label{density} \n\\end{eqnarray}\nAlso, the commutators between $H_{\\rm eff}$ \nand $\\rho_q^{(\\pm)}$ are\n\\begin{equation}\n[H_{\\pm},\\rho_q^{(\\pm)}]\\approx\n\\langle0|[H_{\\pm},\\rho_q^{(\\pm)}]|0\\rangle\n= \\pm v_Fq\\rho^{(\\pm)}_q. \n\\label{denHam}\n\\end{equation}\n(\\ref{density}) and (\\ref{denHam}) describe 1-d \nfree phonons with the sound velocity $v_{s}=v_F$ \n(so we have proved (\\ref{VSF})). Introducing \nnormalized boson annihilation operators \n\\begin{equation}\nb_q=\\sqrt{2\\pi/ qL}\\,\\rho_q^{(+)}, \n\\hspace{.2in} \\tilde{b}_q=\\sqrt{2\\pi/ qL}\\,\n{\\rho}_q^{(-)\\dagger}\n\\end{equation} \nand adding back the zero mode contributions, \nthe bosonized Hamiltonian satisfying \n(\\ref{density}) is given by \n\\begin{equation}\nH_B=v_s\\{ \\sum_{q>0}q(b_q^\\dagger b_q\n+\\tilde{b}_q^\\dagger \\tilde{b}_q)\n+\\frac{1}{2}\\frac{\\pi}{L}[\\lambda M^2\n+\\frac{1}{\\lambda} J^2] \\},\n\\label{bosonH}\n\\end{equation}\nwhich agrees with the bosonized Hamiltonian \n(\\ref{HmB}) in the Luttinger liquid theory.\n\nIn passing, we make a comment on linearization\nof the dressed energy dispersion. When we did \nthis, we changed the ground state energy, \nbecause we assumed that for all $k$ the \nspectrum is linear in $k$. However, we changed \nneither the ground state wave function, nor \nthe low-$T$ physics. On the other hand, the \nlinearized spectrum was valid only for phonon \nexcitations, it has nothing to do with the\nzero-mode excitations. So, after the linearized \nphonon part of the Hamiltonian is bosonized, we \nhad to add back the zero-mode excitations. \n\nThe construction of the bosonized momentum \noperator is a bit more tricky, because \n$c_k^\\dagger$ does not carry a momentum $k$. \nEach term in (\\ref{cdensity}) should carry \nthe same momentum $q$, therefore the fermion \ncreated by $c_{k}^\\dagger$ carries a dressed \nmomentum $p$, which is related to $k$ by\n\\begin{eqnarray}\np(k)=\\Biggl\\{ {\\begin{array}{lll}\n k-k_F+ (k_F/\\lambda),\\;\\;\\; & k>k_F, \\\\\n \\;\\; k/\\lambda, & |k|<k_F,.\\\\\n k+k_F- (k_F/\\lambda), & k<-k_F.\\\\\n\\end{array}}\n\\label{dressP}\n\\end{eqnarray}\nIn terms of this variable, the linearized dressed \nenergy $\\epsilon (p)$ is of a simple form: \n$\\epsilon_{\\pm}(p)=\\pm v_s(p\\mp p_F)$, with\n$p_F=k_F/\\lambda$. The bosonized total momentum \noperator, corresponding to the fermionized \n$P=\\sum_{k} p(k)\\, c_{k}^{\\dagger} c_{k}$, is \n\\begin{equation}\nP=\\displaystyle\\sum_{q>0}q(b_q^\\dagger b_q-\n\\tilde{b}_q^\\dagger \\tilde{b}_q)\n+ \\pi (\\bar{d}_0+M/L)\\,J.\\\\\n\\label{bosonP}\n\\end{equation}\n\nWe see that the fundamental velocity relation, \nthe bosonized Hamiltonian and momentum, and \nthe selection rule of the quantum numbers in \nthe Luttinger liquid theory can all be \nreproduced in IEG if we identify\n\\begin{equation}\n\\lambda\\equiv e^{-2\\varphi}.\n\\label{EQV}\n\\end{equation}\nTo say that IEG can be used to characterize \nthe renormalization group {\\it fixed points} of Luttinger liquids, \nwe still need to check the conformal \ninvariance of the bosonized theory of IEG, \nand to verify the critical properties of \nIEG reproduce those of the Luttinger liquids. \n\n\\subsection{Effective Field Theory and \nConformal Invariance}\n\nTo check conformal invariance, we need \nto rewrite the above bosonized effective \nHalmitonian (\\ref{bosonH}) into a form \nof field theory in coordinate space. \nEmploying the Fourier transformation, \nthe density operator can be written as \n\\begin{eqnarray}\n\\rho(x)&=&\\rho_R(x)+\\rho_L(x),\\nonumber\\\\ \n\\rho_R(x)&=&\\frac{M_R}{L}+\n\\displaystyle\\sum_{q>0}\n\\sqrt{\\frac{q}{2\\pi L\\lambda}}\n(e^{iqx}b_q+e^{-iqx}b_q^\\dagger),\\nonumber\\\\\n\\rho_L(x)&=&\\displaystyle\\frac{M_L}{L}+\n\\displaystyle\\sum_{q>0}\n\\sqrt{\\frac{q}{2\\pi L\\lambda}}\n(e^{-iqx}\\tilde{b}_q+e^{iqx}\n\\tilde{b}_q^\\dagger),\n\\end{eqnarray}\nwhere $M_{R,L}$ are given by $M=M_R+M_L$ \nand $\\tilde{b}_q=b_{-q}$ for $q>0$.\n\nThe boson field $\\phi(x)$, which is \nconjugated to $\\rho(x)$ and satisfies \n\\begin{equation}\n[\\phi(x),\\rho(x')]=i\\delta(x-x'),\n\\end{equation}\nis given by \n\\begin{eqnarray}\n\\phi(x)&=&\\phi_R(x)+\\phi_L(x),\\nonumber\\\\\n\\phi_R(x)&=& \\frac{\\phi_0}{2}\n+\\frac{\\pi J_Rx}{L}+i\\displaystyle \n\\sum_{q>0}\\sqrt{\\frac{\\pi \\lambda}{2qL}}\n(e^{iqx}b_q-e^{-iqx}b_q^\\dagger),\\nonumber\\\\\n\\phi_L(x)&=& \\frac{\\phi_0}{2}\n+\\frac{\\pi J_Lx}{L}+i\\displaystyle \n\\sum_{q>0}\\sqrt{\\frac{\\pi \\lambda}{2qL}}\n(e^{-iqx}\\tilde{b}_q-e^{iqx}\\tilde{b}_q^\\dagger),\n\\end{eqnarray}\nwith $J=J_R+J_L$. We have to assign the quantum \nnumbers such that there are only two independent \neach other in $M_{R,L}$ and $J_{R,L}$. A \nconsistent choice is\n\\begin{equation}\nM_R=J_R,\\hspace{.2in}M_L=-J_L.\n\\end{equation}\nThen,\n\\begin{equation}\nJ=J_R+J_L,\\hspace{.2in}M=J_R-J_L.\n\\end{equation}\nHere $\\phi_0$ is an angular variable\nconjugated to $M$: $[\\phi_0,M]=i$. \nThe Hamiltonian (\\ref{HmB}) becomes\n\\begin{equation}\nH =\\frac{1}{2}\n\\int_0^Ldx\\; [\\pi v_N\\rho(x)^2\n+v_J/\\pi~(\\partial_x\\phi(x))^2],\n\\end{equation}\nor by a field rescaling,\n\\begin{equation}\nH =\\frac{v_s}{2\\pi}\n\\int_0^Ldx\\; [\\Pi(x)^2+(\\partial_xX(x))^2],\n\\label{fieldH}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Pi(x)=\\pi \\lambda^{1/2}\\rho(x),\\hspace{.2in} \nX(x)=\\lambda^{-1/2}\\phi(x).\n\\end{equation}\n\nWith $X(x,t)=e^{iHt}X(x)e^{-iHt}$, the \nLagrangian density reads \n\\begin{equation}\n{\\cal L}=\\frac{v_s}{2\\pi}\\,\\partial_\\alpha \nX(x,t)\\,\\partial^\\alpha X(x,t).\n\\label{LGL}\n\\end{equation}\nThis is the Lagrangian density of a free \nscalar field theory in $1+1$-dimensions.\nWriting the corresponding operators as the \nfunctionals of the scalar field $X(x,t)$, all \ncorrelation functions can be obtained by using \nthe propagators of $X_R(x,t)$ and $X_L(x,t)$,\n\\begin{eqnarray}\n\\langle X_R(x,t)X_R(0,0)\\rangle\n=-\\frac{1}{4}\\ln(x-v_st),\\nonumber\\\\\n\\langle X_L(x,t)X_L(0,0)\\rangle\n=-\\frac{1}{4}\\ln(x+v_st).\n\\label{prop}\n\\end{eqnarray}\nThe statistics of an operator in the \ntheory can also be inferred by the \ncommutators of the scalar fields,\n\\begin{equation}\n[X_{R,L}(x),X_{R,L}(x')]=\n\\pm\\frac{i\\pi}{4}\\theta(x-x').\n\\label{Comm}\n\\end{equation}\n\nWe recognize that $\\cal{L}$ (\\ref{LGL}) is \nthe Lagrangian of a $c=1$ CFT\\cite{CFT}, \nconsistent with the finite-size scaling\n(\\ref{FREE}). Alternatively, it is easy to \ncheck that the theory is invariant under \nthe conformal transformations generated \nby a set of the Virasoro generators\n\\begin{equation}\nL_m=\\frac{1}{2}\\sum_{n=-\\infty}^{\\infty} \n\\alpha_{n-m}\\alpha_n, \\hspace{.2in} \n\\tilde{L}_m=\\frac{1}{2}\\sum_{n=-\\infty}^{\\infty} \n\\tilde{\\alpha}_{n-m}\\tilde{\\alpha}_n,\n\\end{equation}\nwhere the oscillators $\\alpha_m=m^{1/2}b_q$ \nand $\\tilde{\\alpha}_{-m}=m^{1/2}\\tilde b_q^\\dagger$ \nfor $m=qL/2\\pi>0$ being integers. $\\alpha_0=\n(\\pi/2L)^{1/2}[J\\lambda^{-1/2}-M\\lambda^{1/2}]$ \nand $\\tilde{\\alpha}_0=\n(\\pi/2L)^{1/2}[J\\lambda^{-1/2}+M\\lambda^{1/2}]$.\nThe generators obey the Virasoro algebra with \nthe central charge $c=1$,\n\\begin{equation}\n[L^{\\rm tot}_m,L^{\\rm tot}_n]\n=(m-n)L^{\\rm tot}_{m+n}+\\frac{1}{12}(m^3-m)\n\\delta_{m+n,0},\n\\end{equation}\nwith $L^{\\rm tot}_m=L_m+\\tilde{L}_m$.\n\nSince $\\phi_{0}$ is an angular variable, \nthere is a hidden invariance in the theory \nunder $\\phi\\to\\phi+2\\pi$. The field $X$ is \nthus said to be ``compactified'' on a circle, \nwith a radius that is determined by the \nexclusion statistics \\cite{WuYu,sm}: \n\\begin{equation}\nX\\sim X+2\\pi R,\\;\\;\\; R^2=1/\\lambda.\n\\label{INV}\n\\end{equation}\n\nNoting the selection rule (\\ref{select}), \nthe Hamiltonian has a duality\n\\begin{equation}\n\\lambda\\leftrightarrow 1/\\lambda,\n\\hspace{.2in} M\\leftrightarrow J, \n\\end{equation}\nwhich has referred to the particle-hole \nduality \\cite{BerWu,NaWil}. Using the CFT \nterminology, this duality is represented \nas the duality of the compactified radii,\n\\begin{equation}\nR\\leftrightarrow 1/R.\n\\label{dua2}\n\\end{equation}\nWe note that this is different from the\nduality relation $R\\leftrightarrow 2/R$,\nin the usual $c=1$ CFT \\cite{CFT} \ncompactifed on a circle. Actually, \naccording to the standard terminology\nin CFT \\cite{CFT}, our selection rule\n(\\ref{select}) and duality relation \n(\\ref{dua2}) make what we obtained above\na $c=1$ CFT compactified on an {\\it orbifold}\n$S^1/Z_2$, i.e., a circle folded by a \nreflection about a diameter, which \ntopologically is a semi-circle or an interval. \nThis difference can also be seen from the\ngrand partition function: Using the \nidentification between $H_{\\rm eff}$ and \n$L^{\\rm tot}_0$, i.e., $H_{\\rm eff}=v_s L^{\\rm tot}_0$, \nthe grand partition function of IEG (in the \nlow-$T$ limit) can be rewritten as \n\\begin{equation}\nZ_G= Tr_{\\cal H}[q^{L_0} \\bar{q}^{\\tilde{L}_0}],\n\\end{equation}\nwhere $q=e^{iv_s\\tau}$ with $\\tau=i\\beta=i/T$. \nThus, the selection rule (\\ref{select}) \nseverely constrain the allowed values for \nthe eigenvalues of $L_0$ and $\\tilde{L}_0$. \nIt makes the CFT we obtained have an unusual \nspectrum and duality relation, corresponding \nto the $c=1$ orbifold CFT \\cite{CFT}. In next\nsubsection we will see that because of the\ndifference in the selection rules, the \nstatistics of the allowed charge-1 operators \nin the two classes of CFT's are not the same. \n\nWe note that a similar situation happens \nfor the CFT that describes the low-$T$ \nbehavior of the Calogero-Sutherland (C-S) \nmodel \\cite{Cal,Suth}. This model has two \ndifferent versions, with the long-range \ninteractions being among bosons or among \nfermions, respectively. At low temperature, \nthe two versions have different selection \nrules for the zero-mode quantum numbers, \nthus leading to different CFT's: The \nlow-$T$ CFT for the bosonic C-S model is \nthe usual $c=1$ CFT compactified on a \ncircle, which has been studied extensively \nin the literatures \\cite{KY,sm,ISO,ISR,CAR,WuYu}; \nwhile for the fermionic C-S model the low-$T$ \nlimit gives rise to the $c=1$ orbifold CFT. \nThis is because the selection rule for zero \nmodes severely constrains the spectrum of \nthe system, i.e., possible quantum numbers \nof the allowed excitations. (For details, \nsee ref. \\cite{WuYu}.) Thus, only the \nfermionic (not the bosonic) C-S model \nrespects a duality relation $\\lambda\n\\leftrightarrow 1/\\lambda$ that coincides\nwith the particle-hole duality in IEG \n\\cite{BerWu,NaWil}.\n \n\n\\subsection{Correlation Functions}\n\nThe CFT description of the IEG offers a better \nunderstanding for the space of quantum states \nin the theory. States $V[X]|0\\rangle$ or \noperators $V[X]$ are {\\it allowed} only if \nthey respect the invariance (\\ref{INV}),\n\\begin{equation}\nV[X+2\\pi R]\\equiv V[X], \n\\end{equation}\nwith a given boundary condition restriction. \nHere, a Fermi or a Bose operator obeys the \nperiodic boundary condition (PBC). So \nquantum numbers of quasiparticles are \nstrongly constrained, in particular by the \nselection rule for zero-mode quantum numbers. \nFor example, the primary fields obeying the \nPBC in the CFT are given by\n\\begin{eqnarray}\n\\phi_{M,J}(x)&\\sim& f(J,X^0)\n:e^{i(M\\lambda^{1/2}+J/\\lambda^{1/2})X_R(x)}\\nonumber\\\\\n&\\times&e^{i(M\\lambda^{1/2}-J/\\lambda^{1/2})X_L(x)}:,\n\\nonumber\\\\\nf(J,X^0)&=&e^{iJ(\\lambda^{1/2}\n-\\lambda^{-1/2})X^0_R}\ne^{-iJ(\\lambda^{1/2}-\\lambda^{-1/2})X^0_L}\n\\end{eqnarray}\nwhere the prefactor $f(J,X^0)$ makes the fields \nsatisfy the PBC, $M$ and $J$ eigenvalues of the \nnumber and current operators, and $X^0=\\pi Mx/L$. \nThe field carries the charge $M$ and current $J$. \nThe conformal dimensions of the fields are\n\\begin{eqnarray}\nh&=&\\frac{1}{2}[(M\\lambda^{1/2}+J/\\lambda^{1/2})^2\n+(M\\lambda^{1/2}-J/\\lambda^{1/2})^2]\\nonumber\\\\\n&=&M^2\\lambda+J^2\\lambda^{-1}.\n\\end{eqnarray}\nThe statistics of the field can be calculated \nby using (\\ref{Comm}) and the statistics \nfactors are \n\\begin{eqnarray}\n&&\\exp\\{i{\\pi\\over 4}[(M\\lambda^{1/2}\n+J/\\lambda^{1/2})^2-(M\\lambda^{1/2}\n-J/\\lambda^{1/2})^2]\\}\\nonumber\\\\\n&=&(-1)^{MJ}.\n\\end{eqnarray}\nConsider the charge-1 primary fields, with $M=1$. \nTherefore, they can only be fermions since $J=$ \nodd due to the selection rule. The general \ncharge-1 fermion operator is a linear combination \nof the charge-1 primary fields. A careful \nconstruction of the allowed fermion field with \nunit charge leads to \n\\begin{eqnarray}\n\\Psi^\\dagger_F(x,t)=\\rho(x)^{1/2}\n&&\\sum_{m=-\\infty}^{\\infty} e^{iO_m}\n:e^{i(\\lambda^{1/2} + (2m+1)/\\lambda^{1/2})X_{R}(x_-)}:\n\\nonumber \\\\\n&& :e^{i(\\lambda^{1/2}-(2m+1)/\\lambda^{1/2})X_{L}(x_+)}:\n\\; ,\n\\label{fermiPsi}\n\\end{eqnarray}\nwhere the prefactor $f$ has been suppressed and \nthe hermitian, constant-valued operators $O_m$ \nsatisfy \\cite{comm2}\n\\begin{equation} \n[O_m, O_{m'}]=i\\pi(m-m').\n\\end{equation}\n\nThe multi-sector density operator is the \nlinear combination of those primary fields \nwith $M=0$ and $J=$even,\n\\begin{eqnarray}\n\\hat{\\rho}(x)&=&\\Psi_F^\\dagger(x)\n\\Psi_F(x)\\nonumber \\\\\n&=& \\rho(x)\\sum{}_m \n:\\exp\\{i2m[X_R(x)-X_L(x)]/\\lambda^{1/2}\\}:.\n\\end{eqnarray}\n\nAll the secondary fields in the CFT follow by \nconsidering the sound wave contribution to the \nconformal weight of the fields.\n\nThe correlation functions can easily be \ncalculated by using the CFT techniques. For \nexamples, the density-density and single\nparticle correlation functions are as follows,\n\\begin{eqnarray}\n\\langle \\hat{\\rho}(x,t)\\hat{\\rho}(0,0)\\rangle \n&\\approx&\\bar{d}_0^2 \\Biggl[1+\\displaystyle\n\\frac{1}{(2\\pi\\bar{d}_0)^2\\lambda}\\Biggl(\n\\frac{1}{x_R^2}+\\frac{1}{x_L^2}\\Biggr)\\nonumber \\\\\n&+&{\\displaystyle\\sum_{m=1}^{\\infty}} A_m\n\\frac{1}{[x_Rx_L]^{m^2/\\lambda}}\n\\cos(2\\pi\\bar{d}_0mx)\\Biggr],\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n&&G(x,t)\\equiv \\langle \\Psi^\\dagger_F(x,t)\n\\Psi_F(0,0)\\rangle \\nonumber\\\\\n\\approx \\bar{d}_0\\displaystyle\n\\sum_{m=-\\infty}^{\\infty}\n&&B_m\\frac{1}{x_R^{(\\lambda^{1/2}\n+(2m+1)\\lambda^{-1/2})^2/4}}\n\\nonumber\\\\&&\n\\frac{1}{x_L^{(\\lambda^{1/2}\n-(2m+1)\\lambda^{-1/2})^2/4}}\n\\nonumber\\\\&&\ne^{i(2\\pi(m+\\lambda/2)\\bar{d}_0x+\\mu t)},\n\\label{fgreen}\n\\end{eqnarray}\nwhere $x_{R,L}=x\\mp v_st$ and $A_m$ and \n$B_m$ regularization-dependent constants.\n\nUsually a physical quantity, e.g., a boson \nfield, satisfies the periodic boundary \nconditions (PBC). Hence, a charge-1 bosonic \nexcitations are not allowed in the theory,\nbecause it is anti-periodic. However, as we \nknow, an anyon field needn't to obey the PBC. \nSo in the theory, there may be allowed \nanyonic excitations. A charge-1 anyonic \n(or exclusonic) operator is a primary \nfield that does not obey the PBC,\n\\begin{equation}\n\\Psi^\\dagger_{\\lambda}(x)=\n:\\Psi^\\dagger_{F}(x)e^{i(\\lambda^{1/2}\n-\\lambda^{-1/2})(X_R(x)-X_L(x))}:.\n\\end{equation}\nThe anyon commutation relation is easy to \ncheck: \n\\begin{equation}\n\\Psi^\\dagger_{\\lambda}(x)\n\\Psi^\\dagger_{\\lambda}(x')\n-e^{i\\pi\\lambda {\\rm sgn}(x-x')}\n\\Psi^\\dagger_{\\lambda}(x')\n\\Psi^\\dagger_{\\lambda}(x)=0, \\hspace{.2in}\n{\\rm for}~~x\\not= x'.\n\\end{equation} \nIn other words, the anyon field carries \na fractional current. Or by the \n$M\\leftrightarrow J$-duality, the anyon \nwith integer $J$ carries a fractional\ncharge. The correlation function of the \nsingle-anyon reads \\begin{eqnarray}\nG(x,t;\\lambda)&\\equiv& \\langle \n\\Psi^\\dagger_{\\lambda}(x,t)\n\\Psi_{\\lambda}(0,0)\\rangle \\\\\n\\approx \\bar{d}_0\\displaystyle\n\\sum_{m=-\\infty}^{\\infty}\n&B^a_m&\\frac{1}{x_R^{(m+\\lambda)^2/\\lambda}}\n\\frac{1}{x_L^{m^2/\\lambda}}\ne^{i(2\\pi(m+\\lambda/2)x+\\mu t)},\n\\label{green}\n\\end{eqnarray}\nThis correlation function coincides with \nthe asymptotic one \\cite{Ha} in the \nCalogero-Sutherland model. We see that \n\n\\noindent (i) if $\\lambda=1$, (\\ref{green})\nconsists with (\\ref{fgreen}); \n\n\\noindent (ii) there are no boson \nexcitations ($\\lambda=0$) because $G(x,t;0)=0$; \n\n\\noindent (iii) and moreover, $\\lambda>0$ is \nimplied since (\\ref{green}) will diverge \nat the long distance if $\\lambda<0$. \n\n\\noindent (iv) Look at $m=0$. The critical \nexponents can be reads out,\n$$\n\\eta_f=\\lambda+\\lambda^{-1}, \\hspace{.2in} \n\\eta_\\lambda=2\\lambda.\n$$\nThus, \n$$\n\\begin{array}{ll}\n\\eta_f>\\eta_\\lambda,&{\\rm if}~\\lambda<1;\\\\\n\\eta_f<\\eta_\\lambda,&{\\rm if}~\\lambda>1.\n\\end{array}\n$$\n\n\\noindent (v) The multi-sector density operator \nfor exclusons is the same as that of the fermion. \n\nThe single-hole state, i.e.\\, \n$\\Psi^\\dagger_{1/\\lambda} |0\\rangle \\equiv \n\\Psi_\\lambda (\\lambda\\to\\lambda^{-1}) |0\\rangle $, \nwith charge $-1/\\lambda$ alone \nis not allowed. The minimum allowed \nmulti-hole state is given by \n$$\n\\Psi^\\dagger_{1/\\lambda}(x_1)...\n\\Psi^\\dagger_{1/\\lambda}(x_p)|0\\rangle\n$$\nif $\\lambda=p/q$ is rational. \nOne may obtain, e.g.\\ , \n$$\n\\langle [\\Psi^\\dagger_{1/\\lambda}(x,t)]^p\n[\\Psi_{1/\\lambda}(0,0)]^p\\rangle\n\\sim [G(x,t;1/\\lambda)]^p.\n$$\nA more interesting allowed operator is what \ncreates $q$ particle excitations accompanied \nby $p$ hole excitations: \n$$\n\\hat{n}(x,t)=[\\Psi^\\dagger_{\\lambda}(x,t)]^q\n[\\Psi^\\dagger_{1/\\lambda}(x,t)]^p.\n$$\nWe note the similarity of this operator \nto Read's order parameter \\cite{Read} for \nfractional quantum Hall fluids (in bulk). \nIts correlation function can be calculated \nby using Wick's theorem:\n\\begin{equation}\n\\langle \\hat{n}(x,t)\\hat{n}(0,0) \\rangle\\sim\n[G(x,t;\\lambda)]^q[G(x,t;1/\\lambda)]^p.\n\\end{equation}\nIf the contribution from the $m=0$ sector \ndominates, then one gets \n$$\\langle \\hat{n}(x,t)\\hat{n}(0,0) \n\\rangle \\sim (x-v_st)^{-(p+q)}$$. \n\n\n\\section{Two Extensions } \n\nNow we proceed to go beyond IEG. Two \nextensions will be discussed in this \nsection: The one-component GIG with \nthe mutual statistics, and the non-ideal \ngas with the Luttinger-type interactions.\nIn either case, we will show that the \nlow-temperature behavior is that of an \nIEG, controlled by a single \"effective \nstatistics\" parameter $\\lambda_{eff}$,\nwhose value depends on the mutual \nstatistics and the coupling constants\nin the interactions. \n\n\\subsection{Generalized ideal gas with \nmutual statistics} \n\nWe turn to discussing the effects of mutual \nstatistics. Consider a GIG with the statistics \nmatrix (\\ref{stat}) in momentum space given by \n\\begin{equation}\ng(k-k')=\\delta(k-k')+\\Phi(k-k').\n\\end{equation} Here\n$\\Phi(k)=\\Phi(-k)$ is a smooth function. \n$\\Phi(k-k')$ stands for mutual statistics \nbetween particles with different momenta;\nfor IEG $\\Phi(k)=(\\lambda-1)\\delta (k)$. \nThe thermodynamic properties of GIG is \ngiven by eq. (\\ref{Omega}), but now \n$w(k,T)$ satisfies integral equation \n\\cite{Wu,BerWu} which, in terms of the \ndressed energy (\\ref{dressE}), is of the form \n\\begin{equation}\n\\epsilon(k,T)=\\epsilon_0(k)+T\n\\int^{\\infty}_{-\\infty} \\frac{dk'}{2\\pi}\\;\n\\Phi (k-k')\\,\\ln (1+ e^{-\\epsilon(k',T)/T})\\; , \n%\\end{displaymath}\n\\end{equation}\nwhere $\\epsilon_0(k)\\equiv k^{2}-\\mu$. \nIn the low-$T$ limit, it can be proven by \nthe iteration \\cite{WuYu} that\n$\\epsilon(k,T)=\\epsilon(k)+O(T^2/v_s)$, \nwhere $\\epsilon(k)$ is the zero-temperature \ndressed energy given below. At $T=0$, the \nFermi momentum $k_F$ is determined by \n\\begin{equation}\n\\epsilon(\\pm k_F)=0.\n\\end{equation} \nIntroduce \n\\begin{eqnarray}\n(\\alpha\\cdot\\beta) [-k_{F},k_F] \n&\\equiv& \\int^{k_F}_{-k_F}\n\\frac{dk}{2\\pi}\\; \\alpha (k)\\, \\beta (k)\\, ,\\\\\n(\\Phi \\cdot\\alpha) (k;-k_{F},k_F]\n&\\equiv& \\int_{-k_F}^{k_F} \\frac{dk'}{2\\pi}\n\\;\\Phi (k-k') \\,\\alpha (k')\\, .\n\\end{eqnarray}\nThen both $\\rho (k)$ and $\\epsilon(k)$ \nin the ground state satisfy an integral \nequations like\n\\begin{equation}\n\\alpha (k) = \\alpha_0(k)\n-(\\Phi\\cdot \\alpha )\\, (k;-k_F,k_F]\\;.\n\\label{inteq}\n\\end{equation}\nThe dressed momentum $p(k)$ is related \nto $\\rho (k)$ by \n\\begin{equation}\ndp(k)=2\\pi\\rho (k)dk,\\hspace{.2in} \np(k)=-p(-k).\\label{pk} \n\\end{equation}\nThe ground state energy is given by \n\\begin{equation}\nE_0/L=(\\epsilon_0\\cdot \\rho)[-k_F,k_{F}].\n\\end{equation}\nUsing the equation satisfied by $\\rho(k)$, \nit can be expressed by the dressed energy\n\\begin{equation}\nE_0/L =(\\epsilon\\cdot\\rho_0)[-k_F,k_{F}].\n\\end{equation}\n\nThe above equations are of the same form \nas those in the thermodynamic Bethe ansatz \n\\cite{YangYang}, hence the Luttinger-liquid \nrelation \\cite{Hald4},\n$v_{s} = \\sqrt{v_{N}v_{J}}$, remains true. \nA simple proof is sketched as follows. \nThe sound velocity is well-known: \n\\begin{equation}\nv_s= \\partial \\epsilon(p_{F})/\\partial p_{F}.\n\\end{equation} \nThe charge velocity is given by \n\\begin{equation}\nv_N=v_s\\, z(k_F)^{-2},\n\\end{equation}\nwhere the dressed charge $z(k)$ \\cite{Hald4}\nis given by the solution to the integral equation \n\\begin{equation}\nz(k)=1- (\\Phi \\cdot z) (k;-k_F,k_{F}].\n\\end{equation}\nThis relation can be easily derived from the \ndefinitions \n\\begin{eqnarray}\nv_N&=&L \\partial \\mu/\\partial N_{0},\\nonumber\\\\ \nz(k)&=&-\\delta \\epsilon(k)/\\delta \\mu.\n\\end{eqnarray}\nTo create a persistent current, let us boost \nthe Fermi sea by \n\\begin{equation}\n\\pm k_{F} \\to \\pm k_F+\\Delta,\n\\end{equation}\nwhere $\\Delta=z(k_F)/L\\rho(k_F)$. Then the total \nenergy of the state with the persistent current is \n\\begin{eqnarray}\nE_\\Delta/L&=&(\\epsilon_0\\cdot\\rho_\\Delta)[-k_F\n+\\Delta,k_{F}+\\Delta] \\nonumber\\\\\n&=& (\\epsilon_\\Delta\\cdot\\rho_0)\n[-k_F+\\Delta,k_{F}+\\Delta],\n\\end{eqnarray} \\label{515}\nwhere\n\\begin{equation}\n\\rho_{\\Delta} (k)=\\rho_0(k)-\n(\\Phi\\cdot \\rho_\\Delta)(k;-k_F+\\Delta,k_{F}+\\Delta]\n\\end{equation} and \n\\begin{equation}\n\\epsilon_\\Delta(k)=\\epsilon_0(k)-\n(\\Phi \\cdot\\epsilon_\\Delta)(k;-k_F+\\Delta,k_{F}+\\Delta].\n\\end{equation} \nNow, using the last expression for $E_\\Delta$ and \nsubstituting $\\epsilon_\\Delta$ in (\\ref{515}), we have\n\\begin{eqnarray}\n&&E_\\Delta/L=(\\epsilon\\cdot\\rho_0)[-k_F,k_{F}]\n+{\\Delta^2\\over 2}\\epsilon'(k_F)\\nonumber\\\\\n&&\\{\\rho_0(k_F)+(\\rho_0\\cdot 2\\pi F)(k_F;-k_F,k_F]\\}\n\\nonumber\\\\\n&&-{\\Delta^2\\over 2}\\epsilon'(-k_F)\\{\\rho_0(-k_F)\n+(\\rho_0\\cdot 2\\pi F)(-k_F;-k_F,k_F]\\}.\n\\end{eqnarray}\nHere $F(k,k')$ is determined by \n\\begin{equation}\nF(k,k')=\\frac{1}{2\\pi}\\Phi(k,k')\n-\\frac{1}{2\\pi}\\int_{-k_F}^{k_F}dk''\\Phi(k,k'')F(k'',k').\n\\end{equation}\nOn the other hand, we note that the equation for\n$\\rho_0(k)$ can be rewritten as\n\\begin{equation}\n\\rho(k)=\\rho_0(k)-(\\rho_0\\cdot2\\pi F)(k;-k_F,k_F].\n\\end{equation}\nThus, we have\n\\begin{equation}\nE_\\Delta-E_0=L\\Delta^2\n\\epsilon'(k_F)\\rho (k_F)=(2\\pi/L)v_s z(k_F)^{2}.\n\\end{equation} \nThis verifies $v_J=v_s z(k_F)^2$. In view of \neq. (\\ref{Velo}), at low energies, the GIG \nlooks like an IEG with \n\\begin{equation}\n\\lambda_{eff}=z(k_{F})^{-2}.\n\\label{effstat}\n\\end{equation}\n\n\nIt can be shown that it is the effective statistics\n(\\ref{effstat}) that controls the low-$T$ critical \nproperties of GIG, as $\\lambda$ does for IEG\\@. \nLinearization near the Fermi points and bosonization \nof the low-energy effective Hamiltonian go the same\nway as before for IEG\\@. The only difference now\nis that the slope of the linearized dispersion for \nthe dressed energy $\\epsilon_\\pm(k)=\\pm \\epsilon'(k_F)\n(k\\mp k_F)+\\mu = \\pm v_s(p(k)\\mp p_F)+\\mu$, \nis smooth at $k\\sim \\pm k_{F}$. So bosonization is \nstandard and the bosonized Hamiltonian is the same \nas eq. (\\ref{bosonH}) for IEG, only with $\\lambda$ \nreplaced by $\\lambda_{eff}$. However, before going \nto the bosonization we need an effective Hamiltonian \nof the fermions with the dressed energy. Unlike the IEG, \nin the GIG case, $\\epsilon(k,T)=\\epsilon(k)+O(T^2/v_s)$. \nNow, we work out the $T$-expansion of $\\epsilon(k,T)$ \nexplicitly in the low-$T$ limit:\n\\begin{equation}\n\\epsilon(k,T)=\\epsilon(k)+\\tilde{\\epsilon}(k,T)\n+O(T^3/v_s^2).\n\\label{EXP}\n\\end{equation}\nOne finds that\n\\begin{equation}\n\\tilde{\\epsilon}(k,T)=\n\\frac{\\pi T^2}{6\\epsilon'(k_F)}f(k),\n\\end{equation}\nwith the function $f$ determined by \n\\begin{eqnarray}\nf(k)&=&\\Phi(k_F-k)-(\\Phi\\cdot f)(k;-k_F,k_F]\\nonumber\\\\\n&=&\\Phi(k_F-k)-(\\Phi\\cdot \\Phi)(k;-k_F,k_F]\\nonumber\\\\ \n&&+(\\Phi\\cdot (\\Phi\\cdot \\Phi)(k;-k_F,k_F]+...\n\\label{f}\n\\end{eqnarray}\nNote that the equation that $\\rho(k)$ obeys \ncan be rewritten as\n\\begin{eqnarray}\n&&\\frac{\\rho(k)}{\\rho_0}=1-\\int_{-k_F}^{k_F} dk' \n\\{\\Phi(k-k')\\nonumber\\\\\n&&+(\\Phi\\cdot\\Phi)(k';-k_F,k_F]\n(\\Phi\\cdot (\\Phi\\cdot\\Phi)(k';-k_F,k_F]+...\n\\label{rr}\n\\end{eqnarray}\nIntegrating (\\ref{f}) over $k$ and comparing \nwith (\\ref{rr}), one has\n\\begin{equation}\n\\int^{k_F}_{-k_F}\\frac{dk}{2\\pi} f(k)\n=1-\\frac{\\rho(k_F)}{\\rho_0},\n\\end{equation}\nand then\n\\begin{equation}\n\\int^{k_F}_{-k_F}\\frac{dk}{2\\pi} \n\\tilde{\\epsilon}(k,T)=\\frac{\\pi T^2}\n{6\\epsilon'(k_F)}(1-2\\pi\\rho(k_F)).\n\\label{til}\n\\end{equation}\n\nSubstituting (\\ref{EXP}) into the thermodynamic \npotential (\\ref{Omega}), we have\n\\begin{equation}\n{\\Omega(T)\\over L}=-\\frac{T}{2\\pi}\n\\int^\\infty_{-\\infty}dk \\ln(1+e^{-\\epsilon(k)/T})\n+\\frac{1}{2\\pi}\\int^\\infty_{-\\infty}\n\\frac{dk}{1+e^{\\epsilon(k)/T}}\\tilde{\\epsilon}(k,T).\n\\label{OmegaCT}\n\\end{equation}\nIn the low-$T$ limit, the first term in the \nlast equation gives\n$$\n{\\Omega(0)\\over L}-\\frac{\\pi T^2}{6\\epsilon'(k_F)},\n$$\nwith\n$$\n{\\Omega(0)\\over L}=\\frac{1}{2\\pi}\n\\int^{k_F}_{-k_F}dk\\epsilon(k).\n$$\nand the second term is approximately given \nby (\\ref{til}). Thus, \n\\begin{equation}\n{\\Omega(T)\\over L}-{\\Omega(0)\\over L}\n=-\\frac{\\pi T^2}{6v_s},\n\\label{OmegaBA}\n\\end{equation}\nwhich proves the central charge $c=1$ CFT \nbehavior of the theory at the low energy. \n\nWe may also confirm this from the finite size \nscaling in the spatial direction. To see this, \nwe consider the discrete version of the equation \nin which the density $\\rho_L(k_i)$ obeys\n\\begin{equation}\n\\rho_L(k_i)=\\frac{1}{2\\pi}\n-\\sum_{j\\not=i}\\Phi(k_i-k_j).\n\\end{equation}\nUsing the relation between discrete \nsum and integration\n\\begin{eqnarray}\n&&\\frac{1}{L}\\sum_{n=N_1}^{N_2}f(\\frac{I_n}{L})\n=\\int_{(N_1+1/2)/L}^{(N_2-1/2)/L}dx f(x)\n\\nonumber\\\\\n&&+\\frac{1}{24L^2}[f'((N_1-1/2)/L)\n-f'((N_2+1/2)/L)]\\nonumber\\\\ &&\n+O(1/L^3),\\label{FSS}\n\\end{eqnarray}\none has\n\\begin{eqnarray}\n&&\\rho_L(k)\\approx\\frac{1}{2\\pi}\n-(\\Phi\\cdot \\rho_L)(k;-k_F,k_F]\\nonumber\\\\\n&&-\\frac{1}{24L^2}\\frac{1}{\\rho(k_F)}\n\\Biggl[\\frac{\\partial \\Phi(k-k')}{dk'}\\Biggr]_{-k_F}\n\\nonumber\\\\ \n&&+\\frac{1}{24L^2}\\frac{1}{\\rho(k_F)}\n\\Biggl[\\frac{\\partial \\Phi(k-k')}{dk'}\\Biggl]_{k_F}.\n\\end{eqnarray}\nDenote \n\\begin{equation}\n\\rho_L=\\rho+\\rho_1,\n\\end{equation}\nwhere $\\rho(k)$ is of the order $O(1/L^0)$ and \n$\\rho_1(k)$ the order $O(1/L^2)$. Then, \n$\\rho(k)$ is as defined and $\\rho_1(k)$\nis determined by\n\\begin{eqnarray}\n&&\\rho_1(k)=\n-\\frac{1}{24L^2}\\frac{1}{\\rho(k_F)}\n\\Biggl[\\frac{\\partial \\Phi(k-k')}{dk'}\n\\Biggr]_{-k_F}\\nonumber\\\\&&\n+\\frac{1}{24L^2}\\frac{1}{\\rho(k_F)}\n\\Biggl[\\frac{\\partial \\Phi(k-k')}{dk'}\n\\Biggr]_{k_F}\n-(\\Phi\\cdot \\rho_1)(k;-k_F,k_F],\n\\label{rho1}\n\\end{eqnarray}\nThe corresponding thermodynamic potential reads\n\\begin{eqnarray}\n&&\\frac{\\Omega_L(0)}{L}\n=\\frac{1}{L}\\sum_i\\epsilon_0(k(\\frac{I_i}{L}))\n\\nonumber\\\\\n&&=\\int_{-k_F}^{k_F}dk \\rho_L(k)\\epsilon_0(k)\n+\\frac{1}{24L^2\\rho(k_F)} [\\epsilon_0'(k)|_{-k_F}-\\epsilon_0'(k)|_{k_F}]\\nonumber\\\\\n&&=\\int_{-k_F}^{k_F}dk \\rho(k)\\epsilon_0(k)\n+\\int_{-k_F}^{k_F}dk \\rho_1(k)\\epsilon_0(k)\\nonumber\\\\\n&&+\\frac{1}{24L^2\\rho(k_F)}\n[\\epsilon_0'(k)|_{-k_F}-\\epsilon_0'(k)|_{k_F}].\n\\end{eqnarray}\nThe first term of the last equation is \n${\\Omega(0)}/{L}$ and the rest, using \n(\\ref{rho1}), can be written as\n\\begin{eqnarray}\n&&-\\frac{1}{24L^2\\rho(k_F)}\n\\frac{\\partial}{\\partial k}\\biggl(\\epsilon_0(k)\n+(-1)(\\Phi\\cdot \\epsilon_0)\\nonumber\\\\&&\n+(-1)^2 ((\\Phi\\cdot\\Phi)\\cdot\n\\epsilon_0)+...)(k;-k_F,k_F]\\biggr)_{k=k_F}\\nonumber\\\\ \n&&+\\frac{1}{24L^2\\rho(k_F)}\n\\frac{\\partial}{\\partial k}\\biggl(\\epsilon_0(k)\n+(-1)(\\Phi\\cdot \\epsilon_0)\\nonumber\\\\\n&&+(-1)^2 ((\\Phi\\cdot\\Phi)\\cdot\n\\epsilon_0)+...)(k;-k_F,k_F]\\biggr)_{k=-k_F}.\n\\end{eqnarray}\nRecall the equation that $\\epsilon(k)$ obeys, \none has immediately,\n\\begin{equation}\n\\frac{\\Omega_L(0)}{L}-\\frac{\\Omega(0)}{L}\n=-\\frac{\\pi}{12L^2}\\frac{\\epsilon'(k)_{k_F}\n-\\epsilon'(k)_{-k_F}}{2\\pi \\rho(k_F)}\n=-\\frac{\\pi v_s}{6L^2}.\n\\end{equation}\nas desired.\n\nSimilar to the case of IEG, we also could \nhave a fermion representation of the grand \npartition function with the \ntemperature-dependent spectrum. To derive \nthe low-energy effective theory, however, \none rewrites the thermodynamic potential \n(\\ref{OmegaCT}) in the low-$T$ limit as\n\\begin{equation}\n\\frac{\\Omega(T)}{L}\\approx\n\\frac{\\Omega(0)}{L}-2T\\rho(k_F)I(k_F,T),\n\\end{equation}\nwhere \n\\begin{eqnarray}\nI(k_F,T)&=&\\int_{k_F-\\delta}^{k_F+\\delta}dk \n\\ln (1+e^{-|\\epsilon(k)|/T})\n\\nonumber\\\\\n&=&\\int_{p_F-\\delta}^{p_F+\\delta}\n\\frac{dp}{2\\pi}\\rho(p)\n \\ln (1+e^{-|\\epsilon(k((p))|/T}).\n\\end{eqnarray}\nThat is, \n\\begin{eqnarray}\n\\frac{\\Omega(T)}{L}&=&\\int_{-k_F}^{k_F}\n\\frac{dk}{2\\pi}\\epsilon(k)\\nonumber\n\\\\&-&\\frac{T}{2\\pi}\\int_{-p_F+\\delta}\n^{p_F-\\delta}dp\\ln(1+e^{-|\\epsilon(k(p))|/T}),\n\\end{eqnarray}\nwhere $p$ is the physical (dressed) momentum. The \ngrand partition function reads\n\\begin{equation}\nZ_G\\simeq \\prod_{k'}(1+e^{-\\beta\\epsilon(k(k'))}),\n\\end{equation}\nwhere $k'=k$ for $|k|<k_F-\\delta$ and \n$k'=p$ for $|k|>k_F-\\delta$. Now, we can \nhave an effective Hamiltonian because \n${\\epsilon(k)}$ is $T$-independent,\n\\begin{equation}\nH_{\\rm eff}=\\sum_{k'} \\epsilon(k(k'))\nc_{k'}^\\dagger c_{k'}. \n\\label{Heff}\n\\end{equation}\nSimilar to the IEG case, the low-$T$ excitations \ncan be considered by taking the linear \napproximation near the Fermi points, and after \nbosonization the zero-temperature excitations \nshould be added back. The way to bosonize the \nlinear Hamiltonian is also similar to the case \nof IEG. Because the dressed energy is smooth at \nthe Fermi points now, the bosonization is even \nsimpler. The density fluctuation operators are \nsimply given by \n\\begin{eqnarray}\n\\rho_q^{(+)}&=&\\sum_{k\\sim k_F}:c^\\dagger_{p+q}c_p:\n\\nonumber\\\\\n\\rho_q^{(-)}&=&\\sum_{k\\sim -k_F}:c^\\dagger_{p-q}c_p:.\n\\end{eqnarray} \nThe commutators among $\\rho^{(\\pm)}_q$ and $H_\\pm$ \nare\n\\begin{eqnarray}\n[\\rho_q^{(\\pm)},\\rho_{q'}^{(\\pm)\\dagger}]\n&\\approx& \\langle 0|[\\rho_q^{(\\pm)},\n\\rho_q^{(\\pm)\\dagger}]|0\\rangle\n=\\sum_{p_F-q< p < p_F}\\langle 0|c_{p+q}\nc^\\dagger_{p+ q'}|0\\rangle\\nonumber\\\\\n&=&\\delta_{q, q'}\\sum_{p_F-q< p < p_F}1=\n\\frac{L}{2\\pi} q\\delta_{q,q'} \n\\end{eqnarray}\nand\n\\begin{equation}\n[H_{\\pm},\\rho_q^{(\\pm)}]\\approx\n\\langle0|[H_{\\pm},\\rho_q^{(\\pm)}]|0\\rangle\n= \\pm v_Fq\\rho^{(\\pm)}_q. \n\\label{HHH}\n\\end{equation}\nIntroducing the normalized bosonic\nannihilation operators \n\\begin{equation}\nb_q=\\sqrt{2\\pi/ qL}\\,\\rho_q^{(+)}, \n\\hspace{.2in} \\tilde{b}_q=\\sqrt{2\\pi/ qL}\\,\n{\\rho}_q^{(-)\\dagger}\n\\end{equation}\nand adding back the zero-mode contributions, \nthe bosonized Hamiltonian satisfying (\\ref{HHH}) \nis given by \n\\begin{equation}\nH_B=v_s\\{ \\sum_{q>0}q(b_q^\\dagger b_q\n+\\tilde{b}_q^\\dagger \\tilde{b}_q)\n+\\frac{1}{2}\\frac{\\pi}{L}[\\lambda_{\\rm eff} M^2\n+\\frac{1}{\\lambda_{\\rm eff}} J^2] \\},\n\\label{bosonH1}\n\\end{equation}\nwhich agrees with the bosonized Hamiltonian \n(\\ref{HmB}) in the Luttinger liquid theory. \nWe see that with $\\lambda$ replaced by \n$\\lambda_{\\rm eff}$, the bosonized Hamiltonian \nfor the GIG is the same as that for the IEG. \nSo, all consequences we have obtained from the \nbosonized Hamiltonian in the IEG case can be \napplied to the GIG case. Especially, there is \nan (allowed) $\\Psi_{\\lambda_{eff}}^\\dagger$ \ndescribing the particle excitation near the \nFermi surface with both anyon and exclusion \nstatistics being $\\lambda_{eff}$. In this sense, \none may say that the effect of mutual statistics \nis to renormalize the statistics matrix.\n\nHere we remark that in IEG,\n$\\Phi(k,k')=(\\lambda-1)\\,\\delta(k-k')$ is not \nsmooth, so the dressed charge has a jump at \n$k_F$: $z(k_F^+)=1$ and $z(k_F^-)=\\lambda^{-1}$ \nfor $k_F^\\pm=k_F\\pm 0^+$. The general \nLuttinger-liquid relation is of the form \n\\begin{equation}\nv_N=v_s[z(k_F^+)z(k_F^-)]^{-1},\n~~~v_J=v_sz(k_F^+)z(k_F^-).\n\\end{equation} \n\n\\subsection{Non-ideal Gas}\n\nFinally, we examine non-ideal gases, e.g., with \ngeneral Luttinger-type density-density interactions, \n\\begin{eqnarray}\nH&=&H_{\\rm eff}+H_I,\\nonumber\\\\\nH_I&=&\\displaystyle\\frac{\\pi}{L}\n\\sum_{q\\geq 0}[U_{q}(\\rho_q\\rho_q^\\dagger\n+\\tilde{\\rho}_q\\tilde{\\rho}_q^\\dagger)\n+ V_{q}(\\rho_q\\tilde{\\rho}_q^\\dagger\n+\\tilde{\\rho}_q\\rho_q^\\dagger)],\n\\end{eqnarray}\nwhere $H_{\\rm eff}$ is given by \n(\\ref{Heff}) describing a GIG, and $\\rho_q$ \nand $\\tilde{\\rho}_q$ are the excluson density \nfluctuations near $\\pm k_F$ respectively. After \nbosonization, the total Hamiltonian remains \nbilinear in densities:\n\\begin{eqnarray}\nH&=&H_B+H_I\\nonumber\\\\\n&=& \\frac{1}{2}\\sum_{q>0}q[(v_s+U_q)(b_q^\\dagger b_q\n+\\tilde{b}_q^\\dagger \\tilde{b}_q+b_qb_q^\\dagger\n+\\tilde{b}_q\\tilde{b}_q^\\dagger) \\nonumber\\\\\n&&+V_q(b_q^\\dagger \\tilde{b}^\\dagger_q\n+b_q \\tilde{b}_q+\\tilde{b}^\\dagger_qb_q^\\dagger\n+\\tilde{b}_qb_q)] +\\frac{1}{2}\\frac{\\pi}{L}\n[v_N M^2+v_J J^2]\\nonumber\\\\\n&&+\\frac{\\pi}{L}[U_0(M_R^2+M_L^2)+2V_0M_RM_L]-\n\\sum_{q>0}v_sq\n\\end{eqnarray}\nUsing the Bogoliubov transformation, the \nHamiltonian can be easily diagonalized\n\\begin{eqnarray}\nH&=&\\sum_{q>0}\\omega_q(a_q^\\dagger a_q\n+\\tilde{a}_q^\\dagger\\tilde{a}_q)\n+\\frac{1}{2}(\\pi/L)[\\tilde{v}_NM^2\n+\\tilde{v}_JJ^2]+{\\cal E}_0,\\nonumber\\\\\n{\\cal E}_0&=&\\sum_{q>0}(\\omega_q-v_sq),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\na_q^\\dagger&=&\\cosh~\\tilde{\\varphi}_0\n~b^\\dagger_q-\\sinh~\\tilde{\\varphi}_0~\n\\tilde{b}_q^\\dagger,\\nonumber\\\\\n\\tilde{a}_q^\\dagger&=&\\cosh~\\tilde{\\varphi}_0\n~\\tilde{b}^\\dagger_q\n-\\sinh~\\tilde{\\varphi}_0~b_q^\\dagger,\n\\end{eqnarray}\nand the renormalized velocities are\n\\begin{eqnarray} \nv_{s}&\\to&\\tilde{v}_s=\n|(v_s+U_0)^2-V_0^2|^{1/2},\\nonumber\\\\\nv_{N}&\\to&\\tilde{v}_N=\n\\tilde{v}_s e^{-2\\tilde{\\varphi}_0},\n\\nonumber\\\\\nv_{J}&\\to&\\tilde{v}_J=\\tilde{v}_s\ne^{2\\tilde{\\varphi}_0}.\n\\end{eqnarray} \nwith the controlling parameter \n$\\tilde{\\varphi}_0$ determined by\n\\begin{equation}\n\\tanh(2\\tilde{\\varphi}_0)\n=\\frac{v_J-v_N-2V_0}{v_J+v_N+2U_0}.\n\\end{equation}\nThus, the Luttinger-liquid relation \n((\\ref{Velo}) survives with $\\lambda_{eff}$ \nof GIG renormalized to\n\\begin{equation}\n\\tilde{\\lambda}_{eff} = e^{-2\\tilde{\\varphi_0}}.\n\\end{equation}\nNote that the new fixed point depends both \non the position of the Fermi points and \non the interaction parameters $U_{0}$ and $V_{0}$, \nleading to ``non-universal''exponents. \n\n\n\\section{Discussions and conclusions}\n\nIn conclusion, we have shown that 1-d IEG \n(without mutual statistics) exactly reproduces\nthe low-energy and low-$T$ properties of \n(one-component) Luttinger liquids. This gives \nrise to the following physical picture: At \nlow temperature, the Luttinger liquids can be \napproximately thought of as an IEG consisting \nof quasiparticle excitations. Introducing mutual \nstatistics or/ and Luttinger-type interactions \namong these excitations only shifts the value of \n$\\lambda_{eff}$. Thus the essence of Luttinger \nliquids is to have an IEG obeying FES as their \nfixed point. This is our characterization of \nLuttinger liquids in terms of FES. \n\nIn this way, we have explicitly answered the \nthree questions raised in the introduction \nabout Luttinger liquids:\n\\begin{itemize}\n\\item The physical meaning of the Haldane's\ncontrolling parameter is the quasiparticle's\neffective statistics, $\\lambda_{eff}$. \n\\item The Luttinger liquids, more precisely,\nthe IEG, indeed describe the infrared (or \nlow-energy) fixed points in 1-d systems, \nsince their effective field theory at low\nenergy is conformally invariant. However,\nthese fixed points are {\\it not} isolated; \nthey form a fixed-point line. Both the \nchemical potential and coupling constants\nare relevant perturbations that can drive\nthe fixed point to move along the line,\ncorresponding to the \"renormalization\" of\nthe effective statistics $\\lambda_{eff}$\nand leading to \"non-universal\" exponents.\n\\item It is conceivable that some strongly \ncorrelated systems, exhibiting non-Fermi \nliquid behavior, in two or higher dimensions \nmay also be characterized as having a GIG \nwith appropriate statistics matrix as their \nlow-energy or low-temperature fixed point.\nThis is because the concept of exclusion\nstatistics is independent of spatial \ndimeniosnality of the system. \n\\end{itemize}\n\nMoreover, we also showed that the effective \nfield theory of 1-d IEG is a CFT with central \ncharge $c=1$ and compactified radius \n$R=\\sqrt{1/\\lambda}$. The particle-hole duality \nof the exclusons implies the CFT has an unusual \nduality $R\\leftrightarrow 1/R$, meaning that\nthe CFT belongs to a new variant of the $c=1$ \nCFT's, i.e the ones that are compactified on an \n\"orbifold\" $S^1/Z_2$ rather than on a circle. \nPhysically, the differences are due to different \nconstraints on the zero-mode quantum numbers. \nThe CFT explanation makes a better understanding \nof the single-particle operators, especially, \nthe anyonic (or exclusonic ) ones. Also, the CFT \ntechniques provide a systematic way to calculate \nthe correlation functions. \n\nFinally we observe several additional \nimplications of this work: 1) Our bosonization \nand operator derivation of CFT at low energies \nor in low-$T$ limit can be applied to Bethe \nansatz solvable models, including the long-range \n(e.g., Calogero-Sutherland) one \\cite{WuYu}. \n2) Here we have only consider one-species cases, \ni.e., with excitations having no internal \nquantum numbers such as spin. Our bosonization \nand characterization of Luttinger liquids\nare generalizable to GIG with multi-species, \nwith the effective statistics matrix related to \nthe dressed charge matrix \\cite{WuYu}. \n3) The chiral current algebra in eqs. \n(\\ref{density}) and (\\ref{denHam})with \n$\\lambda=1/m$ coincides with that derived \nby Wen\\cite{Wen} for edge states in $\\nu=1/m$ \nfractional quantum Hall fluids. So these edge \nstates and their chiral Luttinger-liquid fixed \npoints can be described in terms of chiral IEG\\@. \n\n\nThis work was supported in part by the U.S. NSF \ngrant PHY-9309458, PHY-9970701 and NSF of China.\n\n\\appendix\n\n\\section{The harmonic fluid description}\n\nIn coordinate space, there is a harmonic fluid \ndescription \\cite{Hald3} of the Luttinger liquid. \nInstead of the $\\theta$-$\\phi$ representation \nthat Haldane originally used, we prefer the \nright-left-moving representation. The density \noperator can be written as the Fourier \ntransformations\n\\begin{eqnarray}\n\\rho(x)&=&\\rho_R(x)+\\rho_L(x),\\nonumber\\\\ \n\\rho_R(x)&=&\\frac{M_R}{L}+\\displaystyle\n\\sum_{q>0}\\sqrt{\\frac{q}{2\\pi L e^{-2\\varphi}}}\n(e^{iqx}b_q+e^{-iqx}b_q^\\dagger),\\nonumber\\\\\n\\rho_L(x)&=&\\displaystyle\\frac{M_L}{L}+\n\\displaystyle\\sum_{q>0}\n\\sqrt{\\frac{q}{2\\pi Le^{-2\\varphi}}}\n(e^{-iqx}\\tilde{b}_q+e^{iqx}\n\\tilde{b}_q^\\dagger),\n\\end{eqnarray}\nwhere $M_{R,L}$ are given by $M=M_R+M_L$ and \n$\\tilde{b}_q=b_{-q}$ for $q>0$.\n\nThe boson field $\\phi(x)$, which is conjugated \nto $\\rho(x)$ and satisfies \n\\begin{equation}\n[\\phi(x),\\rho(x')]=i\\delta(x-x'),\n\\end{equation}\nis given by \n\\begin{eqnarray}\n&&\\phi(x)=\\phi_R(x)+\\phi_L(x),\\nonumber\\\\\n&&\\phi_R(x)= \\frac{\\phi_0}{2}+\\frac{\\pi J_Rx}{L}+i\n\\displaystyle \\sum_{q>0}\\sqrt{\\frac{\\pi e^{-2\\varphi}}\n{2qL}}(e^{iqx}b_q-e^{-iqx}b_q^\\dagger),\\nonumber\\\\\n&&\\phi_L(x)= \\frac{\\phi_0}{2}+\\frac{\\pi J_Lx}{L}+i\n\\displaystyle \\sum_{q>0}\\sqrt{\\frac{\\pi e^{-2\\varphi}}\n{2qL}}(e^{-iqx}\\tilde{b}_q-e^{iqx}\\tilde{b}_q^\\dagger),\n\\nonumber\\\\\n&&\n\\end{eqnarray}\nwith $J=J_R+J_L$. We have to assign the quantum \nnumbers such that there are only two independent\nvariables in $M_{R,L}$ and $J_{R,L}$. A consistent \nchoice is\n\\begin{equation}\nM_R=J_R,\\hspace{.2in}M_L=-J_L.\n\\end{equation}\nThen,\n\\begin{equation}\nJ=J_R+J_L,\\hspace{.2in}M=J_R-J_L.\n\\end{equation}\nHere $\\phi_0$ is an angular variable\nconjugated to $M$: $[\\phi_0,M]=i$. \nThe Hamiltonian (\\ref{HmB}) becomes\n\\begin{equation}\nH =\\frac{1}{2}\n\\int_0^Ldx\\; [\\pi v_N\\rho(x)^2\n+v_J/\\pi~(\\partial_x\\phi(x))^2],\n\\end{equation}\nor by a field rescaling,\n\\begin{equation}\nH =\\frac{v_s}{2\\pi}\n\\int_0^Ldx\\; [\\Pi(x)^2+(\\partial_xX(x))^2],\n\\label{fieldH1}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Pi(x)=\\pi e^{-\\varphi}\\rho(x),\\hspace{.2in} \nX(x)=e^\\varphi\\phi(x).\n\\end{equation}\n\nWith $X(x,t)=e^{iHt}X(x)e^{-iHt}$, the \nLagrangian density reads \n\\begin{equation}\n{\\cal L}=\\frac{v_s}{2\\pi}\\,\\partial_\\alpha \nX(x,t)\\,\\partial^\\alpha X(x,t),\n\\label{LGL1}\n\\end{equation}\nwhich describes a free scalar field theory \nin $1+1$-dimensions.\n\n\n\\begin{references}\n\n%%%%%%%%%%%%%%%%%%%\n\\bibitem{Hald1} F. D. M. Haldane, J. Phys. \n{\\bf C 14}, 2585 (1981).\n%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Hald2} F. D. M. Haldane, Phys. Rev. \nLett. {\\bf 67}, 937 (1991); and in Proc. 16th \nTaniguchi Symposium, eds. N. Kawakami and A. Okiji, \nSpringer Verlag (1994).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Wu} Y. S. Wu, Phys. Rev. Lett. \n{\\bf73}, 922 (1994).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{YangYang} C. N. Yang and C. P. Yang, \nJ. Math. Phys. {\\bf 10}, 1115 (1969).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{BerWu} D. Bernard and Y. S. Wu, \nin Proc. 6th Nankai Workshop, eds. M. L. Ge \nand Y. S. Wu, World Scientific (1995).\n%cond-mat/9404025.\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{NaWil} C. Nayak and F. Wilczek, \nPhys. Rev.~ Lett. {\\bf 73}, 2740 (1994).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Ha} Z. N. C. Ha, Nucl. Phys. \nB{\\bf435}, 604 (1995).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Hatsu} Y. Hatsugai, M. Kohmoto, T. Koma \nand Y. S. Wu, Phys. Rev. B {\\bf 54}, 5358 (1996).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{WuYu} Y. Yu , H. X. Yang and Y. S. Wu, \ncond-mat/9911141.\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Ouvry} A. Dasnierea de Veigy and S. Ouvry, Phys. \nRev. Lett. {\\bf 72},600(1994).\n%%%%%%%%%%%%%%%%%%%\n\\bibitem{HKWY} Y. S. Wu, Y. Yu, Y. Hatsugai \nand M. Kohmoto, Phys. Rev. B {\\bf 57}, 9907 (1998).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{WuYu1} Y. S. Wu and Y. Yu, \nPhys. Rev. Lett. {\\bf 75}, 890 (1995).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{RG} See, e.g.\\ , R. Shankar, \nRev.~Mod. Phys. {\\bf 66}, 129, (1994).\n%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Tomonaga} S. Tomonaga, Prog. Theor. \nPhys. {\\bf 5}, 544 (1950).\n%%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{ML} D. C. Mattis and E. H. Lieb, \nJ. Math. Phys. {\\bf 6}, 304 (1965).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Hald3} F. D. M. Haldane, Phys. Rev. \nLett. {\\bf 47}, 1840 (1981).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Cal} F. Calogero, J. Math. \nPhys. {\\bf 10} 2197 (1967).\n%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Suth}B. Sutherland, J. Math. \nPhys. {\\bf 12}, 246, 251 (1971). \n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{CFT} See, e.g.\\ , R. Dijkgraaf, \nE. Verlinde and H. Verlinde, Commun. Math. \nPhys. {\\bf 115}, 649 (1988). The standard \nduality in usual $c=1$ CFT is \n$R \\leftrightarrow 2/R$. \n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{KY} N. Kawakami and S. K. Yang, \nPhys. Rev. Lett. {\\bf 67}, 2493 (1990).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{sm} R. Shankar and M.V.N.Murthy, Phys. Rev. Lett. {\\bf 72}, 3629 (1994).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{ISO} S. Iso, Nucl. Phys. \nB {\\bf 443}, 581 (1995). \n%%%%%%%%%%%%%%%%%%%%\n\\bibitem{ISR} S. Iso and S. J. Rey, Phys. Lett. \nB {\\bf 352},111 (1995).\n%%%%%%%%%%%%%%%%%%%\n\\bibitem{CAR} R. Caracciolo, A. Lerda and \nG. R. Zemba, Phys. Lett. B {\\bf 352}, 304 (1995).\n%%%%%%%%%%%%%%%%%%%\n\\bibitem{Schulz} See, e.g., H. J. Schulz, in \n\"Proceedings of Les Houches Summer School LXI\", \ned. E. Akkermans, G. Montambaux, J. Pichard, \nand J. Zinn-Justin (Elsevier, Amsterdam, 1995), p.533.\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Lutt}J. M. Luttinger, J. Math. Phys. \n{\\bf 4}, 1154 (1963).\n%%%%%%%%%%%%%%%%%%%%\n\\bibitem{WYS} Y. S. Wu, Invited Lectures in \n``Topics in Theoretical Physics'', (Proceedings of \nthe Second Pacific Winter School for Theoretical \nPhysics; Sorak Mountains, Korea; Jan. 19-24, 1995), \ned. by Y. M. Cho (World Scientific, 1996); pp. 27-59.\n%%%%%%%%%%%%%%%%%%\n\\bibitem{comm0} It is easy to understand why \nit is $\\delta F_0$ rather than $\\delta E_0$ \nthat exactly contains the zero-mode \ncontributions in (\\ref{HmB}): In writing\ndown the Hamiltonian (\\ref{HmB}), we have \nshifted the zero point of the energy to \nthe Fermi surface. \n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{comm1} Alternatively, we may extend \nthe Fermi sea to that for two separate right- \nand left-moving fermions with linear dispersion \nfor $k$ covering the whole real axis in each \nsector. Then our commutation relations are \nexact. (See \\cite{ML}) \n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{comm2} Here operators $O_m$ give \nrise to the Klein factors necessary for \ncorrect commutation relation for $\\Psi_{F}$ \nand $\\Psi_{F}^{\\dagger}$.\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Read} N. Read, Phys. Rev.~ Lett., \n{\\bf 62}, 86 (1988).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Hald4} F. D. M. Haldane, Phys. \nLett. {\\bf 81 A}, 153 (1981);\nN. M. Bogoliubov, A. G. Izergin and \nV. E. Korepin, J. Phys. {\\bf A 20}, 5361 (1987).\n%%%%%%%%%%%%%%%%%%%%%\n\\bibitem{Wen} X.G. Wen, Phys. Rev.~ Lett. \n{\\bf 64 }, 2206 (1990).\n%%%%%%%%%%%%%%%%%%%%%\n\n\n\\end{references}\n\n\n%\\endnarrowtext}\n\\end{document}\n\n\n"
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{
"name": "cond-mat0002063.extracted_bib",
"string": "\\bibitem{Hald1} F. D. M. Haldane, J. Phys. \n{\\bf C 14}, 2585 (1981).\n%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Hald2} F. D. M. Haldane, Phys. Rev. \nLett. {\\bf 67}, 937 (1991); and in Proc. 16th \nTaniguchi Symposium, eds. N. Kawakami and A. Okiji, \nSpringer Verlag (1994).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Wu} Y. S. Wu, Phys. Rev. Lett. \n{\\bf73}, 922 (1994).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{YangYang} C. N. Yang and C. P. Yang, \nJ. Math. Phys. {\\bf 10}, 1115 (1969).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{BerWu} D. Bernard and Y. S. Wu, \nin Proc. 6th Nankai Workshop, eds. M. L. Ge \nand Y. S. Wu, World Scientific (1995).\n%cond-mat/9404025.\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{NaWil} C. Nayak and F. Wilczek, \nPhys. Rev.~ Lett. {\\bf 73}, 2740 (1994).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Ha} Z. N. C. Ha, Nucl. Phys. \nB{\\bf435}, 604 (1995).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Hatsu} Y. Hatsugai, M. Kohmoto, T. Koma \nand Y. S. Wu, Phys. Rev. B {\\bf 54}, 5358 (1996).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{WuYu} Y. Yu , H. X. Yang and Y. S. Wu, \ncond-mat/9911141.\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Ouvry} A. Dasnierea de Veigy and S. Ouvry, Phys. \nRev. Lett. {\\bf 72},600(1994).\n%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{HKWY} Y. S. Wu, Y. Yu, Y. Hatsugai \nand M. Kohmoto, Phys. Rev. B {\\bf 57}, 9907 (1998).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{WuYu1} Y. S. Wu and Y. Yu, \nPhys. Rev. Lett. {\\bf 75}, 890 (1995).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{RG} See, e.g.\\ , R. Shankar, \nRev.~Mod. Phys. {\\bf 66}, 129, (1994).\n%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Tomonaga} S. Tomonaga, Prog. Theor. \nPhys. {\\bf 5}, 544 (1950).\n%%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{ML} D. C. Mattis and E. H. Lieb, \nJ. Math. Phys. {\\bf 6}, 304 (1965).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Hald3} F. D. M. Haldane, Phys. Rev. \nLett. {\\bf 47}, 1840 (1981).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Cal} F. Calogero, J. Math. \nPhys. {\\bf 10} 2197 (1967).\n%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Suth}B. Sutherland, J. Math. \nPhys. {\\bf 12}, 246, 251 (1971). \n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{CFT} See, e.g.\\ , R. Dijkgraaf, \nE. Verlinde and H. Verlinde, Commun. Math. \nPhys. {\\bf 115}, 649 (1988). The standard \nduality in usual $c=1$ CFT is \n$R \\leftrightarrow 2/R$. \n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{KY} N. Kawakami and S. K. Yang, \nPhys. Rev. Lett. {\\bf 67}, 2493 (1990).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{sm} R. Shankar and M.V.N.Murthy, Phys. Rev. Lett. {\\bf 72}, 3629 (1994).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{ISO} S. Iso, Nucl. Phys. \nB {\\bf 443}, 581 (1995). \n%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{ISR} S. Iso and S. J. Rey, Phys. Lett. \nB {\\bf 352},111 (1995).\n%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{CAR} R. Caracciolo, A. Lerda and \nG. R. Zemba, Phys. Lett. B {\\bf 352}, 304 (1995).\n%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Schulz} See, e.g., H. J. Schulz, in \n\"Proceedings of Les Houches Summer School LXI\", \ned. E. Akkermans, G. Montambaux, J. Pichard, \nand J. Zinn-Justin (Elsevier, Amsterdam, 1995), p.533.\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Lutt}J. M. Luttinger, J. Math. Phys. \n{\\bf 4}, 1154 (1963).\n%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{WYS} Y. S. Wu, Invited Lectures in \n``Topics in Theoretical Physics'', (Proceedings of \nthe Second Pacific Winter School for Theoretical \nPhysics; Sorak Mountains, Korea; Jan. 19-24, 1995), \ned. by Y. M. Cho (World Scientific, 1996); pp. 27-59.\n%%%%%%%%%%%%%%%%%%\n\n\\bibitem{comm0} It is easy to understand why \nit is $\\delta F_0$ rather than $\\delta E_0$ \nthat exactly contains the zero-mode \ncontributions in (\\ref{HmB}): In writing\ndown the Hamiltonian (\\ref{HmB}), we have \nshifted the zero point of the energy to \nthe Fermi surface. \n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{comm1} Alternatively, we may extend \nthe Fermi sea to that for two separate right- \nand left-moving fermions with linear dispersion \nfor $k$ covering the whole real axis in each \nsector. Then our commutation relations are \nexact. (See \\cite{ML}) \n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{comm2} Here operators $O_m$ give \nrise to the Klein factors necessary for \ncorrect commutation relation for $\\Psi_{F}$ \nand $\\Psi_{F}^{\\dagger}$.\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Read} N. Read, Phys. Rev.~ Lett., \n{\\bf 62}, 86 (1988).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Hald4} F. D. M. Haldane, Phys. \nLett. {\\bf 81 A}, 153 (1981);\nN. M. Bogoliubov, A. G. Izergin and \nV. E. Korepin, J. Phys. {\\bf A 20}, 5361 (1987).\n%%%%%%%%%%%%%%%%%%%%%\n\n\\bibitem{Wen} X.G. Wen, Phys. Rev.~ Lett. \n{\\bf 64 }, 2206 (1990).\n%%%%%%%%%%%%%%%%%%%%%\n\n\n"
}
] |
cond-mat0002064
|
Depinning and dynamic phases in driven three-dimensional vortex lattices in anisotropic superconductors
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[] |
We use three-dimensional molecular dynamics simulations of magnetically interacting pancake vortices to study the dynamic phases of vortex lattices in highly anisotropic materials such as BSCCO. Our model treats the magnetic interactions of the pancakes exactly, with long-range logarithmic interactions both within and between planes. The pancake vortices decouple at low drives and show two-dimensional plastic flow. The vortex lattice both recouples and reorders as the driving current is increased, eventually forming a recoupled crystalline-like state at high drives. We construct a phase diagram as a function of interlayer coupling and show the relationship between the recoupling transition and the single-layer reordering transitions. \vspace{1pc}
|
[
{
"name": "manuscript.tex",
"string": "\\documentstyle[twoside,fleqn,espcrc2,epsf]{article}\n\\input{epsf}\n\n\\title{Depinning and dynamic phases in driven three-dimensional vortex lattices\nin anisotropic superconductors}\n\\author{C. J. Olson\n\\address{Department of Physics, University of California, Davis, California\n95616}\nand\nN. Gr{\\o}nbech-Jensen\n\\address{Department of Applied Science, University of California, Davis, \nCalifornia 95616}\n\\address{NERSC, Lawrence Berkeley National Laboratory, Berkeley, \nCalifornia 94720}\n}\n\n\\begin{document}\n\n\\begin{abstract}\nWe use three-dimensional molecular dynamics simulations of magnetically\ninteracting pancake vortices to study the dynamic phases of vortex lattices\nin highly anisotropic materials such as BSCCO. Our model treats the\nmagnetic interactions of the pancakes exactly, with long-range logarithmic\ninteractions both within and between planes. The pancake vortices decouple\nat low drives and show \ntwo-dimensional plastic flow. The vortex lattice both recouples\nand reorders as the driving current is increased, eventually forming\na recoupled crystalline-like state at high drives. \nWe construct a phase diagram as a function of interlayer\ncoupling and \nshow the relationship between the recoupling transition and the \nsingle-layer reordering transitions.\n\\vspace{1pc}\n\\end{abstract}\n\n% typeset front matter (including abstract)\n\\maketitle\n\nIn highly anisotropic superconductors such as BSCCO,\nthe vortex lattice is composed of individual pancake vortices that\nmay be either coupled or decoupled between layers depending on \nsuch factors as the material stoichiometry or \nthe magnitude and angle of the applied magnetic field.\nOf particular interest is the possible relationship between\na coupling/decoupling transition and the widely studied second peak or fishtail\neffect \\cite{blatter}.\n\nAs a function of interlayer coupling strength $s$, there are\ntwo limits of vortex behavior in a system containing pointlike disorder. \nFor zero interlayer coupling $s=0$, each plane behaves as\nan independent two-dimensional (2D) system. \nFor infinite interlayer coupling $s=\\infty$,\nthe vortices form perfectly straight three-dimensional (3D) \nlines, and all of the planes move in unison. \nAt finite coupling strength $s\\ne 0$, a transition\nbetween these types of behaviors should\noccur with coupling strength, but it is unclear whether this transition\nis sharp or if an intermediate state of the lattice exists.\nFurthermore, it is known that 2D systems with pointlike pinning\ncan exhibit dynamic reordering under the influence of an applied driving\ncurrent, passing from a liquid-like state at zero drive\nto a recrystallized state at high current \\cite{reorder,olson,kolton}. \nThus, in a 3D system, \na dynamically driven recoupling transition\ncould be expected, but it is unclear where this\ntransition falls in relation to the 2D reordering transitions already\nseen. \n\nTo study the coupling transitions, we have developed a simulation containing\nthe correct magnetic interactions between pancakes \\cite{clem}.\nThis interaction is long range both in and between planes, and is\ntreated according to Ref.\\ \\cite{ngj}. \n\nThe overdamped equation of motion, $T=0$, for vortex $i$ is given by\n$ {\\bf f}_{i} = \\sum_{j=1}^{N_{v}}\\nabla {\\bf U}(\\rho_{i,j},z_{i,j})\n+ {\\bf f}_{i}^{vp} + {\\bf f}_{d}= {\\bf v}_{i}$,\nwhere $N_v$ is the number of vortices, $\\rho$ and $z$ are the distance\nbetween pancakes in cylindrical coordinates.\nThe magnetic energy between pancakes is \n\\begin{eqnarray}\n{\\bf U}(\\rho_{i,j},0)=2d\\epsilon_{0} \n\\left((1-\\frac{d}{2\\lambda})\\ln{\\frac{R}{\\rho}}\n+\\frac{d}{2\\lambda} \nE_{1}(\\rho)\n\\right) \n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\bf U}(\\rho_{i,j},z)=-s\\frac{d^{2}\\epsilon_{0}}{\\lambda}\n\\left(\\exp(-z/\\lambda)\\ln\\frac{R}{\\rho}- \nE_{1}(R)\n\\right) \\nonumber\n\\end{eqnarray}\nwhere\n$R = \\sqrt{z^2 + \\rho^2}$, \n$E_{1}(x) = \\int^{\\infty}_{\\rho}\\exp(-x/\\lambda)/\\rho^{\\prime}$ and\n$\\epsilon_{0} = \\Phi_{0}^{2}/(4\\pi\\xi)^{2}$.\nThe pointlike pins are randomly distributed in each layer and modeled\nby parabolic traps.\nWe vary the relative strength of the interlayer coupling using the\nprefactor $s$.\nWe have simulated a $16\\lambda \\times 16\\lambda$ \nsystem containing 89 vortices and 4 layers, with\na total of 356 pancake vortices. Further work on systems containing\nup to 16 layers will be reported elsewhere \\cite{toappear}.\n\nIn Fig.\\ 1(a) we present \na phase diagram as a function of interlayer coupling strength\n$s$ and driving force $f_d$. \nAt zero drive, we find a recoupling transition for a coupling\nstrength of $s> 4.5$. In samples with $s\\ge 5$, the pancakes\nremain coupled into lines at all drives and show the same\ntransitions seen in previous work \\cite{olson},\nexhibiting plastic flow\nof stiff lines above depinning, and reordering into\nfirst a smectic state and then a recrystallized state of stiff\nlines at higher drives. \n\nFor samples with weaker interlayer coupling, $s<5$, the vortex lattice\nis broken into decoupled planes at zero drive. \nUpon application of a driving current, the samples exhibit 2D\nplastic flow in which each layer moves independently of the others.\nOnce the individual layers reach the driving force at which a\ntransition to a smectic state occurs, the vortices simultaneously\nform the smectic state and recouple, as can be seen from the measure\nshown in Fig.\\ 1(c)\nof the $z$-axis correlation $C_z = 1 - \\langle({\\bf r}_{i,L}-{\\bf r}_{i,L+1})\n\\Theta( a_0/2 - |({\\bf r}_{i,L}-{\\bf r}_{i,L+1})|)\\rangle$,\nwhere \n$a_0$ is the vortex lattice constant.\nThe dynamic recoupling transition line follows the smectic\ntransition line down to $s=2$\nand is associated with a peak in the $dV/dI$ curve\nseen in Fig.\\ 1(b). \nBoth the static and dynamic transition lines between decoupled 2D and \nrecoupled 3D behavior are sharp.\n\nAs a function of the number of layers, we observe the same behavior,\nbut the depinning current\nin the 3D stiff state drops with the number of layers. \nThe depinning current in the 2D decoupled phase is not affected since\nin this case the individual planes behave as isolated entities.\nThe recoupling transition sharpens as the number of\nlayers is increased \\cite{toappear}.\n\nWe acknowledge helpful discussions with L. N. Bulaevskii, A. Kolton,\nC. Reichhardt, R.T. Scalettar, G. T. Zim{\\' a}nyi. \nThis work was supported by CLC and CULAR (LANL/UC) and by\nthe Director, Office of Adv.\\ Scientific\nComp.\\ Res., Div.\\ of Math., Information, and \nComp.\\ Sciences, U.S.\\ DoE contract DE-AC03-76SF00098.\n\n\\begin{thebibliography}{9}\n\n\\bibitem{blatter} G. Blatter {\\it et al.}, Rev. Mod. Phys. 66 (1994)\n1125.\n\n\\bibitem{reorder} T. Giamarchi and P. Le Doussal, Phys. Rev. Lett.\n76 (1996) 3408; L. Balents {\\it et al.}, {\\it ibid} 78 (1997) 751.\n\n\\bibitem{olson} C.J. Olson {\\it et al.}, Phys. Rev. Lett. 81 (1998) 3757. \n\n\\bibitem{kolton} A.B. Kolton {\\it et al.}, Phys. Rev. Lett. 83 (1999)\n3061.\n\n\\bibitem{clem} J.R. Clem, Phys. Rev. B 43 (1990) 7837.\n\n\\bibitem{ngj} N. Gr{\\o}nbech-Jensen, Comp. Phys. Comm. 119 (1999) 115.\n\n\\bibitem{toappear} C.J. Olson and N. Gr{\\o}nbech-Jensen, to be published.\n\n\\end{thebibliography}\n\n\\begin{figure}\n\\centerline{\n%\\epsfxsize=8cm \n\\epsfxsize=7cm \n\\epsfbox{fig1.ps}}\n\\caption{(a) Phase diagram for varying interlayer coupling $s$ and driving\nforce $f_d$. Circles: depinning line; diamonds: decoupling line; \nsquares: smectic transition line; triangles: recrystallization line. \n(b) $V_x$ (circles) and $dV/dI$ (+ signs) for $s=2.0$.\n(c) $C_z$ for $s=2.0$\n(d) $V_x$ (circles) and $dV/dI$ (+ signs) for $s=8.0$.\n(e) $C_z$ for $s=8.0$}\n\\label{fig:fig1}\n\\end{figure}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002064.extracted_bib",
"string": "\\begin{thebibliography}{9}\n\n\\bibitem{blatter} G. Blatter {\\it et al.}, Rev. Mod. Phys. 66 (1994)\n1125.\n\n\\bibitem{reorder} T. Giamarchi and P. Le Doussal, Phys. Rev. Lett.\n76 (1996) 3408; L. Balents {\\it et al.}, {\\it ibid} 78 (1997) 751.\n\n\\bibitem{olson} C.J. Olson {\\it et al.}, Phys. Rev. Lett. 81 (1998) 3757. \n\n\\bibitem{kolton} A.B. Kolton {\\it et al.}, Phys. Rev. Lett. 83 (1999)\n3061.\n\n\\bibitem{clem} J.R. Clem, Phys. Rev. B 43 (1990) 7837.\n\n\\bibitem{ngj} N. Gr{\\o}nbech-Jensen, Comp. Phys. Comm. 119 (1999) 115.\n\n\\bibitem{toappear} C.J. Olson and N. Gr{\\o}nbech-Jensen, to be published.\n\n\\end{thebibliography}"
}
] |
cond-mat0002065
|
Scaling analysis of the magnetic-field-tuned quantum transition in superconducting amorphous In--O films
|
[
{
"author": "M.V.~Golubkov"
},
{
"author": "V.T.~Dolgopolov"
},
{
"author": "G.E.~Tsydynzhapov"
},
{
"author": "and A.A.~Shashkin"
}
] |
We have studied the magnetic-field-tuned superconductor-insulator quantum transition (SIT) in amorphous In--O films with different oxygen content and, hence, different electron density. While for states of the film near the zero-field SIT the two-dimensional scaling behaviour is confirmed, for deeper states in the superconducting phase the SIT scenario changes: in addition to the scaling function that describes the conductivity of fluctuation-induced Cooper pairs, there emerges a temperature-dependent contribution to the film resistance. This contribution can originate from the conductivity of normal electrons.
|
[
{
"name": "fsc1f.tex",
"string": "\\documentstyle[prb,aps,floats,psfig]{revtex}\n\\begin{document}\n\\draft\n\\twocolumn\n\\wideabs{\n\\title{Scaling analysis of the magnetic-field-tuned quantum\ntransition in superconducting amorphous In--O films}\n\\author{V.F.~Gantmakher\\thanks{e-mail: gantm@issp.ac.ru},\nM.V.~Golubkov, V.T.~Dolgopolov, G.E.~Tsydynzhapov, and A.A.~Shashkin}\n\\address{Institute of Solid State Physics, Russian Academy of\nSciences, 142432 Chernogolovka, Russia}\n\\maketitle\n\n\\begin{abstract}\nWe have studied the magnetic-field-tuned superconductor-insulator\nquantum transition (SIT) in amorphous In--O films with different\noxygen content and, hence, different electron density. While for\nstates of the film near the zero-field SIT the two-dimensional\nscaling behaviour is confirmed, for deeper states in the\nsuperconducting phase the SIT scenario changes: in addition to the\nscaling function that describes the conductivity of\nfluctuation-induced Cooper pairs, there emerges a\ntemperature-dependent contribution to the film resistance. This\ncontribution can originate from the conductivity of normal\nelectrons.\n\\end{abstract}\n\\pacs{PACS numbers: 71.30 +h, 74.40 +k}}\n\n\n\nThe scaling analysis is an important experimental tool for studying\nquantum phase transitions. For two-dimensional (2D) disordered\nsuperconductors, alongside with the zero-field\nsuperconductor-insulator transition (SIT) as driven by disorder\nchange in the film, there exists a SIT that is induced by a normal\nmagnetic field. A scenario of the field-induced SIT was proposed in\nRef.~\\onlinecite{Fisher}: at zero temperature the normal magnetic\nfield alters the state of a disordered film from superconducting at\nlow fields, through a metallic one at the critical field $B=B_c$ with\nthe universal sheet resistance $R_c$ close to $h/4e^2\\simeq\n6.4$~k$\\Omega$, to an insulating state at fields $B>B_c$. The SIT\nwas supposed to be continuous with the correlation length $\\xi$ of\nquantum fluctuations, diverging as $\\xi\\propto(B-B_c)^{-\\nu}$, where\nthe critical index $\\nu>1$. At a finite temperature the size of\nquantum fluctuations is restricted by the dephasing length\n$L_\\phi\\propto T^{-1/z}$ with the dynamical critical index $z$, which\ndetermines the characteristic energy $U\\sim\\xi^{-z}$ and\n\nis expected to be equal to $z=1$ for SIT.\n\nThe ratio of these two length parameters defines the\nscaling variable $u$ so that near the transition point ($T=0,B_c$)\nall data $R(T,B)$ as a function of $u$ should fall on a universal\ncurve\n\\begin{equation}\nR(T,B)\\equiv R_cr(u),\\qquad u=(B-B_c)/T^{1/z\\nu}.\\label{x=}\n\\end{equation}\nAlthough small in the scaling region, temperature dependent\ncorrections with the leading quadratic term are expected to the\ncritical resistance $R_c$ \\cite{Fisher,QPT}.\n\n\nThe above theoretical description is based on the concept of electron\npair\nlocalization which has been supported by a recent publication\n\\cite{Larkin}. In that paper it is shown that for 2D superconducting\nfilms with sufficiently strong disorder the region of fluctuation\nsuperconductivity, where the localized\nelectron pairs\n\n(called also boson \\cite{Fisher} and cooperon \\cite{Larkin})\n\noccur, should\nextend down to zero temperature. In this region the unpaired\nelectrons are supposed to be localized because of disorder in a film.\n\n\nSo far, a theory of the field-driven 3D quantum SIT has not been\ncreated. An idea to consider the quantum SIT for 3D disordered\nsystems in zero magnetic field in terms of charged boson localization\n\\cite{golda} was not at first accepted because the fluctuation\nsuperconductivity region was regarded to be small. In fact, as was\nshown later in Ref.~\\onlinecite{Bul}, the fluctuation region enlarges\nas the edge of single electron localization is approached. This gives\nan opportunity to apply the scaling relation deduced for 3D boson\nlocalization \\cite{Fish2} also for the field-induced SIT description\n\n\\begin{equation}\nR(T,u)\\sim T^{-1/z}\\tilde r(u), \\label{e3D}\\end{equation}\nwhere $\\tilde r(u)$ is a universal function and the scaling variable\n$u$ is assumed to have the same form as defined by Eq.~(\\ref{x=}).\n\nFrom Eqs.~(\\ref{x=}) and (\\ref{e3D}) it follows that in the vicinity\nof $B_c$ the isotherms $R(B)$ are straight lines with slopes\n\n\\begin{equation}\n\\frac{\\partial R}{\\partial B}\\propto T^{-(d-2+1/\\nu)/z},\n\\label{sca} \\end{equation}\n\nwhere $d$ is the system dimensionality.\nBecause the behaviours of the resistance in the relations (\\ref{x=})\nand (\\ref{e3D}) are very different, the problem of the film\ndimensionality is of major importance.\n\n\n\nData obtained in experimental studies on $a$-In--O \\cite{HP},\n$a$-Mo--Ge \\cite{Kapit}, and $a$-Mo--Si \\cite{Okuma} followed the 2D\nscaling relation (\\ref{x=}) except for the universality of the $R_c$\nvalue.\n\nThis was regarded as evidence of existence of SIT. The\nfailure in satisfying the scaling relations in ultrathin Bi films\n\\cite{cher} was interpreted as indication of the absence of SIT\nand crossover observation between different flux-flow regimes.\nStudies \\cite{HP,Kapit,Okuma} did not give arguments backing boson\nlocalization.\nAt the first time such arguments are appeared by\ninterpretation of the resistance drop at high fields observed on\n$a$-In--O films\n\\cite{gg2,JETPL}.\n\nHere, we perform the detailed study of the scaling relations near the\nfield-induced SIT for different states of an $a$-In--O film. We find\nthat the 2D scaling relation (\\ref{x=}) holds for film states near\nthe zero-field SIT but progressively fails as the zero-field SIT is\ndeparted from. This failure is manifested by the appearance of an\nextra temperature-dependent term in the film resistance.\n\n\nThe experiments were performed on 200 \\AA\\ thick amorphous In--O\nfilms evaporated by e-gun from high-purity $In_2 O_3$ target onto a\nglass substrate \\cite{InO}.\n\nThis\nmaterial proved to be very useful for\ninvestigations of the transport properties near the SIT\n\\cite{HP,HP1,gg2,ShOv,Kim}. Oxygen deficiency compared to fully\nstoichiometric insulating compound In$_2$O$_3$ causes the film\nconductivity. By changing the oxygen content one can cover the range\nfrom a superconducting material to an insulator with activated\nconductance \\cite{ShOv}. The procedures to change reversibly the film\nstate are described in detail in Ref.~\\onlinecite{gg2}. To reinforce\nthe superconducting properties of our films we used heating in vacuum\nup to a temperature from the interval 70 -- 110$^\\circ$C until the\nsample resistance got saturated. To shift the film state in the\nopposite direction we made exposure to air at room temperature. As\nthe film remains amorphous during these manipulations, it is natural\nto assume that the treatment used results mainly in a change of the\ntotal carrier concentration $n$ and that there is a critical\nconcentration $n_c$ corresponding to the zero-field SIT.\n\n\nThe low-temperature measurements were carried out by a four-terminal\nlock-in technique at a frequency of 10~Hz using two experimental\nsetups: a He$^3$-cryostat down to 0.35~K or Oxford TLM-400 dilution\nrefrigerator in the temperature interval 1.2~K -- 30~mK. The ac\ncurrent was equal to 1~nA and corresponded to the linear regime of\nresponse. The aspect ratio of the samples was close to one.\n\nWe investigated three different homogeneous states of the same\namorphous In--O film \\cite{rem}. We characterize the sample state by\nits room temperature resistance $R_r$. Assuming that the disorder for\nall states is approximately the same, we have for the carrier density\n$n\\propto 1/R_r$, i.e., the smaller $R_r$, the deeper the state in the\nsuperconducting phase and, hence, the larger the value of $B_c$.\n\n\\begin{table}\n\n\\caption{Parameters of the studied states of the sample.}\n\\begin{tabular}{c|cccc}\nState&$R_r$, k$\\Omega$&$R_c$, k$\\Omega$&$B_c$, T&$\\alpha$, K$^{-1}$\\\\\n\\tableline\n1&3.4&7.8&2.2&0\\\\\n2&3.1&8&5.3&-0.1\\\\\n3&3.0&9.2&7.2&-0.6\\\\\n\\end{tabular}\n\\label{t1}\n\\end{table}\n\n\nThe parameters of the investigated states are listed in\nTable~\\ref{t1}. State 1 is the closest to the zero-field SIT and\nstate 3 is the deepest in the superconducting phase.\n\n\\begin{figure}\n\\psfig{file=fig1.eps,width=\\columnwidth,clip=}\n\\smallskip\\caption{Temperature dependences of the resistance of the studied\nstates for different magnetic fields. The separatrices $R_c(T)$ are\nshown by solid lines.}\n\n\\label{f1}\n\n\\end{figure}\n\n\nSets of the isomagnetic curves $R(T)$ for all studied states are\ndepicted in Fig.~\\ref{f1}. For each set the curves can be divided\nroughly into two groups by sign of the second derivative: the\npositive (negative) sign corresponds to the insulating\n(superconducting) behaviour. Henceforth, the boundary isomagnetic\ncurve $R_c(T)$ between superconductor and insulator, which\ncorresponds to the boundary metallic state at $T=0$, will be referred\nto as separatrix. While for state 1 it is easy to identify the\nhorizontal separatrix in accordance with Eq.~(\\ref{x=}), for states 2\nand 3 the fan and separatrix are\n\"tilted\", i.e., each of the curves in the\nlower part of the fan is a maximum at a temperature $T_{\\rm max}$\nwhich shifts with $B$. To determine the separatrix $R_c(T)$ one has\nto extrapolate the maximum position to $T=0$ for which it is good to\nknow the extrapolation law as the accessible temperature range is\nrestricted.\n\n\n\\begin{figure}\n\\psfig{file=fig2.eps,width=\\columnwidth,clip=}\n\n\\smallskip\\caption{Isotherms in the ($B,R$) plane for states 1 and 3. The curve\nintersection region for state 3 is blown up in the inset. The circles\nmark the crossing points of the isotherms with neighbouring\ntemperatures.}\n\n\\label{f2}\n\n\\end{figure}\n\n\nThe absence of a horizontal separatrix for states 2 and 3 can also be\nestablished from the behaviour of isotherms $R(B)$ (Fig.~\\ref{f2}).\nAs seen from the figure, the isotherms of state 1 cross at the same\npoint ($B_c,R_c$) whereas those of state 3 form an envelope.\n\n\nTo determine $B_c$ and $R_c$ for states 2 and 3 we use the simplest\nlinear extrapolation to $T=0$ of the functions $R(T_{\\rm max})$ and\n$B(T_{\\rm max})$, see Fig.~\\ref{f4}. The open symbols correspond to\nthe maximum positions on isomagnetic curves (Fig.~\\ref{f1}) and the\nfilled symbols represent the data obtained from the intersections of\nconsecutive isotherms \\cite{mit} (Fig.~\\ref{f2}):\nif two consecutive isotherms\nat close temperatures $T_1$ and $T_2$ intersect at a point\n($B_i,R_i$), the isomagnetic curve at the field $B_i$ reaches its\nmaximum $\\approx R_i$ at $T_{\\rm max}\\approx (T_1+T_2)/2$.\n\n\\begin{figure}\n\\centerline{\\psfig{file=fig4.eps,width=0.6\\columnwidth,clip=}}\n\n\\smallskip\\caption{Dependences $R(T_{\\rm max})$ and $B(T_{\\rm max})$ as\ndetermined from the data in Fig.~\\protect\\ref{f1}b (open symbols) and\nFig.~\\protect\\ref{f2}b (filled symbols). The values of $R_c$ and\n$B_c$ are obtained with the help of linear extrapolations (solid\nlines). The dotted line is a guide to the eye.}\n\n\\label{f4}\n\n\\end{figure}\n\n\n As seen\nfrom Fig.~\\ref{f4}, the dependence $B(T_{\\rm max})$ is weak and so we\nbelieve that the linear extrapolation is good to extract $B_c$. In\ncontrast, the accuracy of the determination of $R_c$ is poor.\n\n\\begin{figure}\n\\centerline{\\psfig{file=fig3.eps,width=0.6\\columnwidth,clip=}}\n\n\\smallskip\\caption{Behaviour of $\\partial R/\\partial B$ with temperature for\nstates 1 and 3. The values of exponent $z\\nu$ are indicated.}\n\n\\label{f3}\n\n\\end{figure}\n\n\nThe derivative $\\partial R/\\partial B$ near $B_c$ as a function of\ntemperature is shown in Fig.~\\ref{f3}.\nThe exponent turns out to be the same within experimental uncertainty\nfor the film states 1 and 3 and is in agreement with results of\nRefs.~\\onlinecite{HP,Kapit} where authors argued observation of the\nfield-induced 2D SIT for states close to the zero-field SIT. This\nfact is in favour of 2D SIT scenario also for deeper film states in\nthe superconducting phase.\n\n\n\\begin{figure}\n\\psfig{file=fig5.eps,width=\\columnwidth,clip=}\n\n\\smallskip\\caption{Scaling plots for state 1 (a) and for state 3 without (b)\nand with (c) the linear temperature term.}\n\n\\label{f5}\n\n\\end{figure}\n\nKnowing $B_c$ and the scaling exponent we can replot the experimental\ndata as a function of scaling variable $u$ (Fig.~\\ref{f5}). As seen\nfrom Figs.~\\ref{f5}a and b, for state 1 the data collapse onto a\nsingle curve whereas for state 3 we obtain a set of similar curves\nshifted along the vertical axis. Subtracting formally from $R(T,B)$\nthe linear temperature term $R_c\\alpha T$ (where $\\alpha$ is a\nfactor) does reveal the 2D scaling behaviour for state 3\n(Fig.~\\ref{f5}c). We note that the procedure of dividing the\nexperimental data in Fig.~\\ref{f5}b by $R_c(T)$, which corresponds to\nthe formula (\\ref{e3D}) for 3D scaling, does not lead to success.\n\nThus, we find that the 2D scaling holds for states near the\nzero-field SIT while the data for deeper states in the\nsuperconducting phase are best described by the relation (\\ref{x=})\nwith an additive temperature-dependent correction $f(T)$\n\n\\begin{equation}\nR(T,B)\\equiv R_c[r(u)+f(T)]. \\label{f}\n\\end{equation}\n\nTo get a basis for the formal analysis of the experimental data we\nhave to answer two questions: (i) whether our film is really 2D; and\n(ii) what is the physical origin for the temperature dependence of\n$R_c(T)$? In the first case we need to compare the film thickness $h$\nwith characteristic lengths. These are the coherence length\n$\\xi_{sc}=c\\hbar/2eB_{c2}l$ (where $l$ is the mean free path in\nnormal state) in superconducting state and the dephasing length\n$L_\\phi(T)\\simeq\\hbar^2/m\\xi_{sc}T$ \\cite{Fisher,QPT} that restricts\nthe diverging correlation length $\\xi$ in the vicinity of quantum\nSIT. Knowing the normal state film resistance $R\\approx 5$~k$\\Omega$\nat $T\\approx 4$~K and assuming that we deal with the amorphous 3D\nmetal in which the mean free path is normally close to the lowest\npossible value $l\\approx 1/k_F$, we estimate the length\n$l\\approx 8$~\\AA. If we crudely evaluate the field $B_{c2}$ at $B_c=7.2$~T\nas\ndetermined for state 3, we get\n\nupper limit of\n\n$\\xi_{sc}\\sim 500$~\\AA\\ and\n$L_\\phi\\sim 400$~\\AA\\\nat $T=0.5$~K. This supports the 2D scenario of\nquantum SIT although in the normal state the film turns out to be 3D.\n\nWith respect to the temperature-dependent $R_c(T)$, at finite\ntemperatures the conductivity of the film near $B_c$ should include\nthe contribution from localized normal electrons in addition to the\nconductivity defined by the diffusion of fluctuation-induced Cooper\npairs \\cite{Larkin,Kapit}. It is the normal electron conductivity\nthat explains the non-universality of the critical resistance\n\\cite{Kapit} as well as the additional term in Eq.~(\\ref{f}). We\nwrite this term in the general form because the linear extrapolation\nused is likely to break in the vicinity of $T=0$.\n\n\nSo, all of the experimental observations can be reconciled with the\n2D scaling scenario. Intriguingly, the same scaling behaviour has\nbeen established in a parallel magnetic field \\cite{gg5}. Although\nnot in favour of 2D concept, this fact indicates that the\nrestrictions imposed by the theory \\cite{Fisher} may be too severe.\n\n\nWe would like to mention an alternative way to make up for the term\n$f(T)$ in Eq.~(\\ref{f}): to introduce the temperature-dependent field\n$B_c(T)$ defined through the constancy of $R_c$. Formally both ways\nare equivalent and correspond to shifts of the isotherms in\nFig.~\\ref{f2} either along the $R$-axis or along the $B$-axis so that\nin the vicinity of transition a common crossing point is attained. In\ncontrast to the normal behaviour of the critical fields in\nsuperconductors, the so-defined $B_c(T)$ increases with temperature.\nThis can be interpreted in terms of temperature-induced boson\ndelocalization.\n\n\n\nIn summary, in experiments on amorphous In--O films with different\noxygen content we have found a change of the field-driven 2D SIT\nscenario as the film state departs from the zero-field SIT. For deep\nfilm states in the superconducting phase, in addition to the\nuniversal function of scaling variable that describes the\nconductivity of fluctuation-induced Cooper pairs, there emerges a\ntemperature-dependent contribution to the film resistance. This\ncontribution can be attributed to the conductivity of normal\nelectrons.\n\n\n\nWe gratefully acknowledge useful discussions with V.~Dobrosavljevich\nand A.I.~Larkin. This work was supported by Grants RFBR 99-02-16117\nand RFBR-PICS 98-02-22037 and by the Programme \"Statistical Physics\"\nfrom the Russian Ministry of Sciences.\n\n\n\n\\begin{references}\n\n\\bibitem{Fisher} M.P.A.~Fisher, Phys.\\ Rev.\\ Lett.\\ {\\bf 65}, 923\n(1990); M.P.A.~Fisher, G.~Grinshtein, and S.M.~Girvin, Phys.\\ Rev.\\\nLett.\\ {\\bf 64}, 587 (1990).\n\\bibitem{QPT} S.L.~Sondhi, S.M.~Girvin, J.P.~Carini, and D.~Shahar,\nRev.\\ Mod.\\ Phys.\\ {\\bf 69}, 315 (1997).\n\\bibitem{Larkin} A.~Larkin, Ann.\\ Phys.\\ (Leipzig) {\\bf 8}, 785\n(1999).\n\\bibitem{golda} A. Gold, Z.\\ Phys.\\ B\\ {\\bf 52}, 1 (1983).\n\\bibitem{Bul} L.~Bulaevskii, A.~Varlamov, and M.~Sadovskii, Sov.\\\nPhys.\\ Solid\\ State\\ {\\bf 28}, 997 (1986).\n\\bibitem{Fish2} M.P.A.~Fisher, P.B.~Weichman, G.~Grinstein, and\nD.S.~Fisher, Phys.\\ Rev.\\ B {\\bf 40}, 546 (1989).\n\\bibitem{HP} A.F.~Hebard and M.A.~Paalanen, Phys.\\ Rev.\\ Lett.\\ {\\bf\n65}, 927 (1990).\n\\bibitem{Kapit} A.~Yazdani and A.~Kapitulnik, Phys.\\ Rev.\\ Lett.\\\n{\\bf 74}, 3037 (1995).\n\\bibitem{Okuma} S.~Okuma, T.~Terashima, and N.~Kokubo, Solid\\ State\\\nCommun.\\ {\\bf 106}, 529 (1998).\n\\bibitem{cher} J.A.~Chervenak and J.M.~Valles, Jr, cond-mat.\n/9909329/\n\\bibitem{HP1} M.A.~Paalanen, A.F.~Hebard, and R.R.~Ruel,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 69}, 1604 (1992).\n\\bibitem{gg2}\nV.F.~Gantmakher, M.V.~Golubkov, J.G.S.~Lok, and A.K.~Geim, JETP\\ {\\bf\n82}, 951 (1996).\n\\bibitem{JETPL} V.F.~Gantmakher, M.V.~Golubkov,\nV.T.~Dolgopolov, G.E.~Tsydynzhapov, and A.A.~Shashkin, JETP\\ Lett.\\\n{\\bf 68}, 345 (1998).\n\\bibitem{InO} The films were kindly presented\nby A.~Frydman and Z.~Ovadyahu from Jerusalem University.\n\\bibitem{ShOv} D.~Shahar, and Z.~Ovadyahu, Phys.\\ Rev.\\ B\\ {\\bf 46},\n10917 (1992).\n\\bibitem{Kim} J.J.~Kim, J.~Kim, and H.-L.~Lee, Phys.\\ Rev.\\ B\\ {\\bf\n46}, 11709 (1992).\n\\bibitem{rem} Observation of so-called quasireentrant states for the\nfield-driven SIT was reported in Refs.~\\onlinecite{HP,Kim,Gold} and\nexplained by inhomogeneities and single-particle tunneling between\nsuperconducting grains \\cite{Gold}. This interpretation is supported\nin our experiments by the fact that the quasireentrant behaviour\nobserved in some film states disappeared as a result of annealing the\nsample in vacuum for several additional hours after its resistance\nhad already saturated. We do not discuss quasireentrant states in the\npresent paper.\n\\bibitem{Gold} Y.~Liu, D.B.~Haviland, B.~Nease, and A.M.~Goldman,\nPhys.\\ Rev.\\ B\\ {\\bf 47}, 5931 (1993).\n\n\\bibitem{mit} Similar extrapolation procedure for determining $R_c$\nhas been used in Ref.~\\onlinecite{mit1} where metal-insulator\ntransition in 2D system was studied and the carrier density was used\nas the driving parameter.\n\n\\bibitem{mit1} Y.~Hanein, D.~Shahar, J.~Yoon, C.C.~Li, D.C.Tsui, and\nH.~Shtrikman, Phys.\\ Rev.\\ B\\ {\\bf 58}, R7520 (1998).\n\\bibitem{gg5} V.F.~Gantmakher, M.V.~Golubkov, V.T.~Dolgopolov,\nG.E.~Tsydynzhapov, and A.A.~Shashkin, Ann.\\ Phys.\\ (Leipzig) {\\bf 8},\nSI-73 (1999).\n\\end{references}\n\n\n\n\\end{document}\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002065.extracted_bib",
"string": "\\bibitem{Fisher} M.P.A.~Fisher, Phys.\\ Rev.\\ Lett.\\ {\\bf 65}, 923\n(1990); M.P.A.~Fisher, G.~Grinshtein, and S.M.~Girvin, Phys.\\ Rev.\\\nLett.\\ {\\bf 64}, 587 (1990).\n\n\\bibitem{QPT} S.L.~Sondhi, S.M.~Girvin, J.P.~Carini, and D.~Shahar,\nRev.\\ Mod.\\ Phys.\\ {\\bf 69}, 315 (1997).\n\n\\bibitem{Larkin} A.~Larkin, Ann.\\ Phys.\\ (Leipzig) {\\bf 8}, 785\n(1999).\n\n\\bibitem{golda} A. Gold, Z.\\ Phys.\\ B\\ {\\bf 52}, 1 (1983).\n\n\\bibitem{Bul} L.~Bulaevskii, A.~Varlamov, and M.~Sadovskii, Sov.\\\nPhys.\\ Solid\\ State\\ {\\bf 28}, 997 (1986).\n\n\\bibitem{Fish2} M.P.A.~Fisher, P.B.~Weichman, G.~Grinstein, and\nD.S.~Fisher, Phys.\\ Rev.\\ B {\\bf 40}, 546 (1989).\n\n\\bibitem{HP} A.F.~Hebard and M.A.~Paalanen, Phys.\\ Rev.\\ Lett.\\ {\\bf\n65}, 927 (1990).\n\n\\bibitem{Kapit} A.~Yazdani and A.~Kapitulnik, Phys.\\ Rev.\\ Lett.\\\n{\\bf 74}, 3037 (1995).\n\n\\bibitem{Okuma} S.~Okuma, T.~Terashima, and N.~Kokubo, Solid\\ State\\\nCommun.\\ {\\bf 106}, 529 (1998).\n\n\\bibitem{cher} J.A.~Chervenak and J.M.~Valles, Jr, cond-mat.\n/9909329/\n\n\\bibitem{HP1} M.A.~Paalanen, A.F.~Hebard, and R.R.~Ruel,\nPhys.\\ Rev.\\ Lett.\\ {\\bf 69}, 1604 (1992).\n\n\\bibitem{gg2}\nV.F.~Gantmakher, M.V.~Golubkov, J.G.S.~Lok, and A.K.~Geim, JETP\\ {\\bf\n82}, 951 (1996).\n\n\\bibitem{JETPL} V.F.~Gantmakher, M.V.~Golubkov,\nV.T.~Dolgopolov, G.E.~Tsydynzhapov, and A.A.~Shashkin, JETP\\ Lett.\\\n{\\bf 68}, 345 (1998).\n\n\\bibitem{InO} The films were kindly presented\nby A.~Frydman and Z.~Ovadyahu from Jerusalem University.\n\n\\bibitem{ShOv} D.~Shahar, and Z.~Ovadyahu, Phys.\\ Rev.\\ B\\ {\\bf 46},\n10917 (1992).\n\n\\bibitem{Kim} J.J.~Kim, J.~Kim, and H.-L.~Lee, Phys.\\ Rev.\\ B\\ {\\bf\n46}, 11709 (1992).\n\n\\bibitem{rem} Observation of so-called quasireentrant states for the\nfield-driven SIT was reported in Refs.~\\onlinecite{HP,Kim,Gold} and\nexplained by inhomogeneities and single-particle tunneling between\nsuperconducting grains \\cite{Gold}. This interpretation is supported\nin our experiments by the fact that the quasireentrant behaviour\nobserved in some film states disappeared as a result of annealing the\nsample in vacuum for several additional hours after its resistance\nhad already saturated. We do not discuss quasireentrant states in the\npresent paper.\n\n\\bibitem{Gold} Y.~Liu, D.B.~Haviland, B.~Nease, and A.M.~Goldman,\nPhys.\\ Rev.\\ B\\ {\\bf 47}, 5931 (1993).\n\n\n\\bibitem{mit} Similar extrapolation procedure for determining $R_c$\nhas been used in Ref.~\\onlinecite{mit1} where metal-insulator\ntransition in 2D system was studied and the carrier density was used\nas the driving parameter.\n\n\n\\bibitem{mit1} Y.~Hanein, D.~Shahar, J.~Yoon, C.C.~Li, D.C.Tsui, and\nH.~Shtrikman, Phys.\\ Rev.\\ B\\ {\\bf 58}, R7520 (1998).\n\n\\bibitem{gg5} V.F.~Gantmakher, M.V.~Golubkov, V.T.~Dolgopolov,\nG.E.~Tsydynzhapov, and A.A.~Shashkin, Ann.\\ Phys.\\ (Leipzig) {\\bf 8},\nSI-73 (1999).\n"
}
] |
cond-mat0002066
|
$c$-Axis tunneling in YBa$_2$Cu$_3$O$_{7-\delta}$/PrBa$_2$Cu$_3$O$_{7-\delta}$ superlattices
|
[
{
"author": "J.C.Mart\\'\\i nez"
},
{
"author": "A.Schattke"
},
{
"author": "M. Jourdan"
},
{
"author": "G.Jakob"
},
{
"author": "H.Adrian"
}
] |
In this work we report $c$-axis conductance measurements done on a superlattice based on a stack of 2 layers YBa$_2$Cu$_3$O$_{7-\delta}$ and 7 layers PrBa$_2$Cu$_3$O$_{7-\delta}$ (2:7). We find that these quasi-2D structures show no clear superconducting coupling along the $c$-axis. Instead, we observe tunneling with a gap of $\Delta_c=5.0\pm 0.5$ \ meV for the direction perpendicular to the superconducting planes. The conductance spectrum show well defined quasi-periodic structures which are attributed to the superlattice structure. From this data we deduce a low temperature $c$-axis coherence length of $\xi_c=0.24\pm 0.03$\ nm.
|
[
{
"name": "lb7491b.tex",
"string": "% ***********************************\n% *\n% * File: Superl_ppn.tex\n% *\n% *\n\n\\documentstyle[prl, aps, epsf, preprint]{revtex}\n%\\documentstyle[prl, aps, epsf, color, multicol]{revtex}\n\\begin{document}\n\\draft\n\\preprint{Y123/Pr123-v1.4}\n\n\\title{\n$c$-Axis tunneling in YBa$_2$Cu$_3$O$_{7-\\delta}$/PrBa$_2$Cu$_3$O$_{7-\\delta}$\nsuperlattices\n\t}\n\t\n\\author{\n\tJ.C.Mart\\'\\i nez, A.Schattke, M. Jourdan, G.Jakob, H.Adrian\n\t}\n\t\n\\address{\n\tJohannes Gutenberg -University of Mainz; Institute of Physics; 55099\nMainz; Germany\n\t}\n\t\n\\date{ Feb. 17, 1999}\n\n\\maketitle\n\n\\begin{abstract}\nIn this work we report $c$-axis conductance measurements done on a superlattice based on \na stack \nof 2 layers YBa$_2$Cu$_3$O$_{7-\\delta}$ and 7 layers PrBa$_2$Cu$_3$O$_{7-\\delta}$ \n(2:7). We find that these quasi-2D structures show no clear superconducting coupling \nalong the \n$c$-axis. Instead, we observe tunneling with a gap of $\\Delta_c=5.0\\pm 0.5$ \\ meV for the \ndirection perpendicular to the superconducting planes. The conductance spectrum show well \ndefined quasi-periodic structures which are attributed to the superlattice structure. \nFrom this \ndata we deduce a low temperature $c$-axis coherence length of $\\xi_c=0.24\\pm 0.03$\\ nm.\n\\end{abstract}\n\n\\pacs{74.80.Dm,73.61.-r,73.20.Dx}\n\n%\\begin{multicols}{2}\n%\\narrowtext\n\n% ------------------------------------------\n% INTRODUCTION\n%\n\nAs for classical superconductors, tunneling experiments are a direct way of testing the \nlocal \nsuperconducting density of states \\cite{giaevaer}. In particular the co-existence of $s$-\nwave and \n$d$-wave components of the order parameter was directly investigated from $c$-axis planar \ntunneling measurements done in high temperature superconductors (HTS) \\cite{sun,kleiner}. \nHowever tunneling experiments are extremely sensitive to the quality of barriers and \ninterfaces. \nThis aspect explains the extreme care taken by the different groups in the fabrication of \nHTS \ntunnel junctions. One way of getting around this problem is to investigate $c$-axis \ntunneling in \nYBa$_2$Cu$_3$O$_{7-\\delta}$/PrBa$_2$Cu$_3$O$_{7-\\delta}$ superlattices (Y123/Pr123). \nIn those systems it is possible to modify the tunneling properties simply by varying the \nperiodicities of the Y123 and Pr123 layers. Another advantage is that transmission \nelectron \nmicroscope (TEM) studies show atomically flat Y123/Pr123 interfaces in superlattices \n\\cite{jia}.\n\nIn this work we study the influence of the periodicity of an artificial superlattice on \nthe local \nsuperconducting density of states of Y123. This was only possible by a sustained effort \ndivided \nin three main steps: the preparation of high quality Y123/Pr123 superlattices, the \npatterning of \nsuitable mesa structures and the measurement of the $c$-axis transport properties.\n\n% ------------------------------------------\n% PREPARATION TECHNIQUES\n%\n\nY123/Pr123 superlattices are deposited on a similar way as Y123 thin films. The only \ndifference is \nthat after sputtering one to 10 unit cells (u.c.) of Y123 the substrate is turned to a \ncathode \ncontaining the Pr123 target. This process is repeated until a total film thickness of \n200\\ nm. The \nswitching between targets is made with a computer-controlled step motor. To provide low \nohmic contacts the process ends by the in-situ deposition of a protective gold layer with \nthickness from 200 to 400 nm. In order to avoid the formation of pin-holes and reduce the \nsurface roughness the sputtering process was done at 3\\ mbar pressure of a mixture 1 to 2 \nof \nO$_2$ and Ar. \n\nThe crystallographic quality of the superlattices used in this work has been checked by \nx-ray \ndiffraction. Our measurements showed up to third order satellite peaks observed in \n($\\theta -\n2\\theta$) scans. On the other hand TEM studies show steps of only 1\\ u.c. for every 100\\ \nnm. \nThis value is much smaller than the 7 u.c. thick Pr123 barrier. \n\nGiven that a Gold layer covered the superlattices the surface morphology could not be \nchecked \ndirectly on the same samples where the $c$-axis measurements were done. However scanning \ntunneling microscopy done in similar superlattices reveal an average surface modulation \nof \n$\\pm$7.2 nm (6 u.c.) for a length scale of 1.6 $\\mu$m. This roughness is very small when \ncompared to the total thickness of the mesas which was of about 120 nm. Since the \nmeasurements below were performed on 2:7 superlattices and the top layer was always Y123, \nonly the two upper Y123 layers (of a total of 10) are probably affected by the protective \ngold \nlayer. This is expected to have little influence on our experimental results.\n\nBefore making the mesa structures, a ground electrode with 1.2 $\\times$ 10 mm$^2$ was wet \nchemically etched. The mesas were later on prepared by standard UV-photolithography and \nion \nmilling. During the etching process the samples were cooled down to 77\\ K. Later the chip \nwas \ncoated with photoresist, and a window was opened on the top of each mesa by a \nphotolithographic process. The preparation step ends with the deposition of a gold top \nelectrode patterned by wet chemical etching. Because of its smoothness, high homogeneity \nand \nlow defect density the photoresist was directly used for insulating the top contact from \nthe \nground electrode. All measurements were done in a ''three point'' geometry, where the \ntypical \ncontact resistance between the HTS material and gold is $R_S \\approx 3\\times 10^{-5} \n\\Omega$cm$^2$. For the moment, we are limited to mesa structures down to 15$\\times$15 \n$\\mu$m$^2$.\n\n% ------------------------------------------\n% RESULTS\n%\n\nIn Fig.\\ \\ref{fig1} we show a semi-logarithmic plot of the resistances of three mesas \nwith $30\\times \n30$, $40\\times 40$ and $50 \\times 50\\ \\mu$m$^2$ prepared on a Y123/Pr123 superlattice \nwith \n2 layers Y123 and 7 layers Pr123 (2:7). The transition observed at 65\\ K corresponds to \nthe \nsuperconducting transition $T_c$ of the Y123 layers. The reduced $T_c$ is typical for the \nnon-fully developed order parameter in the 2 u.c. thick Y123 layer \\cite{jakob}. Above \n$T_c$, \nthe larger mesas show a temperature dependence that is similar to the one measured in the \n(a,b)-plane. This demonstrates that we measured a non-negligible amount of the (a,b) \ncomponent. However below $T_c$ the superconducting Y123 layers define equipotential \nplanes, and only the $c$-axis component of the resistance can be measured. This is \nconfirmed by \nthe vertical shift existing between the different plots shown in Fig.\\ \\ref{fig1}. The \nlogarithmic y-\naxis shows that the different curves differ below $T_c$ only by a proportionality factor. \nThe \nresistances of the mesas at 20 K were $R_c(30)=168$\\ $\\Omega$, $R_c(40)=68$\\ $\\Omega$ and \n$R_c(50)=63$\\ $\\Omega$ which give ratios of $R_c^{50}/R_c^{30}=0.38$ (expected \n$50^2/30^2=0.36$) and $R_c^{40}/R_c^{30}=0.40$ ($40^2/30^2=0.56$). The discrepancy of \n30\\% observed in the $R_c^{40}/R_c^{30}$ can be attributed to a larger degree of damage \nintroduced during the etching of the $40\\times 40$\\ $\\mu$m mesa. The contact resistance \nof the \ngold top electrode should scale as well with the area of the mesas.\n\n%--------------------------------------\n%\n% Figure 1\n%\n\nWe measured simultaneously the $U$ vs. $I$ characteristics and the differential \nresistance in the \n$30\\times 30$\\ $\\mu$m$^2$ mesa. This was done by using a battery operated current source \nthat superposes above an arbitrary DC current, a small AC signal generated by the \nreference of a \nLock-in amplifier. The DC-and AC-signals were measured with an HP34420A nano-voltmeter \nand a PAR 5210 Lock-in amplifier respectively. In Fig.\\ \\ref{fig2} we show some of our \nresults \non a $30\\times 30 \\mu$m$^2$ mesa done on a 2:7 superlattice for temperatures between 2.0 \nand \n60\\ K. Several features can be identified in Fig.\\ \\ref{fig2}. The first is the parabolic \nbackground \nwhich can be well described by the Simmons model which predicts \\cite{simmons}: \n\\begin{equation}\n\\sigma_b(U)=\\sigma_0\\left( 1+\\zeta U^2\\right)\n\\label{eq1}\n\\end{equation}\nThis model corresponds to a metal-insulator-metal junction with a rectangular barrier of \nwidth $d$ \nand height $\\Phi$. The constant factors are $\\sigma_0=(3/2)(e(2m\\Phi)^{1/2}/h^2d) \\exp(-A \nd \n\\Phi^{1/2})$ and $\\zeta = (Aed)^2/96\\Phi$ where $A=4\\pi(2m)^{1/2}/h$. Considering that we \nhave 10 bi-layers connected in series, we deduced from Eq.\\ \\ref{eq1} that each Pr123 \nbarrier has \nbelow $T=25$\\ K an effective height $\\Phi = 370$\\ meV and an effective width $d=3.5$\\ nm. \nAlthough $d$ is much smaller than the 8.2\\ nm expected from the $7$\\ u.c., we can explain \nthis \nresult by pure geometrical arguments. If we take into account the imperfections in the \nsuperlattice (steps of 1 u.c. per interface) and the fact that superconductivity extends \nto the \nchains (0.5 u.c. per interface), we would have an effective barrier thickness of $\\approx \n4.8$\\ nm \n(4 u.c.). This crude estimation is only $37\\%$ larger than $d$.\nFrom our data we deduce that the values of $d$ and $\\Phi$ were practically temperature \nindependent up to $25$\\ K. Above this temperature the conductivity follows the behavior \ncharacteristic for resonant tunneling via up to two localized states \\cite{glazman}:\n\\begin{equation}\n\\sigma_b(U)=g_0+\\alpha U^{4/3}\n\\label{eq3}\n\\end{equation}\n\nThe two peaks observed at lower temperatures in $\\sigma(U)$ should indeed correspond to a \n$c$-axis superconducting gap. The distance between the two peaks is $U_{pp}=178$\\ mV. \nThis \nparticular mesa was made with a height of about 120 nm estimated from an etching rate \ncalibrated by Atomic Force Microscopy done in different etched films. This gives a stack \nof \n$n=8$ to $10$ bi-layers which present each a $c$-axis superconducting gap of \n$\\Delta_c=U_{pp}/4 n=5.0\\pm 0.5$ \\ meV.\n\n%--------------------------------------\n%\n% Figure 2\n%\n\n\nThis value is in excellent agreement with the value of $\\Delta_c$ given in the literature \nwhich \nscatters between 4 and 6 meV for planar junctions prepared with Pr123, CeO$_2$ and \nSrAlTaO$_6$ barriers \\cite{iguchi,ying,bari,nakajima}. To understand this result we have \nto \nlook at Scanning Tunneling Spectroscopy (STS) data. Typical STS measurements done on both \nY123 thin films and high quality single crystals give gap structures at about 5 and 20 \nmeV \n\\cite{miller,maggio}.These fine structures observed in tunneling spectra were explained \nsuccessfully by Miller at al by considering that Y123 is constituted by a stack of strong \nand \nweak superconducting layers which contribute with different weights to the tunneling \nspectra \n\\cite{miller,tachiki2}. In particular the 5 meV gap has been attributed to the BaO and \nCuO layers \nsituated between the CuO$_2$ blocks. Since these BaO and CuO layers are common to both \nY123 and Pr123 we expect them to constitute the interfaces which are relevant to our \ntunneling \nprocess. Given that $T_c$ in our superlattices is only reduced by 20\\% we do not expect \nthe \nCuO$_2$ superconducting gap to be strongly suppressed. On the other hand given that a 4-6 \nmeV gap structures are observed in measurements done with different barriers we think \nthat the \ngap $\\Delta_c\\approx 5$\\ meV is indeed an intrinsic property of Y123. The existence of \nthis small c-axis gap is consistent with the strong thermal smearing of the gap feature \nat about 50\\ K, which corresponds to an energy of the same magnitude as $\\Delta_c$. \n\nFor the moment it is not clear to us \nwhich will be the influence of the CuO chains to the symmetry of the order parameter. \nHowever \nsince these chains form together with the apical oxygen CuO$_2$ cells oriented along to \nthe $c$-\naxis, it is likely that a $c$-axis gap would exist in Y123 with a $s$-wave like symmetry.\n\n%--------------------------------------\n%\n% Figure 3\n%\n\nTo enhance the other features present in $\\sigma(U)$, we plot in Fig.\\ \\ref{fig3} \n$\\sigma/\\sigma_b$ \nfor temperatures between 2.0 and 65\\ K. For more clarity the data is vertically shifted. \nBelow \n25\\ K the Simmons model is used to calculate $\\sigma_b$ (Eq.\\ \\ref{eq1}). Above that \ntemperature the resonant tunneling expression given by Eq.\\ \\ref{eq3} is employed. We \nwould \nlike to emphasize that for all temperatures the low bias features were included in the \nestimation \nof $\\sigma_b(U)$. \n\nBelow 30\\ K we observe a number of reproducible features which were superposed to the \nsuperconducting gap. For increasing temperatures, the gap and these structures are \nsmeared out \nby thermal fluctuations. At about 30\\ K the only signature of the gap is a soft voltage \ndependence of $\\sigma$ and a zero bias peak which starts to develop, grows up to about \n50\\ K \nand finally disappears near $T_c$.\n\n%--------------------------------------\n%\n% Figure 4\n%\n%--------------------------------------\n%\n% Figure 5\n%\n\nAlthough there is not for the moment a clear explanation for the origin of this zero bias \npeak, its \ndisappearance close to $T_c$ shows that it is clearly related to tunneling of \nsuperconducting \npairs. As suggested by Abrikosov \\cite{abrikosov} this anomaly could be due to resonant \ntunneling through localized states in the barrier. The absence of a zero \nbias peak at lower temperatures is consistent with this picture: the success in fitting \nthe data \nwith a Simmons model indicates that below 25\\ K, the Pr123 layers behave \npredominantly like normal tunneling barriers having one or no resonant states.\n\nThe arrows in Fig.\\ \\ref{fig3} show sharper indentations in $\\sigma(U)$ which can be \nfollowed up \nto 30\\ K. To investigate the additional features present in the low temperature \nconductivity we \nplot in Fig.\\ \\ref{fig4} $\\sigma/\\sigma_b$ for U between 0.1 and 0.5\\ V. The vertical \nlines \ncorrespond to the minima of the oscillations $U_m$ which are particularly visible at \nlower \ntemperatures. To find the periodicity of these oscillations, we plot in Fig.\\ \\ref{fig5} \n$U_m$ vs. \nan integer index $n$. A clear zero crossing of the linear fit is obtained by choosing an \nindex n=9 \nfor the lowest extracted value of $U_m$. From the linear fit we deduce a period of $(11.1 \n\\pm \n0.5)$\\ mV. If we remember that this value corresponds to 8-10 junctions connected in \nseries, a \nsingle junction would show a periodicity of $\\delta U=(1.2\\pm 0.1)$ meV.\n\nThe expected $c$-axis density of states of a superlattice where the superconducting gap \nis a one \ndimensional periodic step has been already calculated by van Gelder in 1969 \n\\cite{van_gelder}. \nWith the HTS materials, the increasing interest on superlattices inspired the work of \nHahn in \nextending the model to the three dimensional case \\cite{hahn}. These models predict the \nopening \nof gaps which were particularly visible in the one dimensional case. In the 3-dimensional \ncase \nthey are smeared out, although still present. The main result from Ref.\\ \\cite{hahn} is \nthat above \nthe superconducting gap these additional gap structures should appear with a periodicity \nof: \n\\begin{equation} \n\\frac{\\xi_c}{s}=\\frac{1}{\\pi^2} \\frac{\\delta U} {\\Delta_c} \n\\label{eq2} \n\\end{equation} \nwhere $s$ is the periodicity of the superlattice, $\\xi_c$ the $c$-axis coherence length, \n$\\Delta_c$ \nthe $c$-axis gap and $\\delta U$ the periodicity of the sub-gap structures. By taking \n$s=10.5$\\ \nnm, and $\\Delta_c=5.0\\pm 0.5$ \\ meV we deduce from Eq.\\ \\ref{eq2} a $c$-axis coherence \nlength \nof $\\xi_c=0.27$ nm \\cite{remark1}. This result is close to the value of $\\xi_c=0.16\\pm \n0.01$\\ nm \ndeduced from an analysis of fluctuation conductivity in Y123/Pr123 superlattices \n\\cite{solovjov}. If we assume for Y123 an anisotropy of $\\gamma \\approx 5$ \\cite{janossi} \nwe \nwould obtain a in-plane coherence length $\\xi_{ab}=\\gamma\\xi_c\\approx 1.4$\\ nm. This \nvalue is \nclose to the generally quoted $\\xi_{ab}=1.5$\\ nm for Y123 \\cite{tinkham}. The larger \nstructures \nindicated by the arrows in Fig.\\ \\ref{fig3}, correspond to a periodicity of $100\\pm 1$ \nmV. It is \ninteresting to notice that the ratio of this periodicity divided by the periodicity of \n$U_m$ is \n$9.0\\pm 0.4$.This value corresponds to the ratio between $s$ and a Y123 unit cell! From \nEq.\\ \n\\ref{eq2} and Fig.\\ \\ref{fig4} we deduce that $\\xi_c$ is practically temperature \nindependent \nbelow 20 K.\n\n% ------------------------------------------\n% CONCLUSION\n%\n\nWe show in this paper that by constructing superlattices it is possible to generate sub-\ngap \nstructures. These features can be directly used to determine in an independent way a $c$-\naxis \ncoherence length $\\xi_c=0.24\\pm 0.03$\\ nm for a 2 u.c. thick Y123. \n\nThe agreement of these results with previous estimations of Y123 coherence lengths shows \nthat \n$\\Delta_c=5.0\\pm 0.5$ \\ meV could be indeed a $c$-axis superconducting gap related to the \nCuO \nchains.\n\nFrom the observed sub-gap structures, we conclude that $\\xi_c(T)$ is practically \ntemperature \nindependence below 20\\ K.\n\nDue to thermal fluctuations, this analysis could not be extended up to higher \ntemperatures.\n\nThe authors would like to thank K. Gray, J.F. Zasadzinski, R.A. Klemm, R. Schilling and \nJ. \nMannhart for valuable and stimulating discussions. This work was supported by the German \nBMBF through Contract 13N6916 and the European Union through the Training and Mobility \nof Researchers (ERBFMBICT972217).\n\n\n\\begin{references}\n\\bibitem{giaevaer} Ivar Giaever, Phys.Rev.Lett. {\\bf 5} 147 (1960)\n\\bibitem{sun} A.G. Sun, D.A. Gajewski, M.B. Maple, R.C. Dynes, Phys.Rev.Lett. {\\bf 72} \n2267 \n(1994)\n\\bibitem{kleiner} R.Kleiner {\\it et al.} Phys.Rev.Lett.{\\bf 72} 593 (1994)\n\\bibitem{jia} C.L.Jia, H.Soltner, G.Jakob, Th.Hahn, H.Adrian, and K.Urban, Physica C {\\bf \n210} 1 \n(1993)\n\\bibitem{jakob} G. Jakob, T. Hahn, C. St{\\\"o}lzel, C.Tom{\\'e}-Rosa, H. Adrian, \nEurophys.Lett. \n{\\bf 19} 135 (1992)\n\\bibitem{simmons} J.G. Simmons, J. Appl. Phys. {\\bf 34} 238 (1963)\n\\bibitem{glazman} L.I. Glazman, K.A. Matveev, Sov.Phys. JETP {\\bf 67} 1276 (1988)\n\\bibitem{iguchi} I. Iguchi, Z. Wen, Physica C {\\bf 178} 1 (1991)\n\\bibitem{ying} Q.Y. Ying, C. Hilbert, Appl.Phys.Lett. {\\bf 65} 3005 (1994)\n\\bibitem{bari} M.A. Bari, F.Baudenbacher, J. Santiso, E. J. Tarte, J.E. Evets, \nM.G.Balmire, \nPhysica C, {\\bf 256} 227 (1996)\n\\bibitem{nakajima} K. Nakajima, T. Arai, S.E. Shafranjuk, T. Yamashita, I. Tanaka, H. \nKojima, \nPhysica C, {\\bf 293} 292 (1997)\n\\bibitem{miller} T.G. Miller, M. McElfresh, R. Reifenberger, Phys.Rev.B {\\bf 48} 7499 \n(1993)\n\\bibitem{maggio} I. Maggio-Aprile, Ch. Renner, A. Erb, E. Walker, \\O Fischer, \nPhys.Rev.Lett. {\\bf \n75} 2754 (1995)\n\\bibitem{tachiki2} M. Tachiki, S. Takahashi, F. Stegkich, H. Adrian, Z. Phys B {\\bf 80} \n161 \n(1990)\n\\bibitem{abrikosov} A.A.Abrikosov, Phys.Rev.B {\\bf 57} 7488 (1998)\n\\bibitem{van_gelder} A.P. van Gelder, Phys.Rev. {\\bf 181} 787 (1969)\n\\bibitem{hahn} A. Hahn, Physica B, {\\bf 165-166} 1065 (1990)\n\\bibitem{remark1} Since $\\xi_c$ is here deduced from the ratio $\\Delta_c/\\delta U$, our \nresults are \nindependent from the number of junctions constituting the stack. \n\\bibitem{solovjov} A.L. Solovjov , V.M. Dmitriev, H.-U. Habermeier, I.E. Trofimov, \nPhys.Rev.B \n{\\bf 55} 8551 (1997)\n\\bibitem{janossi} B. Janossi D. Prost, S. Pekker, L. Fruchter, Physica C {\\bf 181} 51 \n(1991)\n\\bibitem{tinkham} Michael Tinkham, {\\it Introduction to Superconductivity, 2nd ed.} 325 \n(1996) \n\\end{references}\n\n\\newpage\n%--------------------------------------\n%\n% Figure 1\n%\n\\begin{figure}[t]\n\t\\centering\n\t\\epsfxsize = 12 cm\n\t\\epsfbox{\n\t\tFig1.eps}\n\t\\vspace{5 mm}\n\t\\caption{\nResistance of $30\\times 30$, $40\\times 40$ and $50\\times 50$\\ $\\mu$m$^2$ mesas patterned \non a \n2:7 Y123/Pr123 superlattice. The three measurements are normalized to the resistance at \n250\\ K \nand represented with a y-logarithmic axis. \\label{fig1}}\n\\end{figure}\n\n\\newpage\n%--------------------------------------\n%\n% Figure 2\n%\n\n\\begin{figure}[t]\n\t\\centering\n\t\\epsfxsize = 12 cm\n\t\\epsfbox{\n\t\tFig2.eps}\n\t\\vspace{5 mm}\n\t\\caption{\nDifferential conductivity $\\sigma(U)$ of a $30\\times 30 \\mu$m$^2$ mesa for temperatures \nbetween \n2.0 and 60\\ K. The parabolic background can be associated to a barrier with $3.5$ \\ nm \nwidth and 350 meV height (see text).\n\\label{fig2}}\n\\end{figure}\n\n\\newpage\n%--------------------------------------\n%\n% Figure 3\n%\n\n\\begin{figure}[t]\n\t\\centering\n\t\\epsfxsize = 12 cm\n\t\\epsfbox{\n\t\tFig3.eps}\n\t\\vspace{5 mm}\n\t\\caption{\nDifferential conductivity $\\sigma$ divided by the parabolic background $\\sigma_b$ for \n2.0, 10, 15, \n20, 25, 30, 35, 40, 45, 50, 55, 60 and 65\\ K (from bottom to top). The different \nmeasurements \nare shifted vertically for sake of clarity. The arrows indicate sharper features \nnoticeable in the \ntunneling spectrum.\n\\label{fig3}}\n\\end{figure}\n\n\\newpage\n%--------------------------------------\n%\n% Figure 4\n%\n\n\\begin{figure}[t]\n\t\\centering\n\t\\epsfxsize = 12 cm\n\t\\epsfbox{\n\t\tFig4.eps}\n\t\\vspace{5 mm}\n\t\\caption{\nSame data as in Fig.\\ 3 plotted on a larger scale for the indicated temperatures. The \nvertical lines \nindicate the minima between 0.1 and 0.3\\ V observed in $\\sigma/\\sigma_b$ for $T=2$\\ K.\n\\label{fig4}}\n\\end{figure}\n\n\\newpage\n%--------------------------------------\n%\n% Figure 5\n%\n\n\\begin{figure}[t]\n\t\\centering\n\t\\epsfxsize = 12 cm\n\t\\epsfbox{\n\t\tFig5.eps}\n\t\\vspace{5 mm}\n\t\\caption{\nMinima extracted from $\\sigma/\\sigma_b$ plotted as function of an integer index $n$. The \nslope of \n$(11.1 \\pm 0.5)$\\ mV corresponds to a quasi-periodicity of the smaller structures in \nFig.\\ \n\\ref{fig4}.\n\\label{fig5}}\n\\end{figure}\n\n%\\end{multicols}\n\\end{document}\n\n\n"
}
] |
[
{
"name": "cond-mat0002066.extracted_bib",
"string": "\\bibitem{giaevaer} Ivar Giaever, Phys.Rev.Lett. {\\bf 5} 147 (1960)\n\n\\bibitem{sun} A.G. Sun, D.A. Gajewski, M.B. Maple, R.C. Dynes, Phys.Rev.Lett. {\\bf 72} \n2267 \n(1994)\n\n\\bibitem{kleiner} R.Kleiner {\\it et al.} Phys.Rev.Lett.{\\bf 72} 593 (1994)\n\n\\bibitem{jia} C.L.Jia, H.Soltner, G.Jakob, Th.Hahn, H.Adrian, and K.Urban, Physica C {\\bf \n210} 1 \n(1993)\n\n\\bibitem{jakob} G. Jakob, T. Hahn, C. St{\\\"o}lzel, C.Tom{\\'e}-Rosa, H. Adrian, \nEurophys.Lett. \n{\\bf 19} 135 (1992)\n\n\\bibitem{simmons} J.G. Simmons, J. Appl. Phys. {\\bf 34} 238 (1963)\n\n\\bibitem{glazman} L.I. Glazman, K.A. Matveev, Sov.Phys. JETP {\\bf 67} 1276 (1988)\n\n\\bibitem{iguchi} I. Iguchi, Z. Wen, Physica C {\\bf 178} 1 (1991)\n\n\\bibitem{ying} Q.Y. Ying, C. Hilbert, Appl.Phys.Lett. {\\bf 65} 3005 (1994)\n\n\\bibitem{bari} M.A. Bari, F.Baudenbacher, J. Santiso, E. J. Tarte, J.E. Evets, \nM.G.Balmire, \nPhysica C, {\\bf 256} 227 (1996)\n\n\\bibitem{nakajima} K. Nakajima, T. Arai, S.E. Shafranjuk, T. Yamashita, I. Tanaka, H. \nKojima, \nPhysica C, {\\bf 293} 292 (1997)\n\n\\bibitem{miller} T.G. Miller, M. McElfresh, R. Reifenberger, Phys.Rev.B {\\bf 48} 7499 \n(1993)\n\n\\bibitem{maggio} I. Maggio-Aprile, Ch. Renner, A. Erb, E. Walker, \\O Fischer, \nPhys.Rev.Lett. {\\bf \n75} 2754 (1995)\n\n\\bibitem{tachiki2} M. Tachiki, S. Takahashi, F. Stegkich, H. Adrian, Z. Phys B {\\bf 80} \n161 \n(1990)\n\n\\bibitem{abrikosov} A.A.Abrikosov, Phys.Rev.B {\\bf 57} 7488 (1998)\n\n\\bibitem{van_gelder} A.P. van Gelder, Phys.Rev. {\\bf 181} 787 (1969)\n\n\\bibitem{hahn} A. Hahn, Physica B, {\\bf 165-166} 1065 (1990)\n\n\\bibitem{remark1} Since $\\xi_c$ is here deduced from the ratio $\\Delta_c/\\delta U$, our \nresults are \nindependent from the number of junctions constituting the stack. \n\n\\bibitem{solovjov} A.L. Solovjov , V.M. Dmitriev, H.-U. Habermeier, I.E. Trofimov, \nPhys.Rev.B \n{\\bf 55} 8551 (1997)\n\n\\bibitem{janossi} B. Janossi D. Prost, S. Pekker, L. Fruchter, Physica C {\\bf 181} 51 \n(1991)\n\n\\bibitem{tinkham} Michael Tinkham, {\\it Introduction to Superconductivity, 2nd ed.} 325 \n(1996) \n"
}
] |
cond-mat0002067
|
Special Analytical Solutions of the Schr\"odinger Equation for 2 and 3 Electrons in a Magnetic Field and {\em ad hoc} Generalizations to \mbox{N particles}
|
[
{
"author": "by M. Taut"
},
{
"author": "Institut f\\\"ur Festk\\\"orper und Werkstoff- Forschung Dresden"
},
{
"author": "Postfach 270018"
},
{
"author": "01171 Dresden"
},
{
"author": "Germany"
}
] |
We found that the two--dimensional Schr\"odinger equation for 3 electrons in an homogeneous magnetic field (perpendicular to the plane) and a parabolic scalar confinement potential (frequency $\omega_0$) has exact analytical solutions in the limit, where the expectation value of the center of mass vector $R$ is small compared with the average distance between the electrons. These analytical solutions exist only for certain discrete values of the effective frequency $\tilde \omega=\sqrt{\omega_o^2 + ({ \omega_c \over 2} )^2}$. Further, for finite external fields, the total angular momenta must be $M_L=3 m$ with $m=integer$, and spins have to be parallel. The analytically solvable states are always cusp states, and take the components of higher Landau levels into account. These special analytical solutions for 3 particles and the exact solutions for 2 particles \cite{Taut2e} can be written in an unified form. The first set of solutions reads\\ $ \Phi = \prod _{i<k} ({r}_i-{r}_k)^m\; p_{n,m}(|{r}_i-{r}_k|)\;\; \mbox{exp}\biggl(-\frac{1}{2} \; \tilde \omega_{n,m} \sum_l {r}_l^2 \biggr) $\\ where $p_{n,m}(x)$ are certain finite polynomials and $\tilde \omega_{n,m}$ is the spectrum of the fields. The pair angular momentum $m$ has to be an odd integer and the integer $n$ defines the number of terms in the polynomials. For infinite solvable fields $\tilde \omega_1$ there is a second set of the form\\ $ \Phi = {\cal A}_a \;\prod _{i<k} ({r}_i-{r}_k)^{m_{ik}}\;\; \mbox{exp}\biggl(-\frac{1}{2} \; \tilde \omega_1 \sum_l {r}_l^2 \biggr) $\\ where ${\cal A}_a$ is the antisymmetrizer and the pair angular momenta $m_{ik}$ can all be different integers. In both cases the first factor is a short-- hand with the convention ${r}^m=r^{|m|} e^{im\alpha}$. These formulae, when {\em ad hoc} generalized to N coordinates, can be discussed as an ansatz for the wave function of the N--particle system. This ansatz fulfills the following demands: it is exact for two particles and for 3 particles in the limit of small $R$ and for the solvable external fields, and it is an eigenfuncton of the total orbital angular momentum. The Laughlin functions are special cases of this ansatz for infinite solvable fields and equal pair-- angular-- momenta.
|
[
{
"name": "three-dot.tex",
"string": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% This version contains the figures and tables in the text\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% some abbreviations for the tables\n\\def\\a{$\\alpha$}\n\\def\\b{$\\beta$}\n\n\\documentstyle[psfig,12pt,twoside,fleqn]{article}\n\n\\title{Special Analytical Solutions of the Schr\\\"odinger Equation\nfor 2 and 3 Electrons in a Magnetic Field and\n {\\em ad hoc} Generalizations to \\mbox{N particles}} \n\\author{by M. Taut\\\\Institut f\\\"ur Festk\\\"orper und Werkstoff-\n Forschung Dresden\\\\\nPostfach 270018\\\\ 01171 Dresden, Germany\\\\\nemail: m.taut@ifw-dresden.de }\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nWe found that the two--dimensional\nSchr\\\"odinger equation for 3 electrons in an homogeneous\nmagnetic field (perpendicular to the plane) and a parabolic\nscalar confinement potential (frequency $\\omega_0$) \nhas exact analytical solutions in the\nlimit, where the expectation value of the\ncenter of mass vector $\\bf R$ is small compared with the average \ndistance between the electrons.\nThese analytical solutions exist only\nfor certain discrete values of the effective frequency\n$\\tilde \\omega=\\sqrt{\\omega_o^2 + ({ \\omega_c \\over 2} )^2}$.\nFurther, for finite external fields,\nthe total angular momenta must be $M_L=3 m$ with $m=integer$,\nand spins have to be parallel. The analytically solvable states are always\ncusp states, and take the components of higher Landau levels into account. \nThese special\nanalytical solutions for 3 particles \nand the exact solutions for 2 particles \\cite{Taut2e}\ncan be written in\nan unified form.\nThe first set of solutions reads\\\\\n$\n\\Phi = \\prod _{i<k} ({\\bf r}_i-{\\bf r}_k)^m\\;\np_{n,m}(|{\\bf r}_i-{\\bf r}_k|)\\;\\;\n\\mbox{exp}\\biggl(-\\frac{1}{2} \\; \\tilde \\omega_{n,m} \\sum_l {\\bf r}_l^2 \\biggr)\n$\\\\\nwhere $p_{n,m}(x)$ are certain finite \npolynomials and $\\tilde \\omega_{n,m}$ is the spectrum of \nthe fields. \nThe pair angular momentum $m$ has to be an odd integer and the integer \n$n$ defines\nthe number of terms in the polynomials.\nFor infinite solvable fields $\\tilde \\omega_1$\nthere is a second set of the form\\\\\n$\n\\Phi = {\\cal A}_a \\;\\prod _{i<k} ({\\bf r}_i-{\\bf r}_k)^{m_{ik}}\\;\\;\n\\mbox{exp}\\biggl(-\\frac{1}{2} \\; \\tilde \\omega_1 \\sum_l {\\bf r}_l^2 \\biggr)\n$\\\\\nwhere ${\\cal A}_a$ is the antisymmetrizer and the pair angular momenta\n$m_{ik}$\ncan all be different integers.\nIn both cases the first factor is a short-- hand with\nthe convention ${\\bf r}^m=r^{|m|} e^{im\\alpha}$.\nThese formulae, when {\\em ad hoc} generalized to N coordinates,\ncan be discussed as an ansatz for the wave function of the N--particle system.\nThis ansatz fulfills the following demands: it is exact for two particles\nand for 3 particles in\nthe limit of small $\\bf R$ and for the solvable external fields,\nand it is an eigenfuncton of the total\norbital angular momentum. The Laughlin functions are special cases\nof this ansatz\nfor infinite solvable fields and equal pair-- angular-- momenta. \n\\end{abstract}\nPACS classification: \\\\\n 8.30.Vw (Quantum dots etc.), \n 73.30.Hm (Quantum Hall Effect)\\\\\n\n\\newpage\n\n\\section{Introduction}\nMost work on correlated electron systems in a magnetic field \n(and a\nparabolic confinement potential) has been done adopting the following methods:\n Finite particle number ($N<10$) and finite field systems are tackled either by \n{\\em numerical} expansion of the wave functions in\nantisymmetrized products of one-- particle functions \n\\cite{Girvin}--\\cite{McDonald}\nor \n{\\em analytical ad hoc} approaches \nin the high field limit, where only the lowest Landau level (LLL) contributes\n\\cite{Laughlin}--\\cite{Jain-QD}.\nOther main streams are to use the Chern--Simons transformation \n\\cite{Halperin} \\cite{MacDonald-2}\nand hoping that the transformed wave function can be guessed or\napproximated more easily, or to use models for the electron-- electron\ninteraction \\cite{Haldane} \\cite{MacDonald-2} (All the above references are\nmostly reviews).\nIn this paper we are trying another approach: \nWe are looking for {\\em analytical} solutions for few electron systems\nand trying to generalize them {\\em ad hoc} to N particle systems.\nWe use the genuine Coulombic electron-- electron interaction and do {\\em not}\nrestrict ourselves to the lowest Landau level (LLL).\nIn a previous paper \\cite{Taut2e} it has been shown that for $N=2$\n(with Coulomb interaction between the electrons) there\nis a 'spectrum'\nof discrete external field values for which the Schr\\\"odinger equation\ncan be solved\nexactly and analytically. As shown below, these solutions comprise the\nLaughlin states for $N=2$ as special cases.\nThe questions to be addressed in this paper are the following: Do similar\nexact solutions also exist for three electrons? If so, does the corresponding\nfield spectrum agree with the spectrum for $N=2$?\nIs there any connection between the discrete field spectrum for solvability\nand the discrete fields (for given particle density) observed in the \nQuantum Hall effect?\nAre the Laughlin states still among these special states as special cases?\nTo answer one of the questions in advance: We did not find exact analytical \nsolutions for $N=3$. However, if we consider \nthe center of mass coordinate \n${\\bf R} = \\frac{1}{3}\n\\sum_{i=1}^3 {\\bf r}_i$\nversus inter--particle distance as an\nsmall parameter and expand the Hamiltonian in a multi-pole series, \nthe three-- electron-- system can be decomposed into 3 \npair problems which have similar analytical solutions as the\ntwo-- electron-- system.\nThe center of mass vector vanishes exactly in the classical ground state.\nTherefore, one should expect that our expansion works well\nfor weak external fields or for systems, where after a Chern--Simons \ntransformation and a proper mean field approximation\n(for finite systems!) the effective field is weak.\nSurprisingly, also for 3 particles in the small {\\bf R} limit,\nthe Laughlin states, which belong to {\\em infinite} fields,\n are among the analytically solvable solutions.\\\\\n%Therefore one could conjecture,\n%that small $\\bf R$ versus inter--particle distance\n%is a feature of all Quantum Hall states.\\\\\n\n\nThe plan of this \npaper is the following. \nSect.2 gives a survey on the results of the exact solutions of the \nelectron pair problem.\nBecause in this paper the three-- electron-- problem\nis traced back to three pair problems, this seems to be \nhelpful.\nIn Sect.3 we define an orthogonal transformation \nfor the three-- electron-- problem\nwhich contains the center of mass $\\bf R$ as a parameter, and then we \nexpand the transformed \nHamiltonian into a multi-pole series in $\\bf R$.\nIn Sect.4 it is shown that,\nin zero order in $\\bf R$, the transformed Schr\\\"odinger equation\ncan be solved exactly and analytically for a certain set of external fields and\ntotal angular momenta. In Sect.5, the eigenfunctions for 2 and 3 \nparticles are written\nin an unified form and {\\em ad hoc} generalized to arbitrary particle number.\nThis expression is compared with the Laughlin and Jain states.\nIn Sect.6 it is shown that the analytically solvable states are just\nthose states where a cusp appears in the energy versus total \nangular momentum curve.\nThe accuracy of the multi-pole expansion is tested in Sect.7 by calculation of\nthe energy eigenvalues in first order \nperturbation theory in the dipole and the quadrupole term of the \nHamiltonian.\n\n\\newpage\n\n\\section{Exact solutions for two electrons in an \nhomogeneous magnetic field}\nIn this section we summarize the results of \na previous paper \\cite{Taut2e} on the two-- electron-- problem and add some\nimportant subsequent unpublished findings.\nIn particular, we add the asymptotic solutions to our former pattern,\nwhich had not been given the due attention in \\cite{Taut2e},\nand incorporate the electron-- electron coupling constant \n$\\beta$ explicitly, in order to\nbe able to investigate the behavior of the exact solutions in\nvarying the coupling strength.\nCompleting and reviewing the two-- electron-- problem is important, because \nin the present work the three-- electron-- problem is traced back to\nthree two-- electron-- problems.\nIt has been shown in \\cite{Taut2e} that \nthe Schr\\\"odinger equation for two electrons in an homogeneous magnetic field\nplus an external parabolic scalar potential\nhas exact analytical solutions for a certain infinite, but discrete set\nof field values (hereafter referred to as 'solvable fields')\n\\footnote{If we speak of 'fields' without specification\nto a special one, we mean the effective oscillator frequency\n$\\sqrt{\\omega_0^2+(\\frac{\\omega_c}{2})^2}$, which is the relevant\nparameter.}.\nExcept for the asymptotic case of infinite external fields,\nwhich is part of this pattern,\nthere is a one-- to-- one correspondence between exact solutions and\nsolvable fields.\nSuch solutions exist for singlet and triplet states as well as ground and\nexcited states.\nA further qualitative feature is that these solutions occur, whenever\na correlated state (with electron-- electron interaction included)\n is degenerate with\nan uncorrelated one (without electron-- electron interaction) \n\\cite{Taut-unpublished}.\nMoreover, for each total spin and orbital angular momentum\nquantum number as well as for\na given degree of excitation (ground state, first excited state, etc.),\nthere is an infinite series of solvable fields which converges\nto zero. This means that\nthe solvable field values are dense at zero.\\\\\nNow we are going to describe the general analytical form of the \nexact solutions. After introducing relative and center of mass coordinate\n\\begin {equation}\n{\\bf r}={\\bf r}_2-{\\bf r}_1~~~~~;~~~~~{\\bf R}={1\\over 2}({\\bf r}_1+{\\bf\nr}_2)\n\\end{equation}\nthe Hamiltonian (in atomic units $\\hbar=m=e=1$) \n\\begin{equation}\nH=\\sum\\limits^2_{i=1}\\biggl\\{{1\\over 2}\\biggl({\\bf p}_i+\n{1\\over c}{\\bf A}({\\bf r}_i)\\biggl)^2 +\n{1\\over 2}\\omega_o^2r_i^2\\biggl\\}+{\\beta \\over |{\\bf r}_2-{\\bf r}_1|}\n+H_{spin}\n\\end{equation}\ndecouples exactly. \n\\begin{equation}\nH=2 \\;H_r+{1\\over 2} \\;H_R+H_{spin}\n\\end{equation}\n\nWe follow the notation of \\cite{Taut2e}\nas long as not explicitly mentioned. \n $\\omega_o$ is the oscillator frequency of\n the parabolic external\nconfinement potential and ${\\bf A}({\\bf r})=\\frac{1}{2}{\\bf B}\\times {\\bf r}$\nthe vector potential of\nthe external magnetic field.\nThe center of mass\n degree of freedom behaves like a\nquasi-particle in rescaled external fields and the quasi-particle of the\nrelative coordinate is a particle in rescaled external fields plus \na rescaled repulsive Coulomb field \noriginating from the \\mbox{e-- e-- interaction}.\n The Schr\\\"odinger equation of the first problem\nis trivial, the latter problem, which will be considered here,\nis described by the Hamiltonian\n(see eq.(5) in \\cite{Taut2e})\n\\begin {equation}\nH_r={1\\over 2}\\biggl[{\\bf p}+\n{1\\over c}{\\bf A}_r\\biggr]^2+{1\\over\n2}\\omega_r^2r^2+\\frac{\\beta}{2r}\n\\end{equation}\nwhere \\footnote{The index '$r$' and '$R$' refers to the relative \nand c.m. coordinate systems, respectively}\n$\\omega_r={1\\over 2}\\omega_o$, ${\\bf A}_r={1\\over 2}{\\bf A}({\\bf r})$.\nIn polar coordinates ${\\bf r}=(r,\\alpha)$, the following ansatz\nfor the eigenfunction\n\\begin{equation}\n\\phi={e^{im\\alpha}\\over \\sqrt{2\\pi}}~~{u(r)\\over\nr^{1/2}}~~~~~;~~~~~m=0,\\pm 1,\\pm 2,\\ldots\n\\label{ansatz-2e}\n\\end{equation}\nis justified, where the Pauli principle demands that \n$m$ is even or odd in the singlet and triplet state, respectively.\nThe Schr\\\"odinger equation $H_r \\; \\phi({\\bf r})=\\epsilon_r \\; \\phi({\\bf r})$\ngives rise to the radial Schr\\\"odinger equation for $u(r)$\n\\begin{equation}\n\\biggl\\{-{1\\over 2}~{d^2\\over dr^2}+{1\\over 2}\\biggl(m^2-{1\\over 4}\\biggr)\n{1\\over r^2}+{1\\over 2}\\tilde\\omega_r^2 r^2+\\frac{\\beta}{2r}\n\\biggl\\}u(r)=\\tilde \\epsilon_r \\; u(r)\n\\label{rad-SGl}\n\\end{equation}\nwhere $\\tilde\\omega_r={1\\over 2}\\tilde\\omega=\n{1\\over 2}\\sqrt{\\omega_o^2 + ({ \\omega_c \\over 2} )^2}$ , \n$\\tilde \\epsilon_r={1\\over 2}\\tilde \\epsilon=\\epsilon_r-{1\\over 4} m \\omega_c$, \n$\\omega_c={B \\over c}$\nand the solution is subject\nto the normalization condition $\\int\\limits^\\infty_o dr|u(r)|^2=1$.\nIn dimensionless variables $r \\rightarrow \\sqrt{ \\tilde \\omega_r} r$\nand $\\tilde \\epsilon_r \\rightarrow \\tilde \\epsilon_r / \\tilde \\omega_r$ \nthe radial Schr\\\"odinger equation reads\n\\begin{eqnarray}\n\\biggl\\{-{1\\over 2}~{d^2\\over d(\\sqrt{ \\tilde \\omega_r} r)^2}+\n{1\\over 2}\\biggl(m^2-{1\\over 4}\\biggr)\n{1\\over(\\sqrt{ \\tilde \\omega_r} r)^2}+{1\\over 2}(\\sqrt{\\tilde\\omega_r} r)^2+\n\\frac{\\beta}{\\sqrt{ \\tilde \\omega_r}} \\frac{1}{2 ( \\sqrt{ \\tilde \\omega_r} r)}\n\\biggl\\}u(r)= \\nonumber \\\\\n= \\biggl( \\frac{\\tilde \\epsilon_r}{\\tilde \\omega_r} \\biggr) \\; u(r)\n\\label{rad-SGl-dimless}\n\\end{eqnarray}\nThe {\\em exactly solvable} eigenfunctions \nhave the following form\n\\begin{equation}\nu(r)=r^{|m|+\\frac{1}{2}}\\; p(r) \\; e^{-\\frac{1}{2}\\; \\tilde \\omega_r \\; r^2}\n\\label{u}\n\\end{equation}\nwhere $p(r)$ is a {\\em finite} polynomial\n\\begin{equation}\np(r)=\\sum_{\\nu=0}^{(n-1)} a_\\nu \\cdot (\\sqrt{ \\tilde \\omega_r} r)^\\nu\n\\end{equation}\nwith $n$ terms.\nThe soluble fields and the corresponding\neigenvalues are determined by the two requirements \n\\begin{equation}\na_n=0~~~~~;~~~~~a_{n+1}=0 \n\\label{truncation}\n\\end{equation}\nwhich guarantee truncation of the power series and therefore\nnormalizability of the eigenfunctions.\nIt is clear from (\\ref{rad-SGl-dimless}) \n that the two truncation conditions depend from \nthe parameters only in the combination $\\frac{\\beta^2}{\\tilde \\omega_r}$ \nand $\\frac{\\tilde\\epsilon_r}{\\tilde\\omega_r}$. Consequently, we have two\nequations (\\ref{truncation}) for effectively two parameters. \nTechnically, we first calculate the solvable \nfields $\\frac{\\beta^2}{\\tilde\\omega_r}$ \nfrom (16) in \\cite{Taut2e} (with $\\beta$ included) \nand then we get the corresponding eigenvalues\nfrom (17) in \\cite{Taut2e}, which we rewrite here in the form\n\\begin{equation}\n\\frac{\\tilde\\epsilon_r}{\\tilde\\omega_r}= |m|+n\n\\label{eigen}\n\\end{equation}\nNote that (\\ref{eigen}) does not contain $\\beta$ explicitly,\nbut only through the fact that the solvable $\\tilde\\omega_r$ depend on beta.\nThe calculation of the solvable fields is non--trivial\n (and not repeated here in detail), but\nwhen they are found, the calculation of the corresponding eigenvalues is \nsimple through (\\ref{eigen}).\nObserve that the eigenvalues of (\\ref{rad-SGl}) without e-e-interaction read\n\\cite{one-e-dot}\n\\begin{equation}\n\\frac{\\tilde\\epsilon_r}{\\tilde\\omega_r}= |m|+2k+1\n\\label{eigen0}\n\\end{equation}\nwhere $k$ is the node number. Comparison of (\\ref{eigen}) and (\\ref{eigen0})\nshows the above mentioned degeneracies of the interacting system with the\nnoninteracting one. \nWith $n=2k+1$, the interaction-- free solutions fit \ninto a generalized\npattern of all analytical solutions.\nIt can also be shown that equation (16) with (17) in \\cite{Taut2e},\nwhich defines the solvable \n$\\frac{\\beta^2}{\\tilde\\omega_r}$, \nhas always the solution $\\frac{\\beta^2}{\\tilde\\omega_r}=0$ \nwhenever $n$ is odd. As an example, see (20a) in \\cite{Taut2e} for \nthe case $n=3$. (These infinite field solutions are not included in \nTables 1 and 2 in \\cite{Taut2e}.)\nIt is also clear from physical reasoning that for infinite external fields\nthe e--e--interaction has no influence on the eigensolutions, because the\nkinetic energy dominates. Therefore, solutions without e--e--interaction\nare exact for $\\tilde \\omega_r \\rightarrow \\infty$. \nSometimes we shall call these solutions 'asymptotic' solutions.\\\\\nThe completed pattern has the following over-- all-- structure: \nFor $n=1,2$ there is\none solution (ground states), for $n=3,4$ we\nhave two solutions (one ground and one excited state), etc. (see Figure 1).\nGenerally, the soluble fields are the smaller\nthe larger the corresponding $n$ is. For $n \\rightarrow \\infty$ the\ncorresponding soluble $\\tilde\\omega_r$ converge to $0$.\\\\\n%\n\\begin{figure}[!tbh]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=fig1.ps,angle=-90,width=15.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nReduced energies (energy over effective oscillator frequency)\n versus squared coupling constant over\neffective oscillator frequency for relative angular momentum $|m|=1$.\nThe crosses indicate solvable states. The lines connect states with the\nsame node number $k$. The termination index $n$ is the same for all horizontal\nrows of crosses with the same ordinate.\n}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n%\n\n\nNow we consider the eigenfunctions.\nThe series $a_\\nu$ is defined by a \ntwo step recursion relation \nwhich reads for the soluble states \n(insert (17) into (14) in \\cite{Taut2e})\n\\begin{eqnarray}\na_0&=&\\mbox{normalization constant}\\nonumber\\\\\na_1&=&\\frac{1}{(2|m|+1)} \\;\\frac{\\beta}{\\sqrt{ \\tilde \\omega_r}}\\;a_0 \\nonumber\\\\\na_2&=&\\frac{1}{4(|m|+1)} \n\\biggl\\{ \\frac{1}{(2|m|+1)}\\frac{\\beta ^2}{ \\tilde \\omega_r} \n-2(n-1) \\biggr\\} \\; a_0 \n\\nonumber \\\\\n\\cdots \\nonumber \\\\\na_\\nu&=&\\frac{1}{\\nu \\;(\\nu+2|m|)}\n\\biggl\\{ \\frac{\\beta}{\\sqrt{ \\tilde \\omega_r}}\\;\na_{\\nu-1}+2\\;(\\nu-n-1)\\; a_{\\nu-2} \n\\biggr\\}\n\\label{recursion}\n\\end{eqnarray}\nIt produces rather complicated expressions for larger $\\nu$.\nThe recursion can also be started at the other end with $a_{n-1}$ as \na normalization constant.\\\\\nThe eigenfunctions of asymptotic solutions \nfit also into the generalized scheme. \nFor $\\frac{\\beta^2}{\\tilde\\omega_r}=0$, the recursion relation \n(\\ref{recursion}) provides \nonly non--vanishing coefficients with even index, \nmeaning, that $p(r)$ is a function\nof $r^2$. If we insert $n=2k+1$, \nand $\\nu=2 p$ with $p=0,1,2,...$ into (\\ref{recursion}) we obtain the recursion\nrelation \n\\begin{equation}\na_p=\\frac{(p-k-1)}{p(p+|m|)}\\;a_{p-1}\n\\end{equation}\nwhich belongs to the\nGeneralized Laguerre polynomials. \\footnote{We use the \ndefinition in Abramowicz, Stegun; Handbook of Mathematical Functions}\n\\begin{equation}\np_{n=2k+1,m}(r)=L_k^{|m|}(\\tilde\\omega_r\\; r^2)\n\\end{equation}\nConsequently, the \nGeneralized Laguerre polynomials are a special case of our \npolynomials $p_{n,m}(r)$.\nIf we go in Fig.1 along the line for ground states ($k=0$) from the left \nto the right,\nthen the polynomials for the exactly solvable cases \nhave the following form. For $n=1$ ,\n$p_{n=1,m}(r)=L_0^{|m|}(\\tilde\\omega_r\\; r^2)$ is \na constant, for \n$n=2$ (simplest case with finite fields) \n$p_{n=2,m}(r)$ is a linear function without\na positive zero, for $n=3$ we have a quadratic function \nwithout positive zeros, etc.\nAnalogously, the exact solutions \nfor the first excited state are all polynomials\nwith one node, but increasing order.\\\\\n\nAs an overview, we give the formulae for the simplest exact solutions\nof the completed pattern (see also (19) and (20) in \\cite{Taut2e} with\n$\\beta$ included).\n For $n=1$ there is only the\nasymptotic solution\n\\begin{eqnarray}\n\\frac{\\beta^2}{\\tilde \\omega_r}&=&0 \\label{om-N=2-n=1}\\\\\np(r)&=&1\n\\end{eqnarray}\nFor $n=2$ there is one finite-- field solution\n\\begin{eqnarray}\n\\frac{\\beta^2}{\\tilde\\omega_r}&=&2 \\; (2\\; |m|+1) \\label{om-N=2-n=2}\\\\\np(r) &=& \\biggl[ 1+\\frac{\\beta \\;r} {(2\\; |m|+1)} \\biggr]\n\\label{u-N=2-n=2}\n\\end{eqnarray}\nBoth former solutions are ground states.\nFor $n=3$ there is one asymptotic solution, which is a first excited state,\n\\begin{eqnarray}\n\\frac{\\beta^2}{\\tilde \\omega_r}&=&0\\\\\np(r)&=&1-\\frac{\\tilde \\omega_r\\; r^2}{(|m|+1)}\n\\end{eqnarray}\nand one at finite fields, which is a ground state,\n\\begin{eqnarray}\n\\frac{\\beta^2}{\\tilde\\omega_r}&=&4\\; (4\\; |m|+3)\n\\label{om-N=2-n=3}\\\\\np(r) &=& \\biggl[ 1+\\frac{\\beta \\; r} {(2\\; |m|+1)}\n +\\frac{(\\beta \\; r) ^2}\n {2 (2\\; |m|+1)(4\\;|m|+3)} \\biggr]\n\\label{u-N=2-n=3}\n\\end{eqnarray}\nThe corresponding energies follow from (\\ref{eigen}), and\nwe put $a_0=1$ for the constant term of the polynomial $p(r)$ without loss\nof generality.\\\\\nIn order to give an idea in what magnetic field range these exact\nsolutions are located we consider the case without confinement ($\\omega_0=0$).\nThen $\\tilde\\omega_r=\\omega_c/4$, where all solvable frequencies \ngiven above\nare in effective atomic units $a.u.^*$ ($\\hbar=m^*=\\beta=1$). \nOn the other hand, for GaAs we have $B[Tesla]=\\omega_c[a.u.^*]/0.1363$.\nThis means, that the {\\em largest finite} solvable field ( for $n=2$ and $m=0$ )\nis $\\omega_c=2\\;a.u.^*$ and $B=14.7\\; Tesla$. \\\\\n Now we add some words about the limit $\\beta \\rightarrow 0$. The \nwave functions of the asymptotic solutions\n($\\frac{\\beta^2}{\\tilde \\omega_r}=0$) do not depend on $\\beta$ at all,\nindicating that they are robust against a variation of the e-- e-- interaction.\n$\\frac{\\beta^2}{\\tilde \\omega_r}=0$ can be realized either by vanishing e-- e-- interaction ($\\beta \\rightarrow 0$) or infinite fields ($\\tilde \\omega_r\n\\rightarrow \\infty$). On the other hand, \nthe wave functions of the finite-- field-- solutions \n($\\frac{\\beta^2}{\\tilde \\omega_r}=finite$) \nevolve steadily into non-- interacting ones for $\\beta \\rightarrow 0$\nindicating a strong dependence on the form of the e-- e-- interaction.\\\\\n\nIf we insert for the center of mass system the ground state WF, then the total\nspatial WF in the particle coordinates reads (use (\\ref{ansatz-2e}),\n(\\ref{u}), and from \\cite{Taut2e} formula (8))\n\\begin{equation}\n\\Phi({\\bf r}_1,{\\bf r}_2)=({\\bf r}_1-{\\bf r}_2)^m \n\\;p_{n,m}(|{\\bf r}_1-{\\bf r}_2|)\n\\; e^{-\\frac{\\tilde\\omega}{2} (r_1^2+r_2^2)}\n\\label{tot-WF-2e}\n\\end{equation}\nwhere the first factor is a shorthand with the convention \n${\\bf r}^m=r^{|m|} e^{im\\alpha}$. In complex coordinates \n$ z=x+iy$, $\\bar z=x-iy$, \nand with the opposite sign for the angular momentum \n\\footnote{Observe, our $m$ is the angular momentum quantum number itself\n and negative\nfor the LLL.} $\\bar m=-m $\n(compared with Laughlins notation \\cite{Laughlin} a 'bar' has been added), \nthis means\n\\begin{eqnarray}\n{\\bf r}^m&=& z^m ~~~~~ \\mbox{for} ~~~~~m\\ge0\\\\\n &=& \\bar z^{\\bar m} ~~~~~ \\mbox{for} ~~~~~m\\le0 \n\\end{eqnarray}\nTherefore, for $m \\le 0$, (\\ref{tot-WF-2e}) reads in complex coordinates\n\\begin{equation}\n\\Phi(\\bar z_1,\\bar z_2)=(\\bar z_1-\\bar z_2)^{\\bar m}\n\\;p_{n,\\bar m}(|\\bar z_1-\\bar z_2|)\n\\; e^{-\\frac{\\tilde\\omega}{2} (|\\bar z_1|^2+|\\bar z_2|^2)}\n\\label{WF-2e-complex}\n\\end{equation}\nThe solution for $n=1$ (infinite field, $p(x)=a_0=const.$)\nagrees exactly with the Laughlin-- WF, in particular, $\\bar m=1$ is\na determinant of two LLL functions, which corresponds to an uncorrelated \nfull LLL.\\\\\nFor $m \\ge 0$ we have\n\\begin{equation}\n\\Phi(z_1, z_2)=( z_1- z_2)^{m}\n\\;p_{n, m}(|z_1-z_2|)\n\\; e^{-\\frac{\\tilde\\omega}{2} (|z_1|^2+|z_2|^2)}\n\\end{equation}\nwhich is the complex conjugate of (\\ref{WF-2e-complex}) \nand therefore the solution\nfor the opposite direction of the magnetic field, if $\\omega_0=0$. \nIn particular, $n=1$ and $m=1$ is a {\\em determinant} of the \ntwo one-- particle-- states\nwith $k=0$ and $m=0$ and $m=1$, in other words, it corresponds to \nthe uncorrelated solution of two full LLs.\nIn summary, the solutions for $n=1$ comprise the Laughlin WFs and the\nfull LLs. It is tempting to assume that the other solutions \n($n=2,3,...$) have some\nrelation to the other Quantum Hall states, but until now there is no\nprove for it. (see also Sect. 5)\\\\\n\n\n\\newpage\n\n\\section{Approximate decoupling for three electrons}\nThe Hamiltonian for three electrons in an homogeneous magnetic field\n(vector potential ${\\bf A}({\\bf r})$) and a harmonic scalar potential\n(oscillator frequency $\\omega_0$) reads\n\\begin{equation}\nH=\\sum_{i=1}^3\\biggl[{1\\over 2}\\biggl(\\frac{1}{i}{\\bf \\nabla}_i\n+{1\\over c}{\\bf\nA}({\\bf r}_i)\\biggr)^2 +{1\\over 2}\\omega_o^2r_i^2\\biggr]\n+\\sum_{i<k}{\\beta \\over |{\\bf r}_i-{\\bf r}_k|}+H_{spin}\n\\label{h-orig}\n\\end{equation}\nwhere $H_{spin}=g\\; \\sum\\limits_{i=1}^3 \\;{\\bf s}_i \\; \\cdot{\\bf B}$.\nThe goal of the following considerations is the decoupling\nof the Hamiltonian into a sum of independent Hamiltonians for quasi--\n particles.\\\\\nFor {\\em two electrons}\n this happens automatically by introducing the center of mass\nand relative coordinate. This is mainly due to the peculiarity\n that there is only \none interaction term $\\beta \\over {|{\\bf r}_2-{\\bf r}_1|}$ which contains only\none of the new coordinates $\\bf r$ and $\\bf R$. \nThe peculiarity for {\\em three electrons}, which can be taken advantage of,\n is the fact that the number of interaction terms\nis equal to the number of particles.\nThis does not allow for an exact decoupling, but an approximate one into three\nnoninteracting pairs plus a coupling term which is small in the strong\ncorrelation limit.\\\\\nIt is easily shown that an {\\em orthogonal transformation}\n\\footnote{Exactly speaking, for keeping the one-- particle-- terms\nof the Hamiltonian decoupled it suffices that the {\\em row } vectors\nof the matrix in (\\ref{trafo}) are mutually orthogonal. \nHowever, using this more\ngeneral type of transformation does not provide any advantage in our case,\n because the additional freedom destroys the symmetry among the particles.}\n leaves the kinetic energy in a\nhomogeneous magnetic field and the potential energy in an external harmonic\nscalar potential, invariant. On the other hand, the center of mass (c.m.) of\na classical system in the ground state vanishes and \nit is natural to assume that the c.m. in the high correlation limit can\nbe treated as a small expansion parameter.\nTherefore,\nwe look for an orthogonal transformation \nwhich transforms the e-- e-- interaction in such a form\nthat each term depends only on {\\em one} of the new coordinates and the center\nof mass. Additionally, we demand that the center of mass is invariant under \nthe transformation, which guarantees that if it is small in the original \ncoordinates, so it is in the transformed ones.\nThe transformation from the original coordinates ${\\bf r}_i$ to the\nnew ones ${\\bf x}_i$, which fulfills all these requirements, reads\n\\begin{equation}\n \\left[ \\begin{array} {c} {\\bf x}_1\\\\{\\bf x}_2\\\\{\\bf x}_3\\end{array} \\right]= \n\\left[ \\begin{array} {ccc}1/3&a&b\\\\b& 1/3&a\\\\a&b&1/3\\end{array}\\right]\n\\left[ \\begin{array} {c} {\\bf r}_1 \\\\ {\\bf r}_2 \\\\ {\\bf r}_3 \\end{array} \\right]\n\\label{trafo}\n\\end{equation}\nwhere $a=1/3-1/\\sqrt{3}$ and $b=1/3+1/\\sqrt{3}$ and its inverse is\n\\begin{equation}\n \\left[ \\begin{array} {c}{\\bf r}_1\\\\{\\bf r}_2\\\\{\\bf r}_3 \\end{array} \\right] = \n\\left[ \\begin{array} {ccc}1/3&b&a\\\\a& 1/3&b\\\\b&a&1/3\\end{array}\\right]\n\\left[ \\begin{array} {c} {\\bf x}_1 \\\\ {\\bf x}_2 \\\\ {\\bf x}_3 \\end{array} \\right]\n\\label{inverse-trafo}\n\\end{equation}\nFrom (\\ref{inverse-trafo}) it follows that\n\\begin{eqnarray}\n{\\bf r}_1-{\\bf r}_2=\\sqrt{3}\\; \\biggl({\\bf X}-{\\bf x}_3\\biggr)\n \\nonumber \\\\\n{\\bf r}_2-{\\bf r}_3=\\sqrt{3}\\; \\biggl({\\bf X}-{\\bf x}_1\\biggr)\n\\label{diff}\\\\\n{\\bf r}_3-{\\bf r}_1=\\sqrt{3}\\; \\biggl({\\bf X}-{\\bf x}_2\\biggr)\n\\nonumber \n\\end{eqnarray}\nso that the Hamiltonian in the new \ncoordinates is\n\\begin{equation}\nH=\\sum_{i=1}^3\\biggl[{1\\over 2}\\biggl(\\frac{1}{i}{\\bf \\nabla}_i+{1\\over c}{\\bf\nA}({\\bf x}_i)\\biggr)^2 +{1\\over 2} \\; \\omega_o^2 \\; x_i^2\n+\\frac{1}{\\sqrt{3}} \\; {\\beta \\over |{\\bf x}_i-{\\bf X}|} \\biggr]\n+H_{spin}\n\\label{h-trans}\n\\end{equation}\nwhere ${\\bf X} \\equiv \\frac{1}{3} \\sum_{i=1}^3 {\\bf x}_i ={\\bf R}$\n is the \ncenter of mass in the new coordinates.\nIt is possible (but complicated) to take care of\n the Pauli principle in the new coordinates. \nIt is much easier first to\ntransform the wave functions (WF) back to the original coordinates and \nthen do the antisymmetrization . \\\\\nWhile being still exact, (\\ref{h-trans}) is not exactly \ndecoupled because ${\\bf X}$\ncontains all coordinates. For ${\\bf X}$ small compared with ${\\bf x}_i$ , \nthe e-e-interaction term \ncan be expanded in a {\\em multi-pole series }\n\\begin{equation}\nV_{ee}=\\sum_{l=0}^\\infty V_{ee}^{(l)}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nV_{ee}^{(0)}&=& \\frac{\\beta}{\\sqrt{3}} \\; \\sum_{i=1}^3 \\; \\frac{1}{|{\\bf x}_i|}\n \\label{monopole}\\\\\nV_{ee}^{(1)}&=& \\frac{\\beta}{\\sqrt{3}} \\; \\sum_{i=1}^3 \\; \n \\frac{{\\bf X} \\cdot {\\bf x}_i}{|{\\bf x}_i|^3} \\label{dipole}\\\\\nV_{ee}^{(2)}&=& \\frac{\\beta}{\\sqrt{3}} \\; \\frac{1}{2}\\; \\sum_{i=1}^3 \\; \n\\biggl[ 3 \\frac{({\\bf X} \\cdot {\\bf x}_ i)^2} {|{\\bf x}_i|^5}\n - \\frac{({\\bf X})^2 }{|{\\bf x}_i|^3} \\biggr] \\label{quadrupole}\\\\\n\\cdots \\nonumber\n\\end{eqnarray}\nIn zero order in ${\\bf X}$, the Hamiltonian $H^{(0)}$ \nis decoupled and can be solved\nexactly and in many cases even analytically (see below).\nThe general strategy for amending the zero order result\nis to consider the multi-pole terms\nin perturbation theory.\nAs a response to a frequently asked questions, we want to emphasize \nthe following.\nThe approximation ${\\bf X}=0$ does not mean that the new\ncoordinates ${\\bf x}_i$ are no more independent.\nThe one-- particle part of the Hamiltonian is independent of ${\\bf X}$\nand still exact. ${\\bf X}=0$ only means making an approximation\nto the e-- e-- interaction term in $H$.\\\\\nIt might be interesting to note that transformation (\\ref{trafo}),\nif applied to an Hamiltonian with an \nexternal Coulombic potential $\\frac{Z}{r_i}$ (instead of the \noscillator potential ${1\\over 2}\\omega_o^2 r_i^2$),\ntransforms in zero order in $\\bf X$ the e-e-interaction term and\nthe potential energy in the external Coulombic potential into each other,\ni.e. leaves the Hamiltonian in lowest order virtually invariant.\n\n\\newpage\n\n\\section{Exact solution in zero order in the center of mass coordinate}\n\\subsection{Pair Equation}\n\nAfter expanding the kinetic energy, the Hamiltonian in zero order in\n${\\bf X}$ reads\n\\begin{equation}\nH^{(0)}=\\sum_{i=1}^3 h_i\\;+\\;H_{spin}\n\\end{equation}\nwith an effective pair Hamiltonian\n\\begin{equation}\nh_i=-{1\\over 2}{\\bf \\nabla}_i^2\n+{1\\over 2} \\; \\tilde\\omega^2 \\; x_i^2 + {1\\over 2} \\;\\omega_c \\; l_i\n+\\frac{1}{\\sqrt{3}} \\; {\\beta \\over |{\\bf x}_i|} \n\\label{h-i}\n\\end{equation}\nwhere $\\tilde\\omega=\\sqrt{\\omega_0^2+(\\frac{1}{2} \\omega_c)^2}$,\n$\\omega_c=\\frac{B}{c}$ is the cyclotron frequency, and $l_i$\nis the orbital angular momentum operator.\nThis gives rise to the definition of an effective pair equation\n\\begin{equation}\nh_i\\; \\phi_{q_i}({\\bf x}_i) = \\epsilon_{q_i} \\; \\phi_{q_i}({\\bf x}_i)\n\\label{pair-eq}\n\\end{equation}\nwith the normalization condition\n\\begin{equation}\n\\int d^2 {\\bf x}_i \\; |\\phi_{q_i}({\\bf x}_i)|^2 =1.\n\\end{equation}\nThe subscript $q_i$ at eigenvalues and eigenfunctions \ncomprises all quantum numbers\nfor the $i^{th}$ pair.\nIn polar coordinates $(x,\\alpha)$, the pair equation (\\ref{pair-eq}) \nis satisfied by the ansatz\n\\begin{equation}\n\\phi={e^{im\\alpha}\\over \\sqrt{2\\pi}} \\; {u(x)\\over\nx^{1/2}}~~~~~;~~~~~m=0,\\pm 1,\\pm 2,\\ldots\n\\label{ansatz}\n\\end{equation}\n$u(x)$ must satisfy the {\\em radial pair equation}\n\\begin{equation}\n\\biggl\\{-{1\\over 2}~{d^2\\over dx^2}+{1\\over 2}\\biggl(m^2-{1\\over 4}\\biggr)\n{1\\over x^2}+{1\\over 2}\\tilde\\omega^2 x^2\n+{\\beta \\over \\sqrt{3} \\; x}\\biggl\\}u(x)=\\tilde\\epsilon \\; u(x)\n\\label{rad-eq}\n\\end{equation}\nwhere $\\tilde\\epsilon=\\epsilon-{1\\over 2}m \\omega_c$ \nand the solution is subject\nto the normalization condition $\\int\\limits^\\infty_o dx \\;|u(x)|^2=1$.\nIn analogy to the two-- electron-- problem, we now\nuse for the radial part of the pair functions (\\ref{ansatz})\nthe ansatz\n\\begin{equation}\nu(x)=x^{|m|+\\frac{1}{2}}\\; p(x) \\; e^{-\\frac{1}{2}\\; \\tilde \\omega \\; x^2}\n\\end{equation}\nwhere $p(x)$ is a polynomial, which is finite for solvable states.\nIn this way we obtain\nfor the pair function (\\ref{ansatz}) in polar coordinates $(x,\\alpha)$\n\\begin{equation}\n\\phi({\\bf x})=\\frac{{\\bf x}^m }{\\sqrt{2\\pi}} \\; p(x)\\;\ne^{-\\frac{1}{2}\\; \\tilde \\omega \\; x^2}\n\\label{pair-ansatz}\n\\end{equation}\nwhere ${\\bf x}^m$ is a shorthand for $x^{|m|} e^{i m \\alpha}$\n(see also Sect.2).\\\\\n\n\nBecause of the decoupling in zero order,\nthe total eigenvalues and eigenfunctions of $H^{(0)}$ can be obtained from\n\\begin{equation}\nE_{q_1,q_2,q_3}=\\epsilon_{q_1}+\\epsilon_{q_2}+\\epsilon_{q_3}+E_{spin}\n\\label{E-tot}\n\\end{equation}\n\\begin{equation}\n\\Phi_{q_1,q_2,q_3}({\\bf x}_1,{\\bf x}_2,{\\bf x}_3)=\\phi_{q_1}({\\bf x}_1)\\cdot \n\\phi_{q_2}({\\bf x}_2)\\cdot \\phi_{q_3}({\\bf x}_3)\n\\label{WF-tot}\n\\end{equation}\n\nAfter inserting the transformation (\\ref{diff}) into \nthe spatial part of the total WF (\\ref{WF-tot}),\nwe obtain in the original coordinates\n\\begin{eqnarray}\n\\Phi_{q_1,q_2,q_3}&=&\n\\phi_{q_1}\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_2-{\\bf r}_3)\\biggr)\\cdot \\nonumber \\\\\n& &\\phi_{q_2}\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_3-{\\bf r}_1)\\biggr)\\cdot \\nonumber \\\\\n& &\\phi_{q_3}\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_1-{\\bf r}_2)\\biggr)\n\\label{amended}\n\\end{eqnarray}\nIt is obvious that for any total WF the fields \n$\\tilde \\omega_{q_i}=\\tilde \\omega_q$ of the\nthree pairs have to agree.\nIf we \ninsert (\\ref{pair-ansatz}) into (\\ref{amended}),\nwe end up with\n\\begin{eqnarray}\n\\Phi_{q_1,q_2,q_3} &=&\n\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_2-{\\bf r}_3) \\biggr)^{m_1} \\;\np_{q_1}\\biggl(\\biggl|{\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_2-{\\bf r}_3) \\biggr| \\biggr)\\cdot \\; \\nonumber \\\\\n& &\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_3-{\\bf r}_1)\\biggr)^{m_2} \\;\np_{q_2}\\biggl(\\biggl|{\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_3-{\\bf r}_1)\\biggr) \\cdot\\; \\nonumber \\\\\n& &\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_1-{\\bf r}_2)\\biggr)^{m_3} \\;\np_{q_3}\\biggl(\\biggl|{\\bf R}-\\frac{1}{\\sqrt{3}}\n({\\bf r}_1-{\\bf r}_2)\\biggr|\\biggr) \\cdot \\nonumber \\\\\n& & \\mbox{exp}\\biggl(-\\frac{1}{2} \\; \\tilde \\omega_q \\sum_i {\\bf r}_i^2 \\biggr)\n\\label{amended-WF}\n\\end{eqnarray}\nIn order to simplify the exponential factor to the present form\nwe used the special property of our orthogonal transformation \n$\\sum_i {\\bf x}_i^2=\\sum_i {\\bf r}_i^2$.\nThis is still the most general result. \nIt holds for ground and excited state and can be simplified in special cases.\nFor analytically solvable {\\em finite-- field solutions}\nthe quantum numbers of all three pairs have to agree,\nbecause the solvable fields $\\tilde \\omega_{q}$ for each pair depend on all\nquantum numbers contained in $q$. This implies in particular, that\nall angular momenta $m_i$ in (\\ref{amended-WF}) agree.\nFor {\\em asymptotic solutions}, the solvable $\\tilde \\omega$\ndoes not depend on\nthe angular momentum $m$ and the node number $k$\n(or termination index $n=2k+1$) of the corresponding pair.\nTherefore, we can construct analytical solutions from pair states with\ndifferent quantum numbers, i.e. different $m_i$ and $k_i$.\nFor asymptodic ground state \nsolutions, which belong to $n_i=1$, (\\ref{amended-WF})\ncan be simplified by omitting the polynomials, because they are constants.\\\\\n\nIt should be mentioned that the eigenfunctions (\\ref{WF-tot}) of $H^{(0)}$\nform a complete set\nand therefore they can be used as a basis for a numerical solution of the full\nSchr\\\"odinger equation. Their advantage compared with basis functions\nconstructed from one-- particle states \\cite{Girvin} \n\\cite{Maksym} \\cite{Hawrylak} \\cite{Laughlin} is that they contain an e-- e--\ncorrelation cusp. As well known in Quantum Chemistry,\nthe lack of this cusp in the basis functions gives rise to poor convergence\nin CI expansions. In the\npresent paper, however, we consider perturbation theory for improving the zero\norder result beyond the strong correlation limit ${\\bf R}=0$.\n\n\\subsection{Pauli Principle}\nThe WF (\\ref{amended-WF}) does not yet \nfulfill the antisymmetry requirement.\nTaking care of this is particularly important for finding the exclusion\nprinciples for our type of WF. The question is: for what combination of quantum\nnumbers and parameter values does the antisymmetrized WF vanish \nby permutation symmetry? \nIn order to keep formula length under control, we introduce some shorthands\nand conventions. Firstly, the exponential factor in (\\ref{amended-WF}) \nis fully\nsymmetric with respect to permutations and therefore it can be\n simply omitted in the\nantisymmetrization procedure. Further, for the polynomial prefactor in \n(\\ref{pair-ansatz}) we introduce a single symbol\n\\begin{equation}\nt({\\bf x})={\\bf x}^m \\; p(x)\n\\label{def-t}\n\\end{equation}\nThen the spatial total WF can be written as\n\\begin{eqnarray}\n\\Phi_{q_1,q_2,q_3}({\\bf r}_1,{\\bf r}_2,{\\bf r}_3)&=&\nT_{q_1,q_2,q_3}({\\bf r}_1,{\\bf r}_2,{\\bf r}_3) \\cdot\n\\mbox{exp}\\biggl(-\\frac{1}{2} \\; \\tilde \\omega_q \\sum_i {\\bf r}_i^2 \\biggr)\n\\nonumber \\\\\nT_{q_1,q_2,q_3}({\\bf r}_1,{\\bf r}_2,{\\bf r}_3)&=&\nt_{q_1}\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}({\\bf r}_2-{\\bf r}_3)\\biggr)\\cdot \n\\nonumber \\\\\n& &t_{q_2}\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}({\\bf r}_3-{\\bf r}_1)\\biggr)\\cdot \n\\nonumber\\\\\n& &t_{q_3}\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}}({\\bf r}_1-{\\bf r}_2)\\biggr) \n\\label{T-WF}\n\\end{eqnarray}\nNow, only the polynomial prefactor $T({\\bf r}_1,{\\bf r}_2,{\\bf r}_3)$\nhas to be antisymmetrized. \\\\\n\nThe implementation of the antisymmetrization procedure and the\nantisymmetrized WFs can be found in the Appendix. The most\nimportant qualitative result is that a simple \n{\\em exclusion principle} exists only for ${\\bf R}={\\bf 0}$.\nIt states that the WF vanishes by permutation symmetry:\\\\\nin {\\bf quartet} states $S=\\frac{3}{2}$: \\\\\n{\\em if at least two pair functions agree and\nif the total orbital angular momentum $M_L$ is even.}\\\\\nin {\\bf doublet} states $S=\\frac{1}{2}$:\\\\\n {\\em if all three pair functions agree.}\\\\\n\nBecause for analytical {\\em finite-- field-- solutions }\nall quantum numbers of the 3 pairs have to agree, \nit follows from the Pauli principle (see above), that only\nquartet states (with parallel spins) can be given analytically.\nIf $m$ is the angular\nmomentum of the pair functions in (\\ref{amended-WF}) then the total\norbital angular momentum in the solvable states is $M_L=3 m$.\nIn zero order on $\\bf R$, Pauli principle demands that\n$m=odd$. For finite $\\bf R$ there is no restriction\nfrom the Pauli principle and $m=integer$.\nAs will be shown in Section 6, it turns out that these solvable \nstates are states where {\\em kinks} in the curve $E$ versus $M_L$ occur\nand which bear the so called {\\em magic} angular momenta (see Fig. 2).\nIn short, only particular states can be solved analytically,\nbut these states are the most interesting ones.\\\\\nFor {\\em asymptodic solutions} (with $\\beta^2/ \\tilde \\omega=0$) \nall quantum numbers can be different,\nalthough the pair functions belong to the same external fields. This means that\nwe can form total WFs from different pair functions and Pauli principle\napplies as given above. \\\\\n%\n\\begin{figure}[!bth]\n%t=top,h=here,b=bottom\n\\vspace{-2cm}\n\\begin{center}\n{\\psfig{figure=fig2.ps,angle=-90,width=15.cm,bbllx=15pt,bblly=45pt,bburx=580pt,bbury=750pt}}\n\\caption[ ]{\nThe second term of the total energy in second order Taylor approximation\n(\\ref{E-Taylor}) divided by $c_2\\; M_L^2 \\;/3$\nas a function of $M_L$ for the ground states (crosses).\nCrossed circles denote states which are forbidden by Pauli principle\nfor ${\\bf R}=0$\n(Laughlin approximation), but allowed\nfor finite ${\\bf R}$.\nThe states denoted by open circles\nare the ground states for ${\\bf R}=0$ if the states underneath\nare forbidden.\nThe region $M_L <3$ is omitted because it lies partly outside the scale and\nbecause it needs special consideration.\n}\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\n\\subsection{Exact analytical solutions of the pair equation}\nIf we introduce in the radial equation\nfor the relative coordinate in the two-- electron-- case (\\ref{rad-SGl})\nthe same parameters as used for 3 particles (i.e.\n$\\tilde \\omega$ and $\\tilde \\epsilon$), we obtain\n\\begin{equation}\n\\biggl\\{-{1\\over 2}~{d^2\\over dr^2}+{1\\over 2}\\biggl(m^2-{1\\over 4}\\biggr)\n{1\\over r^2}+{1\\over 2} \\biggl( \\frac{\\tilde\\omega^{(N=2)}}{2} \\biggr)^2 r^2\n+{\\beta\\over 2 \\; r}\\biggl\\}u^{(N=2)}(x)=\n\\biggl( \\frac{\\tilde\\epsilon^{(N=2)}}{2} \\biggr) \\; u^{(N=2)}(x)\n\\label{rad-eq2}\n\\end{equation}\nFor avoiding confusion, the parameters and the WF \nin the two-- particle-- problem have been\ngiven the extra superscript $N=2$.\nComparison with the radial pair equation (\\ref{rad-eq}) shows us that\nwe obtain solutions of (\\ref{rad-eq}) from solutions of (\\ref{rad-eq2})\nby simple rescaling.\\\\\nThe 'spectrum' of soluble $\\tilde \\omega$ follows from\n\\begin{equation}\n\\tilde\\omega=\\frac{2}{3}\\;\\tilde\\omega^{(N=2)}\n\\label{sol-om}\n\\end{equation}\nand the corresponding eigenvalues and eigenfunctions from\n\\begin{equation}\n\\tilde\\epsilon=\\frac{2}{3}\\;\\tilde\\epsilon^{(N=2)}\n\\label{sol-eps}\n\\end{equation}\n\\begin{equation}\nu(x)=\\sqrt{\\frac{2}{\\sqrt{3}}} \\;\\;\n u^{(N=2)}\\biggl(r=\\frac{2}{\\sqrt{3}} \\; x\\biggr)\n\\label{sol-wf}\n\\end{equation}\nThe prefactor in (\\ref{sol-wf}) has been chosen to retain normalization\nof the radial pair function, if $u^{(N=2)}(r)$ is normalized.\nIn any solution, the quotient $\\frac{\\tilde\\epsilon}{ \\tilde\\omega}$\nis equal in both problems and given by (see \\cite{Taut2e})\n\\begin{equation}\n\\frac{\\tilde\\epsilon}{\\tilde\\omega}=\n\\frac{\\tilde\\epsilon^{(N=2)}}{\\tilde\\omega^{(N=2)}}=|m|+n\n\\label{sol-quotient}\n\\end{equation}\nwhere $m$ is the angular momentum and $(n-1)$ the highest power\nin the polynomial $p(x)$.\\\\\nIn this way we obtain from (\\ref{om-N=2-n=1} -- \\ref{u-N=2-n=3}) and\n(\\ref{sol-om} -- \\ref{sol-wf})\nthe simplest exact solutions of \n(\\ref{rad-eq}) as follows:\\\\\n For $n=1$ there is only the \nasymptotic solution\n\\begin{eqnarray}\n\\frac{\\beta^2}{\\tilde \\omega}&=&0\\\\\np(x)&=&1\n\\end{eqnarray}\nFor $n=2$ there is one finite-- field-- solution:\n\\begin{eqnarray}\n\\frac{\\beta^2}{\\tilde\\omega}&=&\\frac{3}{2}\\; (2\\; |m|+1) \\label{om-n=2}\\\\\np(x) &=& 1+\\frac{\\frac{2}{\\sqrt{3}}\\; \\beta x} {(2\\; |m|+1)} \n\\label{u-n=2}\n\\end{eqnarray}\nBoth former solutions are ground states.\nFor $n=3$ there is one asymptotic solution, which is a first excited state,\n\\begin{eqnarray}\n\\frac{\\beta^2}{\\tilde \\omega}&=&0\\\\\np(x)&=&1-\\frac{\\tilde \\omega\\; x^2}{(|m|+1)}\n\\end{eqnarray}\nand one at finite fields, which is a ground state,\n\\begin{eqnarray}\n\\frac{\\beta^2}{\\tilde\\omega}&=&3\\; (4\\; |m|+3) \n\\label{om-n=3}\\\\\np(x) &=& 1+\\frac{\\frac{2}{\\sqrt{3}}\\;\\beta x} {(2\\; |m|+1)} \n +\\frac{\\biggl(\\frac{2}{\\sqrt{3}}\\;\\beta x \\biggr)^2} \n {2 (2\\; |m|+1)(4\\;|m|+3)} \n\\label{u-n=3}\n\\end{eqnarray}\nThe corresponding energies follow from (\\ref{sol-quotient}), and\nwe put the normalization constant \n$a_0=1$ without loss\nof generality. The pattern of solvable states agrees qualitatively\nwith Fig.1. Only the abscissa-- values of solvable states are shifted.\nThose, who are not yet convinced in the correctness of these solutions\n are recommended to check them\n by insertion into (\\ref{rad-eq}).\n\n\n\\newpage\n\n\\section{Comparison of our analytical solutions with Quantum Hall States}\nIn this section we confine ourselves to ground states and consider \ntwo cases separately.\nThis will provide two different generalizations of the Laughlin WFs.\nWe want to emphasize, however, that it would be possible \n(but not convenient) to include both cases\nin one formula.\nSecondly, we start with considering the case $\\bf R=0$ (zero order result)\nand add some words on the general case afterwards. \nGenerally, the generalized formulae in this section comprise our\nanalytical results \nfor 2 particles and 3 particles. However, they could {\\em ad hoc} be applied\nto any particle number and considered as trial functions.\\\\\n\nThe {\\em first case} comprises all solutions for three electrons with \nequal pair-- angular-- momenta in (\\ref{amended-WF})\nand the two-- electron result (\\ref{tot-WF-2e}). \nIn other words, it includes all finite-- field solutions and and \nthe asymptodic solutions with equal pair-- angular-- momenta.\nIt can be written in the following unified form\n\\begin{equation}\n\\Phi = \\prod _{i<k} ({\\bf r}_i-{\\bf r}_k)^m\\;\np_{n,m}(|{\\bf r}_i-{\\bf r}_k|)\\;\\;\n\\mbox{exp}\\biggl(-\\frac{1}{2} \\; \\tilde \\omega_{n,m} \\sum_l {\\bf r}_l^2 \\biggr)\n\\label{amended-general}\n\\end{equation}\nIt should be remembered that, apart from the case $n=1$, \nthe soluble field values $\\tilde \\omega_{n,m}$\nand the polynomials $p_{n,m}(x)$\ndepend on the particle number.\\\\\nUsing complex coordinates as defined in Section 2, (\\ref{amended-general})\ncan be reformulated as\n\\begin{eqnarray}\n\\Phi &=& \\prod _{i<k} (\\bar z_i-\\bar z_k)^{\\tilde m} \\;\np_{n,m}(|\\bar z_i-\\bar z_k|) \\; \\chi_1(\\tilde\\omega_{n,m}) \n~~~~~\\mbox{for}~~~m\\le 0 \n\\nonumber \\\\\n &=& \\mbox{complex conj.}~~~~~ \\mbox{for}~~~ m\\ge 0 \n\\label{amended-chi1}\n\\end{eqnarray}\nwhere $\\chi_1(\\tilde\\omega_{n,m})$ is a Slater determinant \nof LLL states for the effective frequency\n$\\tilde\\omega_{n,m}$ and $\\tilde m =|m|-1$. For $m\\le 0$ our WF can \nalso be rewritten using the Laughlin WF \\cite{Laughlin}\n\\begin{equation}\n\\Phi = \\prod _{i<k} \np_{n,m}(|\\bar z_i-\\bar z_k|) \\; \n\\Phi^{Laughlin}_{\\nu=\\frac{1}{\\bar m}}(\\tilde\\omega_{n,m})\n\\label{amended-L}\n\\end{equation}\nwhere the fields $\\tilde\\omega_{n,m}$ occur in the exponential factors\nof the Laughlin function instead of the infinite field used\nin the original Laughlin function.\\\\\nOur solutions (\\ref{amended-general}) and the \nequivalent forms have the following properties:\n$\\Phi$ fulfills the Pauli principle, if $m=odd$ (and $\\tilde m=even$).\nIt is an eigenfunction of the total angular momentum operator with eigenvalue\n$M_L=m \\frac{N(N-1)}{2}$, where $N$ is the electron number.\nApart from the asymptodic case $n=1$, it has components in higher LLs due to the\n$p(x)$--factors.\nIn the case $n=1$ (where $p(x)=const$ and $\\frac{1}{\\tilde \\omega_1}=0$)\n and $m \\le 0$, our WFs agree with the\nLaughlin states.\n\\begin{equation}\n\\Phi^{Laughlin}_{\\nu=\\frac{1}{\\bar m}}(\\tilde\\omega_1)=\n\\prod _{i<k} (\\bar z_i-\\bar z_k)^{\\bar m} \\;\n\\mbox{exp}\\biggl(-\\frac{1}{2} \\; \\tilde \\omega_1 \\sum_l |{\\bar z}_l|^2 \\biggr)\n= \\prod _{i<k} (\\bar z_i-\\bar z_k)^{\\tilde m} \\; \\; \\chi_1(\\tilde\\omega_1)\n\\label{Laughlin}\n\\end{equation}\nFor comparison, we also quote the Jain ansatz \\cite{Jain-FQHE}\n\\begin{equation}\n\\Phi_{\\nu}^{Jain}= {\\cal P}_{LLL} \\; \n\\prod _{i<k} (\\bar z_i-\\bar z_k)^{\\tilde m} \\; \n\\chi_{\\nu^*}(\\tilde\\omega_1)\n\\label{Jain}\n\\end{equation}\nwhere $\\frac{1}{\\nu}=\\pm \\frac{1}{\\nu^*}+\\tilde m$,\n${\\cal P}_{LLL}$ is a projection operator \nonto the LLL and the determinant\n$\\chi_{\\nu^*}(\\tilde\\omega_1)$ is for $\\nu^*$ full LLs and taken at \nthe asymptotically infinite frequency $\\tilde\\omega_1$. \nApart from the special cases discussed above, (\\ref{amended-chi1})\nand (\\ref{Jain}) do not seem to be fully equivalent. At least, \nboth treatments contain the Laughlin WF as a special case, and\nthere is the vague similarity that both have to do with higher LL\ncomponents.\\\\\nThe next property will be discussed using the formulae for $N=3$, but it would\nalso apply to the corresponding generalized trial functions.\nIf we go beyond the ${\\bf R}=0$ approximation, i.e. if we calculate the\n {\\em form} of the pair functions in the transformed \nspace ${\\bf x}_i$ in zero order\nin ${\\bf X}={\\bf R}$, but use the full back-- transformation \nto the original coordinates ${\\bf r}_i$ involving\na {\\em finite} ${\\bf R}$, then the solvable eigenfunctions are those in\n(\\ref{amended-WF}) with $m_1=m_2=m_3=m$. Firstly, we have to remind that \nthis function has to be antisymmetrized as discussed in the Appendix,\nbecause it is not automatically antisymmetric for $m=odd$ as\nin the case ${\\bf R}=0$.\nThis provides a complicated expression. Due to this antisymmetrization\nit has simple zeros wherever two coordinates agree. However, there are no\n$m$-- fold zeros as in the Laughlin WF, because the factors\n$\\biggl({\\bf R}-\\frac{1}{\\sqrt{3}} ({\\bf r}_i-{\\bf r}_k) \\biggr)$\ndo not vanish if two coordinates agree. This holds for all solutions\n(finite and infinite field solutions). This is a feature which agrees \nqualitatively with\nthe Jain functions.\\\\\n\nThe {\\em second case} differs from the first case only for $N>2$.\nFor the asymptodic solutions in (\\ref{amended-WF}), different $m_i$ are allowed,\nand for the ground state the polynomials can be omitted because they are \nconstants. After introducing a new indexing for the angular momenta,\nwhich is more appropriate for the $N$--particle system, \nwe obtain from (\\ref{amended-WF})\n\\begin{eqnarray}\n\\Phi &=& \\prod _{i<k} ({\\bf r}_i-{\\bf r}_k)^{m_{ik}}\\;\\;\n\\mbox{exp}\\biggl(-\\frac{1}{2} \\; \\tilde \\omega_1 \\sum_l {\\bf r}_l^2 \\biggr)\n\\label{amended-different-m}\\\\\n&=&\\prod _{i<k} (\\bar z_i-\\bar z_k)^{{\\bar m}_{ik}} \\;\n\\mbox{exp}\\biggl(-\\frac{1}{2} \\; \\tilde \\omega_1 \\sum_l |{\\bar z}_l|^2 \\biggr)\n\\end{eqnarray}\nThe second equation holds for $m_{ik} \\le 0$.\nFor general $m_{ik}$, this function has to be antisymmetrized. For 3\nparticles, the antisymmetrized result is discussed in Section 4.2 and \ngiven in the Appendix. In this special case Pauli principle\ndemands that $M_L=odd$.\n$\\Phi$ in (\\ref{amended-different-m}) is an eigenfunction of the total\norbital angular momentum with eigenvalue $M_L=\\sum_{i<k} m_{ik}$,\nand lies completely within the LLL.\nIt is apparent that the special case of equal $m_{ik}=m=odd$ agrees\nwith the Laughlin states. The changes produced by the extension \nof the considerations to finite $\\bf R$ are analogous to the first case.\\\\\n\nA more detailed investigation of\nthe applicability of the two generalizations (\\ref{amended-general}) \nand (\\ref{amended-different-m}) to $N$--particle systems and in particular\nQuantum Hall states\nwill be the aim of a separate work. \\\\\n\n\\newpage\n\n\\section{Approximate analytical solution of the pair equation\nand magic angular momenta for three electrons}\nIn order to avoid misunderstandings, it should be told at the very beginning,\n why we are looking for approximate solutions, if there are exact ones.\nThe answer is that the exact solutions\nexist only for certain fields and states,\nwhereas the solutions of this subsection are for all parameters and all states.\nAlthough the exactly soluble states are the most interesting ones, the rest is\nnecessary to prove certain cusp properties of the exactly solvable ones.\nFrom here on we consider only the case $\\beta=1$ and finite fields.\\\\\nAs shown in \\cite{Taut2e}, sect.3.2, the effective potential in\nthe radial pair equation (\\ref{rad-eq}) can be expanded around its minimum\ninto a Taylor series to second order.\nThis cannot be accomplished fully analytically because the minimum position\nresults from the zeros of a forth order polynomial equation.\nIt is possible, however, to establish the approximate effective potential\nto order $r_0^{-1}$, where $r_0^{-1}=(\\frac{9}{8} \\tilde \\omega)^{1/3}$\nis a small parameter in the strong correlation limit.\nThe eigenvalues of the resulting oscillator equation\ncan then be given analytically with the result\n\\begin{equation}\n\\epsilon= c_0 + c_1 \\; m + c_2 \\; m^2 + O(r_0^{-2})\\\\\n\\end{equation}\nwhere\n\\begin{eqnarray}\nc_0&=&\\biggl[ 1-\\frac{1}{4\\;\\; 3^{1/3}} \\tilde\\omega^{2/3}\\biggr]\\;\n\\biggl[ \\frac{1}{2} \\; (3\\; \\tilde \\omega)^{2/3} + \\sqrt{3}\\; \\tilde \\omega\n \\;(k+\\frac{1}{2}) \\biggr] \\\\\nc_1&=&\\frac{1}{2} \\; \\omega_c \\\\\nc_2&=&3^{-1/3}\\; \\tilde\\omega^{2/3}\\;\n\\biggl[ \\frac{1}{2} \\; (3\\; \\tilde \\omega)^{2/3} + \\sqrt{3}\\; \\tilde \\omega\n \\;(k+\\frac{1}{2}) \\biggr]\n\\end{eqnarray}\nwhere $k$ is the node number.\nThis provides a total energy of\n\\begin{equation}\nE=3\\;c_0 + c_1\\; M_L +c_2\\;(m_1^2+m_2^2+m_3^2)\n\\label{E-Taylor}\n\\end{equation}\nwhere $M_L=m_1+m_2+m_3$.\n It is clear that the\nTaylor expansion is the better the more symmetric the effective potential is.\n Therefore it gets poorer with increasing $m$ and increasing $\\tilde \\omega$.\nNevertheless, this formula gives a\nqualitative understanding of the magic angular momenta. If we are interested\nin the ground state {\\em for a given} $M_L$, it is clear \nfrom (\\ref{E-Taylor})\nthat it is formed by\nthat set of $m_i$ for which\nthe sum of the squares of the $m_i$ is minimal for a given sum of the $m_i$.\nThis demand is met if the $m_i$ are 'as equal as possible'.\nAs a example,\nfor $M_L=2$ there are two sets which provide the same $M_L$, namely\n$(002)$ and $(011)$, with the latter forming the ground state.\nIt is also clear that the total momenta of the form $M_L=3 m$ \n(multiple of three) play a special role because all three $m_i$ can be\nequal in this case, but not for the other $M_L$.\nFig. 1 shows the third (and only discontinuous) term \nof (\\ref{E-Taylor}) as a function of $M_L$.\nIt is obvious that this curve has kinks whenever it is possible that\nall three $m_i$ are equal, i.e. for $ M_L= 3 m$ with $m=0,1,2, \\cdots$.\nOn the other hand, we learned, that equal $m_i$ is a prerequisite for\nanalytical solutions. Thus we conclude that {\\em the solvable states are\nalways cusp states}.\\\\\nObserve, that in the limit ${\\bf R}=0$ the Pauli principle \ndemands that even $m$\nare forbidden (see Sect. 4.2). \nWe also want to remind (see Sect. 5) that we obtain the Laughlin WF\nin the ${\\bf R}=0$ limit.\nIf we go beyond this approximation, the total WFs (\\ref{amended}) \ncan be antisymmetrized for {\\em any} values of the $m_i$. \nThese facts elucidate the origin for the well known problem that\nexact diagonalization procedures provide cusps also for those $M_L$, \nwhich correspond to even denominator Laughlin states \n\\cite{Girvin}.\nThis shows that the Laughlin ansatz (if applied to finite systems)\nhas an additional symmetry (produced by putting ${\\bf R}=0$), which\nis {\\em not} present in exact solutions.\nThese conclusions are not in contradiction with the symmetry considerations\nin \\cite{Ruan} and \\cite{Seki}. The latter papers find formula for\nthe cusp (or magic) angular momenta, provided, the states under\ninvestigation are not\nforbidden by Pauli's principle. They do not have (and need) any explicit \nexpression for the wave function. By the way, their general notion on the \neigenstates is consistent with our small--$\\bf R$ expansion.\nThe harmonic approximation used in this section is also related to the\nharmonic approach used in \\cite{Maksym}, but not fully equivalent. However,\nour analytic approximations derived in Sect.s 3 and 4 is not harmonic.\n\n\\newpage\n\n\\section{First order perturbation theory and accuracy of the strong\ncorrelation expansion}\n\nNow we are going to calculate the contributions of the dipole and quadrupole\nterm (\\ref{dipole}) and (\\ref{quadrupole}) of the e-e-interaction\nto the total energy in first order\n\\footnote{By first order we mean first order in\nall multi-pole corrections, but not\nfirst order in $\\bf R$}\nperturbation theory, i.e.\n\\begin{equation}\n\\Delta E^{(l)}= < \\Phi| V_{ee}^{(l)} |\\Phi>\n\\label{E-first-order}\n\\end{equation}\nwhere the zero order wave function is generally given by (\\ref{tot-WF-sym}).\nFor {\\em quartet states}, where the total WF is just a product of spatial and\nspin part and for calculating matrix elements, the total WF in\n(\\ref{E-first-order}) can be replaced by the unsymmetrized spatial part.\n(It is simpler, however, to do the integrations in the\ntransformed ${\\bf x}_i$--coordinates rather than in the ${\\bf r}_i$).\nThe further calculation is straight forward and provides\n\\begin{eqnarray}\n\\Delta E ^{(1)}&=& \\frac{1}{3\\sqrt{3}} \\; \\sum_{k=1}^3 M_{m_k}^{(-1)} \\\\\n\\Delta E ^{(2)}&=& \\frac{1}{9\\sqrt{3}} \\;\\biggl[ \\sum_{k=1}^3 M_{m_k}^{(-1)}-\n\\frac{1}{2}\\; \\sum_{k=1}^3 \\sum_{l(\\neq k)=1}^3 \nM_{m_k}^{(-3)}\\;M_{m_k}^{(2)} \\biggr]\n\\end{eqnarray}\nwhere we defined moments\n\\begin{equation}\nM_m^{(k)}=\\int_0^{\\infty} dx \\; x^k \\; [u_m(x)]^2\n\\label{moment}\n\\end{equation}\nwith $u_m(x)$ being the radial part defined in (\\ref{ansatz}) and given\nexplicitly in special cases using (\\ref{u-n=2}) and (\\ref{u-n=3}).\nIf all three angular momenta agree: \\mbox{$m_1=m_2=m_3\\equiv m$},\nthe result simplifies to\n\\begin{eqnarray}\n\\Delta E ^{(1)}&=& \\frac{1}{\\sqrt{3}} \\; M_{m}^{(-1)}\n\\label{E1}\\\\\n\\Delta E ^{(2)}&=& \\frac{1}{3\\sqrt{3}} \\;\\biggl[ M_{m}^{(-1)}-\n M_{m}^{(-3)}\\;M_{m}^{(2)} \\biggr]\n\\label{E2}\n\\end{eqnarray}\nIt should be noted that the moment $M_{m}^{(-3)}$, appearing in the second\norder contribution, diverges for $m=0$. This is because for $x \\rightarrow 0$\nthe radial pair function goes as\n$u_m(x) \\rightarrow x^{(|m|+\\frac{1}{2})}$ and\nthus the integrand in (\\ref{moment}) behaves\nlike $x^{-2}$ for small $x$. That is why the results for $m=0$ are missing in\nTable 2.\\\\\n\nFor a test of the accuracy of our small--$\\bf R$ expansion\nwe use analytically solvable states only, i.e. quartet states\nwith $M_L=3 m$. We do this \nfor magnetic field only (i.e. $\\omega_0=0$), \nbecause the confinement can be included afterwards by\na simple rescaling of the parameters.\nFor $\\omega_0=0$ it follows from (\\ref{sol-quotient}) and the definitions of\n$\\tilde \\omega$ and $\\tilde \\epsilon$ that the zero order \n(in $\\bf R$) energy per electron\nreads\n\\begin {equation}\n\\frac {E^{(0)}}{3 \\omega_c}=\\frac{(m + |m|)}{2}+\\frac{n}{2}\n\\label{E-exact}\n\\end{equation}\nand, for comparison, the trivial result without e-e-interaction is\n\\cite{one-e-dot}\n\\begin {equation}\n\\frac {E^{(non-int)}}{3 \\omega_c}=\\frac{(m + |m|)}{2}+k+\\frac{1}{2}\n\\end{equation}\nThe formulae for the corrections in first and second order $\\Delta E ^{(1)}$\nand $\\Delta E ^{(2)}$, respectively,\nare given in (\\ref{E1}) and (\\ref{E2}). \nTable 1 and 2 show the results for fixed $m$ and\n%%%%%%% %\nvarying $n$ and fixed $n$ and different $m$, respectively.\n$E^{(Taylor)}$ is the result using the Taylor expansion \nof the effective potential in the radial Schr\\\"odinger equation as\n described in Sect. 6.\nIts agreement with the exact (analytical) solution of the radial \nSchr\\\"odinger equation $E^{(0)}$ gives an account of the accuracy of the Taylor\nexpansion.\\\\\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Table 1\n\\begin{table}[]\n\\caption[]{\nTotal energy per electron in units of $\\omega_c$\nfor analytically solvable quartet states\nwith total orbital angular momentum $M_L=-3$ ($m=-1$)\nfor $\\omega_0=0$ (magnetic\nfield only). $E^{(non-int)}$ is the energy without\nelectron-- electron interaction,\n$E^{(Taylor)}$ is the total energy where the\neffective potential in the radial pair equation is treated\nin second order Taylor expansion (\\ref{E-Taylor}), and\n$E^{(0)}$ is the exact result in zero order in $\\bf R$\nas given in (\\ref{E-exact}).\n $\\Delta E^{(1,2)}$ are the\ndipole and quadrupole contributions\ngiven in (\\ref{E1}) and (\\ref{E2}), and $E^{(2)}$ is\nthe sum of the three former contributions.\nThe Zemann energy is omitted.\n\\vspace{.5cm}}\n\n\\renewcommand{\\baselinestretch}{1.5}\n\\begin{tabular}{|r|r|c|c|l|r|r|r|}\\hline\nn& $\\omega_c$ & $\\frac {E^{(non-int)}}{3 \\omega_c}$ &\n$\\frac {E^{(Taylor)}}{3 \\omega_c}$ &\n$\\frac {E^{(0)}}{3 \\omega_c}$ &\n$\\frac {\\Delta E^{(1)}}{3 \\omega_c}$ &\n$\\frac {\\Delta E^{(2)}}{3 \\omega_c}$ &\n$\\frac {E^{(2)}}{3 \\omega_c}$ \\\\ \\hline\n2 & $\\frac{4}{9}$= 444.444 E-3 & 0.5 & 1.03796\n& 1 & 0.154332 & --0.101713 & 1.052619 \\\\\n3 & $\\frac{2}{21}$=95.2381 E-3 & 0.5& 1.49531\n & 1.5 & 0.290373 & --0.126757 & 1.663616\\\\\n5 & 18.1896 E-3 & 0.5 & 2.49055\n& 2.5 & & & \\\\\n10& 2.20940 E-3 & 0.5 & 4.99381 & 5 & & & \\\\\n15& 0.655360 E-3 & 0.5 & 7.49561 & 7.5 & & & \\\\ \\hline\n\\end{tabular}\n\\end{table}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Table 2\n\\begin{table}[]\n\\caption[]{\nTotal energy per electron in units of $\\omega_c$\nfor analytically solvable quartet states\nfor $\\omega_0=0$ (magnetic field only) as a function of $m$ with\n$M_L=3 m$. ({\\em Only odd $m$ are compatible with the Pauli principle}.)\nAll solutions belong to $n=2$.\nThe meaning of the other column heads is as in Table 1.\n\\vspace{.5cm}}\n\n\\renewcommand{\\baselinestretch}{1.5}\n\\begin{tabular}{|r|r|c|c|r|r|l|r|r|}\\hline\nm& $\\omega_c$ & $\\frac {E^{(non-int)}}{3 \\omega_c}$ &\n$\\frac {E^{(Taylor)}}{3 \\omega_c}$ &\n$\\frac {E^{(0)}}{3 \\omega_c}$ &\n$\\frac {\\Delta E^{(1)}}{3 \\omega_c}$ &\n$\\frac {\\Delta E^{(2)}}{3 \\omega_c}$ &\n$\\frac {E^{(2)}}{3 \\omega_c}$ \\\\ \\hline\n10 & $\\frac{4}{63}$=0.063 & 10.5 & 21.4662\n& 11 & 0.164717 & --0.00834761 & 11.15637 \\\\\n...&&&&&&\\\\\n3 & $\\frac{4}{21}$=0.190 & 3.5 & 5.06036\n & 4 & 0.161018 & --0.0283146 & 4.132703 \\\\\n2 & $\\frac{4}{15}$=0.266 & 2.5 & 3.43552\n & 3 & 0.158918 & --0.0435331 & 3.115385 \\\\\n1 & $\\frac{4}{9}$=0.444 & 1.5 & 2.03796\n& 2 & 0.154332 & --0.101713 & 2.052619 \\\\\n\\hline\n0 & $\\frac{4}{3}$=1.333 & 0.5 & 0.892263\n& 1 & 0.136400 &&\\\\\n\\hline\n--1 & $\\frac{4}{9}$=0.444 & 0.5 &1.03796\n & 1 & 0.154332 & --0.101713 & 1.052619 \\\\\n--2 & $\\frac{4}{15}$=0.266 & 0.5 & 1.43552\n & 1 & 0.158918 & --0.0435331 & 1.115385 \\\\\n--3 & $\\frac{4}{21}$=0.190 & 0.5 & 2.06036\n & 1 & 0.161018 & --0.0283146 & 1.132703 \\\\\n...&&&&&&\\\\\n--10 & $\\frac{4}{63}$=0.063 & 0.5 & 11.4662\n& 1 & 0.164717 & --0.00834761 & 1.156369 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nWe conclude the following.\\\\\ni) Comparison of $E^{(non-int)}$ and $E^{(0)}$ shows the contribution of the\nCoulomb interaction energy to the total energy. As to be expected, \nit grows with\ndecreasing $\\omega_c$ and is tremendous for small $\\omega_c$ (e.g.\nten times larger than the kinetic energy\nfor $n=10$ i.e. $\\omega_c \\approx 10^{-3}$).\\\\\nii) For small $|m|$, the Taylor approximation\nprovides a good tool for solving the radial Schr\\\"odinger equation. \nFor large $|m|$ the effective potential becomes so unsymmetric \nwith respect to its minimum that its result goes fatally wrong. \\\\\niii) The analytical zero order result $E^{(0)}$, which is the main \nachievement of this paper, compares pretty well with the most precise\nresult $E^{(2)}$. This is mainly due to the \ncancellation of $\\Delta E ^{(1)}$ and\n$\\Delta E ^{(2)}$ . The maximum error is of the order of 10 \\%.\n\n\\section{Summary and discussion}\nWe found that the two--dimensional \nSchr\\\"odinger equation for 3 electrons in an homogeneous\nmagnetic field (perpendicular to the plane) and a parabolic \nscalar confinement potential has exact analytical solutions in the\nstrong correlation limit, where the expectation value of the \ncenter of mass vector is small compared with the \naverage distance between the electrons.\nThese analytical solutions exist only\nfor certain discrete values of the external fields\n$\\tilde \\omega$. \nFor finite external fields, \nanalytical solvability demands that\nall three pair-- angular-- momenta agree what leads \nto total angular momenta $M_L=3 m$ with $m=integer$.\nIn zero order in $\\bf R$, Pauli principle allows \nequal pair-- angular-- momenta \nonly for parallel spins and $m=odd$. The analytically solvable states are always\ncusp states, i.e. states where $E(M_L)$ has a cusp.\\\\\nFurther, these special \nanalytical solutions for 3 particles and the exact analytical solutions for \n2 particles could be written in \na unified form, which contains only products over coordinate combinations\n$\\prod _{i<k}$ and sums $\\sum_i$.\nConveniently, instead of using one formula we consider\ntwo cases (\\ref{amended-general}) and\n(\\ref{amended-different-m}). \nThese formulae, when {\\em ad hoc} generalized to N coordinates,\ncan be discussed as ansatzes for the wave function of the N--particle system.\nThese ansatzes fulfill the following demands: they are exact for two particles\nand for 3 particles in\nthe limit of small $\\bf R$,\nand they are eigenfunction of the total\norbital angular momentum. The Laughlin functions are special case,\nor in other words, both formulae provide two different generalizations of the\nLaughlin functions.\\\\\nUntil now we know mainly that our WFs are analytically solvable states\n(within the approximations discussed above). It is also clear that\nstates for an infinitesimally varied $\\tilde \\omega$ look quite different,\ni.e. the finite polynomials have to be replaced by polynomials\nwith an infinite number of terms. For N=2 and 3 these \n'neighboring' states are even explicitly known. It is not yet shown,\nhowever, if these special states have generally something to do with\nthe Quantum Hall states (which show similar singular features), \nor if any physical quantity has any special feature in these \nsoluble states.\nOne encouraging fact is that the Laughlin states are special cases of\nour states. \nIn prospect, it is possibly a good idea to look for \nsimilar exact analytical solutions \nin a spherical geometry instead of the disk geometry used here, because \nit is easier then to attribute a filling factor to each eigensolutions.\nIf there is a connection between our states and Quantum Hall states,\nthis would imply some kind of \ninherent super-symmetry in the Quantum Hall states.\\\\\nNow we want to compare our treatment of three electrons\nwith Laughlins \\cite{Laughlin}. Both approaches are approximate.\nWhereas he forms WF by antisymmetrization of one-- particle states of the\nLLL, which is expected to be good for strong fields, \nwe established an expansion,\nwhich is good in the strong correlation limit and which contains\nin general higher LL components. Consequently, we obtained a richer variety\nof solutions comprising the Laughlin states as special cases. \n\n\\newpage\n\n\\section{Appendix: Pauli Principle}\n\nIn order the obtain familiar looking formula, we rename $T \\rightarrow \\Phi$\nand $t \\rightarrow \\phi$.\nThe question here is, how the properly symmetrized spatial part $\\Phi$ \nhas to be\nsupplemented by an appropriate spin part in order to obtain a wave function\nwhich is eigenfunction of ${\\bf S}^2$ and $S_z$ (with quantum numbers\n$S$ and $M_S$) and which satisfies the Pauli principle.\n${\\bf S}=\\sum_{i=1}^N {\\bf s}_i$ is the total spin operator for all particles.\nFor more than two particles this is\na well established, but non-- trivial procedure.\nThe source of the difficulty is the fact that\nthe spin space can be degenerate, i.e. there is\nmore than one orthogonal spin eigenfunction ${\\it X}_i,\\; (i=1,...f)$ for\ngiven $S$ and $M$ (see Table 3 for N=3).\\\\\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Table 3\n\\begin{table}[hb]\n\\caption[]{\n\nStandard spin eigenfunctions for N=3. $\\alpha$ and $\\beta$ are the\none--particle spin eigenfunctions for spin up and down, respectively.\nThe first factor of a pro\\-duct of one--particle\nfunctions carries the spin variable '1',\nthe second factor carries '2' etc.\\\\}\n\n\\renewcommand{\\baselinestretch}{1.5}\n\\large\n\\begin{tabular}{|r|r|l|l|l|}\\hline\nS& $M_S$ & i & ${\\it X}_i$& permutation sym. \\\\ \\hline\n$\\frac{1}{2}$ & +$\\frac{1}{2}$ & 1 & $\\frac{1}{\\sqrt{6}}$\n[--(\\a \\b+\\b \\a)\\a +2 \\a \\a \\b ]& symmetric for (12) \\\\\n & --$\\frac{1}{2}$ & & $\\frac{1}{\\sqrt{6}}$\n[+(\\b \\a+\\a \\b)\\b --2 \\b \\b \\a ]& \\\\ \\hline\n$\\frac{1}{2}$ &+$\\frac{1}{2}$ & 2 & $\\frac{1}{\\sqrt{2}}$\n(\\a \\b -- \\b \\a) \\a &antisymmetric for (12)\\\\\n &--$\\frac{1}{2}$ & & $- \\frac{1}{\\sqrt{2}}$\n(\\b \\a -- \\a \\b) \\b & \\\\ \\hline\n$\\frac{3}{2}$ & +$\\frac{3}{2}$ & 1 &\n\\a \\a \\a & symmetric for all {\\cal P}\\\\\n & +$\\frac{1}{2}$ & & $\\frac{1}{\\sqrt{3}}$\n[\\a (\\a \\b+\\b \\a)+\\b \\a \\a ] & \\\\\n & --$\\frac{1}{2}$ & & $\\frac{1}{\\sqrt{3}}$\n[\\b (\\b \\a+\\a \\b)+\\a \\b \\b ] & \\\\\n & --$\\frac{3}{2}$ & &\n\\b \\b \\b & \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\normalsize\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nGenerally, the total WF $\\Psi$, which fulfills our demands,\ncan be calculated from (see e.g. \\cite{Pauncz})\n\\begin{equation}\n\\Psi_i(1,2,3)= {\\cal A}_a \\; {\\it X}_i(1,2,3) \\; \\Phi(1,2,3)~~~~~~~~i=(1,... ,f)\n\\label{antisym}\n\\end{equation}\nwith the antisymmetrizer\n\\begin{equation}\n{\\cal A}_a=\\frac{1}{\\sqrt{N!}} \\sum _{\\cal P} (-1)^p \\; {\\cal P}\n\\end{equation}\nand \\cal $\\cal P$ is a permutation operator. $f$ is the dimension\nof the degenerate spin space ($f=1$ and $2$ for\n $S=\\frac{3}{2}$ and $\\frac{1}{2}$,\nrespectively, for N=3) and $(-1)^p$ is the\nparity of the permutation $\\cal P$. The quantum numbers $S$ and $M_S$\n of the spin part ${\\it X}_i$\nas well as the quantum numbers of the spatial part $\\Phi$\nare not indicated. The arguments $(1,2,3)$ are\nspin or spatial coordinates depending on the function in question.\nAlthough being correct, (\\ref{antisym}) does not reveal the inherent\npermutation symmetry of the total WF. An equivalent symmetrized form is\n(see e.g. \\cite{Pauncz})\n\\begin{equation}\n\\Psi_i=\\frac{1}{\\sqrt{f}} \\sum_{k=1}^f {\\it X}_k \\cdot \\Phi^s_{ki}\n\\label{tot-WF-sym}\n\\end{equation}\nwhere the symmetrized spatial function is defined as\n\\begin{equation}\n\\Phi^s_{ki}=\\sqrt{\\frac{f}{N!}}\\sum _{\\cal P} (-1)^p \\; U_{ki}({\\cal P})\\;\n{\\cal P} \\; \\Phi\n\\end{equation}\nand $U_{ki}({\\cal P})$ is an irreducible representation matrix of the\npermutation group for permutation $\\cal P$ given in Table 4.\n% %%%%%%%\nIt is convenient to define column vectors $\\bf \\Phi\\mbox{$^s_i$}$\n(with $i=1,2$) of\nthe matrix $\\Phi^s_{ki}$. Then the $i^{th}$ vector\ncomprises all spatial information\nabout the $i^{th}$ of the orthogonal states $\\Psi_i$.\\\\\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Table 4\n\\begin{table}[]\n\\caption[]{\nIrreducible representation matrices ${\\bf U}({\\cal P})$ for N=3. For\nclass and cyclic permutation symbols see \\cite{Pauncz}.\\\\}\n\n\\renewcommand{\\baselinestretch}{1.5}\n\\large\n\\begin{tabular}{|c|c|c|}\\hline\nclass& {\\cal P} & {\\bf U}({\\cal P}) \\\\ \\hline\n\n[$1^3$] & $\\varepsilon$ &\n$ \\left[ \\begin{array}{cc} 1&0\\\\0&1 \\end{array}\\right] $ \\\\ \\hline\n\n[$2,1$] & (1,2) &\n$ \\left[ \\begin{array}{cc} 1&0\\\\0&-1 \\end{array}\\right] $ \\\\ \\cline{2-3}\n\n & (2,3) &\n$ \\frac{1}{2} \\left[ \\begin{array}{cc} -1& \\sqrt{3} \\\\ \\sqrt{3}&1 \\end{array}\n\\right] $ \\\\\n\\cline{2-3}\n\n & (3,1) &\n$ \\frac{1}{2} \\left[ \\begin{array}{cc} -1&-\\sqrt{3} \\\\ -\\sqrt{3}&1 \\end{array}\n\\right]$\\\\ \\hline\n\n[$3$] & (123) &\n$ \\frac{1}{2} \\left[ \\begin{array}{cc} -1&\\sqrt{3}\\\\-\\sqrt{3}&-1 \\end{array} \n\\right] $ \\\\\n\\cline{2-3}\n\n & (132) &\n$ \\frac{1}{2} \\left[ \\begin{array}{cc} -1&-\\sqrt{3} \\\\ \\sqrt{3} & -1 \\end{array}\n \\right] $ \\\\ \\hline\n\n\\end{tabular}\n\\end{table}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n{\\bf Quartet $S=\\frac{3}{2}$}\\\\\nBecause of $f=1$ and because the spin eigenfunctions are symmetric\nagainst all permutations, the symmetrized spatial function is totally\nantisymmetric and we have\n\\begin{equation}\n\\Psi={\\it X}\\; \\Phi^s ;~~~~~~~~~~~~~~~\\Phi^s={\\cal A}_a \\Phi\n\\end{equation}\nwhich is reminiscent of the permutation\nsymmetry of the triplet state for $N=2$.\nIf we insert the solution (\\ref{T-WF}) for $\\Phi$\nwe obtain a lengthy expression which does not reveal anything.\nFor ${\\bf R}={\\bf 0}$, however,\n(\\ref{def-t}) implies\n$\\phi_{m}(-{\\bf x})=(-1)^m \\; \\phi_{m}({\\bf x})$ or\nin the short hand notation\n\\begin{equation}\n\\phi_{m}(i-k)=(-1)^m \\; \\phi_{m}(k-i)\n\\label{inversion}\n\\end{equation}\n with $m$ the orbital angular momentum of the pair solution, we obtain\n\\begin{eqnarray*}\n\\Phi^s= \\frac{1}{\\sqrt{6}}\n &&\\bigg\\{\\phi_1(2-3) \\; \\biggl[ \\phi_2(3-1)\\; \\phi_3(1-2)\n -(-1)^{M_L} \\phi_2(1-2)\\; \\phi_3(3-1) \\biggr] \\\\\n &&+\\phi_2(2-3) \\; \\biggl[ \\phi_3(3-1)\\; \\phi_1(1-2)\n -(-1)^{M_L} \\phi_3(1-2)\\; \\phi_1(3-2) \\biggr]\\\\\n &&+\\phi_3(2-3) \\; \\biggl[ \\phi_1(3-1)\\; \\phi_2(1-2)\n -(-1)^{M_L} \\phi_1(1-2)\\; \\phi_2(3-1) \\biggr]\n \\bigg\\}\n\\end{eqnarray*}\nwhere $M_L=\\sum_{i=1}^3 m_i$ is the total orbital angular momentum.\nIf we define a matrix of pair functions\n\\begin{equation}\n{\\bf S}=\n\\left[ \\begin{array}{ccc} \\phi_1(2-3) & \\phi_1(3-1) & \\phi_1(1-2)\\\\\n \\phi_2(2-3) & \\phi_2(3-1) & \\phi_2(1-2)\\\\\n \\phi_3(2-3) & \\phi_3(3-1) & \\phi_3(1-2)\n \\end{array} \\right]\n\\end{equation}\nthen our result can be written as a determinant or \npermanent\\footnote{A permanent\n is similar to a determinant without the factors $(-1)^p$}\n of pair functions\n\\begin{eqnarray}\n\\Phi^s &=&\\frac{1}{\\sqrt{6}}\\; \nDet({\\bf S})~~~~~~~~~~~ {\\mbox for} \\;\\; M_L=even \\\\\n\\Phi^s &=&\\frac{1}{\\sqrt{6}}\\; Perm({\\bf S})~~~~~~~~ {\\mbox for} \\;\\; M_L=odd\n\\end{eqnarray}\nFrom this fact we conclude the following important rule\nvalid for ${\\bf R}={\\bf 0}$:\\\\\n{\\em If at least two pair functions agree, $M_L$ has to be odd.}\\\\\nOtherwise the determinant vanishes. This means in particular, that\nthe total ground state cannot be built up from identical\nground state pair functions\nwith $m=0$, which would have the lowest energy without taking\nthe Pauli principle into account.\nTherefore, the ground state\nconsists of one pair with $m=1$ and two pairs with $m=0$ and\nhas a total angular momentum $M_L=1$ in the strong correlation limit.\\\\\nObserve the difference of these conclusions to the Pauli principle for one--\nelectron orbitals. If $\\Phi$ were a product of functions\nwhich depend on one coordinate only (one--particle orbitals),\nwhich is equivalent to a Slater determinant for $\\Phi^s$,\nin the quartet state all three spatial functions must be different, and the\nground state would have a total angular momentum\n$M_L=(m-1)+m+(m+1)=3 m$ where the integer $m$ depends on the strength\nof the magnetic field. This holds if $\\tilde\\omega$ is so large that\nno {\\em excited} one--particle orbitals are involved, i.e. in the\n weak correlation regime .\nIndeed, for strong fields the simple rule that $M_L$ is a multiple\nof 3 is confirmed also by numerical\ncalculations \\cite{Girvin},\\cite{Maksym},\\cite{Hawrylak}.\\\\\n\n{\\bf Doublet $S=\\frac{1}{2}$}\\\\\nBecause of $f=2$ there are two degenerate and orthogonal\nfunctions $\\Psi_i$ which span a\nsubspace. This level does not exist\nfrom symmetry reasons only if both functions vanish.\nWe assume ${\\bf R}={\\bf 0}$ so that (\\ref{inversion}) holds and\nconsider the two\nvectors $\\bf \\Phi \\mbox{$^s_i$}$ one by one. \\\\\nFor $i=1$ and $M_L=even$ we obtain\n\\begin{eqnarray*}\n{\\bf \\Phi \\mbox{$^s_1$}}=\\frac{1}{\\sqrt{3}} \\bigg\\{ \\;\\;\\;\n\\left[ \\begin{array}{c} 1\\\\0 \\end{array} \\right] \\;\n \\phi_3(1-2) \\;\\; D_{12}(2-3,3-1) &&\\\\\n+ \\frac{1}{2} \\left[ \\begin{array}{c} -1\\\\-\\sqrt{3} \\end{array} \\right] \\;\n \\phi_3(2-3) \\;\\; D_{12}(3-1,1-2) &&\\\\\n+ \\frac{1}{2} \\left[ \\begin{array}{c} -1\\\\ \\sqrt{3} \\end{array} \\right] \\;\n \\phi_3(3-1) \\;\\; D_{12}(1-2,2-3) && \\bigg\\}\n\\label{i1}\n\\end{eqnarray*}\nwhere\n\\begin{equation}\nD_{12}(2-3,3-1)=\nDet \\; \\left[ \\begin{array}{cc} \\phi_1(2-3) & \\phi_1(3-1) \\\\\n \\phi_2(2-3) & \\phi_2(3-1) \\\\\n \\end{array} \\right]\n\\label{det}\n\\end{equation}\nand the other determinants are defined analogously: the subscripts refer to the\nquantum numbers of the pair functions involved and the arguments define\nthe arguments of the pair functions. \\\\\nFor $i=1$ and $M_L=odd$ we obtain a similar formula as before but\nwith the {\\em determinants} in (\\ref{det}) replaced by {\\em permanents}\ncalled $P_{12}$.\\\\\nFor $i=2$ and $M_L=odd$ the result is also similar\n but with different column vectors\\\\\n\\begin{eqnarray*}\n{\\bf \\Phi \\mbox{$^s_2$}}=\\frac{1}{\\sqrt{3}} \\bigg\\{ \\;\\;\\;\n\\left[ \\begin{array}{c} 0\\\\1 \\end{array} \\right] \\;\n \\phi_3(1-2) \\;\\; D_{12}(2-3,3-1)&& \\\\\n+ \\frac{1}{2} \\left[ \\begin{array}{c} \\sqrt{3}\\\\-1 \\end{array} \\right] \\;\n \\phi_3(2-3) \\;\\; D_{12}(3-1,1-2) && \\\\\n+ \\frac{1}{2} \\left[ \\begin{array}{c} -\\sqrt{3}\\\\ -1 \\end{array} \\right] \\;\n \\phi_3(3-1) \\;\\; D_{12}(1-2,2-3) && \\bigg\\}\n\\label{i2}\n\\end{eqnarray*}\nwhere $D_{12}$ is defined as in (\\ref{det}).\\\\\nFor $i=2$ and $M_L=even$ the {\\em determinants} in the last formula\n have to be replaced by {\\em permanents}.\\\\\n\nIt is obvious that for $\\phi_1=\\phi_2 (\\neq \\phi_3)$ the determinants\n$D_{12}$ vanish and consequently for\n$M_L=even$, $\\bf \\Phi \\mbox{$_1^s$}$ vanishes and for\n$M_L=odd$, $\\bf \\Phi \\mbox{$_2^s$}$ vanishes, but both do not\nvanish simultaneously.\nTherefore this state is allowed.\nHowever, if all three pair functions agree, also those functions vanish in which\npermanents occur, because the sum of the prefactors sum up to zero.\nConsequently, the Pauli--principle tells us that for ${\\bf R}={\\bf 0}$\nin the doublet state:\\\\\n{\\em All three pair functions must not agree.}\\\\\n This is analogous to the\none--particle model, where two orbitals may agree\n (if they occupy different spin states), but not all three of them.\nThis means that, as for the quartet,\nthe ground state consists of one pair with $m=1$ and two pairs with $m=0$ and\nit has a total angular momentum $M_L=1$. Thus, in our limit and for vanishing\nmagnetic field, the quartet and the doublet ground state energy agree.\nThis degeneracy is lifted by a magnetic field because of the Zemann energy.\n\n\\section{Acknoledgement}\n\nI am indebted to H.Eschrig, W.Weller and their groups as well as\nJ.K.Jain, W.Apel, and U.Z\\\"ulicke for discussion and the\nDeutsche Forschungs-- Gemeinschaft\nfor financial funding.\n\\newpage\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Girvin} S.M. Girvin, T. Jach, Phys. Rev. {\\bf 28}, 4506 (1983)\n\n\\bibitem{Merkt}U. Merkt, J. Huser, M. Wagner, Phys. Rev. B \n {\\bf43}, 7320 (1991);\\\\ \nM. Wagner, U. Merkt, and A.V. Chaplik, Phys. Rev. B {\\bf 45}, 1951 (1992)\n\n\\bibitem{Maksym} P. A. Maksym, T. Chakraborty, Phys. Rev. Lett. \n{\\bf 65}, 108 (1990) and Phys. Rev. B {\\bf 45}, 1947 (1992);\\\\\nP.A. Maksym, Physica B {\\bf 184}, 385 (1993)\n\n\\bibitem{Hawrylak} P. Hawrylak and D. Pfannkuche, Phys. Rev. Lett. {\\bf 70},\n485 (1993);\\\\\nP. Hawrylak, Phys. Rev. Lett. {\\bf 71}, 3347 (1993)\n\n\\bibitem{Dharma} M. W. C. Dharma--wardana, J. Phys. Condens. Matter {\\bf7},\n 4095 (1995) ; and Phys. Rev. B {\\bf 51}, 1653 (1995)\n\n\\bibitem{McDonald} S.--R. E. Yang, A. H. MacDonald, and M. D. Johnson, Phys. Rev. Lett. {\\bf 71}, 3194 (1993); S. Mitra and A. H. MacDonald, Phys. Rev. B\n {\\bf 48}, 2005 (1993)\n \n\\bibitem{Laughlin} R.B. Laughlin, Phys. Rev. {\\bf 27}, 3383 (1983);\\\\\nR. B. Laughlin, in: {\\em The Quantum Hall Effect}, Eds. R.E.Prange and\nS.M.Girvin, Springer Verlag, New York (1987).\n\n\\bibitem{Jain-FQHE} J. K. Jain, in: {\\em Perspectives in Quantum Hall Effects},\nEd.s S. Das Sharma and A. Pinczuk, John Willey \\& Sons,(1996)\n\n\\bibitem{Jain-QD} J. K. Jain and T.Kawamura, Europhys. Lett. {\\bf 29},\n321 (1995); R. K. Kamilla and J. K. Jain, Phys. Rev. B {\\bf 52}, 2798 (1995);\nT.Kawamura and J. K. Jain, J. Phys. Condens. Matter {\\bf 8},2095 (1996)\n\n\\bibitem{Halperin} B.I. Halperin, in: {\\em Perspectives in Quantum Hall \nEffects},\nEd.s S. Das Sharma and A. Pinczuk, John Willey \\& Sons,( 1996)\n\n\\bibitem{Haldane} F. D. M. Haldane, in: {\\em The Quantum Hall Effect},\nSpringer Verlag, New York (1987).\n\n\\bibitem{MacDonald-2} A. H. MacDonald, in: {\\em Mesoscopic Quantum Physics},\nProceedings of the \nLes Houches Summer School , Elsevier (1995)\n\n\\bibitem{Taut2e} M. Taut, J. Phys. A{\\bf 27}, 1045 (1994)\nand erratum J.Phys.A{\\bf27}, 4723 (1994). Additionally, in formula (10)\nin the term containing $\\frac{\\partial}{\\partial\\alpha} $\na factor $\\frac{1}{2}$ is missing , and on the r.h.s. of (19a) and (20a)\n$\\tilde\\omega$ must be replaced by $\\tilde\\omega_r$.\n\n\\bibitem{Taut-unpublished} This fact is unpublished, but it is\ncompletely analogous to the case\nof two electrons in an external oscillator potential in 3 dimensional space.\nSee M.Taut, A.Ernst and H.Eschrig; J. Phys. B {\\bf 31},2689 (1998)\n\n\\bibitem{Ruan} W. Y. Ruan et al., Phys. Rev. B {\\bf 51}, 7942 (1995)\n\n\\bibitem{Seki} T. Seki, Y. Kuramoto, T. Nishino, J. Phys. Soc. Japan {\\bf 65},\n3945 (1996)\n\n\\bibitem{Maksym} P. A. Maksym, Phys. Rev. B {\\bf 53}, 10871 (1996) \n\n\\bibitem{one-e-dot}V. Fock, Z. Phys. {\\bf 47}, 446 (1928) (in German);\\\\\nC.G. Darwin, Proc. Cambridge Philos. Soc. {\\bf 27}, 86 (1930); \\\\\nR.B. Dingle, Proc. Roy. Soc. London A {\\bf 211}, 500 (1952)\n\n\\bibitem{Pauncz}R. Pauncz, {\\em Spin Eigenfunctions}, Plaenum Press,\nNew York and London 1979\n\n\\end{thebibliography}\n\n\\end{document}\n\n"
}
] |
[
{
"name": "cond-mat0002067.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Girvin} S.M. Girvin, T. Jach, Phys. Rev. {\\bf 28}, 4506 (1983)\n\n\\bibitem{Merkt}U. Merkt, J. Huser, M. Wagner, Phys. Rev. B \n {\\bf43}, 7320 (1991);\\\\ \nM. Wagner, U. Merkt, and A.V. Chaplik, Phys. Rev. B {\\bf 45}, 1951 (1992)\n\n\\bibitem{Maksym} P. A. Maksym, T. Chakraborty, Phys. Rev. Lett. \n{\\bf 65}, 108 (1990) and Phys. Rev. B {\\bf 45}, 1947 (1992);\\\\\nP.A. Maksym, Physica B {\\bf 184}, 385 (1993)\n\n\\bibitem{Hawrylak} P. Hawrylak and D. Pfannkuche, Phys. Rev. Lett. {\\bf 70},\n485 (1993);\\\\\nP. Hawrylak, Phys. Rev. Lett. {\\bf 71}, 3347 (1993)\n\n\\bibitem{Dharma} M. W. C. Dharma--wardana, J. Phys. Condens. Matter {\\bf7},\n 4095 (1995) ; and Phys. Rev. B {\\bf 51}, 1653 (1995)\n\n\\bibitem{McDonald} S.--R. E. Yang, A. H. MacDonald, and M. D. Johnson, Phys. Rev. Lett. {\\bf 71}, 3194 (1993); S. Mitra and A. H. MacDonald, Phys. Rev. B\n {\\bf 48}, 2005 (1993)\n \n\\bibitem{Laughlin} R.B. Laughlin, Phys. Rev. {\\bf 27}, 3383 (1983);\\\\\nR. B. Laughlin, in: {\\em The Quantum Hall Effect}, Eds. R.E.Prange and\nS.M.Girvin, Springer Verlag, New York (1987).\n\n\\bibitem{Jain-FQHE} J. K. Jain, in: {\\em Perspectives in Quantum Hall Effects},\nEd.s S. Das Sharma and A. Pinczuk, John Willey \\& Sons,(1996)\n\n\\bibitem{Jain-QD} J. K. Jain and T.Kawamura, Europhys. Lett. {\\bf 29},\n321 (1995); R. K. Kamilla and J. K. Jain, Phys. Rev. B {\\bf 52}, 2798 (1995);\nT.Kawamura and J. K. Jain, J. Phys. Condens. Matter {\\bf 8},2095 (1996)\n\n\\bibitem{Halperin} B.I. Halperin, in: {\\em Perspectives in Quantum Hall \nEffects},\nEd.s S. Das Sharma and A. Pinczuk, John Willey \\& Sons,( 1996)\n\n\\bibitem{Haldane} F. D. M. Haldane, in: {\\em The Quantum Hall Effect},\nSpringer Verlag, New York (1987).\n\n\\bibitem{MacDonald-2} A. H. MacDonald, in: {\\em Mesoscopic Quantum Physics},\nProceedings of the \nLes Houches Summer School , Elsevier (1995)\n\n\\bibitem{Taut2e} M. Taut, J. Phys. A{\\bf 27}, 1045 (1994)\nand erratum J.Phys.A{\\bf27}, 4723 (1994). Additionally, in formula (10)\nin the term containing $\\frac{\\partial}{\\partial\\alpha} $\na factor $\\frac{1}{2}$ is missing , and on the r.h.s. of (19a) and (20a)\n$\\tilde\\omega$ must be replaced by $\\tilde\\omega_r$.\n\n\\bibitem{Taut-unpublished} This fact is unpublished, but it is\ncompletely analogous to the case\nof two electrons in an external oscillator potential in 3 dimensional space.\nSee M.Taut, A.Ernst and H.Eschrig; J. Phys. B {\\bf 31},2689 (1998)\n\n\\bibitem{Ruan} W. Y. Ruan et al., Phys. Rev. B {\\bf 51}, 7942 (1995)\n\n\\bibitem{Seki} T. Seki, Y. Kuramoto, T. Nishino, J. Phys. Soc. Japan {\\bf 65},\n3945 (1996)\n\n\\bibitem{Maksym} P. A. Maksym, Phys. Rev. B {\\bf 53}, 10871 (1996) \n\n\\bibitem{one-e-dot}V. Fock, Z. Phys. {\\bf 47}, 446 (1928) (in German);\\\\\nC.G. Darwin, Proc. Cambridge Philos. Soc. {\\bf 27}, 86 (1930); \\\\\nR.B. Dingle, Proc. Roy. Soc. London A {\\bf 211}, 500 (1952)\n\n\\bibitem{Pauncz}R. Pauncz, {\\em Spin Eigenfunctions}, Plaenum Press,\nNew York and London 1979\n\n\\end{thebibliography}"
}
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cond-mat0002068
|
Dielectric response of charge induced correlated state in the quasi-one-dimensional conductor (TMTTF)$_2$PF$_6$
|
[
{
"author": "F.~Nad$^{1,2}$"
},
{
"author": "P.~Monceau$^1$"
},
{
"author": "C.~Carcel$^3$"
},
{
"author": "and J.M.~Fabre$^3$"
}
] |
Conductivity and permittivity of the quasi-one-dimensionsional organic transfer salt (TMTTF)$_2$PF$_6$ have been measured at low frequencies ($10^3-10^7$~Hz) between room temperature down to below the temperature of transition into the spin-Peierls state. We interpret the huge real part of the dielectric permittivity (up to 10$^6$) in the localized state as the realization in this compound of a charge ordered state of Wigner crystal type due to long range Coulomb interaction.
|
[
{
"name": "Monceau-PRBEditors.tex",
"string": "\n\\documentclass [10pt]{letter}\n\\usepackage{fancyhdr}\n\\thispagestyle{fancy}\n\\renewcommand{\\footrulewidth}{0pt}\n\\renewcommand{\\headrulewidth}{0pt}\n\n\\topmargin -0.2 cm\n\\headheight -2.5 cm\n\\oddsidemargin -1 cm\n\\evensidemargin 0 cm\n\\textwidth 18 cm\n\\textheight 25.8 cm\n\\footskip -2cm\n\n\\begin{document}\n\n\\cfoot{\n\\baselineskip 8pt\n{\\scriptsize{\\bf C\\ e\\ n\\ t\\ r\\ e\\ \\ \\ d\\ e\\ \\ \\ R\\ e\\ c\\ h\\ e\\ r\\ c\\ h\\ \ne\\ s\\ \\ \\ s\\ u\\ r\\ \\ \\ l\\ e\\ s\\ \\ \\ T\\ r\\ \\`e\\ s\\ \\ \\ B\\ a\\ s\\ s\\ \ne\\ s\\ \\ \\ T\\ e\\ m\\ p\\ \\'e\\ r\\ a\\ t\\ u\\ r\\ e\\ s}\\\\\n{\\bf\\em laboratoire associ\\'e \\`a l'Universit\\'e Joseph Fourier de \nGrenoble}\\\\\nC.N.R.S., 25 Avenue des Martyrs, BP 166, 38042 \nGrenoble-Cedex 9, France\\\\\nT�l. +33 (0)4 76 88 11 59 - Fax +33 (0)4 76 87 50 60\\\\\n{\\sf e-mail: monceau@labs.polycnrs-gre.fr - \nhttp://www.polycnrs-gre.fr/crtbt.html}}\\\\\n}\n\n\\bigskip\n\\baselineskip=.3cm\n\\vbox{\n\\hbox{\\hspace{2.3cm}{\\scriptsize {\\bf\\em Centre de Recherches sur les \nTr\\`es Basses Temp\\'eratures}}}\n\\hbox{\\hspace{2.3cm}{\\scriptsize {\\bf Grenoble}}}\n\\bigskip\n\\hbox{\\hspace{2.3cm}{\\scriptsize {\\bf Pierre MONCEAU}}}\n}\n\n\\vskip 0.5cm\n\\hspace{8.5cm}\\baselineskip 14pt\n\\begin{tabular}{l}\n\\mbox{}\\\\\nEditors of Physical Review B\\\\\n\\\\\n\\\\\n\\\\\n\n{}\\\\\n{}\n\\end{tabular}\n%\\noindent Ref: \n%\\noindent Objet:\n\\vspace{0.5cm}\n\n\\hspace{8.5cm} Grenoble, January 12, 2000\n\\vspace{1cm}\n\n\\hspace*{2.2cm}\\begin{minipage}{14cm}\nDear Editors,\n\n\\vspace{0.5cm}\nWe would like to submit our manuscript entitled\n{\\it ``Dielectric response of charge induced correlated state in\nquasi-one-dimensional conductor (TMTTF)$_2$PF$_6$''} for publication in Phys. \nRev. We previously submitted it to Phys. Rev. Lett. (code number LH7335).\n\n\\bigskip\nThe delay for resubmission has been caused by difficulties in\ncommunication between co-authors, one at Moscow, another visiting\nJapan during several weeks. We would like, however, the original\nreceived date to be retained. The manuscript has been amended\naccording to the referee comments as described below :\n\n\\bigskip\n\\noindent {\\bf Referee A}\n\n\\medskip\n\\noindent 1) We modified our description of the extended \nMott-Hubbard model for\nhalf-filled bands (bottom of page 2) taking into account both the\non-site interaction (U) and the near-neighbor interaction (V).\n\n\\medskip\n\\noindent 2) The $2k_F$ lattice fluctuations coupled to the \nlattice are observed at much\nlower temperature (below 60~K) than the large increase of the\ndielectric permittivity. We have rewritten the beginning of the\ndiscussion part, page 6, to make this point more clear.\n\n\\bigskip\n\\noindent {\\bf Referee B}\n\n\\medskip\n\\noindent 1) These organic conductors are very fragile and cracks may reflect\nsignificant strain when the cooling rate is too fast and/or the sample\nholding too tight. We used all the care possible for avoiding cracks\nin our crystals and thus measuring bulk properties.\n\n\\medskip\n\\noindent 2) There is no evidence in (TMTTF)$_2$PF$_6$ of a structural phase\ntransition affecting the main Bragg reflections. The tentative\nexplanation of our data is the occurrence of a superstructure ($4k_F$)\ntransition. Such a transition could be very likely detected from NMR\nexperiments.\n\n\\bigskip\nYours sincerely,\n\n\\vspace{1.5cm}\n\\hspace{8.5cm}{P. MONCEAU}\n\n\\end{minipage}\n\n\n\\end{document}\n\n"
},
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"name": "NAD_TMTTF.tex",
"string": "\\input{epsf}\n\\documentstyle[preprint,aps,amssymb]{revtex}\n%\\documentstyle[amssymb,prl,aps,multicol,epsf]{revtex}\n%\\renewcommand{\\narrowtext}{\\begin{multicols}{2} \\global\\columnwidth20.5pc}\n%\\renewcommand{\\widetext}{\\end{multicols} \\global\\columnwidth42.5pc}\n%\\multicolsep = 8pt plus 4pt minus 3pt\n\\draft\n\n\\begin{document}\n\\title{Dielectric response of charge induced correlated state in the\nquasi-one-dimensional conductor (TMTTF)$_2$PF$_6$}\n\\author{F.~Nad$^{1,2}$, P.~Monceau$^1$, C.~Carcel$^3$, and J.M.~Fabre$^3$}\n\\address{$^1$Centre de Recherches sur les Tr\\`es Basses\nTemp\\'eratures,\\\\\nlaboratoire associ\\'e \\`a l'Universit\\'e Joseph Fourier, CNRS,\nBP 166, 38042 Grenoble cedex 9, France\\\\\n$^2$Institute of Radioengineering and Electronics,\\\\\nRussian Academy of\nSciences, Mokhovaya 11, 103907 Moscow, Russia\\\\\n$^3$Laboratoire de Chimie Structurale Organique,\\\\\nUniversit\\'e de Montpellier, 34095 Montpellier cedex 5, France}\n\\maketitle\n\n\\begin{abstract}\nConductivity and permittivity of the quasi-one-dimensionsional \norganic transfer salt (TMTTF)$_2$PF$_6$ have been measured at low \nfrequencies ($10^3-10^7$~Hz) between room temperature down to below \nthe temperature of transition into the spin-Peierls state. We \ninterpret the huge real part of the dielectric permittivity (up to \n10$^6$) in the localized state as the realization in this compound of \na charge ordered state of Wigner crystal type due to long range \nCoulomb interaction.\n\\end{abstract}\n\n\\pacs{71.30+h, 71.10Hf, 77.22Gm, 75.30Fv}\n\n%\\narrowtext\n\n\\section{Introduction\\protect\\\\}\n\nConductors formed of linear chains of organic mole\\-cules\ntetramethyltetrathiafulvalene (TMTTF) or\ntetramethyltetraselenafulvalene (TMTSF) with a general formula\n(TMTCF)$_2$X with C~= Se, S and X the interchain counterion~= ClO$_4$,\nPF$_6$, Br, \\ldots have been intensively studied these last years\nbecause they exhibit a rich variety of cooperative phenomena\nincluding superconductivity, antiferromagnetism (AF), spin density\nwave (SDW), spin-Peierls state (SP), charge localization\n\\cite{JeromeSSC94}. While the Bechgaard (TMTSF)$_2$X salts\ndisplay a metallic behaviour down to low temperature where a\ntransition in a SDW state occurs below $\\sim 12$~K, (TMTTF)$_2$X salts\nexhibit a charge localization in the temperature range of 100--200~K,\nwith a maximum in conductivity at $T_\\rho$ and a thermally activated\nvariation below $T_\\rho$ \\cite{CoulonJP82,LaversanneJPL84}, revealing\nstrong Coulomb interaction effects in these sulfur salts.\n\nQuasi-one-dimensional conductor (TMTTF)$_2$X consists of\nmolecular chains (along the highest conductivity axis) with two\nelectrons per 4 molecules which corresponds to 1/4 filling in terms\nof holes. These molecular chains are slightly dimerized due to\nintermolecular interaction. As a result, with decreasing temperature\na dimerized gap $\\Delta\\rho$ opens with effective 1/2 filling\nof the upper conduction band. Two intermolecular transfer integrals\nalong molecular stacks t$_1$ and t$_2$ have to be considered\n\\cite{FritschJPI91}. Quantum chemistry calculations show that\nin (TMTTF)$_2$PF$_6$ the dimerization decreases with decreasing\ntemperature and at low temperatures t$_1$/t$_2$ ratio is\nabout 1.1 - 1.2 \\cite{FritschJPI91}.\n\nAs was shown in several of theoretical and experimental works,\nbetween many factors which determine the properties of\n(TMTTF)$_2$X type salts, electron-electron correlation phenomena play\nthe leading part \\cite{JeromeSSC94,FritschJPI91,EmeryPRL82,LeePRB77,%\nPougetPRL76,%\nSchulzIJMPB91,HirschPRB83,CaronPRB84,PencJPIV91,MilaPRB95,SeoJPSJ97}. In\nthis context two types of theoretical models have been essentially\ndeveloped. In the frame of the so-called g-ology models \\cite{SchulzIJMPB91,%\nCaronPRB84,BourbonnaisSM97} the electron-electron correlations are\nconsidered as a perturbation to the one-electron approach. These models have\nbeen used for describing the low energy properties of these salts which\nexhibit the features of a Luttinger liquid rather\nthan a Fermi liquid \\cite{BourbonnaisSM97}.\n\nThe second group of models includes the various versions of the Hubbard model\n\\cite{HirschPRB83,CaronPRB84,PencJPIV91,MilaPRB95,SeoJPSJ97,ClayPRB59}.\nExtended Hubbard model takes\ninto account the interaction between charge carriers on the site of\nhost lattice (on-site interaction) with characteristic energy U as\nwell as the interaction between charge carriers on the neighboring\nsites (near-neighbor interaction) with characteristic energy V.\nIn the case of (TMTTF)$_2$X compounds, at temperatures above the\ntransition into spin ordered states, the electron-electron interaction\nis determined by long range Coulomb interaction and it is stronger\nthan spin interaction. This is one of the reason of spin-charge\nseparation observed in such 1D-conductors \\cite{JeromeSSC94}. In\nthe frame of extended Hubbard model it was shown that the dimerized\nenergy gap is strengthened taking into account the on-site and\nnear-neighbor interactions \\cite{HirschPRB83}. At the same time\nthe spectrum of spin exitations remains gapless which also\ncorresponds to spin-charge separation.\n\nOne important result of the extended Hubbard model approach\nconcerns the formation of a $4k_F$ CDW superlattice of\nWigner crystal type in such 1D compounds with decreasing temperature.\nUsing Monte Carlo technique it was shown that, for large enough U\nand V magnitudes, strictly on-site interaction results only in a weak\n$4k_F$ CDW, while long range Coulomb near-neighbor interaction\ncan produce a CDW singularity at $4k_F$ \\cite{HirschPRB83}.\nAnalogously, using mean field approximation \\cite{SeoJPSJ97}\nit was recently shown that for one-dimensional molecular chain with\nand without dimerization the form of the developed superstructure\ndepends considerably on the magnitude of near-neighbor interaction V:\nat V above some critical value V$_c$ a $4k_F$ CDW superstructure\noccurs with charge disproportionation depending on V.\nEstimations of V/U and V/t$_2$ magnitudes in (TMTTF)$_2$PF$_6$ obtained\nfrom quantum chemistry calculations and from optical conductivity\n\\cite{FritschJPI91,MilaPRB95} yields values for V/U in the range 0.4-0.5\nand for V/t$_2$ in the range of 2-3 manifesting the essential role played\nby the long range Coulomb interaction in this compound. Hereafter we\npresent results of conductivity and dielectric permittivity measurements\nof (TMTTF)$_2$PF$_6$ which provide some evidence of a charge modulated\nstate (analogous to Wigner crystal) resulting from electron-electron\ncharge correlation below $T_\\rho$.\n\n\\section{Experiment}\n\nWe have studied (TMTTF)$_2$PF$_6$ samples originating from two\nbatches. The crystals have been prepared using standard electrochemical\nprocedures \\cite{DelhaesMCLC79}. Electrical contacts were prepared by\nfirst evaporating gold pads on nearly the whole surface of the sample's ends\non which thin gold wires were attached afterwards with silver paste. We\nhave carried out the measurements of complex conductance $G(T,\\omega)$,\nusing an impedance analyser HP~4192A in the frequency range\n10$^3$--10$^7$~Hz and in the temperature range 4.2--295~K. The\namplitude of the ac voltage applied to the sample was within the\nlinear response and typically 30~mV/cm. We have noticed that the\ncooling rate had a significant effect on the results of our\nmeasurements, much more important than in the case of (TMTTF)$_2$Br\nfor instance \\cite{NadEPJB98}: at a cooling rate above 0.5~K/mm,\ncracks appear as seen in jumps in the temperature variation of $G$.\nHowever with a slow cooling rate around 0.2~K/mm and along temperature\nstabilisation before performing measurements, we succeeded in\nrecording the temperature dependences of real and imaginary parts of $G$\nwithout any jumps for 3 samples. These samples have a length of 3--4~mm,\na cross-section about $2\\times 10^{-5}-10^{-4}$~cm$^2$ and a room\ntemperature conductivity $\\sim 40~\\Omega^{-1}$cm$^{-1}$. The results\nobtained for these 3 samples being qualitatively similar and we will\npresent the data for two of them (referred as samples 1 and 2).\n\nFig.\\ref{fig1} shows the variation of the conductance $G(T)$\nof sample~1 normalized by its maximum value $G_m$ as a function of the\ninverse temperature. The detailed $G(T)$ dependence near the room\ntemperature is shown in inset~(a) of Fig.\\ref{fig1}. With\ndecreasing temperature, the conductance of (TMTTF)$_2$PF$_6$ first\ngrows up to a maximum at $T_\\rho$~= 250~K as previously reported in\n\\cite{CoulonJP82,LaversanneJPL84}. Below $T_\\rho$, the decrease of\n$G(T)$ in the temperature range 200--70~K follows an Arrhenius type\nbehavior with an activation energy $\\Delta\\rho\\simeq 300$~K. The value\nof this activation energy is in good agreement with the evaluation of\ncharge gap in \\cite{PencJPIV91}. It was shown\nthat $\\Delta\\rho=1/4(t_1+t_2)$ which provide $\\Delta\\rho\\simeq 300$~K\nfor (TMTTF)$_2$PF$_6$.\nAt the same time this magnitude of energy gap is two times smaller\nthan the value reported in a previous publication \\cite{CoulonJP82}.\nThis difference can be a result of the gap determination in\n\\cite{CoulonJP82} from $G(1/T)$ dependence obtained by averaging of\n$G(1/T)$ dependencies from several different samples with jumps of $G$\ndue to cracks.The jump-like decrease of conduction at every crack\nresults in more steep averaged $G(1/T)$ dependence and accordingly\nin more higher gap magnitude. The small cooling rate enables us to\navoid cracks and to determine more accurately the energy gap\nmagnitude which agrees with theoretical evaluation \\cite{PencJPIV91}.\n\nAt lower temperature we observed a bend on the $G(1/T)$ dependence near\n70~K, i.e. $G$ begins to decrease more faster (activation energy \n$\\approx$~380~K) with a following gradual\ntransition to a new activation regime at temperatures below 25~K. Detailed\nmeasurements of sample 2 show that the decrease of $G$ is thermally\nactivated between 4.2~K and 20~K with an activation energy $\\simeq 36$~K.\nThis energy gap satisfied the BCS-like relation $\\Delta=1.75kT_{\\rm SP}$\nwith $T_{\\rm SP} \\simeq 20$~K as previously shown in the temperature\ndependence of the EPR spin susceptibility and the nuclear relaxation\nrate $T_1^{-1}$ \\cite{WzietekJPI93}.\n\nIt can also be seen in Fig.\\ref{fig1} that the frequency\ndependence of $G(1/T)$ become noticeable below 70~K with a frequency\ndispersion growing at lower temperature. For showing more precisely\nthe particular points on the $G(1/T)$ variation we draw in inset~(b)\nof Fig.\\ref{fig1} the temperature variation of the logarithmic\nderivative which was reproducible for our three samples.\nThe decrease of $G$ near $T\\simeq 70$~K may indicate the beginning \nof a transition into some new ground state.\nTwo additional small minima in $d\\log G/d(1/T)$ are also visible\nnear 140~K and 100~K (inset~(b) in Fig.\\ref{fig1}.\n\nThe temperature variation of the real part of the dielectric\npermittivity $\\epsilon^\\prime(T)$ of sample~1 in the frequency range\n$10^3-5\\times 10^6$~Hz is shown in Fig.\\ref{fig2}.\nBetween 300~K and 220~K, the magnitude of $\\epsilon^\\prime$ is below\nthe background level determined by the resolution of our measurements\nin this temperature range. The growth of $\\epsilon^\\prime$ is\nnoticeable below $\\simeq 200$~K and frequency independent down to\n$T\\simeq 110$~K. Below this temperature a significant frequency\ndispersion occurs as seen in Fig.\\ref{fig2}: at a given frequency\n$\\epsilon^\\prime(T)$ goes through a maximum before falling down. With\ndecreasing frequency, the amplitude of the maximum of\n$\\epsilon^\\prime(T)$ is larger and the maximum position on the\ntemperature scale shifts to lower temperature. This behavior is\nqualitatively similar to critical slowing down phenomena near a phase\ntransition. The $\\epsilon^\\prime(T)$ curves for samples 1 and 2 are\nshown in inset of Fig.\\ref{fig2} in a double logarithmic scale,\nmanifesting their qualitative similarity. In the temperature range\n200--70~K, the $\\epsilon^\\prime(T)$ dependence can be described by\nthe power law: $\\epsilon^\\prime(T)\\sim T^{-\\alpha}$ with\n$\\alpha\\simeq 1/3$. While $\\epsilon^\\prime(T)$\nis decreasing from 70~K down 10~K, a small bump in\n$\\epsilon^\\prime(T)$ can be seen near $\\simeq 35$~K.\n\nThe frequency dependencies of the conductance, of the real\npart $\\epsilon^\\prime$ and of the imaginary part\n$\\epsilon^{\\prime\\prime}$ of the dielectric permittivity of\n(TMTTF)$_2$PF$_6$ have a form similar to those we previously measured\non other 1D organic compounds \\cite{NadEPJB98,NadSSC95}. As usual, the\nfrequency, $f_m$, corresponding to the maximum of\n$\\epsilon^{\\prime\\prime}(T)$\ncorresponds to some mean value of the relaxation time $\\tau=1/2\\pi\nf_m$ of charge polarization. The variation of $\\tau$ with the inverse\ntemperature is drawn in Fig.\\ref{fig3}\nfor sample~1 in the temperature rangee 95--35~K: for $60<T<95$~K,\n$\\tau(1/T)$ is thermally activated with an energy activation of $\\sim\n650$~K; but, at lower $T$, 35~K$<T<$60~K, the activation energy is \nsmaller, $\\approx$~380~K, the same as the activation energy of the \nconductivity in same temperature range.\n\n\\section{Discussion}\n\nAs follows from published data \\cite{JeromeSSC94,WzietekJPI93}\nand from our experimental results, (TMTTF)$_2$PF$_6$ can be\ncharacterized by two distinct energy scales: temperature\n$T_\\rho$~= 250~K corresponding to the conduction maximum and\n$T_{\\rm SP} \\simeq 20$~K corresponding to the transition into\nthe spin-Peierls state. If one will consider the temperatures\n$T_{\\rm SP}<T<T_\\rho$ as a range of simple localization of\ncharge carriers \\cite{JeromeSSC94}, one could try to explain\nthe observed growth of $\\epsilon^\\prime$ as a result of the growth\nof $2k_F$ CDW fluctuations when approaching $T_{\\rm SP}$\n\\cite{PougetJP96}. As was mentioned in \\cite{SchulzIJMPB91}\nthe temperature range of fluctuations near the spin-Peierls\ntransition can be wide enough and reach $\\sim 3T_{\\rm SP}$.\nIt means that in (TMTTF)$_2$PF$_6$ the manifestation of\nthe fluctuations can be noticed beginning from\n$T=T_{\\rm SP} + 3 \\times T_{\\rm SP} \\simeq 80$~K, temperature at\nwhich $2k_F$ fluctuations have started to be really observed\n\\cite{PougetJP96}. However as can be seen from Fig.\\ref{fig2}\nthe $\\epsilon^\\prime$ growth begins nevertheless not from 80~K\nbut from $\\simeq 200$~K. For frequencies above 10$^6$~Hz this\n$\\epsilon^\\prime$ growth is rather achieved at 80~K. This\ndismatching in the temperature ranges for occurence of $2k_F$\nfluctuations and the $\\epsilon^\\prime$ growth makes difficult\nthe explanation of $\\epsilon^\\prime$ growth as a result of\n$2k_F$ CDW fluctuations. In the same temperature range\n50 -- 200~K, EPR susceptibility $\\chi$ decreases monotonously\nwithout any maximum \\cite{WzietekJPI93}.A small maximum on\n$\\chi(T)$ dependence was observed near 40~K with a following\ndecrease of $\\chi$ as a result of the transition to the spin-Peierls\nstate. Such considerable qualitative difference between\n$\\epsilon^\\prime(T)$ and $\\chi(T)$ dependences confirms\nthe existence of spin-charge separation in (TMTTF)$_2$PF$_6$ salts.\n\nWe tentatively ascribe the frequency and temperature dependences\nof $\\epsilon^\\prime$ of (TMTTF)$_2$PF$_6$ in the temperature range\n$T_{\\rm SP}<T<T_\\rho$ as related to the charge induced correlation\nphenomena discussed above. Taking into account only on-site interactions\nin the Hubbard model for half-filled band, the commensurate charge \ninduced superstructure ($4k_F$ CDW) is strictly linked to the host\nlattice. In such a case the possibility of polarization of\nthe superstructure, i.e. its shift with respect to the host lattice,\nis small and consequently one would expect a low magnitude of\nthe dielectric permittivity. Ground state with Mott-Hubbard gap and\nappropriate charge localization have been realized in 3D semiconductors\n\\cite{Mott74}. In such compounds $\\epsilon^\\prime$ is in the order of\n10, a typical value for usual semiconductors, while in our samples of\n(TMTTF)$_2$PF$_6$ the $\\epsilon^\\prime$ magnitudes amount by several\norders of value larger (Fig.\\ref{fig2}).\n\nAs explained above, long range Coulomb interaction of sufficient\nstrength and appropriate charge induced correlation may lead to\na superstructure with charge disproportionation corresponding to a $4k_F$\nCDW as in a Wigner crystal. For (TMTTF)$_2$PF$_6$ this charge\ndisproportionation can be evaluated on the base of calculations in\n\\cite{SeoJPSJ97} taking into account the estimated magnitude of\nthe reduced near-neighbor Coulomb interaction $V/t_2 \\simeq 2$ \\cite\n{FritschJPI91,MilaPRB95}.For such a $V/t_2$ value we estimate the charge\ndisproportionation as about 1:3. Such a degree of disproportionation and\nthe possibility of its variation with temperature \\cite{FritschJPI91},\nprovide some evidence that this charge supertructure is probably more soft,\nmore weakly connected to the host lattice and consequently more easily\npolarizable than in the case of only on-site interaction. We ascribe\nthe large magnitude of the dielectric permittivity, which we have found\nout below $T_\\rho$, to the collective response of such charge superstructure\nof Wigner crystal type with charge disproportionation formed\nin (TMTTF)$_2$PF$_6$. Indeed, as can be seen from Fig.\\ref{fig2} in the\ntemperature range above $T_\\rho$ the $\\epsilon^\\prime$ magnitude does not\nexceed the background level. Its noticable growth begins below 200~K when,\nas we considered, the growing of this Wigner type CDW superstructure\nbegins to determine the kinetic properties of the compound.\nThe $\\epsilon^\\prime$ growth with temperature decreasing is probably\nassociated with the gradual enhancement of the CDW superstructure. Possible\nreasons for such an enhancement can be the growth of intrachain charge induced\ncorrelations as well as the growth of interchain interactions. Both of\nthem favour the three dimensional ordering of the CDW superstructure,\ni.e. the formation of a 3D electronic crystal. The maximum value of\n$\\epsilon^\\prime$ amounts to 10$^5$ -- 10$^6$, 2 or 3 orders of magnitude\nlower than the $\\epsilon^\\prime$ values in incommensurate charge\n\\cite{NadPR} and spin \\cite{NadSSC95} density wave below their\ntransition temperature. However the magnitude of $\\epsilon^\\prime$ in\n(TMTTF)$_2$PF$_6$ is nearly comparable with that in (TMTTF)$_2$Br\n\\cite{NadEPJB98}.\n\nAccording to X-ray measurements \\cite{PougetJP96}, diffuse\n$2k_F$ scattering grows critically in (TMTTF)$_2$PF$_6$ below $\\sim\n80$~K which originates from the gradual enhancement of spin induced\nelectron-electron correlation of antiferromagnetic type. Due to the\ngrowth of the electron-phonon interaction and of the increasing of\nthe interchain interaction, these $2k_F$ spin induced correlations\ndiverge below $T_{\\rm SP}$ resulting in the condensation of an\nordered spin-Peierls state with a $2k_F$ superstructure.\n\nThe opening of the spin-Peierls energy gap and the two-fold commensurability\nof the superlattice lead to the freezing of charge polarization degrees of freedom\nand consequently to the sharp decrease of $\\epsilon^\\prime$. The slowing down\nbehavior of $\\epsilon^\\prime$ and the temperature dependence of the relaxation\ntime (Fig.3) also indicate the lattice involvment (i.e. heavy molecules)\nin the relaxation process.\n\n\\section{Conclusion}\n\nIn conclusion, our measurements of the complex conductivity of \n(TMTTF)$_2$PF$_6$ show the main following\nfeatures: 1)~development of a charge energy gap the magnitude of which\ncorresponds to theoretical evaluations in the frame of extended Hubbard\nmodel; 2)~existence of peculiarities on $G(1/T)$ dependence, for example,\na minimum of the logarithmic derivative near 60~K; 3)~in the same temperature\nrange, finding of a huge maximum of the real part of the dielectric permittivity\n(up to 10$^6$) with a slowing down behavior with decreasing temperature\nwhile the magnetic susceptibility does not show any significant\nvariation, which corresponds to large charge\npolarization simultaneously with spin-charge separation; 4)~considerable\ndifference between the temperature dependence of the dielectric permittivity and\n$2k_F$ diffuse X-ray scattering.\n\nAll these features seem to confirm\nthe possibility of the formation in the temperature range\n$T_{\\rm SP}<T<T_\\rho$ of a charge ordered state with a high\npolarizability. On the basis of our experimental results and some theoretical\napproaches we consider that we consider that the huge amplitude of \nthe real part of the dielectric permittivity of (TMTTF)$_2$PF$_6$ can \nhardly be provided by a Mott-insulator. On the contrary, we argue that \nthis large dielectric polarizability reflects the collective response \nof $4k_F$ charge density wave of Wigner crystal type due to long range\nCoulomb interaction and electron-electron correlation.\nRecently charge disproportionation \\cite{HirakiPRL98} and\n$4k_F$ superlattice \\cite{NogamiSM99} have been reported from NMR and\nX-ray measurements in a 1/4 filled one-dimensional organic compound\n(DI-DCNQI)$_2$Ag without dimerization.\n\n{\\bf Acknowledgments}\n\nWe would like to thank S.~Brazovskii and N.~Kirova for helpful\ndiscussions, and D.~Staresinic for help in the experiment. Part of\nthis work was supported by the Russian Fund for Fundamental Research\n(grant N$^\\circ$~99-02-17364) and the twinning research programme\nN$^\\circ$~19 (grant N$^\\circ$~98-02-22061) between CRTBT-CNRS and\nIRE-RAS.\n\n\\begin{references}\n\n\\bibitem{JeromeSSC94} D.~J\\'erome, Solid State Commun. {\\bf 92},\n89 (1994); D.~Jerome, P.~Auban-Senzier, L.~Balicas {\\it et al.},\nSynth. Metals {\\bf 70}, 719 (1995).\n\n\\bibitem{CoulonJP82} C. Coulon, P. Delhaes, S.~Flandrois {\\it et\nal.}, J. Physique {\\bf 43}, 1059 (1982).\n\n\\bibitem{LaversanneJPL84} R. Laversanne, C. Coulon, B.~Gallois {\\it\net al.}, J. Physique Lett. {\\bf 45}, L393 (1984).\n\n\\bibitem{FritschJPI91} A. Fritsch and L. Ducasse, J. Physique I\n(France) {\\bf 1}, 855 (1991); F.~Castet, A.~Fritsch, and L.~Ducasse,\nJ. Physique I (France) {\\bf 6}, 583 (1996).\n\n\\bibitem{EmeryPRL82} V.J. Emery, R.~Bruisma, and S.~Barisic, Phys.\nRev. Lett {\\bf 48}, 1039 (1982).\n\n\\bibitem{LeePRB77} P.A. Lee, T.M.~Rice, and R.A.~Klemm, Phys. Rev.\nB {\\bf 15}, 2984 (1977).\n\n\\bibitem{PougetPRL76} S.K.~Khanna, J.P. Pouget, R.~Comes, A.F.~Garito, \nand A.J.~Heeger, Phys. Rev. B {\\bf 16},\n1468 (1977); S.~Kagoshima, T.~Ishiguro, and H. Anzai, J. Phys. \nSoc. Jpn {\\bf 41}, 2061 (1976).\n\n\\bibitem{SchulzIJMPB91} H.J. Schulz, Int. J. of Modern Phys. B\n{\\bf 5}, 57 (1991); in {\\it Low-Dimensional Conductors and\nSuperconductors}, Ed. by D.~Jerome and L.~Caron, NATO ASI Series~B:\nPhysics v.155, p.95.\n\n\\bibitem{HirschPRB83} J.E.Hirsch and D.J.~Scalapino, Phys. Rev. B\n{\\bf 27}, 7169 (1983); Phys. Rev. B {\\bf 29}, 5554 (1984).\n\n\\bibitem{CaronPRB84} L.G.~Caron and C.~Bourbonnais, Phys. Rev.\nB {\\bf 29}, 4230 (1984).\n\n\\bibitem{PencJPIV91} K. Penc and F. Mila, J. Physique IV, Col.\n{\\bf C2}, 155 (1991); Phys. Rev. B {\\bf 50}, 11429 (1994).\n\n\\bibitem{MilaPRB95} F. Mila, Phys. Rev. B {\\bf 52}, 4788 (1995).\n\n\\bibitem{SeoJPSJ97} H. Seo and H.~Fukuyama, J. Phys. Soc. Jpn {\\bf\n66}, 1249 (1997).\n\n\\bibitem{BourbonnaisSM97} C. Bourbonnais, Synth. Metals {\\bf\n84}, 19 (1997); C.~Bourbonnais and D.~J\\'erome, in {\\it Advances in \nSynthetic Metals}, Ed. by P.~Bernier, S.~Lefrant, and G.~Bidan, \nElsevier (Lausanne), 1999, p. 206.\n\n\\bibitem{ClayPRB59} R.~Clay, A.~Sandvic and D.~Campbell, Phys. Rev.\n{\\bf 59}, 4665 (1999).\n\n\\bibitem{DelhaesMCLC79} P. Delhaes, C.~Coulon, J.~Amiell,\nS.~Flandrois, E.~Tororeilles, J.M.~Fabre, and L.~Giral, Mol.\nCryst. Liq. Cryst. {\\bf 50}, 43 (1979).\n\n\\bibitem{NadEPJB98} F. Nad, P.~Monceau, and J.M.~Fabre, Eur. Phys.\nJ. B {\\bf 3}, 301 (1998).\n\n\\bibitem{WzietekJPI93} P. Wzietek, F.~Creuzet, C.~Bourbonnais,\nD.~J\\'erome, K.~Bechgaard, and P.~Batail, J. Phys. I (France) {\\bf\n3}, 171 (1993).\n\n\\bibitem{NadSSC95} F. Nad, P.~Monceau, and K.~Bechgaard, Solid\nState Commun. {\\bf 95}, 655 (1995).\n\n\\bibitem{PougetJP96} J.P.Pouget and S.~Ravy, J. Physique I {\\bf 6},\n1501 (1996); Synth. Metals {\\bf 85}, 1523 (1997).\n\n\\bibitem{Mott74} N.F.Mott, {\\it Metal-Isulator Transition} (Taylor\nand Francis Ltd, London 1974).\n\n\\bibitem{NadPR} F. Nad and P.~Monceau, Phys. Rev. B {\\bf 51}, 2052\n(1995).\n\n\\bibitem{HirakiPRL98} K. Hiraki and K.~Kanada, Phys. Rev.\nLett. {\\bf 80}, 4737 (1998).\n\n\\bibitem{NogamiSM99} Y. Nogami, K.~Oshima, K.~Hiraki, and K.~Kanoda, \nJ.~Phys.~IV (France) {\\bf 9}, Pr10--357 (1999).\n\n\\end{references}\n\n\\begin{figure}[t]\n\\centerline{\\epsfxsize=12cm \\epsfbox{NadFig1.eps}}\n\\caption{Variation of the real part of the conductance $G$ (sample~1)\nnormalized to its maximum $G_m$ as a function of the inverse\ntemperature at frequencies (in~kHz): $\\lozenge$~10,\n{\\Large $\\bullet$}~100,\n$\\oplus$~1000. Inset~(a): details of the temperature dependence\nof $G/G_m$ near the maximum. Inset~(b): temperature dependence of the\nlogarithmic derivative $d\\log G/d(1000/T)$.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=14.5cm \\epsfbox{NadFig2.eps}}\n\\caption{Temperature dependence of the real part of the dielectric\npermittivity $\\epsilon^\\prime$ (sample~1) at frequencies (in~kHz):\n$\\oplus$~1, $\\blacktriangle$~10, {\\Large $\\bullet$}~100,\n$\\blacksquare$~1000,\n$\\blacklozenge$~5000. Inset: temperature dependence of\n$\\epsilon^\\prime$ for sample~1 ({\\Large $\\bullet$}) and sample~2~($\\odot$) at\n100~kHz in a double logarithmic scale.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfxsize=14.5cm \\epsfbox{NadFig3.eps}}\n\\caption{Temperature dependence of the relaxation time $\\tau$ of the\ndielectric relaxation (sample~1).}\n\\label{fig3}\n\\end{figure}\n\n\\newpage\n\n%\\widetext\n\n\\end{document}\n\n\n"
}
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[
{
"name": "cond-mat0002068.extracted_bib",
"string": "\\bibitem{JeromeSSC94} D.~J\\'erome, Solid State Commun. {\\bf 92},\n89 (1994); D.~Jerome, P.~Auban-Senzier, L.~Balicas {\\it et al.},\nSynth. Metals {\\bf 70}, 719 (1995).\n\n\n\\bibitem{CoulonJP82} C. Coulon, P. Delhaes, S.~Flandrois {\\it et\nal.}, J. Physique {\\bf 43}, 1059 (1982).\n\n\n\\bibitem{LaversanneJPL84} R. Laversanne, C. Coulon, B.~Gallois {\\it\net al.}, J. Physique Lett. {\\bf 45}, L393 (1984).\n\n\n\\bibitem{FritschJPI91} A. Fritsch and L. Ducasse, J. Physique I\n(France) {\\bf 1}, 855 (1991); F.~Castet, A.~Fritsch, and L.~Ducasse,\nJ. Physique I (France) {\\bf 6}, 583 (1996).\n\n\n\\bibitem{EmeryPRL82} V.J. Emery, R.~Bruisma, and S.~Barisic, Phys.\nRev. Lett {\\bf 48}, 1039 (1982).\n\n\n\\bibitem{LeePRB77} P.A. Lee, T.M.~Rice, and R.A.~Klemm, Phys. Rev.\nB {\\bf 15}, 2984 (1977).\n\n\n\\bibitem{PougetPRL76} S.K.~Khanna, J.P. Pouget, R.~Comes, A.F.~Garito, \nand A.J.~Heeger, Phys. Rev. B {\\bf 16},\n1468 (1977); S.~Kagoshima, T.~Ishiguro, and H. Anzai, J. Phys. \nSoc. Jpn {\\bf 41}, 2061 (1976).\n\n\n\\bibitem{SchulzIJMPB91} H.J. Schulz, Int. J. of Modern Phys. B\n{\\bf 5}, 57 (1991); in {\\it Low-Dimensional Conductors and\nSuperconductors}, Ed. by D.~Jerome and L.~Caron, NATO ASI Series~B:\nPhysics v.155, p.95.\n\n\n\\bibitem{HirschPRB83} J.E.Hirsch and D.J.~Scalapino, Phys. Rev. B\n{\\bf 27}, 7169 (1983); Phys. Rev. B {\\bf 29}, 5554 (1984).\n\n\n\\bibitem{CaronPRB84} L.G.~Caron and C.~Bourbonnais, Phys. Rev.\nB {\\bf 29}, 4230 (1984).\n\n\n\\bibitem{PencJPIV91} K. Penc and F. Mila, J. Physique IV, Col.\n{\\bf C2}, 155 (1991); Phys. Rev. B {\\bf 50}, 11429 (1994).\n\n\n\\bibitem{MilaPRB95} F. Mila, Phys. Rev. B {\\bf 52}, 4788 (1995).\n\n\n\\bibitem{SeoJPSJ97} H. Seo and H.~Fukuyama, J. Phys. Soc. Jpn {\\bf\n66}, 1249 (1997).\n\n\n\\bibitem{BourbonnaisSM97} C. Bourbonnais, Synth. Metals {\\bf\n84}, 19 (1997); C.~Bourbonnais and D.~J\\'erome, in {\\it Advances in \nSynthetic Metals}, Ed. by P.~Bernier, S.~Lefrant, and G.~Bidan, \nElsevier (Lausanne), 1999, p. 206.\n\n\n\\bibitem{ClayPRB59} R.~Clay, A.~Sandvic and D.~Campbell, Phys. Rev.\n{\\bf 59}, 4665 (1999).\n\n\n\\bibitem{DelhaesMCLC79} P. Delhaes, C.~Coulon, J.~Amiell,\nS.~Flandrois, E.~Tororeilles, J.M.~Fabre, and L.~Giral, Mol.\nCryst. Liq. Cryst. {\\bf 50}, 43 (1979).\n\n\n\\bibitem{NadEPJB98} F. Nad, P.~Monceau, and J.M.~Fabre, Eur. Phys.\nJ. B {\\bf 3}, 301 (1998).\n\n\n\\bibitem{WzietekJPI93} P. Wzietek, F.~Creuzet, C.~Bourbonnais,\nD.~J\\'erome, K.~Bechgaard, and P.~Batail, J. Phys. I (France) {\\bf\n3}, 171 (1993).\n\n\n\\bibitem{NadSSC95} F. Nad, P.~Monceau, and K.~Bechgaard, Solid\nState Commun. {\\bf 95}, 655 (1995).\n\n\n\\bibitem{PougetJP96} J.P.Pouget and S.~Ravy, J. Physique I {\\bf 6},\n1501 (1996); Synth. Metals {\\bf 85}, 1523 (1997).\n\n\n\\bibitem{Mott74} N.F.Mott, {\\it Metal-Isulator Transition} (Taylor\nand Francis Ltd, London 1974).\n\n\n\\bibitem{NadPR} F. Nad and P.~Monceau, Phys. Rev. B {\\bf 51}, 2052\n(1995).\n\n\n\\bibitem{HirakiPRL98} K. Hiraki and K.~Kanada, Phys. Rev.\nLett. {\\bf 80}, 4737 (1998).\n\n\n\\bibitem{NogamiSM99} Y. Nogami, K.~Oshima, K.~Hiraki, and K.~Kanoda, \nJ.~Phys.~IV (France) {\\bf 9}, Pr10--357 (1999).\n\n"
}
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cond-mat0002069
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Robust half-metallicity and metamagnetism in Fe$_{x}$Co$_{1-x}$S$_{2}$
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[
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"author": "I.I. Mazin"
}
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The Fe$_{x}$Co$_{1-x}$S$_{2}$ system is predicted, on the basis of density functional calculations, to be a half metal for a large range of concentrations. Unlike most known half metals, the half metallicity in this system should be very stable with respect to crystallographic disorder and other types of defects. The endmember of the series, CoS$_{2},$ is not a half metal, but exhibits interesting and unusual magnetic properties which can, however, be reasonably well understood within the density functional theory, particularly with the help of the extended Stoner model. Calculations suggest strong electron-phonon and electron-magnon coupling in the system, and probably a bad metal behavior at high temperatures.
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[
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"name": "cos2a.tex",
"string": "\\documentstyle[amssymb,multicol,epsfig,aps,prl]{revtex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%TCIDATA{Created=Tue Nov 30 17:45:06 1999}\n%TCIDATA{LastRevised=Thu Jan 06 16:09:50 2000}\n%TCIDATA{Language=American English}\n\n\\ifpreprintsty\\def\\multb{ }\\def\\multe{ } \\else\\def\\multb{ \\begin{multicols}{2}}\\def\\multe{ \\end{multicols}} \\fi\n\n\\begin{document}\n\\draft\n\\title{Robust half-metallicity and metamagnetism\nin Fe$_{x}$Co$_{1-x}$S$_{2}$}\n\\author{I.I. Mazin}\n\\address{Code 6391, Naval Research Laboratory, Washington, DC 20375}\n\\date{January 19, 2000}\n\\maketitle\n\n\\begin{abstract}\nThe Fe$_{x}$Co$_{1-x}$S$_{2}$ system is predicted, on the basis of density\nfunctional calculations, to be a half metal for a large range of\nconcentrations. Unlike most known half metals, the half metallicity in this\nsystem should be very stable with respect to crystallographic disorder and\nother types of defects. The endmember of the series, CoS$_{2},$ is not a half\nmetal, but exhibits interesting and unusual magnetic properties which can,\nhowever, be reasonably well understood within the density functional theory,\nparticularly with the help of the extended Stoner model. Calculations\nsuggest strong electron-phonon and electron-magnon coupling in the system,\nand probably a bad metal behavior at high temperatures.\n\\end{abstract}\n\n\\multb\n\\begin{flushright}\n%Die ganze Zahl schuf der liebe Gott, alles \\\"{u}brige ist Menschenwerk (L.\n{\\it God made the integers, all else is the work of man \\\\\n(L. Kronecker). }\n\\end{flushright}\n\\vskip -.1in\nHalf metals (HM), materials that are metals in one spin channel and\ninsulators in the other, are attracting substantial interest recently, mostly\nbecause of potential application in spintronics devices,\n$e.g.$, spin valves\\cite\n{spintronics}. Unfortunately, although a few dozen various materials have\nbeen predicted to be HM on the basis of band structure calculations, there\nare hardly any that have been convincingly confirmed\nto be such by an experiment. A possible\nexception is CrO$_{2}$ where a spin polarization of up to 90\\% was measured by\nthe Andreev reflection technique\\cite{soulen} (in some other materials there\nis indirect evidence such as integer magnetic moment or optical spectra\nconsistent with half-metallic bands). \nThe usual explanation of such a discrepancy between the theory and\n the experiment is that half-metallicity in\nthese materials is very sensitive to crystallographic disorder and\nstoichiometry (see, e.g., Ref. \\cite{orgassy}). Indeed, no materials have\nbeen predicted to be HM in a wide range of concentrations of the\nconstituents, and insensitive to disorder.\n\nIn this Letter I point out\none such material. Namely, I show that the\npyrite alloys Fe$_{1-x}$Co$_{x}$S$_{2}$ are HM for the most of the\nconcentration range (0.1$\\lesssim x\\lesssim 0.9$), and not sensitive to\nordering of Fe and Co in the metal sublattice. On the other hand,\nat $x\\agt 1$\\cite{x1}, calculations predict magnetic collapse\nunder pressure, and a metamagnetic behavior just before the collapse.\nBoth half metallicity and metamagnetism can be explained by competition \nbetween the kinetic (band) energy, and the Stoner (Hund) interaction.\n\nExperimentally, the system of pyrite solid solutions, (Fe,Co,Ni,Cu)(S,Se)$%\n_{2},$ is amazingly rich. FeS$_{2}$ is a nonmagnetic semiconductor, in\nagreement with the band structure calculations\\cite{eyert,helmut}. With\nas little as 0.1\\% Co, by some data\\cite{jarret}, it becomes a ferromagnetic\nmetal, and remains ferromagnetic\n all the way through CoS$_{2},$ and further on until approximately Co$%\n_{0.9}$Ni$_{0.1}$S$_{2}.$ It was noted that the magnetic moment, $M,$ per Co\natom in Fe$_{1-x}$Co$_{x}$S$_{2}$ solid solutions stays close to 1 $\\mu _{B}$\nin an extremely wide range from $x\\approx 0.1-0.2$\nto $x=0.9-0.95.$ To the best of my knowledge, no\nexplanation of this fact has been suggested till now.\n At larger $x,$ $M$ decreases to $\\approx 0.85$ $\\mu _{B}$\\cite\n{jarret}. Starting from CoS$_{2},$ one can also\nsubstitute S with Se. The Curie temperature, $T_{C}$,\nrapidly decreases, and magnetism disappears at\nSe concentration of 10-12\\%, while $M$ decreases only slightly\\cite{se}. At\nlarger Se concentrations, the material shows metamagnetic behavior with no\nsizeable spontaneous magnetization, but with magnetization of 0.82$-0.85$ $%\n\\mu _{B}$ appearing abruptly when applied magnetic field in exceeds\napproximately $(220x_{Se}-25)$ tesla.\nA very similar behavior was observed in pure\nCoS$_{2}$ under pressure\\cite{se}, suggesting that the magnetic effect of Se\nis just the density of states (DOS) reduction.\n\nAs shown below, \nall these effects are reproduced by the standard\nlocal spin density (LSD) calculations, and find their explanation within the\nextended Stoner model (ESM)\\cite{eStoner}. \nFurther substitution of Co by Ni leads to a Mott-Hubbard transition\ninto an antiferromagnetic state, which is, in\ncontrast to the system considered\nhere, poorly described by the LSD. Further doping by Cu leads to a\nsuperconductivity, presumably due to high-energy sulfur vibrons\\cite{ep}.\n\nTo understand the behavior of the Fe$_{1-x}$Co$_{x}$S$_{2}$ system, I\nperformed several series of density functional LSD calculations: First,\nvirtual crystal approximation (VCA) was used in conjunction with the Linear\nMuffin Tin Orbital (LMTO) method\\cite{lmto}. Then, I did several\ncalculations using rhombohedral supercells of 4 or 8 formula units. Finally,\nI checked the results against more accurate\nfull-potential linear augmented plane wave\ncalculations\\cite{wien} for pure CoS$_{2},$ FeCo$_{3}$S$_{8},$ and (Fe$%\n_{0.25}$Co$_{0.75})$S$_{2}$ in VCA. The results were \nconsistent with the LMTO calculations. All calculations were\nperformed in the experimental crystal structure of CoS$_{2}.$ In\nreality, the S-S bond in FeS$_{2}$ is 4\\%\nlonger. The effect on the band structure is not negligible, particularly\nnear the bottom of the conductivity band\\cite{tbp}. However,\n the difference\nis not important for the purpose of the current Letter, namely the half\nmetallicity of Fe$_{1-x}$Co$_{x}$S$_{2}$ alloys, and understanding the basic\nphysics of its magnetic phase diagram.\nThe resulting\nmagnetic moments are shown in Fig.\\ref{M(x)}. In good\nagreement with the experiment, the magnetic moment per Co is exactly 1 $\\mu\n_{B}$ for the Co concentrations 0.3$\\lesssim x\\lesssim 0.9.$ The same holds\nfor the rhombohedral supercell calculations (Fig.\\ref{M(x)}%\n), demonstrating stability of the HM state with respect to crystallographic\ndisorder. I performed calculations for ordered Fe$_{7}$CoS$_{16},$ Fe$_{3}$%\nCoS$_{8},$ FeCoS$_{4},$ and FeCo$_{3}$S$_{8},$ and found that already Fe$_{7}$CoS$_{16}$\n$(x=0.125)$\nhas magnetic moment of 1 $\\mu _{B}$ per Co. Although the\noriginal paper\\cite{jarret} implied that the total magnetic\nmoment resides on Co,\nthis is not true: for instance, in Fe$_{7}$CoS$_{16}$ less than 30\\% of the\ntotal magnetization (0.45 $\\mu _{B})$ resides on Co. The nearest neighbor Fe\n(6 per cell) carry $\\approx 0.15$ $\\mu _{B}$ each. 8\\% of the total moment\nresides on S, about the same relative amount as in CoS$_{2}$\\cite{S}. So,\none cannot view the low-doping Fe$_{1-x}$Co$_{x}$S$_{2}$ alloys as magnetic\nCo ions embedded in polarizable FeS$_{2}$ background, as for instance Fe in\nPd.\n\nThe best way to understand the physics of this alloy is to start with FeS$_{2}.$ \nFeS$_{2}$ is a\nnonmagnetic semiconductor with a gap between the $t_{2g}$ and $e_{g}$\nstates \\cite{eyert,helmut}.\nThe reason for that is that sulfur forms S$_{2}$\ndimers with the $pp\\sigma $ states split into a bonding and an\nantibonding state. The latter is slightly above the Fe $e_{g}$ states and\nthus empty. The other 5 S $p$ states are below the Fe $t_{2g}$ states, hence\nthe occupation of the Fe $d$ bands is 6, just enough to fill the narrow $%\nt_{2g}$ band. Magnetizing FeS$_{2}$ would require transfer of electrons from\nthe $t_{2g}$ into the $e_{g}$ band, at an energy cost of the band gap $%\n\\Delta $ per electron. The Stoner parameter $I,$ which characterizes the\ngain of Hund energy per one electron transferred from the spin-minority into\nthe spin-majority band, appears to be smaller than $\\Delta \\approx 0.75$ eV\n(For the conductivity band in FeS$_{2},$ due to hybridization with S, $I$ is\nsmaller that in pure Fe\\cite{I} and is $\\approx 0.55$ eV). However, if we\npopulate the same band structure with $x\\ll 1$ additional electrons per\nformula unit, we can either\ndistribute them equally between the two spin subband,\nor place $x^{\\prime }$ in the spin majority band and $x^{\\prime \\prime\n}=x-x^{\\prime }$ in the spin majority band (the total magnetic moment in $%\n\\mu _{B}$ is then $M=x^{\\prime }-x^{\\prime \\prime }$).\nIn the latter case we gain a Hund energy of $-IM^{2}/4,$ but we \nlose kinetic (band structure)\nenergy because some states have only single occupancy and thus\nmore high-energy states have to be occupied. Using effective mass\napproximation for the conductivity band one can express this kinetic\nenergy loss in terms of concentration $x$ and magnetic moment per Co\n$\\beta =M/x$ as $A(x/2)^{5/3}[(1+\\beta )^{5/3}+(1-\\beta\n)^{5/3}]$, where $A=\\frac{3\\hbar ^{2}}{10m^{*}}(\\frac{3}{4\\pi V})^{2/3},$ $%\nm^{*}$ is the effective mass, and $V$ is the volume\nper formula unit. \n\nMinimizing the total energy with respect to $\\beta $ gives the equilibrium\nmagnetic moment in ESM: \n$$\n\\beta =(5A/3I)[(1+\\beta )^{2/3}-(1-\\beta )^{2/3}]/(4x)^{1/3}\n$$\nThe solution is a universal function $\\beta (Ix^{1/3}/A).$ Note that $\\beta\n(z)=0$ for $z<10\\cdot 2^{1/3}/9$,\nand $\\beta (z)=1$ for $z>5/3.$ Correspondingly,\nwith doping the material remains paramagnetic until $x$ reaches $%\nx_{1}=2(10A/9I)^{3},$ and then the magnetic moment per Co atom\ngradually grows until the concentration reaches the\n$x_{2}=(5A/3I)^{3}.$ At larger dopings the material remains\nhalf-metallic with $\\beta =1.$ Eventually, DOS starts to\ndeviate from the effective mass model and the polarization $\\beta $ may\nbecome smaller than 1 again. One can estimate the critical concentrations $%\nx_{1,2}$ using the following values, extracted from the band structure\ncalculations for FeS$_2$: $m^{*}\\approx 0.8,$ $I\\approx 0.55$ eV. This gives $%\nx_{1}\\approx 0.16$ and $x_{2}\\approx 0.26,$ in qualitative agreement with\nthe experiment and LSDA calculations (Fig.\\ref{M(x)}).\n\n\\begin{figure}[tbp]\n\\centerline{\\epsfig{file=cos2a.eps,height=0.85\\linewidth}}\n\\vspace{-.1in} \\setlength{\\columnwidth}{3.2in} \\nopagebreak\n\\caption{ Experimental and calculated magnetic moment per Co\nin Fe$_{1-x}$Co$_{x}$S$_{2}$ alloys. VCA: virtual crystal approximation,\ns/cell: supercell calculations, experimental data are from\n Ref.\\protect\\cite{jarret}.\n} \\label{M(x)}\n\\end{figure}\nThis explains the on the first glance unexpected result: in contrast to most\nknown half metals Fe$_{1-x}$Co$_{x}$S$_{2}$ remains a HM for a large range\nof concentrations, is insensitive to crystallographic disorder, and probably\nnot very sensitive to the state of the surface either: the behavior\nqualitatively described by the universal function $\\beta $ above is\ndetermined by the competition of two large energies: The band gap $\\Delta $,\nwhich is \na measure of crystal field splitting, and the Stoner factor $I$, a\nmeasure of the Hund coupling. As long as $\\Delta >I$ and $x_{1}\\ll 1$,\na large region of half-metallicity exists. Both conditions are related\nprimarily to the atomic characteristic of constituents and gross features of\nthe crystal structure, and are not sensitive to details.\n\nThe same competition between the band energy and the Stoner energy\nleads to very different magnetic properties\nin case of stoichiometric CoS$_{2}\n$, and of CoS$_{2}$ doped with Ni ($x>1$ region in Fig.1). The\nexperimental moment in the stoichiometric compound is 0.85-0.9 $\\mu\n_{B}$, in perfect agreement with full potential LSDA calculations, and in\nreasonable agreement with LMTO results as well. External pressure, or\nsubstituting S by Se, rapidly reduces magnetic moment\\cite{se}. As shown in\nRef.\\cite{Sebands}, Se doping increases the width of the conductivity\nband (because the Se$_{2}$ dimer has smaller $pp\\sigma $\nsplitting than S$_{2}$, and therefore the Co $e_{g}$ states are better aligned\nwith the chalcogen $pp\\sigma ^{*}$ states\\cite{tbp}). Thus Se doping has the\nsame effect as applying pressure. Doping with Ni has similar effect and\nNi concentrations of the order of 10\\% make the system non-magnetic.\nThe large magnetic moment of CoS$_{2}$ suggests that the ferromagnetism in this\nsystem is robust and the fact that it disappears so rapidly with pressure\nand doping seems surprising.\n\\begin{figure}[tbp]\n\\centerline{\\epsfig{file=dos.eps,width=0.9\\linewidth}}\n\\vspace{0.1in} \\setlength{\\columnwidth}{3.2in} \\nopagebreak\n\\caption{Density of states for {\\it paramagnetic} CoS$_2$, calculated\nby LAPW method. $E-E_F=0.35$ eV corresponds to Co$_{0.9}$Ni$_{0.1}$S$_2$\nin the rigid band approximation.\n} \\label{CoS2-DOS}\n\\end{figure}\nSince all these effects are reproduced very well\nby regular LSDA calculations (Fig.1), ESM\ncalculations again provide valuable physical insight.\nUnlike the parabolic band case considered above, now one has to\ntake into account specific structure of DOS of CoS$_2$ (Fig.\\ref{CoS2-DOS}).\nThe loss of the one-electron energy can be\nexpressed\\cite{N(M)} in terms of the\n average DOS, $\\tilde{N}(M),$ defined as $\nM/(\\mu _{B}H_{xc})$, where $(\\mu _{B}H_{xc})$ is the (rigid) exchange splitting\nproducing magnetic moment $M$:\n$\\Delta E=\\int_{0}^{M}MdM /2\\tilde{N}(M)$.\n The best visualization of the ESM is via plotting $%\n\\tilde{N}(M)$ as a function of $M$. Wherever this curve crosses the line $1/I\n$, one has an extremum of the total energy. If the slope of the curve is\npositive at the intersection point, the extremum is a maximum, and the state is\nunstable, otherwise it\nis a minimum and indicates a (meta)stable magnetic state. The ESM plot for\nCoS$_{2},$ based on the LAPW DOS (Fig.\\ref{CoS2-DOS}) is shown on Fig.\\ref\n{esm}. At normal pressure, there are potentially two metastable states: a\nlow spin, of the order of 0.3 $\\mu _{B}$, and a high spin, of the order of\n0.85 $\\mu _{B}$. For Stoner parameter $I<0.63$ eV neither state is\n(meta)stable, for $0.63<I<0.71$ eV only the low-spin state is, for $%\n0.70<I<0.73$ eV there are two metastable states, and for yet larger $I$ only\nthe high spin state may be realized. For $I>0.77$ eV ESM produces a HM. One can\nestimate $I$ for a compound using the procedure described in Ref.\\cite{MS},\nor deduce it by comparing the ESM to fixed moment calculations. The latter\nmethod leads to $I=0.68$ eV for LMTO and $I=0.72$ eV for full potential\ncalculations. The complicated structure of $\\tilde{N}(M)$ can be traced down\nto the structure of DOS near the Fermi level (Fig.\\ref{CoS2-DOS}):\n after the initial drop of $\\tilde{N%\n}(M),$ it starts to increase again when the initially fully occupied \npeak at $E-E_{F}\\approx -0.1$ eV becomes magnetically \npolarized. The high-spin solution\ncorresponds to the situation where this peak is fully polarized.\n\n\\begin{figure}[tbp]\n\\centerline{\\epsfig{file=esm.eps,height=0.7\\linewidth}}\n\\vspace{0.1in} \\setlength{\\columnwidth}{3.2in} \\nopagebreak\n\\caption{Extended Stoner plot for CoS$_2$, showing the effective \n(averaged) density of states $\\tilde N(E_F)$ as a function of\nmagnetic moment $M$. The horizontal line corresponds to a Stoner factor $\nI=0.72$ eV. The two crossing points near $M\\approx 0.3$ $\\mu_B$, and\n$M\\approx 0.8$ $\\mu_B$, show two (meta)stable states in\nthe rigid band approximation.\n} \\label{esm}\n\\end{figure}\nLet me now consider the effect of pressure.\nRoughly speaking, applying pressure amounts to rescaling the\nband structure proportionally to squared inverse lattice parameter.\nCorrespondingly, the density of states is rescaled proportional to $%\n(V_{0}/V)^{2/3}.$ The total energy in the ESM will include that as \n\\begin{eqnarray}\nE(V,M)&\\approx& E_{0}+\\left( \\frac{V}{V_{0}}\\right) ^{2/3}\\int_{0}^{M}\\frac{MdM%\n}{2\\tilde{N}(M,V_{0})} \\nonumber\\\\\n&-&\\frac{IM^{2}}{4}+\\frac{B(V-V_{0})^{2}}{2V_{0}},\n\\label{EOS}\n\\end{eqnarray}\nwhere $V_{0},$ $E_{0},$ and $B$ are the equilibrium volume, energy, and the\nbulk modulus, respectively, of the paramagnetic phase. The magnetic energy\nshifts the energy minimum towards larger volumes, a standard\nmagnetostriction effect. What is unusual about the equation of states (\\ref\n{EOS}) with $\\tilde{N}(M,V_{0})$ from Fig.\\ref{esm} is that for some range\nof values of $I$\n there\nare two local minima, a high spin state with a larger volume, and a low spin\nstate with a smaller volume. \nThe actual $I$ seems to fall into this range.\nThis suggests a first order phase transition\nwith pressure, and there are indications\\cite{1O}\n that it has been observed in the\nexperiment\\cite{se}, although the low spin state, which has $M\\approx\n0.2$ $\\mu _{B}$ in ESM and $\\approx\n0.1$ $\\mu _{B}$ in fixed moment calculations,\nwas reported in Ref. \\cite{se}\nto have no or very small magnetic moment. \nAnother consequence of the physical\npicture outlined here is metamagnetism: In the low spin state close to the\ncritical pressure the system can be switched over to the high-spin state by\nan external magnetic field defined by the energy and magnetic\nmoment difference between the\ntwo states. Again, metamagnetic\nbehavior has been observed in Co(S,Se)$_{2},$ in Co$_{1-x}$Ni$_{x}$S$_{2},$\nand in compressed CoS$_{2}.$ Finally, the theory predicts rapid increase of\nthe equilibrium magnetization with {\\it negative} pressure: in the ESM an\nexpansion of 3-4\\% in volume already increases $M$ to nearly 1 $\\mu _{B}.$\nThis suggests that the spin fluctuations at high temperature (above $T_{C}$)\nmay have larger amplitude than the ordered moments at zero temperature. This\nkind of behavior has also been observed\\cite{M(T)}, and discussed in the\nliterature\\cite{Ogawa83}.\n\nThe family of pyrite materials formed by 3d transition metals and chalcogens\nis incredibly rich. It shows various kinds of magnetism and metamagnetism,\nmetal-insulator transitions of different types, superconductivity, and half\nmetallicity. Except for the vicinity of a Mott-Hubbard transition, that is,\nclose to NiS$_{2},$ the physics of these materials can be rather well\nunderstood within the local spin density functional theory. In particular,\nthe extended Stoner formalism provides considerable insight into the magnetic\nbehavior of this system. The most important conclusions from the\ncalculations are:\n\n1. The Fe$_{1-x}$Co$_{x}$S$_{2}$ alloy is predicted to be a half metal in a\nlarge range of concentration. Unlike most other known half metals, and very\nimportantly for the applications, the half-metallicity should be robust with\nrespect to defects and crystallographic disorder.\n\n2. CoS$_{2}$ is an itinerant ferromagnet which has, due to a\ncomplicated structure of its density of states, two magnetic states: a \nhigh-spin one, with the moment $M\\sim 0.8$ $\\mu _{B},$ and a low-spin one,\nwith $M\\alt0.1$ $\\mu _{B}.$ The first order transition to the low spin state\ncan be induced by external pressure, doping with Se, or with Ni. In the low\nspin state the material exhibits metamagnetic properties. The structure of\nthe density of states also manifests itself via an unusual temperature\ndependence of the magnitude of magnetic fluctuations.\n\n3. The closeness of CoS$_{2}$ to a magnetic phase\ntransition also leads to strong coupling between electronic, lattice, and\nmagnetic degrees of freedom, making it a relative of such ``bad metals'' as\nSrRuO$_{3}$ and magnetoresistive manganites. Therefore interesting transport\nproperties are to be expected in the system at and near stoichiometric\ncomposition, including large magnetoresistance and violation of the\nYoffe-Regel limit at high temperatures.\n\nFor such an interesting, the 3d pyrites seem to be unusually\nlittle studied either experimentally or theoretically. Hopefully this paper\nwill encourage further investigations.\n\n\\begin{references}\n\\vspace{-.4in}\n\\bibitem{spintronics} G.A. Prinz, Science, {\\bf 282}, 1660 (1998).\n\n\\bibitem{soulen} R. J. Soulen et al, Science {\\bf 282}, 85 (1998).\n\n\\bibitem{orgassy} D. Orgassa {\\it et al}, Phys. Rev.{\\bf B 60} 13237 (1999).\n\\bibitem{x1} Here and below I speak about $x>1$, meaning CoS$_2$ doped with Ni\ninstead of Fe.\n\n\\bibitem{eyert} V. Eyert, K.H. Hock, S. Fiechter, and H. Tributsch , Phys.\nRev.{\\bf B 57}, 6350 (1998).\n\n\\bibitem{helmut} Opahle I, Koepernik K, Eschrig H, Phys. Rev.{\\bf B60},\n14035 (1999), and references therein.\n\n\\bibitem{jarret} H.S. Jarret et al, Phys. Rev. Lett., {\\bf 21}, 617 (1968).\n\n\\bibitem{se} T. Goto et al, Phys. Rev. {\\bf B56}, 14019 (1997).\n\n\\bibitem{eStoner} G. L. Krasko, Phys. Rev. {\\bf B 36}, 8565 (1987); O.K.\nAndersen {\\it et al}, Physica {\\bf 86-88B}, 249 (1977).\n\n\\bibitem{ep} CuS$_{2}$ has considerable density of states at the Fermi\nlevel, 1 st./eV spin\nf.u., and substantial electron-phonon interaction with high-frequency S$_{2}$\nvibrons. Our calculations for CuS$_{2}$ will be published elsewhere. Note\nalso that a very large electronic specific heat coefficient of 19.5 mJ/mole K%\n$^{2}$ was reported (S. Waki and S. Ogawa, J. Phys. Soc. Jap. {\\bf 32}, 284,\n1972), corresponding to a renormalization $(1+\\lambda )\\approx 3.$ Of\ncourse, part of this renormalization should be coming from electron-magnon\ninteraction, similar to, say, SrRuO$_{3}$ \\cite{MS}. \n\n\\bibitem{lmto} The Stuttgart package LMTO-TB 4.7\n(http://www.mpi-stuttgart.mpg.de/ANDERSEN/LMTODOC/LMTODOC\\\\\n.html) was employed.\nSix automatically generated empty spheres\nwere used per formula unit, and the result agreed reasonably\nwell with the full-potential calculations (better than\nthe alternative setup recommended in Ref.\\protect\\cite{eyert})\nSomewhat unusual, the visual agreement between the band dispersions in LAPW\nand in LMTO is rather good, while the calculated magnetic moments differ by\nmore than 20\\% (0.65 $\\mu _{B}$ in LMTO {\\it vs. }0.85 $\\mu _{B}$ in LAPW).\nThis is another indication that, as discussed below in the text, the\nstoichiometric system is close to a high spin -- low spin transition.\n\n\\bibitem{wien} The WIEN-97 package was used (P. Blaha, K. Schwarz, and J.\nLuitz, Vienna University of Technology, 1997, improved and updated version\nof the code published by P. Blaha {\\it et al,} Comp. Phys. Commun., {\\bf 59}%\n, 399, 1990).\n\n\\bibitem{tbp} I.I. Mazin, to be published elsewhere.\n\n\n\\bibitem{S} Noticeable magnetization of S has been confirmed experimentally\n(A. Ohsawa {\\it et al}, J. Phys. Soc. Jap. {\\bf 40}, 992, 1976).\n\n\\bibitem{I} Regarding calculations of the Stoner factor $I$ for compounds,\nthe reader is referred, for instance, to Ref.\\cite{MS}\n\n\\bibitem{Sebands} H. Yamada, K. Terao, and M. Aoki, J. Mag. Mag. Mater. \n{\\bf 177}, 607 (1998).\n\n\\bibitem{N(M)}This formula follows from the fact the \nderivative of the one-electron energy with respect to $M$ is \n$(\\mu _{B}H_{xc})/2=(M/2)/\\tilde{N} (M)$.\n\\bibitem{MS} I.I. Mazin and D.J. Singh, Phys. Rev. {\\bf B56}, 2556 (1997).\n\n\\bibitem{1O} Experimentally, the transition at the Curie temperature\nbecomes first order at $P\\agt0.4$ GPa.\n\n\\bibitem{M(T)} N. Inoue and H. Yasuoka, Sol. State Comm. {\\bf 30}, 341\n(1979).\n\n\\bibitem{Ogawa83} S. Ogawa, J. Mag. Mag. Mater. {\\bf 31-34}, 269 (1983).\n\\end{references}\n\\multe\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002069.extracted_bib",
"string": "\\bibitem{spintronics} G.A. Prinz, Science, {\\bf 282}, 1660 (1998).\n\n\n\\bibitem{soulen} R. J. Soulen et al, Science {\\bf 282}, 85 (1998).\n\n\n\\bibitem{orgassy} D. Orgassa {\\it et al}, Phys. Rev.{\\bf B 60} 13237 (1999).\n\n\\bibitem{x1} Here and below I speak about $x>1$, meaning CoS$_2$ doped with Ni\ninstead of Fe.\n\n\n\\bibitem{eyert} V. Eyert, K.H. Hock, S. Fiechter, and H. Tributsch , Phys.\nRev.{\\bf B 57}, 6350 (1998).\n\n\n\\bibitem{helmut} Opahle I, Koepernik K, Eschrig H, Phys. Rev.{\\bf B60},\n14035 (1999), and references therein.\n\n\n\\bibitem{jarret} H.S. Jarret et al, Phys. Rev. Lett., {\\bf 21}, 617 (1968).\n\n\n\\bibitem{se} T. Goto et al, Phys. Rev. {\\bf B56}, 14019 (1997).\n\n\n\\bibitem{eStoner} G. L. Krasko, Phys. Rev. {\\bf B 36}, 8565 (1987); O.K.\nAndersen {\\it et al}, Physica {\\bf 86-88B}, 249 (1977).\n\n\n\\bibitem{ep} CuS$_{2}$ has considerable density of states at the Fermi\nlevel, 1 st./eV spin\nf.u., and substantial electron-phonon interaction with high-frequency S$_{2}$\nvibrons. Our calculations for CuS$_{2}$ will be published elsewhere. Note\nalso that a very large electronic specific heat coefficient of 19.5 mJ/mole K%\n$^{2}$ was reported (S. Waki and S. Ogawa, J. Phys. Soc. Jap. {\\bf 32}, 284,\n1972), corresponding to a renormalization $(1+\\lambda )\\approx 3.$ Of\ncourse, part of this renormalization should be coming from electron-magnon\ninteraction, similar to, say, SrRuO$_{3}$ \\cite{MS}. \n\n\n\\bibitem{lmto} The Stuttgart package LMTO-TB 4.7\n(http://www.mpi-stuttgart.mpg.de/ANDERSEN/LMTODOC/LMTODOC\\\\\n.html) was employed.\nSix automatically generated empty spheres\nwere used per formula unit, and the result agreed reasonably\nwell with the full-potential calculations (better than\nthe alternative setup recommended in Ref.\\protect\\cite{eyert})\nSomewhat unusual, the visual agreement between the band dispersions in LAPW\nand in LMTO is rather good, while the calculated magnetic moments differ by\nmore than 20\\% (0.65 $\\mu _{B}$ in LMTO {\\it vs. }0.85 $\\mu _{B}$ in LAPW).\nThis is another indication that, as discussed below in the text, the\nstoichiometric system is close to a high spin -- low spin transition.\n\n\n\\bibitem{wien} The WIEN-97 package was used (P. Blaha, K. Schwarz, and J.\nLuitz, Vienna University of Technology, 1997, improved and updated version\nof the code published by P. Blaha {\\it et al,} Comp. Phys. Commun., {\\bf 59}%\n, 399, 1990).\n\n\n\\bibitem{tbp} I.I. Mazin, to be published elsewhere.\n\n\n\n\\bibitem{S} Noticeable magnetization of S has been confirmed experimentally\n(A. Ohsawa {\\it et al}, J. Phys. Soc. Jap. {\\bf 40}, 992, 1976).\n\n\n\\bibitem{I} Regarding calculations of the Stoner factor $I$ for compounds,\nthe reader is referred, for instance, to Ref.\\cite{MS}\n\n\n\\bibitem{Sebands} H. Yamada, K. Terao, and M. Aoki, J. Mag. Mag. Mater. \n{\\bf 177}, 607 (1998).\n\n\n\\bibitem{N(M)}This formula follows from the fact the \nderivative of the one-electron energy with respect to $M$ is \n$(\\mu _{B}H_{xc})/2=(M/2)/\\tilde{N} (M)$.\n\n\\bibitem{MS} I.I. Mazin and D.J. Singh, Phys. Rev. {\\bf B56}, 2556 (1997).\n\n\n\\bibitem{1O} Experimentally, the transition at the Curie temperature\nbecomes first order at $P\\agt0.4$ GPa.\n\n\n\\bibitem{M(T)} N. Inoue and H. Yasuoka, Sol. State Comm. {\\bf 30}, 341\n(1979).\n\n\n\\bibitem{Ogawa83} S. Ogawa, J. Mag. Mag. Mater. {\\bf 31-34}, 269 (1983).\n"
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cond-mat0002070
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LPM-#1-LT#2
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[
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"author": "P Lajk\\'o\\dag\\ddag\\ and L Turban\\dag"
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[
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"name": "cond.tex",
"string": "%\\magnification=\\magstep1\n\\input iopppt-modif.tex\n\\input xref\n\\input epsf\n\\eqnobysec\n%\n%\\def\\ps{\\noalign{\\vskip3pt}}\n% positive space in tables, matrices \n% \\ns for negative space\n\\def\\received#1{\\insertspace \n \\parindent=\\secindent\\ifppt\\textfonts\\else\\smallfonts\\fi \n \\hang{Received #1}\\rm } \n%\\def\\appendix{\\goodbreak\\beforesecspace \n% \\noindent\\textfonts{\\bf Appendix}\\secspace} \n%\\headline={\\ifodd\\pageno{\\ifnum\\pageno=\\firstpage\\titlehead\n% \\else\\rrhead\\fi}\\else\\lrhead\\fi} \n%\\def\\lpm#1#2{LPM-#1-LT#2}\n%\\def\\figure#1{\\global\\advance\\figno by 1\\gdef\\labeltype{\\figlabel}% \n% {\\parindent=\\secindent\\smallfonts\\hang \n% {\\bf Figure \\ifappendix\\applett\\fi\\the\\figno.} \\rm #1\\par}} \n%\\def\\endtable{\\parindent=\\textind\\textfonts\\rm\\bigskip} \n%\\def\\rrhead{\\textfonts\\hskip\\secindent\\it \n% \\shorttitle\\hfill\\folio} \n%\\def\\lrhead{\\textfonts\\hbox to\\secindent{\\folio\\hss}% \n% \\it\\aunames\\hss} \n%\\footline={\\ifnum\\pageno=\\firstpage\n%\\smallfonts cond-mat/\\hfil\\textfonts\\folio\\fi} \n%\\def\\titlehead{\\smallfonts \n%\\hfil\\lpm{}{}} \n\n%\\firstpage=1\n%\\pageno=1\n\n%\\pptstyle\n\\jnlstyle\n\n\\jl{1}\n \n\\overfullrule=0pt\n\n\n\\title{Percolation and conduction in restricted geometries}[Percolation\nand conduction in restricted geometries]\n\n\\author{P Lajk\\'o\\dag\\ddag\\ and L Turban\\dag}[P Lajk\\'o and L Turban]\n\n\\address{\\dag Laboratoire de Physique des \nMat\\'eriaux, Universit\\'e Henri Poincar\\'e (Nancy~I),\nBP~239, F--54506~Vand\\oe uvre l\\`es Nancy Cedex, France}\n\n\\address{\\ddag Institute of Theoretical Physics, Szeged University, \nH--6720 Szeged, Hungary}\n\n\\received{11 October 1999}\n\n\\abs\nThe finite-size scaling behaviour for percolation and conduction is studied\nin two-dimensional triangular-shaped random resistor networks at the\npercolation threshold. The numerical simulations are\nperformed using an efficient star-triangle algorithm. The percolation\nexponents, linked to the critical behaviour at corners, are in\ngood agreement with the conformal results. The conductivity exponent,\n$t'=\\zeta/\\nu$, is found to be independent of the shape of the system. Its\nvalue is very close to recent estimates for the surface and\nbulk conductivity exponents. \n\\endabs \n\n%\\vglue1cm\n\n\\pacs{05.70.Jk, 64.60.Ak}\n\n\\submitted\n\n\\date\n\\section{Introduction}\nSince the paper of Cardy \\cite{cardy83} we know that, at a second-order\nphase transition, the local critical behaviour can be influenced by the\nshape of the system. Furthermore, in the case of conformally\ninvariant two-dimensional (2D) systems, the tools of conformal invariance\ncan be used, at the critical point, to relate the local critical behaviour\nat a corner to the surface critical behaviour \\cite{cardy84,barber84}. The\ncorner shape, which is scale invariant, leads to local exponents varying\ncontinuously with the opening angle. This marginal local critical behaviour\nhas been indeed observed numerically for different systems\n(see~\\cite{igloi93} for a review) and, more recently, analytical results\nhave been obtained for the Ising model [5--8]. \n\nIn this paper, we present a numerical study\nof the critical behaviour of 2D random resistor networks with triangular\nshapes. This problem involves two sets of exponents, namely the percolation\nexponents and the conductivity exponents [9--11]. Although our main\ninterest concerns the conduction exponents, our simulations\nallow us, as a by-product, to check the conformal predictions for the\ncorner exponents of the percolation problem.\n\nThe percolation exponents are known exactly, both in the bulk and at the\nsurface, through a correspondance with the limit $q\\to1$ of the $q$-state\nPotts model [12--14]. The conformal aspects of the critical percolation\nproblem in finite geometries have been extensively studied in\n\\cite{langlands94}, following the work of~\\cite{cardy92}.\nConformal invariance has been also verified in a transfer-matrix\ncalculation of the surface percolation exponent, using the gap-exponent\nrelation \\cite{dequeiroz95}. The critical behaviour at surface and corners\nhas been considered in~\\cite{wolf90} where the conduction problem\nis addressed briefly.\n\nA recent series expansion study \\cite{essam96} suggests that\nthe surface conductivity has the same critical behaviour as the\nbulk one (see \\cite{grassberger99} and references therein). The main purpose\nof this work is to examine, with high numerical accuracy, wether the shape\nof a finite system may have some influence on the scaling behaviour of the\nconductivity.\n\nWe study the finite-size-scaling behaviour of the conductance\nand percolation probability between points located at the corners of a\ntriangle, either on the triangular or on the square lattice. As in the\nLobb-Frank algorithm \\cite{lobb84}, the numerical technique involves a\nsuccession of triangle-star and star-triangle transformations, which allows\none to reduce the triangular resistor network to a star network in a finite\nnumber of steps. \n\nIn section~2 we introduce the different correlation functions of the\npercolation and conduction problems and define the associated exponents. We\nalso review the conformal results for the corner exponents. In section~3 we\ngive a detailed explanation the triangle-star star-triangle algorithm. The\nfinite-size scaling simulation results are presented in section 4 and\ndiscussed in section 5. \n \n\n\\section{Percolation and conduction correlation functions}\nWe consider a random resistor network for which each lattice bond has\na probability $p$ to have a unit conductance and $1-p$ to be an\ninsulator. Let the connectedness characteristic function $c_{ij}$ be\ndefined as \n$$\nc_{ij}=\\cases{$1$ & if sites $i$ and $j$ are connected\\cr\n$0$ & otherwise.}\n\\label{e2.1}\n$$\nWith $[\\cdots]_{\\rm av}$ denoting a configurational average, the\npercolation correlation function \n$$\nP_{ij}=[c_{ij}]_{\\rm av}\n\\label{e2.2}\n$$\ngives the probability that sites $i$ and $j$ belong to the same cluster of\nconducting bonds. The average conductance given by\n$$\nG_{ij}=[g_{ij}]_{\\rm av}\n\\label{e2.3}\n$$\nwhere $g_{ij}$ is the conductance of the system between sites $i$ and\n$j$, plays the role of a correlation function, or non-local conductive\nsusceptibility, for the conduction problem \\cite{fish78}. Let $N$ be\nthe number of samples taken into account in the configurational average and\n$N_{\\rm con}$ the corresponding number of samples for which sites $i$ and\n$j$ are connected. The correlation functions in equations~\\ref{e2.2}\nand~\\ref{e2.3} can be rewritten explicitly as \n$$\n\\eqalign{\n&P_{ij}=\\lim_{N\\to\\infty}{1\\over N}\\sum_{\\alpha=1}^N c_{ij}^\\alpha\n=\\lim_{N\\to\\infty}{N_{\\rm con}\\over N}\\cr\n&G_{ij}=\\lim_{N\\to\\infty}{1\\over N}\\sum_{\\alpha=1}^N \ng_{ij}^\\alpha=\\lim_{N\\to\\infty}{N_{\\rm con}\\over\nN}{1\\over N_{\\rm con}}\\sum_{\\alpha=1}^{N_{\\rm con}}[g_{ij}^{\\rm\ncon}]^\\alpha\\,.\\cr } \\label{e2.4}\n$$\nThus one can define the reduced conduction correlation function\n$$\n\\Gamma_{ij}={G_{ij}\\over P_{ij}}=\\lim_{N\\to\\infty}{1\\over N_{\\rm\ncon}} \\sum_{\\alpha=1}^{N_{\\rm con}}[g_{ij}^{\\rm con}]^\\alpha\n\\label{e2.5}\n$$\nwhich gives the average conductance between two points $i$ and $j$\nwhen they belong to the same percolation cluster.\n\nIn an infinite system, for two points at a distance $r_{ij}=r$, the\ncorrelation functions display a power law decay \n$$\nP_{ij}\\sim r^{-2x}\\qquad \\Gamma_{ij}\\sim r^{-\\zeta/\\nu}\n\\label{e2.6}\n$$\nat the percolation threshold $p=p_{\\rm c}$. The exponents $x$ and $\\nu$ are\nthe scaling dimension of the bulk order parameter and the correlation\nlength exponent for the percolation problem, respectively. The\nconductivity exponent $\\zeta$ governs the behaviour of the\nmacrosopic conductivity near the percolation threshold\nwhere \\cite{degennes76} \n$$\n\\Sigma\\sim (p-p_{\\rm c})^t\\qquad t=\\nu t'=\\zeta+(d-2)\\nu\\qquad p>p_{\\rm\nc}\\,. \n\\label{e2.7}\n$$\nNote that $\\zeta/\\nu=t'$ in two dimensions.\n\nAs mentioned in the introduction, the percolation exponents are exactly\nknown in two dimensionsthrough a correspondance with the $q$-state Potts\nmodel \\cite{wu82,cardy87}: \n$$ \n\\nu=\\case{4}{3}\\qquad x=\\case{5}{48}\\qquad\nx_{\\rm s}=\\case{1}{3} \n\\label{e2.8}\n$$\nwhere $x_{\\rm s}$ is the scaling dimension of the surface order parameter\nat the ordinary transition. \n\nRecent high statistics simulations led to the following accurate\nestimate for the conductivity exponent in two dimensions\n\\cite{grassberger99}: \n$$\nt'={\\zeta\\over\\nu}=0.9825\\pm0.0008\\qquad \\zeta=1.3100\\pm0.0011\\,.\n\\label{e2.9}\n$$\nLow-density series expansion results \\cite{essam96} are\nconsistent with a surface conductivity exponent $\\zeta_{\\rm s}$\nkeeping its bulk value $\\zeta$. \n\nFor a triangular-shaped system of size $L$,\nwhen the points $i$ and $j$ are located at corners with opening angles\n$\\theta_i$ and $\\theta_j$, according to finite-size scaling, one expects the\nfollowing behaviour at criticality: \n$$\n\\eqalign{\n&P_{ij}=P(\\theta_i,\\theta_j;L)\\sim\nL^{-\\eta(\\theta_i,\\theta_j)}\\sim L^{-x(\\theta_i)-x(\\theta_j)}\\cr\n&\\Gamma_{ij}=\\Gamma(\\theta_i,\\theta_j;L) \\sim\nL^{-\\zeta(\\theta_i,\\theta_j)/\\nu}\\,.\\cr} \n\\label{e2.10}\n$$\nHere $x(\\theta_i)$ is the scaling dimension of the local order parameter at a\ncorner with opening angle $\\theta_i$. In the following, we shall also consider\nthe three-point correlation function $P_{ijk}$, which gives the probability that\nthe points $i$, $j$ and $k$, located at corners with opening angles $\\theta_i$,\n$\\theta_j$ and $\\theta_k$, belong to the same cluster. This quantity scales like\nthe product of the corresponding local order parameters, i.e., as \n$$\nP_{ijk}=P(\\theta_i,\\theta_j,\\theta_k;L)\\sim\nL^{-\\eta(\\theta_i,\\theta_j,\\theta_k)}\\sim\nL^{-x(\\theta_i)-x(\\theta_j)-x(\\theta_k)}\n\\label{e2.11} \n$$\nat the percolation threshold.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%% figure 1\n{\\par\\begingroup\\medskip\n\\epsfxsize=11truecm\n\\topinsert\n\\centerline{\\epsfbox{fig1-cond.eps}}\n\\smallskip\n\\figure{Reduction of a triangular-shaped system to a star\nthrough a succession of triangle-star and star-triangle\ntransformations.\\label{fig1}} \n\\endinsert \n\\endgroup\n\\par}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figures 2 and 3\n{\\par\\begingroup\\medskip\n\\epsfxsize=6.5truecm\n\\topinsert\n\\centerline{\\epsfbox{fig2-cond.eps}}\n\\smallskip\n\\figure{In the triangle-star transformation, up-pointing triangles\nwith con\\-duc\\-tances $g_\\alpha^{(p)}(i,j)$ are transformed into\nstars with conductances $\\gamma_\\alpha^{(p)}(i,j)$ given by\nequation~(3.1).\\label{fig2}} \n\\epsfxsize=9.5truecm\n\\centerline{\\epsfbox{fig3-cond.eps}}\n\\smallskip \n\\figure{The conductances $g_\\alpha^{(p+1)}(i,j)$ on up-pointing triangles\nat step $p+1$, given in equation~(3.2), follow from star-triangle\ntransformations on down-pointing stars. The conductances of the star involved in\nthe construction of $g_1^{(p+1)}(i,j)$ are indicated. They originate from\ndifferent up-pointing triangles at step $p$.\\label{fig3}} \n\\endinsert \n\\endgroup \n\\par}\n\nA dependance of the local exponents on the opening angles is\ngenerally expected since a wedge is a scale-invariant geometry and the angles\nare marginal variables for the local critical behaviour. The critical 2D\nPotts model being conformally invariant, one obtains the local critical\nbehaviour for the percolation problem at a corner using the conformal\ntransformation $w=z^{\\theta/\\pi}$, which maps the half-space onto a wedge\nwith opening angle $\\theta$ \\cite{cardy84,barber84}. This leads to the\nfollowing expression for the scaling dimension of the order parameter:\n$$\nx(\\theta)={\\pi\\over\\theta}\\, x_{\\rm s}\\,.\n\\label{e2.12}\n$$\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figure 4\n{\\par\\begingroup\\medskip\n\\epsfxsize=11truecm\n\\topinsert\n\\centerline{\\epsfbox{fig4-cond.eps}}\n\\smallskip\n\\figure{A triangular-shaped system on the square lattice is obtained\nthrough a deformation of the triangular lattice when the\nconductances are set to zero in one of the principal\ndirections.\\label{fig4}} \n\\endinsert \n\\endgroup \n\\par}\n\n\\section{The star-triangle transformation}\nLet us consider a finite random resistor network with the shape of an\nequilateral triangle of side $L$ on the triangular lattice. Through a succession\nof triangle-star and star-triangle transformations, the original system can be\nprogressively transformed into a star with 3 branches of length $L$ as\nshown in figure~\\ref{fig1}.\n\nAt step $p$, in the first part of the lattice transformation, up-pointing\ntriangles with coordinates $(i,j)$ and bond conductances\n$g_\\alpha^{(p)}(i,j)$ ($\\alpha=1,2,3$) are replaced by stars with\nbond conductances \n$$\n\\fl\n\\gamma_\\alpha^{(p)}(i,j)\\!=\\!{g_1^{(p)}(i,j)\\, \ng_2^{(p)}(i,j)\\!+\\! g_2^{(p)}(i,j)\\, g_3^{(p)}(i,j)\\!+\\! g_3^{(p)}(i,j)\\,\ng_1^{(p)}(i,j)\\over g_\\alpha^{(p)}(i,j)}\\quad\\alpha=1,2,3\n\\label{e3.1}\n$$\nas shown in figure~\\ref{fig2}. In the second part, down-pointing\nstars are transformed into down-pointing triangles and\nup-pointing triangles are relabelled as indicated in\nfigure~\\ref{fig3}. Thus one obtains the bond conductances for\nup-pointing triangles at step $p+1$ as\n$$\n\\eqalign{\n&g_1^{(p+1)}(i,j)={\\gamma_2^{(p)}(i,j)\\,\\gamma_3^{(p)}(i+1,j)\\over\n\\gamma_1^{(p)}(i+1,j-1)+\\gamma_2^{(p)}(i,j)+\\gamma_3^{(p)}(i+1,j)}\\cr \n&g_2^{(p+1)}(i,j)={\\gamma_3^{(p)}(i,j+1)\\,\\gamma_1^{(p)}(i,j)\\over\n\\gamma_1^{(p)}(i,j)+\\gamma_2^{(p)}(i-1,j+1)+\\gamma_3^{(p)}(i,j+1)}\\cr\n&g_3^{(p+1)}(i,j)={\\gamma_1^{(p)}(i+1,j)\\,\\gamma_2^{(p)}(i,j+1)\\over\n\\gamma_1^{(p)}(i+1,j)+\\gamma_2^{(p)}(i,j+1)+\\gamma_3^{(p)}(i+1,j+1)}\\,.\\cr\n}\n\\label{e3.2}\n$$\nNote that surface bonds in up-pointing triangles at step\n$p+1$ result from the transformation of incomplete down-pointing stars.\nThe expressions given in~\\ref{e3.2} still apply in this case, provided\nthe conductances associated with the missing bonds are set equal to\nzero, i.e., with the boundary conditions\n$$\n\\eqalign{\n&\\gamma_1^{(p)}(i,0)=0\\qquad\\gamma_2^{(p)}(0,j)=0\\qquad\ni,j=2,L-p\\cr\n&\\gamma_3^{(p)}(i,j)=0\\qquad i=2,L-p\\qquad j=L-p-i+2\\,.\\cr\n}\n\\label{e3.3}\n$$\n\nAt step $p=L-1$, the final star configuration is obtained after the\ntriangle-star transformation has been performed. Then, for example, the\nconductance between points $A$ and $B$ on a sytem with size $L$ is\ngiven by \n$$\ng_{AB}=\\left[\\sum_{p=0}^{L-1}{1\\over\\gamma_1^{(p)}(1,L-p)}\n+{1\\over\\gamma_2^{(p)}(L-p,1)}\\right]^{-1}\\,. \\label{e3.4}\n$$\nThe same transformation, with all the conductances $g_3^{(0)}(i,j)$\nset to zero in the initial configuration, can be used to reduce to a star\na triangular-shaped system on the square lattice as shown in\nfigure~\\ref{fig4}.\n\n\\section{Finite-size scaling results} \n\nWe have studied the finite-size scaling behaviour of the\npercolation and reduced conduction correlation functions between corners on\ntriangular-shaped sytems. We worked at the percolation threshold, either on the\ntriangular lattice ($p_{\\rm c}=2\\sin(\\pi/18)$) or on the square\nlattice ($p_{\\rm c}=1/2$) \\cite{shante71}. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figures 5, 6 and 7\n{\\par\\begingroup\\medskip\n\\epsfxsize=7.5truecm\n\\topinsert\n\\centerline{\\epsfbox{fig5-cond.eps}}\n\\smallskip\n\\figure{Percolation on the triangular\nlattice: effective decay exponents for the two-point (\\opentri)\nand three-point (\\opentridown) correlation functions plotted\nagainst $L^{-\\omega}$ and their extrapolated values\n(\\fullcirc).\\label{fig5}} \n\\medskip\n\\epsfxsize=7.5truecm\n\\centerline{\\epsfbox{fig6-cond.eps}}\n\\smallskip\n\\figure{Percolation on the square\nlattice: effective decay exponents for the two-point (\\opensqr, \\opendiamond)\nand three-point (\\opentridown) correlation functions plotted\nagainst $L^{-\\omega}$ and their extrapolated values (\\fullcirc).\\label{fig6}} \n\\endinsert \n\\endgroup \n\\par}\n\n\nOn the triangular lattice (figure~\\ref{fig1}), we calculated the two-point\nfunctions $P(\\case{\\pi}{3},\\case{\\pi}{3};L)$, $\\Gamma(\\case{\\pi}{3},\\case{\\pi}{3};L)$ and the three-point\nfunction $P(\\case{\\pi}{3},\\case{\\pi}{3},\\case{\\pi}{3};L)$ whereas on the square lattice\n(figure \\ref{fig4}), we studied the two-point functions $P(\\case{\\pi}{2},\\case{\\pi}{4};L)$,\n$P(\\case{\\pi}{4},\\case{\\pi}{4};L)$, $\\Gamma(\\case{\\pi}{2},\\case{\\pi}{4};L)$, $\\Gamma(\\case{\\pi}{4},\\case{\\pi}{4};L)$ and the\nthree-point function $P(\\case{\\pi}{2},\\case{\\pi}{4},\\case{\\pi}{4};L)$. \n\nThe initial bond configurations were generated using two types of random number\ngenerators, a simple shift register algorithm and the ranlux97 generator. We\nchecked that both generators led to consistent results within the statistical\nerrors. On the square lattice, in order to increase the number of percolating\nsamples, the external bonds connecting $A$ and $B$ to the rest of the\nsystem in figure \\ref{fig4} were always assumed to be conducting.\n \n{\\par\\begingroup\\medskip\n\\epsfxsize=7.5truecm\n\\topinsert\n\\centerline{\\epsfbox{fig7-cond.eps}} \\smallskip\n\\figure{Conduction in triangular-shaped\nsystems: effective decay exponents of the corner-to-corner\nconduction correlation functions on the triangular (\\opentri) and\nthe square (\\opensqr, \\opendiamond) lattices, plotted against $L^{-\\omega}$.\nThe dotted lines are the best linear fits used for the\nextrapolations and the arrow indicates Grassberger's\nresult for the the bulk conductivity exponent.\\label{fig7}} \n\\medskip\n\\table{Exponents governing the decay of the percolation correlation\nfunctions ($\\eta$) and the conduction correlation functions ($\\zeta/\\nu$)\nfor triangular-shaped systems at criticality on the triangular and\nsquare lattices. The numerical values obtained for the percolation problem are\ncompared to the values expected from conformal invariance. The\nlast two lines give the effective correction-to-scaling\nexponents $\\omega$ used in the extrapolation process.\\label{tab1}}[f]\n\\lineup \n\\align\\L{#}&&\\L{#}\\cr \n\\br\n&\\centre{2}{Triangular lattice}&\\centre{3}{Square lattice}\\cr\\ns\n\\crule{1}&\\crule{2}&\\crule{3}\\cr\nAngles& $(\\case{\\pi}{3},\\case{\\pi}{3})$ & $(\\case{\\pi}{3},\\case{\\pi}{3},\\case{\\pi}{3})$ & $(\\case{\\pi}{2},\\case{\\pi}{4})$ &\n$(\\case{\\pi}{4},\\case{\\pi}{4})$ & $(\\case{\\pi}{2},\\case{\\pi}{4},\\case{\\pi}{4})$ \\cr \n$\\eta$ numerical & $1.99\\pm0.02$ &$3.0\\pm0.3$ & $2.01\\pm0.05$ & $2.65\\pm0.07$ &\n$3.32\\pm0.16$ \\cr \n$\\eta$ conf. inv. & $2$ & $3$ & $2$ & $\\case{8}{3}$ & $\\case{10}{3}$ \\cr \n$\\zeta/\\nu$ numerical & $0.9827\\pm0.0017$ & --- & $0.9829\\pm0.0012$ &\n$0.979\\pm0.018$ & --- \\cr \n$\\omega$ percolation & $1.05$ & $0.76$ & $1.14$ & $1.06$ & $1.13$ \\cr\n$\\omega$ conduction & $0.81$ & --- & $0.43$ &$1.02$ & --- \\cr \n\\br\n\\endalign\n\\endtable\n\\endinsert \n\\endgroup \n\\par}\n\nWe used system sizes of the form $L=2^k$ up to $L=256$. The star-triangle\nalgorithm described in section~3 led to a computation time scaling roughly\nas $L^2 \\ln L$. The number of samples generated was $N=2\\times 10^7$ for the\ntriangular lattice, except for the largest size where $N=3\\times 10^7$. On\nthe square lattice, $4\\times 10^7$ samples were generated for all sizes. The\npercolation and conduction correlation functions $P(L)$ and $\\Gamma(L)$ were\nstored as $n$ independent averages over groups of $10^6$ samples in order to\nevaluate the statistical errors. More precisely, as defined in\nequation~\\ref{e2.5}, the conduction correlation function is averaged over\nthat part of the $10^6$ samples for which the two points are connected. \n\nEffective exponents at size $L$ for the percolation and conduction problems were\nobtained using the two-points approximants\n$$\n\\fl\n\\eta_{\\rm eff}(L)={\\ln P(L/2)-\\ln P(2L)\\over\\ln4}\\qquad\n\\zeta/\\nu\\vert_{\\rm\neff}(L)={\\ln\\Gamma(L/2)-\\ln\\Gamma(2L)\\over\\ln4}\\,. \\label{e4.1}\n$$\nThe central value of the approximants are calculated by averaging $P(L)$ and\n$\\Gamma(L)$ over all the samples. The error bars were deduced\nfrom the mean-square deviations $\\sigma^2$ of the approximants, obtained\nwith the $n$ statistically independent averages, as\n$\\pm\\sqrt{\\sigma^2/(n-1)}$.\n\nThe finite-size results for the percolation exponents are shown in\nfigure~\\ref{fig5} for the triangular lattice and figure~\\ref{fig6} for the\nsquare lattice. The results for the conductivity exponents are shown in\nfigure~\\ref{fig7} for both lattices. \n\nThe finite-size results were extrapolated in the following way. Assuming a single\ncorrection-to-scaling exponent $\\omega$, in each case we looked for the\nvalue of $\\omega$ leading to the best linear variation at large size for\nthe central value of the effective exponent as a function of $L^{-\\omega}$.\nThe intercept with the vertical axis gives the extrapolated values\n$\\eta=\\eta_{\\rm eff}(\\infty)$ and $\\zeta/\\nu=\\zeta/\\nu\\vert_{\\rm eff}(\\infty)$. The\nstatistical error on the extrapolated values were deduced, as above, from the\nmean-square deviation of the extrapolated exponents deduced from $n$\nstatistically independent sets of approximants, taking the same value of\n$\\omega$ for the linear fit. To the statistical error, we added a systematic\nerror, linked to the deviation from the asymptotic regime. It was taken as\nthe difference between the extrapolated values when one takes or not into\naccount the largest size in the extrapolation process. Doing so, we probably\noverestimate the systematic error. \n\nDue to the non-monotonous behaviour of the effective exponent for the\nconductance between corners with opening angle $\\case{\\pi}{4}$, the error\nbar on the extrapolated exponent had to be estimated differently. It was\nobtained through an extrapolation of the extreme values of the effective\nexponents using the same method as for the central value. \n\nThe extrapolated exponents for the percolation and conduction problems are shown\nin table~\\ref{tab1}.\n\n\\section{Discussion}\n\nLet us first consider the percolation problem. The decay exponents of the\ntwo- and three-point correlation functions, following from conformal\ninvariance, are easily obtained using equations \\ref{e2.8} and\n(2.10)--(2.12). Our numerical results in table \\ref{tab1} are in good\nagreement with the expected ones, although with a lower precision for the\nthree-point exponents, due to larger statistical errors. \n\nThe percolation probability between two corners $P(\\theta_i,\\theta_j;L)$ can\nbe written as the sum of two contributions. The leading one is the\nprobability that $i$ and $j$ are connected without being connected to the\nthird corner $k$ which scales as $L^{-\\eta(\\theta_i,\\theta_j)}$. The second\nis the probability that the three corners are connected and it decays as\n$L^{-\\eta(\\theta_i,\\theta_j,\\theta_k)}$. Thus we have a\ncorrection-to-scaling exponent equal to $x(\\theta_k)$ for the two-point\npercolation probability. Another correction is due to the leading\nirrelevant operator with scaling dimension $-1$ at the surface\n\\cite{dequeiroz95}. This explains the value, close to $1$, of the effective\ncorrection exponent $\\omega$ for two-point percolation, since the\namplitude of the first correction is small. \n\nOur main result concerns the decay exponent $\\zeta/\\nu$ which is\nequal to $t'$ in two dimensions. The numerical values given in table\n\\ref{tab1} for the different geometries are all quite close to the value of\n$t'$ for the bulk, which is given in equation \\ref{e2.9}. These results,\ntogether with the surface ones \\cite{essam96}, strongly support the\nexistence of a single conductivity scale, independent of the sample\ngeometry. The influence of the opening angles can be seen in the amplitudes\nand perhaps also in the correction-to-scaling exponents. \n\nOne may notice that, apart from the case of the diagonal direction on the\nsquare lattice, the approximants for the conductivity exponents display a\nmonotonous behaviour at large size. This is to be compared to the\nnon-monotonous behaviour observed in reference \\cite{grassberger99} for bond\npercolation conductivity on the square lattice. It makes the\nextrapolation of the exponents more easy, thus reducing the computational\neffort necessary for a given precision on the extrapolated values. \n\nExcept for the diagonal direction on the square lattice, our error bars are\nabout the double of the ones of Grassberger, for a maximum system size\n$L=256$ instead of $L=4096$ in \\cite{grassberger99}. If one accepts the\nuniversality of the conductivity exponent, which is strongly suggested by\nour results, the triangular geometry used here appears as a\npotentially efficient tool to further improve the precision on the value of\n$\\zeta/\\nu$.\n\n\\ack The Laboratoire de Physique des Mat\\'eriaux is Unit\\'e Mixte de\nRecherche CNRS No 7556. The authors are indebted to F Igl\\'oi for useful\ndiscussions. This work has been supported by the Hungarian National\nResearch Fund under grants Nos OTKA F/7/026004, M028418 and by the\nHungarian Ministery of Education under grant No FKFP 0596/1999. PL\nthanks the Soros Foundation, Budapest, for a travelling grant.\n\n\\numreferences\n\\bibitem{cardy83} {Cardy J L 1983}\\ {\\JPA}\\ {\\bf 16}\\ {3617}\n\\bibitem{cardy84} {Cardy J L 1984}\\ {\\NP\\it B}\\ {\\bf 240}\\ {514}\n\\bibitem{barber84} {Barber M N, Peschel I and Pearce P A 1984}\\ {\\it J.\nStat. Phys.}\\ {\\bf 37}\\ {497} \n\\bibitem{igloi93} {Igl\\'oi F, Peschel I and Turban L 1993}\\ {\\it Adv. \nPhys.}\\ {\\bf 42}\\ {683}\n\\bibitem{abraham94} {Abraham D B and Latr\\'emoli\\`ere F T 1994}\\ {\\PR\\\nE}\\ {\\bf 50}\\ {R9} \n\\bibitem{abraham95} {Abraham D B and Latr\\'emoli\\`ere F T 1995}\\ {\\it J.\nStat. 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Phys}\\\n{\\bf 20}\\ {325} \n \n\n\\vfill\\eject\\bye\n"
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"string": "\\hsize=31pc \n\\vsize=49pc \n% \n\\lineskip=0pt \n\\parskip=0pt plus 1pt \n\\hfuzz=1pt \n\\vfuzz=2pt \n\\pretolerance=2500 \n\\tolerance=5000 \n\\vbadness=5000 \n\\hbadness=5000 \n\\widowpenalty=500 \n\\clubpenalty=200 \n\\brokenpenalty=500 \n\\predisplaypenalty=200 \n\\voffset=-1pc \n\\nopagenumbers \n% \n\\catcode`@=11 \n% \n\\newif\\ifams \n\\amsfalse %\\amstrue si � suivant utilise\n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% The following section may be commented out and % \n% \\ifams set to either \\amstrue to use the AMS fonts % \n% or \\amsfalse if they are not available % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n%\\def\\Yesreply{Y } \n%\\def\\Noreply{N } \n%\\def\\yesreply{y } \n%\\def\\noreply{n } \n%\\newif\\ifnotyorn \n%\\message{Do you want to use AMSfonts, msam and msbm? Y or N: }% \n%\\loop \n%\\read-1 to \\reply \n%\\ifx\\reply\\yesreply\\global\\amstrue\\notyornfalse \n%\\else\\ifx\\reply\\Yesreply\\global\\amstrue\\notyornfalse \n%\\else\\ifx\\reply\\noreply\\global\\amsfalse\\notyornfalse \n%\\else\\ifx\\reply\\Noreply\\global\\amsfalse\\notyornfalse \n%\\else\\notyorntrue \n%\\message{Please type y or Y (Yes) or n or N (No)}\\fi\\fi\\fi\\fi \n%\\ifnotyorn\\repeat \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\newfam\\bdifam \n\\newfam\\bsyfam \n\\newfam\\bssfam \n\\newfam\\msafam \n\\newfam\\msbfam \n% \n\\newif\\ifxxpt \n\\newif\\ifxviipt \n\\newif\\ifxivpt \n\\newif\\ifxiipt \n\\newif\\ifxipt \n\\newif\\ifxpt \n\\newif\\ifixpt \n\\newif\\ifviiipt \n\\newif\\ifviipt \n\\newif\\ifvipt \n\\newif\\ifvpt \n% \n% Headings in 20pt, 17pt or 14pt \n% \n\\def\\headsize#1#2{\\def\\headb@seline{#2}% \n \\ifnum#1=20\\def\\HEAD{twenty}% \n \\def\\smHEAD{twelve}% \n \\def\\vsHEAD{nine}% \n \\ifxxpt\\else\\xdef\\f@ntsize{\\HEAD}% \n \\def\\m@g{4}\\def\\s@ze{20.74}% \n \\loadheadfonts\\xxpttrue\\fi \n \\ifxiipt\\else\\xdef\\f@ntsize{\\smHEAD}% \n \\def\\m@g{1}\\def\\s@ze{12}% \n \\loadxiiptfonts\\xiipttrue\\fi \n \\ifixpt\\else\\xdef\\f@ntsize{\\vsHEAD}% \n \\def\\s@ze{9}% \n \\loadsmallfonts\\ixpttrue\\fi \n \\else \n \\ifnum#1=17\\def\\HEAD{seventeen}% \n \\def\\smHEAD{eleven}% \n \\def\\vsHEAD{eight}% \n \\ifxviipt\\else\\xdef\\f@ntsize{\\HEAD}% \n \\def\\m@g{3}\\def\\s@ze{17.28}% \n \\loadheadfonts\\xviipttrue\\fi \n \\ifxipt\\else\\xdef\\f@ntsize{\\smHEAD}% \n \\loadxiptfonts\\xipttrue\\fi \n \\ifviiipt\\else\\xdef\\f@ntsize{\\vsHEAD}% \n \\def\\s@ze{8}% \n \\loadsmallfonts\\viiipttrue\\fi \n \\else\\def\\HEAD{fourteen}% \n \\def\\smHEAD{ten}% \n \\def\\vsHEAD{seven}% \n \\ifxivpt\\else\\xdef\\f@ntsize{\\HEAD}% \n \\def\\m@g{2}\\def\\s@ze{14.4}% \n \\loadheadfonts\\xivpttrue\\fi \n \\ifxpt\\else\\xdef\\f@ntsize{\\smHEAD}% \n \\def\\s@ze{10}% \n \\loadxptfonts\\xpttrue\\fi \n \\ifviipt\\else\\xdef\\f@ntsize{\\vsHEAD}% \n \\def\\s@ze{7}% \n \\loadviiptfonts\\viipttrue\\fi \n \\ifnum#1=14\\else \n \\message{Header size should be 20, 17 or 14 point \n will now default to 14pt}\\fi \n \\fi\\fi\\headfonts} \n% \n% Text in 12pt, 11pt or 10pt \n% \n\\def\\textsize#1#2{\\def\\textb@seline{#2}% \n \\ifnum#1=12\\def\\TEXT{twelve}% \n \\def\\smTEXT{eight}% \n \\def\\vsTEXT{six}% \n \\ifxiipt\\else\\xdef\\f@ntsize{\\TEXT}% \n \\def\\m@g{1}\\def\\s@ze{12}% \n \\loadxiiptfonts\\xiipttrue\\fi \n \\ifviiipt\\else\\xdef\\f@ntsize{\\smTEXT}% \n \\def\\s@ze{8}% \n \\loadsmallfonts\\viiipttrue\\fi \n \\ifvipt\\else\\xdef\\f@ntsize{\\vsTEXT}% \n \\def\\s@ze{6}% \n \\loadviptfonts\\vipttrue\\fi \n \\else \n \\ifnum#1=11\\def\\TEXT{eleven}% \n \\def\\smTEXT{seven}% \n \\def\\vsTEXT{five}% \n \\ifxipt\\else\\xdef\\f@ntsize{\\TEXT}% \n \\def\\s@ze{11}% \n \\loadxiptfonts\\xipttrue\\fi \n \\ifviipt\\else\\xdef\\f@ntsize{\\smTEXT}% \n \\loadviiptfonts\\viipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsTEXT}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\else\\def\\TEXT{ten}% \n \\def\\smTEXT{seven}% \n \\def\\vsTEXT{five}% \n \\ifxpt\\else\\xdef\\f@ntsize{\\TEXT}% \n \\loadxptfonts\\xpttrue\\fi \n \\ifviipt\\else\\xdef\\f@ntsize{\\smTEXT}% \n \\def\\s@ze{7}% \n \\loadviiptfonts\\viipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsTEXT}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\ifnum#1=10\\else \n \\message{Text size should be 12, 11 or 10 point \n will now default to 10pt}\\fi \n \\fi\\fi\\textfonts} \n% \n% Small sized material in 10pt, 9pt or 8pt \n% \n\\def\\smallsize#1#2{\\def\\smallb@seline{#2}% \n \\ifnum#1=10\\def\\SMALL{ten}% \n \\def\\smSMALL{seven}% \n \\def\\vsSMALL{five}% \n \\ifxpt\\else\\xdef\\f@ntsize{\\SMALL}% \n \\loadxptfonts\\xpttrue\\fi \n \\ifviipt\\else\\xdef\\f@ntsize{\\smSMALL}% \n \\def\\s@ze{7}% \n \\loadviiptfonts\\viipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsSMALL}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\else \n \\ifnum#1=9\\def\\SMALL{nine}% \n \\def\\smSMALL{six}% \n \\def\\vsSMALL{five}% \n \\ifixpt\\else\\xdef\\f@ntsize{\\SMALL}% \n \\def\\s@ze{9}% \n \\loadsmallfonts\\ixpttrue\\fi \n \\ifvipt\\else\\xdef\\f@ntsize{\\smSMALL}% \n \\def\\s@ze{6}% \n \\loadviptfonts\\vipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsSMALL}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\else \n \\def\\SMALL{eight}% \n \\def\\smSMALL{six}% \n \\def\\vsSMALL{five}% \n \\ifviiipt\\else\\xdef\\f@ntsize{\\SMALL}% \n \\def\\s@ze{8}% \n \\loadsmallfonts\\viiipttrue\\fi \n \\ifvipt\\else\\xdef\\f@ntsize{\\smSMALL}% \n \\def\\s@ze{6}% \n \\loadviptfonts\\vipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsSMALL}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\ifnum#1=8\\else\\message{Small size should be 10, 9 or \n 8 point will now default to 8pt}\\fi \n \\fi\\fi\\smallfonts} \n% \n\\def\\F@nt{\\expandafter\\font\\csname} \n\\def\\Sk@w{\\expandafter\\skewchar\\csname} \n\\def\\@nd{\\endcsname} \n\\def\\@step#1{ scaled \\magstep#1} \n\\def\\@half{ scaled \\magstephalf} \n\\def\\@t#1{ at #1pt} \n% \n% For 14, 17 and 20 point fonts use \\loadheadfonts \n% \n\\def\\loadheadfonts{\\bigf@nts \n\\F@nt \\f@ntsize bdi\\@nd=cmmib10 \\@t{\\s@ze}% \n\\Sk@w \\f@ntsize bdi\\@nd='177 \n\\F@nt \\f@ntsize bsy\\@nd=cmbsy10 \\@t{\\s@ze}% \n\\Sk@w \\f@ntsize bsy\\@nd='60 \n\\F@nt \\f@ntsize bss\\@nd=cmssbx10 \\@t{\\s@ze}} \n% \n% For 12 point fonts use \\loadxiiptfonts \n% \n\\def\\loadxiiptfonts{\\bigf@nts \n\\F@nt \\f@ntsize bdi\\@nd=cmmib10 \\@step{\\m@g}% \n\\Sk@w \\f@ntsize bdi\\@nd='177 \n\\F@nt \\f@ntsize bsy\\@nd=cmbsy10 \\@step{\\m@g}% \n\\Sk@w \\f@ntsize bsy\\@nd='60 \n\\F@nt \\f@ntsize bss\\@nd=cmssbx10 \\@step{\\m@g}} \n% \n% For 11 point fonts use \\loadxiptfonts \n% \n\\def\\loadxiptfonts{% \n\\font\\elevenrm=cmr10 \\@half \n\\font\\eleveni=cmmi10 \\@half \n\\skewchar\\eleveni='177 \n\\font\\elevensy=cmsy10 \\@half \n\\skewchar\\elevensy='60 \n\\font\\elevenex=cmex10 \\@half \n\\font\\elevenit=cmti10 \\@half \n\\font\\elevensl=cmsl10 \\@half \n\\font\\elevenbf=cmbx10 \\@half \n\\font\\eleventt=cmtt10 \\@half \n\\ifams\\font\\elevenmsa=msam10 \\@half \n\\font\\elevenmsb=msbm10 \\@half\\else\\fi \n\\font\\elevenbdi=cmmib10 \\@half \n\\skewchar\\elevenbdi='177 \n\\font\\elevenbsy=cmbsy10 \\@half \n\\skewchar\\elevenbsy='60 \n\\font\\elevenbss=cmssbx10 \\@half} \n% \n% For 10 point fonts use \\loadxptfonts \n% \n\\def\\loadxptfonts{% \n\\font\\tenbdi=cmmib10 \n\\skewchar\\tenbdi='177 \n\\font\\tenbsy=cmbsy10 \n\\skewchar\\tenbsy='60 \n\\ifams\\font\\tenmsa=msam10 \n\\font\\tenmsb=msbm10\\else\\fi \n\\font\\tenbss=cmssbx10}% \n% \n% For 8 and 9 point fonts use \\loadsmallfonts \n% \n\\def\\loadsmallfonts{\\smallf@nts \n\\ifams \n\\F@nt \\f@ntsize ex\\@nd=cmex\\s@ze \n\\else \n\\F@nt \\f@ntsize ex\\@nd=cmex10\\fi \n\\F@nt \\f@ntsize it\\@nd=cmti\\s@ze \n\\F@nt \\f@ntsize sl\\@nd=cmsl\\s@ze \n\\F@nt \\f@ntsize tt\\@nd=cmtt\\s@ze} \n% \n% For 7 point fonts use \\loadviiptfonts \n% \n\\def\\loadviiptfonts{% \n\\font\\sevenit=cmti7 \n\\font\\sevensl=cmsl8 at 7pt \n\\ifams\\font\\sevenmsa=msam7 \n\\font\\sevenmsb=msbm7 \n\\font\\sevenex=cmex7 \n\\font\\sevenbsy=cmbsy7 \n\\font\\sevenbdi=cmmib7\\else \n\\font\\sevenex=cmex10 \n\\font\\sevenbsy=cmbsy10 at 7pt \n\\font\\sevenbdi=cmmib10 at 7pt\\fi \n\\skewchar\\sevenbsy='60 \n\\skewchar\\sevenbdi='177 \n\\font\\sevenbss=cmssbx10 at 7pt}% \n% \n% For 6 point fonts use \\loadviptfonts \n% \n\\def\\loadviptfonts{\\smallf@nts \n\\ifams\\font\\sixex=cmex7 at 6pt\\else \n\\font\\sixex=cmex10\\fi \n\\font\\sixit=cmti7 at 6pt} \n% \n% For 5 point fonts use \\loadvptfonts \n% \n\\def\\loadvptfonts{% \n\\font\\fiveit=cmti7 at 5pt \n\\ifams\\font\\fiveex=cmex7 at 5pt \n\\font\\fivebdi=cmmib5 \n\\font\\fivebsy=cmbsy5 \n\\font\\fivemsa=msam5 \n\\font\\fivemsb=msbm5\\else \n\\font\\fiveex=cmex10 \n\\font\\fivebdi=cmmib10 at 5pt \n\\font\\fivebsy=cmbsy10 at 5pt\\fi \n\\skewchar\\fivebdi='177 \n\\skewchar\\fivebsy='60 \n\\font\\fivebss=cmssbx10 at 5pt} \n% \n\\def\\bigf@nts{% \n\\F@nt \\f@ntsize rm\\@nd=cmr10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize i\\@nd=cmmi10 \\@step{\\m@g}% \n\\Sk@w \\f@ntsize i\\@nd='177 \n\\F@nt \\f@ntsize sy\\@nd=cmsy10 \\@step{\\m@g}% \n\\Sk@w \\f@ntsize sy\\@nd='60 \n\\F@nt \\f@ntsize ex\\@nd=cmex10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize it\\@nd=cmti10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize sl\\@nd=cmsl10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize bf\\@nd=cmbx10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize tt\\@nd=cmtt10 \\@step{\\m@g}% \n\\ifams \n\\F@nt \\f@ntsize msa\\@nd=msam10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize msb\\@nd=msbm10 \\@step{\\m@g}\\else\\fi} \n% \n\\def\\smallf@nts{% \n\\F@nt \\f@ntsize rm\\@nd=cmr\\s@ze \n\\F@nt \\f@ntsize i\\@nd=cmmi\\s@ze \n\\Sk@w \\f@ntsize i\\@nd='177 \n\\F@nt \\f@ntsize sy\\@nd=cmsy\\s@ze \n\\Sk@w \\f@ntsize sy\\@nd='60 \n\\F@nt \\f@ntsize bf\\@nd=cmbx\\s@ze \n\\ifams \n\\F@nt \\f@ntsize bdi\\@nd=cmmib\\s@ze \n\\F@nt \\f@ntsize bsy\\@nd=cmbsy\\s@ze \n\\F@nt \\f@ntsize msa\\@nd=msam\\s@ze \n\\F@nt \\f@ntsize msb\\@nd=msbm\\s@ze \n\\else \n\\F@nt \\f@ntsize bdi\\@nd=cmmib10 \\@t{\\s@ze}% \n\\F@nt \\f@ntsize bsy\\@nd=cmbsy10 \\@t{\\s@ze}\\fi \n\\Sk@w \\f@ntsize bdi\\@nd='177 \n\\Sk@w \\f@ntsize bsy\\@nd='60 \n\\F@nt \\f@ntsize bss\\@nd=cmssbx10 \\@t{\\s@ze}}% \n% \n% Fonts for headings \n% \n\\def\\headfonts{% \n\\textfont0=\\csname\\HEAD rm\\@nd \n\\scriptfont0=\\csname\\smHEAD rm\\@nd \n\\scriptscriptfont0=\\csname\\vsHEAD rm\\@nd \n\\def\\rm{\\fam0\\csname\\HEAD rm\\@nd \n\\def\\sc{\\csname\\smHEAD rm\\@nd}}% \n% \n\\textfont1=\\csname\\HEAD i\\@nd \n\\scriptfont1=\\csname\\smHEAD i\\@nd \n\\scriptscriptfont1=\\csname\\vsHEAD i\\@nd \n% \n\\textfont2=\\csname\\HEAD sy\\@nd \n\\scriptfont2=\\csname\\smHEAD sy\\@nd \n\\scriptscriptfont2=\\csname\\vsHEAD sy\\@nd \n% \n\\textfont3=\\csname\\HEAD ex\\@nd \n\\scriptfont3=\\csname\\smHEAD ex\\@nd \n\\scriptscriptfont3=\\csname\\smHEAD ex\\@nd \n% \n\\textfont\\itfam=\\csname\\HEAD it\\@nd \n\\scriptfont\\itfam=\\csname\\smHEAD it\\@nd \n\\scriptscriptfont\\itfam=\\csname\\vsHEAD it\\@nd \n\\def\\it{\\fam\\itfam\\csname\\HEAD it\\@nd \n\\def\\sc{\\csname\\smHEAD it\\@nd}}% \n% \n\\textfont\\slfam=\\csname\\HEAD sl\\@nd \n\\def\\sl{\\fam\\slfam\\csname\\HEAD sl\\@nd \n\\def\\sc{\\csname\\smHEAD sl\\@nd}}% \n% \n\\textfont\\bffam=\\csname\\HEAD bf\\@nd \n\\scriptfont\\bffam=\\csname\\smHEAD bf\\@nd \n\\scriptscriptfont\\bffam=\\csname\\vsHEAD bf\\@nd \n\\def\\bf{\\fam\\bffam\\csname\\HEAD bf\\@nd \n\\def\\sc{\\csname\\smHEAD bf\\@nd}}% \n% \n\\textfont\\ttfam=\\csname\\HEAD tt\\@nd \n\\def\\tt{\\fam\\ttfam\\csname\\HEAD tt\\@nd}% \n% \n\\textfont\\bdifam=\\csname\\HEAD bdi\\@nd \n\\scriptfont\\bdifam=\\csname\\smHEAD bdi\\@nd \n\\scriptscriptfont\\bdifam=\\csname\\vsHEAD bdi\\@nd \n\\def\\bdi{\\fam\\bdifam\\csname\\HEAD bdi\\@nd}% \n% \n\\textfont\\bsyfam=\\csname\\HEAD bsy\\@nd \n\\scriptfont\\bsyfam=\\csname\\smHEAD bsy\\@nd \n\\def\\bsy{\\fam\\bsyfam\\csname\\HEAD bsy\\@nd}% \n% \n\\textfont\\bssfam=\\csname\\HEAD bss\\@nd \n\\scriptfont\\bssfam=\\csname\\smHEAD bss\\@nd \n\\scriptscriptfont\\bssfam=\\csname\\vsHEAD bss\\@nd \n\\def\\bss{\\fam\\bssfam\\csname\\HEAD bss\\@nd}% \n% \n\\ifams \n\\textfont\\msafam=\\csname\\HEAD msa\\@nd \n\\scriptfont\\msafam=\\csname\\smHEAD msa\\@nd \n\\scriptscriptfont\\msafam=\\csname\\vsHEAD msa\\@nd \n% \n\\textfont\\msbfam=\\csname\\HEAD msb\\@nd \n\\scriptfont\\msbfam=\\csname\\smHEAD msb\\@nd \n\\scriptscriptfont\\msbfam=\\csname\\vsHEAD msb\\@nd \n\\else\\fi \n% \n\\normalbaselineskip=\\headb@seline pt% \n\\setbox\\strutbox=\\hbox{\\vrule height.7\\normalbaselineskip \ndepth.3\\baselineskip width0pt}% \n\\def\\sc{\\csname\\smHEAD rm\\@nd}\\normalbaselines\\bf} \n% \n% Fonts for text \n% \n\\def\\textfonts{% \n\\textfont0=\\csname\\TEXT rm\\@nd \n\\scriptfont0=\\csname\\smTEXT rm\\@nd \n\\scriptscriptfont0=\\csname\\vsTEXT rm\\@nd \n\\def\\rm{\\fam0\\csname\\TEXT rm\\@nd \n\\def\\sc{\\csname\\smTEXT rm\\@nd}}% \n% \n\\textfont1=\\csname\\TEXT i\\@nd \n\\scriptfont1=\\csname\\smTEXT i\\@nd \n\\scriptscriptfont1=\\csname\\vsTEXT i\\@nd \n% \n\\textfont2=\\csname\\TEXT sy\\@nd \n\\scriptfont2=\\csname\\smTEXT sy\\@nd \n\\scriptscriptfont2=\\csname\\vsTEXT sy\\@nd \n% \n\\textfont3=\\csname\\TEXT ex\\@nd \n\\scriptfont3=\\csname\\smTEXT ex\\@nd \n\\scriptscriptfont3=\\csname\\smTEXT ex\\@nd \n% \n\\textfont\\itfam=\\csname\\TEXT it\\@nd \n\\scriptfont\\itfam=\\csname\\smTEXT it\\@nd \n\\scriptscriptfont\\itfam=\\csname\\vsTEXT it\\@nd \n\\def\\it{\\fam\\itfam\\csname\\TEXT it\\@nd \n\\def\\sc{\\csname\\smTEXT it\\@nd}}% \n% \n\\textfont\\slfam=\\csname\\TEXT sl\\@nd \n\\def\\sl{\\fam\\slfam\\csname\\TEXT sl\\@nd \n\\def\\sc{\\csname\\smTEXT sl\\@nd}}% \n% \n\\textfont\\bffam=\\csname\\TEXT bf\\@nd \n\\scriptfont\\bffam=\\csname\\smTEXT bf\\@nd \n\\scriptscriptfont\\bffam=\\csname\\vsTEXT bf\\@nd \n\\def\\bf{\\fam\\bffam\\csname\\TEXT bf\\@nd \n\\def\\sc{\\csname\\smTEXT bf\\@nd}}% \n% \n\\textfont\\ttfam=\\csname\\TEXT tt\\@nd \n\\def\\tt{\\fam\\ttfam\\csname\\TEXT tt\\@nd}% \n% \n\\textfont\\bdifam=\\csname\\TEXT bdi\\@nd \n\\scriptfont\\bdifam=\\csname\\smTEXT bdi\\@nd \n\\scriptscriptfont\\bdifam=\\csname\\vsTEXT bdi\\@nd \n\\def\\bdi{\\fam\\bdifam\\csname\\TEXT bdi\\@nd}% \n% \n\\textfont\\bsyfam=\\csname\\TEXT bsy\\@nd \n\\scriptfont\\bsyfam=\\csname\\smTEXT bsy\\@nd \n\\def\\bsy{\\fam\\bsyfam\\csname\\TEXT bsy\\@nd}% \n% \n\\textfont\\bssfam=\\csname\\TEXT bss\\@nd \n\\scriptfont\\bssfam=\\csname\\smTEXT bss\\@nd \n\\scriptscriptfont\\bssfam=\\csname\\vsTEXT bss\\@nd \n\\def\\bss{\\fam\\bssfam\\csname\\TEXT bss\\@nd}% \n% \n\\ifams \n\\textfont\\msafam=\\csname\\TEXT msa\\@nd \n\\scriptfont\\msafam=\\csname\\smTEXT msa\\@nd \n\\scriptscriptfont\\msafam=\\csname\\vsTEXT msa\\@nd \n% \n\\textfont\\msbfam=\\csname\\TEXT msb\\@nd \n\\scriptfont\\msbfam=\\csname\\smTEXT msb\\@nd \n\\scriptscriptfont\\msbfam=\\csname\\vsTEXT msb\\@nd \n\\else\\fi \n% \n\\normalbaselineskip=\\textb@seline pt \n\\setbox\\strutbox=\\hbox{\\vrule height.7\\normalbaselineskip \ndepth.3\\baselineskip width0pt}% \n\\everymath{}% \n\\def\\sc{\\csname\\smTEXT rm\\@nd}\\normalbaselines\\rm} \n% \n% Fonts for small material (captions, footnotes etc) \n% \n\\def\\smallfonts{% \n\\textfont0=\\csname\\SMALL rm\\@nd \n\\scriptfont0=\\csname\\smSMALL rm\\@nd \n\\scriptscriptfont0=\\csname\\vsSMALL rm\\@nd \n\\def\\rm{\\fam0\\csname\\SMALL rm\\@nd \n\\def\\sc{\\csname\\smSMALL rm\\@nd}}% \n% \n\\textfont1=\\csname\\SMALL i\\@nd \n\\scriptfont1=\\csname\\smSMALL i\\@nd \n\\scriptscriptfont1=\\csname\\vsSMALL i\\@nd \n% \n\\textfont2=\\csname\\SMALL sy\\@nd \n\\scriptfont2=\\csname\\smSMALL sy\\@nd \n\\scriptscriptfont2=\\csname\\vsSMALL sy\\@nd \n% \n\\textfont3=\\csname\\SMALL ex\\@nd \n\\scriptfont3=\\csname\\smSMALL ex\\@nd \n\\scriptscriptfont3=\\csname\\smSMALL ex\\@nd \n% \n\\textfont\\itfam=\\csname\\SMALL it\\@nd \n\\scriptfont\\itfam=\\csname\\smSMALL it\\@nd \n\\scriptscriptfont\\itfam=\\csname\\vsSMALL it\\@nd \n\\def\\it{\\fam\\itfam\\csname\\SMALL it\\@nd \n\\def\\sc{\\csname\\smSMALL it\\@nd}}% \n% \n\\textfont\\slfam=\\csname\\SMALL sl\\@nd \n\\def\\sl{\\fam\\slfam\\csname\\SMALL sl\\@nd \n\\def\\sc{\\csname\\smSMALL sl\\@nd}}% \n% \n\\textfont\\bffam=\\csname\\SMALL bf\\@nd \n\\scriptfont\\bffam=\\csname\\smSMALL bf\\@nd \n\\scriptscriptfont\\bffam=\\csname\\vsSMALL bf\\@nd \n\\def\\bf{\\fam\\bffam\\csname\\SMALL bf\\@nd \n\\def\\sc{\\csname\\smSMALL bf\\@nd}}% \n% \n\\textfont\\ttfam=\\csname\\SMALL tt\\@nd \n\\def\\tt{\\fam\\ttfam\\csname\\SMALL tt\\@nd}% \n% \n\\textfont\\bdifam=\\csname\\SMALL bdi\\@nd \n\\scriptfont\\bdifam=\\csname\\smSMALL bdi\\@nd \n\\scriptscriptfont\\bdifam=\\csname\\vsSMALL bdi\\@nd \n\\def\\bdi{\\fam\\bdifam\\csname\\SMALL bdi\\@nd}% \n% \n\\textfont\\bsyfam=\\csname\\SMALL bsy\\@nd \n\\scriptfont\\bsyfam=\\csname\\smSMALL bsy\\@nd \n\\def\\bsy{\\fam\\bsyfam\\csname\\SMALL bsy\\@nd}% \n% \n\\textfont\\bssfam=\\csname\\SMALL bss\\@nd \n\\scriptfont\\bssfam=\\csname\\smSMALL bss\\@nd \n\\scriptscriptfont\\bssfam=\\csname\\vsSMALL bss\\@nd \n\\def\\bss{\\fam\\bssfam\\csname\\SMALL bss\\@nd}% \n% \n\\ifams \n\\textfont\\msafam=\\csname\\SMALL msa\\@nd \n\\scriptfont\\msafam=\\csname\\smSMALL msa\\@nd \n\\scriptscriptfont\\msafam=\\csname\\vsSMALL msa\\@nd \n% \n\\textfont\\msbfam=\\csname\\SMALL msb\\@nd \n\\scriptfont\\msbfam=\\csname\\smSMALL msb\\@nd \n\\scriptscriptfont\\msbfam=\\csname\\vsSMALL msb\\@nd \n\\else\\fi \n% \n\\normalbaselineskip=\\smallb@seline pt% \n\\setbox\\strutbox=\\hbox{\\vrule height.7\\normalbaselineskip \ndepth.3\\baselineskip width0pt}% \n\\everymath{}% \n\\def\\sc{\\csname\\smSMALL rm\\@nd}\\normalbaselines\\rm}% \n% \n\\everydisplay{\\indenteddisplay \n \\gdef\\labeltype{\\eqlabel}}% \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Macros to define extra maths symbols % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\hexnumber@#1{\\ifcase#1 0\\or 1\\or 2\\or 3\\or 4\\or 5\\or 6\\or 7\\or 8\\or \n 9\\or A\\or B\\or C\\or D\\or E\\or F\\fi} \n\\edef\\bffam@{\\hexnumber@\\bffam} \n\\edef\\bdifam@{\\hexnumber@\\bdifam} \n\\edef\\bsyfam@{\\hexnumber@\\bsyfam} \n% \n\\def\\undefine#1{\\let#1\\undefined} \n\\def\\newsymbol#1#2#3#4#5{\\let\\next@\\relax \n \\ifnum#2=\\thr@@\\let\\next@\\bdifam@\\else \n \\ifams \n \\ifnum#2=\\@ne\\let\\next@\\msafam@\\else \n \\ifnum#2=\\tw@\\let\\next@\\msbfam@\\fi\\fi \n \\fi\\fi \n \\mathchardef#1=\"#3\\next@#4#5} \n\\def\\mathhexbox@#1#2#3{\\relax \n \\ifmmode\\mathpalette{}{\\m@th\\mathchar\"#1#2#3}% \n \\else\\leavevmode\\hbox{$\\m@th\\mathchar\"#1#2#3$}\\fi} \n% \n\\def\\bcal{\\fam\\bsyfam\\relax} \n\\def\\bi#1{{\\fam\\bdifam\\relax#1}} \n% \n% If file amsmacro is not in current directory \n% or somewhere with set path add path before \n% file name in following line \n% \n\\ifams\\input amsmacro\\fi \n% \n% Bold italic Greek characters \n% \n\\newsymbol\\bitGamma 3000 \n\\newsymbol\\bitDelta 3001 \n\\newsymbol\\bitTheta 3002 \n\\newsymbol\\bitLambda 3003 \n\\newsymbol\\bitXi 3004 \n\\newsymbol\\bitPi 3005 \n\\newsymbol\\bitSigma 3006 \n\\newsymbol\\bitUpsilon 3007 \n\\newsymbol\\bitPhi 3008 \n\\newsymbol\\bitPsi 3009 \n\\newsymbol\\bitOmega 300A \n\\newsymbol\\balpha 300B \n\\newsymbol\\bbeta 300C \n\\newsymbol\\bgamma 300D \n\\newsymbol\\bdelta 300E \n\\newsymbol\\bepsilon 300F \n\\newsymbol\\bzeta 3010 \n\\newsymbol\\bfeta 3011 \n\\newsymbol\\btheta 3012 \n\\newsymbol\\biota 3013 \n\\newsymbol\\bkappa 3014 \n\\newsymbol\\blambda 3015 \n\\newsymbol\\bmu 3016 \n\\newsymbol\\bnu 3017 \n\\newsymbol\\bxi 3018 \n\\newsymbol\\bpi 3019 \n\\newsymbol\\brho 301A \n\\newsymbol\\bsigma 301B \n\\newsymbol\\btau 301C \n\\newsymbol\\bupsilon 301D \n\\newsymbol\\bphi 301E \n\\newsymbol\\bchi 301F \n\\newsymbol\\bpsi 3020 \n\\newsymbol\\bomega 3021 \n\\newsymbol\\bvarepsilon 3022 \n\\newsymbol\\bvartheta 3023 \n\\newsymbol\\bvaromega 3024 \n\\newsymbol\\bvarrho 3025 \n\\newsymbol\\bvarzeta 3026 \n\\newsymbol\\bvarphi 3027 \n\\newsymbol\\bpartial 3040 \n\\newsymbol\\bell 3060 \n\\newsymbol\\bimath 307B \n\\newsymbol\\bjmath 307C \n% \n\\mathchardef\\binfty \"0\\bsyfam@31 \n\\mathchardef\\bnabla \"0\\bsyfam@72 \n\\mathchardef\\bdot \"2\\bsyfam@01 \n\\mathchardef\\bGamma \"0\\bffam@00 \n\\mathchardef\\bDelta \"0\\bffam@01 \n\\mathchardef\\bTheta \"0\\bffam@02 \n\\mathchardef\\bLambda \"0\\bffam@03 \n\\mathchardef\\bXi \"0\\bffam@04 \n\\mathchardef\\bPi \"0\\bffam@05 \n\\mathchardef\\bSigma \"0\\bffam@06 \n\\mathchardef\\bUpsilon \"0\\bffam@07 \n\\mathchardef\\bPhi \"0\\bffam@08 \n\\mathchardef\\bPsi \"0\\bffam@09 \n\\mathchardef\\bOmega \"0\\bffam@0A \n% \n\\mathchardef\\itGamma \"0100 \n\\mathchardef\\itDelta \"0101 \n\\mathchardef\\itTheta \"0102 \n\\mathchardef\\itLambda \"0103 \n\\mathchardef\\itXi \"0104 \n\\mathchardef\\itPi \"0105 \n\\mathchardef\\itSigma \"0106 \n\\mathchardef\\itUpsilon \"0107 \n\\mathchardef\\itPhi \"0108 \n\\mathchardef\\itPsi \"0109 \n\\mathchardef\\itOmega \"010A \n% \n\\mathchardef\\Gamma \"0000 \n\\mathchardef\\Delta \"0001 \n\\mathchardef\\Theta \"0002 \n\\mathchardef\\Lambda \"0003 \n\\mathchardef\\Xi \"0004 \n\\mathchardef\\Pi \"0005 \n\\mathchardef\\Sigma \"0006 \n\\mathchardef\\Upsilon \"0007 \n\\mathchardef\\Phi \"0008 \n\\mathchardef\\Psi \"0009 \n\\mathchardef\\Omega \"000A \n% \n% Counter definitions \n% \n\\newcount\\firstpage \\firstpage=1 % start page no \n\\newcount\\jnl % journal no \n\\newcount\\secno % section number \n\\newcount\\subno % number of subsection \n\\newcount\\subsubno % number of subsubsection \n\\newcount\\appno % appendix number \n\\newcount\\tabno % table number \n\\newcount\\figno % figure number \n\\newcount\\countno % equation numbers \n\\newcount\\refno % reference number \n\\newcount\\eqlett \\eqlett=97 % equation letter \n% \n\\newif\\ifletter \n\\newif\\ifwide \n\\newif\\ifnotfull \n\\newif\\ifaligned \n\\newif\\ifnumbysec \n\\newif\\ifappendix \n\\newif\\ifnumapp \n\\newif\\ifssf \n\\newif\\ifppt \n% \n\\newdimen\\t@bwidth \n\\newdimen\\c@pwidth \n\\newdimen\\digitwidth %character width \n\\newdimen\\argwidth %argument width \n\\newdimen\\secindent \\secindent=5pc %indentation of maths \n\\newdimen\\textind \\textind=16pt %indentation of text \n\\newdimen\\tempval %temporary value \n\\newskip\\beforesecskip \n\\def\\beforesecspace{\\vskip\\beforesecskip\\relax} \n\\newskip\\beforesubskip \n\\def\\beforesubspace{\\vskip\\beforesubskip\\relax} \n\\newskip\\beforesubsubskip \n\\def\\beforesubsubspace{\\vskip\\beforesubsubskip\\relax} \n\\newskip\\secskip \n\\def\\secspace{\\vskip\\secskip\\relax} \n\\newskip\\subskip \n\\def\\subspace{\\vskip\\subskip\\relax} \n\\newskip\\insertskip \n\\def\\insertspace{\\vskip\\insertskip\\relax} \n% \n\\def\\sp@ce{\\ifx\\next*\\let\\next=\\@ssf \n \\else\\let\\next=\\@nossf\\fi\\next} \n\\def\\@ssf#1{\\nobreak\\secspace\\global\\ssftrue\\nobreak} \n\\def\\@nossf{\\nobreak\\secspace\\nobreak\\noindent\\ignorespaces} \n% \n\\def\\subsp@ce{\\ifx\\next*\\let\\next=\\@sssf \n \\else\\let\\next=\\@nosssf\\fi\\next} \n\\def\\@sssf#1{\\nobreak\\subspace\\global\\ssftrue\\nobreak} \n\\def\\@nosssf{\\nobreak\\subspace\\nobreak\\noindent\\ignorespaces} \n% \n\\beforesecskip=24pt plus12pt minus8pt \n\\beforesubskip=12pt plus6pt minus4pt \n\\beforesubsubskip=12pt plus6pt minus4pt \n\\secskip=12pt plus 2pt minus 2pt \n\\subskip=6pt plus3pt minus2pt \n\\insertskip=18pt plus6pt minus6pt% \n\\fontdimen16\\tensy=2.7pt \n\\fontdimen17\\tensy=2.7pt \n% \n% Labels etc for cross referencing macros \n% \n\\def\\eqlabel{(\\ifappendix\\applett \n \\ifnumbysec\\ifnum\\secno>0 \\the\\secno\\fi.\\fi \n \\else\\ifnumbysec\\the\\secno.\\fi\\fi\\the\\countno)} \n\\def\\seclabel{\\ifappendix\\ifnumapp\\else\\applett\\fi \n \\ifnum\\secno>0 \\the\\secno \n \\ifnumbysec\\ifnum\\subno>0.\\the\\subno\\fi\\fi\\fi \n \\else\\the\\secno\\fi\\ifnum\\subno>0.\\the\\subno \n \\ifnum\\subsubno>0.\\the\\subsubno\\fi\\fi} \n\\def\\tablabel{\\ifappendix\\applett\\fi\\the\\tabno} \n\\def\\figlabel{\\ifappendix\\applett\\fi\\the\\figno} \n% \n\\def\\gac{\\global\\advance\\countno by 1} \n% \n% Redefinition of footnote macros to lose rule and remove indentation \n% \n\\def\\footnoterule{} \n\\def\\vfootnote#1{\\insert\\footins\\bgroup \n\\interlinepenalty=\\interfootnotelinepenalty \n\\splittopskip=\\ht\\strutbox % top baseline for broken footnotes \n\\splitmaxdepth=\\dp\\strutbox \\floatingpenalty=20000 \n\\leftskip=0pt \\rightskip=0pt \\spaceskip=0pt \\xspaceskip=0pt% \n\\noindent\\smallfonts\\rm #1\\ \\ignorespaces\\footstrut\\futurelet\\next\\fo@t} \n% \n% Redefinition of endinsert to give more controllable \n% space around tables and figures \n% \n\\def\\endinsert{\\egroup \n \\if@mid \\dimen@=\\ht0 \\advance\\dimen@ by\\dp0 \n \\advance\\dimen@ by12\\p@ \\advance\\dimen@ by\\pagetotal \n \\ifdim\\dimen@>\\pagegoal \\@midfalse\\p@gefalse\\fi\\fi \n \\if@mid \\insertspace \\box0 \\par \\ifdim\\lastskip<\\insertskip \n \\removelastskip \\penalty-200 \\insertspace \\fi \n \\else\\insert\\topins{\\penalty100 \n \\splittopskip=0pt \\splitmaxdepth=\\maxdimen \n \\floatingpenalty=0 \n \\ifp@ge \\dimen@=\\dp0 \n \\vbox to\\vsize{\\unvbox0 \\kern-\\dimen@}% \n \\else\\box0\\nobreak\\insertspace\\fi}\\fi\\endgroup} \n% \n% special macros for display equations \n% \n% for indentation of turned over lines in mathematics \n% \n\\def\\ind{\\hbox to \\secindent{\\hfill}} \n% \n% for turned over equals sign to left of maths indent \n% \n\\def\\eql{\\llap{${}={}$}} \n\\def\\lsim{\\llap{${}\\sim{}$}} \n\\def\\lsimeq{\\llap{${}\\simeq{}$}} \n\\def\\lequiv{\\llap{${}\\equiv{}$}} \n% \n% for other signs to left of maths indent \n% \n\\def\\lo#1{\\llap{${}#1{}$}} \n% \n% displayed equation indented \n% \n\\def\\indeqn#1{\\alignedfalse\\displ@y\\halign{\\hbox to \\displaywidth \n {$\\ind\\@lign\\displaystyle##\\hfil$}\\crcr #1\\crcr}} \n% \n% displayed equation indented with alignments \n% \n\\def\\indalign#1{\\alignedtrue\\displ@y \\tabskip=0pt \n \\halign to\\displaywidth{\\ind$\\@lign\\displaystyle{##}$\\tabskip=0pt \n &$\\@lign\\displaystyle{{}##}$\\hfill\\tabskip=\\centering \n &\\llap{$\\@lign\\hbox{\\rm##}$}\\tabskip=0pt\\crcr \n #1\\crcr}} \n% \n\\def\\fl{{\\hskip-\\secindent}} \n% \n\\def\\indenteddisplay#1$${\\indispl@y{#1 }} \n\\def\\indispl@y#1{\\disptest#1\\eqalignno\\eqalignno\\disptest} \n\\def\\disptest#1\\eqalignno#2\\eqalignno#3\\disptest{% \n \\ifx#3\\eqalignno \n \\indalign#2% \n \\else\\indeqn{#1}\\fi$$} \n% \n% Roman small caps (if in Roman \\sc gives small caps) \n% \n\\def\\rmsc{\\rm\\sc} \n% \n% Italic small caps (if in italic \\sc gives italic small caps) \n% \n\\def\\itsc{\\it\\sc} \n% \n% Bold small caps (if in bold \\sc gives bold small caps) \n% \n\\def\\bfsc{\\bf\\rm} \n% \n% Small caps in maths \n% \n\\def\\msc#1{\\hbox{\\sc #1}} \n% \n% Miscellaneous definitions \n% \n\\def\\ms{\\noalign{\\vskip3pt plus3pt minus2pt}} \n\\def\\bs{\\noalign{\\vskip6pt plus3pt minus2pt}} \n\\def\\ns{\\noalign{\\vskip-3pt}} \n\\def\\bk{\\noalign{\\break}} \n\\def\\nb{\\noalign{\\nobreak}} \n% \n\\def\\hfb{\\hfill\\break} \n% \n% Bold h bar \n% \n\\def\\bhbar{\\rlap{\\kern1pt\\raise.4ex\\hbox{\\bf\\char'40}}\\bi{h}} \n% \n\\def\\case#1#2{{\\textstyle{#1\\over#2}}} \n\\def\\dash{---{}--- } \n\\let\\du=\\d \n\\def\\d{{\\rm d}} \n\\def\\e{{\\rm e}} \n\\def\\etal{{\\it et al\\/}\\ } \n\\def\\frac#1#2{{#1\\over#2}} \n\\ifams \n\\def\\lap{\\lesssim} \n\\def\\gap{\\gtrsim} \n\\let\\le=\\leqslant \n\\let\\lequal=\\leq \n\\let\\leq=\\leqslant \n\\let\\ge=\\geqslant \n\\let\\gequal=\\geq \n\\let\\geq=\\geqslant \n\\else \n\\let\\lequal=\\le \n\\let\\gequal=\\ge \n\\def\\gap{\\;\\lower3pt\\hbox{$\\buildrel > \\over \\sim$}\\;}% \n\\def\\lap{\\;\\lower3pt\\hbox{$\\buildrel < \\over \\sim$}\\;}\\fi \n\\def\\i{{\\rm i}} \n\\chardef\\ii=\"10 \n\\def\\tqs{\\hbox to 25pt{\\hfil}} \n\\let\\lc=\\lowercase \n\\let\\uc=\\uppercase \n\\def\\Or{\\mathop{\\rm O}\\nolimits} \n\\def\\Tr{\\mathop{\\rm Tr}\\nolimits} \n\\def\\tr{\\mathop{\\rm tr}\\nolimits} \n\\def\\Cal#1{{\\cal #1}} \n\\def\\Bbbone{1\\kern-.22em {\\rm l}} \n% \n% Primes to display summations and products \n% which also have sub or superscripts \n% \n\\def\\rp{\\raise8pt\\hbox{$\\scriptstyle\\prime$}} \n% \n% then use \\sum^{...}_{...}\\rp or \\prod^{...}_{...}\\rp. \n% \n% Shadow brackets \n% \n% Single brackets for normal size only \n% \n\\def\\lshad{\\lbrack\\!\\lbrack} \n\\def\\rshad{\\rbrack\\!\\rbrack} \n% \n% Variable size for display style \n% \n\\def\\[#1\\]{\\setbox0=\\hbox{$\\dsty#1$}\\argwidth=\\wd0 \n \\setbox0=\\hbox{$\\left[\\box0\\right]$}\\advance\\argwidth by -\\wd0 \n \\left[\\kern.3\\argwidth\\box0\\kern.3\\argwidth\\right]} \n% \n% Variable size for text style \n% \n\\def\\lsb#1\\rsb{\\setbox0=\\hbox{$#1$}\\argwidth=\\wd0 \n \\setbox0=\\hbox{$\\left[\\box0\\right]$}\\advance\\argwidth by -\\wd0 \n \\left[\\kern.3\\argwidth\\box0\\kern.3\\argwidth\\right]} \n% \n\\def\\pounds{{\\tenu \\$}} \n% \n% Square for end of theorems \n% \n\\def\\sqr{\\hfill\\hbox{$\\square$}} \n% \n\\def\\pt(#1){({\\it #1\\/})} \n\\def\\ts{{\\thinspace}}% \n\\def\\co{$^{\\rm c}\\!/\\!_{\\rm o}$} \n% \n\\let\\dsty=\\displaystyle \n\\let\\tsty=\\textstyle \n\\let\\ssty=\\scriptstyle \n\\let\\sssty=\\scriptscriptstyle \n% \n% Definition for Nuclear Physics Keyword abstract \n% \n\\def\\reactions#1{\\vskip 12pt plus2pt minus2pt% \n\\vbox{\\hbox{\\kern\\secindent\\vrule\\kern12pt% \n\\vbox{\\kern0.5pt\\vbox{\\hsize=24pc\\parindent=0pt\\smallfonts\\rm NUCLEAR \nREACTIONS\\strut\\quad #1\\strut}\\kern0.5pt}\\kern12pt\\vrule}}} \n% \n% Definition for slashed characters \n% \n\\def\\slashchar#1{\\setbox0=\\hbox{$#1$}\\dimen0=\\wd0% \n\\setbox1=\\hbox{/}\\dimen1=\\wd1% \n\\ifdim\\dimen0>\\dimen1% \n\\rlap{\\hbox to \\dimen0{\\hfil/\\hfil}}#1\\else \n\\rlap{\\hbox to \\dimen1{\\hfil$#1$\\hfil}}/\\fi} \n% \n% Redefine \\textindent for use in \\item \n% \n\\def\\textindent#1{\\noindent\\hbox to \\parindent{#1\\hss}\\ignorespaces} \n% \n% Symbols and curves for use in figure captions \n% \n\\def\\opencirc{\\raise1pt\\hbox{$\\scriptstyle{\\bigcirc}$}} \n\\def\\fullcirc{\\hbox{\\headfonts\\rm$\\scriptstyle\\bullet$}} \n\\ifams \n\\def\\opensqr{\\hbox{$\\square$}} \n\\def\\opentri{\\hbox{$\\vartriangle$}} \n\\def\\opentridown{\\hbox{$\\triangledown$}} \n\\def\\opendiamond{\\hbox{$\\lozenge$}} \n\\def\\fullsqr{\\hbox{$\\blacksquare$}} \n\\def\\fulldiamond{\\hbox{$\\blacklozenge$}} \n\\def\\fulltri{\\hbox{$\\blacktriangle$}} \n\\def\\fulltridown{\\hbox{$\\blacktriangledown$}} \n\\else \n\\def\\opensqr{\\vbox{\\hrule height.4pt\\hbox{\\vrule width.4pt height3.5pt \n \\kern3.5pt\\vrule width.4pt}\\hrule height.4pt}} \n\\def\\opentri{\\hbox{$\\scriptstyle\\bigtriangleup$}} \n\\def\\opentridown{\\raise1pt\\hbox{$\\scriptstyle\\bigtriangledown$}} \n\\def\\opendiamond{\\hbox{$\\diamond$}} \n\\def\\fullsqr{\\vrule height3.5pt width3.5pt} \n\\def\\fulldiamond{\\opendiamond} % These produce the \n\\def\\fulltri{\\opentri} % equivalent open character \n\\def\\fulltridown{\\opentridown} % to be filled in. \n\\fi \n\\def\\dotted{\\hbox{${\\mathinner{\\cdotp\\cdotp\\cdotp\\cdotp\\cdotp\\cdotp}}$}} \n\\def\\dashed{\\hbox{-\\ts -\\ts -\\ts -}} \n\\def\\broken{\\hbox{-- -- --}} \n\\def\\longbroken{\\hbox{--- --- ---}} \n\\def\\chain{\\hbox{--- $\\cdot$ ---}} \n\\def\\dashddot{\\hbox{--- $\\cdot$ $\\cdot$ ---}} \n\\def\\full{\\hbox{------}} \n% \n% Redefinition of \\cases \n% \n\\def\\m@th{\\mathsurround=0pt} \n% \n% Displaystyle now used for first term \n% \n\\def\\cases#1{% \n\\left\\{\\,\\vcenter{\\normalbaselines\\openup1\\jot\\m@th% \n \\ialign{$\\displaystyle##\\hfil$&\\rm\\tqs##\\hfil\\crcr#1\\crcr}}\\right.}% \n% \n% Original version of cases now called \\oldcases \n% \n\\def\\oldcases#1{\\left\\{\\,\\vcenter{\\normalbaselines\\m@th \n \\ialign{$##\\hfil$&\\rm\\quad##\\hfil\\crcr#1\\crcr}}\\right.} \n% \n% Cases with number at end each line (using automatic numbering) \n% \n\\def\\numcases#1{\\left\\{\\,\\vcenter{\\baselineskip=15pt\\m@th% \n \\ialign{$\\displaystyle##\\hfil$&\\rm\\tqs##\\hfil \n \\crcr#1\\crcr}}\\right.\\hfill \n \\vcenter{\\baselineskip=15pt\\m@th% \n \\ialign{\\rlap{$\\phantom{\\displaystyle##\\hfil}$}\\tabskip=0pt&\\en \n \\rlap{\\phantom{##\\hfil}}\\crcr#1\\crcr}}} \n% \n\\def\\ptnumcases#1{\\left\\{\\,\\vcenter{\\baselineskip=15pt\\m@th% \n \\ialign{$\\displaystyle##\\hfil$&\\rm\\tqs##\\hfil \n \\crcr#1\\crcr}}\\right.\\hfill \n \\vcenter{\\baselineskip=15pt\\m@th% \n \\ialign{\\rlap{$\\phantom{\\displaystyle##\\hfil}$}\\tabskip=0pt&\\enpt \n \\rlap{\\phantom{##\\hfil}}\\crcr#1\\crcr}}\\global\\eqlett=97 \n \\global\\advance\\countno by 1} \n% \n% for equation numbers instead of \\eqno \n% \n\\def\\eq(#1){\\ifaligned\\@mp(#1)\\else\\hfill\\llap{{\\rm (#1)}}\\fi} \n\\def\\ceq(#1){\\ns\\ns\\ifaligned\\@mp\\fi\\eq(#1)\\cr\\ns\\ns} \n\\def\\eqpt(#1#2){\\ifaligned\\@mp(#1{\\it #2\\/}) \n \\else\\hfill\\llap{{\\rm (#1{\\it #2\\/})}}\\fi} \n\\let\\eqno=\\eq \n% \n% Automatic numbering of equations \n% \n\\countno=1 \n\\def\\eqnobysec{\\numbysectrue} \n\\def\\aleq{&\\rm(\\ifappendix\\applett \n \\ifnumbysec\\ifnum\\secno>0 \\the\\secno\\fi.\\fi \n \\else\\ifnumbysec\\the\\secno.\\fi\\fi\\the\\countno} \n\\def\\noaleq{\\hfill\\llap\\bgroup\\rm(\\ifappendix\\applett \n \\ifnumbysec\\ifnum\\secno>0 \\the\\secno\\fi.\\fi \n \\else\\ifnumbysec\\the\\secno.\\fi\\fi\\the\\countno} \n\\def\\@mp{&} \n\\def\\en{\\ifaligned\\aleq)\\else\\noaleq)\\egroup\\fi\\gac} \n\\def\\cen{\\ns\\ns\\ifaligned\\@mp\\fi\\en\\cr\\ns\\ns} \n\\def\\enpt{\\ifaligned\\aleq{\\it\\char\\the\\eqlett})\\else \n \\noaleq{\\it\\char\\the\\eqlett})\\egroup\\fi \n \\global\\advance\\eqlett by 1} \n\\def\\endpt{\\ifaligned\\aleq{\\it\\char\\the\\eqlett})\\else \n \\noaleq{\\it\\char\\the\\eqlett})\\egroup\\fi \n \\global\\eqlett=97\\gac} \n% \n% abbreviations for Institute of Physics Publishing journals \n% \n\\def\\CQG{{\\it Class. Quantum Grav.}} \n\\def\\IP{{\\it Inverse Problems\\/}} \n\\def\\JPA{{\\it J. Phys. A: Math. Gen.}} \n\\def\\JPB{{\\it J. Phys. B: At. Mol. Phys.}} %1968-87 \n\\def\\jpb{{\\it J. Phys. B: At. Mol. Opt. Phys.}} %1988 and onwards \n\\def\\JPC{{\\it J. Phys. C: Solid State Phys.}} %1968--1988 \n\\def\\JPCM{{\\it J. Phys.: Condens. Matter\\/}} %1989 and onwards \n\\def\\JPD{{\\it J. Phys. D: Appl. Phys.}} \n\\def\\JPE{{\\it J. Phys. E: Sci. Instrum.}} \n\\def\\JPF{{\\it J. Phys. F: Met. Phys.}} \n\\def\\JPG{{\\it J. Phys. G: Nucl. Phys.}} %1975--1988 \n\\def\\jpg{{\\it J. Phys. G: Nucl. Part. Phys.}} %1989 and onwards \n\\def\\MSMSE{{\\it Modelling Simulation Mater. Sci. Eng.}} \n\\def\\MST{{\\it Meas. Sci. Technol.}} %1990 and onwards \n\\def\\NET{{\\it Network\\/}} \n\\def\\NL{{\\it Nonlinearity\\/}} \n\\def\\NT{{\\it Nanotechnology}} \n\\def\\PAO{{\\it Pure Appl. Optics\\/}} \n\\def\\PMB{{\\it Phys. Med. Biol.}} \n\\def\\PSST{{\\it Plasma Sources Sci. Technol.}} \n\\def\\QO{{\\it Quantum Opt.}} \n\\def\\RPP{{\\it Rep. Prog. Phys.}} \n\\def\\SST{{\\it Semicond. Sci. Technol.}} \n\\def\\SUST{{\\it Supercond. Sci. Technol.}} \n\\def\\WRM{{\\it Waves Random Media\\/}} \n% \n% Other commonly quoted journals \n% \n\\def\\AC{{\\it Acta Crystallogr.}} \n\\def\\AM{{\\it Acta Metall.}} \n\\def\\AP{{\\it Ann. Phys., Lpz.}} \n\\def\\APNY{{\\it Ann. Phys., NY\\/}} \n\\def\\APP{{\\it Ann. Phys., Paris\\/}} \n\\def\\CJP{{\\it Can. J. Phys.}} \n\\def\\GRG{{\\it Gen. Rel. Grav.}} \n\\def\\JAP{{\\it J. Appl. Phys.}} \n\\def\\JCP{{\\it J. Chem. Phys.}} \n\\def\\JJAP{{\\it Japan. J. Appl. Phys.}} \n\\def\\JP{{\\it J. Physique\\/}} \n\\def\\JPhCh{{\\it J. Phys. Chem.}} \n\\def\\JMMM{{\\it J. Magn. Magn. Mater.}} \n\\def\\JMP{{\\it J. Math. Phys.}} \n\\def\\JOSA{{\\it J. Opt. Soc. Am.}} \n\\def\\JPSJ{{\\it J. Phys. Soc. Japan\\/}} \n\\def\\JQSRT{{\\it J. Quant. Spectrosc. Radiat. Transfer\\/}} \n\\def\\NC{{\\it Nuovo Cimento\\/}} \n\\def\\NIM{{\\it Nucl. Instrum. Methods\\/}} \n\\def\\NP{{\\it Nucl. Phys.}} \n\\def\\PL{{\\it Phys. Lett.}} \n\\def\\PR{{\\it Phys. Rev.}} \n\\def\\PRL{{\\it Phys. Rev. Lett.}} \n\\def\\PRS{{\\it Proc. R. Soc.}} \n\\def\\PS{{\\it Phys. Scr.}} \n\\def\\PSS{{\\it Phys. Status Solidi\\/}} \n\\def\\PTRS{{\\it Phil. Trans. R. Soc.}} \n\\def\\RMP{{\\it Rev. Mod. Phys.}} \n\\def\\RSI{{\\it Rev. Sci. Instrum.}} \n\\def\\SSC{{\\it Solid State Commun.}} \n\\def\\ZP{{\\it Z. Phys.}} \n% \n\\headline={\\ifodd\\pageno{\\ifnum\\pageno=\\firstpage\\hfill \n \\else\\rrhead\\fi}\\else\\lrhead\\fi} \n% \n\\def\\rrhead{\\textfonts\\hskip\\secindent\\it \n \\shorttitle\\hfill\\rm\\folio} \n% \n\\def\\lrhead{\\textfonts\\hbox to\\secindent{\\rm\\folio\\hss}% \n \\it\\aunames\\hss} \n% \n\\footline={\\ifnum\\pageno=\\firstpage \\hfill\\textfonts\\rm\\folio\\fi} \n% \n% \n\\def\\@rticle#1#2{\\vglue.5pc \n {\\parindent=\\secindent \\bf #1\\par} \n \\vskip2.5pc \n {\\exhyphenpenalty=10000\\hyphenpenalty=10000 \n \\baselineskip=18pt\\raggedright\\noindent \n \\headfonts\\bf#2\\par}\\futurelet\\next\\sh@rttitle}% \n% \n\\def\\title#1{\\gdef\\shorttitle{#1} \n \\vglue4pc{\\exhyphenpenalty=10000\\hyphenpenalty=10000 \n \\baselineskip=18pt \n \\raggedright\\parindent=0pt \n \\headfonts\\bf#1\\par}\\futurelet\\next\\sh@rttitle} \n\\let\\paper=\\title \n% \n\\def\\article#1#2{\\gdef\\shorttitle{#2}\\@rticle{#1}{#2}} \n% \n\\def\\review#1{\\gdef\\shorttitle{#1}% \n \\@rticle{REVIEW \\ifpbm\\else ARTICLE\\fi}{#1}} \n\\def\\topical#1{\\gdef\\shorttitle{#1}% \n \\@rticle{TOPICAL REVIEW}{#1}} \n\\def\\comment#1{\\gdef\\shorttitle{#1}% \n \\@rticle{COMMENT}{#1}} \n\\def\\note#1{\\gdef\\shorttitle{#1}% \n \\@rticle{NOTE}{#1}} \n\\def\\prelim#1{\\gdef\\shorttitle{#1}% \n \\@rticle{PRELIMINARY COMMUNICATION}{#1}} \n\\def\\letter#1{\\gdef\\shorttitle{Letter to the Editor}% \n \\gdef\\aunames{Letter to the Editor} \n \\global\\lettertrue\\ifnum\\jnl=7\\global\\letterfalse\\fi \n \\@rticle{LETTER TO THE EDITOR}{#1}} \n% \n\\def\\sh@rttitle{\\ifx\\next[\\let\\next=\\sh@rt \n \\else\\let\\next=\\f@ll\\fi\\next} \n% \n\\def\\sh@rt[#1]{\\gdef\\shorttitle{#1}} \n\\def\\f@ll{} \n% \n\\def\\author#1{\\ifletter\\else\\gdef\\aunames{#1}\\fi\\vskip1.5pc \n {\\parindent=\\secindent \n \\hang\\textfonts \n \\ifppt\\bf\\else\\rm\\fi#1\\par} \n \\ifppt\\bigskip\\else\\smallskip\\fi \n \\futurelet\\next\\@unames} \n% \n\\def\\@unames{\\ifx\\next[\\let\\next=\\short@uthor \n \\else\\let\\next=\\@uthor\\fi\\next} \n\\def\\short@uthor[#1]{\\gdef\\aunames{#1}} \n\\def\\@uthor{} \n% \n\\def\\address#1{{\\parindent=\\secindent \n \\exhyphenpenalty=10000\\hyphenpenalty=10000 \n\\ifppt\\textfonts\\else\\smallfonts\\fi\\hang\\raggedright\\rm#1\\par}% \n \\ifppt\\bigskip\\fi} \n% \n\\def\\jl#1{\\global\\jnl=#1} \n\\jl{0}% \n% \n\\def\\journal{\\ifnum\\jnl=1 J. Phys.\\ A: Math.\\ Gen.\\ \n \\else\\ifnum\\jnl=2 J. Phys.\\ B: At.\\ Mol.\\ Opt.\\ Phys.\\ \n \\else\\ifnum\\jnl=3 J. Phys.:\\ Condens. Matter\\ \n \\else\\ifnum\\jnl=4 J. Phys.\\ G: Nucl.\\ Part.\\ Phys.\\ \n \\else\\ifnum\\jnl=5 Inverse Problems\\ \n \\else\\ifnum\\jnl=6 Class. Quantum Grav.\\ \n \\else\\ifnum\\jnl=7 Network\\ \n \\else\\ifnum\\jnl=8 Nonlinearity\\ \n \\else\\ifnum\\jnl=9 Quantum Opt.\\ \n \\else\\ifnum\\jnl=10 Waves in Random Media\\ \n \\else\\ifnum\\jnl=11 Pure Appl. Opt.\\ \n \\else\\ifnum\\jnl=12 Phys. Med. Biol.\\ \n \\else\\ifnum\\jnl=13 Modelling Simulation Mater.\\ Sci.\\ Eng.\\ \n \\else\\ifnum\\jnl=14 Plasma Phys. Control. Fusion\\ \n \\else\\ifnum\\jnl=15 Physiol. Meas.\\ \n \\else\\ifnum\\jnl=16 Sov.\\ Lightwave Commun.\\ \n \\else\\ifnum\\jnl=17 J. Phys.\\ D: Appl.\\ Phys.\\ \n \\else\\ifnum\\jnl=18 Supercond.\\ Sci.\\ Technol.\\ \n \\else\\ifnum\\jnl=19 Semicond.\\ Sci.\\ Technol.\\ \n \\else\\ifnum\\jnl=20 Nanotechnology\\ \n \\else\\ifnum\\jnl=21 Meas.\\ Sci.\\ Technol.\\ \n \\else\\ifnum\\jnl=22 Plasma Sources Sci.\\ Technol.\\ \n \\else\\ifnum\\jnl=23 Smart Mater.\\ Struct.\\ \n \\else\\ifnum\\jnl=24 J.\\ Micromech.\\ Microeng.\\ \n \\else Institute of Physics Publishing\\ \n \\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi \n \\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi} \n% \n\\def\\beginabstract{\\insertspace \n \\parindent=\\secindent\\ifppt\\textfonts\\else\\smallfonts\\fi \n \\hang{\\bf Abstract. }\\rm } \n% \n\\let\\abs=\\beginabstract \n% \n\\def\\cabs{\\hang\\quad\\ } \n% \n\\def\\endabstract{\\par \n \\parindent=\\textind\\textfonts\\rm \n \\ifppt\\vfill\\fi} \n\\let\\endabs=\\endabstract \n% \n\\def\\submitted{\\ifppt\\noindent\\textfonts\\rm Submitted to \\journal\\par \n \\bigskip\\fi} \n% \n\\def\\today{\\number\\day\\ \\ifcase\\month\\or \n January\\or February\\or March\\or April\\or May\\or June\\or \n July\\or August\\or September\\or October\\or November\\or \n December\\fi\\space \\number\\year} \n% \n\\def\\date{\\ifppt\\noindent\\textfonts\\rm \n Date: \\today\\par\\goodbreak\\bigskip\\fi} \n% \n% Physics Abstracts classification numbers \n% \n\\def\\pacs#1{\\ifppt\\noindent\\textfonts\\rm \n PACS number(s): #1\\par\\bigskip\\fi} \n% \n\\def\\ams#1{\\ifppt\\noindent\\textfonts\\rm \n AMS classification scheme numbers: #1\\par\\bigskip\\fi} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Sections, subsections, etc % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\section#1{\\ifppt\\ifnum\\secno=0\\eject\\fi\\fi \n \\subno=0\\subsubno=0\\global\\advance\\secno by 1 \n \\gdef\\labeltype{\\seclabel}\\ifnumbysec\\countno=1\\fi \n \\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf \\the\\secno. #1}\\par\\futurelet\\next\\sp@ce} \n% \n\\def\\subsection#1{\\subsubno=0\\global\\advance\\subno by 1 \n \\gdef\\labeltype{\\seclabel}% \n \\ifssf\\else\\goodbreak\\beforesubspace\\fi \n \\global\\ssffalse\\nobreak \n \\noindent{\\it \\the\\secno.\\the\\subno. #1\\par}% \n \\futurelet\\next\\subsp@ce} \n% \n\\def\\subsubsection#1{\\global\\advance\\subsubno by 1 \n \\gdef\\labeltype{\\seclabel}% \n \\ifssf\\else\\goodbreak\\beforesubsubspace\\fi \n \\global\\ssffalse\\nobreak \n \\noindent{\\it \\the\\secno.\\the\\subno.\\the\\subsubno. #1}\\null. \n \\ignorespaces} \n% \n\\def\\nosections{\\ifppt\\eject\\else\\vskip30pt plus12pt minus12pt\\fi \n \\noindent\\ignorespaces} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Appendices % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\numappendix#1{\\ifappendix\\ifnumbysec\\countno=1\\fi\\else \n \\countno=1\\figno=0\\tabno=0\\fi \n \\subno=0\\global\\advance\\appno by 1 \n \\secno=\\appno\\gdef\\applett{A}\\gdef\\labeltype{\\seclabel}% \n \\global\\appendixtrue\\global\\numapptrue \n \\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf Appendix \\the\\appno. #1\\par}% \n \\futurelet\\next\\sp@ce} \n% \n\\def\\numsubappendix#1{\\global\\advance\\subno by 1\\subsubno=0 \n \\gdef\\labeltype{\\seclabel}% \n \\ifssf\\else\\goodbreak\\beforesubspace\\fi \n \\global\\ssffalse\\nobreak \n \\noindent{\\it A\\the\\appno.\\the\\subno. #1\\par}% \n \\futurelet\\next\\subsp@ce} \n% \n\\def\\@ppendix#1#2#3{\\countno=1\\subno=0\\subsubno=0\\secno=0\\figno=0\\tabno=0 \n \\gdef\\applett{#1}\\gdef\\labeltype{\\seclabel}\\global\\appendixtrue \n \\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf Appendix#2#3\\par}\\futurelet\\next\\sp@ce} \n% \n\\def\\Appendix#1{\\@ppendix{A}{. }{#1}} \n\\def\\appendix#1#2{\\@ppendix{#1}{ #1. }{#2}} \n\\def\\App#1{\\@ppendix{A}{ }{#1}} \n\\def\\app{\\@ppendix{A}{}{}} \n% \n\\def\\subappendix#1#2{\\global\\advance\\subno by 1\\subsubno=0 \n \\gdef\\labeltype{\\seclabel}% \n \\ifssf\\else\\goodbreak\\beforesubspace\\fi \n \\global\\ssffalse\\nobreak \n \\noindent{\\it #1\\the\\subno. #2\\par}% \n \\nobreak\\subspace\\noindent\\ignorespaces} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Acknowledgments, notes added and foreign abstracts % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\@ck#1{\\ifletter\\bigskip\\noindent\\ignorespaces\\else \n \\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf Acknowledgment#1\\par}% \n \\nobreak\\secspace\\noindent\\ignorespaces\\fi} \n% \n\\def\\ack{\\@ck{s}} \n\\def\\ackn{\\@ck{}} \n% \n\\def\\n@ip#1{\\goodbreak\\beforesecspace\\nobreak \n \\noindent\\smallfonts{\\it #1}. \\rm\\ignorespaces} \n\\def\\naip{\\n@ip{Note added in proof}} \n\\def\\na{\\n@ip{Note added}} \n\\def\\endnaip{\\par\\textfonts\\rm} \n% \n% \\resume and \\zus in Physics in Medicine and Biology only \n% \n\\def\\resume#1{\\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf R\\'esum\\'e\\par}% \n \\nobreak\\secspace\\parindent=0pt\\smallfonts\\rm #1\\par \n \\vskip7.5pt} \n% \n\\def\\zus#1{\\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf Zusammenfassung\\par}% \n \\nobreak\\secspace\\parindent=0pt\\smallfonts\\rm #1\\par \n \\vskip7.5pt} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Tables % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\Tables{\\vfill\\eject\\global\\appendixfalse\\textfonts\\rm \n \\everypar{}\\noindent{\\bf Tables and table captions}\\par \n \\bigskip} \n% \n\\def\\apptabs#1{\\global\\appendixtrue\\global\\tabno=0\\gdef\\applett{#1}} \n% \n\\def\\table#1{\\tablecaption{#1}} \n\\def\\tablecont{\\topinsert\\global\\advance\\tabno by -1 \n \\tablecaption{(continued)}} \n% \n\\def\\tablecaption#1{\\gdef\\labeltype{\\tablabel}\\global\\widefalse \n \\leftskip=\\secindent\\parindent=0pt \n \\global\\advance\\tabno by 1 \n \\smallfonts{\\bf Table \\ifappendix\\applett\\fi\\the\\tabno.} \\rm #1\\par \n \\smallskip\\futurelet\\next\\t@b} \n% \n\\def\\endtable{\\vfill\\goodbreak} \n% \n\\def\\t@b{\\ifx\\next*\\let\\next=\\widet@b \n \\else\\ifx\\next[\\let\\next=\\fullwidet@b \n \\else\\let\\next=\\narrowt@b\\fi\\fi \n \\next} \n% \n\\def\\widet@b#1{\\global\\widetrue\\global\\notfulltrue \n \\t@bwidth=\\hsize\\advance\\t@bwidth by -\\secindent} \n\\def\\fullwidet@b[#1]{\\global\\widetrue\\global\\notfullfalse \n \\leftskip=0pt\\t@bwidth=\\hsize} \n\\def\\narrowt@b{\\global\\notfulltrue} \n% \n\\def\\align{\\catcode`?=13\\ifnotfull\\moveright\\secindent\\fi \n \\vbox\\bgroup\\halign\\ifwide to \\t@bwidth\\fi \n \\bgroup\\strut\\tabskip=1.2pc plus1pc minus.5pc} \n% \n\\def\\endalign{\\egroup\\egroup\\catcode`?=12} \n\\let\\Lbar=\\L \n% \n% Use \\L{#}, \\R{#} and \\C{#} to specify left, right or centred \n% columns immediately after \\table. For example \n% \\align\\L{#}&&\\L{#}\\cr gives the preamble for a table with \n% all columns aligned left, \\align\\L{#}&\\C{#}&\\R{#}\\cr \n% gives a table with 3 columns, the first aligned left, the second \n% centred and the third aligned right. \n% \n\\def\\L#1{#1\\hfill} \n\\def\\R#1{\\hfill#1} \n\\def\\C#1{\\hfill#1\\hfill} \n% \n% Rules for tables \\br at top and bottom \n% \\mr to separate headings from entries \n% \n\\def\\br{\\noalign{\\vskip2pt\\hrule height1pt\\vskip2pt}} \n\\def\\mr{\\noalign{\\vskip2pt\\hrule\\vskip2pt}} \n% \n\\def\\tabnote#1{\\vskip-\\lastskip\\noindent #1\\par} \n% \n% Definitions for centring headings over several columns \n% \\centre{4}{Results for helium} will centre \n% Results for helium over four columns \n% \\crule{4} will produce a rule centred over four columns \n% to go below a centred heading \n% \n\\def\\centre#1#2{\\multispan{#1}{\\hfill#2\\hfill}} \n\\def\\crule#1{\\multispan{#1}{\\hrulefill}} \n% \n\\def\\qm{\\catcode`?=12?\\catcode`?=13} \n\\catcode`?=13 \n\\def\\lineup{\\setbox0=\\hbox{\\smallfonts\\rm 0}% \n \\digitwidth=\\wd0% \n \\def?{\\kern\\digitwidth}% \n \\def\\\\{\\hbox{$\\phantom{-}$}}% \n \\def\\-{\\llap{$-$}}} \n\\catcode`?=12 \n% \n% Macros for two parts of a table of equal width side by side \n% \\table{caption}[w] \n% \\sidetable{first part}{second part} \n% \\endtable \n% Use \\table preamble for tables of 31picas width \n% \n\\def\\sidetable#1#2{\\hbox{\\ifppt\\hsize=18pc\\t@bwidth=18pc \n \\else\\hsize=15pc\\t@bwidth=15pc\\fi \n \\parindent=0pt\\vtop{\\null #1\\par}% \n \\ifppt\\hskip1.2pc\\else\\hskip1pc\\fi \n \\vtop{\\null #2\\par}}} \n% \n\\def\\lstable#1#2{\\everypar{}\\tempval=\\hsize\\hsize=\\vsize \n \\vsize=\\tempval\\hoffset=-3pc \n \\global\\tabno=#1\\gdef\\labeltype{\\tablabel}% \n \\noindent\\smallfonts{\\bf Table \\ifappendix\\applett\\fi \n \\the\\tabno.} \\rm #2\\par \n \\smallskip\\futurelet\\next\\t@b} \n% \n\\def\\endlstable{\\vfill\\eject} \n% \n\\def\\inctabno{\\global\\advance\\tabno by 1} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Figures % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\Figures{\\vfill\\eject\\global\\appendixfalse\\textfonts\\rm \n \\everypar{}\\noindent{\\bf Figure captions}\\par \n \\bigskip} \n% \n\\def\\appfigs#1{\\global\\appendixtrue\\global\\figno=0\\gdef\\applett{#1}} \n% \n\\def\\figure#1{\\figc@ption{#1}\\bigskip} \n% \n\\def\\figc@ption#1{\\global\\advance\\figno by 1\\gdef\\labeltype{\\figlabel}% \n {\\parindent=\\secindent\\smallfonts\\hang \n {\\bf Figure \\ifappendix\\applett\\fi\\the\\figno.} \\rm #1\\par}} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Reference lists % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\refHEAD{\\goodbreak\\beforesecspace \n \\noindent\\textfonts{\\bf References}\\par \n \\let\\ref=\\rf \n \\nobreak\\smallfonts\\rm} \n% \n\\def\\references{\\refHEAD\\parindent=0pt \n \\everypar{\\hangindent=18pt\\hangafter=1 \n \\frenchspacing\\rm}% \n \\secspace} \n% \n\\def\\numreferences{\\refHEAD\\parindent=30pt \n \\everypar{\\hang\\noindent\\frenchspacing\\rm} \n \\secspace} \n% \n\\def\\rf#1{\\par\\noindent\\hbox to 21pt{\\hss #1\\quad}\\ignorespaces} \n% \n\\def\\refjl#1#2#3#4{\\noindent #1 {\\it #2 \\bf #3} #4\\par} \n% \n\\def\\refbk#1#2#3{\\noindent #1 {\\it #2} #3\\par} \n% \n% reference to a journal article in numerical system \n% \n\\def\\numrefjl#1#2#3#4#5{\\par\\rf{#1}#2 {\\it #3 \\bf #4} #5\\par} \n% \n% reference to a book or report in numerical system \n% \n\\def\\numrefbk#1#2#3#4{\\par\\rf{#1}#2 {\\it #3} #4\\par} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Theorems, lemmas, etc % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\proclaim#1{\\bigbreak\\noindent{\\it #1}.\\quad\\ignorespaces} \n\\def\\Remark{\\bigbreak\\noindent{\\it Remark}.\\quad\\ignorespaces} \n\\def\\Lemma{\\bigbreak\\noindent{\\it Lemma}.\\quad\\ignorespaces} \n\\def\\Theorem{\\bigbreak\\noindent{\\it Theorem}.\\quad\\ignorespaces} \n\\def\\Comment{\\bigbreak\\noindent{\\it Comment}.\\quad\\ignorespaces} \n\\def\\Proof{\\bigbreak\\noindent{\\it Proof}.\\quad\\ignorespaces} \n\\def\\Corollary{\\bigbreak\\noindent{\\it Corollary}.\\quad\\ignorespaces} \n\\def\\Note{\\bigbreak\\noindent{\\it Note}.\\quad\\ignorespaces} \n\\def\\Definition{\\bigbreak\\noindent{\\it Definition}.\\quad\\ignorespaces} \n\\def\\remark#1{\\bigbreak\\noindent{\\it Remark #1}.\\quad\\ignorespaces} \n\\def\\lemma#1{\\bigbreak\\noindent{\\it Lemma #1}.\\quad\\ignorespaces} \n\\def\\theorem#1{\\bigbreak\\noindent{\\it Theorem #1}.\\quad\\ignorespaces} \n\\def\\proof#1{\\bigbreak\\noindent{\\it Proof #1}.\\quad\\ignorespaces} \n\\def\\corollary#1{\\bigbreak\\noindent{\\it Corollary #1}.\\quad\\ignorespaces} \n% \n% NB \\note#1 is used to give a Note (as opposed to a paper or letter) \n% in PMB therefore use commands \\notes#1 for numbered Note \n% instead of \\note \n% \n\\def\\notes#1{\\bigbreak\\noindent{\\it Note #1}.\\quad\\ignorespaces} \n\\def\\definition#1{\\bigbreak\\noindent{\\it Definition #1}.\\quad\\ignorespaces} \n\\def\\endproclaim{\\par\\bigbreak} \n\\let\\linespace=\\endproclaim \n% \n\\catcode`\\@=12 \n% \n% Parameter values for `Preprint' style \n% \n\\def\\pptstyle{\\ppttrue\\headsize{17}{24}% \n\\textsize{12}{16}% \n\\smallsize{10}{12}% \n\\hsize=37.2pc\\vsize=56pc \n\\textind=20pt\\secindent=6pc} \n% \n% Parameter values for `Journal' style \n% \n\\def\\jnlstyle{\\pptfalse\\headsize{14}{18}% \n\\textsize{10}{12}% \n\\smallsize{8}{10} \n\\textind=16pt} \n% \n% Parameter values for `Eleven point' style \n% \n\\def\\mediumstyle{\\ppttrue\\headsize{14}{18}% \n\\textsize{11}{13}% \n\\smallsize{9}{11}% \n\\hsize=37.2pc\\vsize=56pc% \n\\textind=18pt\\secindent=5.5pc} \n% \n% Parameter values for `Large size' style \n% \n\\def\\largestyle{\\ppttrue\\headsize{20}{28}% \n\\textsize{12}{16}% \n\\smallsize{10}{12}% \n\\hsize=37.2pc\\vsize=56pc% \n\\textind=20pt\\secindent=6pc} \n% \n\\parindent=\\textind \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%"
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{
"name": "xref.tex",
"string": "\\catcode`@=11 \n\\newwrite\\auxfile \n\\newwrite\\xreffile \n\\newif\\ifxrefwarning \\xrefwarningtrue \n\\newif\\ifauxfile \n\\newif\\ifxreffile \n\\def\\testforxref{\\begingroup \n \\immediate\\openin\\xreffile = \\jobname.xrf\\space \n \\ifeof\\xreffile\\global\\xreffilefalse \n \\else\\global\\xreffiletrue\\fi \n \\immediate\\closein\\xreffile \n \\endgroup} \n\\def\\testforaux{\\begingroup \n \\immediate\\openin\\auxfile = \\jobname.aux\\space \n \\ifeof\\auxfile\\global\\auxfilefalse \n \\else\\global\\auxfiletrue\\fi \n \\immediate\\closein\\auxfile \n \\endgroup} \n\\def\\openreffile{\\immediate\\openout\\auxfile = \\jobname.aux}% \n\\def\\readreffile{% \n \\testforxref \n \\testforaux \n \\ifxreffile \n \\begingroup \n \\@setletters \n \\input \\jobname.xrf \n \\endgroup \n \\else \n\\message{No cross-reference file existed, some labels may be undefined}% \n \\fi\\openreffile}% \n\\def\\@setletters{% \n \\catcode`_=11 \\catcode`+=11 \n \\catcode`-=11 \\catcode`@=11 \n \\catcode`0=11 \\catcode`1=11 \n \\catcode`2=11 \\catcode`3=11 \n \\catcode`4=11 \\catcode`5=11 \n \\catcode`6=11 \\catcode`7=11 \n \\catcode`8=11 \\catcode`9=11 \n \\catcode`(=11 \\catcode`)=11 \n \\catcode`:=11 \\catcode`'=11 \n \\catcode`&=11 \\catcode`;=11 \n \\catcode`.=11}% \n\\gdef\\el@b{\\eqlabel} \n\\gdef\\sl@b{\\seclabel} \n\\gdef\\tl@b{\\tablabel} \n\\gdef\\fl@b{\\figlabel} \n\\def\\l@belno{\\ifx\\labeltype\\el@b \n \\let\\labelno=\\en\\def\\@label{\\eqlabel}% \n \\else\\let\\labelno=\\ignorespaces \n \\ifx\\labeltype\\sl@b \\def\\@label{\\seclabel}% \n \\else\\ifx\\labeltype\\tl@b \\def\\@label{\\tablabel}% \n \\else\\ifx\\labeltype\\fl@b \\def\\@label{\\figlabel}% \n \\else\\def\\@label{\\seclabel}\\fi\\fi\\fi\\fi} \n\\def\\label#1{\\l@belno\\expandafter\\xdef\\csname #1@\\endcsname{\\@label}% \n \\immediate\\write\\auxfile{\\string \n \\gdef\\expandafter\\string\\csname @#1\\endcsname{\\@label}}% \n \\labelno}% \n\\def\\ref#1{% \n \\expandafter \\ifx \\csname @#1\\endcsname\\relax \n \\message{Undefined label `#1'.}% \n \\expandafter\\xdef\\csname @#1\\endcsname{(??)}\\fi \n \\csname @#1\\endcsname}% \n\\readreffile \n% \n%%%%%%%%%%% \n% \n\\def\\bibitem#1{\\global\\advance\\refno by 1% \n \\immediate\\write\\auxfile{\\string \n \\gdef\\expandafter\\string\\csname #1@\\endcsname{\\the\\refno}}% \n \\rf{[\\the\\refno]}}% \n\\def\\bitem[#1]#2{\\immediate\\write\\auxfile{\\string \n \\gdef\\expandafter\\string\\csname #2@\\endcsname{#1}}% \n \\rf{[#1]}}% \n% \n\\def\\cite#1{\\hbox{[\\splitarg{#1}]}}% \n\\def\\splitarg#1{\\@pt#1,\\@ptend}% \n\\def\\@pt#1,#2\\@ptend{\\ifempty{#1}\\else \n \\@pttwo #1\\@pttwoend \n \\ifempty{#2}\\else\\sp@cer\\@pt#2\\@ptend\\fi\\fi}% \n\\def\\@pttwo#1\\@pttwoend{\\expandafter \n \\ifx \\csname#1@\\endcsname\\@pttwoend\\else \n \\@ifundefined{#1}{{\\bf ?}% \n \\message{Undefined citation `#1' on page \n \\the\\pageno}}{\\csname#1@\\endcsname}\\fi}% \n\\def\\@pttwoend{@@@@@}% \n\\def\\sp@cer{,\\nobreak\\thinspace}% \n\\def\\ifempty#1{\\@ifempty #1\\@xx\\@xxx}% \n\\def\\@ifempty#1#2\\@xxx{\\ifx #1\\@xx}% \n\\def\\@xx{@@@@}% \n\\def\\@xxx{@@@@}% \n\\long\\def\\@ifundefined#1#2#3{\\expandafter\\ifx\\csname \n #1@\\endcsname\\relax#2\\else#3\\fi}% \n% \n\\catcode`@=12 \n\\endinput \n \n% 28/10/93 Added \\bitem definition for non numeric cross referencing \n"
}
] |
[
{
"name": "cond-mat0002070.extracted_bib",
"string": "\\bibitem{cardy83} {Cardy J L 1983}\\ {\\JPA}\\ {\\bf 16}\\ {3617}\n\n\\bibitem{cardy84} {Cardy J L 1984}\\ {\\NP\\it B}\\ {\\bf 240}\\ {514}\n\n\\bibitem{barber84} {Barber M N, Peschel I and Pearce P A 1984}\\ {\\it J.\nStat. Phys.}\\ {\\bf 37}\\ {497} \n\n\\bibitem{igloi93} {Igl\\'oi F, Peschel I and Turban L 1993}\\ {\\it Adv. \nPhys.}\\ {\\bf 42}\\ {683}\n\n\\bibitem{abraham94} {Abraham D B and Latr\\'emoli\\`ere F T 1994}\\ {\\PR\\\nE}\\ {\\bf 50}\\ {R9} \n\n\\bibitem{abraham95} {Abraham D B and Latr\\'emoli\\`ere F T 1995}\\ {\\it J.\nStat. Phys.}\\ {\\bf 81}\\ {539} \n\n\\bibitem{abraham96} {Abraham D B and Latr\\'emoli\\`ere F T 1996}\\ {\\PRL}\\\n{\\bf 76}\\ {4813} \n\n\\bibitem{davies97} {Davies B and Peschel I 1997}\\ {\\AP}\\\n{\\bf 6}\\ {187} \n\n\\bibitem{kirkpatrick73} {Kirkpatrick S 1973}\\ {\\RMP}\\ {\\bf 45}\\ {574}\n\n\\bibitem{bunde91} {Bunde A and Havlin S 1991}\\ {\\it Fractals and\nDisordered Systems}\\ {(Berlin: Springer)}\\ {p 97} \n\n\\bibitem{stauffer92} {Stauffer D and Aharony A 1992}\\\n{\\it Introduction to Percolation Theory}\\ {(London: Taylor \\& Francis)}\\ {p\n89} \n\n\\bibitem{kasteleyn69} {Kasteleyn P W and Fortuin C M 1969}\\\n{\\it J. Phys. Soc. Japan. (Suppl.)}\\ {\\bf 26}\\ {11} \n\n\\bibitem{stephen77} {Stephen M J 1977}\\ {\\PR\\ B}\\ {\\bf 15}\\ {5674} \n\n\\bibitem{wu78} {Wu F Y 1978}\\ {\\it J. Stat. Phys.}\\ {\\bf 18}\\ {115} \n\n\\bibitem{langlands94} {Langlands R, Pouliot P and Saint-Aubin Y 1994}\\ {\\it \nBull. Am. Math. Soc.}\\ {\\bf 30}\\ {1}\n\n\\bibitem{cardy92} {Cardy J L 1992}\\ {\\JPA}\\ {\\bf 25}\\ {L201}\n\n\\bibitem{dequeiroz95} {de Queiroz S L A 1995}\\ {\\JPA}\\ {\\bf 27}\\ {L363}\n\n\\bibitem{wolf90} {Wolf T, Blender R and Dietrich W 1990}\\ {\\JPA}\\ {\\bf 23}\\\n{L153} \n\\bibitem{essam96} {Essam J W, Lookman T and De'Bell K 1996}\\ {\\JPA}\\ \n{\\bf 29}\\ {L143}\n\n\\bibitem{grassberger99} {Grassberger P 1999}\\ {\\it Physica A}\\ {\\bf 262}\\\n{251} \n\n\\bibitem{lobb84} {Lobb C J and Frank D J 1984}\\ {\\PR\\ B}\\ {\\bf 30}\\ {4090}\n\n\\bibitem{fish78} {Fish R and Harris A B 1978}\\ {\\PR\\ B}\\ {\\bf 18}\\ {416} \n\n\\bibitem{degennes76} {de Gennnes P G 1976}\\ {\\it J. Physique Lett.}\\ {\\bf\n37}\\ {L1} \n\n\\bibitem{wu82} {Wu F Y 1982}\\ {\\RMP}\\ {\\bf 54}\\ {235}\n\n\\bibitem{cardy87} {Cardy J L 1987}\\ {\\it Phase transitions and critical\nphenomena}\\ {vol 11, ed C Domb and J L Lebowitz (New-York: Academic)}\\ {p\n55}\n\n\\bibitem{shante71} {Shante V K S and Kirkpatrick S 1971}\\ {\\it Adv. Phys}\\\n{\\bf 20}\\ {325} \n \n\n\\vfill\\eject\\bye\n\n\\hsize=31pc \n\\vsize=49pc \n% \n\\lineskip=0pt \n\\parskip=0pt plus 1pt \n\\hfuzz=1pt \n\\vfuzz=2pt \n\\pretolerance=2500 \n\\tolerance=5000 \n\\vbadness=5000 \n\\hbadness=5000 \n\\widowpenalty=500 \n\\clubpenalty=200 \n\\brokenpenalty=500 \n\\predisplaypenalty=200 \n\\voffset=-1pc \n\\nopagenumbers \n% \n\\catcode`@=11 \n% \n\\newif\\ifams \n\\amsfalse %\\amstrue si � suivant utilise\n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% The following section may be commented out and % \n% \\ifams set to either \\amstrue to use the AMS fonts % \n% or \\amsfalse if they are not available % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n%\\def\\Yesreply{Y } \n%\\def\\Noreply{N } \n%\\def\\yesreply{y } \n%\\def\\noreply{n } \n%\\newif\\ifnotyorn \n%\\message{Do you want to use AMSfonts, msam and msbm? Y or N: }% \n%\\loop \n%\\read-1 to \\reply \n%\\ifx\\reply\\yesreply\\global\\amstrue\\notyornfalse \n%\\else\\ifx\\reply\\Yesreply\\global\\amstrue\\notyornfalse \n%\\else\\ifx\\reply\\noreply\\global\\amsfalse\\notyornfalse \n%\\else\\ifx\\reply\\Noreply\\global\\amsfalse\\notyornfalse \n%\\else\\notyorntrue \n%\\message{Please type y or Y (Yes) or n or N (No)}\\fi\\fi\\fi\\fi \n%\\ifnotyorn\\repeat \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\newfam\\bdifam \n\\newfam\\bsyfam \n\\newfam\\bssfam \n\\newfam\\msafam \n\\newfam\\msbfam \n% \n\\newif\\ifxxpt \n\\newif\\ifxviipt \n\\newif\\ifxivpt \n\\newif\\ifxiipt \n\\newif\\ifxipt \n\\newif\\ifxpt \n\\newif\\ifixpt \n\\newif\\ifviiipt \n\\newif\\ifviipt \n\\newif\\ifvipt \n\\newif\\ifvpt \n% \n% Headings in 20pt, 17pt or 14pt \n% \n\\def\\headsize#1#2{\\def\\headb@seline{#2}% \n \\ifnum#1=20\\def\\HEAD{twenty}% \n \\def\\smHEAD{twelve}% \n \\def\\vsHEAD{nine}% \n \\ifxxpt\\else\\xdef\\f@ntsize{\\HEAD}% \n \\def\\m@g{4}\\def\\s@ze{20.74}% \n \\loadheadfonts\\xxpttrue\\fi \n \\ifxiipt\\else\\xdef\\f@ntsize{\\smHEAD}% \n \\def\\m@g{1}\\def\\s@ze{12}% \n \\loadxiiptfonts\\xiipttrue\\fi \n \\ifixpt\\else\\xdef\\f@ntsize{\\vsHEAD}% \n \\def\\s@ze{9}% \n \\loadsmallfonts\\ixpttrue\\fi \n \\else \n \\ifnum#1=17\\def\\HEAD{seventeen}% \n \\def\\smHEAD{eleven}% \n \\def\\vsHEAD{eight}% \n \\ifxviipt\\else\\xdef\\f@ntsize{\\HEAD}% \n \\def\\m@g{3}\\def\\s@ze{17.28}% \n \\loadheadfonts\\xviipttrue\\fi \n \\ifxipt\\else\\xdef\\f@ntsize{\\smHEAD}% \n \\loadxiptfonts\\xipttrue\\fi \n \\ifviiipt\\else\\xdef\\f@ntsize{\\vsHEAD}% \n \\def\\s@ze{8}% \n \\loadsmallfonts\\viiipttrue\\fi \n \\else\\def\\HEAD{fourteen}% \n \\def\\smHEAD{ten}% \n \\def\\vsHEAD{seven}% \n \\ifxivpt\\else\\xdef\\f@ntsize{\\HEAD}% \n \\def\\m@g{2}\\def\\s@ze{14.4}% \n \\loadheadfonts\\xivpttrue\\fi \n \\ifxpt\\else\\xdef\\f@ntsize{\\smHEAD}% \n \\def\\s@ze{10}% \n \\loadxptfonts\\xpttrue\\fi \n \\ifviipt\\else\\xdef\\f@ntsize{\\vsHEAD}% \n \\def\\s@ze{7}% \n \\loadviiptfonts\\viipttrue\\fi \n \\ifnum#1=14\\else \n \\message{Header size should be 20, 17 or 14 point \n will now default to 14pt}\\fi \n \\fi\\fi\\headfonts} \n% \n% Text in 12pt, 11pt or 10pt \n% \n\\def\\textsize#1#2{\\def\\textb@seline{#2}% \n \\ifnum#1=12\\def\\TEXT{twelve}% \n \\def\\smTEXT{eight}% \n \\def\\vsTEXT{six}% \n \\ifxiipt\\else\\xdef\\f@ntsize{\\TEXT}% \n \\def\\m@g{1}\\def\\s@ze{12}% \n \\loadxiiptfonts\\xiipttrue\\fi \n \\ifviiipt\\else\\xdef\\f@ntsize{\\smTEXT}% \n \\def\\s@ze{8}% \n \\loadsmallfonts\\viiipttrue\\fi \n \\ifvipt\\else\\xdef\\f@ntsize{\\vsTEXT}% \n \\def\\s@ze{6}% \n \\loadviptfonts\\vipttrue\\fi \n \\else \n \\ifnum#1=11\\def\\TEXT{eleven}% \n \\def\\smTEXT{seven}% \n \\def\\vsTEXT{five}% \n \\ifxipt\\else\\xdef\\f@ntsize{\\TEXT}% \n \\def\\s@ze{11}% \n \\loadxiptfonts\\xipttrue\\fi \n \\ifviipt\\else\\xdef\\f@ntsize{\\smTEXT}% \n \\loadviiptfonts\\viipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsTEXT}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\else\\def\\TEXT{ten}% \n \\def\\smTEXT{seven}% \n \\def\\vsTEXT{five}% \n \\ifxpt\\else\\xdef\\f@ntsize{\\TEXT}% \n \\loadxptfonts\\xpttrue\\fi \n \\ifviipt\\else\\xdef\\f@ntsize{\\smTEXT}% \n \\def\\s@ze{7}% \n \\loadviiptfonts\\viipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsTEXT}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\ifnum#1=10\\else \n \\message{Text size should be 12, 11 or 10 point \n will now default to 10pt}\\fi \n \\fi\\fi\\textfonts} \n% \n% Small sized material in 10pt, 9pt or 8pt \n% \n\\def\\smallsize#1#2{\\def\\smallb@seline{#2}% \n \\ifnum#1=10\\def\\SMALL{ten}% \n \\def\\smSMALL{seven}% \n \\def\\vsSMALL{five}% \n \\ifxpt\\else\\xdef\\f@ntsize{\\SMALL}% \n \\loadxptfonts\\xpttrue\\fi \n \\ifviipt\\else\\xdef\\f@ntsize{\\smSMALL}% \n \\def\\s@ze{7}% \n \\loadviiptfonts\\viipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsSMALL}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\else \n \\ifnum#1=9\\def\\SMALL{nine}% \n \\def\\smSMALL{six}% \n \\def\\vsSMALL{five}% \n \\ifixpt\\else\\xdef\\f@ntsize{\\SMALL}% \n \\def\\s@ze{9}% \n \\loadsmallfonts\\ixpttrue\\fi \n \\ifvipt\\else\\xdef\\f@ntsize{\\smSMALL}% \n \\def\\s@ze{6}% \n \\loadviptfonts\\vipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsSMALL}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\else \n \\def\\SMALL{eight}% \n \\def\\smSMALL{six}% \n \\def\\vsSMALL{five}% \n \\ifviiipt\\else\\xdef\\f@ntsize{\\SMALL}% \n \\def\\s@ze{8}% \n \\loadsmallfonts\\viiipttrue\\fi \n \\ifvipt\\else\\xdef\\f@ntsize{\\smSMALL}% \n \\def\\s@ze{6}% \n \\loadviptfonts\\vipttrue\\fi \n \\ifvpt\\else\\xdef\\f@ntsize{\\vsSMALL}% \n \\def\\s@ze{5}% \n \\loadvptfonts\\vpttrue\\fi \n \\ifnum#1=8\\else\\message{Small size should be 10, 9 or \n 8 point will now default to 8pt}\\fi \n \\fi\\fi\\smallfonts} \n% \n\\def\\F@nt{\\expandafter\\font\\csname} \n\\def\\Sk@w{\\expandafter\\skewchar\\csname} \n\\def\\@nd{\\endcsname} \n\\def\\@step#1{ scaled \\magstep#1} \n\\def\\@half{ scaled \\magstephalf} \n\\def\\@t#1{ at #1pt} \n% \n% For 14, 17 and 20 point fonts use \\loadheadfonts \n% \n\\def\\loadheadfonts{\\bigf@nts \n\\F@nt \\f@ntsize bdi\\@nd=cmmib10 \\@t{\\s@ze}% \n\\Sk@w \\f@ntsize bdi\\@nd='177 \n\\F@nt \\f@ntsize bsy\\@nd=cmbsy10 \\@t{\\s@ze}% \n\\Sk@w \\f@ntsize bsy\\@nd='60 \n\\F@nt \\f@ntsize bss\\@nd=cmssbx10 \\@t{\\s@ze}} \n% \n% For 12 point fonts use \\loadxiiptfonts \n% \n\\def\\loadxiiptfonts{\\bigf@nts \n\\F@nt \\f@ntsize bdi\\@nd=cmmib10 \\@step{\\m@g}% \n\\Sk@w \\f@ntsize bdi\\@nd='177 \n\\F@nt \\f@ntsize bsy\\@nd=cmbsy10 \\@step{\\m@g}% \n\\Sk@w \\f@ntsize bsy\\@nd='60 \n\\F@nt \\f@ntsize bss\\@nd=cmssbx10 \\@step{\\m@g}} \n% \n% For 11 point fonts use \\loadxiptfonts \n% \n\\def\\loadxiptfonts{% \n\\font\\elevenrm=cmr10 \\@half \n\\font\\eleveni=cmmi10 \\@half \n\\skewchar\\eleveni='177 \n\\font\\elevensy=cmsy10 \\@half \n\\skewchar\\elevensy='60 \n\\font\\elevenex=cmex10 \\@half \n\\font\\elevenit=cmti10 \\@half \n\\font\\elevensl=cmsl10 \\@half \n\\font\\elevenbf=cmbx10 \\@half \n\\font\\eleventt=cmtt10 \\@half \n\\ifams\\font\\elevenmsa=msam10 \\@half \n\\font\\elevenmsb=msbm10 \\@half\\else\\fi \n\\font\\elevenbdi=cmmib10 \\@half \n\\skewchar\\elevenbdi='177 \n\\font\\elevenbsy=cmbsy10 \\@half \n\\skewchar\\elevenbsy='60 \n\\font\\elevenbss=cmssbx10 \\@half} \n% \n% For 10 point fonts use \\loadxptfonts \n% \n\\def\\loadxptfonts{% \n\\font\\tenbdi=cmmib10 \n\\skewchar\\tenbdi='177 \n\\font\\tenbsy=cmbsy10 \n\\skewchar\\tenbsy='60 \n\\ifams\\font\\tenmsa=msam10 \n\\font\\tenmsb=msbm10\\else\\fi \n\\font\\tenbss=cmssbx10}% \n% \n% For 8 and 9 point fonts use \\loadsmallfonts \n% \n\\def\\loadsmallfonts{\\smallf@nts \n\\ifams \n\\F@nt \\f@ntsize ex\\@nd=cmex\\s@ze \n\\else \n\\F@nt \\f@ntsize ex\\@nd=cmex10\\fi \n\\F@nt \\f@ntsize it\\@nd=cmti\\s@ze \n\\F@nt \\f@ntsize sl\\@nd=cmsl\\s@ze \n\\F@nt \\f@ntsize tt\\@nd=cmtt\\s@ze} \n% \n% For 7 point fonts use \\loadviiptfonts \n% \n\\def\\loadviiptfonts{% \n\\font\\sevenit=cmti7 \n\\font\\sevensl=cmsl8 at 7pt \n\\ifams\\font\\sevenmsa=msam7 \n\\font\\sevenmsb=msbm7 \n\\font\\sevenex=cmex7 \n\\font\\sevenbsy=cmbsy7 \n\\font\\sevenbdi=cmmib7\\else \n\\font\\sevenex=cmex10 \n\\font\\sevenbsy=cmbsy10 at 7pt \n\\font\\sevenbdi=cmmib10 at 7pt\\fi \n\\skewchar\\sevenbsy='60 \n\\skewchar\\sevenbdi='177 \n\\font\\sevenbss=cmssbx10 at 7pt}% \n% \n% For 6 point fonts use \\loadviptfonts \n% \n\\def\\loadviptfonts{\\smallf@nts \n\\ifams\\font\\sixex=cmex7 at 6pt\\else \n\\font\\sixex=cmex10\\fi \n\\font\\sixit=cmti7 at 6pt} \n% \n% For 5 point fonts use \\loadvptfonts \n% \n\\def\\loadvptfonts{% \n\\font\\fiveit=cmti7 at 5pt \n\\ifams\\font\\fiveex=cmex7 at 5pt \n\\font\\fivebdi=cmmib5 \n\\font\\fivebsy=cmbsy5 \n\\font\\fivemsa=msam5 \n\\font\\fivemsb=msbm5\\else \n\\font\\fiveex=cmex10 \n\\font\\fivebdi=cmmib10 at 5pt \n\\font\\fivebsy=cmbsy10 at 5pt\\fi \n\\skewchar\\fivebdi='177 \n\\skewchar\\fivebsy='60 \n\\font\\fivebss=cmssbx10 at 5pt} \n% \n\\def\\bigf@nts{% \n\\F@nt \\f@ntsize rm\\@nd=cmr10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize i\\@nd=cmmi10 \\@step{\\m@g}% \n\\Sk@w \\f@ntsize i\\@nd='177 \n\\F@nt \\f@ntsize sy\\@nd=cmsy10 \\@step{\\m@g}% \n\\Sk@w \\f@ntsize sy\\@nd='60 \n\\F@nt \\f@ntsize ex\\@nd=cmex10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize it\\@nd=cmti10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize sl\\@nd=cmsl10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize bf\\@nd=cmbx10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize tt\\@nd=cmtt10 \\@step{\\m@g}% \n\\ifams \n\\F@nt \\f@ntsize msa\\@nd=msam10 \\@step{\\m@g}% \n\\F@nt \\f@ntsize msb\\@nd=msbm10 \\@step{\\m@g}\\else\\fi} \n% \n\\def\\smallf@nts{% \n\\F@nt \\f@ntsize rm\\@nd=cmr\\s@ze \n\\F@nt \\f@ntsize i\\@nd=cmmi\\s@ze \n\\Sk@w \\f@ntsize i\\@nd='177 \n\\F@nt \\f@ntsize sy\\@nd=cmsy\\s@ze \n\\Sk@w \\f@ntsize sy\\@nd='60 \n\\F@nt \\f@ntsize bf\\@nd=cmbx\\s@ze \n\\ifams \n\\F@nt \\f@ntsize bdi\\@nd=cmmib\\s@ze \n\\F@nt \\f@ntsize bsy\\@nd=cmbsy\\s@ze \n\\F@nt \\f@ntsize msa\\@nd=msam\\s@ze \n\\F@nt \\f@ntsize msb\\@nd=msbm\\s@ze \n\\else \n\\F@nt \\f@ntsize bdi\\@nd=cmmib10 \\@t{\\s@ze}% \n\\F@nt \\f@ntsize bsy\\@nd=cmbsy10 \\@t{\\s@ze}\\fi \n\\Sk@w \\f@ntsize bdi\\@nd='177 \n\\Sk@w \\f@ntsize bsy\\@nd='60 \n\\F@nt \\f@ntsize bss\\@nd=cmssbx10 \\@t{\\s@ze}}% \n% \n% Fonts for headings \n% \n\\def\\headfonts{% \n\\textfont0=\\csname\\HEAD rm\\@nd \n\\scriptfont0=\\csname\\smHEAD rm\\@nd \n\\scriptscriptfont0=\\csname\\vsHEAD rm\\@nd \n\\def\\rm{\\fam0\\csname\\HEAD rm\\@nd \n\\def\\sc{\\csname\\smHEAD rm\\@nd}}% \n% \n\\textfont1=\\csname\\HEAD i\\@nd \n\\scriptfont1=\\csname\\smHEAD i\\@nd \n\\scriptscriptfont1=\\csname\\vsHEAD i\\@nd \n% \n\\textfont2=\\csname\\HEAD sy\\@nd \n\\scriptfont2=\\csname\\smHEAD sy\\@nd \n\\scriptscriptfont2=\\csname\\vsHEAD sy\\@nd \n% \n\\textfont3=\\csname\\HEAD ex\\@nd \n\\scriptfont3=\\csname\\smHEAD ex\\@nd \n\\scriptscriptfont3=\\csname\\smHEAD ex\\@nd \n% \n\\textfont\\itfam=\\csname\\HEAD it\\@nd \n\\scriptfont\\itfam=\\csname\\smHEAD it\\@nd \n\\scriptscriptfont\\itfam=\\csname\\vsHEAD it\\@nd \n\\def\\it{\\fam\\itfam\\csname\\HEAD it\\@nd \n\\def\\sc{\\csname\\smHEAD it\\@nd}}% \n% \n\\textfont\\slfam=\\csname\\HEAD sl\\@nd \n\\def\\sl{\\fam\\slfam\\csname\\HEAD sl\\@nd \n\\def\\sc{\\csname\\smHEAD sl\\@nd}}% \n% \n\\textfont\\bffam=\\csname\\HEAD bf\\@nd \n\\scriptfont\\bffam=\\csname\\smHEAD bf\\@nd \n\\scriptscriptfont\\bffam=\\csname\\vsHEAD bf\\@nd \n\\def\\bf{\\fam\\bffam\\csname\\HEAD bf\\@nd \n\\def\\sc{\\csname\\smHEAD bf\\@nd}}% \n% \n\\textfont\\ttfam=\\csname\\HEAD tt\\@nd \n\\def\\tt{\\fam\\ttfam\\csname\\HEAD tt\\@nd}% \n% \n\\textfont\\bdifam=\\csname\\HEAD bdi\\@nd \n\\scriptfont\\bdifam=\\csname\\smHEAD bdi\\@nd \n\\scriptscriptfont\\bdifam=\\csname\\vsHEAD bdi\\@nd \n\\def\\bdi{\\fam\\bdifam\\csname\\HEAD bdi\\@nd}% \n% \n\\textfont\\bsyfam=\\csname\\HEAD bsy\\@nd \n\\scriptfont\\bsyfam=\\csname\\smHEAD bsy\\@nd \n\\def\\bsy{\\fam\\bsyfam\\csname\\HEAD bsy\\@nd}% \n% \n\\textfont\\bssfam=\\csname\\HEAD bss\\@nd \n\\scriptfont\\bssfam=\\csname\\smHEAD bss\\@nd \n\\scriptscriptfont\\bssfam=\\csname\\vsHEAD bss\\@nd \n\\def\\bss{\\fam\\bssfam\\csname\\HEAD bss\\@nd}% \n% \n\\ifams \n\\textfont\\msafam=\\csname\\HEAD msa\\@nd \n\\scriptfont\\msafam=\\csname\\smHEAD msa\\@nd \n\\scriptscriptfont\\msafam=\\csname\\vsHEAD msa\\@nd \n% \n\\textfont\\msbfam=\\csname\\HEAD msb\\@nd \n\\scriptfont\\msbfam=\\csname\\smHEAD msb\\@nd \n\\scriptscriptfont\\msbfam=\\csname\\vsHEAD msb\\@nd \n\\else\\fi \n% \n\\normalbaselineskip=\\headb@seline pt% \n\\setbox\\strutbox=\\hbox{\\vrule height.7\\normalbaselineskip \ndepth.3\\baselineskip width0pt}% \n\\def\\sc{\\csname\\smHEAD rm\\@nd}\\normalbaselines\\bf} \n% \n% Fonts for text \n% \n\\def\\textfonts{% \n\\textfont0=\\csname\\TEXT rm\\@nd \n\\scriptfont0=\\csname\\smTEXT rm\\@nd \n\\scriptscriptfont0=\\csname\\vsTEXT rm\\@nd \n\\def\\rm{\\fam0\\csname\\TEXT rm\\@nd \n\\def\\sc{\\csname\\smTEXT rm\\@nd}}% \n% \n\\textfont1=\\csname\\TEXT i\\@nd \n\\scriptfont1=\\csname\\smTEXT i\\@nd \n\\scriptscriptfont1=\\csname\\vsTEXT i\\@nd \n% \n\\textfont2=\\csname\\TEXT sy\\@nd \n\\scriptfont2=\\csname\\smTEXT sy\\@nd \n\\scriptscriptfont2=\\csname\\vsTEXT sy\\@nd \n% \n\\textfont3=\\csname\\TEXT ex\\@nd \n\\scriptfont3=\\csname\\smTEXT ex\\@nd \n\\scriptscriptfont3=\\csname\\smTEXT ex\\@nd \n% \n\\textfont\\itfam=\\csname\\TEXT it\\@nd \n\\scriptfont\\itfam=\\csname\\smTEXT it\\@nd \n\\scriptscriptfont\\itfam=\\csname\\vsTEXT it\\@nd \n\\def\\it{\\fam\\itfam\\csname\\TEXT it\\@nd \n\\def\\sc{\\csname\\smTEXT it\\@nd}}% \n% \n\\textfont\\slfam=\\csname\\TEXT sl\\@nd \n\\def\\sl{\\fam\\slfam\\csname\\TEXT sl\\@nd \n\\def\\sc{\\csname\\smTEXT sl\\@nd}}% \n% \n\\textfont\\bffam=\\csname\\TEXT bf\\@nd \n\\scriptfont\\bffam=\\csname\\smTEXT bf\\@nd \n\\scriptscriptfont\\bffam=\\csname\\vsTEXT bf\\@nd \n\\def\\bf{\\fam\\bffam\\csname\\TEXT bf\\@nd \n\\def\\sc{\\csname\\smTEXT bf\\@nd}}% \n% \n\\textfont\\ttfam=\\csname\\TEXT tt\\@nd \n\\def\\tt{\\fam\\ttfam\\csname\\TEXT tt\\@nd}% \n% \n\\textfont\\bdifam=\\csname\\TEXT bdi\\@nd \n\\scriptfont\\bdifam=\\csname\\smTEXT bdi\\@nd \n\\scriptscriptfont\\bdifam=\\csname\\vsTEXT bdi\\@nd \n\\def\\bdi{\\fam\\bdifam\\csname\\TEXT bdi\\@nd}% \n% \n\\textfont\\bsyfam=\\csname\\TEXT bsy\\@nd \n\\scriptfont\\bsyfam=\\csname\\smTEXT bsy\\@nd \n\\def\\bsy{\\fam\\bsyfam\\csname\\TEXT bsy\\@nd}% \n% \n\\textfont\\bssfam=\\csname\\TEXT bss\\@nd \n\\scriptfont\\bssfam=\\csname\\smTEXT bss\\@nd \n\\scriptscriptfont\\bssfam=\\csname\\vsTEXT bss\\@nd \n\\def\\bss{\\fam\\bssfam\\csname\\TEXT bss\\@nd}% \n% \n\\ifams \n\\textfont\\msafam=\\csname\\TEXT msa\\@nd \n\\scriptfont\\msafam=\\csname\\smTEXT msa\\@nd \n\\scriptscriptfont\\msafam=\\csname\\vsTEXT msa\\@nd \n% \n\\textfont\\msbfam=\\csname\\TEXT msb\\@nd \n\\scriptfont\\msbfam=\\csname\\smTEXT msb\\@nd \n\\scriptscriptfont\\msbfam=\\csname\\vsTEXT msb\\@nd \n\\else\\fi \n% \n\\normalbaselineskip=\\textb@seline pt \n\\setbox\\strutbox=\\hbox{\\vrule height.7\\normalbaselineskip \ndepth.3\\baselineskip width0pt}% \n\\everymath{}% \n\\def\\sc{\\csname\\smTEXT rm\\@nd}\\normalbaselines\\rm} \n% \n% Fonts for small material (captions, footnotes etc) \n% \n\\def\\smallfonts{% \n\\textfont0=\\csname\\SMALL rm\\@nd \n\\scriptfont0=\\csname\\smSMALL rm\\@nd \n\\scriptscriptfont0=\\csname\\vsSMALL rm\\@nd \n\\def\\rm{\\fam0\\csname\\SMALL rm\\@nd \n\\def\\sc{\\csname\\smSMALL rm\\@nd}}% \n% \n\\textfont1=\\csname\\SMALL i\\@nd \n\\scriptfont1=\\csname\\smSMALL i\\@nd \n\\scriptscriptfont1=\\csname\\vsSMALL i\\@nd \n% \n\\textfont2=\\csname\\SMALL sy\\@nd \n\\scriptfont2=\\csname\\smSMALL sy\\@nd \n\\scriptscriptfont2=\\csname\\vsSMALL sy\\@nd \n% \n\\textfont3=\\csname\\SMALL ex\\@nd \n\\scriptfont3=\\csname\\smSMALL ex\\@nd \n\\scriptscriptfont3=\\csname\\smSMALL ex\\@nd \n% \n\\textfont\\itfam=\\csname\\SMALL it\\@nd \n\\scriptfont\\itfam=\\csname\\smSMALL it\\@nd \n\\scriptscriptfont\\itfam=\\csname\\vsSMALL it\\@nd \n\\def\\it{\\fam\\itfam\\csname\\SMALL it\\@nd \n\\def\\sc{\\csname\\smSMALL it\\@nd}}% \n% \n\\textfont\\slfam=\\csname\\SMALL sl\\@nd \n\\def\\sl{\\fam\\slfam\\csname\\SMALL sl\\@nd \n\\def\\sc{\\csname\\smSMALL sl\\@nd}}% \n% \n\\textfont\\bffam=\\csname\\SMALL bf\\@nd \n\\scriptfont\\bffam=\\csname\\smSMALL bf\\@nd \n\\scriptscriptfont\\bffam=\\csname\\vsSMALL bf\\@nd \n\\def\\bf{\\fam\\bffam\\csname\\SMALL bf\\@nd \n\\def\\sc{\\csname\\smSMALL bf\\@nd}}% \n% \n\\textfont\\ttfam=\\csname\\SMALL tt\\@nd \n\\def\\tt{\\fam\\ttfam\\csname\\SMALL tt\\@nd}% \n% \n\\textfont\\bdifam=\\csname\\SMALL bdi\\@nd \n\\scriptfont\\bdifam=\\csname\\smSMALL bdi\\@nd \n\\scriptscriptfont\\bdifam=\\csname\\vsSMALL bdi\\@nd \n\\def\\bdi{\\fam\\bdifam\\csname\\SMALL bdi\\@nd}% \n% \n\\textfont\\bsyfam=\\csname\\SMALL bsy\\@nd \n\\scriptfont\\bsyfam=\\csname\\smSMALL bsy\\@nd \n\\def\\bsy{\\fam\\bsyfam\\csname\\SMALL bsy\\@nd}% \n% \n\\textfont\\bssfam=\\csname\\SMALL bss\\@nd \n\\scriptfont\\bssfam=\\csname\\smSMALL bss\\@nd \n\\scriptscriptfont\\bssfam=\\csname\\vsSMALL bss\\@nd \n\\def\\bss{\\fam\\bssfam\\csname\\SMALL bss\\@nd}% \n% \n\\ifams \n\\textfont\\msafam=\\csname\\SMALL msa\\@nd \n\\scriptfont\\msafam=\\csname\\smSMALL msa\\@nd \n\\scriptscriptfont\\msafam=\\csname\\vsSMALL msa\\@nd \n% \n\\textfont\\msbfam=\\csname\\SMALL msb\\@nd \n\\scriptfont\\msbfam=\\csname\\smSMALL msb\\@nd \n\\scriptscriptfont\\msbfam=\\csname\\vsSMALL msb\\@nd \n\\else\\fi \n% \n\\normalbaselineskip=\\smallb@seline pt% \n\\setbox\\strutbox=\\hbox{\\vrule height.7\\normalbaselineskip \ndepth.3\\baselineskip width0pt}% \n\\everymath{}% \n\\def\\sc{\\csname\\smSMALL rm\\@nd}\\normalbaselines\\rm}% \n% \n\\everydisplay{\\indenteddisplay \n \\gdef\\labeltype{\\eqlabel}}% \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Macros to define extra maths symbols % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\hexnumber@#1{\\ifcase#1 0\\or 1\\or 2\\or 3\\or 4\\or 5\\or 6\\or 7\\or 8\\or \n 9\\or A\\or B\\or C\\or D\\or E\\or F\\fi} \n\\edef\\bffam@{\\hexnumber@\\bffam} \n\\edef\\bdifam@{\\hexnumber@\\bdifam} \n\\edef\\bsyfam@{\\hexnumber@\\bsyfam} \n% \n\\def\\undefine#1{\\let#1\\undefined} \n\\def\\newsymbol#1#2#3#4#5{\\let\\next@\\relax \n \\ifnum#2=\\thr@@\\let\\next@\\bdifam@\\else \n \\ifams \n \\ifnum#2=\\@ne\\let\\next@\\msafam@\\else \n \\ifnum#2=\\tw@\\let\\next@\\msbfam@\\fi\\fi \n \\fi\\fi \n \\mathchardef#1=\"#3\\next@#4#5} \n\\def\\mathhexbox@#1#2#3{\\relax \n \\ifmmode\\mathpalette{}{\\m@th\\mathchar\"#1#2#3}% \n \\else\\leavevmode\\hbox{$\\m@th\\mathchar\"#1#2#3$}\\fi} \n% \n\\def\\bcal{\\fam\\bsyfam\\relax} \n\\def\\bi#1{{\\fam\\bdifam\\relax#1}} \n% \n% If file amsmacro is not in current directory \n% or somewhere with set path add path before \n% file name in following line \n% \n\\ifams\\input amsmacro\\fi \n% \n% Bold italic Greek characters \n% \n\\newsymbol\\bitGamma 3000 \n\\newsymbol\\bitDelta 3001 \n\\newsymbol\\bitTheta 3002 \n\\newsymbol\\bitLambda 3003 \n\\newsymbol\\bitXi 3004 \n\\newsymbol\\bitPi 3005 \n\\newsymbol\\bitSigma 3006 \n\\newsymbol\\bitUpsilon 3007 \n\\newsymbol\\bitPhi 3008 \n\\newsymbol\\bitPsi 3009 \n\\newsymbol\\bitOmega 300A \n\\newsymbol\\balpha 300B \n\\newsymbol\\bbeta 300C \n\\newsymbol\\bgamma 300D \n\\newsymbol\\bdelta 300E \n\\newsymbol\\bepsilon 300F \n\\newsymbol\\bzeta 3010 \n\\newsymbol\\bfeta 3011 \n\\newsymbol\\btheta 3012 \n\\newsymbol\\biota 3013 \n\\newsymbol\\bkappa 3014 \n\\newsymbol\\blambda 3015 \n\\newsymbol\\bmu 3016 \n\\newsymbol\\bnu 3017 \n\\newsymbol\\bxi 3018 \n\\newsymbol\\bpi 3019 \n\\newsymbol\\brho 301A \n\\newsymbol\\bsigma 301B \n\\newsymbol\\btau 301C \n\\newsymbol\\bupsilon 301D \n\\newsymbol\\bphi 301E \n\\newsymbol\\bchi 301F \n\\newsymbol\\bpsi 3020 \n\\newsymbol\\bomega 3021 \n\\newsymbol\\bvarepsilon 3022 \n\\newsymbol\\bvartheta 3023 \n\\newsymbol\\bvaromega 3024 \n\\newsymbol\\bvarrho 3025 \n\\newsymbol\\bvarzeta 3026 \n\\newsymbol\\bvarphi 3027 \n\\newsymbol\\bpartial 3040 \n\\newsymbol\\bell 3060 \n\\newsymbol\\bimath 307B \n\\newsymbol\\bjmath 307C \n% \n\\mathchardef\\binfty \"0\\bsyfam@31 \n\\mathchardef\\bnabla \"0\\bsyfam@72 \n\\mathchardef\\bdot \"2\\bsyfam@01 \n\\mathchardef\\bGamma \"0\\bffam@00 \n\\mathchardef\\bDelta \"0\\bffam@01 \n\\mathchardef\\bTheta \"0\\bffam@02 \n\\mathchardef\\bLambda \"0\\bffam@03 \n\\mathchardef\\bXi \"0\\bffam@04 \n\\mathchardef\\bPi \"0\\bffam@05 \n\\mathchardef\\bSigma \"0\\bffam@06 \n\\mathchardef\\bUpsilon \"0\\bffam@07 \n\\mathchardef\\bPhi \"0\\bffam@08 \n\\mathchardef\\bPsi \"0\\bffam@09 \n\\mathchardef\\bOmega \"0\\bffam@0A \n% \n\\mathchardef\\itGamma \"0100 \n\\mathchardef\\itDelta \"0101 \n\\mathchardef\\itTheta \"0102 \n\\mathchardef\\itLambda \"0103 \n\\mathchardef\\itXi \"0104 \n\\mathchardef\\itPi \"0105 \n\\mathchardef\\itSigma \"0106 \n\\mathchardef\\itUpsilon \"0107 \n\\mathchardef\\itPhi \"0108 \n\\mathchardef\\itPsi \"0109 \n\\mathchardef\\itOmega \"010A \n% \n\\mathchardef\\Gamma \"0000 \n\\mathchardef\\Delta \"0001 \n\\mathchardef\\Theta \"0002 \n\\mathchardef\\Lambda \"0003 \n\\mathchardef\\Xi \"0004 \n\\mathchardef\\Pi \"0005 \n\\mathchardef\\Sigma \"0006 \n\\mathchardef\\Upsilon \"0007 \n\\mathchardef\\Phi \"0008 \n\\mathchardef\\Psi \"0009 \n\\mathchardef\\Omega \"000A \n% \n% Counter definitions \n% \n\\newcount\\firstpage \\firstpage=1 % start page no \n\\newcount\\jnl % journal no \n\\newcount\\secno % section number \n\\newcount\\subno % number of subsection \n\\newcount\\subsubno % number of subsubsection \n\\newcount\\appno % appendix number \n\\newcount\\tabno % table number \n\\newcount\\figno % figure number \n\\newcount\\countno % equation numbers \n\\newcount\\refno % reference number \n\\newcount\\eqlett \\eqlett=97 % equation letter \n% \n\\newif\\ifletter \n\\newif\\ifwide \n\\newif\\ifnotfull \n\\newif\\ifaligned \n\\newif\\ifnumbysec \n\\newif\\ifappendix \n\\newif\\ifnumapp \n\\newif\\ifssf \n\\newif\\ifppt \n% \n\\newdimen\\t@bwidth \n\\newdimen\\c@pwidth \n\\newdimen\\digitwidth %character width \n\\newdimen\\argwidth %argument width \n\\newdimen\\secindent \\secindent=5pc %indentation of maths \n\\newdimen\\textind \\textind=16pt %indentation of text \n\\newdimen\\tempval %temporary value \n\\newskip\\beforesecskip \n\\def\\beforesecspace{\\vskip\\beforesecskip\\relax} \n\\newskip\\beforesubskip \n\\def\\beforesubspace{\\vskip\\beforesubskip\\relax} \n\\newskip\\beforesubsubskip \n\\def\\beforesubsubspace{\\vskip\\beforesubsubskip\\relax} \n\\newskip\\secskip \n\\def\\secspace{\\vskip\\secskip\\relax} \n\\newskip\\subskip \n\\def\\subspace{\\vskip\\subskip\\relax} \n\\newskip\\insertskip \n\\def\\insertspace{\\vskip\\insertskip\\relax} \n% \n\\def\\sp@ce{\\ifx\\next*\\let\\next=\\@ssf \n \\else\\let\\next=\\@nossf\\fi\\next} \n\\def\\@ssf#1{\\nobreak\\secspace\\global\\ssftrue\\nobreak} \n\\def\\@nossf{\\nobreak\\secspace\\nobreak\\noindent\\ignorespaces} \n% \n\\def\\subsp@ce{\\ifx\\next*\\let\\next=\\@sssf \n \\else\\let\\next=\\@nosssf\\fi\\next} \n\\def\\@sssf#1{\\nobreak\\subspace\\global\\ssftrue\\nobreak} \n\\def\\@nosssf{\\nobreak\\subspace\\nobreak\\noindent\\ignorespaces} \n% \n\\beforesecskip=24pt plus12pt minus8pt \n\\beforesubskip=12pt plus6pt minus4pt \n\\beforesubsubskip=12pt plus6pt minus4pt \n\\secskip=12pt plus 2pt minus 2pt \n\\subskip=6pt plus3pt minus2pt \n\\insertskip=18pt plus6pt minus6pt% \n\\fontdimen16\\tensy=2.7pt \n\\fontdimen17\\tensy=2.7pt \n% \n% Labels etc for cross referencing macros \n% \n\\def\\eqlabel{(\\ifappendix\\applett \n \\ifnumbysec\\ifnum\\secno>0 \\the\\secno\\fi.\\fi \n \\else\\ifnumbysec\\the\\secno.\\fi\\fi\\the\\countno)} \n\\def\\seclabel{\\ifappendix\\ifnumapp\\else\\applett\\fi \n \\ifnum\\secno>0 \\the\\secno \n \\ifnumbysec\\ifnum\\subno>0.\\the\\subno\\fi\\fi\\fi \n \\else\\the\\secno\\fi\\ifnum\\subno>0.\\the\\subno \n \\ifnum\\subsubno>0.\\the\\subsubno\\fi\\fi} \n\\def\\tablabel{\\ifappendix\\applett\\fi\\the\\tabno} \n\\def\\figlabel{\\ifappendix\\applett\\fi\\the\\figno} \n% \n\\def\\gac{\\global\\advance\\countno by 1} \n% \n% Redefinition of footnote macros to lose rule and remove indentation \n% \n\\def\\footnoterule{} \n\\def\\vfootnote#1{\\insert\\footins\\bgroup \n\\interlinepenalty=\\interfootnotelinepenalty \n\\splittopskip=\\ht\\strutbox % top baseline for broken footnotes \n\\splitmaxdepth=\\dp\\strutbox \\floatingpenalty=20000 \n\\leftskip=0pt \\rightskip=0pt \\spaceskip=0pt \\xspaceskip=0pt% \n\\noindent\\smallfonts\\rm #1\\ \\ignorespaces\\footstrut\\futurelet\\next\\fo@t} \n% \n% Redefinition of endinsert to give more controllable \n% space around tables and figures \n% \n\\def\\endinsert{\\egroup \n \\if@mid \\dimen@=\\ht0 \\advance\\dimen@ by\\dp0 \n \\advance\\dimen@ by12\\p@ \\advance\\dimen@ by\\pagetotal \n \\ifdim\\dimen@>\\pagegoal \\@midfalse\\p@gefalse\\fi\\fi \n \\if@mid \\insertspace \\box0 \\par \\ifdim\\lastskip<\\insertskip \n \\removelastskip \\penalty-200 \\insertspace \\fi \n \\else\\insert\\topins{\\penalty100 \n \\splittopskip=0pt \\splitmaxdepth=\\maxdimen \n \\floatingpenalty=0 \n \\ifp@ge \\dimen@=\\dp0 \n \\vbox to\\vsize{\\unvbox0 \\kern-\\dimen@}% \n \\else\\box0\\nobreak\\insertspace\\fi}\\fi\\endgroup} \n% \n% special macros for display equations \n% \n% for indentation of turned over lines in mathematics \n% \n\\def\\ind{\\hbox to \\secindent{\\hfill}} \n% \n% for turned over equals sign to left of maths indent \n% \n\\def\\eql{\\llap{${}={}$}} \n\\def\\lsim{\\llap{${}\\sim{}$}} \n\\def\\lsimeq{\\llap{${}\\simeq{}$}} \n\\def\\lequiv{\\llap{${}\\equiv{}$}} \n% \n% for other signs to left of maths indent \n% \n\\def\\lo#1{\\llap{${}#1{}$}} \n% \n% displayed equation indented \n% \n\\def\\indeqn#1{\\alignedfalse\\displ@y\\halign{\\hbox to \\displaywidth \n {$\\ind\\@lign\\displaystyle##\\hfil$}\\crcr #1\\crcr}} \n% \n% displayed equation indented with alignments \n% \n\\def\\indalign#1{\\alignedtrue\\displ@y \\tabskip=0pt \n \\halign to\\displaywidth{\\ind$\\@lign\\displaystyle{##}$\\tabskip=0pt \n &$\\@lign\\displaystyle{{}##}$\\hfill\\tabskip=\\centering \n &\\llap{$\\@lign\\hbox{\\rm##}$}\\tabskip=0pt\\crcr \n #1\\crcr}} \n% \n\\def\\fl{{\\hskip-\\secindent}} \n% \n\\def\\indenteddisplay#1$${\\indispl@y{#1 }} \n\\def\\indispl@y#1{\\disptest#1\\eqalignno\\eqalignno\\disptest} \n\\def\\disptest#1\\eqalignno#2\\eqalignno#3\\disptest{% \n \\ifx#3\\eqalignno \n \\indalign#2% \n \\else\\indeqn{#1}\\fi$$} \n% \n% Roman small caps (if in Roman \\sc gives small caps) \n% \n\\def\\rmsc{\\rm\\sc} \n% \n% Italic small caps (if in italic \\sc gives italic small caps) \n% \n\\def\\itsc{\\it\\sc} \n% \n% Bold small caps (if in bold \\sc gives bold small caps) \n% \n\\def\\bfsc{\\bf\\rm} \n% \n% Small caps in maths \n% \n\\def\\msc#1{\\hbox{\\sc #1}} \n% \n% Miscellaneous definitions \n% \n\\def\\ms{\\noalign{\\vskip3pt plus3pt minus2pt}} \n\\def\\bs{\\noalign{\\vskip6pt plus3pt minus2pt}} \n\\def\\ns{\\noalign{\\vskip-3pt}} \n\\def\\bk{\\noalign{\\break}} \n\\def\\nb{\\noalign{\\nobreak}} \n% \n\\def\\hfb{\\hfill\\break} \n% \n% Bold h bar \n% \n\\def\\bhbar{\\rlap{\\kern1pt\\raise.4ex\\hbox{\\bf\\char'40}}\\bi{h}} \n% \n\\def\\case#1#2{{\\textstyle{#1\\over#2}}} \n\\def\\dash{---{}--- } \n\\let\\du=\\d \n\\def\\d{{\\rm d}} \n\\def\\e{{\\rm e}} \n\\def\\etal{{\\it et al\\/}\\ } \n\\def\\frac#1#2{{#1\\over#2}} \n\\ifams \n\\def\\lap{\\lesssim} \n\\def\\gap{\\gtrsim} \n\\let\\le=\\leqslant \n\\let\\lequal=\\leq \n\\let\\leq=\\leqslant \n\\let\\ge=\\geqslant \n\\let\\gequal=\\geq \n\\let\\geq=\\geqslant \n\\else \n\\let\\lequal=\\le \n\\let\\gequal=\\ge \n\\def\\gap{\\;\\lower3pt\\hbox{$\\buildrel > \\over \\sim$}\\;}% \n\\def\\lap{\\;\\lower3pt\\hbox{$\\buildrel < \\over \\sim$}\\;}\\fi \n\\def\\i{{\\rm i}} \n\\chardef\\ii=\"10 \n\\def\\tqs{\\hbox to 25pt{\\hfil}} \n\\let\\lc=\\lowercase \n\\let\\uc=\\uppercase \n\\def\\Or{\\mathop{\\rm O}\\nolimits} \n\\def\\Tr{\\mathop{\\rm Tr}\\nolimits} \n\\def\\tr{\\mathop{\\rm tr}\\nolimits} \n\\def\\Cal#1{{\\cal #1}} \n\\def\\Bbbone{1\\kern-.22em {\\rm l}} \n% \n% Primes to display summations and products \n% which also have sub or superscripts \n% \n\\def\\rp{\\raise8pt\\hbox{$\\scriptstyle\\prime$}} \n% \n% then use \\sum^{...}_{...}\\rp or \\prod^{...}_{...}\\rp. \n% \n% Shadow brackets \n% \n% Single brackets for normal size only \n% \n\\def\\lshad{\\lbrack\\!\\lbrack} \n\\def\\rshad{\\rbrack\\!\\rbrack} \n% \n% Variable size for display style \n% \n\\def\\[#1\\]{\\setbox0=\\hbox{$\\dsty#1$}\\argwidth=\\wd0 \n \\setbox0=\\hbox{$\\left[\\box0\\right]$}\\advance\\argwidth by -\\wd0 \n \\left[\\kern.3\\argwidth\\box0\\kern.3\\argwidth\\right]} \n% \n% Variable size for text style \n% \n\\def\\lsb#1\\rsb{\\setbox0=\\hbox{$#1$}\\argwidth=\\wd0 \n \\setbox0=\\hbox{$\\left[\\box0\\right]$}\\advance\\argwidth by -\\wd0 \n \\left[\\kern.3\\argwidth\\box0\\kern.3\\argwidth\\right]} \n% \n\\def\\pounds{{\\tenu \\$}} \n% \n% Square for end of theorems \n% \n\\def\\sqr{\\hfill\\hbox{$\\square$}} \n% \n\\def\\pt(#1){({\\it #1\\/})} \n\\def\\ts{{\\thinspace}}% \n\\def\\co{$^{\\rm c}\\!/\\!_{\\rm o}$} \n% \n\\let\\dsty=\\displaystyle \n\\let\\tsty=\\textstyle \n\\let\\ssty=\\scriptstyle \n\\let\\sssty=\\scriptscriptstyle \n% \n% Definition for Nuclear Physics Keyword abstract \n% \n\\def\\reactions#1{\\vskip 12pt plus2pt minus2pt% \n\\vbox{\\hbox{\\kern\\secindent\\vrule\\kern12pt% \n\\vbox{\\kern0.5pt\\vbox{\\hsize=24pc\\parindent=0pt\\smallfonts\\rm NUCLEAR \nREACTIONS\\strut\\quad #1\\strut}\\kern0.5pt}\\kern12pt\\vrule}}} \n% \n% Definition for slashed characters \n% \n\\def\\slashchar#1{\\setbox0=\\hbox{$#1$}\\dimen0=\\wd0% \n\\setbox1=\\hbox{/}\\dimen1=\\wd1% \n\\ifdim\\dimen0>\\dimen1% \n\\rlap{\\hbox to \\dimen0{\\hfil/\\hfil}}#1\\else \n\\rlap{\\hbox to \\dimen1{\\hfil$#1$\\hfil}}/\\fi} \n% \n% Redefine \\textindent for use in \\item \n% \n\\def\\textindent#1{\\noindent\\hbox to \\parindent{#1\\hss}\\ignorespaces} \n% \n% Symbols and curves for use in figure captions \n% \n\\def\\opencirc{\\raise1pt\\hbox{$\\scriptstyle{\\bigcirc}$}} \n\\def\\fullcirc{\\hbox{\\headfonts\\rm$\\scriptstyle\\bullet$}} \n\\ifams \n\\def\\opensqr{\\hbox{$\\square$}} \n\\def\\opentri{\\hbox{$\\vartriangle$}} \n\\def\\opentridown{\\hbox{$\\triangledown$}} \n\\def\\opendiamond{\\hbox{$\\lozenge$}} \n\\def\\fullsqr{\\hbox{$\\blacksquare$}} \n\\def\\fulldiamond{\\hbox{$\\blacklozenge$}} \n\\def\\fulltri{\\hbox{$\\blacktriangle$}} \n\\def\\fulltridown{\\hbox{$\\blacktriangledown$}} \n\\else \n\\def\\opensqr{\\vbox{\\hrule height.4pt\\hbox{\\vrule width.4pt height3.5pt \n \\kern3.5pt\\vrule width.4pt}\\hrule height.4pt}} \n\\def\\opentri{\\hbox{$\\scriptstyle\\bigtriangleup$}} \n\\def\\opentridown{\\raise1pt\\hbox{$\\scriptstyle\\bigtriangledown$}} \n\\def\\opendiamond{\\hbox{$\\diamond$}} \n\\def\\fullsqr{\\vrule height3.5pt width3.5pt} \n\\def\\fulldiamond{\\opendiamond} % These produce the \n\\def\\fulltri{\\opentri} % equivalent open character \n\\def\\fulltridown{\\opentridown} % to be filled in. \n\\fi \n\\def\\dotted{\\hbox{${\\mathinner{\\cdotp\\cdotp\\cdotp\\cdotp\\cdotp\\cdotp}}$}} \n\\def\\dashed{\\hbox{-\\ts -\\ts -\\ts -}} \n\\def\\broken{\\hbox{-- -- --}} \n\\def\\longbroken{\\hbox{--- --- ---}} \n\\def\\chain{\\hbox{--- $\\cdot$ ---}} \n\\def\\dashddot{\\hbox{--- $\\cdot$ $\\cdot$ ---}} \n\\def\\full{\\hbox{------}} \n% \n% Redefinition of \\cases \n% \n\\def\\m@th{\\mathsurround=0pt} \n% \n% Displaystyle now used for first term \n% \n\\def\\cases#1{% \n\\left\\{\\,\\vcenter{\\normalbaselines\\openup1\\jot\\m@th% \n \\ialign{$\\displaystyle##\\hfil$&\\rm\\tqs##\\hfil\\crcr#1\\crcr}}\\right.}% \n% \n% Original version of cases now called \\oldcases \n% \n\\def\\oldcases#1{\\left\\{\\,\\vcenter{\\normalbaselines\\m@th \n \\ialign{$##\\hfil$&\\rm\\quad##\\hfil\\crcr#1\\crcr}}\\right.} \n% \n% Cases with number at end each line (using automatic numbering) \n% \n\\def\\numcases#1{\\left\\{\\,\\vcenter{\\baselineskip=15pt\\m@th% \n \\ialign{$\\displaystyle##\\hfil$&\\rm\\tqs##\\hfil \n \\crcr#1\\crcr}}\\right.\\hfill \n \\vcenter{\\baselineskip=15pt\\m@th% \n \\ialign{\\rlap{$\\phantom{\\displaystyle##\\hfil}$}\\tabskip=0pt&\\en \n \\rlap{\\phantom{##\\hfil}}\\crcr#1\\crcr}}} \n% \n\\def\\ptnumcases#1{\\left\\{\\,\\vcenter{\\baselineskip=15pt\\m@th% \n \\ialign{$\\displaystyle##\\hfil$&\\rm\\tqs##\\hfil \n \\crcr#1\\crcr}}\\right.\\hfill \n \\vcenter{\\baselineskip=15pt\\m@th% \n \\ialign{\\rlap{$\\phantom{\\displaystyle##\\hfil}$}\\tabskip=0pt&\\enpt \n \\rlap{\\phantom{##\\hfil}}\\crcr#1\\crcr}}\\global\\eqlett=97 \n \\global\\advance\\countno by 1} \n% \n% for equation numbers instead of \\eqno \n% \n\\def\\eq(#1){\\ifaligned\\@mp(#1)\\else\\hfill\\llap{{\\rm (#1)}}\\fi} \n\\def\\ceq(#1){\\ns\\ns\\ifaligned\\@mp\\fi\\eq(#1)\\cr\\ns\\ns} \n\\def\\eqpt(#1#2){\\ifaligned\\@mp(#1{\\it #2\\/}) \n \\else\\hfill\\llap{{\\rm (#1{\\it #2\\/})}}\\fi} \n\\let\\eqno=\\eq \n% \n% Automatic numbering of equations \n% \n\\countno=1 \n\\def\\eqnobysec{\\numbysectrue} \n\\def\\aleq{&\\rm(\\ifappendix\\applett \n \\ifnumbysec\\ifnum\\secno>0 \\the\\secno\\fi.\\fi \n \\else\\ifnumbysec\\the\\secno.\\fi\\fi\\the\\countno} \n\\def\\noaleq{\\hfill\\llap\\bgroup\\rm(\\ifappendix\\applett \n \\ifnumbysec\\ifnum\\secno>0 \\the\\secno\\fi.\\fi \n \\else\\ifnumbysec\\the\\secno.\\fi\\fi\\the\\countno} \n\\def\\@mp{&} \n\\def\\en{\\ifaligned\\aleq)\\else\\noaleq)\\egroup\\fi\\gac} \n\\def\\cen{\\ns\\ns\\ifaligned\\@mp\\fi\\en\\cr\\ns\\ns} \n\\def\\enpt{\\ifaligned\\aleq{\\it\\char\\the\\eqlett})\\else \n \\noaleq{\\it\\char\\the\\eqlett})\\egroup\\fi \n \\global\\advance\\eqlett by 1} \n\\def\\endpt{\\ifaligned\\aleq{\\it\\char\\the\\eqlett})\\else \n \\noaleq{\\it\\char\\the\\eqlett})\\egroup\\fi \n \\global\\eqlett=97\\gac} \n% \n% abbreviations for Institute of Physics Publishing journals \n% \n\\def\\CQG{{\\it Class. Quantum Grav.}} \n\\def\\IP{{\\it Inverse Problems\\/}} \n\\def\\JPA{{\\it J. Phys. A: Math. Gen.}} \n\\def\\JPB{{\\it J. Phys. B: At. Mol. Phys.}} %1968-87 \n\\def\\jpb{{\\it J. Phys. B: At. Mol. Opt. Phys.}} %1988 and onwards \n\\def\\JPC{{\\it J. Phys. C: Solid State Phys.}} %1968--1988 \n\\def\\JPCM{{\\it J. Phys.: Condens. Matter\\/}} %1989 and onwards \n\\def\\JPD{{\\it J. Phys. D: Appl. Phys.}} \n\\def\\JPE{{\\it J. Phys. E: Sci. Instrum.}} \n\\def\\JPF{{\\it J. Phys. F: Met. Phys.}} \n\\def\\JPG{{\\it J. Phys. G: Nucl. Phys.}} %1975--1988 \n\\def\\jpg{{\\it J. Phys. G: Nucl. Part. Phys.}} %1989 and onwards \n\\def\\MSMSE{{\\it Modelling Simulation Mater. Sci. Eng.}} \n\\def\\MST{{\\it Meas. Sci. Technol.}} %1990 and onwards \n\\def\\NET{{\\it Network\\/}} \n\\def\\NL{{\\it Nonlinearity\\/}} \n\\def\\NT{{\\it Nanotechnology}} \n\\def\\PAO{{\\it Pure Appl. Optics\\/}} \n\\def\\PMB{{\\it Phys. Med. Biol.}} \n\\def\\PSST{{\\it Plasma Sources Sci. Technol.}} \n\\def\\QO{{\\it Quantum Opt.}} \n\\def\\RPP{{\\it Rep. Prog. Phys.}} \n\\def\\SST{{\\it Semicond. Sci. Technol.}} \n\\def\\SUST{{\\it Supercond. Sci. Technol.}} \n\\def\\WRM{{\\it Waves Random Media\\/}} \n% \n% Other commonly quoted journals \n% \n\\def\\AC{{\\it Acta Crystallogr.}} \n\\def\\AM{{\\it Acta Metall.}} \n\\def\\AP{{\\it Ann. Phys., Lpz.}} \n\\def\\APNY{{\\it Ann. Phys., NY\\/}} \n\\def\\APP{{\\it Ann. Phys., Paris\\/}} \n\\def\\CJP{{\\it Can. J. Phys.}} \n\\def\\GRG{{\\it Gen. Rel. Grav.}} \n\\def\\JAP{{\\it J. Appl. Phys.}} \n\\def\\JCP{{\\it J. Chem. Phys.}} \n\\def\\JJAP{{\\it Japan. J. Appl. Phys.}} \n\\def\\JP{{\\it J. Physique\\/}} \n\\def\\JPhCh{{\\it J. Phys. Chem.}} \n\\def\\JMMM{{\\it J. Magn. Magn. Mater.}} \n\\def\\JMP{{\\it J. Math. Phys.}} \n\\def\\JOSA{{\\it J. Opt. Soc. Am.}} \n\\def\\JPSJ{{\\it J. Phys. Soc. Japan\\/}} \n\\def\\JQSRT{{\\it J. Quant. Spectrosc. Radiat. Transfer\\/}} \n\\def\\NC{{\\it Nuovo Cimento\\/}} \n\\def\\NIM{{\\it Nucl. Instrum. Methods\\/}} \n\\def\\NP{{\\it Nucl. Phys.}} \n\\def\\PL{{\\it Phys. Lett.}} \n\\def\\PR{{\\it Phys. Rev.}} \n\\def\\PRL{{\\it Phys. Rev. Lett.}} \n\\def\\PRS{{\\it Proc. R. Soc.}} \n\\def\\PS{{\\it Phys. Scr.}} \n\\def\\PSS{{\\it Phys. Status Solidi\\/}} \n\\def\\PTRS{{\\it Phil. Trans. R. Soc.}} \n\\def\\RMP{{\\it Rev. Mod. Phys.}} \n\\def\\RSI{{\\it Rev. Sci. Instrum.}} \n\\def\\SSC{{\\it Solid State Commun.}} \n\\def\\ZP{{\\it Z. Phys.}} \n% \n\\headline={\\ifodd\\pageno{\\ifnum\\pageno=\\firstpage\\hfill \n \\else\\rrhead\\fi}\\else\\lrhead\\fi} \n% \n\\def\\rrhead{\\textfonts\\hskip\\secindent\\it \n \\shorttitle\\hfill\\rm\\folio} \n% \n\\def\\lrhead{\\textfonts\\hbox to\\secindent{\\rm\\folio\\hss}% \n \\it\\aunames\\hss} \n% \n\\footline={\\ifnum\\pageno=\\firstpage \\hfill\\textfonts\\rm\\folio\\fi} \n% \n% \n\\def\\@rticle#1#2{\\vglue.5pc \n {\\parindent=\\secindent \\bf #1\\par} \n \\vskip2.5pc \n {\\exhyphenpenalty=10000\\hyphenpenalty=10000 \n \\baselineskip=18pt\\raggedright\\noindent \n \\headfonts\\bf#2\\par}\\futurelet\\next\\sh@rttitle}% \n% \n\\def\\title#1{\\gdef\\shorttitle{#1} \n \\vglue4pc{\\exhyphenpenalty=10000\\hyphenpenalty=10000 \n \\baselineskip=18pt \n \\raggedright\\parindent=0pt \n \\headfonts\\bf#1\\par}\\futurelet\\next\\sh@rttitle} \n\\let\\paper=\\title \n% \n\\def\\article#1#2{\\gdef\\shorttitle{#2}\\@rticle{#1}{#2}} \n% \n\\def\\review#1{\\gdef\\shorttitle{#1}% \n \\@rticle{REVIEW \\ifpbm\\else ARTICLE\\fi}{#1}} \n\\def\\topical#1{\\gdef\\shorttitle{#1}% \n \\@rticle{TOPICAL REVIEW}{#1}} \n\\def\\comment#1{\\gdef\\shorttitle{#1}% \n \\@rticle{COMMENT}{#1}} \n\\def\\note#1{\\gdef\\shorttitle{#1}% \n \\@rticle{NOTE}{#1}} \n\\def\\prelim#1{\\gdef\\shorttitle{#1}% \n \\@rticle{PRELIMINARY COMMUNICATION}{#1}} \n\\def\\letter#1{\\gdef\\shorttitle{Letter to the Editor}% \n \\gdef\\aunames{Letter to the Editor} \n \\global\\lettertrue\\ifnum\\jnl=7\\global\\letterfalse\\fi \n \\@rticle{LETTER TO THE EDITOR}{#1}} \n% \n\\def\\sh@rttitle{\\ifx\\next[\\let\\next=\\sh@rt \n \\else\\let\\next=\\f@ll\\fi\\next} \n% \n\\def\\sh@rt[#1]{\\gdef\\shorttitle{#1}} \n\\def\\f@ll{} \n% \n\\def\\author#1{\\ifletter\\else\\gdef\\aunames{#1}\\fi\\vskip1.5pc \n {\\parindent=\\secindent \n \\hang\\textfonts \n \\ifppt\\bf\\else\\rm\\fi#1\\par} \n \\ifppt\\bigskip\\else\\smallskip\\fi \n \\futurelet\\next\\@unames} \n% \n\\def\\@unames{\\ifx\\next[\\let\\next=\\short@uthor \n \\else\\let\\next=\\@uthor\\fi\\next} \n\\def\\short@uthor[#1]{\\gdef\\aunames{#1}} \n\\def\\@uthor{} \n% \n\\def\\address#1{{\\parindent=\\secindent \n \\exhyphenpenalty=10000\\hyphenpenalty=10000 \n\\ifppt\\textfonts\\else\\smallfonts\\fi\\hang\\raggedright\\rm#1\\par}% \n \\ifppt\\bigskip\\fi} \n% \n\\def\\jl#1{\\global\\jnl=#1} \n\\jl{0}% \n% \n\\def\\journal{\\ifnum\\jnl=1 J. Phys.\\ A: Math.\\ Gen.\\ \n \\else\\ifnum\\jnl=2 J. Phys.\\ B: At.\\ Mol.\\ Opt.\\ Phys.\\ \n \\else\\ifnum\\jnl=3 J. Phys.:\\ Condens. Matter\\ \n \\else\\ifnum\\jnl=4 J. Phys.\\ G: Nucl.\\ Part.\\ Phys.\\ \n \\else\\ifnum\\jnl=5 Inverse Problems\\ \n \\else\\ifnum\\jnl=6 Class. Quantum Grav.\\ \n \\else\\ifnum\\jnl=7 Network\\ \n \\else\\ifnum\\jnl=8 Nonlinearity\\ \n \\else\\ifnum\\jnl=9 Quantum Opt.\\ \n \\else\\ifnum\\jnl=10 Waves in Random Media\\ \n \\else\\ifnum\\jnl=11 Pure Appl. Opt.\\ \n \\else\\ifnum\\jnl=12 Phys. Med. Biol.\\ \n \\else\\ifnum\\jnl=13 Modelling Simulation Mater.\\ Sci.\\ Eng.\\ \n \\else\\ifnum\\jnl=14 Plasma Phys. Control. Fusion\\ \n \\else\\ifnum\\jnl=15 Physiol. Meas.\\ \n \\else\\ifnum\\jnl=16 Sov.\\ Lightwave Commun.\\ \n \\else\\ifnum\\jnl=17 J. Phys.\\ D: Appl.\\ Phys.\\ \n \\else\\ifnum\\jnl=18 Supercond.\\ Sci.\\ Technol.\\ \n \\else\\ifnum\\jnl=19 Semicond.\\ Sci.\\ Technol.\\ \n \\else\\ifnum\\jnl=20 Nanotechnology\\ \n \\else\\ifnum\\jnl=21 Meas.\\ Sci.\\ Technol.\\ \n \\else\\ifnum\\jnl=22 Plasma Sources Sci.\\ Technol.\\ \n \\else\\ifnum\\jnl=23 Smart Mater.\\ Struct.\\ \n \\else\\ifnum\\jnl=24 J.\\ Micromech.\\ Microeng.\\ \n \\else Institute of Physics Publishing\\ \n \\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi \n \\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi\\fi} \n% \n\\def\\beginabstract{\\insertspace \n \\parindent=\\secindent\\ifppt\\textfonts\\else\\smallfonts\\fi \n \\hang{\\bf Abstract. }\\rm } \n% \n\\let\\abs=\\beginabstract \n% \n\\def\\cabs{\\hang\\quad\\ } \n% \n\\def\\endabstract{\\par \n \\parindent=\\textind\\textfonts\\rm \n \\ifppt\\vfill\\fi} \n\\let\\endabs=\\endabstract \n% \n\\def\\submitted{\\ifppt\\noindent\\textfonts\\rm Submitted to \\journal\\par \n \\bigskip\\fi} \n% \n\\def\\today{\\number\\day\\ \\ifcase\\month\\or \n January\\or February\\or March\\or April\\or May\\or June\\or \n July\\or August\\or September\\or October\\or November\\or \n December\\fi\\space \\number\\year} \n% \n\\def\\date{\\ifppt\\noindent\\textfonts\\rm \n Date: \\today\\par\\goodbreak\\bigskip\\fi} \n% \n% Physics Abstracts classification numbers \n% \n\\def\\pacs#1{\\ifppt\\noindent\\textfonts\\rm \n PACS number(s): #1\\par\\bigskip\\fi} \n% \n\\def\\ams#1{\\ifppt\\noindent\\textfonts\\rm \n AMS classification scheme numbers: #1\\par\\bigskip\\fi} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Sections, subsections, etc % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\section#1{\\ifppt\\ifnum\\secno=0\\eject\\fi\\fi \n \\subno=0\\subsubno=0\\global\\advance\\secno by 1 \n \\gdef\\labeltype{\\seclabel}\\ifnumbysec\\countno=1\\fi \n \\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf \\the\\secno. #1}\\par\\futurelet\\next\\sp@ce} \n% \n\\def\\subsection#1{\\subsubno=0\\global\\advance\\subno by 1 \n \\gdef\\labeltype{\\seclabel}% \n \\ifssf\\else\\goodbreak\\beforesubspace\\fi \n \\global\\ssffalse\\nobreak \n \\noindent{\\it \\the\\secno.\\the\\subno. #1\\par}% \n \\futurelet\\next\\subsp@ce} \n% \n\\def\\subsubsection#1{\\global\\advance\\subsubno by 1 \n \\gdef\\labeltype{\\seclabel}% \n \\ifssf\\else\\goodbreak\\beforesubsubspace\\fi \n \\global\\ssffalse\\nobreak \n \\noindent{\\it \\the\\secno.\\the\\subno.\\the\\subsubno. #1}\\null. \n \\ignorespaces} \n% \n\\def\\nosections{\\ifppt\\eject\\else\\vskip30pt plus12pt minus12pt\\fi \n \\noindent\\ignorespaces} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Appendices % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\numappendix#1{\\ifappendix\\ifnumbysec\\countno=1\\fi\\else \n \\countno=1\\figno=0\\tabno=0\\fi \n \\subno=0\\global\\advance\\appno by 1 \n \\secno=\\appno\\gdef\\applett{A}\\gdef\\labeltype{\\seclabel}% \n \\global\\appendixtrue\\global\\numapptrue \n \\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf Appendix \\the\\appno. #1\\par}% \n \\futurelet\\next\\sp@ce} \n% \n\\def\\numsubappendix#1{\\global\\advance\\subno by 1\\subsubno=0 \n \\gdef\\labeltype{\\seclabel}% \n \\ifssf\\else\\goodbreak\\beforesubspace\\fi \n \\global\\ssffalse\\nobreak \n \\noindent{\\it A\\the\\appno.\\the\\subno. #1\\par}% \n \\futurelet\\next\\subsp@ce} \n% \n\\def\\@ppendix#1#2#3{\\countno=1\\subno=0\\subsubno=0\\secno=0\\figno=0\\tabno=0 \n \\gdef\\applett{#1}\\gdef\\labeltype{\\seclabel}\\global\\appendixtrue \n \\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf Appendix#2#3\\par}\\futurelet\\next\\sp@ce} \n% \n\\def\\Appendix#1{\\@ppendix{A}{. }{#1}} \n\\def\\appendix#1#2{\\@ppendix{#1}{ #1. }{#2}} \n\\def\\App#1{\\@ppendix{A}{ }{#1}} \n\\def\\app{\\@ppendix{A}{}{}} \n% \n\\def\\subappendix#1#2{\\global\\advance\\subno by 1\\subsubno=0 \n \\gdef\\labeltype{\\seclabel}% \n \\ifssf\\else\\goodbreak\\beforesubspace\\fi \n \\global\\ssffalse\\nobreak \n \\noindent{\\it #1\\the\\subno. #2\\par}% \n \\nobreak\\subspace\\noindent\\ignorespaces} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Acknowledgments, notes added and foreign abstracts % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\@ck#1{\\ifletter\\bigskip\\noindent\\ignorespaces\\else \n \\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf Acknowledgment#1\\par}% \n \\nobreak\\secspace\\noindent\\ignorespaces\\fi} \n% \n\\def\\ack{\\@ck{s}} \n\\def\\ackn{\\@ck{}} \n% \n\\def\\n@ip#1{\\goodbreak\\beforesecspace\\nobreak \n \\noindent\\smallfonts{\\it #1}. \\rm\\ignorespaces} \n\\def\\naip{\\n@ip{Note added in proof}} \n\\def\\na{\\n@ip{Note added}} \n\\def\\endnaip{\\par\\textfonts\\rm} \n% \n% \\resume and \\zus in Physics in Medicine and Biology only \n% \n\\def\\resume#1{\\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf R\\'esum\\'e\\par}% \n \\nobreak\\secspace\\parindent=0pt\\smallfonts\\rm #1\\par \n \\vskip7.5pt} \n% \n\\def\\zus#1{\\goodbreak\\beforesecspace\\nobreak \n \\noindent{\\bf Zusammenfassung\\par}% \n \\nobreak\\secspace\\parindent=0pt\\smallfonts\\rm #1\\par \n \\vskip7.5pt} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Tables % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\Tables{\\vfill\\eject\\global\\appendixfalse\\textfonts\\rm \n \\everypar{}\\noindent{\\bf Tables and table captions}\\par \n \\bigskip} \n% \n\\def\\apptabs#1{\\global\\appendixtrue\\global\\tabno=0\\gdef\\applett{#1}} \n% \n\\def\\table#1{\\tablecaption{#1}} \n\\def\\tablecont{\\topinsert\\global\\advance\\tabno by -1 \n \\tablecaption{(continued)}} \n% \n\\def\\tablecaption#1{\\gdef\\labeltype{\\tablabel}\\global\\widefalse \n \\leftskip=\\secindent\\parindent=0pt \n \\global\\advance\\tabno by 1 \n \\smallfonts{\\bf Table \\ifappendix\\applett\\fi\\the\\tabno.} \\rm #1\\par \n \\smallskip\\futurelet\\next\\t@b} \n% \n\\def\\endtable{\\vfill\\goodbreak} \n% \n\\def\\t@b{\\ifx\\next*\\let\\next=\\widet@b \n \\else\\ifx\\next[\\let\\next=\\fullwidet@b \n \\else\\let\\next=\\narrowt@b\\fi\\fi \n \\next} \n% \n\\def\\widet@b#1{\\global\\widetrue\\global\\notfulltrue \n \\t@bwidth=\\hsize\\advance\\t@bwidth by -\\secindent} \n\\def\\fullwidet@b[#1]{\\global\\widetrue\\global\\notfullfalse \n \\leftskip=0pt\\t@bwidth=\\hsize} \n\\def\\narrowt@b{\\global\\notfulltrue} \n% \n\\def\\align{\\catcode`?=13\\ifnotfull\\moveright\\secindent\\fi \n \\vbox\\bgroup\\halign\\ifwide to \\t@bwidth\\fi \n \\bgroup\\strut\\tabskip=1.2pc plus1pc minus.5pc} \n% \n\\def\\endalign{\\egroup\\egroup\\catcode`?=12} \n\\let\\Lbar=\\L \n% \n% Use \\L{#}, \\R{#} and \\C{#} to specify left, right or centred \n% columns immediately after \\table. For example \n% \\align\\L{#}&&\\L{#}\\cr gives the preamble for a table with \n% all columns aligned left, \\align\\L{#}&\\C{#}&\\R{#}\\cr \n% gives a table with 3 columns, the first aligned left, the second \n% centred and the third aligned right. \n% \n\\def\\L#1{#1\\hfill} \n\\def\\R#1{\\hfill#1} \n\\def\\C#1{\\hfill#1\\hfill} \n% \n% Rules for tables \\br at top and bottom \n% \\mr to separate headings from entries \n% \n\\def\\br{\\noalign{\\vskip2pt\\hrule height1pt\\vskip2pt}} \n\\def\\mr{\\noalign{\\vskip2pt\\hrule\\vskip2pt}} \n% \n\\def\\tabnote#1{\\vskip-\\lastskip\\noindent #1\\par} \n% \n% Definitions for centring headings over several columns \n% \\centre{4}{Results for helium} will centre \n% Results for helium over four columns \n% \\crule{4} will produce a rule centred over four columns \n% to go below a centred heading \n% \n\\def\\centre#1#2{\\multispan{#1}{\\hfill#2\\hfill}} \n\\def\\crule#1{\\multispan{#1}{\\hrulefill}} \n% 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\n\\def\\endlstable{\\vfill\\eject} \n% \n\\def\\inctabno{\\global\\advance\\tabno by 1} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Figures % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\Figures{\\vfill\\eject\\global\\appendixfalse\\textfonts\\rm \n \\everypar{}\\noindent{\\bf Figure captions}\\par \n \\bigskip} \n% \n\\def\\appfigs#1{\\global\\appendixtrue\\global\\figno=0\\gdef\\applett{#1}} \n% \n\\def\\figure#1{\\figc@ption{#1}\\bigskip} \n% \n\\def\\figc@ption#1{\\global\\advance\\figno by 1\\gdef\\labeltype{\\figlabel}% \n {\\parindent=\\secindent\\smallfonts\\hang \n {\\bf Figure \\ifappendix\\applett\\fi\\the\\figno.} \\rm #1\\par}} \n% \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% % \n% Reference lists % \n% % \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% \n\\def\\refHEAD{\\goodbreak\\beforesecspace \n \\noindent\\textfonts{\\bf References}\\par \n \\let\\ref=\\rf 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used to give a Note (as opposed to a paper or letter) \n% in PMB therefore use commands \\notes#1 for numbered Note \n% instead of \\note \n% \n\\def\\notes#1{\\bigbreak\\noindent{\\it Note #1}.\\quad\\ignorespaces} \n\\def\\definition#1{\\bigbreak\\noindent{\\it Definition #1}.\\quad\\ignorespaces} \n\\def\\endproclaim{\\par\\bigbreak} \n\\let\\linespace=\\endproclaim \n% \n\\catcode`\\@=12 \n% \n% Parameter values for `Preprint' style \n% \n\\def\\pptstyle{\\ppttrue\\headsize{17}{24}% \n\\textsize{12}{16}% \n\\smallsize{10}{12}% \n\\hsize=37.2pc\\vsize=56pc \n\\textind=20pt\\secindent=6pc} \n% \n% Parameter values for `Journal' style \n% \n\\def\\jnlstyle{\\pptfalse\\headsize{14}{18}% \n\\textsize{10}{12}% \n\\smallsize{8}{10} \n\\textind=16pt} \n% \n% Parameter values for `Eleven point' style \n% \n\\def\\mediumstyle{\\ppttrue\\headsize{14}{18}% \n\\textsize{11}{13}% \n\\smallsize{9}{11}% \n\\hsize=37.2pc\\vsize=56pc% \n\\textind=18pt\\secindent=5.5pc} \n% \n% Parameter values for `Large size' style \n% \n\\def\\largestyle{\\ppttrue\\headsize{20}{28}% \n\\textsize{12}{16}% \n\\smallsize{10}{12}% \n\\hsize=37.2pc\\vsize=56pc% \n\\textind=20pt\\secindent=6pc} \n% \n\\parindent=\\textind \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\catcode`@=11 \n\\newwrite\\auxfile \n\\newwrite\\xreffile \n\\newif\\ifxrefwarning \\xrefwarningtrue \n\\newif\\ifauxfile \n\\newif\\ifxreffile \n\\def\\testforxref{\\begingroup \n \\immediate\\openin\\xreffile = \\jobname.xrf\\space \n \\ifeof\\xreffile\\global\\xreffilefalse \n \\else\\global\\xreffiletrue\\fi \n \\immediate\\closein\\xreffile \n \\endgroup} \n\\def\\testforaux{\\begingroup \n \\immediate\\openin\\auxfile = \\jobname.aux\\space \n \\ifeof\\auxfile\\global\\auxfilefalse \n \\else\\global\\auxfiletrue\\fi \n \\immediate\\closein\\auxfile \n \\endgroup} \n\\def\\openreffile{\\immediate\\openout\\auxfile = \\jobname.aux}% \n\\def\\readreffile{% \n \\testforxref \n \\testforaux \n \\ifxreffile \n \\begingroup \n \\@setletters \n \\input \\jobname.xrf \n \\endgroup \n \\else \n\\message{No cross-reference file existed, some labels may be undefined}% \n \\fi\\openreffile}% \n\\def\\@setletters{% \n \\catcode`_=11 \\catcode`+=11 \n \\catcode`-=11 \\catcode`@=11 \n \\catcode`0=11 \\catcode`1=11 \n \\catcode`2=11 \\catcode`3=11 \n \\catcode`4=11 \\catcode`5=11 \n \\catcode`6=11 \\catcode`7=11 \n \\catcode`8=11 \\catcode`9=11 \n \\catcode`(=11 \\catcode`)=11 \n \\catcode`:=11 \\catcode`'=11 \n \\catcode`&=11 \\catcode`;=11 \n \\catcode`.=11}% \n\\gdef\\el@b{\\eqlabel} \n\\gdef\\sl@b{\\seclabel} \n\\gdef\\tl@b{\\tablabel} \n\\gdef\\fl@b{\\figlabel} \n\\def\\l@belno{\\ifx\\labeltype\\el@b \n \\let\\labelno=\\en\\def\\@label{\\eqlabel}% \n \\else\\let\\labelno=\\ignorespaces \n \\ifx\\labeltype\\sl@b \\def\\@label{\\seclabel}% \n \\else\\ifx\\labeltype\\tl@b \\def\\@label{\\tablabel}% \n \\else\\ifx\\labeltype\\fl@b \\def\\@label{\\figlabel}% \n \\else\\def\\@label{\\seclabel}\\fi\\fi\\fi\\fi} \n\\def\\label#1{\\l@belno\\expandafter\\xdef\\csname #1@\\endcsname{\\@label}% \n \\immediate\\write\\auxfile{\\string \n \\gdef\\expandafter\\string\\csname @#1\\endcsname{\\@label}}% \n \\labelno}% \n\\def\\ref#1{% \n \\expandafter \\ifx \\csname @#1\\endcsname\\relax \n \\message{Undefined label `#1'.}% \n \\expandafter\\xdef\\csname @#1\\endcsname{(??)}\\fi \n \\csname @#1\\endcsname}% \n\\readreffile \n% \n%%%%%%%%%%% \n% \n\\def"
}
] |
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cond-mat0002071
|
Generalization properties of finite size polynomial Support Vector Machines
|
[
{
"author": "Sebastian Risau-Gusman and Mirta B. Gordon"
},
{
"author": "DRFMC/SPSMS CEA Grenoble"
},
{
"author": "17 av. des Martyrs"
},
{
"author": "38054 Grenoble cedex 09"
},
{
"author": "France"
}
] |
\parbox{14cm}{ The learning properties of finite size polynomial Support Vector Machines are analyzed in the case of realizable classification tasks. The normalization of the high order features acts as a squeezing factor, introducing a strong anisotropy in the patterns distribution in feature space. As a function of the training set size, the corresponding generalization error presents a crossover, more or less abrupt depending on the distribution's anisotropy and on the task to be learned, between a fast-decreasing and a slowly decreasing regime. This behaviour corresponds to the stepwise decrease found by Dietrich et al.~[1] in the thermodynamic limit. The theoretical results are in excellent agreement with the numerical simulations. }
|
[
{
"name": "paper.tex",
"string": "\\documentstyle[floats,aps,psfig]{revtex}\n%\\documentstyle[floats,aps,psfig,draft]{revtex}\n%\\documentstyle[floats,aps,psfig,twocolumn]{revtex}\n\\begin{document}\n\n\\draft\n%\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname\n%@twocolumnfalse\\endcsname\n\n\\title{\nGeneralization properties of finite size polynomial Support Vector Machines \n} \n\n\n\\author{Sebastian Risau-Gusman and Mirta B. Gordon\\\\ \nDRFMC/SPSMS CEA Grenoble, 17 av. des Martyrs\\\\ \n38054 Grenoble cedex 09, France \n} \n\n\\date{\\today}\n \n\\maketitle \n \n\\begin{center}\n\\begin{abstract}\n \n\\parbox{14cm}{\nThe learning properties of finite size polynomial \nSupport Vector Machines are analyzed in the case \nof realizable classification tasks. The normalization \nof the high order features acts as a squeezing \nfactor, introducing a strong anisotropy in the patterns \ndistribution in feature space. As a function of the training \nset size, the corresponding generalization \nerror presents a crossover, more or less abrupt depending on \nthe distribution's anisotropy and on the task to be \nlearned, between a fast-decreasing and \na slowly decreasing regime. This behaviour corresponds to the stepwise \ndecrease found by Dietrich et al.~[1] in the thermodynamic \nlimit. The theoretical results are in excellent agreement \nwith the numerical simulations. \n} \n\\end{abstract}\n\\end{center}\n\\pacs{PACS numbers : 87.10.+e, 02.50.-r, 05.20.-y}\n\n%]\n\n\\section{Introduction}\n\nIn the last decade, the typical properties of neural networks \nthat learn classification tasks from a set of examples have been \nanalyzed using the approach of Statistical Mechanics. In the \ngeneral setting, the value of a binary output neuron represents \nwhether the input vector, describing a particular pattern, \nbelongs or not to the class to be recognized. Manuscript character \nrecognition and medical diagnosis are examples of such \nclassification problems. The process of inferring the rule \nunderlying the input-output mapping given a set of examples \nis called learning. The aim is to predict correctly the class \nof novel data, i.e. to generalize.\n\nIn the simplest neural network, the perceptron, the inputs \nare directly connected to a single output neuron. The output \nstate is given by the sign of the weighted sum of the inputs. Then, \nlearning amounts to determine the weights of the connexions in \norder to obtain the correct outputs to the training examples. \nConsidering the weights as the components of a vector, \nthe network classifies the input vectors according to whether \ntheir projections onto the weight vector are positive or negative. \nThus, patterns of different classes are separated by the hyperplane \northogonal to the weight vector. Beyond these linear separations, \ntwo different learning schemes have been suggested. Either the input \nvectors are mapped by linear hidden units to so called internal \nrepresentations that must be linearly separable by the output \nneuron, or a more powerful output unit is defined, \nable to perform more complicated functions than just the weighted \nsum of its inputs. \n\nThe first solution is implemented using \nfeedforward layered neural networks. The classification of the \ninternal representations, performed by the output neuron, \ncorresponds in general to a complicated separation surface \nin input space. However, the relation between the number of \nhidden units of a network and the class of rules it can infer is \nstill an open problem. In practice, the number of hidden \nneurons is either guessed or determined through constructive \nheuristics. \n\nA solution that uses a more complex output unit, the \nSupport Vector Machine (SVM)~\\cite{Vapnik}, has been \nrecently proposed. The input patterns are transformed into\nhigh dimensional feature vectors whose components \nmay include the original input together with specific \nfunctions of its coordinates selected a priori, with the aim \nthat the learning set be {\\it linearly separable} \nin feature space. In that case the learning problem \nis reduced to that of training a simple perceptron. For example, \nif the feature space includes all the pairwise products of \nthe input vector, the SVM may implement any classification rule \ncorresponding to a quadratic separating surface in input space. \nHigher order polynomial SVMs and other types of SVMs \nmay be defined by introducing the corresponding features.\nA big advantage is that learning a linearly separable rule \nis a convex optimization problem. The difficulties of having \nmany local minima, that hinder the process of training \nmultilayered neural networks, are thus circumvented.\nOnce the adequate feature space is defined, the \nSVM selects the particular hyperplane called Maximal Margin \n(or Maximal Stability) Hyperplane (MMH), which lies at the largest \ndistance to its closest patterns in the training set. These patterns are \ncalled Support Vectors (SV). The MMH solution has \ninteresting properties~\\cite{OKKN}. In particular, \nthe fraction of learning patterns that belong to the \nSVs provides an upper bound~\\cite{Vapnik} to the \ngeneralization error, that is, to the probability of \nincorrectly classifying a new input. It has been \nshown~\\cite{OK} that the perceptron \nweights are a linear combination of the SVs, an interesting \nproperty in high dimensional feature spaces, as their \nnumber is bounded. \n\nA perceptron can learn with very high probability any set of \nexamples, regardless of the underlying classification rule, \nprovided that their number does not exceed twice its input \nspace dimension~\\cite{Cover}. However, this simple rote \nlearning does not capture the rule underlying the \nclassification. As it may arise that the feature space \ndimension of the SVM is comparable to, or even larger \nthan, the number of available training patterns, we would \nexpect that SVMs have a poor generalization performance. \nSurprisingly, this seems not to be the case in the \napplications~\\cite{CV}. \n\nTwo theoretical papers~\\cite{DOS,BG} have recently addressed this \ninteresting question. They determined the typical properties \nof a family of polynomial SVMs in the limit of large dimensional \nspaces, reaching completely different results in spite of the \nseemingly innocuous differences between the models. Both papers \nconsider polynomial SVMs in which the input vectors \n${\\bf x} \\in \\mbox{I\\hspace{-0.18cm}R}^N$ are \nmapped onto quadratic features $\\bbox \\Phi$. More \nprecisely, the {\\it normalized} mapping ${\\bbox \\Phi}_n ({\\bf x})=({\\bf x}, \nx_1{\\bf x}/\\sqrt{N}, x_2{\\bf x}/\\sqrt{N}, \\cdots, \nx_N{\\bf x}/\\sqrt{N})$ has been considered in~\\cite{DOS}. \nThe {\\it non-normalized} mapping \n${\\bbox \\Phi}_{nn} ({\\bf x})=({\\bf x},x_1{\\bf x}, \nx_2{\\bf x}, \\cdots, x_{k}{\\bf x})$ has been studied in~\\cite{BG} \nas a function of $k$, the number of quadratic features. For $k=N$ \nthe dimension of both feature spaces is the same, corresponding to \na {\\it linear subspace} of dimension $N$, and a {\\it quadratic subspace} \nof dimension $N^2$. The mappings only differ in the distributions \nof the quadratic components in feature space. Due to the \nnormalization, those of ${\\bbox \\Phi}_n$ \nare squeezed by a {\\it normalizing factor} $a=1/\\sqrt{N}$ \nwith respect to those of ${\\bbox \\Phi}_{nn}$. In the case of \nlearning a linearly separable rule with \nthe non-normalized mapping ${\\bbox \\Phi}_{nn}$, the generalization \nerror at any given learning set size increases dramatically \nwith the number $k$ of quadratic features included~\\cite{BG}.\nOn the contrary, in the case of mapping ${\\bbox \\Phi}_{n}$, \nthe generalization error exhibits an interesting stepwise \ndecrease, also found within the Gibbs learning \nparadigm in a quadratic feature space~\\cite{YO}. \nIf the number of training patterns scales with $N$, \nthe dimension of the linear subspace, it decreases \nup to an asymptotic lower bound. If the number \nof examples scales proportionally to $N^2$, it \nvanishes asymptotically. In particular, \nif the rule to be inferred is linearly separable in the \ninput space, learning in the feature space with the mapping \n${\\bbox \\Phi}_{n}$ is harmless, as the decrease of the \ngeneralization error with the number of training patterns \npresents a slight slow-down with respect to that of a \nsimple perceptron learning in input space. \n\nAs this stepwise learning is {\\it exclusively} related \nto the fact that the normalizing factor of the quadratic \nfeatures vanishes in the thermodynamic limit $N \\rightarrow \n\\infty$, in the present paper we determine the influence of \nthe normalizing factor on the typical generalization \nperformance of finite size SVMs. To this end, we introduce two \nparameters, $\\sigma$ and $\\Delta$, caracterizing the mapping of \nthe $N$-dimensional input patterns onto the feature \nspace. The {\\it variance} $\\sigma$ reflects the \nwidth of the high-order features distribution and is\nrelated to the normalizing factor $a$. The {\\it inflation \nfactor} $\\Delta$ accounts for the proportion of quadratic \nfeatures with respect to the input space \ndimension $N$. Actual quadratic SVMs are caracterized \nby different values of $\\Delta$ and $\\sigma$, \ndepending on $N$ and $a$. Keeping $\\sigma$ and \n$\\Delta$ fixed in the thermodynamic limit allows \nus to determine the typical properties \nof actual SVMs, which have finite compressing \nfactors and inflation ratios. \n\nIn fact, the behaviour of the SVMs is the \nsame as that of a simple perceptron learning a \ntraining set with patterns drawn from a highly \nanisotropic probability distribution, such that \na {\\it macroscopic} fraction of components have a \ndifferent variance from the others. Not surprisingly, \nwe find that the asymptotic behaviour corresponding \nto both the small and large training set size limits, \nis the same as the one of the perceptron's MMH. \nOnly the prefactors depend on the mapping used by the SVM. \n\nAs expected, the stepwise learning obtained with \nthe normalized mapping in the thermodynamic limit \nbecomes a crossover. Upon increasing the \nnumber of training patterns, the generalization \nerror first present an abrupt decrease, that corresponds \nto learning the weight components in the linear subspace, \nfollowed by a slower decrease corresponding to the learning \nof the quadratic components. The steepness of the crossover \nnot only depends on $\\Delta$ and $\\sigma$, but also on \nthe task to be learned. The agreement between our analytic \nresults and numerical simulations is excellent. \n\nThe paper is organized as follows: in section \\ref{sec:model} \nwe introduce the model and the main steps of the Statistical \nMechanics calculation. Numerical simulation results are compared \nto the corresponding theoretical predictions\nin section \\ref{sec:results}. The two \nregimes of the generalization error and the \nasymptotic behaviours are discussed in section \n\\ref{sec.discussion}. The conclusion \nis left to section \\ref{sec:conclusion}. \n \n\\section{The model} \n\\label{sec:model} \n\nWe consider the problem of learning a binary classification \ntask from examples with a SVM in polynomial feature spaces. \nThe learning set contains $M$ patterns $({\\bf x}^\\mu, \n\\tau^\\mu)$ ($\\mu=1,\\cdots,M$) where ${\\bf x}^\\mu$ \nis an input vector in the $N$-dimensional input space, \nand $\\tau^\\mu \\in \\{-1,1\\}$ \nis its class. We assume that the components $x_i^\\mu$ ($i=1, \n\\cdots, N$) are independent identically distributed (i.i.d.) \nrandom variables drawn from gaussian distributions \nhaving zero-mean and unit variance:\n\n\\begin{equation}\n\\label{eq.pdex} \nP({\\bf x}) = \\prod_{i=1}^{N} \\frac{1}{\\sqrt{2 \\pi}} \\exp \\left( - \\frac{ x_i^2} {2} \\right). \n\\end{equation} \n\n\\noindent In the following we concentrate on quadratic feature \nspaces, although our conclusions are more general, and may be \napplied to higher order polynomial SVMs, as discussed in section \n\\ref{sec.discussion}. The mappings \n${\\bbox \\Phi}_{nn} ({\\bf x})=({\\bf x},x_1{\\bf x}, x_2{\\bf x}, \n\\cdots, x_N{\\bf x})$ and ${\\bbox \\Phi}_n ({\\bf x})=({\\bf x}, \nx_1{\\bf x}/\\sqrt{N}, x_2{\\bf x}/\\sqrt{N}, \\cdots, \nx_N{\\bf x}/\\sqrt{N})$ are particular instances of mappings \nof the form ${\\bbox \\Phi} ({\\bf x})= (\\phi_1, \n\\phi_2, \\cdots, \\phi_N, \\phi_{11}, \n\\phi_{12},\\cdots, \\phi_{NN})$ where $\\phi_i=x_i$, and \n$\\phi_{ij}=a \\, x_i x_j$, where $a$ is the normalizing \nfactor of the quadratic components: $a=1$ \nfor mapping ${\\bbox \\Phi}_{nn}$ and $a=1/\\sqrt{N}$ for \n${\\bbox \\Phi}_{n}$. The patterns probability \ndistribution in feature-space is:\n\n\\begin{equation}\n\\label{eq.pdephi}\nP\\left({\\mbox{$\\bbox \\Phi$}}\\right) = \\int \\, \\prod_{i=1}^{N} \n\\frac{dx_i}{\\sqrt{2 \\pi}} \\exp \\left( - \\frac{ x_i^2} {2} \\right) \n\\delta(\\phi_i - x_i)\n\\prod_{j=1}^{N} \\delta \\left(\\phi_{ij}-a \\, x_i x_j \\right). \n\\end{equation}\n \n\\noindent Clearly, the components of ${\\bbox \\Phi}$ are not independent \nrandom variables. For example, a number $O(N^3)$ of triplets of the \nform $\\phi_{ij} \\phi_{jk} \\phi_{ki}$ have positive correlations. \nThese contribute to the third order moments, which should vanish \nif the features were gaussian. Moreover, the fourth order connected \ncorrelations~\\cite{Monasson} do not vanish in the thermodynamic \nlimit. Nevertheless, in the following we will neglect these \nand higher order connected moments. This approximation, used \nin~\\cite{BG} and implicit in~\\cite{DOS}, is equivalent to \nassuming that all the components in feature space are \nindependent gaussian variables. Then, the only difference \nbetween the mappings $\\bbox \\Phi_{n}$ and $\\bbox \\Phi_{nn}$ \nlies in the variance of the quadratic \ncomponents distribution. The results obtained using this \nsimplification are in excellent agreement with the \nnumerical tests described in the next section.\n\nSince, due to the symmetry of the \ntransformation, only $N(N+1)/2$ among the $N^2$ quadratic \nfeatures are different, hereafter we restrict the \nfeature space and only consider the non redundant \ncomponents, that we denote \n${\\bbox \\xi} =({\\bbox \\xi}_u,{\\bbox \\xi}_\\sigma)$. \nIts first $N$ components \n${\\bbox \\xi}_u = (\\xi_1, \\cdots, \\xi_N)$ hereafter called \n{\\it u-components}, represent the input pattern of unit variance, \nlying in the linear subspace. \nThe remaining components ${\\bbox \\xi}_\\sigma= (\\xi_{N+1}, \n\\cdots, \\xi_{\\tilde N})$ stand for the {\\it non redundant} \nquadratic features, of variance $\\sigma$, hereafter \ncalled $\\sigma$-{\\it components}. $\\tilde N$ is \nthe dimension $\\tilde N = N(1+\\Delta)$ of the \nrestricted feature space, where the \n{\\it inflation ratio} $\\Delta$ is \nthe relative number of non-redundant quadratic features \nper input space dimension. The quadratic mapping has \n$\\Delta=(N+1)/2$. \n\nAccording to the preceding \ndiscussion, we assume that learning $N$-dimensional \npatterns selected with the isotropic distribution \n(\\ref{eq.pdex}) with a quadratic SVM is equivalent \nto learning the MMH with a simple perceptron in an \n$\\tilde N$-dimensional space where the patterns are \ndrawn using the following \nanisotropic distribution, \n\n\\begin{equation}\n\\label{eq.pdexi}\nP\\left(\\mbox{$\\bbox \\xi$}\\right) = \\prod_{i=1}^{N} \n\\frac{1}{\\sqrt{2 \\pi}} \\exp \\left( - \\frac{ \\xi_i^2} {2} \\right) \\;\n\\prod_{j=N+1}^{\\tilde N} \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left( - \\frac{ \\xi_j^2} {2 \\sigma^2} \\right). \n\\end{equation} \n\n\\noindent The second moment of the {\\it u}-features \nis $\\langle {\\bbox \\xi}_u^2 \\rangle = N$ and that of the \n$\\sigma$-features is $\\langle {\\bbox \\xi}_\\sigma^2 \\rangle = N \n\\Delta \\sigma^2$. If $\\sigma^2 \\Delta=1$, we get \n$\\langle {\\bbox \\xi}_\\sigma^2 \\rangle = \\langle \n{\\bbox \\xi}_u^2 \\rangle$, which is the relation satisfied \nby the normalized mapping considered in~\\cite{DOS}. The \nnon-normalized mapping corresponds to $\\sigma^2 \\Delta=N$. \nIn the following, instead of selecting either of these possibilities \na priori, we consider $\\Delta$ and $\\sigma$ as independent \nparameters, that are kept constant when taking the thermodynamic \nlimit. \n\nSince the rules to be inferred are assumed to be linear \nseparations in feature space, we represent them by the \nweights ${\\bf w}^*=(w_1^*,w_2^*,\\cdots,w_{\\tilde N}^*)$ \nof a {\\it teacher perceptron}, so that the class \nof the patterns is $\\tau ={\\rm sign}(\\mbox{$\\bbox \\xi$} \n\\cdot {\\bf w}^*)$. Without \nany loss of generality we consider normalized teachers: \n${\\bf w}^* \\cdot {\\bf w}^* = \\tilde N$. The training set \nin feature space is then ${\\mathcal L}_M=\n\\{ ({\\bbox \\xi}^\\mu, \\tau^\\mu) \\}_{\\mu=1,\\cdots,M}$. \n\nIn the following we study the typical properties of polynomial \nSVMs learning realizable classification tasks, using the tools \nof Statistical Mechanics. If ${\\bf w}=(w_1, \\cdots, w_{\\tilde N})$ \nis the student perceptron weight vector, $\\gamma^\\mu = \n\\tau^\\mu {\\bbox \\xi}^\\mu \\cdot \\bf w / \\sqrt{{\\bf w} \\cdot {\\bf w}}$ \nis the {\\it stability} of pattern $\\mu$ in feature space. \nThe pertinent cost function is : \n\n\\begin{equation} \n\\label{eq.cost} \nE({\\bf w}, \\kappa; {\\mathcal L}_M)=\\sum_{\\mu=1}^M \\Theta(\\kappa-\\gamma^\\mu).\n\\end{equation} \n\n\\noindent $\\kappa$, the smallest allowed distance \nbetween the hyperplane and the training patterns, is \ncalled the margin. The MMH corresponds to the weights \nwith vanishing cost (\\ref{eq.cost}) that maximize $\\kappa$. \n\nThe typical properties of cost (\\ref{eq.cost}) in the case \nof isotropic pattern distributions have been exhaustively \nstudied~\\cite{OKKN,GG}. The case of a single anisotropy axis \nhas also been investigated~\\cite{MBS}. Here we study the \ncase of the anisotropic distribution (\\ref{eq.pdexi}), \nwhere a {\\it macroscopic} fraction of components have \ndifferent variance from the others, which is pertinent for \nunderstanding the properties of the SVM.\n\nConsidering the cost (\\ref{eq.cost}) as an energy, the partition \nfunction at temperature $1/\\beta$ writes\n\n\\begin{equation}\n\\label{eq.partition}\nZ(\\kappa, \\beta; {\\mathcal L}_M)=\\int \\exp [-\\beta E({\\bf w}, \\kappa; {\\mathcal L}_M) ] \\; p({\\bf w}) \\, d{\\bf w}.\n\\end{equation}\n\n\\noindent Without any loss of generality, we assume that the a \npriori distribution of the student weights is uniform over the \nhypersphere of radius $\\tilde N^{1/2}$, i.e. \n$p({\\bf w})=\\delta({\\bf w} \\cdot {\\bf w} - \\tilde N)$, \nmeaning that the student weights are normalized in feature \nspace. In the limit $\\beta \\rightarrow \\infty$, the \ncorresponding free energy $f(\\kappa, \\beta; {\\mathcal L}_M) \n= - (1/\\beta N) \\ln Z(\\kappa, \\beta; {\\mathcal L}_M)$ is \ndominated by the weights that minimize the cost (\\ref{eq.cost}).\n\nThe typical properties of the MMH are obtained by looking for \nthe largest value of $\\kappa$ for which the quenched average \nof the free energy over the patterns distribution, in the zero \ntemperature limit $\\beta \\rightarrow \\infty$, vanishes. This \naverage is calculated by the replica method, using the identity\n\n\\begin{equation}\nf(\\kappa, \\beta)=-\\frac{1}{N \\beta} \\, \\overline{\\ln Z(\\kappa,\\beta;{\\mathcal L}_M)} = -\\frac{1}{N \\beta} \\, \\lim_{n \\rightarrow 0} \n\\frac{\\ln \\overline{Z^n(\\kappa,\\beta;{\\mathcal L}_M)} }{n},\n\\end{equation}\n\n\\noindent where the overline represents the average over \n${\\mathcal L}_M$, composed of patterns selected according \nto (\\ref{eq.pdexi}). \n\nWe obtain the typical properties of the MMH corresponding to \ngiven values of $\\Delta$ and $\\sigma$ by taking \nthe thermodynamic limit $N \\rightarrow \\infty$, \n$M \\rightarrow \\infty$, with $\\alpha \\equiv M/N$, \n$\\Delta$ and $\\sigma$ constant. Notice that the relation \nbetween the number \nof training examples and the {\\it feature} space dimension, \n$\\tilde \\alpha \\equiv M/\\tilde N = \\alpha/(1+\\Delta)$, is \nfinite. Thus, not only are we able to study the dependence \nof the learning properties as a function of the training \nset size as usual, but also of the inflation factor that \ncharacterizes the SVM, as well as of the variance of the \nquadratic components. As we only consider realizable rules, \ni.e. classification tasks that are linearly separable in \nfeature space, the energy (\\ref{eq.cost}) is a convex \nfunction of the weights ${\\bf w}$, and replica symmetry \nholds.\n\nFor any $\\kappa < \\kappa_{max}$, there are a macroscopic \nnumber of weights that minimize the cost function \n(\\ref{eq.cost}). In particular, in the case of $\\kappa=0$, \nthe cost is the number of training errors, \nand is minimized by any weight vector that classifies \ncorrectly the training set. The typical properties of such \nsolution, called Gibbs learning, may be expressed in terms \nof several order parameters~\\cite{RG}. Among them, $q_u^{ab}= \n\\sum_{i=1}^N \\langle \\overline{w_i^a w_i^b} \\rangle/\\tilde N$, \n$q_\\sigma^{ab}= \\sum_{i=N+1}^{\\tilde N} \\langle\n\\overline{w_i^a w_i^b} \\rangle/\\tilde N$ and $Q^a = \n\\sum_{i=N+1}^{\\tilde N}\\langle \\overline{w_i^a w_i^a} \n\\rangle/\\tilde N$, where $a \\neq b$ are replica indices and \n$\\langle \\cdots \\rangle$ stands for the usual \nthermodynamic average (with Boltzmann factor corresponding \nto the partition function (\\ref{eq.partition})). $q_u^{ab}$ \nand $q_\\sigma^{ab}$ represent the overlaps between different \nsolutions in the {\\it u}- and the $\\sigma$- subspaces respectively. \n$\\tilde N \\, Q^a$ is the typical norm of the \n$\\sigma$-components of replica $a$. \nBecause of replica symmetry we have $Q^a$=$Q^b=Q$, \n$q_\\sigma^{ab}=q_\\sigma$ and $q_u^{ab}=q_u$ for all $a$, \n$b$. Upon increasing $\\kappa$, the volume of the error-free \nsolutions in weight space shrinks, and vanishes when \n$\\kappa$ is maximized. Correspondingly, $q_u \\rightarrow 1-Q$ \nand $q_\\sigma \\rightarrow Q$, with $x \\equiv lim_{\\kappa \n\\rightarrow \\kappa_{max}} (1-q_u/(1-Q))/(1-q_\\sigma/Q)$ \nfinite. In the limit of $\\kappa \\rightarrow \\kappa_{max}$, \nthe properties of the MMH may be expressed in \nterms of $x$, $\\kappa_{max}$ and the following order parameters,\n\n\\begin{eqnarray} \n\\label{eq.Q}\nQ &=& \\frac{1}{\\tilde N} \\sum_{i = N+1}^{\\tilde N} \\overline{\\langle w_i^2 \\rangle},\n\\\\\nR_u &=& \\frac{1}{\\sqrt{(1-Q)(1-Q^*)}} \\frac{1}{\\tilde N} \\sum_{i=1}^N \\overline{\\langle w_i w^*_i \\rangle}, \n\\\\\n\\label{eq.Rsigma}\nR_\\sigma &=& \\frac{1}{\\sqrt{Q \\, Q^*}} \\frac{1}{\\tilde N} \\sum_{i=N+1}^{\\tilde N} \\overline{\\langle w_i w^*_i \\rangle},\n\\end{eqnarray}\n\n\\noindent where $Q^*= \\sum_{i = N+1}^{\\tilde N} (w^*_i)^2/\\tilde N$ \nis the teacher's squared weight vector in the \n$\\sigma$-subspace. $Q$ is the corresponding typical value \nfor the student. $R_u$ and $R_\\sigma$ are proportional to \nthe overlaps between the student and the teacher weights in \nthe {\\it u}- and the $\\sigma$- subspaces respectively. \nThe factors in the denominators arise because the weights are \nnot normalized in each subspace.\n\nThe saddle point equations corresponding to the extremum \nof the free energy for the MMH are\n\n\\begin{eqnarray}\n\\label{eq.sp1}\n2 \\frac{\\alpha}{\\Delta} \\Delta_\\sigma I_1 &=& (1-R_\\sigma^2) \\,\n\\frac{(x + \\Delta_\\sigma)^2}{1+\\Delta_\\sigma}, \n\\\\\n\\label{eq.sp2}\n2 \\frac{\\alpha}{\\Delta} I_2 \n&=& \\sqrt{\\frac{1+\\Delta^*_\\sigma}{\\Delta^*_\\sigma}} \\, R_\\sigma \\,\n\\frac{x + \\Delta_\\sigma}{\\sqrt{\\Delta_\\sigma (1+\\Delta_\\sigma)}}, \n\\\\\n\\label{eq.sp3}\n2 \\frac{\\alpha}{\\Delta} Q (1-\\sigma^2) I_3 &=& \n\\left(1- x \\frac{1-R_\\sigma^2}{1-R_u^2}\\right) \\, \n\\frac{x + \\Delta_\\sigma}{1+\\Delta_\\sigma},\n\\\\\n\\label{eq.sp4} \n\\frac{R_\\sigma^2}{1-R_\\sigma^2} &=& \\frac{\\Delta^*_\\sigma}{\\Delta} \\frac{R^2_u}{1-R^2_u},\n\\\\\n\\label{eq.sp5}\n\\frac{R_u}{R_\\sigma} &=& x \\frac{\\Delta}{\\sqrt{\\Delta_\\sigma \\Delta^*_\\sigma}}.\n\\end{eqnarray}\n\n\\noindent where $\\Delta_\\sigma \\equiv \\sigma^2 Q/(1-Q)$ \nand $\\Delta^*_\\sigma \\equiv \\sigma^2 Q^*/(1-Q^*)$. The \nintegrals in the left hand side of equations \n(\\ref{eq.sp1}-\\ref{eq.sp3}) are\n\n\\begin{eqnarray}\nI_1 &=& \\int_{-\\tilde \\kappa}^\\infty Dt \\, (t+{\\tilde \\kappa})^2 \\, \nH\\left(\\frac{t R}{\\sqrt{1-R^2}}\\right),\n\\\\\nI_2 &=& \\frac{1}{\\sqrt{2 \\pi}} \\left[\\frac{\\sqrt{1-R^2} \n\\exp(-{\\tilde \\kappa^2}/(2 (1-R^2)))} {\\sqrt{2 \\pi}} + \n{\\tilde \\kappa} H\\left(\\frac{-\\tilde \\kappa}{\\sqrt{1-R^2}}\\right)\\right],\n\\\\\nI_3 &=& \\int_{-\\tilde \\kappa}^\\infty Dt \\, \\tilde \\kappa \\, (t+{\\tilde \\kappa}) \\, H\\left(\\frac{t R}{\\sqrt{1-R^2}}\\right),\n\\end{eqnarray}\n\n\\noindent with $Dt \\equiv dt \\, \\exp{(-t^2/2)}/\\sqrt{2 \\pi}$, \n$H(x)=\\int_x^\\infty Dt$, and \n\n\\begin{eqnarray}\n\\tilde \\kappa &=& \\frac{\\kappa_{max}}{\\sqrt{(1-Q) (1+\\Delta_\\sigma)}},\n\\\\\n\\label{eq.R}\nR &=& \\frac{R_u + \\sqrt{\\Delta_\\sigma \\Delta^*_\\sigma} R_\\sigma} \n{\\sqrt{(1+\\Delta_\\sigma) (1+\\Delta^*_\\sigma)}}.\n\\end{eqnarray}\n\n\\noindent The value of $R$ determines the generalization \nerror through $\\epsilon_g=(1/\\pi) \\arccos(R)$.\n\nAfter solving the above equations for $Q$, $R_u$, $R_\\sigma$, \n$x$ and $\\tilde \\kappa$, it is straightforward to determine $\\rho_{SV}$, \nthe fraction of training patterns that belong to the subset of SV~\\cite{O,GG,BG}: \n\n\\begin{equation}\n\\rho_{SV} = 2 \\int_{-\\infty}^{\\tilde \\kappa} H\\left(-tR/\\sqrt{1-R^2}\\right) Dt.\n\\end{equation}\n\nIn summary of this section, instead of considering \na particular scaling of the fraction of high \norder features components and their \nnormalization with $N$, we analyzed the more \ngeneral case where these quantities are kept as \nfree parameters. We determined \nthe saddle point equations that define the typical \nproperties of the corresponding SVM. This approach \nallows us to consider several learning scenarios, \nand more interestingly, to study the crossover \nbetween the different generalization regimes.\n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=Q0.eps,height= 7 cm}}\n\\caption{Order parameters of SVMs for purely linear \nteacher rules, $Q^*=0$. Symbols are experimental \nresults for input space dimension $N=50$, corresponding \nto the two kinds of quadratic mappings, $\\Phi_n$ \nwith $a=1/\\sqrt{N}$ (full symbols) and $\\Phi_{nn}$ with \nnormalizing factor $a=1$ (open symbols) respectively. \nError bars are smaller than the symbols. The \nlines are solutions of equations ({\\protect {\\ref{eq.sp1}-\\ref{eq.sp5}}}), \nfor $\\Delta=(N+1)/2$ and $\\sigma^2=N a^2/\\Delta$ with $N=50$, \nand $a$ corresponding to each mapping.}\n\\label{fig.Q*0}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=Q1.eps,height= 7 cm}}\n\\caption{Order parameters of SVMs for purely quadratic \nteacher rules, $Q^*=1$. Definitions are the same \nas in figure {\\protect {\\ref{fig.Q*0}}}.}\n\\label{fig.Q*1}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=Q096.eps,height= 7 cm}}\n\\caption{Order parameters of SVMs for isotropic \nteacher rules, $Q^*_{iso}=\\Delta/(1+\\Delta)$. Definitions are the same \nas in figure {\\protect {\\ref{fig.Q*0}}}.}\n\\label{fig.Q*51/53}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{figure=Q05.eps,height= 7 cm}}\n\\caption{Order parameters of SVMs for a general \nteacher rule, $Q^*=0.5$. Definitions are the same \nas in figure {\\protect {\\ref{fig.Q*0}}}.}\n\\label{fig.Q*05}\n\\end{figure}\n\nWe describe first the experimental data, obtained \nwith quadratic SVMs, using both mappings, ${\\bbox \\Phi}_{nn}$ and \n${\\bbox \\Phi}_n$, which have normalizing factors $a=1$ and \n$a=1/\\sqrt{N}$ respectively, where $N$ is the \ninput space dimension. The $M= \\alpha N$ random input \nexamples of each training set \nwere selected with probability (\\ref{eq.pdex}) and \nlabelled by teachers of normalized weights ${\\bbox w}^* \\equiv \n({\\bbox w}^*_l,{\\bbox w}^*_q)$ drawn at random. \n${\\bbox w}^*_l$ are the $N$ components in the linear \nsubspace and ${\\bbox w}^*_q$ are the $N^2$ components \nin the quadratic subspace. Notice that, \nbecause of the symmetry of the mappings, teachers having \nthe same value of the symmetrized weights in the \nquadratic subspace, $(w^*_{q,ij}+w^*_{q,ji})/2$, are \nall equivalent. The teachers are characterized by the proportion of \n(squared) weight components in the quadratic subspace, \n$Q^*={\\bbox w}^*_q \\cdot {\\bbox w}^*_q/{\\bbox w}^* \\cdot {\\bbox w}^*$. \nIn particular, $Q^*=0$ and $Q^*=1$ correspond to a purely \nlinear and a purely quadratic teacher respectively. \n\nThe experimental student weights ${\\bbox w} \\equiv \n({\\bbox w}_l,{\\bbox w}_q)$ were obtained by \nsolving numerically the dual problem~\\cite{CV,V}, using the \nQuadratic Optimizer for Pattern Recognition program~\\cite{AS}, \nthat we adapted to the case without threshold treated in this paper. \nWe determined $Q$, and the overlaps $R_l$ and $R_q$ \nin the linear and the quadratic subspaces, respectively. \nFor each value of $M$, averages were performed over \na large enough number of different teachers and training \nsets to get the precision shown in the figures. \n\nExperiments were carried out for \n$N=50$. The corresponding feature space dimension \nis $N(N+1)=2550$. The restricted feature space \nconsidered in our model is composed of the $N$ (linear) \ninput components, which define the {\\it u}-subspace \nof the feature space, and the $N \\Delta$ \nnon redundant quadratic components of the \n$\\sigma$-subspace. For the sake of comparison \nwith the theoretical results determined in the \nthermodynamic limit, we caracterize the actual \nSVM by its (finite size) inflation factor \n$\\Delta=(N+1)/2$, and the variance \n$\\sigma^2$ of the components in the $\\sigma$-subspace, \nrelated to the normalizing factor $a$ of the new features \nthrough $\\sigma^2 = N a^2 / \\Delta$. In our case, \nsince $N=50$, $\\Delta=25.5$ and $\\sigma^2 = 1.960784 a^2$, \nthat is $\\sigma^2 = 1.960784$ for the non-normalized mapping \nand $\\sigma^2 = 0.039216$, for the normalized one. \n\n\nThe values of $Q$, the fraction of squared student weights in the \n$\\sigma$-subspace, and the teacher-student overlaps $R_u$ and \n$R_\\sigma$, normalized within the corresponding sub-space, \nare represented on figures \\ref{fig.Q*0} \nto \\ref{fig.Q*05} as a function of $\\alpha \\equiv M/N$, \nusing full and open symbols for the mappings ${\\bbox \\Phi}_n$ \nand ${\\bbox \\Phi}_{nn}$ respectively. Notice that the \nabscissas correspond to the fraction of training patterns \nper {\\it input} space dimension. Error bars are smaller \nthan the symbols' size. The lines are {\\it not} fits, but the \ntheoretical curves corresponding to the same classes of \nteachers as the experimental results. The excellent \nagreement with the experimental data is striking. \nThus, the high order correlations of the \nfeatures, neglected in the theoretical models, \nare indeed negligible. \n\nFig. \\ref{fig.Q*0} corresponds to a purely linear \nteacher ($Q^*=0$), i.e. to a quadratic SVM learning a \nrule linearly separable in input space. As in this case \n$R_\\sigma=0$, only $R_u$ and $Q$ \nare represented. In the case of a purely quadratic rule, \n$Q^*=1$, represented on fig. \\ref{fig.Q*1}, $R_u=0$. Notice \nthat the corresponding overlaps, $R_u$ and $R_\\sigma$, \ndo not have a similar behaviour, as the latter increases \nmuch slower than the former, irrespective of the mapping. \nThis happens because, as the number of quadratic components \nscales like $N \\Delta$, a number of examples of the \norder of $N \\Delta$ are needed to learn them. Indeed, \n$R_u$ reaches a value \nclose to $1$ with $\\alpha \\sim O(1)$ while $R_\\sigma$ \nneeds $\\alpha \\sim O(\\Delta)$ to reach similar values.\n\nFig. \\ref{fig.Q*51/53} shows the results corresponding \nto the isotropic teacher, having $Q^*=Q^*_{iso} \\equiv \n\\Delta/(1+\\Delta)$. For $\\Delta=25.5$ we have $Q^*_{iso}=0.962$\nA particular case of such a teacher has all its weight \ncomponents of equal absolute value, i.e. $(w_i^*)^2 = \n1/{\\tilde N}$, and was studied in~\\cite{YO} and~\\cite{DOS}. \nFinally, the results corresponding to a general \nrule, with $Q^*=0.5$, are shown in fig. \\ref{fig.Q*05}. \nNotice that at fixed $\\alpha$, $R_u$ decreases and \n$R_\\sigma$ increases with $Q^*$ at a rate that \ndepends on the mapping. These quantities \ndetermine the student's generalization error through \nthe combination (\\ref{eq.R}). The fact that they increase \nas a function of $\\alpha$ with different speed is a signature of \nhierarchical learning. \n\n\\begin{figure}\n\\centerline{\\psfig{figure=eg.eps,height= 7 cm}}\n\\caption{Learning curves of SVMs for different teacher \nrules $Q^*$. Definitions are the same as in figure \n{\\protect {\\ref{fig.Q*0}}}. The inset is an \nenlargement of the smalll $\\alpha$ region.}\n\\label{fig.epsg}\n\\end{figure}\n\nThe generalization error $\\epsilon_g$ corresponding \nto the different rules is plotted against \n$\\alpha$ on fig. \\ref{fig.epsg}, for both mappings. \nAt any fixed $\\alpha$, the performance obtained \nwith the normalized mapping is better the smaller the value \nof $Q^*$. The non-normalized mapping shows the opposite \ntrend: its performance for a purely linear teacher is \nextremely bad, but it improves for increasing values of \n$Q^*$ and slightly overrides that of the normalized \nmapping in the case of a purely quadratic teacher. \nThese results reflect the competition on learning the \nanisotropically distributed features. In the case of \nthe normalized mapping, the $\\sigma$-components are \ncompressed ($\\sigma^2=0.039$) with respect to the {\\it u}-components, \nwhich have unit variance. This is advantageous whenever \nthe linear components carry the most significant information, which \nis the case for $Q^* \\ll 1$. When $Q^*=1$, the linear components \nonly introduce noise that hinders the learning process. As the \nnumber of linear components is much smaller than the \nnumber of quadratic ones, their pernicious effect should be \nmore conspicuous the smaller the value of $\\Delta$. Conversely, \nthe non-normalized mapping has $\\sigma^2=1.96$, meaning that the \ncompressed components are those of the {\\it u}-subspace. Therefore, \nthis mapping is better when most of the information is \ncontained in the $\\sigma$-subspace, which is the case for \nteachers with large $Q^*$ and, in particular, with $Q^*=1$.\n\n\\begin{figure}\n\\centerline{\\psfig{figure=rhoSV.eps,height= 7 cm}}\n\\caption{Fraction of learning patterns that belong to the subset of Support Vectors.}\n\\label{fig.rhoSV}\n\\end{figure} \n\nFinally, for the sake of completeness, the fraction of \nsupport vectors $\\rho_{SV} \\equiv M_{SV}/M$, where \n$M_{SV}$ is the number of training patterns with maximal \nstability, is represented on figure \\ref{fig.rhoSV}. This \nfraction is an upper bound to the generalization error. \nNotice that these curves present qualitatively the same \ntrends as $\\epsilon_g$. Interestingly, $\\rho_{SV}$ \nis smaller for the normalized mapping than for the \nnon-normalized one for most of the rules. Since the student's \nweights can be expressed as a linear combination of SVs~\\cite{Vapnik}, \nthis result is of practical interest. \n\n\\section{Discussion}\n\\label{sec.discussion}\n\nIn order to understand the results obtained in the \nprevious section, we first analyze the relative \nbehaviour of $R_u$ and $R_\\sigma$, which can be \ndeduced from equation (\\ref{eq.sp4}). \nIf $\\Delta^*_\\sigma \\ll \\Delta$, which is the case \nfor sufficiently small $Q^*$, we get that $R_\\sigma \\ll R_u$. \nThis means that the quadratic components are more difficult \nto learn than the linear ones. On the other \nhand, if the teacher lies mainly in the quadratic subspace, \n$\\Delta^*_\\sigma \\gg \\Delta$, and then $R_\\sigma > R_u$. \nThe crossover between these different behaviours occurs at \n$\\Delta^*_\\sigma = \\Delta$, for which equation \n(\\ref{eq.sp4}) gives $R_\\sigma = R_u$. For $N=50$, \nwhich is the case in our simulations, this \narises for $Q^*_n=0.998$ or $Q^*_{nn}=0.929$, depending on whether \nwe use the normalized or the non-normalized mapping. \nIn the particular case of the isotropic teacher and the \nnon-normalized mapping, $Q^* > Q^*_{nn}$, so that \n$R_\\sigma > R_u$, as shown on figure \\ref{fig.Q*51/53}.\nThese considerations alone are not sufficient to understand \nthe behaviour of the generalization error, which depends on the \nweighted sum of $R_\\sigma$ and $R_u$ (see equation \n(\\ref{eq.R})).\n\nThe behaviour at small $\\alpha$ is useful to understand the \nonset of hierarchical learning. A close inspection \nof equations (\\ref{eq.sp1}-\\ref{eq.sp4}) \nshows that in the limit $\\alpha \\rightarrow 0$, \n$x=\\sigma^2$ and $Q \\simeq \\Delta \\sigma^2 / (\\Delta \\sigma^2 +1)$ \nto leading order in $\\alpha$. This results may be understood \nwith the following simple argument: if \nthere is only one training pattern, clearly it is a \nSV and the student's weight vector is proportional to \nit. As a typical example has $N$ components of \nunit length in the {\\it u}-subspace and $N \\Delta$ components \nof length $\\sigma$ in the $\\sigma$-subspace, we have \n$Q=N\\Delta \\sigma^2 / (N \\Delta \\sigma^2 +N)$. With the \nnormalized mapping, $\\lim_{\\alpha \\rightarrow 0} Q = 1/2$. \nIn the case of the non normalized one $\\lim_{\\alpha \\rightarrow 0} Q \n= (2 \\Delta - 1)/2 \\Delta$, which depends on the inflation factor \nof the SVM. In this limit, we obtain:\n\n\\begin{eqnarray}\n\\kappa_{max} &\\simeq& \n\\frac{1+\\sigma^2 \\Delta}{\\sqrt{1+\\sigma^4 \\Delta}} \\frac{1}{\\sqrt{\\alpha}}, \\\\\nR_u &\\simeq& \\sqrt{\\frac{2}{\\pi}} \\frac{1}{\\sqrt{1+\\Delta^*_\\sigma}} \n\\sqrt{\\alpha}, \\\\\nR_\\sigma &\\simeq& \\sqrt{\\frac{2}{\\pi}} \\, \\sqrt{\\frac{\\Delta^*_\\sigma}{1+\\Delta^*_\\sigma}} \\, \n\\sqrt{\\frac{\\alpha}{\\Delta}}.\n\\end{eqnarray}\n\n\\noindent Therefore, $R \\sim \\sqrt{\\alpha}$, like for the simple \nperceptron MMH~\\cite{GG}, but with a prefactor that depends \non the mapping and the teacher. \n\nIn our model, we expect that hierarchical learning correspond to \na fast increase of $R$ at small $\\alpha$, mainly\ndominated by the contribution of $R_u$. As in the limit \n$\\alpha \\rightarrow 0$,\n\n\\begin{equation}\nR \\simeq \\frac{R_u + R_\\sigma \\sqrt{\\sigma^4 \\Delta \\Delta^*_\\sigma}}\n{\\sqrt{1+\\sigma^4 \\Delta} \\sqrt{1+\\Delta^*_\\sigma}},\n\\end{equation}\n\n\\noindent we expect hierarchical learning if \n$\\sigma^4 \\Delta \\ll 1$ and $\\Delta^*_\\sigma \\lesssim 1$. \nThe first condition establishes a constraint on the mapping, \nwhich is only satisfied by the normalized one. The second \ncondition, that ensures that $R_\\sigma < R_u$ holds, gives \nthe range of teachers for which this hierarchical \ngeneralization takes place. Under these conditions, \n$R$ grows fast and the contribution \nof $R_\\sigma$ is negligible because it is weighted \nby $\\sqrt{\\sigma^4 \\Delta \\Delta^*_\\sigma}$. The effect \nof hierarchical learning is more important the smaller \n$\\Delta^*_\\sigma$. The most dramatic effect arises for $Q^*=0$, \ni.e. for a quadratic SVM learning a linearly separable rule. \n\nOn the other hand, if \n$\\sigma^4 \\Delta \\gg 1$, which is the case for the non \nnormalized mapping, both $R_u$ and $R_\\sigma$ contribute to \n$R$ with comparable weights. Notice that, if the normalized \nmapping is used, the condition \n$\\Delta^*_\\sigma \\lesssim 1$ implies that $Q^* < Q^*_{iso} \\equiv\n\\Delta/(1+\\Delta)$, where $Q^*_{iso}$ corresponds to the isotropic \nteacher. A straightforward calculation shows that a fraction of $47.5 \\%$ \nof teachers satisfies this constraint for $N=50$. In fact, the \ndistribution of teachers as a function of $Q^*$ has its maximum \nat $Q^*_{iso}$. When $N \\rightarrow \\infty$, the distribution \nbecomes $\\delta(Q^* - Q^*_{iso})$, and $Q^*_{iso}$ tends to the \nmedian, meaning that in this limit, only about $50 \\%$ of the \nteachers give raise to hierarchical learning when using \nthe normalized mapping.\n\nIn the limit $\\alpha \\rightarrow \\infty$, all the \ngeneralization error curves converge to the same asymptotic \nvalue as the simple perceptron MMH learning in the feature \nspace, namely $\\epsilon_g=0.500489 (1+\\Delta)/ \\alpha$, \nindependently of $\\sigma$ and $Q^*$. Thus, $\\epsilon_g$ vanishes \nslower the larger the inflation factor $\\Delta$.\n\nFinally, it is worth to point out that for $\\sigma = 1$, \nwhich would correspond to a normalizing factor $a=\\sqrt{\\Delta/N}$, \nthe pattern distribution in feature space is isotropic. \nIrrespective of $Q^*$, the corresponding \ngeneralization error is exactly the same as that of a simple \nperceptron learning the MMH with isotropically \ndistributed examples in feature space.\n\n\n\\begin{figure}\n\\centerline{\\psfig{figure=limitTD.eps,height= 7 cm}}\n\\caption{Generalization error of a SVM corresponding to \ndifferent thermodynamic limits. See the text for the definition \nof $\\alpha$ in each regime.}\n\\label{fig.thermolim}\n\\end{figure}\n\nSince the inflation factor $\\Delta$ of the SVM feature space \nin our approach is a free parameter, it does not diverge\nin the thermodynamic limit $N \\rightarrow \\infty$ . As a \nconsequence, $\\epsilon_g$ does not present any stepwise \nbehaviour, but just a crossover between a fast decrease \nat small $\\alpha$ followed by a slower decrease regime at \nlarge $\\alpha$. The results of Dietrich et al.~\\cite{DOS} \nfor the {\\it normalized} mapping, that corresponds to $\\sigma^2 \n\\Delta=1$ in our model, can be deduced by taking appropriately \nthe limits before solving our saddle point equations. \nThe regime where the number of training patterns $M= \\alpha N$ \nscales with $N$, is straightforward. It is obtained by taking \nthe limit $\\sigma \\rightarrow 0$ and $\\Delta \\rightarrow \\infty$ keeping \n$\\sigma^2 \\Delta=1$ in our equations, with $\\alpha$ finite. The \nregime where the number of training patterns \n$M= \\alpha N$ scales with $N \\Delta$, the number of quadratic \nfeatures, obtained by keeping \n$\\tilde \\alpha \\equiv \\alpha/(1+ \\Delta)$ finite whilst taking, \nhere again, the limit $\\sigma \\rightarrow 0$, \n$\\Delta \\rightarrow \\infty$ with $\\sigma^2 \\Delta=1$. \nThe corresponding curves are represented on figure \n\\ref{fig.thermolim} for the case of an isotropic teacher. \nIn order to make the comparisons with our results at \nfinite $\\Delta$, the regime where $\\tilde \\alpha$ is \nfinite is represented as a function of $\\alpha = \n(1+\\Delta) \\tilde \\alpha$ using the value of $\\Delta$ \ncorresponding to our numerical simulations, namely, \n$\\Delta = 25.5$. In the same figure we represented the \ngeneralization error $\\epsilon_g=(1/\\pi) \\arccos(R)$ \nwhere $R$, given by eq. (\\ref{eq.R}), is obtained after \nsolving the saddle point equations with parameter \nvalues $\\sigma^2=0.039$ and $\\Delta=25.5$.\n\nThese results, obtained for quadratic SVMs, are easily \ngeneralizable to higher order polynomial SVMs. The \ncorresponding saddle point equations are cumbersome, \nand will not be given here. We expect a cascade of \nhierarchical generalization behaviour, \nin which successively more and more compressed features are \nlearned. This may be understood by considering the \nset of saddle point equations that generalize \nequation (\\ref{eq.sp4}). These equations relate the \nteacher-student overlaps in the successive subspaces. \nThe sequence of different feature subspaces generalized by the \nSVM depends on the relative complexity of the teacher and \nthe student. This is contained in the factors \n$\\Delta^*_{\\sigma_m}/\\Delta_m$ corresponding to the \n$m^{th}$ subspace, that appear in the set of equations \nthat generalize eq. (\\ref{eq.sp4}).\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe introduced a model that clarifies some aspects of the \ngeneralization properties of polynomial Support Vector \nMachines (SVMs) in high dimensional feature spaces. To \nthis end, we focused on quadratic SVMs. The quadratic \nfeatures, which are the pairwise products of input \ncomponents, may be scaled by a {\\it normalizing factor}. \nDepending on its value, the generalization \nerror presents very different behaviours in the \nthermodynamic limit~\\cite{DOS,BG}. \n\nIn fact, a finite size SVM may be caracterized by two \nparameters: $\\Delta$ and $\\sigma$. The \n{\\it inflation factor} $\\Delta$ is the \nratio between the quadratic and the linear \nfeatures dimensions. Thus, it is proportional \nto the input space dimension $N$. The {\\it variance} \n$\\sigma$ of the quadratic features is related to \nthe corresponding normalizing factor. Usually, either\n$\\sigma \\sim 1/\\sqrt{N}$ (normalized mapping) or \n$\\sigma \\sim 1$ (non normalized mapping). \nIn previous studies, not only the input space dimension \ndiverges in the thermodynamic limit $N \\rightarrow \\infty$, \nbut also $\\Delta$ and $\\sigma$ are correspondingly scaled.\n\nIn our model, neither the proportion of quadratic features \n$\\Delta$ nor their variance $\\sigma$ are necessarily related \nto the input space dimension $N$. They are considered as \nparameters caracterizing the SVMs. Since we keep them constant \nwhen taking the thermodynamic limit, we can study the learning \nproperties of actual SVMs with finite inflation ratios and \nnormalizing factors, as a function of $\\alpha \\equiv M/N$, \nwhere $M$ is the number of training examples. Our theoretical \nresults were obtained neglecting the correlations among the \nquadratic features. The agreement between our computer \nexperiments with actual SVMs and the theoretical \npredictions is excellent. The effect of the correlations \ndoes not seem to be important, as there is almost no \ndifference between the theoretical curves and the numerical \nresults.\n\nWe find that the generalization error $\\epsilon_g$ \ndepends on the type of rule to be inferred through \n$Q^*$, the (normalized) sum of the teacher's squared weight \ncomponents in the quadratic subspace. If \n$Q^*$ is small enough, the quadratic components \nneed more patterns to be learned than the linear \nones. However, only if the quadratic features \nare normalized, $\\epsilon_g$ is dominated by \nthe high rate learning of the linear components\nat small $\\alpha$. Then, on increasing $\\alpha$, \nthere is a crossover to a regime \nwhere the decrease of $\\epsilon_g$ becomes much slower. \nThe crossover between these two behaviours is smoother \nfor larger values of $Q^*$, and this effect of \nhierarchical learning disappears for large enough $Q^*$. \nOn the other hand, if the features are not normalized, \nthe contributions of both the linear and the quadratic \ncomponents to $\\epsilon_g$ are of the same order, and \nthere is no hierarchical learning at all. \n\nIn the case of the normalized mapping, if the limits \n$\\Delta \\sim N \\rightarrow \\infty$ \nand $\\sigma^2 \\sim 1/N \\rightarrow 0$\nare taken together with the thermodynamic limit, the \nhierarchical learning effect gives raise to the two different \nregimes, corresponding to $M \\sim N$ or $M \\sim N^2$, described previously~\\cite{YO,DOS}.\n \nIt is worth to point out that if the rule to be learned \nallows for hierarchical learning, the generalization \nerror of the normalized mapping is much smaller than \nthat of the non normalized one. In fact, \nthe teachers corresponding to such rules are those \nwith $Q^* \\lesssim Q^*_{iso}$, where $Q^*_{iso}$ \ncorresponds to the isotropic teacher, the one \nhaving all its weights components equal. For the others, both \nthe normalized mapping and the non normalized one \npresent similar performances. If the weights of the \nteacher are selected at random on a hypersphere \nin feature space, the most probable \nteachers have precisely $Q^*=Q^*_{iso}$, and the fraction \nof teachers with $Q^* \\leq Q^*_{iso}$ represent of the order of \n$50\\%$ of the inferable rules. Thus, from a practical \npoint of view, without having any prior knowledge \nabout the rule underlying a set of examples, the \nnormalized mapping should be preferred.\n \n\\section*{Acknowledgements}\n\nIt is a pleasure to thank Arnaud Buhot for a careful \nreading of the manuscript, and Alex Smola for \nproviding us the Quadratic \nOptimizer for Pattern Recognition program~\\cite{AS}. \nThe experimental results were obtained with the Cray-T3E \ncomputer of the CEA (project 532/1999).\n\nSR-G acknowledges economic support from the EU-research contract \nARG/B7-3011/94/97. \n\nMBG is member of the CNRS.\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{Vapnik} V. Vapnik (1995) The nature of statistical learning\ntheory. Springer Verlag, New York.\n\n\\bibitem{OKKN} M. Opper, W. Kinzel, J. Kleinz and R. Nehl (1990) \nJ. Phys. A: Math. Gen. 23, L-581.\n\n\\bibitem{OK} M. Opper and W. Kinzel (1995) in {\\it Models of Neural \nNetworks III}, E. Domany, J.L. van Hemmen, K. Schulten (Eds.), pp. 151-209.\n\n\\bibitem{Cover} T. Cover (1965) IEEE Trans. Electron. Comput. 14, 326-334.\n\n\\bibitem{CV} C. Cortes and V. Vapnik (1995) Machine Learning 20, 273-297.\n\n\\bibitem{DOS} R. Dietrich, M. Opper, and H. Sompolinsky (1999) Phys. Rev. Lett. 82, 2975-2978.\n\n\\bibitem{BG} A. Buhot and M. B. Gordon (1999) {\\it ESANN'99-European \nSymposium on Artificial Neural Networks}. Proceedings, \nMichel Verleysen ed., pp. 201-206; A. Buhot and M. B. Gordon (1998) cond-mat/9802179. \n\n\\bibitem{YO} H. Yoon and J.-H. Oh (1998) J. Phys. A: Math. Gen. 31, 7771-7784.\n\n\\bibitem{Monasson} R. Monasson (1993) J. Phys. A 25, 3701.\n\n\\bibitem{MBS} C. Marangi, M. Biehl and S. Solla (1995) Europhys. Lett. 30, 117-122.\n\n\\bibitem{RG} S. Risau-Gusman and M. B. Gordon (2000) in {\\it Advances in Neural Information Processing Systems} 12, edited by S. A. Solla, T. K. Leen, K-R. Muller (MIT Press), to be published.\n\n\\bibitem{GG} M. B. Gordon and D. R. Grempel (1995) Europhys. Lett. 29, 257-262.\n\n\\bibitem{O} M. Opper (1988), Phys. Rev. A 38, 3824-3826.\n\n\\bibitem{V} R. J. Vanderbei (1998) Technical Report SOR-94-15, Princeton University.\n\n\\bibitem{AS} Program available upon request to http://svm.first.gmd.de.\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n~\\footnote{More precisely, to \nthe generalization error averaged over all the training \nsets of the same size, which is precisely \nthe quantity determined within the Statistical Mechanics framework}\n"
}
] |
[
{
"name": "cond-mat0002071.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{Vapnik} V. Vapnik (1995) The nature of statistical learning\ntheory. Springer Verlag, New York.\n\n\\bibitem{OKKN} M. Opper, W. Kinzel, J. Kleinz and R. Nehl (1990) \nJ. Phys. A: Math. Gen. 23, L-581.\n\n\\bibitem{OK} M. Opper and W. Kinzel (1995) in {\\it Models of Neural \nNetworks III}, E. Domany, J.L. van Hemmen, K. Schulten (Eds.), pp. 151-209.\n\n\\bibitem{Cover} T. Cover (1965) IEEE Trans. Electron. Comput. 14, 326-334.\n\n\\bibitem{CV} C. Cortes and V. Vapnik (1995) Machine Learning 20, 273-297.\n\n\\bibitem{DOS} R. Dietrich, M. Opper, and H. Sompolinsky (1999) Phys. Rev. Lett. 82, 2975-2978.\n\n\\bibitem{BG} A. Buhot and M. B. Gordon (1999) {\\it ESANN'99-European \nSymposium on Artificial Neural Networks}. Proceedings, \nMichel Verleysen ed., pp. 201-206; A. Buhot and M. B. Gordon (1998) cond-mat/9802179. \n\n\\bibitem{YO} H. Yoon and J.-H. Oh (1998) J. Phys. A: Math. Gen. 31, 7771-7784.\n\n\\bibitem{Monasson} R. Monasson (1993) J. Phys. A 25, 3701.\n\n\\bibitem{MBS} C. Marangi, M. Biehl and S. Solla (1995) Europhys. Lett. 30, 117-122.\n\n\\bibitem{RG} S. Risau-Gusman and M. B. Gordon (2000) in {\\it Advances in Neural Information Processing Systems} 12, edited by S. A. Solla, T. K. Leen, K-R. Muller (MIT Press), to be published.\n\n\\bibitem{GG} M. B. Gordon and D. R. Grempel (1995) Europhys. Lett. 29, 257-262.\n\n\\bibitem{O} M. Opper (1988), Phys. Rev. A 38, 3824-3826.\n\n\\bibitem{V} R. J. Vanderbei (1998) Technical Report SOR-94-15, Princeton University.\n\n\\bibitem{AS} Program available upon request to http://svm.first.gmd.de.\n\n\\end{thebibliography}"
}
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cond-mat0002072
|
EFFECTIVE POTENTIAL APPROACH TO QUANTUM DISSIPATION IN CONDENSED MATTER SYSTEMS
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[
{
"author": "Alessandro Cuccoli$^{a,c}$"
},
{
"author": "Andrea Fubini$^{a,c}$"
},
{
"author": "Valerio Tognetti$^{a,c}$"
},
{
"author": "Ruggero Vaia$^{b,c}$"
}
] |
The effects of dissipation on the thermodynamic properties of nonlinear quantum systems are approached by the path-integral method in order to construct approximate classical-like formulas for evaluating thermal averages of thermodynamic quantities. Explicit calculations are presented for one-particle and many-body systems. The effects of the dissipation mechanism on the phase diagram of two-dimensional Josephson arrays is discussed.
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[
{
"name": "gallipo.tex",
"string": "%====================================================================%\n% sprocl.tex 27-Feb-1995 %\n% This latex file rewritten from various sources for use in the %\n% preparation of the standard proceedings Volume, latest version %\n% by Susan Hezlet with acknowledgments to Lukas Nellen. %\n% Some changes are due to David Cassel. %\n%====================================================================%\n\n\\documentstyle[sprocl,amsmath,psfig]{article}\n\n%\\bibliographystyle{unsrt} %for BibTeX - sorted numerical labels by\n %order of first citation.\n\\arraycolsep1.5pt\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%BEGINNING OF TEXT\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{document}\n\n\\title{EFFECTIVE POTENTIAL APPROACH\n TO QUANTUM DISSIPATION IN CONDENSED MATTER SYSTEMS}\n\n\\author{Alessandro Cuccoli$^{a,c}$, Andrea Fubini$^{a,c}$,\n Valerio Tognetti$^{a,c}$,\\\\ Ruggero Vaia$^{b,c}$}\n\n\\address{$^a$~Dipartimento di Fisica dell'Universit\\`a di Firenze,\\\\\n Largo E. Fermi~2, I-50125 Firenze, Italy. \\\\\n $^b$~Istituto di Elettronica Quantistica\n del Consiglio Nazionale delle Ricerche, \\\\\n via Panciatichi~56/30, I-50127 Firenze, Italy.\\\\\n $^c$~Istituto Nazionale di Fisica della Materia (INFM),\n Unit\\`a di Firenze.}\n\n\n\\maketitle\n\n\\abstracts{\nThe effects of dissipation on the thermodynamic properties of nonlinear\nquantum systems are approached by the path-integral method in order\nto construct approximate classical-like formulas for evaluating thermal\naverages of thermodynamic quantities. Explicit calculations are\npresented for one-particle and many-body systems. The effects of the\ndissipation mechanism on the phase diagram of two-dimensional\nJosephson arrays is discussed.}\n\n\\section{Introduction}\n\nThe usefulness of the improved~\\cite{GTall} effective potential\napproach,~\\cite{FeynmanH65} has been proven by several applications\nto condensed matter systems. However, open systems were not\nimmediately treated and previous studies were confined to obtain a\nclassical-like expression for the free energy.~\\cite{Weiss99,BaoZW95}\n\nIn fact, the effective-potential method, also called {\\em pure-quantum}\n{\\em self-con\\-sistent} {\\em harmonic approximation} (PQSCHA) after its\ngeneralization to phase-space Hamiltonians,~\\cite{CTVV92ham,CGTVV95}\nis able~\\cite{CRTV97} to give the density matrix of a nonlinear\nsystem interacting with a dissipation bath through the\nCaldeira-Leggett (CL) model.~\\cite{CaldeiraL83}\nFor a better understanding of the method let us first consider\none single degree of freedom. In the general case\nthe CL model starts from the Hamiltonian\n\\begin{equation}\n \\hat{H}={\\hat {p}^2\\over2m}+V(\\hat {q}) +{1\\over2}\\sum_i\\bigg\\{{\\hat\n {p}_i^2\\over m_i} + m_i\\omega_i^2\\Big[\\hat {q}_i-F_i(\\hat {q})\\Big]^2\n \\bigg\\}~, \\label{e.HCL}\n\\end{equation}\nwhere $(\\hat{p},\\hat{q})$ and $(\\hat{p}_i,\\hat{q}_i)$ are the momenta\nand coordinates of the system and of an environment (or bath) of\nharmonic oscillators.\nWhen $F_i(\\hat{q})=\\hat{q}$ the dissipation is said to be {\\em linear}\nand we will restrict ourselves to this case in the following. The bath\ncoordinates can be integrated out exactly from the corresponding path\nintegral and the CL Euclidean action is obtained in the form:\n\\begin{equation}\n S[q(u)] =\\!\\int_0^{\\beta} \\!\\!\\! du \\left[ {m\\over2}\\dot q^2(u) {+}\n V\\big(q(u)\\big) \\right] +\\! \\int_0^{\\beta} \\! {du\\over2} \\!\n \\int_0^{\\beta}\\!\\!\\!\\! du'\\,k(u{-}u')\\,q(u)\\,q(u')\\,.\n \\label{e.S}\n\\end{equation}\nTo make contact with the classical concept of dissipation, one may\ntake the classical counterpart of Eq.~(\\ref{e.HCL}) and get the\nLangevin equation of motion:\n\\[\n m\\ddot q + m\\int_0^\\infty\\!\\! dt'\\,\\gamma(t')\\, \\dot q(t{-}t') +\n V'(q) = 0~;\n\\]\nthen the Laplace transform $\\gamma(z)$ of the {\\em damping function}\n$\\gamma(t)$, as expressed in terms of the spectral density of the\nenvironmental coupling,~\\cite{Weiss99} can be related to the Matsubara\ncomponents of the {\\em damping kernel} $k(u)$ ($\\nu_n=2\\pi{n}/\\beta$)\\,:\n\\begin{equation}\n k(u)\\equiv m\\beta^{-1}~{\\textstyle{\\sum}_n}~e^{i\\nu_nu}\\,k_n~,\n ~~~~~~~~~~~ k_n=|\\nu_n|\\,\\gamma\\big(z{=}|\\nu_n|\\big)~,\n \\label{e.kgamma}\n\\end{equation}\nWe will consider two cases:\n\\begin{eqnarray*}\n &{\\rm Ohmic:~~~~~~} &\\gamma(t)=\\gamma\\,\\delta(t-0)~, ~~~~~~~\\,\n \\gamma(z)=\\gamma~,\n\\\\\n &{\\rm Drude:~~~~~~}\n &\\gamma(t)=\\gamma\\,\\omega_{\\rm{D}}\\,e^{-\\omega_{\\rm{D}}t}~, ~~~~~~\n \\gamma(z)={\\gamma\\,\\omega_{\\rm{D}}/(\\omega_{\\rm{D}}+z)}~,\n\\end{eqnarray*}\nwhere the dissipation strength $\\gamma$ and the bath bandwidth\n$\\omega_{\\rm{D}}$ characterize the environmental coupling;\n$\\omega_{\\rm{D}}\\to\\infty$ gives the Ohmic (or Markovian) case.\nIn order to interpolate between different regimes and to take\ninto account memory effects we can also assume:\n\\begin{equation}\n \\gamma(z)\\propto z^s~,\n\\label{gamma}\n\\end{equation}\nwith $-1<s<1$ and $s>0$ ($s<0$) is said the {\\em super-} ({\\em sub-})\nOhmic case.\n\n\\section{Effective potential in presence of dissipation \\label{PQSCHA}}\n\\medskip\n\nThe PQSCHA approximation consists in taking as trial action $S_0$, the\nmost general quadratic functional with the same linear dissipation of $S$,\nnamely,\n\\begin{eqnarray}\n S_0[q(u)] &=& \\int_0^{\\beta} du \\left[ {m\\over2}\\dot q^2(u) + w\n + m\\omega^2\\,\\big(q(u){-}\\bar{q}\\big)^2 \\right]\n\\nonumber\\\\\n & & \\hspace{20mm} +\\! \\int_0^{\\beta} \\! {du\\over2} \\!\n \\int_0^{\\beta}\\!\\!\\!\\! du'\\,k(u{-}u')\\,q(u)\\,q(u')\\,. \\label{e.S0}\n\\end{eqnarray}\nThe quantity $\\bar{q}=\\beta^{-1}\\int{q(u)}\\,du$ is the average point\nof paths and\n\\begin{equation}\n w=w(\\bar{q})~,~~~~~~~\n \\omega^2=\\omega^2(\\bar{q})~,\n\\end{equation}\nare parameters to be determined by minimizing the right-hand side of\nthe {\\em Feynman inequality},\n\\begin{equation}\nF\\leq{F_0}+\\beta^{-1}\\langle S-S_0\\rangle_{S_0}~.\n\\end{equation}\nFor any observable $\\hat{\\cal O}(\\hat{p},\\hat{q})$ the $S_0$-average\ncan be expressed~\\cite{CGTVV95,CRTV97} in terms of its Weyl\nsymbol~\\cite{Berezin80} ${\\cal O}(p,q)$. As final result we obtain the\nclassical-like form\n\\begin{equation}\n \\langle\\hat{{\\cal O}}\\rangle =\\frac1{{\\cal Z}_0}\n \\sqrt{m\\over{2\\pi\\hbar^2\\beta}} \\int\\,d\\bar{q}\n ~\\big\\langle\\!\\big\\langle{\\cal O}(p,\\bar{q}+\\xi)\\big\\rangle\\!\\big\\rangle\n ~e^{-\\beta\\,V_{\\rm eff}(\\bar{q})}~,\n\\label{e.aveOpq}\n\\end{equation}\nwhere $\\langle\\!\\langle{\\cdots}\\rangle\\!\\rangle$ is a Gaussian average\noperating over $p$ and $\\xi$ with moments\n\\begin{eqnarray}\n \\langle\\!\\langle{\\xi^2}\\rangle\\!\\rangle \\equiv \\alpha(\\bar{q}) &=&\n {2\\over\\beta m}~ \\sum\\limits_{n=1}^{\\infty}\n {1\\over \\nu_n^2{+}\\omega^2(\\bar{q}){+}k_n}\n ~~\\xrightarrow[k\\to{0}]~~\n {1\\over2m\\omega}\\Big(\\coth f{-}{1\\over f}\\Big)~,\n\\label{e.alpha}\n\\\\\n \\langle\\!\\langle{p^2}\\rangle\\!\\rangle \\equiv \\lambda(\\bar{q}) &=&\n {m\\over\\beta}\\,\\sum\\limits_{n=-\\infty}^{\\infty}\n {\\omega^2(\\bar{q})+k_n \\over \\nu_n^2{+}\\omega^2(\\bar{q}){+}k_n}\n ~~\\xrightarrow[k\\to{0}]~~\n {m\\omega\\over2}\\coth f ~,\n\\label{e.lambda}\n\\end{eqnarray}\nwhere $f\\equiv\\beta\\omega/2$.\nThe effective potential is defined as $ V_{\\rm eff}(\\bar{q})\\equiv\nw(\\bar{q})+\\sigma(\\bar{q})$~, with\n\\begin{equation}\n \\sigma(\\bar{q})={1\\over\\beta}\\sum_{n=1}^{\\infty}\n \\ln\\,{\\nu_n^2{+}\\omega^2(\\bar{q}){+}k_n \\over \\nu_n^2}\n ~~\\xrightarrow[k\\to{0}]~~\n {1\\over\\beta}\\,\\ln{\\sinh f\\over f} ~.\n\\label{e.sigma}\n\\end{equation}\nThe r.h.s. of Eqs.~(\\ref{e.alpha}-\\ref{e.sigma}) show that the well known\nnon-dissipative limits are recovered for ${k\\to{0}}$. The variational\nparameters can be self-consistently calculated and the explicit\nexpressions are the following\n\\begin{equation}\n w(\\bar{q})=\\big\\langle\\!\\big\\langle V(\\bar{q}+\\xi)\\big\\rangle\\!\\big\\rangle\n -{\\textstyle \\frac 12} m\\omega^2\\alpha~,\n\\hspace{10mm}\n m\\,\\omega^2(\\bar{q}) = \\big\\langle\\!\\big\\langle V''(\\bar{q}+\\xi)\\big\\rangle\\!\\big\\rangle~.\n\\end{equation}\n\n\n\\section{Applications}\n\\medskip\n\nIn order to understand how this approximation scheme works, let us\ntake first a very simple system: one particle in a double-well\npotential with Ohmic dissipation. A typical result for this system\nis shown in Fig.~\\ref{f.1}: the dissipation quenches the quantum\nfluctuations of the coordinate. However, it must be noted from\nEq.(\\ref{e.lambda}) that those of the momentum are infinite.\n\n\\begin{figure}[b!]\n\\centerline{\\psfig{bbllx=16mm,bblly=80mm,bburx=192mm,bbury=200mm,%\nfigure=f1.eps,width=70mm,angle=0}}\n\\caption{Configuration density $P(x)=\\langle\\delta(\\hat{x}-x)\\rangle$ of\nthe double-well quartic potential for the fixed coupling $g=5$, the\nreduced temperature $t=1$, and different values of the Ohmic damping\nparameter $\\Gamma=\\gamma/\\omega_0$, being $\\omega_0$ the\ncharacteristic frequency of the system. The filled circles are the\nexact result for $\\Gamma=0$; the dotted curve at $\\Gamma=\\infty$\ncorresponds to the classical limit.\n\\label{f.1}}\n\\end{figure}\n\nLet us now turn to the many-body case. The first application is a\nquantum $\\varphi^4$-chain of particles with Drude-like\ndissipation:~\\cite{CFTV99} the undamped system is described by the\nfollowing action\n\\begin{equation}\n S_{\\varphi^4}={3\\over{2Q}}\\int_0^\\beta du\\,a\\,\\sum_i\\left[{\\dot\n q_i^2\\over2} +{(q_i-q_{i-1})^2\\over2a^2} + {\\Omega^2\\over8}\n \\big(1-q_i^2\\big)^2 \\right]~,\n\\label{e.Sphi4}\n\\end{equation}\nwhere $a$ is the chain spacing, $\\Omega$ is the gap of the bare\ndispersion relation, and $Q$ is the quantum coupling. The classical\ncontinuum model supports kink excitations of characteristic width\n$\\Omega^{-1}$ and static energy\n$\\varepsilon_{\\rm{K}}=\\Omega/Q$. This energy is used as the energy\nscale in defining the reduced temperature\n$t\\equiv(\\beta\\varepsilon_{\\rm{K}})^{-1}$. We also use the kink length\nin lattice units $R\\equiv(a\\Omega)^{-1}$ ($R\\to\\infty$ in the\ncontinuum limit). We assume uncorrelated identical CL baths for each\ndegree of freedom, and use the low-coupling\napproximation~\\cite{CGTVV95} (LCA) in order to deal with the effective\npotential. The partition function turns out to be written as\n\\begin{eqnarray}\n {\\cal{Z}}&=& ({3t\\over4\\pi RQ^2})^{N\\over2}\\!\\!\\!\\int\\! d^N \\! q\\;\n e^{-\\beta V_{\\rm eff}} ~,\n\\label{e.Z}\\\\\n \\beta V_{\\rm eff}&=&\\frac{3}{2Rt} {\\sum}_i\\left[{R^2\\over2}\n (q_i-q_{i-1})^2 + v_{\\rm eff}(q_i) \\right]~,\n\\label{e.Vfphi4}\\\\\n v_{\\rm eff}(q)&=&{1\\over8}\\big(1-3D-q^2\\big)^2 + {3\\over4}D^2 +{Rt\\over\n N}{\\sum}_k\\sigma_k~.\n\\label{e.vfphi4}\n\\end{eqnarray}\n\nThe renormalization parameter $D=D(t;Q,R;\\gamma,\\omega_{\\rm{D}})$\ngeneralizes Eq.~(\\ref{e.alpha}) and is the solution of the\nself-consistent equations:\n\\begin{eqnarray}\n D&=&{4Rt\\over3N}{\\sum}_k{\\sum}_n\n {\\Omega^2\\over\\nu_n^2+\\omega_k^2+k_n},\n\\\\\n \\omega_k^2(t)&=&\\Omega^2\\big[(1-3D)+4R^2\\sin^2(ka/2)\\big]~,\n\\end{eqnarray}\nwhere $k_n=\\gamma\\omega_{\\rm{D}}\\,\\nu_n/(\\omega_{\\rm{D}}+\\nu_n)$ and\nhere $\\nu_n/\\Omega=2\\pi{n}t/Q$.\n\nAgain, we observe that the fluctuations of\ncoordinate-dependent observables are quenched by increasing the\ndissipation strength $\\gamma$, while those of momentum-dependent ones\nare enhanced due to the momentum exchanges with the environment. The\nresult of these opposite behaviors is a non-trivial dependence on\ndissipation of ``mixed'' quantities like the specific heat as is shown\nin Fig.~\\ref{f.2}\\,.\n\\begin{figure}[b!]\n\\begin{minipage}[t]{119mm}\n\\centerline{\\psfig{bbllx=20mm,bblly=20mm,bburx=188mm,bbury=238mm,%\nfigure=f2a.eps,width=59mm,angle=270}\n\\hfill\n\\psfig{bbllx=20mm,bblly=20mm,bburx=188mm,bbury=238mm,%\nfigure=f2b.eps,width=59mm,angle=270}}\n\\caption{Total specific heat $c(t)$ and the kinetic and interaction\nparts of the specific heat, namely\n$c_{{}_{\\rm{K}}}(t)=\\partial_t\\langle\\hat{K}\\rangle$ and\n$c_{{}_{\\rm{V}}}(t)=\\partial_t\\langle\\hat{V}\\rangle$ vs. reduced\ntemperature $t$, for different values of the damping strength\n$\\Gamma=\\gamma/\\Omega$. Note that for $\\Gamma\\to\\infty$, $c(t)$ tends\nto $c_{\\rm{cl}}-1/2$: this behavior can be explained by observing\nthat, in the strong damping limit,\n$c_{{}_{\\rm{K}}}(t)\\to{0}$.\\label{f.2}}\n\\end{minipage}\n\\end{figure}\n\\medskip\n\nThe last system we consider is the dissipative quantum XY\nmodel. Much interest in such system is due to its close relation with\n2D granular superconductors and Josephson junction arrays (JJA). In the\nlast 15 years much attention has been devoted to the theoretical\nand experimental study~\\cite{jja} of the phase ordering in JJA\nand of how it is influenced by the environmental coupling.\nThe undamped system is described by the action\n\\begin{equation}\n S_{\\rm JJA}=\\int_0^\\beta du \\bigg\\{\\frac12 \\sum_{ij}{\\dot \\phi_i(u)\n \\frac{C_{ij}}{4 e^2} \\dot \\phi_j(u)} - J \\sum_{<ij>}\n \\cos[\\phi_i(u)-\\phi_j(u)] \\bigg\\} ~,\n\\label{e.SJJA}\n\\end{equation}\nwhere $\\phi_i$ is the superconducting phase of the $i$-th island,\n${<}ij{>}$ restricts the sum over nearest-neighbor bonds, $J$ is the\nJosephson coupling,\n$C_{ij}=(C_0+4C_1)\\delta_{ij}-C_1\\delta_{ij}^{({\\rm{nn}})}$,\n$C_0$ and $C_1$ are the self and mutual capacitance between\nsuperconducting islands and $\\delta_{ij}^{({\\rm{nn}})}=1$ for\nnearest neighbors, zero otherwise.\n\nFrom Eq.~(\\ref{e.SJJA}) two contributions to the energy in this system\ncan be observed: the kinetic one, due to the charging energy between\nislands ($E_{\\rm{c}} =q^2/2C$, $q=2e$), and the potential energy, due\nto the Josephson coupling in the superconducting junctions. When\n$E_{\\rm{c}}{\\ll}J$, the charges on each island fluctuate independently\nfrom the phases $\\phi_i$\\,: the latter have then a classical XY behavior\nand the associated Berezinskii-Kosterlitz-Thouless (BKT) phase transition\ntakes place at temperature $T^{\\rm (cl)}_{{}_{\\rm{BKT}}}\\sim{0.892}\\,J$.\nIn the opposite limit, the energy cost to transfer charges between\nneighboring islands is too high, so the charges tend to be localized\nand the phase ordering tends to be suppressed.\n\nIn this scenario dissipation is supposed to have an important\nrole. The environmental interaction tends to suppress the quantum\nfluctuations~\\cite{shon/tink} of $\\phi_i$ and to restore an almost\nclassical BKT phase transition. Nevertheless, it is not clear which is\nthe physical mechanism of the dissipation. For the single Josephson\njunction with Ohmic dissipation the classical ``resistively and\ncapacitively shunted junction'' (RCSJ) model is\nrecovered~\\cite{shon/tink}, but in the case of many degrees of freedom\nthe environmental interaction is much more complicated, e.g.\nnon-exponential memory effects and then non-Ohmic damping can\nappear.\nThe dissipation model we assume consists in independent\nenvironmental baths, one for each junction (or bond).\nThe dissipative part of the action is\n\\begin{equation}\n S_{\\rm D}={\\textstyle \\frac12}\\, {\\sum}_{n}\\, {^t\\boldsymbol{\\phi}_n} \\boldsymbol{K}_n\n \\boldsymbol{\\phi}_n \\,,\n\\label{e.SD}\n\\end{equation}\nwhere the Fourier transform of the CL kernel matrix is given by\n\\begin{equation}\nK_{n,ij} = \\frac{\\gamma}{2\\pi}\\,\n\\bigg(\\frac{|\\nu_n|}{\\omega_0}\\bigg)^{1+s}\n(4~\\delta_{ij} - \\delta_{ij}^{\\rm (nn)}) ~,\n\\label{e.Kn}\n\\end{equation}\nand $\\omega_0$ is a characteristic frequency, that we\nchoose as the Debye frequency\n\\begin{equation}\n \\omega_0 \\equiv \\omega_{\\pi,\\pi}\n = 4 \\bigg(\\frac{q^2}{2 C_0} \\frac{J}{1+8\\eta}\\bigg)^{1/2}~,\n ~~~~~\\eta\\equiv\\frac{C_1}{C_0}.\n\\end{equation}\nNow it is possible to define the quantum coupling parameter, that\nmeasures the ``quanticity'' of the system as the ratio\nbetween the characteristic quantum and classical energy scales\n$g=\\hbar\\omega_0/J$. The effective potential is calculated\nusing the extension to the many-body case~\\cite{CFTV99} of the\nscheme of Sec.~\\ref{PQSCHA} and, apart from uniform terms, is given by\n\\begin{equation}\n V_{\\rm eff}=-J_{\\rm{eff}}(T) \\sum_{<ij>} \\cos(\\phi_i-\\phi_j) ~,\n\\label{e.Veff}\n\\end{equation}\nwhere $J_{\\rm{eff}}(T)=J\\,e^{-D_1(T)/2}$ and $D_1(T)$ is the renormalization\nparameter that measures the pure-quantum contribution to the fluctuations\nof the nn relative superconducting phases.\nThe phase diagram of the system is calculated starting from the classical\neffective potential (\\ref{e.Veff}), i.e. by solving $T_{{}_{\\rm{BKT}}}\n=J_{\\rm{eff}}(T_{{}_{\\rm{BKT}}})\\,T^{\\rm (cl)}_{{}_{\\rm{BKT}}}$\\,,\nand is shown in Fig.~\\ref{f.3}, for different values of the\nparameters that characterize the dissipation.\n\n\\newpage\n\n\\begin{figure}[b!]\n\\begin{minipage}[t]{119mm}\n\\centerline{\\psfig{bbllx=45mm,bblly=26mm,bburx=188mm,bbury=238mm,%\nfigure=f3a.eps,width=59mm,angle=270}\n\\hfill\n\\psfig{bbllx=45mm,bblly=26mm,bburx=188mm,bbury=238mm,%\nfigure=f3b.eps,width=59mm,angle=270}}\n\\caption{Phase diagram in the $(g,t)$ plane, $t\\equiv{T/J}$, for\ndifferent values of the damping parameters, $\\Gamma=\\gamma/\\omega_0$\nand $s$. On the left, the case of Ohmic dissipation: the critical temperature\n$T_{{}_{\\rm{BKT}}}$ tends to $T^{\\rm (cl)}_{{}_{\\rm{BKT}}}$ for increasing\ndamping strength. On the right, for different values of $s$: the\ncases $s=1$ and $s=-1$ are nondissipative, since they correspond to\na variation of the capacitance- and of the frequency spectrum, respectively.\n\\label{f.3}}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{thebibliography}{10}\n\n\\bibitem{GTall}\nR. Giachetti and V. Tognetti, Phys. Rev. Lett. {\\bf 55}, 912 (1985);\nPhys. Rev. B {\\bf 33}, 7647 (1986). R.~P.~Feynman and H.~Kleinert,\nPhys. Rev. A {\\bf 34}, 5080 (1986).\n\n\\bibitem{FeynmanH65}\nR.P.~Feynman and A.R.~Hibbs, {\\em Quantum Mechanics and Path\nIntegrals} (Mc Graw Hill, New York, 1965).\n\n\\bibitem{Weiss99}\nU.~Weiss, {\\em Quantum Dissipative Systems} (World Scientific,\nSingapore, 2nd edition, 1999).\n\n\\bibitem{BaoZW95}\nJ.D.~Bao, Y.Z.~Zhuo, and X.~Z. Wu, Phys. Rev. E {\\bf 52}, 5656\n(1995).\n\n\\bibitem{CTVV92ham}\nA.~Cuccoli, V.~Tognetti, P.~Verrucchi, and R.~Vaia, Phys. Rev. A {\\bf\n45}, 8418 (1992).\n\n\\bibitem{CGTVV95}\nA.~Cuccoli, R.~Giachetti, V.~Tognetti, R.~Vaia and P.~Verrucchi,\nJ. Phys.: Condens. Matter {\\bf 7}, 7891 (1995).\n\n\\bibitem{CRTV97}\nA.~Cuccoli, A.~Rossi, V.~Tognetti, and R.~Vaia, Phys. Rev. E {\\bf 55},\n4849 (1997).\n\n\\bibitem{CaldeiraL83}\nA.O.~Caldeira and A.J.~Leggett, Ann. of Phys. {\\bf 149}, 374 (1983).\n\n\\bibitem{Berezin80}\nF.A.~Berezin, Sov. Phys. Usp. {\\bf 23}, 763 (1980).\n\n\\bibitem{CFTV99}\nA.~Cuccoli, A.~Fubini, V.~Tognetti, and R.~ Vaia, Phys. Rev. E {\\bf\n60}, 231 (1999).\n\n\\bibitem{jja}\n{\\em Proceedings of the ICTP Workshop on Josephson Junction Arrays},\nedited by H.A.~Cerdeira and S.R.~Shenoy,\nPhysica B {\\bf 222(4)}, 253-406 (1996) and references therein.\n\n\\bibitem{shon/tink}\nG.~Sch\\\"on and A.D.~Zaikin, Phys. Rep. {\\bf 198}, 237 (1990);\nM.~Tinkham, {\\em Introdution to superconductivity}\n(McGraw-Hill, New York, 1996).\n\n\\end{thebibliography}\n\n\\end{document}\n"
}
] |
[
{
"name": "cond-mat0002072.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\n\\bibitem{GTall}\nR. Giachetti and V. Tognetti, Phys. Rev. Lett. {\\bf 55}, 912 (1985);\nPhys. Rev. B {\\bf 33}, 7647 (1986). R.~P.~Feynman and H.~Kleinert,\nPhys. Rev. A {\\bf 34}, 5080 (1986).\n\n\\bibitem{FeynmanH65}\nR.P.~Feynman and A.R.~Hibbs, {\\em Quantum Mechanics and Path\nIntegrals} (Mc Graw Hill, New York, 1965).\n\n\\bibitem{Weiss99}\nU.~Weiss, {\\em Quantum Dissipative Systems} (World Scientific,\nSingapore, 2nd edition, 1999).\n\n\\bibitem{BaoZW95}\nJ.D.~Bao, Y.Z.~Zhuo, and X.~Z. Wu, Phys. Rev. E {\\bf 52}, 5656\n(1995).\n\n\\bibitem{CTVV92ham}\nA.~Cuccoli, V.~Tognetti, P.~Verrucchi, and R.~Vaia, Phys. Rev. A {\\bf\n45}, 8418 (1992).\n\n\\bibitem{CGTVV95}\nA.~Cuccoli, R.~Giachetti, V.~Tognetti, R.~Vaia and P.~Verrucchi,\nJ. Phys.: Condens. Matter {\\bf 7}, 7891 (1995).\n\n\\bibitem{CRTV97}\nA.~Cuccoli, A.~Rossi, V.~Tognetti, and R.~Vaia, Phys. Rev. E {\\bf 55},\n4849 (1997).\n\n\\bibitem{CaldeiraL83}\nA.O.~Caldeira and A.J.~Leggett, Ann. of Phys. {\\bf 149}, 374 (1983).\n\n\\bibitem{Berezin80}\nF.A.~Berezin, Sov. Phys. Usp. {\\bf 23}, 763 (1980).\n\n\\bibitem{CFTV99}\nA.~Cuccoli, A.~Fubini, V.~Tognetti, and R.~ Vaia, Phys. Rev. E {\\bf\n60}, 231 (1999).\n\n\\bibitem{jja}\n{\\em Proceedings of the ICTP Workshop on Josephson Junction Arrays},\nedited by H.A.~Cerdeira and S.R.~Shenoy,\nPhysica B {\\bf 222(4)}, 253-406 (1996) and references therein.\n\n\\bibitem{shon/tink}\nG.~Sch\\\"on and A.D.~Zaikin, Phys. Rep. {\\bf 198}, 237 (1990);\nM.~Tinkham, {\\em Introdution to superconductivity}\n(McGraw-Hill, New York, 1996).\n\n\\end{thebibliography}"
}
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cond-mat0002073
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Toward a systematic $1/d$ expansion: Two particle properties
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[
{
"author": "Gergely Zar{\\'a}nd$^{1,2}$"
},
{
"author": "Daniel L. Cox$^1$"
},
{
"author": "and Avraham Schiller$^3$"
}
] |
We present a procedure to calculate $1/d$ corrections to the two-particle properties around the infinite dimensional dynamical mean field limit. Our method is based on a modified version of the scheme of Ref.~\protect{\onlinecite{SchillerIngersent}}. To test our method we study the Hubbard model at half filling within the fluctuation exchange approximation (FLEX), a selfconsistent generalization of iterative perturbation theory. Apart from the inherent unstabilities of FLEX, our method is stable and results in causal solutions. We find that $1/d$ corrections to the local approximation are relatively small in the Hubbard model.
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[
{
"name": "oneover3.tex",
"string": "\\documentstyle[twocolumn,prl,aps,epsfig,floats]{revtex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{document}\n\\draft\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Title\n%Abstract\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%%2col\n\\twocolumn[\\hsize\\textwidth\\columnwidth\\hsize\\csname @twocolumnfalse\\endcsname\n%% start of wide text\n%%2col\n\n\\newcommand{\\ibf}{\\mbox{\\boldmath $f$}}\n\\title{\nToward a systematic $1/d$ expansion: Two particle properties\n}\n\n\\author{\nGergely Zar{\\'a}nd$^{1,2}$, Daniel L. Cox$^1$, and Avraham Schiller$^3$\n}\n\n\\address{\n$^1$Department of Physics, University of California Davis, CA 95616\\\\ \n$^2$Research Group of the Hungarian Academy of Sciences, Institute of Physics,\nTU Budapest, H-1521\n$^3$Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel\n}\n\\date{\\today}\n\\maketitle\n\n\\begin{abstract}\nWe present a procedure to calculate $1/d$ corrections to the\ntwo-particle properties around the infinite dimensional dynamical\nmean field limit. Our method is based on a modified version\nof the scheme of Ref.~\\protect{\\onlinecite{SchillerIngersent}}.\nTo test our method we study the Hubbard model at half filling \nwithin the fluctuation exchange approximation (FLEX), \na selfconsistent generalization of iterative perturbation theory. \nApart from the inherent unstabilities of FLEX, our method is stable \nand results in causal solutions. We find that $1/d$ corrections \nto the local approximation are relatively small \nin the Hubbard model.\n\\end{abstract}\n\n\\pacs{PACS numbers: 71.27.+a, 75.20.Hr}\n%% 71.10.Hf -- Non-Fermi-liquid ground states, electron phase diagrams\n%% and phase transitions in model systems\n%% 71.27.+a -- Strongly correlated electron systems; heavy fermions.\n%% 72.15.Qm -- Scattering mechanisms and Kondo effect\n%% 75.20.Hr -- Local moment in compounds and alloys; Kondo effect,\n%% valence fluctuations, heavy fermions.\n\n%%2col\n%% end of wide text\n]\n\\narrowtext\n%%2col\n%Section 1\n%Introduction\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nDuring the past few years dynamical mean field theory (DMFT) \nbecame one of the most popular methods to study strongly \ncorrelated systems \\cite{Kotliar}. \nDMFT developed from the path-breaking \nobservation \\cite{MetznerVollhard} that in the limit $d\\to\\infty$\nof a $d$-dimensional lattice model with suitably rescaled hopping \nparameters, spatial fluctuations are completely suppressed\nand the self-energy becomes local. As a consequence, the self-energy \ncan be written as a functional of the on-site Green's function \nof the electrons and the lattice problem reduces to a quantum impurity\nproblem, where the impurity is embedded in a selfconsistently\ndetermined environment.\nThe main virtue of this method is that it \ncaptures all local {\\em time-dependent} correlations \nand makes possible to study, e.g. the Mott-Hubbard \ntransition or the phase diagram of different Kondo lattices in \ndetail. \n\n\n\nWhile in the case of the Mott-Hubbard transition the transition \nseems to be driven by the above-mentioned local fluctuations, in \nmany cases correlated hopping \\cite{Avi} or inter-site interaction \neffects \\cite{Vlad,intersite} may play a crucial role as well, \nand while some of these effects can be qualitatively captured by a \nnatural extension of the DMFT, others are beyond the scope of it and \nwould only appear as $1/d$ corrections. Furthermore, in order \nto check the quality of the local approximation for a finite \ndimensional system of interest, it is very important to compare it with \nthe size of the appearing $1/d$ corrections as well.\n\n\n\nSeveral attempts have been made to partially restore some of \nthe spatial correlations lost in the DMFT. One of the most successful ones \nis the cluster approximation proposed by Jarrell et al \\cite{Jarrell}. \nThis method has the advantage of being causal, however, it requires\nconsiderable numerical prowess and it is not systematic in the \nsmall parameter $1/d$. Another method based on the systematic expansion \nof the generating functional has been suggested by\nSchiller and Ingersent \\cite{SchillerIngersent}. However, despite of its technical and conceptual \nsimplicity, this method has not been used very extensively because \nit seemed to be somewhat unstable and in some cases gave \nartificial non-causal solutions. \n\nIn the present work we first show, that the method of Schiller and \nIngersent (SI) can be considerably stabilized by a minor, however\ncrucial change in the algorithm, assuring that the contributions of some \nunwanted spurious diagrams exactly cancel. The price for this \nstability is a somewhat increased computation time, since in \neach cycle of the original algorithm an additional subcycle\nis needed to assure cancellation. With this change the SI \nmethod can then be safely extended to the calculation \nof two-particle properties. Here the main difficulties \n%%We circumvent the appearing new difficulties \nare connected to the inversion involved in the solution of the \nBethe-Salpeter equation and the non-locality of the irreducible \nvertex functions. These difficulties are cicrcumvented by introducing \nbond variables for the two-particle propagators. Finally, we test the \ngeneral formalism with the fluctuation exchange approximation \n(FLEX) \\cite{Bickers}. \n\n\nAlthough the method presented applies to arbitrary lattice structures\nand various models with nearest neighbor interactions,\nfor concreteness, let us consider the Hubbard model on a $d$-dimensional \nhypercubic lattice at half filling:\n\\begin{equation}\nH = {t\\over \\sqrt{d}} \\sum_{<i,j>, \\sigma} \nc^{\\dagger}_{i\\sigma} c_{j\\sigma} + U \\sum_i (n_{i\\uparrow}-{1\\over\n2}) ( n_{i\\downarrow}-{1\\over 2})\\;.\n\\end{equation}\nHere the dynamics of the conduction electrons $c_{i\\sigma}$ is driven \nby the hopping $t$ between nearest neighbor sites,\n$n_{i\\sigma} = c^{\\dagger}_{i\\sigma} c_{i\\sigma}$ is the occupation \nnumber, and the electrons interact via the on-site Coulomb repulsion $U$. \n\n\nIn the SI formalism one considers the following \nsingle ($n=1$) and a two-impurity $(n=2)$ imaginary \ntime effective functionals to generate $1/d$ corrections: \n\\begin{eqnarray}\nS^{(n)} &=& \\sum_\\sigma \\sum_{\\alpha,\\beta = 1}^n \n\\int d\\tau \\int d\\tau' {\\bar c}_{\\alpha \\sigma}(\\tau) \n\\bigl[{\\cal G}^{(n)}\\bigr]^{-1}_{\\alpha,\\beta} (\\tau-\\tau') c_{\\beta \\sigma}(\\tau') \\nonumber \\\\\n& +& \\sum_{\\alpha = 1}^n U \\int d\\tau n_{\\alpha \\uparrow}(\\tau) \nn_{\\alpha \\downarrow}(\\tau)\\;.\n\\end{eqnarray}\nHere, as usually, ${\\bar c}_{\\alpha \\sigma}(\\tau)$ and \n${ c}_{\\alpha \\sigma}(\\tau)$ denote Grassman fields, and the\nindices $\\alpha$ and $\\beta$ label the sites for $n=2$ while\nthey are redundant for $n=1$.\nThe 'medium propagators' ${\\cal G}^{(1)}$ and ${\\cal G}^{(2)}$ must be\n chosen in such a way that the dressed impurity propagators $G^{(1)}$ and \n$G^{(2)}$ coincide with the full on-site and nearest neighbor \nlattice propagators, $G^{\\rm latt}_{00}$ and $G^{\\rm latt}_{01}$:\n\\begin{equation}\nG^{(1)} = G^{(2)}_{11} = G^{\\rm latt}_{00}\\; \\phantom{n},\n \\phantom{nnn} G^{(2)}_{12} = G^{\\rm latt}_{01}\\;.\n\\end{equation}\nIn this case one can easily show that --- restricting oneself to \nskeleton diagrams of the order of ${\\cal O}(1/d)$ ---\nthe impurity self energies $\\Sigma^{(1)}$ and $\\Sigma^{(2)}_{\\alpha\\beta}$ \nand the diagonal and off-diagonal lattice self energies, \n$\\Sigma^{\\rm latt}_0$ and $\\Sigma^{\\rm latt}_1$ are related by \n\\cite{SchillerIngersent}\n\\begin{eqnarray}\n\\Sigma^{\\rm latt}_0 &=& \\Sigma^{(1)} + 2d(\\Sigma^{(2)}_{11} - \\Sigma^{(1)})\\;,\n\\label{eq:Sigma0} \\\\\n\\Sigma^{\\rm latt}_1 &=& \\Sigma^{(2)}_{12}\\;.\\label{eq:Sigma12}\n\\end{eqnarray}\nKnowing the lattice self energy the lattice Green function \ncan then be expressed as \n\\begin{equation} \nG_{lm}^{\\rm latt}(i\\omega) = {1\\over1 +\\sqrt{d} \\; \\Sigma^{\\rm latt}_1}\nG^{0}_{lm}\\left({i\\omega - \\Sigma^{\\rm latt}_0(i\\omega) \\over\n1 +\\sqrt{d} \\; \\Sigma^{\\rm latt}_1(i\\omega)}\\right)\\;,\n\\end{equation}\nwhere $G^{0}_{lm}(z)$ and $G_{lm}^{\\rm latt}(z)$ denote the unperturbed \nand dressed lattice propagators between sites $l$ and $m$, \nrespectively.\n\n\nBased on the relations above SI suggested the following simple iterative \nprocedure to obtain a solution that includes ${\\cal O}(1/d)$ corrections: \n$$\n{\\cal G}^{(1,2)} \\Rightarrow {\\Sigma}^{(1,2)} \\Rightarrow \n\\Sigma^{\\rm latt} , G^{\\rm latt} \\Rightarrow {\\cal G}^{(1,2)}\\;.\n$$ \nA careful analysis shows, however, that the second step in \nthis scheme is extremely unstable. To understand this it is \nenough to notice that the second term \nof Eq.~(\\ref{eq:Sigma0}) is constructed in such a way that {\\em at the \nfixed point} all {\\em completely local } skeleton diagrams in the \nexpansion of $\\Sigma^{(2)}_{11}$ are canceled by the subtracted \n$\\Sigma^{(1)}$ term. \nHowever, this cancellation only happens under the condition that \nthe dressed Green's functions $G^{(2)}_{11}$ and $G^{(1)}$\nare exactly the same.\nIf $G^{(2)}_{11}$ and $G^{(1)}$ differ by a term of ${\\cal O}(1/d)$ \nin a given iteration step, the cancellation above\nis not exact, and an error of the order of \n$2d\\times {\\cal O}(1/d)\\sim 1$ is generated immediately. \nMoreover, the generated erroneous term is typically acausal \nbecause of the subtraction procedure involved in\nEq.~(\\ref{eq:Sigma0}), \nand may drive the iteration towards some more stable\nbut unphysical fixed point of the integral equations. \nWe suggest to replace the critical steps ${\\cal G}^{(1,2)} \n\\Rightarrow {\\Sigma}^{(1,2)} \\Rightarrow \\Sigma^{\\rm latt}$ by the \nfollowing procedure: \n(1) Calculate $G^{(2)}$ from ${\\cal G}^{(2)}$, \n(2) Determine ${\\cal G}^{(1)}$ selfonsistently in such a way that \n${G}^{(1)} \\equiv {G}^{(2)}_{11}$ be satisfied, (3) Determine \n$\\Sigma^{(1,2)}$ and from them $\\Sigma^{\\rm latt}$. \nStep (2) above is crucial to guarantee that unwanted terms in \nEq.~(\\ref{eq:Sigma0}) exactly cancel. \n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Fig. 3\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\psfig{figure=twopart.eps,width=8.5cm}\n\\end{center}\n\\vspace{0.05cm}\n\\caption{\n\\label{fig:bethe}\nGraphical representation of the Bethe-Salpeter equation: \nDouble lines indicate dressed Fermion propagators, while \nboxes denote particle-hole irreducible vertex diagrams.\n}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\nThe two-particle properties can be investigated in a way similar \nto Ref.~\\onlinecite{SchillerIngersent}. To this end we introduce \nthe lattice particle-hole irreducible vertex function\n${\\bf \\tilde \\Gamma}^{\\rm latt}$, which is connected to the \nfull lattice propagator ${\\bf L}^{\\rm latt} $ by the Bethe-Salpeter \nequation (see Fig.~\\ref{fig:bethe}): \n\\begin{equation}\n{\\bf L}^{\\rm latt}(\\omega) = {\\bf L}_0^{\\rm latt}(\\omega) ( {\\bf 1} + \n{\\bf \\tilde \\Gamma}^{\\rm latt}(\\omega) {\\bf L}_0^{\\rm latt}(\\omega) )^{-1},\n\\label{eq:bethe}\n\\end{equation}\nwhere $\\omega$ denotes the transverse frequency and \na tensor notation has been introduced in the spatial, spin and\nother frequency indices:\n$L(\\omega)^{\\sigma_1,\\sigma_2; \\sigma_3,\\sigma_4}_{ i_1,i_2,\\omega_1; \ni_3,i_4,\\omega_3}\n\\to {\\bf L}(\\omega)$. The 'vertex-free' propagator \n$ {\\bf L}_0^{\\rm latt}(\\omega)$ is defined as \n$L_0^{\\rm latt}(\\omega)^{\\sigma_1,\\sigma_2; \\sigma_3,\\sigma_4}_{ i_1,i_2,\n\\omega_1; i_3,i_4,\\omega_3} =$ $\\delta_{\\sigma_1\\sigma_3}\n\\delta_{\\sigma_2\\sigma_4} \\delta_{\\omega_1\\omega_3}$\n$G_{i_1,i_3}^{\\rm latt}(i\\omega_1)$ $G_{i_4,i_2}^{\\rm latt}(i(\\omega_1+\\omega))$.\n\nA detailed analysis shows that up to \n$1/d$ order the only non-zero matrix elements of \n${\\bf \\tilde \\Gamma}^{\\rm latt}(\\omega)$ are those \nwhere the indices $ i_1,i_2,i_3,i_4$ belong to the same or \ntwo nearest neighbor lattice sites, i.e. a bond. \nA thorough investigation of the corresponding skeleton diagrams \nshows that ${\\bf \\tilde \\Gamma}^{\\rm latt}(\\omega)$\ncan be expressed similarly to \nEqs.~(\\ref{eq:Sigma0}) and (\\ref{eq:Sigma12}) as:\n\\begin{equation}\n{\\bf \\tilde \\Gamma}^{\\rm latt} = \n\\left\\{\n\\mbox{\n\\begin{tabular}{ll }\n${\\bf \\tilde \\Gamma}^{(1)} + 2d ({\\bf \\tilde \\Gamma}^{(2)} - {\\bf \\tilde \\Gamma}^{(1)} )$ &\n\\phantom{n} $i_1=i_2=i_3 = i_4$, \\\\\n${\\bf \\tilde \\Gamma}^{(2)}$ &\n\\phantom{n} $i_k\\in$ bond, \\\\\n0 & \\phantom{n}otherwise, \n\\end{tabular}\n}\\right. \n\\label{eq:Gamma_latt}\n\\end{equation}\nwhere in the second line it is implicitely assumed that \nthe $i_k$'s are not all equal. \nHere the particle-hole irreducible one- and two-impurity \nvertex functions, ${\\bf \\tilde \\Gamma}^{(1)}$, \nand ${\\bf \\tilde \\Gamma}^{(2)}$ are defined similarly to \n${\\bf \\tilde \\Gamma}^{\\rm latt}$, and satisfy the impurity\nBethe-Salpeter equation:\n \\begin{equation}\n{\\bf L}^{(n)}(\\omega) = {\\bf L}_0^{(n)}(\\omega) ( {\\bf 1} + \n{\\bf \\tilde \\Gamma}^{(n)}(\\omega) {\\bf L}_0^{(n)}(\\omega) )^{-1},\n\\label{eq:bethe12}\n\\end{equation}\nwith $n=1,2$. Of course, in the impurity case the spatial indices\nof the propagators ${\\bf L}_0^{(n)}(\\omega) $ and ${\\bf L}^{(n)}(\\omega)$ \nare restricted to the impurity sites, but apart from this\nthe ${\\bf L}_0^{(n)}(\\omega) $'s are equal to the lattice \npropagator ${\\bf L}_0^{\\rm latt}(\\omega)$. \n\nFrom the considerations above it immediately follows that \nthe $1/d$ corrections to the two-particle properties can be \ncalculated in the following way: (1) Find the solution of the \nsingle particle iteration scheme, (2) Determine the one- and two-impurity \ncorrelators, (3) Invert Eq.~(\\ref{eq:bethe12}) to obtain \n${\\bf \\tilde \\Gamma}^{(1)}$ and ${\\bf \\tilde \\Gamma}^{(2)}$, \n(4) Calculate ${\\bf \\tilde \\Gamma}^{\\rm latt}$ using \nEq.~(\\ref{eq:Gamma_latt}), and \n(5) Solve the Bethe-Salpeter equation (\\ref{eq:bethe}) for\n ${\\bf L}^{\\rm latt}$ and calculate two-particle response functions from it.\n\nThe major difficulties in the procedure above are associated \nwith the inversion appearing in Eqs.~(\\ref{eq:bethe}), since\nthe propagator ${\\bf L}_0^{\\rm latt}$ connects any four lattice \nsites and has an infinite number of frequency indices. The first \ndifficulty can be resolved by observing that ${\\bf \\tilde \\Gamma}^{\\rm latt}$ \nconnects only neighboring sites. Therefore with a introduction of \n{\\em bond variables} a partial Fourier transformation can be carried out in \nthese, and summations over all pairs of lattice sites \nreduce to a summation over $d$ 'bond direction indices' \nand two additional indices specifying the position of the \nelectron and the hole within a given bond. \nFurthermore, to avoid overcounting, the vertex function\n${\\bf \\tilde \\Gamma}^{\\rm latt}$ must be replaced by a slightly \nmodified 'bond vertex function' \\cite{ZarCoxScill2}.\nA further reduction of the \nmatrices involved can be achieved by diagonalizing the \npropagators in the spin labels. Finally, to carry out the \nsummations and inversions over the infinite omega variables\nwe introduced a frequency cutoff $\\omega_c$ and extrapolated \nthe $\\omega_c=\\infty$ result from a finite size scaling analysis \nin this cutoff \\cite{Pruschke}, thereby reducing the numerical error of\nour calculations below one percent.\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Fig. 1\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\psfig{figure=phi2.eps,width=7.2cm}\n\\end{center}\n\\vspace{0.05cm}\n\\caption{\n\\label{fig:diagrams}\nThe generating functional for FLEX: Double lines denote \nsingle- and two-impurity dressed cavity propagators $G^{(1)}(\\tau,\\tau')$\nand $G^{(2)}(\\tau,\\tau')$. Position and spin indices are not \nshown. The corresponding self-energy diagrams are obtained as\n$\\Sigma(\\tau-\\tau') = -\\delta \\phi[G(\\tau,\\tau')] / \\delta G(\\tau',\n\\tau)$, and the particle-hole irreducible vertex is given by \na similar second order functional derivative.\n}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nTo test the procedure above one is tempted to try to generalize the\niterative perturbation theory (IPT) applied remarkably successfully\nfor the $d=\\infty$ case \\cite{IPT}, however, it is clear from the\ndiscussion above that within IPT it is impossible to satisfy the\ncondition ${G}^{(1)} \\equiv {G}^{(2)}_{11}$ (which explains why\nearlier attempts to generalize IPT to order $1/d$ failed\n\\cite{Georges}). We therefore applied the so-called\nfluctuation exchange approximation (FLEX) \\cite{Bickers}. While this\nmethod is unable to capture the metal insulator transition, it is\nable to reproduce the Kondo resonance in the metallic\nphase\\cite{BickersSC}, has been successfully used to calculate\nweak and intermediate coupling properties of the 2-dimensional\nHubbard model \\cite{BickersSC}, and it has the important property of\nbeing formulated in terms of the {\\em dressed} single particle\nGreen's functions. In this approach the interactions between\nparticles are mediated by fluctuations in the particle-particle and\nparticle-hole channels, and the self-energies and the particle-hole\n(particle-particle) irreducible vertex functions are generated from\nthe generating $\\Phi$ functionals built in terms of the dressed\nGreen's functions, depicted in Fig.~\\ref{fig:diagrams}. A further\nadvantage of FLEX is that due to the special structure of the\ndiagrams involved a fast Fourier transform algorithm can be exploited\nto increase the speed and precision of the calculation\nsubstantially.\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Fig. 2\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\psfig{figure=sigmas.eps,width=6.5cm}\n\\end{center}\n\\vspace{0.05cm}\n\\caption{\n\\label{fig:self}\nTop: Calculated diagonal and off-diagonal lattice self-energies \nfor $U = 4$ and $T = 0.002$. All energies are measured in units of \n$t$. Bottom: Local spectral functions for $U=0$ and $U=4$ at \n$T = 0.002$. \n}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\nThe calculated three-dimensional diagonal and off-diagonal \nlattice self-energies are shown in Fig.~\\ref{fig:self} together with \nthe local spectral function. These have been obtained \nby means of a Pade approximation to carry out the \nanalytic continuation from the imaginary to the real axis. \nThough in the spectral function a well-developed Kondo peak \nis observed, \nthe FLEX is unable to reproduce the depletion of spectral weight \nin the neighborhood of it\ndue to the 'over-regularization'\ncharacteristic to most selfconsistent perturbative schemes.\nRemarkably, we experienced no convergence problems similar \nto those of Ref.~\\onlinecite{SchillerIngersent}, apart \nfrom the ones inherent in FLEX \\cite{Bickers}. We checked that the \nspectral functions integrate to one within numerical precision\nand the solutions obtained are causal. The typical values of \n$\\Sigma^{\\rm latt}_1$ are nearly an order of magnitude \nsmaller \\cite{sig_1footn} than $\\Sigma^{\\rm latt}_0$, indicating that \nthe local approximation gives surprisingly good results \nand $1/d$ corrections are indeed small as anticipated in \nRef.~\\onlinecite{Kotliar} and also in agreement with the results of\nRef.~\\onlinecite{SchillerIngersent}. To get some further information about \nthe quality of local approximation in Fig.~\\ref{fig:rhok}\nwe plotted the momentum dependent spectral functions \nat different points of the Brillouin zone. The $1/d$ \ncontributions give typically a 10-20 percent correction, \nbut none of the generic properties is modified in the \nparamagnetic phase. \n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Fig. 4\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\psfig{figure=rhoomk.eps,width=7.2cm}\n\\end{center}\n\\vspace{0.05cm}\n\\caption{\n\\label{fig:rhok}\nMomentum dependent spectral functions along the line \n$k = \\alpha (\\pi,\\pi,\\pi)$ within the local (dashed lines) and \n$1/d$ calculations (continuous lines). The alpha values \nused were $\\alpha = 0.497$, $\\alpha = 0.495$, and $\\alpha = 0.49$\nfrom left to right. The energy of the quasiparticles has been \nrenormalized approximately by a factor of four compared to the \nbare electron energies.\n}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nOnce convergence is reached at the single particle level, \none can turn to the two-particle properties.\nWithin FLEX this is somewhat simpler, because ---\nalthough many rather complicated diagrams are generated \\cite{Bickers,ZarCoxScill2}\n--- ${\\bf \\tilde \\Gamma}^{(n)}$ can be built up directly in terms of\nthe full lattice Green's functions. We find that similarly to\nthe off-diagonal self-energy the off-diagonal \nelements of ${\\bf \\tilde \\Gamma}^{\\rm latt}$ are rather small.\nHaving solved Eq.~(\\ref{eq:bethe}) one can calculate various \ncorrelation functions.\nAs an example, in Fig.~\\ref{fig:chi} we show the momentum dependent \nsusceptibility of the half-filled Hubbard model in its paramagnetic \nphase for two different temperatures along the $(1,1,1)$\ndirection, obtained from the FLEX calculations. The susceptibility \ndevelops a peak at $(\\pi,\\pi,\\pi)$\nat low temperatures, as a sign of unstability \ntoward antiferromagnetic phase transition.\n\n\n\nWe also determined the transition \ntemperature at several values of $U$ and compared our results with \nexisting Monte Carlo data\\cite{Scalettar}. We found \na critical temperature $T_c$ typically by a factor of three lower \nthan that of Ref.~\\onlinecite{Scalettar}. This difference is a result \nof the overregularization of the interaction vertex by FLEX.\n Indeed, replacing $ {\\bf \\tilde \\Gamma}^{\\rm latt}(\\omega)$ by the \n{\\em bare} particle-hole vertex in Eq.~(\\ref{eq:bethe}) the order-parameter\nfluctuations become larger (see Fig.~\\ref{fig:chi}) and $T_c$ is in \nexcellent agreement with the Monte Carlo data. \n\n\nIn conclusion, we presented an extended version of the SI method \nto calculate $1/d$ corrections to the two-particle properties. \nWe tested the new procedure by FLEX. No convergence problems \nand no violation of causality appeared in our method, although \nthis is not generally guaranteed within the present scheme. \nOur method should be tested on other models\nand with other, more time-consuming methods in the future \nas well. \n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%Fig. 5\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{figure}\n\\begin{center}\n\\psfig{figure=chi2.eps,width=7.2cm}\n\\end{center}\n\\vspace{0.05cm}\n\\caption{\n\\label{fig:chi}\nMomentum dependent susceptibilities. The susceptibility \ndevelops a peak at $(\\pi,\\pi,\\pi)$ as a precursor of the \nantiferromagnetic phase transition.\n}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nThe authors are grateful to N. Bickers for valuable discussions. \nThis research has been supported by \nthe U.S - Hungarian Joint Fund No. 587, grant No. DE-FG03-97ER45640 of the\nU.S DOE Office of Science, Division of Materials Research, \nand Hungarian Grant Nr. OTKA T026327, OTKA F29236, and \nOTKA T029813.\n\n\\vspace{-0.5cm} \n\\begin{thebibliography}{99}\n\n\\bibitem{SchillerIngersent} \nA. Schiller and K. Ingersent, Phys. Rev. Lett. {\\bf 75}, 113 (1995).\n\\bibitem{Kotliar} For reviews see A. Georges, G. Kotliar, W. Krauth, and \nM. Rozenberg, Rev. Mod. Phys. {\\bf 68}, 13 (1996).\n\\bibitem{MetznerVollhard} W. Metzner and D. Vollhardt, Phys. Rev. Lett. \n{\\bf 62}, 324 (1989).\n\\bibitem{Avi} A. Schiller (unpublished).\n\\bibitem{Vlad} A. Pastor and V. Dobrosavljevic, Cond-Mat 9903272.\n\\bibitem{intersite} E. M\\\"uller-Hartmann, Z. Phys. B {\\bf 74}, 507 (1989);\nP.G.J. van Dongen, Phys. Rev. B {\\bf 50}, 14016 (1994).\n\\bibitem{Jarrell} M. Hettler et al., Rapid Comm. PRB 58, 7475 (1998); \nand Cond-Mat 9902267.\n\\bibitem{Bickers} N. E. Bickers and D.J. Scalapino, \nAnnals of Phys. {\\bf 193}, 206 (1989).\n\\bibitem{ZarCoxScill2} G. Zar\\'and, D.L. Cox, and A. Schiller\n(under preparation).\n\\bibitem{Pruschke} T. Pruschke, private communications, 1999. \n\\bibitem{IPT} A. Georges and G. Kotliar, Phys. Rev. B {\\bf 45}, 6479.\n\\bibitem{Georges} A. Georges (unpublished).\n\\bibitem{BickersSC} N.E. Bickers and C-X Chen, Solid State Comm. {\\bf\n82}, 311 (1992). \n%\\bibitem{BickersSC} \nN.E. Bickers, D.J. Scalapino and S.R. White, Phys. Rev. Lett. {\\bf 62},\n961 (1989); C.-H. Pao and N. E. Bickers\nPhys. Rev. Lett {\\bf 72}, 1870 (1994).\n\\bibitem{sig_1footn} In reality, $\\Sigma_1^{\\rm latt}$ appears with a \nprefactor $\\sqrt{d}$ in the Green's function which, however, does not\nchange this conclusion.\n\\bibitem{Scalettar} R. Scalettar, {\\it et al.}, Phys. Rev. B{\\bf 39},\n4711 (1989).\n\n\n\\end{thebibliography}\n\\end{document}\n\n\n\n\n\n\n\n"
}
] |
[
{
"name": "cond-mat0002073.extracted_bib",
"string": "\\begin{thebibliography}{99}\n\n\\bibitem{SchillerIngersent} \nA. Schiller and K. Ingersent, Phys. Rev. Lett. {\\bf 75}, 113 (1995).\n\\bibitem{Kotliar} For reviews see A. Georges, G. Kotliar, W. Krauth, and \nM. Rozenberg, Rev. Mod. Phys. {\\bf 68}, 13 (1996).\n\\bibitem{MetznerVollhard} W. Metzner and D. Vollhardt, Phys. Rev. Lett. \n{\\bf 62}, 324 (1989).\n\\bibitem{Avi} A. Schiller (unpublished).\n\\bibitem{Vlad} A. Pastor and V. Dobrosavljevic, Cond-Mat 9903272.\n\\bibitem{intersite} E. M\\\"uller-Hartmann, Z. Phys. B {\\bf 74}, 507 (1989);\nP.G.J. van Dongen, Phys. Rev. B {\\bf 50}, 14016 (1994).\n\\bibitem{Jarrell} M. Hettler et al., Rapid Comm. PRB 58, 7475 (1998); \nand Cond-Mat 9902267.\n\\bibitem{Bickers} N. E. Bickers and D.J. Scalapino, \nAnnals of Phys. {\\bf 193}, 206 (1989).\n\\bibitem{ZarCoxScill2} G. Zar\\'and, D.L. Cox, and A. Schiller\n(under preparation).\n\\bibitem{Pruschke} T. Pruschke, private communications, 1999. \n\\bibitem{IPT} A. Georges and G. Kotliar, Phys. Rev. B {\\bf 45}, 6479.\n\\bibitem{Georges} A. Georges (unpublished).\n\\bibitem{BickersSC} N.E. Bickers and C-X Chen, Solid State Comm. {\\bf\n82}, 311 (1992). \n%\\bibitem{BickersSC} \nN.E. Bickers, D.J. Scalapino and S.R. White, Phys. Rev. Lett. {\\bf 62},\n961 (1989); C.-H. Pao and N. E. Bickers\nPhys. Rev. Lett {\\bf 72}, 1870 (1994).\n\\bibitem{sig_1footn} In reality, $\\Sigma_1^{\\rm latt}$ appears with a \nprefactor $\\sqrt{d}$ in the Green's function which, however, does not\nchange this conclusion.\n\\bibitem{Scalettar} R. Scalettar, {\\it et al.}, Phys. Rev. B{\\bf 39},\n4711 (1989).\n\n\n\\end{thebibliography}"
}
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cond-mat0002074
|
Two time scales and FDT violation in a Finite Dimensional Model for Structural Glasses
|
[
{
"author": "Federico Ricci-Tersenghi$^1$"
},
{
"author": "Daniel A. Stariolo$^2$ and Jeferson J. Arenzon$^2$"
}
] |
We study the breakdown of fluctuation-dissipation relations between time dependent density-density correlations and associated responses following a quench in chemical potential in the Frustrated Ising Lattice Gas. The corresponding slow dynamics is characterized by two well separated time scales which are characterized by a constant value of the fluctuation-dissipation ratio. This result is particularly relevant taking into account that activated processes dominate the long time dynamics of the system.
|
[
{
"name": "fdt_flg.tex",
"string": "\\documentstyle[aps,prl,multicol,epsfig]{revtex}\n\n\\begin{document}\n\n\\newcommand \\qea {\\mbox{$q_{\\scriptscriptstyle {\\rm EA}}$}}\n\n\\title{Two time scales and FDT violation in a Finite Dimensional Model\nfor Structural Glasses}\n\n\\author{Federico Ricci-Tersenghi$^1$, Daniel A. Stariolo$^2$ and\nJeferson J. Arenzon$^2$}\n\n\\address{\n$^1$Abdus Salam International Center for Theoretical Physics,\nCondensed Matter Group\\\\\nStrada Costiera 11, P.O. Box 586, 34100 Trieste, Italy\\\\\n$^2$Instituto de F{\\'\\i}sica, Universidade Federal do Rio Grande do Sul\\\\ \nCP 15051, 91501-970 Porto Alegre RS, Brazil\\\\\nE-mails: {\\tt riccife@ictp.trieste.it, stariolo@if.ufrgs.br,\narenzon@if.ufrgs.br}}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\nWe study the breakdown of fluctuation-dissipation relations between\ntime dependent density-density correlations and associated responses\nfollowing a quench in chemical potential in the Frustrated Ising\nLattice Gas. The corresponding slow dynamics is characterized by two\nwell separated time scales which are characterized by a constant value\nof the fluctuation-dissipation ratio. This result is particularly\nrelevant taking into account that activated processes dominate the\nlong time dynamics of the system.\n\\end{abstract}\n\n\\pacs{PACS numbers: 75.10.Nr, 05.50.+q, 75.40.Gb, 75.40.Mg}\n \n\\begin{multicols}{2}\n\\narrowtext\n\nIn recent years considerable progress has been achieved in the\ntheoretical description of the glassy state of matter. A scenario for\nthe observed slow dynamics of glass forming materials has emerged\nthrough detailed analysis of mean field (MF) spin glass models\n~\\cite{young}. The equations describing the off-equilibrium dynamics\nof these MF spin glasses simplify, above the transition, to the single\nequation for the Mode Coupling Theory for supercooled\nliquids~\\cite{gotze}. These approaches have been successful in\nexplaining history dependence or aging effects and the nature of the\ntwo characteristic relaxations in glasses, the short time or\n$\\beta$-relaxation and the structural long time\n$\\alpha$-relaxation. In the $\\alpha$-relaxation the system falls out\nof local equilibrium, as a consequence Fluctuation-Dissipation Theorem\n(FDT) breaks down and can be replaced by the more general relation\n\\begin{equation}\nR(t,t_w) = \\frac{X(t,t_w)}{T}\\frac{\\partial C(t,t_w)}{\\partial t_w}\\ ,\n\\label{gfdt}\n\\end{equation}\nwhere $C(t,t_w)$ is a two times correlation function and $R(t,t_w)$\nwith $t\\!>\\!t_w$ is the associated response. $T$ is the heat bath\ntemperature and $X(t,t_w)$ is a function that measures the departure\nfrom FDT: at equilibrium $X=1$ and the usual FDT is recovered, while\nin the out of equilibrium regime $X<1$. In MF\napproximation~\\cite{cuku} the function $X$, called\n``fluctuation-dissipation ratio''(FDR), turns out to depend on both\ntimes only through $C(t,t_w)$. Moreover in MF models of\nglasses $X$ is a constant (when different from 1). This scenario\nreflects the existence of only two well separated time scales, the\nequilibrium or FDT scale and a longer one where the system is out of\nequilibrium. The FDR has been interpreted as an effective temperature\nand it has been demonstrated that it is exactly the temperature that a\nthermometer would measure if it would be coupled to the slowly\nrelaxing modes of the system~\\cite{luca}. Recently the first\nexperimental determination of the FDR has been done in\nglycerol~\\cite{grigera}.\n\n\nThis scenario, while appealing, is essentially based on an analogy\nbetween the physics of some MF spin glasses and the behavior\nof real finite structural glasses (being the formal connection valid\nonly in the high temperature region). At this point it seems crucial to\ntest the link between MF theories and realistic models in the\nglassy phase. In particular, we still do not know which will be the role\nof activated processes in realistic models. Activated processes are\nabsent in completely connected models in the thermodynamic limit while\non the contrary they dominate the relaxation dynamics below the glass\ntransition temperature $T_g$ in real glasses.\n\nThe goal of the present letter is to go beyond the MF like\ndescription of structural glasses by considering a finite dimensional\nmodel, the frustrated Ising lattice gas (FILG)~\\cite{mario,nos}, which\npresents most of the relevant features of glass forming materials, in\nparticular activated processes at low temperatures. Here we analyze\nthe violation of FDT in the FILG in three dimensions through Monte\nCarlo simulations. The (very precise) results confirm the qualitative\nscenario of MF models of a constant FDR with large separation\nof time scales and set the stage for a detailed investigation of\nactivated processes in realistic models of glasses.\n\nThe FILG is defined by the Hamiltonian:\n\\begin{equation}\nH = -J \\sum_{<ij>} (\\varepsilon_{ij} \\sigma_i \\sigma_j - 1) n_i n_j -\n\\mu \\sum_i n_i\\ .\n\\end{equation}\nAt each site of the lattice there are two different dynamical\nvariables: local density (occupation) variables $n_i=0,1$ ($i=1 \\ldots\nN$) and internal degrees of freedom, $\\sigma_i=\\pm1$. The usually\ncomplex spatial structure of the molecules of glass forming liquids,\nwhich can assume several spatial orientations, is in part responsible\nfor the geometric constraints on their mobility. Here we are in the\nsimplest case of two possible orientations, and the steric effects\nimposed on a particle by its neighbors are felt as restrictions on its\norientation due to the quenched random variables $\\varepsilon_{ij}=\\pm\n1$. The first term of the Hamiltonian implies that when $J\\rightarrow\n\\infty$ any frustrated loop in the lattice will have at least one hole\nand then the density will be $\\rho<1$, preventing the system from\nreaching the close packed configuration. The system will then present\n``geometric frustration''. Finally, $\\mu$ represents a chemical\npotential ruling the system density (at fixed volume).\n\nThe system presents a slow ``aging'' dynamics after a quench from a\nsmall value of $\\mu$ characteristic of the liquid phase to a large\n$\\mu$ corresponding to the glassy phase~\\cite{nos}, what is equivalent\nto a sudden compression.\nIn the present numerical experiments we always let the system evolve\nafter a quench in $\\mu$ with $J$ and $T$ fixed. The origin of the\ntimes is set on the quench time. After the quench the density slowly relaxes\nup to a critical value near $\\rho_c \\approx 0.675$ (further\ndetails will be given in~\\cite{nos3}). After a waiting time $t_w$ we\nfix the density to the value $\\rho=\\rho(t_w)$ (for technical reasons\nexplained below) and a small random perturbation ($\\mu_i=\\pm1$) is\napplied~\\cite{note1}:\n\\begin{equation}\nH'(t) = H(t) - \\epsilon(t) \\sum_i \\mu_i\\, n_i(t)\\ .\n\\label{pert_ham}\n\\end{equation}\nIn all our numerical experiments the field is switched on at time\n$t_w$ and kept fixed for later times, that is $\\epsilon(t) =\n\\epsilon\\, \\theta(t-t_w)$. Then we measure the density-density\nautocorrelation function\n\\begin{equation}\nC(t,t_w) = \\frac{1}{N\\rho}\\sum_i \\overline{\\langle n_i(t) n_i(t_w)\n\\rangle}\\ ,\n\\label{corr}\n\\end{equation}\nwhere $\\langle\\ \\cdot\\ \\rangle$ and $\\overline{\\ \\cdot\\ }$ are the\naverages over thermal histories and disorder\nrealizations~\\cite{note2}. At the same time we measure the associated\nresponse function integrated over the time and divided by the\nperturbing field intensity, which defines the off-equilibrium\ncompressibility\n\\begin{equation}\n\\kappa(t,t_w) = \\frac1\\epsilon \\int_{-\\infty}^{t} R(t,s) \\epsilon(s)\n{\\mathrm d}s = \\int_{t_w}^{t} R(t,s) {\\mathrm d}s\\ ,\n\\label{compre1}\n\\end{equation}\nwhere, as usual, the response is defined as\n\\begin{equation}\nR(t,t') = \\frac{1}{N\\rho} \\sum_i \\frac{\\partial \\overline{\\langle\nn_i(t) \\rangle}}{\\partial \\epsilon(t')}\\ .\n\\label{response}\n\\end{equation}\nIn the large times limit ($t,t_w\\to\\infty$), $X(t,t_w)$ depends\non both times only through the correlation $C(t,t_w)$.\nThen integrating Eq.(\\ref{gfdt}) from $t_w$ to $t$ we obtain a useful\nrelation linking the correlation and the compressibility in the out of\nequilibrium regime\n\\begin{equation}\nT \\kappa(t,t_w) = \\int_{C(t,t_w)}^1 X(C) {\\mathrm d}C\\ .\n\\label{main}\n\\end{equation}\nThis is the key relation used in order to extract the FDR.\n\nIn our case the perturbing term in the Hamiltonian, shown in\nEq.(\\ref{pert_ham}), gives to the integrated response the following form\n\\begin{equation}\n\\kappa(t,t_w) = \\frac{1}{N\\rho} \\left( \\sum_i \\overline{[\\langle \\mu_i\nn_i(t) \\rangle]_{\\rm av}} - \\sum_i \\overline{[\\langle \\mu_i n_i(t_w)\n\\rangle]_{\\rm av}} \\right)\\ ,\n\\label{compre2}\n\\end{equation}\nwhere $[\\ \\cdot\\ ]_{\\rm av}$ is the average over the random $\\mu_i$\nrealizations~\\cite{note2}. The second term can be ignored because the\n$\\mu_i$ are random and completely uncorrelated from the configuration\nat time $t_w$.\n\nPerforming a parametric plot of the compressibility (or the integrated\nresponse) versus the correlation is a useful way of getting\ninformation about the different dynamical regimes present in the model\nand in particular the time scales structure of the system. In fact\nfrom Eq.(\\ref{main}) it is easy to see that, plotting $T\n\\kappa(t,t_w)$ vs. $C(t,t_w)$, the FDR can be simply obtained as minus\nthe derivative of the curve, i.e.\n\\begin{equation}\nX(C') = -\\left.{\\partial[T\\kappa(t,t_w)]}\\over{\\partial C(t,t_w)}\n\\right|_{C(t,t_w)=C'}\\ .\n\\end{equation}\nThere is already a considerable literature on this kind of analysis in\nsystems with and without quench disorder~\\cite{fdt_varie}.\nParticularly relevant to the present discussion are\nreferences~\\cite{fdt_vetri}, where a constant FDR was found\nin model glasses with interactions of the Lennard-Jones type and in a\npurely kinetic lattice gas. The FILG has the advantage of being a\nHamiltonian lattice model with short range interactions and, in this\nsense, it is more realistic than purely kinetic models and more\naccessible analytically and computationally than Lennard-Jones\nsystems. Moreover, it is a valid on-lattice model for structural\nglasses and it may be simple enough to apply statistical mechanics\ntechniques. \n\nWe have simulated the FILG in 3D for linear sizes $L=30$ and\n$60$. Fixing the coupling constant $J=1$ and a temperature $T=0.1$\nthe system presents a glass transition around $\\mu=0.5$~\\cite{mario}. \n\nIn all our numerical experiments we have prepared the system in an\ninitial state with low density, characteristic of the liquid phase and\nthen, at time zero, we have performed a sudden quench in $\\mu$ to a\nvalue deep in the glass phase (the data presented here refer to\n$\\mu=1$). As already explained above, after a time $t_w$ a\nperturbation in the form of a random small chemical potential has been\napplied and the density-density autocorrelation, Eq.(\\ref{corr}), and\nthe corresponding integrated response, Eq.(\\ref{compre2}), have been\nrecorded. At time $t_w$ the density has been fixed to the value\n$\\rho=\\rho(t_w)$ for the following reason: the perturbing term in the\nHamiltonian [see Eq.(\\ref{pert_ham})] favors roughly half of the sites\n(those with $\\mu_i=1$) and the remaining half to be empty. When the\nperturbation is switched on, the first half starts to be more filled\nthan the second one and this increases the integrated response, as it\nshould. On very long times, however, because the density reached at\ntime $t_w$ is still a bit below the asymptotic value, the number of\nparticles continues growing and new particles are added with higher\nprobability on the sites with $\\mu_i=-1$, which are more empty. Then\nthe response becomes negative because of systematic errors. We avoid\nthe negative responses fixing the density at time $t_w$. In the\nlimit $t_w\\!\\to\\!\\infty$ we recover the right behavior in any case,\nhowever with our choice the extrapolation is safer.\n\nIn all our simulations we have verified that the strength of the\nperturbation $\\epsilon=0.02$ is small enough in order to be in the\nlinear response regime. We have also checked that the system\nthermalizes at temperatures $T \\ge 0.3$ (being always $\\mu=1$) and\nsatisfies FDT. Further details will be given in~\\cite{nos3}.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=plot2.eps,width=8cm}}\n\\caption{The plot of the integrated response times the temperature\nversus the correlation gives clear evidence for the existence of two\nwell separated time scales, together with a constant FDR in both\nregimes. The dashed line is $T\\kappa=1-C$. The errors are of the order\nof the symbols and have been estimated from sample to sample\nfluctuations.}\n\\label{fig1}\n\\end{figure} \n\nIn Fig.~\\ref{fig1} we show the main result of this letter: a\nparametric plot of the integrated response versus the density\nautocorrelation for different waiting times. The behavior of the\ncurves is exactly the one predicted by MF theories for a glass former.\nTwo distinct regimes can be perfectly recognized. In the FDT or\nquasi-equilibrium regime, $t-t_w \\ll t_w$ ($\\beta$-relaxation), the\npoints lie on the straight line given by\n\\begin{equation}\nT \\kappa(t,t_w) = 1 - C(t,t_w)\\ .\n\\end{equation}\nThis first time regime corresponds then to a FDR $X(t,t_w)=1$\nindependent of $t$ and $t_w$. In this regime the system is in\nquasi-equilibrium, with the particles moving inside the cages formed\nby nearly frozen neighbors and the temperature measured from\nparticles fluctuations is that of the heat bath. When $t-t_w \\ge t_w$\nthe system falls out of equilibrium, entering the aging regime, and\nthe data in Fig.~\\ref{fig1} depart from the FDT line. From the figure\nit is clear that the FDR is still a constant, but now $0 < X(t,t_w) <\n1$. The constancy of the FDR in the out of equilibrium regime is one\nof the central predictions of MF approaches. Here we see that\nthis is still valid for a finite dimensional system.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=plot5.eps,width=8cm}}\n\\caption{Same as Fig.~\\ref{fig1}, now with different volumes in order\nto show the absence of finite size effects. The linear fit to the data\nis very good and gives a value for the FDR $X \\simeq 0.65$. For the EA\nmodel (inset) the linear fit is far from good.}\n\\label{fig2}\n\\end{figure} \n\nIn Fig.~\\ref{fig2} we compare the results for the FDR measured on two\nlarge systems, whose sizes are $N=30^3$ and $N=60^3$. Both curves\ncorrespond to $t_w=10^4$ and no finite size effects are evident.\nFitting the data in the out of equilibrium regime to the straight line\n\\begin{equation}\nT \\kappa(t,t_w) = X(t_w) [\\qea(t_w)-C(t,t_w)] + [1-\\qea(t_w)]\\ ,\n\\label{fit}\n\\end{equation}\nallows us to compute the $t_w$-dependent FDR $X(t_w)$ and the\nEdwards-Anderson order parameter $\\qea(t_w)$. In the large times\nlimit they should converge to the corresponding equilibrium \nvalues~\\cite{fmpp,mpv}. \n\n%$x$\n%and \\qea~\\cite{fmpp}, giving information on the pure states\n%structure~\\cite{mpv}: $1-\\qea$ measures the pure state width or\n%equivalently the size of connected regions in configurational space\n%and $x$ is the probability of two configurations being in the same\n%state.\n\nWe report the linear fit in Fig.~\\ref{fig2} in order to show how well\nthe data can be fitted with the formula in Eq.(\\ref{fit}). As one can\nsee from Fig.~\\ref{fig1}, the slope $X(t_w)$ changes very little with\n$t_w$ and it takes the same value (within the error) for the two\nlargest waiting times. The results of our fits give $X=0.64(3)$ and\n$\\qea=0.92(1)$. In comparison with other works the correlation range\nwe are exploring may seem quite small. However it should be kept in\nmind that we are using non-connected correlations functions which tend\nin the large times limit to $\\rho^2 \\simeq 0.44$. So we are actually\nspanning half of the allowed range.\n\nIn the inset of Fig.~\\ref{fig2} we present the same kind of data\n($t_w\\!=\\!10^5,10^6$) for the Edwards-Anderson (EA) model, which is\nexpected to have more than two time scales. It is clear that in the EA\nmodel the slope changes along the curve and a straight line is not\nable to fit the whole set of data. Moreover higher temperatures\ndata~\\cite{fdt_varie} suggest that the slope still have to decrease\n(in modulus) for smaller correlations, making the linear fit even\npoorer.\n\nAn important difference between the present model and MF approaches\nshould be evident: activated processes present in finite-dimensional\nsystems should dominate the asymptotic dynamics and as $t_w\\to\\infty$\nequilibrium dynamics should be restored (a similar behavior can be\nobserved in MF models by considering small systems~\\cite{felix}).\nHowever the time a macroscopic system needs to reach such an\nequilibrium (thermalization time, $t_{\\rm eq}$) increases very rapidly\nwith the system size and then the use of large sizes (here we have\n$L=30$ and $60$) prevents the system from reaching equilibrium in\naccessible time scales. In other words, we expect that the FDR\nexplicitly depends on $t_w$ and it tends to 1 in the limit\n$t_w\\to\\infty$. However in the range $1 \\ll t_w \\ll t_{\\rm eq}$\n(where we actually are) the FDR should relax into some very long\nplateau. It is very remarkable that finite times effects are very well\ndescribed by MF theories.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=plot6.eps,width=8cm}}\n\\caption{Same as Fig.~\\ref{fig1} for different temperatures in the\nglassy phase.}\n\\label{fig3}\n\\end{figure}\n\nIn Fig.~\\ref{fig3} we show the usual integrated response versus\ncorrelation plot for different temperatures, being the chemical\npotential always equal to $\\mu=1$. We estimated, both from the\ndensity measurements and from the FDR, the glassy transition to be\nlocated around $T_g \\simeq 0.2$. So all the data refer to the glassy\nphase. The main result to be noted is the good parallelism between all\nthe curves in the aging regime. They can be perfectly fitted with the\nsame value for $X$ and different values for \\qea, that increases\nlowering the temperature. As $T$ approaches $T_g$ from below, \nFDT is recovered. Also, as\n$\\qea(T)$ decreases, $X$ remains constant within the numerical precision,\nwhat is in contrast with MF models where the FDR is\nusually proportional to the temperature ($X \\propto T$). The\nexact behavior near the transition, that is, whether\n$X$ (and $\\qea$) are continuous or not~\\cite{mf}, \nis difficult to establish numerically and will be addressed in a future work.\n%Maybe the\n%model has a dynamical transition at $T_g$ and, for any $T < T_g$, we\n%are actually measuring the properties of the threshold states.\n\nIn summary, we have found that the glassy phase of a realistic model\nof structural glass presents two well separated time scales, as found\nin MF models of spin glasses. The out of equilibrium long time\ndynamics can be characterized by a constant value of the fluctuation\ndissipation ratio $X$ which is, with a very good approximation,\nindependent of temperature in the glass phase. This last observation\ndoes not agree with MF predictions. Being a model in 3D, the\nfrustrated lattice gas is an interesting test ground for performing a\nsystematic study of activated processes, a main ingredient absent in\nMF models.\n\nThis work was partly supported by Brazilian agencies CNPq and FAPEMIG. JJA\nacknowledges the Abdus Salam ICTP (Trieste) for support during his stay, \nwhere part of this work was done.\n\n\\begin{thebibliography}{20}\n\n\\bibitem{young} {\\it Spin Glasses and Random Fields}, Ed. A.P. Young,\n(World Scientific, Singapore, 1997).\n\n\\bibitem{gotze} W. G{\\\"o}tze and L. Sj{\\\"o}gren, Rep. Prog. Phys. {\\bf\n55}, 241 1992. W. G{\\\"o}tze in {\\it Liquids, Freezing and the Glass\nTransition}, Les Houches 1989, Eds. J.P. Hansen, D. Levesque and\nJ. Zinn-Justin (North Holland, Amsterdam, 1991).\n\n\\bibitem{cuku} L.F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. {\\bf\n71}, 173 (1993); J. Phys. A {\\bf 27}, 5749 (1994); Phil. Mag. {\\bf\n71}, 501 (1995).\n\n\\bibitem{luca} L.F. Cugliandolo, J. Kurchan and L. Peliti, Phys.\nRev. E {\\bf 55}, 3898 (1997).\n\n\\bibitem{grigera}T.S. Grigera and N.E. Israeloff,\nPhys. Rev. Lett. {\\bf 83}, 5038 (1999).\n\n\\bibitem{mario} M. Nicodemi and A. Coniglio, J. Phys. A {\\bf 30}, L187\n(1997); Phys. Rev. E {\\bf 57}, R39 (1998).\n\n\\bibitem{nos} D.A. Stariolo and J.J. Arenzon, Phys. Rev. E {\\bf 59},\nR4762 (1999).\n\n\\bibitem{nos3} J.J. Arenzon, F. Ricci-Tersenghi and D.A. Stariolo, in\npreparation.\n\n\\bibitem{note1} For simplicity we do not make explicit the parametric\ndependence of $\\epsilon(t)$ on $t_w$.\n\n\\bibitem{note2} All the observables we measure are self-averaging\nquantities and so every average can be neglected provided one uses\nvery large systems.\n\n\\bibitem{fdt_varie} S. Franz and H. Rieger, J. Stat. Phys. {\\bf\n79}, 749 (1995). E. Marinari, G. Parisi, F. Ricci-Tersenghi and\nJ.J. Ruiz-Lorenzo, J. Phys. A {\\bf 31}, 2611 (1998). A. Barrat,\nPhys. Rev. E {\\bf 57}, 3629 (1998). G. Parisi, F. Ricci-Tersenghi and\nJ.J. Ruiz-Lorenzo, Eur. Phys. J. B {\\bf 11}, 317 (1999).\n\n\\bibitem{fdt_vetri}G. Parisi, Phys. Rev. Lett. {\\bf 79}, 3660 (1997).\nJ.L. Barrat and W. Kob, Europhys. Lett. {\\bf 46}, 637 (1999).\nM. Sellitto, Eur. Phys. J. B {\\bf 4}, 135 (1998). R. Di Leonardo,\nL. Angelani, G. Parisi and G. Ruocco, {\\tt cond-mat/0001311}.\n\n\\bibitem{fmpp} S. Franz, M. M\\'ezard, G. Parisi and L. Peliti,\nPhys. Rev. Lett. {\\bf 81}, 1758 (1998).\n\n\\bibitem{mpv} M. M\\'ezard, G. Parisi and M.A. Virasoro, {\\it Spin\nGlass Theory and Beyond}, World Scientific (Singapore, 1987).\n\n\\bibitem{felix} A. Crisanti and F. Ritort, {\\tt cond-mat/9911226}.\n\n\\bibitem{mf} J.J. Arenzon, M. Nicodemi, and M. Sellitto, J. de Physique\n {\\bf 6}, 1143 (1996); M. Sellitto, M. Nicodemi, and J.J. Arenzon\n J. de Physique {\\bf 7}, 45 (1997).\n\\end{thebibliography}\n\n\\end{multicols}\n\\end{document}\n"
}
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[
{
"name": "cond-mat0002074.extracted_bib",
"string": "\\begin{thebibliography}{20}\n\n\\bibitem{young} {\\it Spin Glasses and Random Fields}, Ed. A.P. Young,\n(World Scientific, Singapore, 1997).\n\n\\bibitem{gotze} W. G{\\\"o}tze and L. Sj{\\\"o}gren, Rep. Prog. Phys. {\\bf\n55}, 241 1992. W. G{\\\"o}tze in {\\it Liquids, Freezing and the Glass\nTransition}, Les Houches 1989, Eds. J.P. Hansen, D. Levesque and\nJ. Zinn-Justin (North Holland, Amsterdam, 1991).\n\n\\bibitem{cuku} L.F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. {\\bf\n71}, 173 (1993); J. Phys. A {\\bf 27}, 5749 (1994); Phil. Mag. {\\bf\n71}, 501 (1995).\n\n\\bibitem{luca} L.F. Cugliandolo, J. Kurchan and L. Peliti, Phys.\nRev. E {\\bf 55}, 3898 (1997).\n\n\\bibitem{grigera}T.S. Grigera and N.E. Israeloff,\nPhys. Rev. Lett. {\\bf 83}, 5038 (1999).\n\n\\bibitem{mario} M. Nicodemi and A. Coniglio, J. Phys. A {\\bf 30}, L187\n(1997); Phys. Rev. E {\\bf 57}, R39 (1998).\n\n\\bibitem{nos} D.A. Stariolo and J.J. Arenzon, Phys. Rev. E {\\bf 59},\nR4762 (1999).\n\n\\bibitem{nos3} J.J. Arenzon, F. Ricci-Tersenghi and D.A. Stariolo, in\npreparation.\n\n\\bibitem{note1} For simplicity we do not make explicit the parametric\ndependence of $\\epsilon(t)$ on $t_w$.\n\n\\bibitem{note2} All the observables we measure are self-averaging\nquantities and so every average can be neglected provided one uses\nvery large systems.\n\n\\bibitem{fdt_varie} S. Franz and H. Rieger, J. Stat. Phys. {\\bf\n79}, 749 (1995). E. Marinari, G. Parisi, F. Ricci-Tersenghi and\nJ.J. Ruiz-Lorenzo, J. Phys. A {\\bf 31}, 2611 (1998). A. Barrat,\nPhys. Rev. E {\\bf 57}, 3629 (1998). G. Parisi, F. Ricci-Tersenghi and\nJ.J. Ruiz-Lorenzo, Eur. Phys. J. B {\\bf 11}, 317 (1999).\n\n\\bibitem{fdt_vetri}G. Parisi, Phys. Rev. Lett. {\\bf 79}, 3660 (1997).\nJ.L. Barrat and W. Kob, Europhys. Lett. {\\bf 46}, 637 (1999).\nM. Sellitto, Eur. Phys. J. B {\\bf 4}, 135 (1998). R. Di Leonardo,\nL. Angelani, G. Parisi and G. Ruocco, {\\tt cond-mat/0001311}.\n\n\\bibitem{fmpp} S. Franz, M. M\\'ezard, G. Parisi and L. Peliti,\nPhys. Rev. Lett. {\\bf 81}, 1758 (1998).\n\n\\bibitem{mpv} M. M\\'ezard, G. Parisi and M.A. Virasoro, {\\it Spin\nGlass Theory and Beyond}, World Scientific (Singapore, 1987).\n\n\\bibitem{felix} A. Crisanti and F. Ritort, {\\tt cond-mat/9911226}.\n\n\\bibitem{mf} J.J. Arenzon, M. Nicodemi, and M. Sellitto, J. de Physique\n {\\bf 6}, 1143 (1996); M. Sellitto, M. Nicodemi, and J.J. Arenzon\n J. de Physique {\\bf 7}, 45 (1997).\n\\end{thebibliography}"
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cond-mat0002075
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Finite-time singularity in the dynamics \\of the world population, economic and financial indices. \\ {\normalsize Running title: Finite-time singularity in world population growth}
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[
{
"author": "Anders Johansen$^1$ and Didier Sornette$^{1,2,3}$\\footnote{corresponding author}"
},
{
"author": "Los Angeles"
},
{
"author": "California 90095"
},
{
"author": "$^2$ Department of Earth and Space Science"
},
{
"author": "$^3$ Laboratoire de Physique de la Mati\\`{e}re Condens\\'{e}e"
},
{
"author": "CNRS UMR6622 and Universit\\'{e} de Nice-Sophia Antipolis"
},
{
"author": "B.P. 71"
},
{
"author": "Parc Valrose"
},
{
"author": "06108 Nice Cedex 2"
},
{
"author": "France"
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Contrary to common belief, both the Earth's human population and its economic output have grown faster than exponential, i.e., in a super-Malthusian mode, for most of the known history. These growth rates are compatible with a spontaneous singularity occuring at the {same} critical time $2052 \pm 10$ signaling an abrupt transition to a new regime. The degree of abruptness can be infered from the fact that the maximum of the world population growth rate was reached in $1970$, i.e., about $80$ before the predicted singular time, corresponding to approximately $4\%$ of the studied time interval over which the acceleration is documented. This rounding-off of the finite-time singularity is probably due to a combination of well-known finite-size effects and friction and suggests that we have already entered the transition region to a new regime. In theoretical support, a multivariate analysis coupling population, capital, R\&D and technology shows that a dramatic acceleration in the population during most of the timespan can occur even though the isolated dynamics do not exhibit it. Possible scenarios for the cross-over and the new regime are discussed.
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"string": "%\\documentstyle[preprint,prl,aps,epsfig]{revtex}\n\\documentstyle[11pt,epsfig]{article}\n\\setlength{\\topmargin}{-1cm}\n%\\setlength{\\headsep}{1.6cm}\n\\setlength{\\evensidemargin}{-1cm}\n\\setlength{\\oddsidemargin}{-1cm}\n\\setlength{\\textheight}{24cm}\n\\setlength{\\textwidth}{18cm}\n\\newcommand \\be{\\begin{equation}}\n\\newcommand \\bea{\\begin{eqnarray}}\n\\newcommand \\ee{\\end{equation}}\n\\newcommand \\eea{\\end{eqnarray}}\n\\newcommand{\\lp}{\\left(}\n\\newcommand{\\rp}{\\right)}\n\n\n\\begin{document}\n\n\\title{Finite-time singularity in the dynamics \\\\of the world population,\neconomic and financial indices. \\\\ \n{\\normalsize Running title: Finite-time singularity in world population growth}}\n\n\\author{Anders Johansen$^1$ and Didier Sornette$^{1,2,3}$\\footnote{corresponding\nauthor}\\\\\n$^1$ Institute of Geophysics and\nPlanetary Physics\\\\ University of California, Los Angeles, California 90095\\\\\n$^2$ Department of Earth and Space Science\\\\\nUniversity of California, Los Angeles, California 90095\\\\\n$^3$ Laboratoire de Physique de la Mati\\`{e}re Condens\\'{e}e\\\\ CNRS UMR6622 and\nUniversit\\'{e} de Nice-Sophia Antipolis\\\\ B.P. 71, Parc\nValrose, 06108 Nice Cedex 2, France \\\\\ne-mails: anders@moho.ess.ucla.edu and sornette@moho.ess.ucla.edu}\n\n\n\n\\date{\\today}\n\\maketitle\n\n\n\n\\abstract{Contrary to common belief, both the Earth's human population and its \neconomic output have grown faster than exponential, i.e., in a\nsuper-Malthusian mode, for most of the known\nhistory. These growth rates are compatible with a spontaneous singularity \noccuring at the {\\it same} critical time $2052 \\pm 10$ signaling an abrupt \ntransition to a new regime. The degree of abruptness can be infered from\nthe fact that the maximum of the world population growth rate was reached\nin $1970$, i.e., about $80$ before the predicted singular time, \ncorresponding to approximately $4\\%$ of the studied time interval over which\nthe acceleration is documented. This rounding-off of the finite-time \nsingularity is probably due to a combination of well-known finite-size \neffects and friction and suggests that we have already entered the \ntransition region to a new regime. \nIn theoretical support, a multivariate analysis coupling \npopulation, capital, R\\&D and technology shows that a dramatic acceleration in the \npopulation during most of the timespan\ncan occur even though the isolated dynamics do not exhibit it. \nPossible scenarios for the cross-over and the new regime are \ndiscussed.}\n\n\\vskip 1cm\nPhysica A 294 (3-4), 465-502 (15 May 2001)\n\n%\\thispagestyle{empty}\n\n\\newpage\n\n\\setcounter{page}{1}\n\n\\section{Introduction}\n\nBoth the world economy as well as the human population have grown at a \ntremendous \npace especially during the last two centuries. It is estimated that 2000 \nyears ago the population of the world was approximately 300 million and for \na long time the world population did not grow significantly, since periods \nof growth were followed by periods of decline. It took more than 1600 years \nfor the world population to double to 600 million and since then the growth \nhas accelerated. It reached 1 billion in 1804 (204 years later), 2 billion in \n1927 (123 years later), 3 billion in 1960 (33 years later), 4 billion in 1974 \n(14 years later), 5 billion in 1987 (13 years later) and 6 billion in 1999\n(12 years later). This rapidly accelerating growth has raised sincere worries \nabout its sustainability as well as concerns that we humans as a result might \ncause severe and {\\it irreversible} damage to eco-systems, global weather \nsystems etc \\cite{Cohenscience,Hern}. At, what one may say the other \nextreme, the optimists expect that the innovative spirit of mankind will be \nable to solve the problems associated with a continuing increase in the \ngrowth rate \\cite{vonFoerster,Simon}. Specifically,\nthey believe that the world economic development will continue as a successive\nunfolding of revolutions, {\\it e.g.}, the Internet, bio-technological and \nother yet\nunknown innovations, replacing the prior agricultural, industrial, medical\nand information\nrevolutions of the past. Irrespective of the interpretation, the important\npoint is the\npresence\nof an {\\it acceleration} in the {\\it growth rate}. Here, it is first shown \nthat,\ncontrary to common belief, both the Earth human population as well as its\neconomic output have grown faster that exponential for most of the known \nhistory and most strikingly so in the last centuries. Furthermore, we will \nshow that both the population growth rate and the economic growth rate\nare consistent with a spontaneous singularity at the {\\it same} critical\ntime $2052 \\pm 10$ and with the same characteristic self-similar geometric\npatterns (defined below as log-periodic oscillations). Multivariate dynamical\nequations coupling population, capital and R\\&D and technology can indeed\nproduce such an ``explosion'' in the population even though the isolated \ndynamics do not. In particular, this interplay provides an explanation of \nour finding of the same value of the critical time $t_c (\\approx 2052 \\pm 10)$\nboth for the population and the economic indices. As a consequence,\neven the optimistic view has to be revised, since the acceleration of the\ngrowth rate\ncontains endogenously its own limit in the shape of a finite-time\nsingularity to be\ninterpreted as a transition to a qualitatively new behaviour. Close to the\nmathematical\nsingularity, finite-size effects will smoothen the transition and it is \nquite possible that Mankind may already have entered this transition phase. \nPossible scenarios for the cross-over\nand the new regime are discussed.\n\n\n\n\\subsection{The logistic equation and finite-time singularities}\n\nAs a standard model of population growth, Malthus' model assumes that the size\nof a population increases by a fixed proportion $\\tau$ over a given period of\ntime independently of the size of the population and thus gives an exponential\ngrowth. The logistic equation attempts to correct for the resulting unbounded\nexponential growth by assuming a finite carrying capacity $K$ such that the\npopulation instead evolves according to\n\\be\n{d p \\over dt} = r p(t) \\left[K - p(t)\\right]~. \\label{ajdak}\n\\ee\nCohen and others (see \\cite{Cohenscience} and references therein) have put\nforward idealised models taking into account interaction between the human\npopulation $p(t)$ and the corresponding carrying capacity $K(t)$ by assuming\nthat $K(t)$ increases with $p(t)$ due to technological progress such as the\nuse of tools and fire, the development of agriculture, the use of fossil fuels,\nfertilisers {\\it etc.} as well an expansion into new habitats and the removal\nof limiting factors by the development of vaccines, pesticides, antibiotics,\n{\\it etc.} If $K(t) > p(t)$, then $p(t)$\nexplodes to infinity after a finite time creating a singularity. In this case,\nthe limiting factor $-p(t)$ can be dropped out and, assuming a simple power\nlaw relationship $K \\propto p^{\\delta}$ with $\\delta >1$, (\\ref{ajdak})\nbecomes\n\\be\n{d p \\over dt} = r [p(t)]^{1+\\delta}~, \\label{aafajdak}\n\\ee\nwhere the growth rate accelerates with time according to\n$r [p(t)]^{\\delta}$.\nThe generic consequence of a power law acceleration in the growth rate is the\nappearance of singularities in finite time:\n\\be \\label{pow}\np(t) \\propto (t_c - t)^z, ~~{\\rm with}~z=-{1 \\over \\delta}~~ \\mbox{ and $t$\nclose to $t_c$}.\n\\ee\nEquation (\\ref{aafajdak}) is said to have a ``spontaneous'' or ``movable''\nsingularity at the critical time $t_c$ \\cite{benderorszag}, the critical time\n$t_c$ being determined by the constant of integration, {\\it i.e.}, the initial\ncondition $p(t=0)$. One can get an intuitive understanding of such\nsingularities by looking at the function $p(t) = \\exp \\left( t p(t) \\right)$\nwhich corresponds to\nreplacing $\\tau K$ by $p$ in Malthus' exponential solution\n$p(t) = p(0) \\exp [\\tau K t]$. $p$ is then the solution of\n$dp/dt = p^2/(1 - t p)$ \\cite{benderorszag}\nleading to an ever increasing growth with the explicit solution\n\\be\np(t) = e \\left( 1 - C \\sqrt{t_c - t}\\right)~,\n\\ee\nwhere $t_c=1/e=0.368$, $C$ is a numerical factor and the exponent $z=1/2$. In\nthis case, the finite time spontaneous singularity does not lead to a\ndivergence of the population at the critical time $t_c$; only the growth\nrate diverges at $(t_c-t)^{-1/2}$. Spontaneous singularities in ODE's and\nPDE's are quite common and have been found in many well-established models\nof natural systems either at special points in space such as in the Euler\nequations of inviscid fluids \\cite{Pumiersiggia} or in the equations of\nGeneral Relativity coupled to a mass field leading to the formation of\nblack holes \\cite{Choptuik}, in models of micro-organisms aggregating to form\nfruiting bodies \\cite{Rascle}, or to the more prosaic rotating coin (Euler's\ndisk) \\cite{Moffatt}, see \\cite{reviewsor} for a review. Some of the\nmost prominent, as well as more controversial, examples due to their impact on\nhuman society are models of rupture and material failure \\cite{failure,faicri},\nearthquakes \\cite{earthquake} and stock market crashes \\cite{crash,nasdaq}.\n\n\\subsection{Data sets and methodology}\n\nHere, we examine several data sets expressing the\ndevelopment of mankind on Earth\nin term of size and economic impact,\nto test the hypothesis that our history might be compatible with a\nfuture finite-time singularity. These data sets are as follows.\n\\begin{itemize}\n\\item The human population data from 0 to 1998 was retrieved from the\nweb-site of The United Nations Population Division\nDepartment of Economic and Social Affairs (http://www.popin.org/pop1998/).\n\n\\item The GDP of the world from 0 to 1998, estimated by J. Bradford DeLong\nat the Department of Economics, U.C. Berkeley \\cite{Bradford}, was\ngiven to us by R. Hanson \\cite{Hanson}.\n\n\\item The financial data series include the Dow Jones index from 1790\n\\cite{UShistory} to 2000, the Standard \\& Poor (S\\&P) index from 1871 to 2000,\nas well as a number of regional and global indices since 1920. The Dow Jones\nindex was constructed by The Foundation for the Study of Cycles \\cite{cycles}.\nIt is the Dow Jones index back to 1896, which has been extrapolated back to\n1790 and further. The other indices are from Global Financial Data\n\\cite{global}.\nThese indices are constructed as follows. For the S\\&P, the data from 1871\nto 1918\nare from the Cowles commission, which back-calculated the data using the\n{\\it Commercial and Financial Chronicle}. From 1918, the data is the Standard\nand Poor's Composite index (S\\&P) of stocks. The other indices uses Global\nFinancial Data's indices from 1919 through 1969 and Morgan Stanley Capital\nInternational's indices from 1970 through 2000. The EAFE Index includes\nEurope,\nAustralia and the Far East. The Latin America Index includes Argentina,\nBrazil,\nChile, Colombia, Mexico, Peru and Venezuela.\n\\end{itemize}\n\nDemographers usually construct population projections in \na disaggregated manner, filtering the data by age, stage of development, region, etc.\nDisaggregating and controlling for such variables are thought to be crucial for\ndemographic development and for any reliable population prediction.\nHere, we propose a different strategy based on aggregated data, which is\njustified by the following concept: in order to get a meaningful prediction\nat an aggregate level, it is often more relevant to study aggregate\nvariables than ``local'' variables that can miss the whole picture in \nfavor of special idiosyncrasies.\n To take an example from material sciences, the prediction of the\nfailure of heterogeneous materials subjected to stress can be performed\naccording to two methodologies. Material scientists often analyse in \nexquisite details the wave forms of the acoustic emissions or other\nsignatures of damage resulting from\nmicro-cracking within the material. However, this is of very little help\nto predict the overall failure which is often a cooperative global\nphenomenon \\cite{Herroux} resulting from \nthe interactions and interplay between the \nmany different micro-cracks nucleating, growing and fusing within the \nmaterials. In this example, it has been shown indeed that aggregating\nall the acoustic emissions in a single aggregated variable is much better\nfor prediction purpose \\cite{faicri}.\n\n\n\n\\subsection{Content of the paper}\n\nIn the next section, we first show that the exponential model is utterly\ninadequate in\ndescribing\nthe population growth as well as the growth in the World GDP and the global and\nregional financial indices. We then present the alternative model\nconsisting of a power law\ngrowth ending at a critical time $t_c$. We first give a non-parametric approach\ncomplemented by a fitting procedure. Section 3 proposes a first generalization\nof power laws with complex exponents, leading to so-called log-periodic\noscillations\ndecorating the overall power law acceleration. The fitting procedure is\ndescribed\nas well as a non-parametric test of the existence of the log-periodic\npatterns for\nthe world population. Section 4 presents a second-order generalization of\nthe power law\nmodel, which allows for a frequency modulation in the log-periodic structure.\nThis extended formula is used to fit the extended Dow Jones Industrial average.\nSection 5 summarizes what has been achieved and compares our results with\nprevious work.\nIn particular, we give the explicit solutions of multivariate dynamical\nequations for\nseveral coupled variables, such as population, technology and capital, to\nshow that\nthe same finite-time singularity can emerge from the interplay of these\nfactors while\neach of them individually is not enough to create the singularity. Section\n6 concludes by\ndiscussing a set of scenarios for mankind close to and beyond the critical\ntime.\n\n\n\\section{Singular Growth Rate}\n\n\\subsection{Tests of exponential growth}\n\n\\subsubsection{Human population and world GDP}\n\nA faster than exponential growth is clearly observed in the human population\ndata from year 0 up to 1970, at which the estimated annual rate of\nincrease of\nthe global population reached its (preliminary?) all-time peak of $2.1\\%$.\nFigure \\ref{semilogpop} shows the logarithm of the estimated world population\nas a function of (linear) time, such that an exponential growth rate would be\nqualified by a linear increase. In contrast, one clearly observes a strong\nupward\ncurvature characterising a ``super-exponential'' behaviour.\nA faster than exponential growth is also clearly observed\nin the estimated GDP (Gross Domestic Product) of the World,\nshown in figure \\ref{semilogwgdp} for the year 0 up to 2000.\n\n\\subsubsection{Financial indices}\n\nOver a shorter time period, a faster than exponential growth is also\nobserved in figures \\ref{semilogdj} to\n\\ref{semilogwindex} for a number of\neconomic indicators such as the Dow Jones Average since the establishment\nof the\nU.S.A. in 1790 \\cite{UShistory}, the S\\&P since 1871, as well for a number\nof regional and global indices since 1920, including the Latin American\nindex, the\nEuropean index, the EAFE index and the World index. In all these figures,\nthe logarithm of the index is plotted\nas a function of (linear) time, such that an exponential growth rate would be\nqualified by a linear increase. In all cases, one clearly observes in contrast\na significant upward\ncurvature characterising a ``super-exponential'' behaviour.\n\n\n\\subsection{A first test of power law growth}\n\n\\subsubsection{Procedure}\n\nAs shown in the derivation of equation (\\ref{aafajdak}), it is enough that\nthe growth rate\nincreases with any arbitrarily small {\\it positive} power of $p(t)$ for\na finite-time singularity to develop with the characteristic power law\ndependence\n(\\ref{pow}). Can such a behaviour explain the super-exponential behaviour\ndocumented in the figures \\ref{semilogpop}-\\ref{semilogwindex}?\n\nThe small number of data points in these time series\nand the presence of large fluctuations prevent the use of a direct fitting\nprocedure with\n(\\ref{pow}). Indeed, such a fit, which typically attempts to minimise the\nroot-mean-square (r.m.s.)\ndifference\nbetween the theoretical formula and the data, is highly degenerate: many\nsolutions are found which differ by variations of at most a few percent of\nthe root-mean-square (r.m.s.) of the errors. Such\ndifferences in r.m.s. are not significant, especially considering the\nstrongly non-Gaussian nature of the fluctuations in these data sets. Maximum\nlikelihood methods\nare similarly limited. To address this problem of degeneracy, we turn to a\nnon-parametric approach,\nconsisting in fixing $t_c$ and plotting the logarithm of the data as a\nfunction of\n$\\log (t_c -t)$. In such a plot, a linear behaviour qualifies the power law\n(\\ref{pow}),\nand the slope gives the exponent $z$ which then can be\ndetermined visually or, better, by a fit but now with $t_c$ fixed. This\nprocedure\nis not plagued by the previously discussed degeneracy and provides reliable\nand unique results.\n\n\\subsubsection{World population}\n\nIn figures \\ref{pop2030}-\\ref{pop2050},\nthe world population in logarithm scale is shown as a function of $t_c - t$\nalso in\nlogarithmic scale for three choices 2030, 2040 and 2050, respectively for\n$t_c$.\nEven though the fits\nwith equation (\\ref{pow}) for three cases varies in quality, they all\ncapture the acceleration\nin the second half of the data on a logarithmic scale. The curvature seen\nin the data far from $t_c$ can be modeled by including a constant\nterm in equation (\\protect\\ref{pow}) embodying for instance the effect of\nan initial\ncondition, as we discuss below. Changing $t_c$ from 2030 to 2050\nhas two competing effects observed in figures \\ref{pop2030}-\\ref{pop2050}:\na larger value\nof $t_c$ provides a better fit in the latter time period while\ndeteriorating somewhat the\nfit to the data in the early time periods.\n\n\\subsubsection{World GDP}\n\nAs discussed in the introduction, the human population is strongly coupled\nwith its outputs and with the Earth's carrying capacity, and can partly be\nmeasured by its\neconomic production. Hence, we should expect a close relationship between\nthe size of the human population and its GDP.\nFigures \\ref{wgdp2040}-\\ref{wgdp2060} show the logarithm\nof the estimated World GDP as a function of\n $t_c - t$, both in log-log coordinates, where $t_c$ has been chosen to\n2040, 2050\nand 2060, respectively. The equation (\\ref{pow}) is again parameterising\nthe data\nquite satisfactorily.\n\nWe stress that we use the logarithm of the World\nGDP as well as the logarithm of the national, regional or global indices\npresented below\nas the ``bare'' data on which we test the power law hypothesis. This means\nthat we plot\nthe logarithm of the GDP or of the indices in logarithmic scale,\nwhich effectively amounts to taking\nthe logarithm of the logarithm of the GDP as a function of $t_c-t$,\nitself also\n in logarithmic scale in order to test for the power law (\\ref{pow}).\nThis is done in an attempt to minimise the effect of inflation and other\nsystematic drifts,\nand in accordance with standard economic practice that only\nrelative changes should be considered. Removing an average inflation of 4\\%\ndoes not change\nthe results qualitatively but the corresponding results are not\nquantitatively reliable\nas the inflation has varied\nsignificantly over US history with quantitative impacts that are difficult\nto estimate.\n\n\n\\subsubsection{Financial indices}\n\nFurther support for a singular power law behaviour of the economy\ncan be found by analysing in a similar way the national, regional or global\nindices shown in figures\n\\ref{semilogdj}-\\ref{semilogwindex}. The results are shown in figures\n\\ref{dj2040}-\\ref{windex2060}.\nEquation (\\ref{pow}) is again\nperfectly compatible with the data and much better than any exponential\nmodel.\nAs shown in Table \\ref{ztable}, the fits of all six indices are found to be\nconsistent with similar\nvalues for the exponent $z \\approx -1$, the absolute value of\nthe exponent increasing with $t_c$.\n\nThe results presented in this section on the world population, on the world\nGDP and on\nsix financial indices suggest that the power law (\\ref{pow}) is an adequate\nmodel. It is also\nparsimonious since the same simple mathematical expression, approximately\nthe same critical time\n$t_c$ and same exponent are found consistently for all time series.\nThese results confirm and extend the analysis presented forty years earlier\nfor the\nworld population only\n\\cite{Doomsdaypaper}, which concluded at a $t_c=2026$. The results shown in\nthe figures\n\\ref{pop2030}-\\ref{pop2050}, with the sensitivity analysis provided by\nvarying $t_c$ from 2030 to 2050, illustrate the large uncertainty in its\ndetermination.\nIt is thus worthwhile to attempt quantifying further the observed power law\ngrowth and\ntest how well $t_c$ is constrained.\n\n\\subsection{Quantitative fits to a power law}\n\nIn the derivation of (\\ref{pow}), a key assumption was to neglect the\nlimiting negative\nterm in (\\ref{ajdak}), which is warranted sufficiently close to $t_c$. Far\nfrom $t_c$, this\nanalysis and more general considerations lead us to expect the existence of\ncorrections to the pure\npower law (\\ref{pow}). Furthermore, it may be necessary to include\nhigher order terms as well as generalise the exponent as we will see in the\nnext\nsection.\n\nThe simplest extension of equation (\\ref{pow}) is\n\\be\n\\label{eq:solution}\np(t) = A + B (t_c-t)^{z}~.\n\\ee\nIn order to make a first quantitative estimate of the acceleration in the\ngrowth rate, determined\nby the exponent $z$ and the position $t_c$ of the singularity,\nwe now let $t_c$ be a free parameter. In figure \\ref{powlpfits},\nthe equation (\\ref{eq:solution}) is fitted to the world\npopulation from 0 to 1998. The parameter values of the fit are\n$A\\approx 0$, $B \\approx 22120$, $t_c \\approx 2078$ and $z \\approx -1.9$.\nThe negative value of the exponent is compatible with $A\\approx 0$. The\nnegative\nexponent $z \\approx -1.9$ obtained in the fit means that equation\n(\\ref{eq:solution})\nhas a singularity at $t=t_c$ corresponding to an infinite population. This is\nclearly impossible on a finite Earth. The point to be extracted from this\nanalysis\nis that the world population has until very recently grown\nat an accelerating growth rate in good agreement with a singular behaviour.\nSingularities are always mathematical idealisations of natural phenomena:\nthey are\nnot present in reality but foreshadow an important transition or change of\nregime.\nIn the present context, they must be interpreted as a kind of ``critical\npoint'' signaling a fundamental and abrupt change of regime similar to what\noccurs\nin phase transitions \\cite{critical}.\n\nAs already discussed in relation with equation (\\ref{ajdak}) and in the\nprevious section, the world population growth cannot\nbe separated from that of its evolving carrying\ncapacity. As a first attempt to quantify this variable in an independent way,\nwe analyse quantitatively the two largest data sets among all the\nfinancial indices and GDP: due to the large fluctuations of the financial\nindices compared to\nthe number of points, only the S\\&P and the Dow Jones\nAverage gave reliable results when $t_c$ is a free parameter.\nFigure \\ref{powlogdjsp} shows\nthe corresponding fits with equation (\\ref{eq:solution}). The parameter values\nof the best fits are $A\\approx-14$, $B\\approx 71$, $z\\approx-0.27$ and\n$t_c\\approx 2068$ for the Dow Jones Average and $A\\approx 0$, $B\\approx 1693$,\n$z\\approx-1.3$ and $t_c\\approx 2067$ for the S\\&P.\nThe fit with equation (\\ref{eq:solution}) exemplifies the acceleration of the\ngrowth rate, which is our main message. However, the location of the critical\npoint is still not very reliable when based on simple power fits\nof very noisy data \\cite{DombGreen}. This motivates us to extend this analyses\nin the following sections.\n\n\\section{Beyond a simple power law}\n\nThe results shown in the figures\n\\ref{pop2030}-\\ref{windex2060}, with the sensitivity analysis provided by\nvarying $t_c$ from 2030 or 2040 to 2050 or 2060, illustrate the large\nuncertainty in the determination of the critical time. The direct fit with\n(\\ref{eq:solution}) still gives a very large uncertainty. As can be seen\nfrom the figures, an important reason lies in the existence of large\nfluctuations around the average power law behaviour. In the next section,\nwe will see that this variability might be genuine and not simply noise.\nFurthermore, adding an extra degree of freedom will certainly improve\na parametrisation of the data.\n\n\n\\subsection{Generalisation to power laws with complex exponents:\nlog-periodicity}\n\nThe idea is to generalise the real exponent $z$ to a complex exponent\n$\\beta + i \\omega$, such that a power law is changed into\n$(t_c-t)^{\\beta + i \\omega}$, whose real part is\n$(t_c-t)^{\\beta} \\cos\\left(\\omega \\ln (t_c-t)\\right)$ \\cite{reviewsor}.\nThe cosine will decorate the average power law behaviour with so-called\nlog-periodic oscillations, the name steming from the fact the oscillations\nare periodic in $\\ln(t_c -t)$ and not in $t$. As we shall see,\nthese log-periodic\noscillations can account for a large part of the observed variability around\nthe power law. Thus, taking them into account provides a better parametrisation\nof the data and hence better constraints on the parameters of the power law\n$\\beta$ and $t_c$.\n\nThere are fundamental reasons for introducing log-periodic corrections.\nSingularities often exhibit genuine log-periodic corrections that result from\nspecific mechanisms \\cite{reviewsor}:\nsingularities in the Euler equations with complex exponents have been found\nto result from a cascade of Rayleigh-Taylor instabilities leading to\nlog-periodic oscillatory structures around singular vortices organised\naccording to discrete self-similar pancakes\\cite{Pumiersiggia}; in the\nprocess of formation of black holes, the matter field solution oscillates\nperiodically in the logarithm of the difference between time and time of\nthe formation of the singularity \\cite{Choptuik};\nthe phase separation kinetics of a binary\nmixture subjected to an uniform shear flow quenched from a disordered to a\nhomogeneous\nordered phase exhibits log-periodic oscillations due to a cyclical\nmechanism of stretching\nand break-up of domains, which allows to store and dissipate elastic energy\nin the system\n\\cite{Corberi2}; material failure occurs\nafter intermittent damage acceleration and quiescent phases that are\nwell-described by log-periodic structures decorating an overall power law\nsingularity \\cite{failure}; stock market crashes preceded by\nspeculative bubbles \\cite{crash,nasdaq} provide an highly relevant analogy to\nthe question of sustainability in the growth rate of the human population.\nMore generally, from the point of view of field theory as a tool-box for\nconstructing theories of complex systems, we should expect generically the\nexistence of complex exponents and their associated log-periodic\ncorrections \\cite{salsor}. We suggest that the\npresence of log-periodic oscillations deriving from general theoretical\nconsiderations can provide a first step to account for the ubiquitous\nobservation of cycles in\npopulation dynamics and in economics.\n\n\\subsection{Log-periodic fit of the World population}\n\n\\subsubsection{Results}\n\nGuided by the recent progress in the understanding of complex systems and\nthe possibility of complex exponents discussed in the previous section,\nwe have also fitted the world population data with the following equation\n\\be \\label{lppow}\np(t) \\approx A_1 + B_1 (t_c-t)^{\\beta}\n+C_1(t_c-t)^{\\beta}\\cos\\lp\\omega \\ln(t_c-t)+\\phi \\rp~,\n\\ee\nas shown in figure \\ref{powlpfits}. We obtained two solutions, the best\nhaving $A\\approx 0$,\n$B\\approx 1624$, $C\\approx -127$, $z\\approx -1.4$, $t_c \\approx 2056$, $\\omega\n\\approx 6.3$ and $\\phi \\approx 5.1$. The second solution has $A\\approx 0.25$,\n$B\\approx 1624$, $C\\approx -127$, $z \\approx -1.7$,\n$t_c \\approx 2079$, $\\omega \\approx 6.9$ and $\\phi \\approx -4.4$.\nIn this extension of equation (\\ref{eq:solution}), the cosine term embodies a\ndiscrete scale invariance \\cite{Dubrulle} decorating the overall\nacceleration with a geometrical scaling ratio $\\lambda = \\exp \\left( 2\\pi /\n\\omega \\right)$: the local maxima of the oscillations are converging to\n$t_c$ with the\ngeometrical ratio $1/\\lambda$.\n\n\\subsubsection{Sensitivity analysis}\n\nDue to the small number of points in the population data set, the\nrobustness of the fit with equation (\\ref{lppow}) was investigated with \nrespect to fluctuations in the important physical parameters $t_c$, $\\beta$ \nand $\\omega$ \\cite{comment}. The method we used was as follows. Together \nwith the data set (data set 1) obtained\nfrom the United Nations Population Division\nDepartment of Economic and Social Affairs (see the introduction section),\nwhich covers the period $\\left[ 0:1998\\right]$, seven other\ndata sets where analysed in an identical manner. These first three data\nsets were\ngenerated by removing the first point (data set 2), the two first points\n(data set 3) and the 3 first points (data set 4). Hence, those three data\nsets cover the periods $\\left[ 1000:1998\\right]$, $\\left[1250:1998\\right]$\nand $\\left[ 1500:1998\\right]$. A fifth data set (data set 5) was constructed by\nincluding the UN estimate that the world's population would reach 6 billion in\nOctober 1999 to the original data set (data set 1). Three additional data sets\nwere created by removing points in the other end from the original data set\n(data set 1), {\\it i.e}, by removing the last point (data set 6), the two\nlast points (data set 7) and the three last points (data set 8). Hence, those\nthree data sets cover the periods $\\left[0:1990\\right]$, $\\left[0:1980\\right]$\nand $\\left[ 0:1970\\right]$.\n\nThe differences between the results obtained for the first five data sets\nare minor,\nas can be seen in Table \\ref{tablepop} showing the values corresponding to the\nbest fits. Data set 6 and 7 are also compatible with the previous 5 whereas\nthe fit to data set 8 exhibit a significant discrepancy. For $t_c$, this gives\nthe window\n2052 $\\pm$ 10 years, which is rather well-constrained.\nFurthermore, the values obtained for $\\omega \\approx 6 \\pm 0.5$\n(again except for data set 8) are also quite compatible with previous results.\nThe corresponding fluctuations in the fundamental parameters $z \\approx 1.35\n\\pm 0.11$ and $\\lambda \\approx 2.8 \\pm 0.3$ are also within reasonable bounds.\nNote that it is difficult to obtain a better\nresolution in time as world population statistics in past centuries\nare all generated by using some sort of statistical regression model. This\nmight explain the relatively low value of the spectral peak obtained\nfor data set 5, see below.\nFurthermore, the peak clearly stands out against the background for seven\nout of eight spectra as we now discuss. Another encouraging observation is the\nnotable amplitude of\nthe log-periodic oscillations quantified by $C$, approximately $5-10\\%$ of\nthe pure power law acceleration quantified by $B$, as seen in the caption\nof figure\n\\ref{powlpfits}.\n\n\\subsubsection{Non-parametric tests of log-periodicity}\n\nWe also present a non-parametric test for the existence of the log-periodic\noscillations decorating the spontaneous\nsingularity, obtained by eliminating the leading trend using the transformation\n\\be \\label{residue}\np\\lp t\\rp \\rightarrow \\frac{p\\lp t\\rp - A_1 -\nB_1(t_c-t)^{\\beta}}{C_1(t_c-t)^{\\beta}}~ .\n\\ee\nThis transformation should produce a pure $\\cos\\lp \\omega \\ln(t_c-t)+\\phi\\rp$\n{\\it if} equation (\\ref{lppow}) was a perfect description. In figure\n\\ref{reslombpop}, we show the residual defined by (\\ref{residue})\nfor data 3 and data 5 as a function of $\\ln (t_c - t)$ as well as\ntheir Lomb periodograms which provide a power spectrum analysis for\nunevenly sampled data: the approximately regular oscillations in\n$\\ln (t_c - t)$ give a significant spectral peak at a log-angular frequency\n$\\omega \\approx 5.8-6.1$ compatible with the fit of equation (\\ref{lppow}),\nsee Table \\ref{tablepop}.\n\n\\subsection{Summary}\n\nTo sum up the evidence obtained so far, the comparison between the\nsemi-logarithmic\nplots in figures \\ref{semilogpop}-\\ref{semilogwindex} and the log-log plots\nin figures\n\\ref{pop2030}-\\ref{windex2060} validate the power law model (\\ref{pow}) at the\nexpense of the exponential model: there is no doubt that the world\npopulation and major economic and financial indices on average have grown much\nfaster than\nexponentially. The second message is that the rather large fluctuations\ndecorating\nan average power law acceleration can be remarkably well described by a simple\ngeneralisation of the power law in terms of a complex exponent:\nnot only do we see a good agreement between the spectral analysis\nand the fits with equation (\\ref{lppow}), in addition\nthe small fluctuations\nin the values for $t_c$, $\\beta$ and $\\omega$ for the 7 of the 8 data sets\nmake\nthe analysis credible for the world population. Of course,\nthis does not prove that equation (\\ref{lppow})\nis the correct description and equation (\\ref{eq:solution}) is a wrong\ndescription.\nHowever, since the r.m.s. of the fits with the two equations differs by a\nfactor\nof $\\approx 4$, there is no doubt that equation (\\ref{lppow}) does a better\njob\nof parameterising the data. This is the numerical argument. The theoretical\njustification has already been given above. The two combined certainly makes\nthe case stronger. For the financial indices, the use of equation\n(\\ref{lppow})\ndoes not lead to a significant improvement, and this leads us to examine\nthe relevance of the next order\nof the expansion of corrections to the power law.\n\n\n\\section{To second order}\n\n\\subsection{Next order of the log-periodic expansion}\n\nThe data set containing the Dow Jones Average consists of $\\approx 2500$\nmonthly\nquotes for the period $\\left[ 1790:1999.9 \\right]$.\nWe propose that it is representative of the capitalistic growth of the\nU.S.A. The time span and the sampling rate of this data set makes it reasonable\nto use the generalisation (\\ref{2feq}) of (\\ref{lppow}) to second order\nwhich allows\nfor a continuous shift in the angular log-frequency $\\omega$ \\cite{SJ97} in\nwhat effectively corresponds to a Landau or renormalisation group expansion\ndepending on the prefered framework.\n\nWe briefly summarize the method.\nUsing the renormalization group (RG) formalism on a financial index $I$\namounts\nto assuming that the index at a given time $t$ is related to that at\nanother time $t'$ by the transformations\n\\be\nx' = \\phi(x) ~ ,\n\\label{firstt}\n\\ee\n\\be\nF(x) = g(x) + \\frac{1}{\\mu} F\\biggl(\\phi(x)\\biggl)~,\n\\label{secondd}\n\\ee\nwhere $x=t_c-t$. $t_c$ is the critical time and $\\phi$ is called\nthe RG flow map. Here,\n$$\nF(x)=I(t_c)-I(t)~,\n$$\nsuch that $F=0$ at the critical point and $\\mu$ is a constant describing\nthe scaling of the index evolution upon a rescaling of time (\\ref{firstt}).\nThe function $g(x)$ represents the non-singular part of the function\n$F(x)$. We assume as usual that the function $F(x)$ is\ncontinuous and that $\\phi(x)$ is differentiable.\nIn order to use this formalism to constrain the possible time dependence of\nthe index, we notice that the solution in terms of a power law\nof the RG equation (\\ref{secondd}) together with (\\ref{firstt}) and the\nlinear approximation $\\phi(x) = \\lambda x$ valid close to the critical point\ncan be rewritten as\n\\be\n{d F(x) \\over d\\ln x} = \\alpha F(x) ~.\n\\label{ertf}\n\\ee\nThis states simply that a power law is nothing but a linear relationship when\nexpressed in the variables $\\ln F(x)$ and $\\ln x$. A critical point is\ncharacterized by observables which have an invariant description with\nrespect to scale transformations on $x$. We can exploit this and\nthe expression (\\ref{ertf}) to propose the structure\nof the leading corrections to the power law with log-periodicity. Hence, we\nnotice\nthat (\\ref{ertf}) can be interpreted as a bifurcation equation for the\nvariable $F$\nas a function of a fictitious ``time'' ($\\ln x$) as a function of the\n``control\nparameter'' $\\alpha$. When $\\alpha > 0$, $F(x)$ increases with $\\ln x$\nwhile it\ndecreases for $\\alpha < 0$. The special value $\\alpha = 0$ separating the\ntwo regimes\ncorresponds to a bifurcation. Once we have recognized the\nstructure of the expression (\\ref{ertf}) in terms of a bifurcation, we can\nuse the\ngeneral reduction theorem telling us that the structure of the\nequation for $F$ close to the bifurcation can only take a universal {\\it\nnon-linear} form given by\n\\be\n{dF(x) \\over d\\ln x} = (\\alpha + i \\omega) F(x)\n+ (\\eta + i \\kappa) |F(x)|^2 + {\\cal O}(F^3) .\n\\label{azepo}\n\\ee\nwhere $\\alpha>0$, $\\omega$, $\\eta$ and $\\kappa$ are real coefficients and\n${\\cal O}(F^3)$ means that higher order terms are neglected. The\ngenerality of this expression stems from the fact that it is nothing but a\nTaylor's expansion of a general functional form ${dF(x) \\over d\\ln x} =\n{\\cal\nF}(F(x))$. Such expansions are known in the physics literature as Landau\nexpansions. We stress that this expression represents a non-trivial\naddition to the theory, constrained uniquely by symmetry laws.\nGoing up to second order included, equation $(\\ref{lppow})$ becomes \\cite{SJ97}\n\\be\n\\ln\\lp I\\lp t \\rp \\rp = A_2 + B_2 {\\lp t_c- t \\rp ^{\\beta} \\over \\sqrt{ 1 +\n\\lp { t_c- t \\over \\tau} \\rp^{2 \\beta}}} \\left[ 1+ C_2 \\cos \\biggl(\n\\omega \\ln \\lp t_c-t \\rp + {\\Delta \\omega \\over 2 \\beta} \\ln \\lp 1 +\n\\lp {t_c- t \\over \\tau} \\rp^{2\\beta} - \\phi \\rp \\biggl) \\right]~.\n\\label{2feq}\n\\ee\nThis extension has been found useful in order to account for the behaviour of\nstock market prices before large crashes over extended period of times up to\n8 years. The present analysis thus constitutes a major generalisation as it\nincludes over 200 years of data. Previous work have established\na robust and universal signature preceding large crashes occuring in major\nfinancial stock markets, namely accelerated price increase decorated by large\nscale log-periodic oscillations culminating in a spontaneous singularity\n(critical point). The previously reported cases, which are well-described by\nequation $(\\ref{lppow})$, comprise the Oct. 1929 US crash, the Oct. 1987 world\nmarket crash, the Jan. 1994 and Oct. 1997 Hong-Kong crashes, the Aug.~1998\nglobal market event, the April 2000 Nasdaq crash,\nthe 1985 Forex event on the US dollar, the correction on the US dollar\nagainst the Canadian dollar and the Japanese Yen starting in Aug.~1998, as\nwell as the bubble on the Russian market and its ensuing collapse in June 1997\n\\cite{crash,nasdaq}. Furthermore, twenty-one significant bubbles followed\nby large\ncrashes or by severe corrections in the stock markets indices of the South\nAmerican and Asian countries, which exhibit log-periodic\nsignatures decorating an average power law acceleration,\nhave also been identified \\cite{emerg}. In all these\nanalyses, the time scales have been restricted to 1 to 8 years. In\ncontrast, the general renormalisation group theory of such spontaneous\nsingularities allow for an hierarchy of critical points at all scales\n\\cite{erzan,SSS}. The results given below suggest that singularities do indeed\ncascade in a robust way up to the largest time scales or conversely from the\nlargest scale to the smallest scales \\cite{Drozdz}.\n\n\\subsection{Second order fit of the Dow Jones Average index}\n\n\\subsubsection{Methodology}\n\nWe fit the logarithm of the extended Dow Jones index to equation (\\ref{2feq}).\nAs mentioned,\ntaking the logarithm provides in our opinion the simplest and most\nrobust way to account for inflation. Furthermore, taking the\nlogarithm embodies the notion that only relative changes are important.\nAnother more subtle reason can be given in terms of the magnitude of the\ncrash following the singularity: a simple model of rational expectations\n\\cite{bubmodel}\nshows that if the loss during a crash is proportional to the maximum price,\nthen the relevant quantity is the logarithm of the price in accordance\nwith the standard economic notion that only relative changes should be\nrelevant.\n\nFitting equation (\\ref{2feq}) to some data set is difficult even with a large\ndata set (for noisy data with only a few hundred points or less, it becomes\nquite\nimpossible), due to the degenerate r.m.s. landscape corresponding to the\nexistence of many local minima as a function of the free parameters $t_c$,\n$\\beta$, $\\omega$, $\\tau$ $\\Delta \\omega$ and $\\phi$. This means that the\nr.m.s. alone is not a good measure of the quality of the fit and additional\n{\\em physical} constraint are needed as discriminators. This has been\ndiscussed\nat length in \\cite{JLS}. In brief, we will demand that the value of\n$\\beta$ and $\\omega$ are compatible with what has been found previously for\nlarge crashes and that the value of the transition time $\\tau$ between the\ntwo competing frequencies is compatible with the time window $t_c - t_0$,\nwhere $t_0$ is the date of the first data point and $t_c$ the date of the\nsingularity in the first derivative. Unfortunately, we have no means\nto impose a criterion on the frequency-shift $\\Delta \\omega$.\n\n Specifically,\nwe will demand that $0.2 < \\beta < 0.7$, $4.5 < \\omega < 9$ and $\\lp t_c -\nt_0 \\rp/3 < \\tau \\stackrel{<}{\\sim} \\lp t_c - t_0 \\rp$ and the more the\nparameters fall in the mid-range, the higher confidence is attributed to the\nfit. These constraints are similar to what was used in \\cite{JLS} except for\nthe constrain on $\\tau$ which upper limit has been made stricter here. The\nreason\nfor this is simply that, whereas in the cases of the 1929 and 1987 stock market\ncrashes on Wall Street, it was not obvious to decide the starting date of the\nbubble, it is now objectively determined by a historical event being the\ncreation of the U.S.A. as an independent nation \\cite{UShistory}.\nThe parameter values of the\nfive qualifying fits is shown in Table \\ref{tabledj}. We stress that the\nmajority\nof the fits were discarded due to rather large values for either $\\omega$ or\n$\\tau$ or negative values for $\\omega$. We see that the best fit in terms of\nthe r.m.s. also has the most reasonable parameter values for $\\beta$, $\\omega$\nand $\\tau$ in terms of the discussion above.\n\n\\subsubsection{Results}\n\nThe best fit of equation (\\ref{2feq})) to the $210$ years of monthly quotes is\nshown in figure \\ref{djfit} and its parameter values are given in the caption.\nNote that the value of the angular log-frequency $\\omega \\approx 6.5$ compared\nto $\\omega \\approx 6.3$ as well as the value for the position of the\nsingularity\n$t_c \\approx 2053$ compared $t_c \\approx 2056$ are in close agreement with the\nvalues found for the analysis of the world population. Furthermore,\nthe cross-over time scale $\\tau \\approx 171$ years is perfectly compatible with\nthe total time window of 210 years. In figure \\ref{reslombdj}, the relative\nerror\nbetween the fit and the data is shown. We see that the error fluctuates nicely\naround zero as it should. Furthermore, the error is decreasing from left\nto right clearly showing that the acceleration in the data is better\nand better modeled by equation (\\ref{2feq})) as we approach the present.\nThis behaviour is in fact to be expected from an equation such as\n(\\ref{aafajdak}) allowing for an additive noise term to describe other sources\nof uncertainties: using\nthe Fokker-Planck formalism, one can show that, as the singularity at $t_c$ is\napproached, the noise term becomes negligible and the acceleration of the data\nshould approach better and better a pure power law. This can also be seen\ndirectly from (\\ref{aafajdak}) with an additive noise: the\ndivergence of $p(t)$ dwarves any bound noise contribution.\n\n\\subsubsection{Discussion}\n\nThe inset of figure \\ref{djfit} shows the extrapolation of the\nfit up to the critical time $t_c=2053$. It suggests that the Dow Jones\nindex will\nclimb to impressive values in the coming decades from its present level\naround 11,000 at the beginning of year 2000.\nIt is interesting that this resonates with a series of claims that the Dow\nJones\nwill climb to\n36,000 \\cite{Glassman}, 40,000 \\cite{Elias} or even 100,000 \\cite{Kadlec}\n in the next two or three decades.\nGlassman, an investing columnist for\nthe Washington Post, and Hassett, a former senior economist with the Federal\nReserve, develop the argument that stocks have been\nundervalued for decades and that, for the next few years, investors can\nexpect a\ndramatic one-time upward adjustment in stock prices \\cite{Glassman}.\nElias, a financial advisor and author, believes that\nforces such as direct foreign investment, domestic savings, and cooperative\ncentral-banking policies will drive the vigorous market, as will the\ndynamics of\nthe New Economy, which allows for the coexistence of high economic growth, low\ninterest rates, and low inflation. In his view, the Dow Jones could reach\n40,000\naround 2016 \\cite{Elias}. Kadlec, chief investment strategist for Seligman\nAdvisors Inc.\npredicts that the Dow\nJones Industrial Average will end up at 100,000 in the year 2020\n\\cite{Kadlec}.\nWe find that equation (\\ref{2feq}) predicts that the level 36,000-40,000\nwill be reached in 2018-2020 A.D. and the level 100,000 in 2026 A.D, not far\nfrom these claims! Of course, the extrapolation of this growth\ncloser to the singularity becomes unreliable due to standard limitations, such\nas finite size effects, and must be taken with a ``hand-full of salt''.\n\nIn the academic financial literature, a time series such as the Dow Jones\nshown in figure \\ref{djfit} has been argued to exhibit an anomalously large\nreturn, averaging $6\\%$ per year over the 1889-1978 period \\cite{Mehra},\nwhich cannot be explained by any reasonable risk aversion coefficient.\nA solution for this puzzle is that infrequent large crashes occur or even a\nmajor still untriggered crash is looming over us; in this interpretation,\nthe ``anomalous'' return becomes the normal remuneration for the risk to\nstay invested in the market \\cite{Ritz}. Our analysis suggests that the\nsituation\nis even worse than this: not only the market has a large growth rate but\nthis growth rate is {\\it accelerating} such that the market is growing as\na power law towards a spontaneous singularity.\n\n\n\\section{Synthesis and theoretical discussion}\n\n\\subsection{Summary}\n\nThe fact, that both the human world population over two thousand years, the\nGDP of the world and six national, regional and world financial indices over\nmost of their lifespan agree both in i) the prediction of a spontaneous\nsingularity, ii) the approximate location of the critical time and iii) the\napproximate self-similar\npatterns decorating the singularity is quite remarkable to say the least. This\nsuggests that they may have a closely correlated dynamics, in fact more than\nthe coupling between population $p(t)$ and carrying capacity $K(t)$ written\nin equations such as $(\\ref{ajdak})$ would make us believe. The outstanding\nscientific question is whether the rate of innovations fueling the economic\ngrowth is a random process on which industrial and population selection\noperates or if it is driven by the pressing needs of the growing population.\nThe main message of this study is that, whatever the answer and irrespective\nof one's optimistic or pessimistic view of the world sustainability, these\nimportant pieces of data all point to the existence of an end to the\npresent era, which will be irreversible and cannot be overcome by any novel\ninnovation of the preceding kind, {\\it e.g.}, a new technology that makes\nthe final conquest of the Oceans and the vast mineral resources there\npossible. This, since any new innovation is deeply embedded in the very\nexistence of a singularity, in fact it {\\it feeds} it. As a result, a\nfuture transition of mankind towards a qualitatively new level is quite\npossible.\n\nThe reader not familiar with critical phenomena and singularities\n\\cite{DombGreen,critical,Dubrulle}\nmay dismiss our approach without further ado on the basis that\nall demographic insights show that the population\ngrowth is now decelerating rather than accelerating. \nIndeed, many developed countries show a substantial reduction in fertility.\nHowever, ``the tree should not hide the forest'' as the proverb says, in other words\nthis deceleration is compatible with the concept of a finite-time \nsingularity in the presence of so-called ``finite size effects''\n\\cite{finitesizeeffect}.\nNamely, it is well-known that nature does not have pure singularities in the\nmathematical sense of the term. Such critical points are\nalways rounded off or\nsmoothed out by the existence of friction and dissipation and by the finiteness\nof the system. This is a well-known feature of critical points \n\\cite{finitesizeeffect}. Finite-time singularities are similarly rounded-off\nby frictional effects, A clear example is provided by Euler's disk\n\\cite{Moffatt}, a rotating coin settling to rest in finite time after, in \nprinciple, an infinite number of rotations. In reality, the rotational speed\naccelerates until a point when friction due to air drag and solid contact with the\nsupport saturate this acceleration and stop the rotation abruptly. \nThe upshot here is that finite-size effect and friction do not prevent the \neffect we document here to be present, namely the acceleration of the growth rate,\nup to a point where the proximity to the critical point makes finite size\neffects and dissipation-like effects to take over. The fact that these\n``imperfections'' become relevant in the ultimate stage of the trajectory\ndoes not change the validity of the conclusions. The change of regime to a new\nphase subsists. Only its absolute abruptness is replaced by a somewhat\nsmoother transition, albeit still rather sharp on the time scale of the \ntotal time span. In the present context,\nthe observed very recent deceleration of the growth rate can be taken as\na signature that mankind is entering in the critical region towards a \ntransition to a new regime. Since the world population growth rate topped\nin 1970, this corresponds to approximately 80 years from the predicted\ncritical point, or only $4\\%$ of the total \ntimespan of the investigated time series.\n\n\n\n\n\\subsection{Related work}\n\nOther authors have documented a super-exponential acceleration of human\nactivity.\nKapitza has recently analysed the dynamical evolution of the human\npopulation \\cite{Kapitza},\nboth aggregated and regionally and also documents a consistent overall\nacceleration\nuntil recent times. He introduces a saturation effect to limit the blow-up and\ndiscuss different scenarios. Using data from the Cambridge encyclopedia,\nhe argues that epochs of characteristic evolutions\nor changes shrink as a geometrical series. In other words, the epoch sizes\nare approximately\nequidistant in the logarithm of the time to present.\nIn a study of an important human activity,\nvan Raan has found that the scientific production since the 16th century\nin Europe has accelerated much faster than\nexponentially \\cite{Raan}.\nUsing the data of DeLong \\cite{Bradford},\nHanson finds that the history of the world economic production since\nprehistoric times\ncan only be accounted for by adding\nthree exponentials, each one being interpreted as a new ``revolution''\n\\cite{Hanson}:\nhunting followed by farming and then by industry. He finds that each\nexponential mode grew over one hundred times\nfaster than its predecessor. He also plots the logarithm of the\nworld product as a function of the logarithm of $t_c-t$ with $t_c =2050$\nand find a\nreasonable straight line decorated by oscillations marking the different\ntransitions.\n\nMacro-economic models have been developed that predict the possibility of\naccelerated growth\n\\cite{Romer}. Maybe the simplest model is that of Kremer \\cite{Kremer} who\nnotes that,\nover almost all human history, technological progress has led mainly to\nan increase in\npopulation rather than an increase in output per person. In his model,\nthe economic output\nper person $Y(t)/L(t)$, where $Y(t)$ is the total output comprising all\nartifacts\n and $L(t)$ is the\ntotal population,\nis thus set equal to the subsistence level ${\\bar y}$ which is assumed fixed:\n\\be\n{Y(t) \\over L(t)} = {\\bar y}~. \\label{fjak}\n\\ee\nThe output is supposed to depend on technology and knowledge\n$A(t)$ and labour (proportional to $L(t)$):\n\\be\nY(t) = Y_0 \\left[ A(t) L(t) \\right]^{1-\\alpha}~, \\label{fjaka}\n\\ee\nwhere $0 < \\alpha < 1$. The growth rate of knowledge and technology is taken\nproportional to population and to knowledge:\n\\be\n{dA \\over dt} = B L(t) A(t)~, \\label{fbnkala}\n\\ee\nembodying the concept that a larger population offers more opportunities\nfor finding\nexceptionally talented-people who will make important innovations and that\nnew knowledge\nis obtained by leveraging existing knowledge.\nEliminating $Y(t)$ and $A(t)$ between (\\ref{fjak}-\\ref{fbnkala}) gives the\nequation for the\ntotal population:\n\\be\n{dL \\over dt} = {1-\\alpha \\over \\alpha} B~ [L(t)]^2~. \\label{fakal}\n\\ee\nThis is the case $\\delta = 1$ of equation (\\ref{aafajdak}), showing that\nthe population\nand its output develop a finite-time singularity (\\ref{pow}) with the exponent\n$z=-1$. Kremer tested this prediction by using population estimates extending\nback to 1 million B.C., constructed by archaeologists and anthropologists:\nhe showed\nthat the population growth rate is approximately linearly increasing with\nthe population\n\\cite{Kremer},\nin agreement with (\\ref{fakal}). Our result $z\\approx -1.9$ for the human\npopulation\nexaggerates the singularity. On the other hand, as shown in Table 1, we find\na remarkable consistent value $z \\approx -1$ for all financial indices. Our\nrefinements with the log-periodic formulas in order to account for the\nsignificant\nstructures decorating the average power laws necessary lead to deviations\nfrom this\n``mean-field'' value, which should be considered as an approximation\nneglecting the effect\nof fluctuations.\nThis theory also predicts, in agreement with historical\nfacts, that in the historical times when regions were separated,\n technological progress was faster in regions with larger population, thus\nexplaining\n the differences between Eurasia-Africa, the Americas, Australia and Tasmania.\n\n\\subsection{Multivariate finite-time singularities}\n\nKremer's model is only one of a general class of growth models \\cite{Romer}.\nWe briefly recall the general framework developed by Romer \\cite{Romer2},\nwhich allows us to generalise the concept of\nfinite-time singularities to multivariate dynamics and to\nexhibit the structure of its solution and follow \\cite{Romer} in our\nexposition.\nThe model involves four variables,\nlabour $L$, capital $K$,\ntechnology $A$ and output $Y$. There are two sectors, a goods-producing\nsector where\noutput is produced and an R\\&D sector where additions to the stock of\nknowledge are made.\nThe fraction $a_L$ of the labour force is used in the R\\&D sector and the\nfraction $1-a_L$ in\nthe goods-producing sector; similarly, the fraction $a_K$ of the capital\nstock is used in R\\&D\nand the rest in goods production. Both sectors use the full stock of\nknowledge. The quantity\nof output produced at time $t$ is defined as\n\\be\nY(t) = \\left[ (1-a_K) K(t) \\right]^{\\alpha}~ \\left[A(t)(1-a_L)\nL(t)\\right]^{1-\\alpha}~,\n\\label{fjakala}\n\\ee\nwith $0 < \\alpha < 1$. Expression (\\ref{fjakala}) uses the so-called\nCobb-Douglas functional\nform with power law relationships which imply constant returns to capital\nand labour: within\na given technology, doubling the inputs doubles the amount that can be\nproduced. Expression (\\ref{fjakala}) writes that the economic output increases\nwith invested capital, with technology and R\\&D and with labor.\n\nThe production of innovation is written as\n\\be\n{dA \\over dt} = B \\left[ a_K K(t)\\right]^{\\beta}~\\left[ a_L\nL(t)\\right]^{\\gamma}~\n\\left[A(t)\\right]^{\\theta}~,~~~~~B>0,~~\\beta \\geq 0, ~~ \\gamma \\geq 0~.\n\\label{fjaklaaa}\n\\ee\nThe growth of knowledge is thus controlled by the pre-existing knowledge,\nby capital\ninvestment in research and by the size of the population of innovators.\n\nAs in the Solow model \\cite{Romer}, the saving rate $s$ is exogenous and\nconstant and depreciation\nis set to zero for simplicity so that\n\\be\n{dK \\over dt} = s Y(t) = s \\left[ (1-a_K) K(t) \\right]^{\\alpha}~\n\\left[A(t)(1-a_L) L(t)\\right]^{1-\\alpha}~. \\label{fqqoloa}\n\\ee\n\nLet us consider (\\ref{fjaklaaa}). If $K$ and $L$ are constant, it reduces\nto an equation of\nthe form (\\ref{aafajdak}), which exhibits a finite-time singularity only\nfor $\\theta > 1$.\nIn the presence of the coupling to the other growing dynamical variables\n$K$ and $L$,\na finite-time singularity may occur even in the situation $\\theta <1$.\n\nAs a first example, let us consider the case of a fixed population $L(t)\n=$ constant.\nEquations (\\ref{fjaklaaa}) and (\\ref{fqqoloa}) can be rewritten as\n\\bea\n{dA \\over dt} &=& b A^{\\theta} K^{\\beta}~. \\label{nfncbbzb}\\\\\n{dK \\over dt} &=& a A^{1-\\alpha} K^{\\alpha} ~, \\label{vbxz}\n\\eea\nWe look for the condition on the exponents such that $A(t)$ and $K(t)$ exhibit\na finite-time singularity. We thus look for solutions of the form\n\\bea\nA(t) &=& A_0 (t_c-t)^{-\\delta}~, \\label{bcvccz}\\\\\nK(t) &=& K_0 (t_c-t)^{-\\kappa}~, \\label{nbeq}\n\\eea\nwith $\\delta$ and $\\kappa$ positive.\nInserting these expressions in (\\ref{vbxz}) and (\\ref{nfncbbzb}) leads to\ntwo equations for the two exponents $\\delta$ and $\\kappa$ obtained from the\nconditions that\nthe powers of $(t_c-t)$ are the same on the r.h.s. and l.h.s. of\n(\\ref{vbxz}) and (\\ref{nfncbbzb}).\nTheir solution is\n\\bea\n\\delta &=& {1+\\beta - \\alpha \\over (1-\\alpha)(\\theta+\\beta-1)}~,\\\\\n\\kappa &=& {2-\\theta - \\alpha \\over (1-\\alpha)(\\theta+\\beta-1)}~.\n\\eea\nThe condition that both $\\delta$ and $\\kappa$ are positive enforce that\n$\\theta+\\beta > 1$, which is the condition replacing $\\theta >1$ for\nthe existence of a finite-time singularity in the monovariate case.\nThis shows that the combined effect of past innovation and capital has\nthe possibility of creating an explosive growth rate even when {\\it each}\nof these factors in isolation does not. Note that inequality $\\theta+\\beta\n> 1$ ensures that $\\delta > \\kappa$, {\\it i.e.}, the growth of the\ntechnological stock is faster than that of the capital.\n\nThere are many ways to reinsert the dynamical evolution of the population.\nLet us here\nconsider the simplest one used by Kremer \\cite{Kremer}, which consists in\nassuming that\n$L(t)$ is proportional to $K(t)$ as given by (\\ref{fjak}). Then,\nexpressions (\\ref{fjaklaaa})\nand(\\ref{fqqoloa}) give\n\\bea\n{dA \\over dt} &=& a' \\left[L(t) \\right]^{\\beta +\n\\gamma}~\\left[A(t)\\right]^{\\theta}~,\n~~~~~a'>0,~~\\beta \\geq 0, ~~ \\gamma \\geq 0~, \\label{fjakelaawa} \\\\\n{dL \\over dt} &=& b' L(t) \\left[A(t) \\right]^{1-\\alpha}~. \\label{fqasfqoloa}\n\\eea\nLooking for solutions of the form (\\ref{bcvccz}) and (\\ref{nbeq}) gives\n\\bea\n\\delta &=& {1 \\over 1-\\alpha}~,\\\\\n\\kappa &=& {2-\\theta - \\alpha \\over \\beta + \\gamma}~.\n\\eea\nIt is interesting to find that the technology growth exponent $\\delta$ is\nnot at all controlled by $\\theta$ nor $\\beta$ and $\\gamma$. This\nillustrates that a finite-time singularities can be created\nfrom the interplay of several growing variables resulting in a non-trivial\nbehaviour. In the present context, it means that the interplay between\ndifferent quantities, such as capital and technology, may produce an\n``explosion'' in the population even though the individual dynamics\ndo not. In particular, this interplay provides an explanation of our finding\nof the same value of the critical time $t_c (\\approx 2052 \\pm 10)$ both for\nthe\npopulation and economic indices.\n\n\n\\section{Possible scenarios}\n\nWe now attempt to guess what could be the possible scenarios for mankind\nclose to and beyond the critical time $t_c$.\n\nA gloomy scenario is that humanity will enter a\nsevere recession fed by the slow death of its host (the Earth), in the\nspirit of\nthe analogy \\cite{Hern} proposed between the human species and cancer.\nThis worry about human population size and growth is shared by many\nscientists, including\nthe Union of Concerned Scientists (comprising 99 Nobel Prize winners) which\nasks\nnations to\n``stabilise population.'' Representatives of national\nacademies of science from throughout the world met in New Delhi, 24-27 October\n1993, at a ``Science Summit'' on World Population. The participants issued a\nstatement, signed by representatives of 58 academies on population issues\nrelated to\ndevelopment, notably on the determinants of fertility and concerning the effect\nof demographic growth on the environment and the quality of life.\nThe statement finds that ``continuing population growth poses a great\nrisk to humanity,'' and proposes a demographic goal: ``In our judgment,\nhumanity's ability to\ndeal successfully with its social, economic, and environmental problems will\nrequire the achievement of zero population growth within the lifetime of our\nchildren'' and ``Humanity is\napproaching a crisis point with respect to the interlocking issues of\npopulation,\nenvironment and development because the Earth is finite''\n\\cite{statementAcads}.\nPossible scenarios involve a systematic development of terrorism and the\nsegregation of\nmankind into at least two groups, a minority of wealthy communities hiding\nbehind\nfortresses from the crowd of ``barbarians'' roaming outside, as discussed\nin a recent seminar at the US National Academy of Sciences. Such a scenario\nis also quite possible for the relation between developed and developing\ncountries.\n\n\nOn a more positive note, it may be that ``ecological'' actions of the kind\nmentioned above will grow\nin the next decades, leading to a smooth transition towards an\necologically-integrated\nindustry and humanity. Some signs may give indications of this path: during the\n1990s, wind power has been growing at a rate of 26\\% a year and solar\nphoto-voltaic power\nat 17\\% compared to the growth in coal and oil under 2\\%; governments have\n``ratified'' more\nthan 170 international environmental treaties, on everything from fishing\nto decertification\n\\cite{stateofworld}. However, there are serious resistances \\cite{Nego},\nin particular because there is no consensus on the seriousness of the\nsituation:\nfor instance, the economist J.L. Simon writes that ``almost every\nmeasure of material and environmental human welfare in the United States\nand in the World\nshows improvement rather than deterioration'' \\cite{Simon}.\nIt may be that the strikingly\nsimilar explosive trend in population and GDP would not necessarily\npersist in the future when taking the differences between\nregional developments into account. Perhaps what is needed to avoid the\nfinite-time\nsingularity is a massive transfer of resources from developed to developing\ncountries. The recent discussions at the G7/8 summit indicates that\nthe developed world is becoming increasingly aware of the discrepancy.\n\nExtrapolating further, the evolution from a growth regime to a balanced\nsymbiosis with nature\nand with the Earth's resources requires the transition to a\nknowledge-based society, in which\nknowledge, intellectual, artistic and humanistic values replace the quest\nfor material wealth.\nIndeed, the main economic difference is that ``knowledge'' is non-rival\n\\cite{Romer2}:\nthe use of an idea or of a piece of knowledge in one place does not prevent\nit from being\nused elsewhere; in contrast, say an item of clothing by an individual\nprecludes its\nsimultaneous use by someone else. Only the emphasis on non-rival goods will\nlimit\nultimately the plunder of the planet. Some so-called\n``primitive'' societies seem to have been able to evolve\ninto such a state \\cite{Gunsdiam}.\n\nThe race for growth could however continue or even be enhanced if fundamentally\nnew discoveries at a different level of the hierarchy witnessed until present\nenabled mankind to start the colonisation of other planets. The conditions for\nthis are rather drastic, since novel modes of much faster propulsions are\nrequired as\nwell as revolutions in our control of the adverse biological effects of\nspace on\nhumans. It may be that some evolved form of humans\nwill appear who are more adapted to the hardship of space.\nThis could lead to a new era of renewed accelerated growth after a period of\nconsolidation, culminating in a new finite-time singularity, probably\ncenturies in the future.\n\n\n{\\bf Acknowledgement:} We thank P. Kendall and R. Prechter for help in\nproviding the\nfinancial data from the Foundation For The Study Of Cycles, R. Hanson for\nthe world GDP data and useful discussions,\nB. Taylor of Global Financial Data for the permission to use their data,\nM. Lagier, D. Zajdenweber for discussions, U. Frisch and D. Stauffer for a\ncritical reading of\nthe manuscript and for useful suggestions.\n\n\\vskip 0.5cm\n{Note Added in Proofs}:\nNottale, Chaline and Grou \\cite{Chaline2,Chaline3} have recently \nindependently applied a log-periodic analysis to the main crises of different\ncivilisation. They first noticed that historical events seem to accelerate. This was\nactually anticipate by Meyer who used a primitive for of \nlog-periodic acceleration analysis \\cite{Meyer1,Meyer2}. Grou \\cite{Grou} has\ndemonstrated that the economic evolution since the neolithic can be described in\nterms of various dominating poles which are subjected to an accelerating crisis/\nno-crisis pattern. Their quantitative analysis on the\nmedian dates of the main periods of economic crisis in the history of Western\ncivilization (as listed in \\cite{Grou,Braudel,Gilles} are as follows (the dominating pole\nand the date are given in years / JC): \\{Neolithic: -6500\\}, \\{Egypt: -3000\\},\\{Egypt: -900\\},\n\\{Grece: -100\\}, \\{Rome: +400\\}, \\{Byzance: +800\\}, \\{Arab expansion: +1100\\}, \\{Southern\nEurop: +1400\\}, \\{Netherland:+1650\\}, \\{Great-Britain: +1775\\}, \\{Great-Britain: +1830\\},\n\\{Great-Britain: +1880\\}, \\{Great-Britain: +1935\\}, \\{United-States: +1975\\}.\nLog-periodic acceleration with scale factor $\\lambda = 1.32 \\pm 0.018$ occurs towards \n$t_c = 2080 \\pm 30$. Agreement between the data and the log-periodic law is\nstatistically highly significant ($t_{\\rm student} = 145$, Proba $<< 10^{-4}$).\nIt is striking that this independent analysis based on a different data set gives \na critical time which is compatible with our own estimate $2052 \\pm 10$.\n\n\n\\newpage\n\n\n\n\\begin{thebibliography}{}\n\n\n\\bibitem{benderorszag} Bender C, Orszag S.A (1978)\npage 147 in {\\it Advanced Mathematical Methods for Scientists and Engineers.}\nMcGraw-Hill, New York. \n\n\\bibitem{Bradford} J. Bradford DeLong (1998) \nEstimating World GDP, One Million B.C. - Present. Working paper available at \nhttp://econ161.berkeley.edu/TCEH/1998\\_Draft/World\\_GDP/Estimating\\_World\\_GDP.html\n\n\\bibitem{stateofworld} Brown LR, Flavin C (1999) \n{\\it State of the World, Millenium edition}. A Worldatch Institute report on \nProgress towards a sustainable society, W.W. 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Available at:\nhttp://www.iisd.ca/linkages/download/pdf/enb1534e.pdf.\n\n\\bibitem{UShistory} Even though U.S.A was recognised as a nation by \nthe Paris Treaty in 1783, a number of events point to the fact it was not \n{\\it established} as a nation before $\\approx$ 1790. They are as follows. \n1) The constitution went into effect in March 1789, having been ratified by \nNew Hampshire as the ninth state on June 21, 1788. 2) The last of the thirteen\nstates, Rhode Island, first approved it on May 29, 1790. 3) The first census \nin the U.S. was made in 1790. 4) The Naturalisation Act of 1790 grants the \nright of U.S. citizenship to all ``free white persons.'' 5) In 1790, the\nFederal Government declared that it was redeeming the SCRIP MONEY that was\nissued during the Revolutionary War. 6) At about this time, the Government\nannounced the creation of the first bank of the United States in conjunction \nwith the sale of \\$10,000,000 dollars in shares of stock.\n\n\\bibitem{Chaline2} Nottale L., Chaline J., Grou P. (2000) Les arbres de l'\\'evolution\n(Hachette Litterature, Paris) 379 p.\n\n\\bibitem{Chaline3} Nottale L., Chaline J., Grou P. (2000) in ``Fractals 2000 in Biology\nand Medicine'', Proceedings of Third International Symposium, Ascona,\nSwitzerland, March 8-11, 2000, Ed. G. Losa, Birkh�user Verlag.\n\n\\bibitem{Meyer1} Meyer F. (1947) L'acc\\'el\\'eration \\'evolutive.\nEssai sur le rythme \\'evolutif et son \ninterpr\\'etation quantique. Librairie des Sciences et des Arts, Paris, 67p.\n\n\\bibitem{Meyer2} [14] Meyer F. (1954) Probl\\'ematique de l'\\'evolution. P.U.F., 279p.\n\n\\bibitem{Grou} Grou P. (1987,1995) L'aventure \\'economique. L'Harmattan, Paris, 160 p.\n\n\\bibitem{Braudel} Braudel F. (1979) Civilisation mat\\'erielle, \\'economie et capitalisme. \nA. Colin \n\n\\bibitem{Gilles} Gilles B. (1982) Histoire des techniques. Gallimard\n\n\n\\end{thebibliography}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=semilogpopadd.eps,height=8cm,width=8.5cm}\n\\caption{\\label{semilogpop}Semi-logarithmic plot of World population\nfrom year 0 until Oct. 1999.\nIn this representation, a linear increase would qualify an exponential\ngrowth. Note in contrast\nthe super-exponential behavior.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=semilogwgdp.eps,height=8cm,width=8.5cm}\n\\caption{\\label{semilogwgdp}Semi-logarithmic plot of World GDP from\nyear 1 until 2000.}}\n%\\end{center}\n%\\end{figure}\n\n\\vspace{1.5cm}\n\n%\\begin{figure}\n%\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=semilogdj.eps,height=8cm,width=8.5cm}\n\\caption{\\label{semilogdj}Semi-logarithmic plot of the Dow Jones from 1790\nuntil 2000.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=semilogsp.eps,height=8cm,width=8.5cm}\n\\caption{\\label{semilogsp}Semi-logarithmic plot of the S\\&P from 1871 until\n2000. }}\n\\end{center}\n\\end{figure}\n\n%\\mbox{ } \\pagebreak\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=semiloglatin.eps,height=8cm,width=8.5cm}\n\\caption{\\label{semiloglat}Semi-logarithmic plot of the Latin American\nindex from 1938 until 2000.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=semilogeur.eps,height=8cm,width=8.5cm}\n\\caption{\\label{semilogeur}Semi-logarithmic plot of the European index from\n1920 until 2000.}}\n%\\end{center}\n%\\end{figure}\n\n\\vspace{1.5cm}\n\n%\\begin{figure}\n%\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=semilogeafe.eps,height=8cm,width=8.5cm}\n\\caption{\\label{semilogeafe}Semi-logarithmic plot of the EAFE index from\n1920 until 2000.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=semilogwindex.eps,height=8cm,width=8.5cm}\n\\caption{\\label{semilogwindex}Semi-logarithmic plot of the World index from\n1920 until 2000.}}\n\\end{center}\n\\end{figure}\n\n%\\mbox{ } \\pagebreak\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitpopadd2030.eps,height=8cm,width=8.5cm}\n\\caption{\\label{pop2030} World population as a function of $t_c - t$\nwith $t_c=2030$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with a fixed $t_c=2030$, see\nTable \\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitpopadd2040.eps,height=8cm,width=8.5cm}\n\\caption{\\label{pop2040} World population as a function of $t_c - t$\nwith $t_c=2040$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2040$ fixed, see Table\n\\protect\\ref{ztable}.}}\n%\\end{center}\n%\\end{figure}\n\n\\vspace{1.5cm}\n\n%\\begin{figure}\n%\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitpopadd2050.eps,height=8cm,width=8.5cm}\n\\caption{\\label{pop2050} World population as a function of $t_c - t$\nwith $t_c=2050$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2050$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitlogwgdp2040-3.eps,height=8cm,width=8.5cm}\n\\caption{\\label{wgdp2040} World GDP as a function of $t_c - t$ with\n$t_c=2040$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2040$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\end{center}\n\\end{figure}\n\n\n%\\mbox{ } \\pagebreak\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitlogwgdp2050-3.eps,height=8cm,width=8.5cm}\n\\caption{\\label{wgdp2050} World GDP as a function of $t_c - t$ with\n$t_c=2050$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2050$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitlogwgdp2060-3.eps,height=8cm,width=8.5cm}\n\\caption{\\label{wgdp2060}World GDP as a function of $t_c - t$ with\n$t_c=2060$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2060$ fixed, see Table\n\\protect\\ref{ztable}.}}\n%\\end{center}\n%\\end{figure}\n\n\\vspace{1.5cm}\n\n%\\begin{figure}\n%\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitlogdj2040.eps,height=8cm,width=8.5cm}\n\\caption{\\label{dj2040}The Dow Jones as a function of $t_c - t$ with\n$t_c=2040$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2040$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitlogdj2050.eps,height=8cm,width=8.5cm}\n\\caption{\\label{dj2050}The Dow Jones as a function of $t_c - t$ with\n$t_c=2050$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2050$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\end{center}\n\\end{figure}\n\n%\\mbox{ } \\pagebreak\n\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitlogdj2060.eps,height=8cm,width=8.5cm}\n\\caption{\\label{dj2060}The Dow Jones as a function of $t_c - t$ with\n$t_c=2060$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2060$ fixed, Tsee able\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fit2040logsp1871.eps,height=8cm,width=8.5cm}\n\\caption{\\label{sp2040}The S\\&P as a function of $t_c - t$ with $t_c=2040$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2040$ fixed, see Table\n\\protect\\ref{ztable}.}}\n%\\end{center}\n%\\end{figure}\n\n\\vspace{1.5cm}\n\n%\\begin{figure}\n%\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fit2050logsp1871.eps,height=8cm,width=8.5cm}\n\\caption{\\label{sp2050}The S\\&P as a function of $t_c - t$ with $t_c=2050$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2050$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fit2060logsp1871.eps,height=8cm,width=8.5cm}\n\\caption{\\label{sp2060}The S\\&P as a function of $t_c - t$ with $t_c=2060$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2060$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\end{center}\n\\end{figure}\n\n%\\mbox{ } \\pagebreak\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitloglat2040.eps,height=8cm,width=8.5cm}\n\\caption{\\label{lat2040}The Latin America index as a function of $t_c - t$\nwith\n$t_c=2040$. The straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2040$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitloglat2050.eps,height=8cm,width=8.5cm}\n\\caption{\\label{lat2050}The Latin America index as a function of $t_c - t$ with\n$t_c=2050$. The straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2050$ fixed, see\nTable \\protect\\ref{ztable}.}}\n%\\end{center}\n%\\end{figure}\n\n\\vspace{1.5cm}\n\n%\\begin{figure}\n%\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitloglat2060.eps,height=8cm,width=8.5cm}\n\\caption{\\label{lat2060}The Latin America index as a function of $t_c - t$\nwith $t_c=2060$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2060$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitlogeur2040.eps,height=8cm,width=8.5cm}\n\\caption{\\label{eur2040}The European index as a function of $t_c - t$ with\n$t_c=2040$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2040$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\end{center}\n\\end{figure}\n\n\n%\\mbox{ } \\pagebreak\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitlogeur2050.eps,height=8cm,width=8.5cm}\n\\caption{\\label{eur2050}The European index as a function of $t_c - t$ with\n$t_c=2050$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2050$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitlogeur2060.eps,height=8cm,width=8.5cm}\n\\caption{\\label{eur2060}The European index as a function of $t_c - t$ with\n$t_c=2060$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2060$ fixed, see Table\n\\protect\\ref{ztable}.}}\n%\\end{center}\n%\\end{figure}\n\n\\vspace{1.5cm}\n\n%\\begin{figure}\n%\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitlogeafe2040.eps,height=8cm,width=8.5cm}\n\\caption{\\label{eafe2040}The EAFE index as a function of $t_c - t$ with\n$t_c=2040$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2040$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitlogeur2050.eps,height=8cm,width=8.5cm}\n\\caption{\\label{eafe2050}The EAFE index as a function of $t_c - t$ with\n$t_c=2050$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2050$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\end{center}\n\\end{figure}\n\n\n%\\mbox{ } \\pagebreak\n\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitlogeafe2060.eps,height=8cm,width=8.5cm}\n\\caption{\\label{eafe2060}The EAFE index as a function of $t_c - t$ with\n$t_c=2060$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2060$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitlogwindex2040.eps,height=8cm,width=8.5cm}\n\\caption{\\label{windex2040}The World index as a function of $t_c - t$ with\n$t_c=2040$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2040$ fixed, see Table\n\\protect\\ref{ztable}.}}\n%\\end{center}\n%\\end{figure}\n\n\\vspace{1.5cm}\n\n%\\begin{figure}\n%\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=fitlogwindex2050.eps,height=8cm,width=8.5cm}\n\\caption{\\label{windex2050}The World index as a function of $t_c - t$ with\n$t_c=2050$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2050$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitlogwindex2060.eps,height=8cm,width=8.5cm}\n\\caption{\\label{windex2060}The World index as a function of $t_c - t$ with\n$t_c=2060$.\nThe straight line is the\nfit with a power law $p(t) = a(t_c - t)^z$ with $t_c=2060$ fixed, see Table\n\\protect\\ref{ztable}.}}\n\\end{center}\n\\end{figure}\n\n%\\mbox{ } \\pagebreak\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=powlppopfits.eps,height=8cm,width=12cm}\n\\caption{\\protect\\label{powlpfits} The dotted line is the best fit with\nequation $(\\protect\\ref{eq:solution})$ to data set 5, see text. The fit\ngives $\\mbox{r.m.s.}=0.111$, $A\\approx 0$, $B \\approx 22120$, $t_c \\approx\n2078$\nand $z \\approx -1.9$. The full line is the best fit with equation\n$(\\protect\\ref{lppow})$ and gives $\\mbox{r.m.s.}=0.030$, $A\\approx 0$,\n$B\\approx 1624$, $C\\approx -127$, $z\\approx -1.4$, $t_c \\approx 2056$, $\\omega\n\\approx 6.3$ and $\\phi \\approx 5.1$.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=11-2nspow252001logdj1790.eps,height=8cm,width=8.5cm}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=fitpowlogsp1871.eps,height=8cm,width=8.5cm}}\n\\caption{\\label{powlogdjsp}Left: The Dow Jones Average fitted with equation\n\\protect\\ref{eq:solution}. The values of the fit are $A\\approx-14$,\n$B\\approx 71$,\n$z\\approx-0.27$ and $t_c\\approx2068$. Right: The S\\&P fitted with equation\n\\protect\\ref{eq:solution}. The values of the fit are $A\\approx 0$,\n$B\\approx 1693$,\n$z\\approx -1.3$ and $t_c\\approx 2067$.}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\parbox[l]{8.5cm}{\n\\epsfig{file=datset3-5.res.eps,height=8cm,width=8.5cm}}\n\\hspace{5mm}\n\\parbox[r]{8.5cm}{\n\\epsfig{file=datset3-5.fp.eps,height=8cm,width=8.5cm}}\n\\caption{\\protect\\label{reslombpop}Left: Residue between best fit and data sets\n3 and 5, as defined by equation $(\\protect\\ref{residue})$. Right: Spectrum of\nresidue using a Lomb periodogram. The position of the peak corresponds to\n$\\omega \\approx 5.8$, which should be compared with $\\omega \\approx 6.5$ for\nthe fit with equation $(\\protect\\ref{lppow})$ for data set 5. For data set 3,\nthe peak corresponds to $\\omega \\approx 6.1$, which should be compared with\n$\\omega \\approx 6.5$ for the fit.}\n%\\end{center}\n%\\end{figure}\n\n\n\\vspace{0.5cm}\n%\\pagebreak\n\n%\\begin{figure}\n%\\begin{center}\n\\epsfig{file=FigUSmarket.eps,height=10cm,width=14cm}\n\\caption{\\protect\\label{djfit} The circles are the logarithm of the yearly\nquotes from Dec. 1790 to Dec. 1999, which are shown instead of the monthly\nquotes used in the fit to better show the two fit curves.\nThe upward trending full line is the best fit with equation\n$(\\protect\\ref{eq:solution})$ with $\\mbox{r.m.s.} = 0.307$, $A \\approx -13.7$,\n$B \\approx 70.8$, $z\\approx -0.27$ and $t_c \\approx 2068$. The full\noscillating line is the best fit with\nequation $(\\protect\\ref{2feq})$ to the extended Dow Jones for the period\n$\\left[ 1790:1999.75 \\right]$. The fit gives $\\mbox{r.m.s}=0.236$,\n$A\\approx 25.1$, $B\\approx -4.13$, $C\\approx -0.055$, $\\beta \\approx 0.39$,\n$t_c \\approx 2053$, $\\omega \\approx 6.5$, $\\tau \\approx 171$, $\\Delta \\omega\n\\approx -58$ and $\\phi \\approx -5.8$. The inset shows the extrapolation of the\nfit up to the critical time $t_c \\approx 2053$.}\n\\end{center}\n\\end{figure}\n\n%\\mbox{ }\n%\\newpage\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=errbestfitlogdj1790.eps,height=8cm,width=8.5cm}\n\\caption{\\protect\\label{reslombdj} Relative error between the fit\nwith equation $(\\protect\\ref{2feq})$ and the data as a function of time.}\n\\vspace{12cm}\n\\end{center}\n\\end{figure}\n\n\n\n\\mbox{ }\n\\newpage\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nIndex & Year & $z$ \\\\ \\hline\nDJ & 2040 & $-0.68$ \\\\ \\hline\nDJ & 2050 & $-0.77$ \\\\ \\hline\nDJ & 2060 & $-0.86$ \\\\ \\hline\nS\\&P & 2040 & $-1.10$ \\\\ \\hline\nS\\&P & 2050 & $-1.25$ \\\\ \\hline\nS\\&P & 2060 & $-1.40$ \\\\ \\hline\nLatin Am & 2040 & $-0.89$ \\\\ \\hline\nLatin Am & 2050 & $-1.04$ \\\\ \\hline\nLatin Am & 2060 & $-1.18$ \\\\ \\hline\nEurope & 2040 & $-0.89$ \\\\ \\hline\nEurope & 2050 & $-1.05$ \\\\ \\hline\nEurope & 2060 & $-1.20$ \\\\ \\hline\nEAFE & 2040 & $-0.87$ \\\\ \\hline\nEAFE & 2050 & $-1.00$ \\\\ \\hline\nEAFE & 2060 & $-1.13$ \\\\ \\hline\nWorld & 2040 & $-0.88$ \\\\ \\hline\nWorld & 2050 & $-1.01$ \\\\ \\hline\nWorld & 2050 & $-1.14$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{ztable}Values for the exponent $z$ from the fits with\nequation (\\ref{pow}) shown in figures \\ref{pop2030} - \\ref{windex2060} }\n\\end{table}\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \\hline\ndata & number of points & time period & $t_c$ & $\\beta$ & $\\omega$ &\n$\\lambda$ & $\\omega_{\\rm spectrum}$ & Peak power \\\\ \\hline\nset 1 & $18$ & $\\left[ 0:1998\\right]$ & $2056$ & $-1.39$ & $6.3$ & $2.7$ &\n$5.7$ & $4.3$\n\\\\ \\hline\nset 2 & $17$ & $\\left[ 1000:1998\\right]$ & $2053$ & $-1.35$ & $6.2$ & $2.8$\n& $5.8$ & $5.0$\n\\\\ \\hline\nset 3 & $16$ & $\\left[ 1250:1998\\right]$ & $2059$ & $-1.45$ & $6.5$ & $2.6$\n& $5.8$ & $5.9$\n\\\\ \\hline\nset 4 & $15$ & $\\left[ 1500:1998\\right]$ & $2058$ & $-1.43$ & $6.5$ & $2.6$\n& $5.9$ & $6.1$\n\\\\ \\hline\nset 5 & $19$ & $\\left[ 0:1999.75\\right]$ & $2062$ & $-1.46$ & $6.5$ & $2.6$\n& $6.1$ & $4.5$ \\\\ \\hline\nset 6 & $17$ & $\\left[ 0:1990\\right]$ & $2043$ & $-1.24$ & $5.8$ & $2.9$\n& $5.3$ & $3.4$ \\\\ \\hline\nset 7 & $16$ & $\\left[ 0:1980\\right]$ & $2043$ & $-1.25$ & $5.5$ & $3.1$\n& $5.2$ & $3.4$ \\\\ \\hline\nset 8 & $15$ & $\\left[ 0:1970\\right]$ & $2034$ & $-1.20$ & $4.9$ & $3.9$\n& $5.1$ & $3.4$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tablepop} $t_c$ is the critical time predicted\nfrom the fit of the world population data to equation (\\ref{lppow}). The\nother physical parameters $\\beta$ and $\\omega$ of the fit are also shown.\n$\\lambda = \\exp \\left( 2\\pi / \\omega \\right)$ is the prefered scale\nratio of the underlying dynamics.\n$\\omega_{\\rm spectrum}$ is the angular log-frequency obtained from the\nnon-parametric spectral analysis of the log-periodic oscillations.}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|} \\hline\nminima & $t_c$ & $\\beta$ & $\\omega$ & $\\tau$ & r.m.s. \\\\ \\hline\nfirst & $2053$ & $0.39$ & $6.5$ & $171$ & $0.23582$ \\\\ \\hline\nsecond & $2046$ & $0.36$ & $5.3$ & $240$ & $0.23584$ \\\\ \\hline\nthird & $2067$ & $0.42$ & $6.8$ & $122$ & $0.23644$ \\\\ \\hline\nfourth & $2009$ & $0.61$ & $5.5$ & $188$ & $0.27459$ \\\\ \\hline\nfifth & $2007$ & $0.62$ & $4.8$ & $206$ & $0.27461$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tabledj} $t_c$ is the critical time predicted\nfrom the fit of the logarithm of the extended Dow Jones for the period\n$\\left[ 1790:1999.75 \\right]$ to equation(\\ref{2feq}). The other\nphysical parameters $\\beta$, $\\omega$ and $\\tau$ of the fits are also shown.}\n\\end{table}\n\n\n\n\n\\end{document}\n\n\n\nDate: Sun, 08 Oct 2000 19:04:48 -0400\nFrom: Robin Hanson <rhanson@gmu.edu>\nOrganization: George Mason University\nTo: anders@moho.ess.ucla.edu, sornette@moho.ess.ucla.edu\nCC: hal@finney.org\nSubject: Finite Time Singularity\n\nHi. I thought I'd offer some comments on your revised paper, now titled\n\"Finite-time Singularity ...\" Btw, I'd be interested to be on your list of\npeople\nyou tell when you get new papers/drafts on this topic. I only accidentally\n\nheard of this new draft of yours. FYI, my paper has just be revised:\nhttp://hanson.gmu.edu/longgrow.pdf\n\nIn it I added a footnote about fitting your log-periodic model to my\ndata (it did badly). I didn't try the second order model yet, but do you\nreally think it would do much better?\n\nSection 2.2.3 says that you took the log of log GDP vs. log (tc - t) in\norder to\n\"minimize the effect of inflation and other systematic drifts.\" You do the\nsame\nfor your stock price stats. But the GDP data has already been constructed\nto\neliminate inflation and systematic drifts. And you could easily do the\nsame for\nthe stock price stats. Also, why do you not care about systematic drifts\nin\npopulation estimates?\n\nWorse, ln(ln(s*GDP) = ln(ln(s)+ln(GDP), and so whether this appears as a\nstraight line when plotted vs. ln(tc-t) is not invariant to the choice of\na\nscale s. At a minimum I think you should make it clear to readers whether\n\nyou searched over the degree of freedom of scale s in doing your fits.\nAlso, if there were a systematic drift of s(t), I really don't see how this\nform\nfixes the problem. You really need to say more about this - it is a central\npoint.\n\nYou seem to have arbitrarily cut off your population and GDP time series\nat 0 AD. You might want to explain to readers why. I would think the more\n\ndata the better.\n\nYou seem to fundamentally equivocate about how close we are to the point\nwhen the model no longer applies. For example, your abstract labels the\nupcoming transition \"abrupt\", you never say if this means 50 years, 5\nyears,\nor 5 days. But of course this is the big central question to ask, if we\nbelieve\nyour model. If GDP or population are to grow by less than a factor of two\n\nbefore the model stops applying, then we are very near the transition, and\n\"abrupt\" means more like 50 years. If abrupt means less than one year,\nthen we could have many orders of magnitude of growth to go before the\ntransition.\n\nIn your \"Discussion\" section you seem willing to entertain the idea that\nthe Dow will reach 100K in 2026, but you entertain no such large increases\nin GDP or population. Your earlier mention of fitting your model to\nearlier\nstock market bubbles and crashes suggests to the reader that you are only\nwilling to believe stock prices might go that high because you think it\nwould be a bubble.\n\nThis impression that you think that little economic or population growth\nwill occur before the transition is also supported by your \"Possible\nscenario\" section which seems to implicitly assume that the post transition\n\nworld will similar to today's world. If the economy was to grow by a\nfactor of 10,000 between now and the transition, in contrast, there would\nbe\nlittle more basis for comparing the post transition world to today's world\nthan for comparing it to the world of 0AD.\n\nYou don't fit any parameters in your models describing a move away from\nthe the models to some other regime, so your parameters don't give you\nany basis for saying how close to the transition we are. You should be\nclearer with your readers about what you are assuming and what the basis\nfor that is.\n\nYou emphasize in the paper that new technologies cannot avoid the\ntransition,\nbecause they just drive it faster. But then you suggest at the end that a\nmassive\ntransfer of resources might avoid the singularity. This just seems to\nundercut\nyour earlier point, however. If transfers could help us avoid it, why not\ntechnologies? Your argument seems clearer if you just insist\nthat nothing can avoid it.\n\nYou also suggest that space colonization might delay the transition by\nseveral centuries. But this doesn't make any sense either. The whole\npoint\nis that an infinite amount of growth could happen according to the same\naccelerating growth schedule and it would be done by about 2050. Where\nit happens is irrelevant.\n\nBtw, we are already a knowledge-based society, and even if knowledge\nbecomes more valued for its own right, that won't prevent the use of more\nand more material resources as technology allows it. All it takes is for\nmaterial resources to be a useful input into the production of knowledge.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"
}
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[
{
"name": "cond-mat0002075.extracted_bib",
"string": "\\begin{thebibliography}{}\n\n\n\\bibitem{benderorszag} Bender C, Orszag S.A (1978)\npage 147 in {\\it Advanced Mathematical Methods for Scientists and Engineers.}\nMcGraw-Hill, New York. \n\n\\bibitem{Bradford} J. Bradford DeLong (1998) \nEstimating World GDP, One Million B.C. - Present. Working paper available at \nhttp://econ161.berkeley.edu/TCEH/1998\\_Draft/World\\_GDP/Estimating\\_World\\_GDP.html\n\n\\bibitem{stateofworld} Brown LR, Flavin C (1999) \n{\\it State of the World, Millenium edition}. A Worldatch Institute report on \nProgress towards a sustainable society, W.W. Norton \\&Co. and Worldwatch \nInstitute.\n\n\\bibitem{finitesizeeffect} Cardy, J.L. editor (1988)\nFinite-size scaling (Amsterdam; New York:\nNorth-Holland; New York, NY, USA; Elsevier Science Pub. Co).\n\n\\bibitem{Choptuik} Choptuik M.W (1999) Universality and \nscaling in gravitational collapse of a massless scalar. {\\it Physical Review \nLetters} 70: 9-12 \\& Critical behaviour in gravitational collapse. {\\it \nProgress of Theoretical Physics} Supplement 136: 353-365.\n\n\\bibitem{Cohenscience} Cohen J.E. (1995) Population growth and \nEarth's human carrying capacity. {\\it Science} 269: 341-346.\n\n\\bibitem{Corberi2} Corberi F, Gonnella G, \nLamura A (2000) Structure and rheology of binary mixtures in shear flow \n{\\it Physical Review E} 61: 6621-6631.\n\n\\bibitem{erzan} Derrida B, Eckmann JP, Erzan A\n(1983) Renormalisation groups with periodic and aperiodic orbits. \n{\\it Journal of Physics A} 16: 893-906 (1983).\n\n\\bibitem{Gunsdiam} Diamond JM (1997) {\\it Guns, germs, and \nsteel : the fates of human societies.} W.W. Norton \\& Co. New York.\n\n\\bibitem{DombGreen} Domb C, Green M.S. (1976)\n{\\it Phase transitions and critical phenomena.} Academic Press, London, \nNew York.\n\n\\bibitem{Drozdz} Drozdz S, Ruf F, Speth J \nWojcik M (1999) Imprints of log-periodic self-similarity in the stock market.\n{\\it European Physics Journal B} 10: 589-593.\n\n\\bibitem{Dubrulle} Dubrulle B, Graner F,\nSornette D (1997) {\\it Scale invariance and beyond.} EDP Sciences and \nSpringer, Berlin.\n\n\\bibitem{Elias} Elias D (1999) {\\it Dow 40,000 : Strategies for \nProfiting from the Greatest BullMarket in History.} McGraw-Hill ?\n\n\\bibitem{comment} The fits have been performed using \nthe ``amoeba-search''\nalgorithm (see {\\it Numerical Recipes} by W.H. Press, B.P. Flannery, S.A.\nTeukolsky and W.T. Vetterling, Cambridge University Press, Cambridge\nUK, 1992) minimizing the variance of the fit to the data. We stress that all\nthree linear variables $A$, $B$ and $C$ are slaved to the other nonlinear\nvariables\nby imposing the condition that, at a local minimum, the variance has zero first\nderivative with respect these variables. Hence, they should not be regarded as\nfree parameters, but are calculated solving three linear equations using\nstandard techniques including pivoting. Note in addition that the phase $\\phi$\nin (\\ref{lppow}) is just a (time) unit as are the coefficients $A$, $B$ and\n$C$. The key physical variables are thus $t_c$, $\\beta$ and $\\omega$.\n\n\\bibitem{vonFoerster} von Foerster \nH, Mora P.M, Amiot L. W (1961) Population Density and Growth. {\\it Science}\n133: 1931-1937\n\n\\bibitem{Doomsdaypaper} von \nFoerster H, Mora P.M, Amiot L.W (1960) Doomsday: Friday 13 November\nA.D. 2026. {\\it Science} 132: 1291-1295.\n\n\\bibitem{cycles} More information \nabout the foundation can be found at http://www.cycles.org/cycles.htm. However,\nit seems that the foundation is not very active presently.\n\n\\bibitem{Glassman} Glassman JK, Hassett KA (1999)\n{\\it DOW 36,000: The New Strategy for Profiting from the Coming Rise in the \nStock Market.} Times Books ?\n\n\n\\bibitem{global} Global Financial Data, Freemont \nVillas, Los Angeles, CA 90042. The data use are free samples available at\nhttp://www.globalfindata.com/.\n\n\\bibitem{critical} Hahne F.J (1983) {\\it Critical Phenomena, \nLecture Notes in Physics 186} page 209. Springer, Berlin, Heidelberg.\n\n\\bibitem{Hanson} Hanson R (2000) Could it happen again? \nLong-term growth as a sequence of exponential modes. Working paper \navailable at http://hanson.gmu.edu/longgrow.html.\n\n\\bibitem{Hern} Hern W.M (1993) Is human culture carcinogenic for \nuncontrolled population growth and ecological destruction? \n{\\it BioScience} 43: 768-773. He concludes that the sum of human activities, \nviewed over the past tens of thousand of years, exhibits all four major \ncharacteristics of a malignant process: rapid uncontrolled growth; invasion \nand destruction of adjacent tissues (ecosystems, in this case); metastasis \n(colonization and urbanization, in this case); and dedifferentiation (loss\nof distinctiveness in individual components as well as communities throughout \nthe planet).\n\n\\bibitem{Herroux} Herrmann, H.J. and Roux, S., editors (1990)\nStatistical models for the fracture of disordered media (Amsterdam; \nNew York: North-Holland ; New York, N.Y., U.S.A.)\n\n\\bibitem{failure} Johansen A, Sornette D (1998)\nEvidence of discrete scale invariance by canonical averaging. {\\it \nInternational Journal of Modern Physics C} 9: 433-447 and references therein.\n\n\\bibitem{bubmodel} Johansen A, Sornette D (1999)\nCritical crashes. {\\it Risk Magazine} 12: 91-94.\n\n\\bibitem{crash} Johansen A, Sornette D\nLedoit 0 (1999) Predicting Financial Crashes using discrete scale invariance.\n{Journal of Risk} 1: 5-32 and references therein.\n\n\\bibitem{JLS} Johansen A, Ledoit O, \nSornette (2000) Crashes as critical points. {\\it International Journal of \nTheoretical and Applied Finance} 3: 219-255.\n\n\\bibitem{faicri} Johansen A, Sornette D (2000)\nCritical ruptures. {\\it European Physics Journal B} 18: 163-181\n(e-print at http://arXiv.org/abs/cond-mat/0003478)\n\n\\bibitem{earthquake} Johansen A, Saleur H, \nSornette D (2000) New Evidence of Earthquake Precursory Phenomena in the 17 \nJan. 1995 Kobe Earthquake, Japan. {\\it European Physics Journal B} 15: 551-555\nand references therein.\n\n\\bibitem{emerg} Johansen A, Sornette D (2000) \nLog-periodic power law bubbles in Latin-American and Asian markets and \ncorrelated anti-bubbles in Western stock markets: An empirical study. \nin press in Int. J. Theor. Appl. Finance. Available at \nhttp://arXiv.org/abs/cond-mat/9907270 \n\n\\bibitem{nasdaq} Johansen A, Sornette D (2000)\nThe Nasdaq crash of April 2000: Yet another example of log-periodicity in a \nspeculative bubble ending in a crash. {\\it European Physics Journal \nB} 17,: 319-328\n(e-print at http://arXiv.org/abs/cond-mat/0004263).\n\n\\bibitem{Kadlec} Kadlec CW (1999) {\\it Dow 100,000: Fact or \nFiction} Prentice Hall Press ?.\n\n\\bibitem{Kapitza} Kapitza SP (1996), Phenomenological theory \nof world population growth. {\\it Uspekhi Fizichskikh Nauk} 166: 63-80.\n\n\\bibitem{Kremer} Kremer M (1993) Population growth and \ntechnological change: One million B.C. to 1990. {\\it Quarterly Journal of \nEconomics} 108: 681-716.\n\n\\bibitem{Mehra} Mehra R, Prescott E (1985) Title ?\n{\\it Journal of Monetary Economics 15}: 145-161.\n\n\\bibitem{Moffatt} Moffatt H.K (2000) Euler's disk and its \nfinite-time singularity. {\\it Nature} 404: 833-834.\n\n\\bibitem{Pumiersiggia} Pumir A, Siggia E.D (1992) \nVortex morphology and Kelvin theorem. {\\it Physical Review A} 45: R5351-5354.\n\n\\bibitem{Raan} van Raan AFJ (2000) On growth, ageing and \nfractal differentitation of science. {\\it Scientometrics} 47: 347-362.\n\n\\bibitem{Rascle} Rascle M, Ziti C (1995) Finite-time \nblow-up in some models of chemotaxis. {Journal of Mathematical Biology} 33: \n388-414.\n\n\\bibitem{Ritz} Rietz TA, Mehra R,\n Prescott EC (1988) The Equity Risk Premium: A Solution?\n{\\it Journal of Monetary Economics} 22: 117-136.\n\n\\bibitem{Romer} Romer D(1996) {\\it Advanced macroeconomics.}\nMcGraw-Hill Companies New York.\n\n\\bibitem{Romer2} Romer PM (1990) Endogeneous technological change.\n{\\it Journal of Political Economy} 98: S71-S102.\n\n\\bibitem{salsor} Saleur, Sornette (1996)\nComplex exponents and log-periodic corrections in frustrated systems.\n{\\it Journal de Physique I France} 6: 327-355.\n\n\\bibitem{SSS} Saleur H, Sammis CG, Sornette D\n(1996) Renormalization group theory of earthquakes. {\\it Nonlinear Processes\nin Geophysics} 3: 102-109.\n\n\\bibitem{statementAcads} \nScience Summit on World Population: A Joint Statement by 58 of the \nWorld's Scientific Academies (1994) {\\it Population and Development Review}\n20: 233-238.\n\n\\bibitem{Simon} Simon J.L (1996) {\\it The Ultimate Resource 2?} \nPrinceton University Press, Princeton, NJ.\n\n\\bibitem{SJ97} Sornette D, Johansen A (1997) Large \nfinancial crashes. {\\it Physica A} 245: 411-422.\n\n\\bibitem{reviewsor} Sornette D (1998) Discrete scale \ninvariance and complex dimensions. {\\it Physics Reports} 297: 239-270.\n\n\\bibitem{Nego} {\\it Earth negotiation \nBulletin} Vol. 15, No. 34 March 27 2000. Available at:\nhttp://www.iisd.ca/linkages/download/pdf/enb1534e.pdf.\n\n\\bibitem{UShistory} Even though U.S.A was recognised as a nation by \nthe Paris Treaty in 1783, a number of events point to the fact it was not \n{\\it established} as a nation before $\\approx$ 1790. They are as follows. \n1) The constitution went into effect in March 1789, having been ratified by \nNew Hampshire as the ninth state on June 21, 1788. 2) The last of the thirteen\nstates, Rhode Island, first approved it on May 29, 1790. 3) The first census \nin the U.S. was made in 1790. 4) The Naturalisation Act of 1790 grants the \nright of U.S. citizenship to all ``free white persons.'' 5) In 1790, the\nFederal Government declared that it was redeeming the SCRIP MONEY that was\nissued during the Revolutionary War. 6) At about this time, the Government\nannounced the creation of the first bank of the United States in conjunction \nwith the sale of \\$10,000,000 dollars in shares of stock.\n\n\\bibitem{Chaline2} Nottale L., Chaline J., Grou P. (2000) Les arbres de l'\\'evolution\n(Hachette Litterature, Paris) 379 p.\n\n\\bibitem{Chaline3} Nottale L., Chaline J., Grou P. (2000) in ``Fractals 2000 in Biology\nand Medicine'', Proceedings of Third International Symposium, Ascona,\nSwitzerland, March 8-11, 2000, Ed. G. Losa, Birkh�user Verlag.\n\n\\bibitem{Meyer1} Meyer F. (1947) L'acc\\'el\\'eration \\'evolutive.\nEssai sur le rythme \\'evolutif et son \ninterpr\\'etation quantique. Librairie des Sciences et des Arts, Paris, 67p.\n\n\\bibitem{Meyer2} [14] Meyer F. (1954) Probl\\'ematique de l'\\'evolution. P.U.F., 279p.\n\n\\bibitem{Grou} Grou P. (1987,1995) L'aventure \\'economique. L'Harmattan, Paris, 160 p.\n\n\\bibitem{Braudel} Braudel F. (1979) Civilisation mat\\'erielle, \\'economie et capitalisme. \nA. Colin \n\n\\bibitem{Gilles} Gilles B. (1982) Histoire des techniques. Gallimard\n\n\n\\end{thebibliography}"
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cond-mat0002076
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\input{abstract}
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{
"name": "abstract.tex",
"string": "We investigate small-world networks from the point of view of their origin. \nWhile the characteristics of small-world networks are now fairly well \nunderstood, there is as yet no work on what drives the emergence of such a\nnetwork architecture. In situations such as neural or transportation \nnetworks, where a physical distance between the nodes of the network exists,\nwe study whether the small-world topology arises as a consequence of a \ntradeoff between maximal connectivity and minimal wiring. Using simulated\nannealing, we study the properties of a randomly rewired network as the\nrelative tradeoff between wiring and connectivity is varied. When the\nnetwork seeks to minimize wiring, a regular graph results. At the other\nextreme, when connectivity is maximized, a near random network is obtained. \nIn the intermediate regime, a small-world network is formed. However, \nunlike the model of Watts and Strogatz (Nature {\\bf 393}, 440 (1998)), \nwe find an alternate route to small-world behaviour through the formation\nof hubs, small clusters where one vertex is connected to a large number\nof neighbours. \n"
},
{
"name": "references.tex",
"string": "\\bibitem{3:watts1} Watts, D. J. and Strogatz, S. H. {\\sl Collective\nDynamics of `small world' networks. } Nature, 393:440-442, 1998.\n\\bibitem{3:barabasi} Barab\\'{a}si, A.-L. and Albert, R. {\\sl Emergence of\nScaling in Random Networks.} cond-mat/9910332.\n\\bibitem{3:kasturi} Kasturirangan, R. {\\sl Multiple Scales in Small-World\nGraphs.} cond-mat/9904055.\n\\bibitem{3:press} Press, W. H., Teukolsky, S. A., Vetterling, W. T.\nand Flannery, B. P. {\\sl Numerical Recipes in C: The Art of Scientific\nComputing.} Cambridge University Press, Second Edition, 1988.\n\\bibitem{3:watts2} Watts, D. J. {\\sl Small Worlds: The Dynamics of Networks\nbetween Order and Randomness.} Princeton University Press, 1999.\n\\bibitem{3:barrat} Barrat, A. and Weigt, M. {\\sl On the properties of\nsmall-world network models.} cond-mat/9903411.\n\\bibitem{3:kleinberg} Kleinberg, J. M., Kumar, R., Raghavan, P., \nRajagopalan, S.\nand Tomkins, A. S. {\\sl The Web as a Graph: Measurements, Models, and Methods.}\nAsano, T. et al. (eds.): COCOON '99, LNCS 1627, 1-17, Springer-Verlag Berlin, Heidelberg, 1999.\n\\bibitem{3:matthews} Matthews, R. {\\sl Get connected.} New Scientist, 4\nDecember 1999.\n\n"
},
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"name": "smallw.tex",
"string": "\\documentstyle[11pt,twoside,fleqn,psfig]{article}\n\\pagestyle{myheadings}\n\\textwidth=6.25in\n\\oddsidemargin=0.25in\n\\evensidemargin=0.25in\n\\topmargin=-0.1in\n\\footskip=0.8in\n\\parindent=0.0cm\n\\parskip=0.3cm\n\\textheight=8.00in\n\\mathindent=0.0in\n\\setcounter{tocdepth}{3}\n\\setcounter{secnumdepth}{3}\n\\newcounter{mycounter}\n\\sloppy\n\n\\newenvironment{my_item}{\\begin{list}{$\\bullet$}{\\setlength{\\itemsep}{0.1mm}\n\\setlength{\\parsep}{0.1mm} \\setlength{\\topsep}{0.1mm}\n\\setlength{\\rightmargin}{\\leftmargin}}}{\\end{list}}\n\n\\newenvironment{my_enum}{\\begin{list}{\\themycounter.}{\\usecounter{mycounter}\n\\setlength{\\itemsep}{0.1mm} \\setlength{\\parsep}{0.1mm}\n\\setlength{\\topsep}{0.1mm}}}{\\end{list}}\n\n\\font\\BbbNormal = msbm10\n\\font\\BbbScript = msbm7\n\\newfam\\Bbbfam \\def\\Bbb{\\fam\\Bbbfam\\BbbNormal}\n\\textfont\\Bbbfam=\\BbbNormal\n\\scriptfont\\Bbbfam=\\BbbScript\n\n\\begin{document}\n\\begin{center}\n{\\huge \\bf Small-worlds: How and why}\n\\end{center}\n\n\\begin{center}\n{\\large Nisha Mathias$^{1}$ and Venkatesh Gopal$^{2}$}\n\\end{center}\n\n\\begin{center}\n$^{1}${\\it Department of Computer Science and Automation, \nIndian Institute of Science,\\\\ Bangalore 560012, INDIA; email:\nnisha@csa.iisc.ernet.in}\n\\end{center}\n\\vspace{-1cm}\n\\begin{center}\n$^{2}${\\it Raman Research Institute, Sadashivanagar, Bangalore\n560080, INDIA; email: vgopal@rri.ernet.in}\n\\end{center}\n\\date{}\n\n\\begin{abstract}\n\\input{abstract}\n\\end{abstract}\n\\input{text}\n\n\\begin{center}\n{\\bf Acknowledgements}\n\\end{center}\nN.M. thanks V. Vinay for very useful discussions.\n\n\\begin{thebibliography}{10}\n\\input{references}\n\\end{thebibliography}\n\n\\end{document}\n"
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"name": "text.tex",
"string": "\\section{Introduction}\nCoupled systems may be modelled as networks or graphs, where the\nvertices represent the elements of the system, and the edges represent\nthe interactions between them. The topology of these networks\ninfluences their dynamics. Network topologies may be random, where\neach node or vertex is randomly wired to any other node; or they may\nbe regular, with each vertex being connected to a fixed number of\nneighbouring nodes. Watts and Strogatz \\cite{3:watts1} showed that\nbetween these two extremes lay another regime of connectivity, which\nthey called a {\\em small-world} network. Such networks are `almost'\nregular graphs, but with a few long range connections.\n\nWhat does it mean to have a `long-range' connection? Consider a few\nexamples of networks: neurons in the brain, transportation and social\nnetworks, citations of scientific papers and the world wide web.\nThere is a difference between the elements of this list. Social networks, \npaper citations and the internet, are networks where the\nlinks have no physical distance. For example, a link between two\nwebsites physically far apart is no different from one between two\nmachines that are next to one another. Neural and transportation networks\nhowever, have a well defined physical distance between their nodes.\nIn this paper, we investigate how placing a {\\em cost} on the length of an\nedge affects the connectivity of the network.\n\nWe now briefly describe the small-world model of Watts and Strogatz\n(WS) and also introduce the notation that we shall\nuse. WS considered a ring lattice; $n$ sites arranged at regular\nintervals on a ring, with each vertex connected to\n$k$ nearest neighbours. Disorder\nis introduced into the graph by randomly rewiring \neach of the edges with a probability $p$. While at $p = 0$, the graph \nremains $k$-regular, at $p = 1$, a random graph results.\nThey quantified the structural properties of this lattice by two \nparameters, $L$ and $C$. $L$, the {\\em characteristic path length} \nreflects the average connectivity of the network, while $C$, the \n{\\em clustering coefficient} measures the extent to which neighbours\nof a vertex are neighbours of each other. Networks exhibiting\nsmall-world behaviour are characterized by low characteristic path length, and\nhigh clustering coefficient. Finally, we point out\nthat there are two kinds of distances in a graph. One is the {\\em graph}\ndistance, the minimal number of links between any two vertices\nof the graph. The other is the Euclidean or {\\em physical} distance\nbetween these vertices.\n\nAlthough recent work has shown small-worlds to be pervasive in a \nrange of networks that arise from both natural and man-made technology \n\\cite{3:watts1,3:barabasi}, the hows and whys of this ubiquity \nhave not been explained. The fact\nthat small-worlds seem to be one of nature's `architectural'\nprinciples, leads us to ask what constraints might force networks to\nchoose a small-world topology. We attempt to understand the emergence\nof the small-world topology in networks where the physical distance is\na criterion that cannot be ignored.\n\n\\section{Can small-worlds arise as the result of an Optimization?}\n\nConsider a toy model of the brain. Let us assume that it consists of\nlocal processing units, connected by wires. What constraints act on\nthis system? On the one hand, one would want the highest connectivity\nbetween the local processing units so that information could be\nexchanged as fast as possible. On the other, it is wasteful to wire\neverything to everything else. The energy requirements are higher,\nmore heat is generated, and more material needs to be used, and\nconsequently, more space is occupied. Unrealistic though this model\nis, it motivated us to examine whether small-worlds would emerge as\nthe result of these constraints.\n\nThe concept of multiple scales was introduced by Kasturirangan\n\\cite{3:kasturi}, where he asserted that the fundamental mechanism\nbehind the small-world phenomena is not disorder or randomness, but\nthe presence of edges of many different length scales. The {\\em length\nscale} of a newly introduced edge $e_{ij}$, is defined to be the graph\ndistance between vertices $i$ and $j$ {\\em before} the\nedge was introduced. He argued that the distribution of length scales\nof the new edges is significantly more important than whether the new\nedges are long, medium or short range.\n\n\\begin{figure}[!htbp]\n\\centerline{\\psfig{figure=/figs/ws.250.4.0.125000.eps,width=7.3cm,height=7.3cm} \n \\psfig{figure=/figs/ws.250.4.1.000000.eps,width=7.3cm,height=7.3cm}}\n\\caption{Edge length scale distribution at (a) $p$=0.125, and (b) $p$=1.00; \n$p$ being the degree of disorder introduced into the $n$=250, $k$=4 \nregular network using the WS rewiring procedure to introduce small-world\nbehaviour. The inset in\n(a) displays the distribution of all length scales other than the unit scale.\nBoth plots are averaged over 25 samples.} \n\\label{fig:small-world and random network}\n\\end{figure}\n\nWe obtain the edge scale distribution by binning the length scales of all the\nedges in a graph, with respect to its corresponding regular graph.\nStarting with a $k$-regular graph, and using the WS\nrewiring procedure, \nwe study the edge scale distribution at various degrees of disorder. \nFigure~\\ref{fig:small-world and random network} shows the edge scale\ndistribution at two degrees of disorder. Figure~\\ref{fig:small-world and \nrandom network}(a) shows the edge\nscale distribution in the small-world regime, (p = 0.125). Due to\nintroduction of a small amount of disorder, a few edges are rewired to\nbecome far and consequently have a large length scale. However,\nthey are too few in number to significantly alter the length scale\ndistribution and hence, the edges of unit length scale dominate the\ndistribution. Figure~\\ref{fig:small-world and \nrandom network}(b) shows the edge scale distribution at $p=1$,\na random graph. Here, the edges are uniformly distributed over the\nentire length scale range, that is, from 1 to $n/k$. The network\nstill retains a slight bias towards the unit length scale. At both\nthese degrees of randomness however, the characteristic path length\nscales logarithmically with $n$. There thus appears to be some factor\nthat constrains the distribution of edge length scales to (a) and not\n(b), namely, restricting the rewiring to just a few far edges. We\nquestion whether the association of a cost to each edge, proportional\nto its length, serves to work as this constraint.\n\n\\section{Optimization model}\n\nWe use the method of simulated annealing\n\\cite{3:press} to find the network which results\nin the best optimization of the objective function $E$, whose\nminimization is the goal of the procedure. The network used in the\nmodel is that of vertices arranged symmetrically along a ring. The\nsize of the network, $n$, as well as the total number of edges are\nfixed. So also are the positions of the vertices, which are equally\nspaced along the circumference of the circle. Initially, the network\nis $k$-regular, similar to the WS model. The configuration has an\nassociated energy $E$, a function of both its wiring cost and the\naverage degree of separation between its vertices. The\nobjective function $E$ is taken to be, \n\\begin{eqnarray*} E &= &\\lambda L + (1-\\lambda)W, \\end{eqnarray*} \na linear combination of the\nnormalized characteristic path length $L$, and the normalized wiring\ncost $W$. The characteristic path length $L$, as defined by Watts and\nStrogatz, is the average distance between all pairs of vertices, given\nby \\begin{eqnarray*} L &= &\\frac{1}{n(n-1)} \\sum \\limits_{i \\neq j}\nd_{ij} , \\end{eqnarray*} where $d_{ij}$ is the number of links along\nthe shortest path between vertices $i$ and $j$. It is therefore a\nmeasure based on graph distance, and reflects the global\nconnectivity among all vertices in the graph. The wiring cost $W$, in\ncontrast, is a measure of the physical distance between\nconnected vertices. The cost of wiring an edge $e_{ij}$, is taken to\nbe the Euclidean distance between the vertices $i$ and $j$. Hence,\nthe total wiring cost is \n\\begin{eqnarray*} W &= &\\sum \\limits_{e_{ij}}\n\\sqrt{(x_i - x_j)^2 + (y_i - y_j)^2 }, \\end{eqnarray*} \nwhere\n$(x_i,y_i)$ are the coordinates of vertex $i$ on the ring lattice.\nThe characteristic path length $L$ is normalized by $L(0)$, the path\nlength in the $k$-regular network; while $W$ is normalized by the\ntotal wiring cost that results when the edges at each vertex are the\nlongest possible, namely, when each vertex is connected to its\ndiametrically opposite vertex, and to the vertices surrounding it.\nThe parameter $\\lambda$ is varied depending on the relative importance\nof the minimization of $L$ and $W$. One can regard $(1-\\lambda)$\nas the wiring cost per unit length, and $W$ as the length of\nwiring required.\n\nStarting from the initial regular network, a standard Monte Carlo\nscheme \\cite{3:press} is used to search for the energy minimum.\nSimilar to the WS model, duplicate edges and loops were not allowed,\nand it was ensured that the rewiring did not result in isolated\nvertices. The starting value for $T$, the annealing `temperature',\nwas initially chosen to be the initial energy, $E$, itself. The \ntemperature was then lowered in steps, each amounting to a 10\npercent decrease in $T$. Each value of $T$ was held constant for $150$\nreconfigurations, or for $15$ successful reconfigurations, whichever\nwas earlier.\n\n\\section{Optimized Networks: Results}\n\nSince minimum characteristic path length, and minimum wiring cost are\ncontradictory goals, the optimization of either one or the other will\nresult in networks at the two ends of the randomization spectrum. As\nexpected, at $\\lambda=0$, when the optimization function concentrates\nonly on minimizing the cost of wiring edges, a regular network emerges with\nuniform connectivity and high characteristic path length ($L \\sim n$).\nThe edge scale distribution shows all edges to be concentrated almost\nentirely within the unit length scale, as shown in\nFig.\\ \\ref{fig:opt0-1} (a). At $\\lambda=1$, when only the\ncharacteristic path length is to be minimized, again of no surprise,\nthe optimization results in a near random network ($L \\sim \\ln n$). The\nedge scale distribution shown in Fig.\\ \\ref{fig:opt0-1} (b) has edges\nhaving lengths distributed uniformly over the entire length scale\nrange.\n\n\\begin{figure}[!htbp]\n\\centerline{\n \\psfig{figure=/figs/250.4.edgeScaleDist.0.000000.eps,width=7.3cm,height=7.3cm} \n \\psfig{figure=/figs/250.4.edgeScaleDist.1.000000.eps,width=7.3cm,height=7.3cm}}\n\\caption{Edge scale distribution resulting from optimization at (a)\n$\\lambda=0$, and (b) $\\lambda=1$ \nfor a network having $n=250$, $k=4$. Both distribution plots are averaged\nover 25 simulations.} \n\\label{fig:opt0-1}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\centerline{\n \\psfig{figure=/figs/edgeScaleDist.0.000000.eps,width=5.0cm,height=5.0cm}\n \\psfig{figure=/figs/edgeScaleDist.0.000500.eps,width=5.0cm,height=5.0cm}\n \\psfig{figure=/figs/edgeScaleDist.0.005000.eps,width=5.0cm,height=5.0cm}}\n\\centerline{\n \\psfig{figure=/figs/edgeScaleDist.0.012500.eps,width=5.0cm,height=5.0cm}\n \\psfig{figure=/figs/edgeScaleDist.0.025000.eps,width=5.0cm,height=5.0cm}\n \\psfig{figure=/figs/edgeScaleDist.0.050000.eps,width=5.0cm,height=5.0cm}} \\centerline{\n \\psfig{figure=/figs/edgeScaleDist.0.125000.eps,width=5.0cm,height=5.0cm}\n \\psfig{figure=/figs/edgeScaleDist.0.250000.eps,width=5.0cm,height=5.0cm}\n \\psfig{figure=/figs/edgeScaleDist.0.500000.eps,width=5.0cm,height=5.0cm}}\n\\centerline{\n \\psfig{figure=/figs/edgeScaleDist.0.750000.eps,width=5.0cm,height=5.0cm}\n \\psfig{figure=/figs/edgeScaleDist.0.850000.eps,width=5.0cm,height=5.0cm}\n \\psfig{figure=/figs/edgeScaleDist.1.000000.eps,width=5.0cm,height=5.0cm}}\n\\caption{Edge scale distribution for an $n$=100, $k$=4 network at various\n$\\lambda$: \n(a) 0.0, (b) $5 \\times 10^{-4}$, (c) $5 \\times 10^{-3}$, \n(d) $1.25 \\times 10^{-2}$, (e) $2.5 \\times 10^{-2}$, (f) $5 \\times 10^{-2}$, \n(g) $1.25 \\times 10^{-1}$, (h) $2.5 \\times 10^{-1}$, (i) $5 \\times 10^{-1}$,\n(j) $7.5 \\times 10^{-1}$, (k) $8.5 \\times 10^{-1}$, (l) 1.0. \nThe inset in each plot shows the distribution of all scales with the unit \nlength scale excluded. Each plot is an average over 40 simulations.} \n\\label{fig:optEdgeScaleDist}\n\\end{figure} \n\n\\subsection{The emergence of hubs}\n\nAt intermediate values of $\\lambda$, the optimization model results in\n{\\em hubs}, that is, a group of nodes connected to a single node. Due\nto the constraint which seeks to minimize physical distance between\nconnected vertices, hubs are formed by vertices close to one another.\nIn addition, the minimization of graph distance ensures the existence\nof connections between hub centres, enabling whole hubs to communicate\nwith each other. The edges at any hub centre therefore, span a wide\nrange of length scales. Hubs emerge due to the\ncontribution of $L$ to the optimization function. The formation of\nhubs, en route to the emergence of a small-world network, has so far not been reported in the literature.\n\nThe extreme situation is a `universal' hub: a single node, \nwith all other nodes having connections to it. However,\nexcept for the situations when the cost of wiring is negligible, \n we find that the optimization does not result in a\nuniversal hub. This is apparent, since a universal hub requires all\nthe remaining $n-1$ vertices to have connections to the vertex at the\ncentre of the hub, resulting in length scales which span the entire\nscale range, long connections being prohibitively expensive. A real\nworld example of such a universal hub network is unlikely since a\nlarge hub is a bottleneck to traffic through it, resulting in\novercrowding at the hubs \\cite{3:kasturi}. Hence, the need for\nmultiple, and consequently smaller, hubs.\n\nWatts \\cite{3:watts2} defines the {\\em significance} of a vertex $v$, as the \ncharacteristic path length of its neighbourhood $\\Gamma(v)$, in its absence. \nHub centres are significant since they\ncontract distances {\\em between} every pair of vertices within the\nhub. Thus, vertex pairs although not directly connected, are\nconnected via the single common vertex. Hence, the average\nsignificance, a measure which reflects the number of contractions, is\nconsiderable. Thus, in contrast to the WS model, where networks\nbecome small due to shortcuts, here smallness can be attributed to the\nsmall fraction of highly significant vertices.\n\nThe formation of the universal hub at sufficiently large $\\lambda$ is\nnot surprising, since it can be shown that for a network that\nminimizes $L$ and employs only rewirings, a universal hub will effect\nthe largest minimization. The formation of {\\em multiple} hubs\nhowever is due to the role played by $W$ in the optimization, which is\nto constrain the physical length of edges, and therefore, the size of\nhubs. As the hubs grow, whenever the cost of edges from the hub\ncentre to farthest nodes become high, the edges break away resulting\nin multiple hubs. Thus, high wiring cost prevents the formation of\nvery large hubs, and controls both the size and number of hubs.\nFigures~\\ref{fig:hub variation} and \\ref{fig:2d hub variation},\ndemonstrate the evolution of hubs in an $n=100, k=4$ optimized network\nas $\\lambda$ is varied between 0 and 1. While Fig.\\ \\ref{fig:hub variation}\nuses ring-lattice displays to illustrate the evolution, \nFig.\\ \\ref{fig:2d hub variation} illustrates the same\nnetworks as 2d-displays. In the ring-lattice displays, vertices are fixed\nsymmetrically around the lattice, with hub centres and long-range inter-hub\nlinks being clearly visible. The 2d-displays are generated by a graph drawer \nwhich uses a spring embedder to clearly demonstrate vertex interconnectivity.\nNow, with vertices no longer fixed along a ring-lattice, short-range inter-hub\nlinks can be distinguished apart from local connectivity. \n\n\\begin{figure}[!htbp]\n\\centerline{\\psfig{figure=/figs/matrix.0.000000.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.000500.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.005000.eps,width=4.8cm,height=4.8cm}}\n\\centerline{\\psfig{figure=/figs/matrix.0.012500.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.025000.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.050000.eps,width=4.8cm,height=4.8cm}} \n\\centerline{\\psfig{figure=/figs/matrix.0.125000.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.250000.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.500000.eps,width=4.8cm,height=4.8cm}}\n\\centerline{\\psfig{figure=/figs/matrix.0.750000.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.850000.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.1.000000.eps,width=4.8cm,height=4.8cm}} \\caption{The ring lattice displays illustrate the evolution of hubs as \n$\\lambda$ is varied over the [0,1] range for an $n=100, k=4$ optimized\nnetwork. Very short inter-hub links cannot be distinguished apart from \nlocal vertex connectivity, however longer range inter-hub links are \nclearly visible. Distinct hub centres illustrate the presence of hubs, as\nwell as their variation in size and number. The single hub centre at \nthe universal hub limit is clearly illustrated. $\\lambda$:\n(a) 0.0, (b) $5 \\times 10^{-4}$, (c) $5 \\times 10^{-3}$, \n(d) $1.25 \\times 10^{-2}$, (e) $2.5 \\times 10^{-2}$, (f) $5 \\times 10^{-2}$, \n(g) $1.25 \\times 10^{-1}$, (h) $2.5 \\times 10^{-1}$, (i) $5 \\times 10^{-1}$,\n(j) $7.5 \\times 10^{-1}$, (k) $8.5 \\times 10^{-1}$, (l) 1.0.} \n\\label{fig:hub variation}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\centerline{\\psfig{figure=/figs/matrix.0.000000.2d.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.000500.2d.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.005000.2d.eps,width=4.8cm,height=4.8cm}}\n\\centerline{\\psfig{figure=/figs/matrix.0.012500.2d.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.025000.2d.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.050000.2d.eps,width=4.8cm,height=4.8cm}} \n\\centerline{\\psfig{figure=/figs/matrix.0.125000.2d.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.250000.2d.eps,width=5.5cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.500000.2d.eps,width=4.8cm,height=4.8cm}}\n\\centerline{\\psfig{figure=/figs/matrix.0.750000.2d.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.0.850000.2d.eps,width=4.8cm,height=4.8cm}\n \\psfig{figure=/figs/matrix.1.000000.2d.eps,width=4.8cm,height=4.8cm}}\n\\caption{Illustrates the evolution of hubs for an $n=100, k=4$ optimized \nnetwork as $\\lambda$ is varied over the same [0,1] range as the previous\nfigure. The same networks are displayed as 2d-graphs using a graph generator \nwith a spring embedder. Now since vertices are not displayed as\nbeing fixed along a ring lattice, vertex interconnectivity, as well as the\nemergence of hubs and their variation in size and number is well illustrated. \n$\\lambda$:\n(a) 0.0, (b) $5 \\times 10^{-4}$, (c) $5 \\times 10^{-3}$, \n(d) $1.25 \\times 10^{-2}$, (e) $2.5 \\times 10^{-2}$, (f) $5 \\times 10^{-2}$, \n(g) $1.25 \\times 10^{-1}$, (h) $2.5 \\times 10^{-1}$, (i) $5 \\times 10^{-1}$,\n(j) $7.5 \\times 10^{-1}$, (k) $8.5 \\times 10^{-1}$, (l) 1.0.}\n\\label{fig:2d hub variation}\n\\end{figure}\n\n\n\\subsection{Hub evolution}\n\nWe now detail the evolution of hubs using the edge scale distribution\nshown in Fig.\\ \\ref{fig:optEdgeScaleDist}, and hub variation described \nby Figs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}. \nAll three figures show the same $n=100$ and $k=4$ network at various \n$\\lambda$.\n\nIn Figs.\\ \\ref{fig:optEdgeScaleDist} and \\ref{fig:2d hub variation}(a),\nthe optimization results in a near regular network, with hardly any\nhubs. When the cost reduces slightly to allow for an increase in edge\nwiring, small hubs are formed. For increasing, but very small\n$\\lambda$, Figs.\\ \\ref{fig:optEdgeScaleDist}(b-c) show the edges\nto be almost entirely concentrated in the unit length scale, with very\nfew longer edges. The non-unit length scale edges account for very few and\nvery small hubs, as illustrated in Figs.\\ \\ref{fig:2d hub variation}(b-c). \nDue to their small size, and very short inter-hub links,\nthe hubs are indistinguishable from local vertex connectivity in the\nring lattice displays in Figs.\\ \\ref{fig:hub variation}(a-d).\n\nA slight fall in the wiring cost permits an increased number of hubs.\nThe high cost of wiring constrains hubs to be still rather small. \nHence, the distribution of scales in\nFig.\\ \\ref{fig:optEdgeScaleDist}(d) still shows only two length\nscales. However, there is a marked increase in edges of the second\nlength scale. The effort towards minimizing $L$, ensures that the few\nhubs are bunched close together so that short inter-hub links can be\nused to enable the maximum distance contraction possible \n(Fig.\\ \\ref{fig:2d hub variation}(d)).\n\nWhen further reduction in cost permits increased wiring, it is mostly\nthe inter-hub links that take advantage of the reduced cost to enable\nhubs to be scattered over the entire network.\nFigure~\\ref{fig:optEdgeScaleDist}(e) shows clearly the multiple\nlength scales generated by inter-hub links. The marked increase in\nthe range of the inter-hub links (Fig.\\ \\ref{fig:hub variation}(e)),\nallows them for the first time to be visible in the ring-lattice\nplots. Figure~\\ref{fig:2d hub variation}(e) shows that there is not\nmuch variation in the hub size, except for the longer range of the\ninter-hub links.\n\nFigures~\\ref{fig:optEdgeScaleDist} and \\ref{fig:hub variation}(f-h)\ndemonstrate that as $\\lambda$ increases further, the length and number\nof far edges are progressively less constrained, and the extended length \npermits larger and many more hubs. Vertices lose their local nearest-neighbour\ninterconnectivity as hubs centres dominate in connectivity \n(Figs.\\ \\ref{fig:2d hub variation}(f-h)). However, as the size of hubs \nincreases, they are consequently reduced in number. In \nFigs.\\ \\ref{fig:2d hub variation}(i-k) one observes efforts towards \na uniform reduced local connectivity. The number of inter-hub\nlinks increases to yield greater inter-hub distance contraction.\n\nFigures~\\ref{fig:2d hub variation}(i-j) are marked by a sharp reduction in the \nnumber of hubs as the hubs balloon in size. This evolution culminates in the\nemergence of the universal hub, (Fig.\\ \\ref{fig:2d hub variation}(k)), a \nsingle hub of connectivity. \nThe formation of \nedges between the hub centre, and\nall the other $n-1$ vertices, as illustrated in Fig.\\ \\ref{fig:hub\nvariation}(k), results in a uniform distribution of non-unit length\nscale edges. Wiring, which is still associated with a cost, albeit\nsmall, ensures that the remainder of the edges are entirely local, as\ncan be observed from the distribution in Fig.\\ \\ref{fig:optEdgeScaleDist}(k).\n\nFigure~\\ref{fig:hub variation}(l) demonstrates that when $\\lambda = 1$,\nthe universal hub is retained. However, due to the absence of any effort\ntowards minimal wiring, edges are uniformly distributed across the entire\nlength scale range as shown in Fig.\\ \\ref{fig:optEdgeScaleDist}(l).\nThe loss in local connectivity can be clearly seen in comparison to\nFigs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}(k). \nOptimization towards minimizing \nonly $L$, results in the {\\em re-emergence} of multiple hubs, but the\nremoval of the constraint on wiring allows hubs to be composed of \nlargely non-adjacent vertices.\n\nIn conclusion, during the evolution of hubs illustrated in figures\n(a-l), as the cost of wiring is decreased, the following sequence is seen :\n\\begin{my_item} \n\\item Hubs emerge, and grow in size and number\n\\item Increase in the range and number of inter-hub links\n\\item Subsequent reduction in the number of hubs \n\\item Formation of a universal hub \n\\item Hubs re-emerge accompanied by a loss in local vertex interconnectivity,\nwhile the universal hub remains \n\\end{my_item}\n\n\\section{Optimization and the WS model: Some comparisons}\n\nFor the remaining part of this section, we present further results,\nbut against the backdrop of the WS model. To define small-world\nbehaviour, two ingredients were used by Watts and Strogatz. The first\nwas the characteristic path length, a global property of the graph,\nwhile the second, the clustering coefficient, $C$, is a local property\nwhich quantifies neighbourhood `cliquishness'. Associated with each \nvertex $v$, is its neighbourhood, $\\Gamma_v$, the $k_v$ vertices to which it \nis directly connected, and among which there can be a maximum of \n$k_v(k_v-1)/2$ connections. $C_v$, the clustering\ncoefficient of $v$, denotes the fraction of the links actually present \namong its neighbours, defined as\n\\begin{eqnarray*}\nC_v = \\frac{|E(\\Gamma_v)|}{\\scriptsize \\left( \\begin{array}{c} k_v \\\\\n2 \\end{array} \\right)}, \\end{eqnarray*} \nwhile $C$ is $C_v$ averaged over all $v$.\n\nThe WS and optimization models are compared with respect to their \nnormalized small-world characteristics. \nIn addition, we study their \ndifferent behaviours with respect to normalized wiring and degree. All \nresults are obtained using an $n=100, k=4$ network. \nEach plot is the result of averaging over 40 simulation runs.\n\n\\subsection{Characteristic path length}\n\nWe begin our comparison with the characteristic path length, the\nparameter whose smallness gives these networks their name.\nFigure~\\ref{fig:wsOptL} compares $L$ for the WS and optimized\nmodels. The control parameters in the two models, $\\lambda$ the\noptimization parameter, and $p$ the WS parameter, are similar in that\nthey both control the introduction of far edges. It should be\nremembered though, that while $p$ controls only the {\\em number} of\nfar edges, allowing their length scales to be uniformly distributed\nacross the entire range, $\\lambda$ constrains not only the number,\nbut also the physical {\\em length} of far edges.\n\nIn both cases, $L$ shows a sharp drop that signifies the onset of\nsmall-world behaviour. However, in contrast to the gradual drop effected by \nthe random assortment of rewired edges in the WS model, \nthe drop due to hub formation is much sharper.\nAlthough its initial reduction is \nsmaller due to the additional constraint on edge length, \nits final value is much lower than the WS random graph limit.\n\nThe variation in $L$ resulting from optimization, can be understood \nfrom the role played by the hub centres in contracting distance between pairs of\nvertices. \nBefore the cliff, the hubs being few and very small, effect a very slight\ndistance contraction. The tip of the cliff forms due to a marked increase in\nhubs, while the sharp drop occurs when extended range inter-hub links yield\na pronounced distance contraction between many distant hubs and \ntheir widely separated neighbourhoods.\nThe transition from many, small hubs to much larger and consequently\nfewer hubs, results in the gradual reduction in $L$. Finally, on the \nemergence of the universal hub, which has no counterpart in the WS\nmodel, the single hub centre contracts the\ndistance between {\\em every} pair of vertices, resulting in an average \ndistance less than 2. \n\n\\begin{figure}[!htbp]\n\\centerline{\\psfig{figure=/figs/wsOptL.eps,width=7.6cm,height=7.5cm}}\n\\caption{Variation in the normalized characteristic path length, $L/L(0)$,\nversus $p$ and $\\lambda$, for the WS model and optimization model\nrespectively.}\n\\label{fig:wsOptL}\n\\end{figure}\n\n\\subsection{Clustering coefficient}\n\nFigure \\ref{fig:wsOptC}, which compares the variation in clustering coefficient \nfor the two\nmodels, shows far more interesting behaviour. The drop in local\nconnectivity that is seen in the WS model does not occur {\\em at all}\nfor the optimized network because of the formation of hubs. Although\nthe clustering coefficient was not a characteristic that was sought to\nbe maximized, high cliquishness emerges. Figure~\\ref{fig:wsOptC}\nshows that the formation of hubs sustains the clustering coefficient\nat a value higher than that for the corresponding regular graph,\nunlike the WS model. Thus, the similarity between $p$ and $\\lambda$\nas control parameters is only valid for $L$.\n\n\\begin{figure}[!htbp]\n\\centerline{\\psfig{figure=/figs/wsOptC.eps,width=7.6cm,height=7.5cm}}\n\\caption{Comparison between the WS model and optimization model with respect to\nthe variation in their normalized clustering coefficient, $C/C(0)$.}\n\\label{fig:wsOptC}\n\\end{figure}\n\nBefore a more detailed analysis of Fig.\\ \\ref{fig:wsOptC}, we discuss the\nclustering\ncoefficient further. For a vertex $v$, its neighbourhood size $k_v$,\nplays a significant role. The smaller is $k_v$, \nthe smaller the number of possible intra-neighbourhood edges. Hence,\nvertices which lose in connectivity, gain in\ncliquishness. In a similar manner, vertices which gain in\nconnectivity, lose in cliquishness because of their larger\nneighbourhood size. This is because, although the vertices have a\nlarger number of intra-neighbourhood edges, they form a smaller\nfraction of the total number of possible edges. At the universal hub\nlimit, the hub centre has the least clustered neighbourhood owing to\nthe fact that all the remaining $n-1$ vertices form its neighbourhood.\nThe clustering coefficient can be shown to be approximately\n$(k-2)/n$. Although the average degree remains unchanged, the varying hub size\nand number can influence which neighbourhoods dominate the average clustering\ncoefficient.\n\nIn addition, a factor which influences the clustering within\nneighbourhoods, is the inclusion of a hub centre to a neighbourhood.\nThe effect on clustering differs depending on the range of the link\nbetween the vertex and the hub centre. If the range of the link is\nlarge and the vertex lies outside the hub, then a far away vertex is\nbeing included into an otherwise locally connected neighbourhood. The\nhub centre has little or no association with the remaining neighbours\nand so it lowers the average cliquishness. However, if the vertex\nlies within the hub, it amounts to including a node which is connected\nto all, or a large fraction of its neighbours. Hence, its\nneighbourhood becomes more clustered. This effect is more pronounced\nwhen (1) the neighbourhood size is small, (2) the vertex includes more than one\nhub centre in its neighbourhood, and (3) the hub whose centre is being included\nis composed of largely local neighbours.\nThus, unlike $L$ which is tuned by a single parameter, $C$ is\ncontrolled by many more, a point that we will return to shortly in the\nanalysis of Fig.\\ \\ref{fig:wsOptC}.\n\nFrom Fig.\\ \\ref{fig:wsOptC} we see as expected for the WS model, that \n$C(p)$ falls as\napproximately $C(0)(1-p)^3$ \\cite{3:barrat}, and eventually drops\nalmost to zero for the completely randomized graph. This is due to\nthe increasing number of inclusions of random far nodes, into\notherwise locally connected neighbourhoods. In contrast, the presence\nof hubs in the optimization model ensures that $C(\\lambda)$ never falls below\n$C(0)$; reaching its maximum when the network converges to a universal\nhub.\n\nKeeping in mind the evolution of the optimized network shown in \nFigs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}\nwe can understand qualitatively the behaviour of $C(\\lambda)$ in \nFig.\\ \\ref{fig:wsOptC}. When $\\lambda$ is small, the network is\ndominated by regular neighbourhoods (Fig.\\ (a-c)). As described\nearlier, vertices adjacent to hub centres gain in cliquishness due to their\nreduced neighbourhood sizes. Despite there being just a few small hubs, since\nthere are more reduced connectivity vertices than hub centres, the average $C$\nis raised slightly above that of a regular graph.\n\nWith a slight increase in $\\lambda$, \nFigs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}(d) shows a sharp increase in $C$. At\nthis point, the marked increase in hubs, with only a slight increase\nin size, results in a pronounced increase in the number of reduced\nconnectivity vertices. Many of these vertices have neighbourhoods which are \ncompletely clustered (called cliques), since in addition to their reduced \nsize, they include one or more hub centres into their neighbourhood.\nCliques, not surprisingly, dominate the average resulting in the large\njump in $C$. However, the emergence of long range inter-hub links in \nFigs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}(e-f) results in lowering $C$. Their introduction causes: (1) hub \ncentres to\nhave lowered cliquishness owing to the inclusion of distant nodes into their\nneighbourhoods, and (2) some reduced connectivity neighbourhoods to no \nlonger be complete cliques due to the inclusion of the centre of a hub, which\nhas lost local neighbours to inter-hub neighbours.\n\nAcross Figs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}(g-h), $C$ rises again due to increased cliques generated\nnot only \nby the increased number and size of hubs, but also because their larger\nsize allows for the inclusion of local neighbours once again.\nHowever, in Figs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}(i-k), while the few hubs gain in connectivity (and\nconsequently lose in cliquishness),\nthe remaining vertices veer towards uniformity in reduced connectivity. \nThe resulting marked reduction in the number of cliques, accounts for the\nslight drop at Figs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}(i), while the near uniform reduced connectivity serves to\nfurther raise $C$. \nFinally, in the universal hub limit, (Figs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}(k)), \nall vertices have a uniform reduced vertex connectivity, at the expense \nof the single hub centre. Having gained in cumulative connectivity, the \nhub centre has a very low clustering coefficient of approximately $(k-2)/n$. \nHowever, the remaining reduced sized neighbourhoods and their inclusion of the \nhub centre, ensures the average $C$ shoots up to its maximum. \n\nIn Figs.\\ \\ref{fig:hub variation} and \\ref{fig:2d hub variation}(l), the average clustering falls due to the non-uniformity in vertex\nconnectivity. However the inclusion of the universal hub centre into every\nneighbourhood ensures $C$ does not drop too much. \nThe multiple hubs result in a variation in vertex \nconnectivity, with hub centres gaining in connectivity at the expense\nof others. This leaves a few vertices having a {\\em single} connection. \nWith a neighbourhood of only 1, and no intra-neighbourhood connectivity,\nthese vertices are totally unclustered\\footnote{Such vertices have $k_v=1$, \nand $|E(\\Gamma_v)|=0$, which results in an invalid definition of $C_v$. \nTheir clustering coefficient can be taken to be either 0 or 1. To be\nnoted, is that rather than 0, a value of 1 would result in the average\nclustering coefficient being higher than that at the universal hub.} which \naccounts for the drop in $C$.\n\nThus, we see an interesting inter-play between neighbourhood size, hub \ncentre inclusions, and the number and range of inter-hub links. However, \nthe data for the variation of $C$ in the optimized model is noisy,\nmainly because $k$ is very small. Constraints in computational resources\nhave forced us to work with small $n$. Further, to maintain the sparseness \ncondition of $n \\gg k$, a low $k$ was used, which \ndoes not really satisfy the WS condition that $k \\gg 1$. Due to the small $k$,\neven a small loss in connectivity, can cause neighbourhood\ncliquishness to rise sharply. Although different factors come into\nplay during the $C$ variation, the spikes are due to the pronounced\neffect of reduced connectivity neighbourhoods, and in particular to\nthose of cliques. The cliques serve to maintain the entire\n$C$ variation higher than would probably result for higher $k$. \nWork is in progress to obtain data using large $k$ networks.\n\n\\subsection{Wiring cost and Degree}\n\nFigure~\\ref{fig:optWD}(a) displays the increase in the cost of wiring, or\nalternatively, the amount of wiring, with $p$ and $\\lambda$.\nThe comparison between the optimization model and WS model\nillustrates clearly the difference made by the inclusion of the\nminimal wiring constraint. For small\n$\\lambda$, both models exhibit similar wiring cost. At larger\n$\\lambda$ however, the absence of a similar constraint in the WS model\nresults in a much greater amount of wiring. The clear advantage exhibited by\nthe optimized networks persists until $\\lambda=1$, when optimization \nneglects the minimization of wiring cost entirely. At this point, the optimized\nnetwork uses greater wiring than its WS counterpart, but only slightly.\n\n\\begin{figure}[!htbp]\n\\centerline{\\psfig{figure=/figs/wsOptW.eps,width=7.6cm,height=7.5cm}\n \\psfig{figure=/figs/wsOptD.eps,width=7.6cm,height=7.5cm}}\n\\caption{Comparison between the WS and optimization models versus $p$ and\n$\\lambda$ respectively. (a) Variation in the wiring cost, $W$, normalized \nby the optimized value of $W(1)$. (b) Variation in the maximum degree, $D$, \nnormalized by network size, $n$. $D$ is equivalent to the size of the largest\nhub in the optimization model.} \n\\label{fig:optWD}\n\\end{figure}\n\nIn contrast to the WS model, a constraint on degree is not maintained\nin the optimized model. Figure~\\ref{fig:optWD}(b) shows how the\nmaximum degree increases with $\\lambda$ and $p$, for the two models.\nThe maximum degree, $D$, is normalized by the network size, $n$.\nFor the optimized network, $D$ is equivalent to the size of the\nlargest hub. At small $\\lambda$, there is no difference between the\ntwo models, but once hubs begin to emerge, $D$ increases sharply for\nthe optimized network. At both the universal hub, and the near random\ngraph limit, the maximum degree is $(n-1)$, the size of the universal\nhub. In contrast, each edge being rewired only once in the WS model \nallows for only a slight variation in degree.\nOne also observes a similarity in the variations\nof $W$ and $D$ for the optimized networks, since $W$ controls the\nsize of hubs, and hence $D$.\n\n\\subsection{Edge scale distribution}\n\nSince the WS rewiring mechanism exercises no restraint on the length\nscales of the rewired edges, the rewired edges are correspondingly\nuniformly distributed over the entire length scale range. In\ncontrast, for the minimally wired networks, lower length scales occur\nwith a higher probability.\n\nFigure~\\ref{fig:power-laws} shows plots of the edge scale probability\ndistribution on a log-log scale, where a power-law behaviour is seen.\nFigure~\\ref{fig:power-laws}(a) and (b) illustrate the edge scale\ndistributions at varying $\\lambda$, while \\ref{fig:power-laws}(c) is\na combined plot which demonstrates the variation in the power-law\ndistributions with $\\lambda$. Each distribution is displayed along\nwith its associated linear least-squares fit. The variation in their \nexponents, as obtained from the linear least-squares fit to the data \nagainst $\\lambda$, is shown in Fig.\\ \\ref{fig:power-laws}(d).\n\n\\begin{figure}[!htbp]\n\\centerline{\n \\psfig{figure=/figs/edgeScaleProbDistFit.0.001250.eps,width=6.5cm,height=6.5cm}\n \\psfig{figure=/figs/edgeScaleProbDistFit.0.125000.eps,width=6.5cm,height=6.5cm}}\n\\centerline{\n \\psfig{figure=/figs/edgeScaleProbDistFit.eps,width=6.5cm,height=6.5cm}\n \\psfig{figure=/figs/E.eps,width=6.6cm,height=6.5cm}}\n\\caption{Log-log plot of the edge scale probability distribution at (a)\n$\\lambda=0.00125$, and (b) $\\lambda=0.125$, for an $n=100, k=4$ optimized \nnetwork. (c) Combined plot of the probability distributions, each with its\nassociated linear least-squares fit. (d) Variation in the power-law \nexponents with $\\lambda$. Each distribution plot is averaged over 40\nrealizations.}\n\\label{fig:power-laws}\n\\end{figure}\n\nThe variation in the power law exponents with $\\lambda$, can be clearly \ndemarcated into two regions. The first, spanning two orders of magnitude \nvariation in $\\lambda$, exhibits a very slight exponent variation.\nFigure~\\ref{fig:power-laws}(a) illustrates the typical probability\ndistribution in this regime. Just two points emerge, since the high\nwiring cost constrains almost all edges to have a unit length scale,\nwith a very slight probability of a higher length scale. A sharp jump\nin the exponent marks the beginning of the second regime.\nFigure~\\ref{fig:power-laws}(b) illustrates the typical probability\ndistribution in this regime. It is seen that a straight line is a\nreasonably good fit to the data over a wide edge scale range. Finally, \nwhen $\\lambda = 1$ and a near random network is achieved, a flat distribution\nof length scales results with each scale being equally probable.\nThe combined plot of all the distributions with their associated\nleast-squares fits, although noisy, illustrates the behaviour of the \ndata (Fig.\\ \\ref{fig:power-laws}(c)).\n\nThe exponent variation clearly reveals two regimes of behaviour. The first jump\nin the exponent corresponds to the onset of small-world behaviour, the \nfirst perceptible reduction in $L$ that is seen in Fig.\\ \\ref{fig:wsOptL}. \nThis marks the beginning of the multiple\nscale regime, and is also a signature of hub formation. As we have\nmentioned previously, due to computational constraints we were unable\nto investigate larger networks. We believe that the noise in the data\nis due to the small size of the networks that we have studied.\n\nFinally, we wish to comment upon Kasturirangan's multiple scale\nhypothesis. Figure~\\ref{fig:power-laws}(d) demonstrates clearly the\nconnection between the onset of\nsmall-world behaviour and the emergence of multiple length scales in\nthe network. This clearly supports the claim in \\cite{3:kasturi} that\nsmall-worlds arise as a result of the network having connections that\nspan many length scales, and forms the first \nquantitative support of his hypothesis. It is to\nbe noted that multiple scales contribute to reducing $L$ in the WS\nmodel as well. However, since no restriction on the length of edges\nexists, {\\em any} non-zero $p$ will result in multiple scales. Hence,\nthe onset of small-world behaviour appears with a smooth reduction in $L$.\n\nWe have also observed a power law tail for vertex connectivity. We found \nthat most vertices had a small degree, and some were well short\nof the average degree, with vertices at hub centres gaining at their expense.\nOwing to the small network size however, the scaling range was rather limited, \nand so we have not included these results.\n\n\\section{Do similar networks exist?}\n\nAny efficient transportation network works under a similar underlying\nprinciple of maximizing connectivity while ensuring that the cost is\nminimized. Our results seem to indicate that any efficient\ntransportation network will be a small-world, and in addition will\nexhibit a similar hub connectivity. In a clear illustration of the underlying\nprinciple, any map of airline routes, or roadways shows big cities as being\nhubs of connectivity. This is hardly surprising though, since in\nsuch networks, a conscious effort is made toward such a minimization.\nHowever, the same philosophy may well be at work in natural transportation and\nother biological networks.\n\nWe would also like to point out that our observed hub structure can be seen in\na number of large complex networks ranging from fields as diverse as\nthe world wide web to the world of actors.\nKleinberg et al. \\cite{3:kleinberg} have observed the following\nrecurrent phenomena on the web: For any particular topic, there tend\nto be a set of ``authoritative'' pages focused on the topic, and a set\nof ``hub'' pages, each containing links to useful, relevant pages on\nthe topic. Also, it has been noted in \\cite{3:matthews} that the \nsmall-world phenomenon in the world of actors arises due to \n``linchpins'': hubs of connectivity \nin the acting industry that transcend genres and eras. In addition,\nBarab\\'{a}si and Albert \\cite{3:barabasi} explore several\nlarge databases describing the topology of large networks that span a\nrange of fields. They observe that independent of the system and the\nidentity of its constituents, the probability $P(k)$ that a vertex in\nthe network interacts with $k$ other vertices decays as a power-law,\nfollowing $P(k) \\sim k^{-\\gamma}$. The power law for the network\nvertex connectivity indicates that highly connected vertices (large\n$k$) have a large chance of occurring, dominating the connectivity;\nhence demonstrating the presence of hubs in these networks. Thus,\nhubs seem to constitute an integral structural component of a number\nof large and complex random networks, both natural and man-made. \n\n\\section{Conclusions}\n\nWatts and Strogatz showed that small-worlds capture the best of both\ngraph-worlds: the regular and the random. There has however been no\nwork citing reasons for their ubiquitous emergence. Our work is an\nstep is this direction, questioning whether small-worlds can\narise as a tradeoff between optimizing the average degree of\nseparation between nodes in a network, as well as the total cost of\nwiring.\n\nPrevious work has concentrated on small-world behaviour that arises as a\nresult of the random rewiring of a few edges with no constraint of the\nlength of the edges. On introducing this constraint, we have shown\nthat an alternate route to small-world behaviour is through the\nformation of hubs. The vertex at each hub centre contracts the\ndistance between every pair of vertices within the hub, yielding a\nsmall characteristic path length. In addition, the introduction of a\nhub centre into each neighbourhood serves to sustain the clustering\ncoefficient at its initially high value. We find that the optimized\nnetworks have $C \\ge C_{regular}$, and $L \\le L_{random}$ and thus\ndo better than those described by Watts and Strogatz.\n\nIn summary, our work lends support to the idea that a competitive\nminimization principle may underly the formation of a small-world\nnetwork. Also, we observe that hubs could constitute an integral\nstructural component of any small-world network, and that power-laws\nin edge length scale, and vertex connectivity may be signatures of\nthis principle in many complex and diverse systems.\n\nFinally, in future work, we will be studying larger networks that were\ncomputationally inaccessible to us at present. We are also\ninvestigating the application of the small-world architecture in the\nbrain, and also, a dynamic model will be considered to understand the\nemergence of small-worlds in social networks.\n\n"
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"name": "cond-mat0002076.extracted_bib",
"string": "\\begin{thebibliography}{10}\n\\input{references}\n\\end{thebibliography}"
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cond-mat0002077
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Modeling of Tunneling Spectroscopy in HTSC
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[
{
"author": "Yu. M. Shukrinov~$^{a,b}$"
},
{
"author": "A. Namiranian~$^{a}$"
},
{
"author": "A. Najafi~$^{a}$"
}
] |
The tunneling density of states of HTSC is calculated taking into account tight binding band structure, group velocity, and tunneling directionality for s-wave and d-wave gap symmetry. The characteristic density of states has quasiparticle peaks' asymmetry, flat s-wave and cusplike d-wave case subgap behavior, and asymmetric background. We consider that the underlying asymmetry of the conductance peaks is primarily due to the features of quasiparticle energy spectrum, and the d-wave symmetry enhances the degree of the peaks' asymmetry. Increasing of the lifetime broadening factor changes the degree of tunneling conductance peaks' asymmetry, and leads to the confluence of the quasiparticle and van Hove singularity peaks.
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"name": "modeling.tex",
"string": "%\\documentstyle[aps,multicol,epsf]{revtex}\n\\documentstyle[multicol,aps,prb,tabularx,epsf]{revtex}\n%\\documentstyle[preprint,aps]{revtex}\n\\def\\d{{\\rm d}}\n\\begin{document}\n\\title{Modeling of Tunneling Spectroscopy in HTSC}\n\\author{Yu. M. Shukrinov~$^{a,b}$, A. Namiranian~$^{a}$,A. Najafi~$^{a}$}\n\\address{$^{a}$ Institute for Advanced Studies in Basic Sciences, Gava Zang,\nZanjan 45195-159, Iran\\\\\n$^{b}$Physical Technical Institute of Tajik Academy of\nSciences\\\\\n 299/1 Aini Str., Dushanbe, 734063, Tajikistan, C.I.S.}\n\\date{\\today}\n\\maketitle\n\\begin{abstract}\nThe tunneling density of states of HTSC is calculated taking\ninto account tight binding band structure, group\nvelocity, and tunneling directionality for s-wave and d-wave gap symmetry.\nThe characteristic density of states has quasiparticle peaks' asymmetry, flat\ns-wave and cusplike d-wave case subgap behavior, and asymmetric background.\nWe consider that the underlying asymmetry of the conductance peaks\nis primarily due to the features of quasiparticle energy spectrum, and the\nd-wave symmetry enhances the degree of the peaks' asymmetry. Increasing of the\nlifetime broadening factor changes the degree of tunneling conductance peaks'\nasymmetry, and leads to the confluence of the quasiparticle and van Hove singularity\npeaks.\n\n\\end{abstract}\n\\pacs{74.50, 74.80.F}\n%%\n\\begin{multicols}{2}\n%%\n\n\n\\section{ INTRODUCTION}\nTunneling measurements on HTSC have revealed a rich variety of properties\nand characteristics [1-4]. They may be classified according to their low and high energy\nfeatures. With the low energy features we may attribute: (i) variable subgap\nshape of conductance, ranging from sharp, cusplike, to a flat, BCS-like\nfeature [1]; (ii) voltage and temperature dependence of quasiparticle\nconductivity [5,6]; (iii) subgap structure [2]; (iv) zero bias conductance\npeak (ZBCP) [7]. The high energy features include: (i) asymmetry of conductance\npeaks [1],(ii) van Hove singularity (VHS), (iii) conductance shape outside of the gap region (background (BG)) and\nits asymmetry [1]; (iv) dip feature [8]; (v) hump feature [8].These features are collected in schematic Fig 1 .\n\n\n\n\n\\begin{figure}\n\\epsfysize=2.0in\n\\centerline{\\epsffile{1.eps}}\n{\\bf Fig.1} Schematic $\\frac{dI}{dV}$-characteristic of NIS-structure with the main features.\n\\label{Fig1}\n\\end{figure}\nWhile the tunneling spectroscopy on\nconventional superconductors allows directly to find the energy gap of\nsuperconductor, the same measurements in HTSC are not as easily interpreted. Some times the same\nexperiments on the same samples show different results [9] : cusplike or flat subgap feature, symmetric or asymmetric\nconductance peaks. Usually the sharpest gap features are obtained when BG\nis weakly decreasing. A quantitative measure of it is the ratio\nof the conductance peak height (PH) to the background conductance:\n PHB=PH/BG.\nWhen the BG conductance is decreasing, the $PHB>2$, but when\nBG conductance is linearly increasing ($\\sim$V),\n$PHB<2$ . Kouznetsov and Coffey [10] and Kirtly and Scalapino [11]\nsuggested that the linearly increasing BG is arising from inelastic tunneling.\nAs was mentioned in [1], the conductance is dominated by quasiparticle tunneling and that the effect of Andreev reflection is not significant.\nA theoretical model for tunneling spectroscopy employing tight-binding\nband structure, $d_{x^2-y^2}$ gap symmetry, group velocity and tunneling\ndirectionality was studied by Z. Yusof, J. F. Zasadzinski, L. Coffey and\nN. Miyahawa [1]. An angle resolved photoemission spectroscopy (ARPES)\nband structure specific to optimally-doped BSCCO (Bi-2212) was used to\ncalculate the tunneling density of states for a direct comparison to the\nexperimental tunneling conductance. This model produces an asymmetric,\ndecreasing conductance background, asymmetric conductance peaks and\nvariable subgap shape, ranging from sharp, cusplike to a flat, BCS-like\nfeature.\nA standard technique in analyzing the tunneling conductance\nis to use a smeared BCS function\n\\begin{equation}\nN(E)=N(0)\\frac{E-i\\Gamma}{\\sqrt{(E-i\\Gamma)^2-\\Delta^2}}\n\\end{equation}\nin which a scattering rate parameter (lifetime broadening factor) $\\Gamma$ is used to take into account\nany broadening of the gap region in the DOS. Fig. 2 shows the DOS\ncalculated by formula (1) at $\\Delta=46 $ meV and $\\Gamma=9$ meV (a),\n$\\Gamma=3$ meV (b) and $\\Gamma=0$ (c). The characteristic features\nof the DOS is the flat subgap structure at small\n$\\Gamma$. This method can not explain the asymmetry of the conductance peaks observed in the\ntunneling experiments.\n\nIn the case of d-wave symmetry we have\n\\end{multicols}\n\n\\begin{figure}\n\\nonumber\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=2.4in\n\\epsffile{2a.eps}\n\\epsfxsize=2.3in\n\\epsffile{2b.eps}\n\\epsfxsize=2.3in\n\\epsffile{2c.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.8cm}\n{\\bf Fig.2} s-wave DOS at $\\Delta=46$ meV and different values of $\\Gamma$, calculated by formula (1).\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=2.3in\n\\epsffile{3a.eps}\n\\epsfxsize=2.3in\n\\epsffile{3b.eps}\n\\epsfxsize=2.4in\n\\epsffile{3c.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.8cm}\n{\\bf Fig.3} d-wave DOS at $\\Delta=46$ meV and different values of $\\Gamma$, calculated by formula (2).\n\\vspace{0.3cm}\n\\begin{multicols}{2}\n\\begin{equation}\nN(E)=N(0)Re\\int_{0}^{2\\pi}\\frac{d\\phi}{2\\pi}\n\\frac{E-i\\Gamma}{\\sqrt{(E-i\\Gamma)^2-\\Delta_{0}^2cos^{2}(2\\phi)}}\n\\end{equation}\n\n\\noindent and DOS calculated by this formula are presented in Fig. 3.\nThe characteristic features of the DOS is the cusplike subgap structure. As was mentioned\nin [1] this standard technique requires that the comparison be made with\nnormalized tunneling conductance date, and since HTSC tunneling\nconductance can exhibit a varied and complex background shape, this\nprocedure may \"filter out\" too much information from the conductance\ndata. An alternative is to simply normalize the data by a constant.\\par\nIn [8] the tunneling data were first normalized by constructing a\n\"normal state\" conductance obtained by fitting the high bias data to a\nthird order polynomial. The normalized conductance date were compared\nto a weighted momentum averaged d-wave DOS\n\\begin{equation}\nN(E)=\\int f(\\phi)\\frac{E-i\\Gamma}\n{\\sqrt{(E-i\\Gamma)^2-\\Delta_{0}^2cos^{2}(2\\phi)}}d\\phi\n\\end{equation}\nHere $f(\\phi)$ is an angular weighting function, which allows for a better fit\nwith the experimental date in the gap region. A weighting function\n$f(\\phi)=1+0.4cos(4\\phi)$ was used which imposed a preferential angular\nselection of the DOS along the absolute maximum of the d-wave gap and\ntapers off towards the nodes of the gap. This is a rather weak directional\nfunction since the minimum of $f(0)$ along the nodes of the d-wave\ngap is still none-negligible [8].\\par\n\nA. J. Fedro and D. Koelling [12] have done the modeling of the normal state\nand superconducting DOS of HTSC, using tight-binding band structure,\nincluding the next nearest neighbors\n\\begin{equation}\n\\xi_k=-2t(cos(k_xa)+cos(k_ya))+4t'cos(k_xa)cos(k_ya)-\\mu\n\\end{equation}\n\nThe calculation showed two singularities in DOS: a van Hove\nsingularity in the center of energy band due to saddle point near ($\\pi$,0) at\n$t'=0$ and another at the lower edge of the energy band due to extra flattening\nout at (0,0).\nAs extended s-wave and d-wave superconducting DOS were considered\nin case of hole-doped situation ($\\mu<0$) for different hole concentration.\nThe Fermi surface for $t'=0$ and $t'=0.45t$ at the same concentration,\ncorresponding $\\mu/2t=-0.187$ which was used in [12] are presented in\nFig. 4a. It is needed to move up from the Fermi surface (set as the zero energy)\nto reach the point ($\\pi$,0) in case $t'=0$ and move down in case\n$t'=0.45t$. So, for $t'=0$ the Fermi energy lies to the left of the van Hove\nsingularity and will move away from it with increased hole doping while for\n$t'=0.45t$ it lies to the right and will move towards to with increased hole\ndoping (See Fig. 4b where the DOS for $t'=0$ and $t'=0.45$ are presented ).\n For calculation of the superconducting DOS Fedro and Koelling used formula\n\\begin{equation}\nN(E)=\\frac{1}{2}\\sum_{k}(1+\\frac{\\xi_k}{E_k})\\delta(E-E_k)+\n(1-\\frac{\\xi_k}{E_k})\\delta(E+E_k)\n\\end{equation}\n\nThis formula is the limit of the expression for tunneling\ndensity of states (6) at $\\Gamma=0$ and $|T_k|^2=1$, where $T_k$ is tunneling matrix element.\n\\end{multicols}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{4a.eps}\n\\hspace{0.5cm}\n\\epsfxsize=3.0in\n\\epsffile{4b.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.4} Fermi surfaces (left) and DOS (right) for $t'$=0 (solid lines) and $t'$=0.45t (dashed lines) in formula (4) at $\\frac{\\mu}{2t}$=-0.187 which corresponds\nhole - doped situation.\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{5s.eps}\n\\hspace{0.3cm}\n\\epsfxsize=3.0in\n\\epsffile{5d.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.5} DOS for $t'$=0 at different $\\Gamma$ for s-wave symmetry (a) and\nd-wave symmetry (b), calculated by formula (6).\n\\begin{multicols}{2}\nThe Fig.5a shows the result of calculation\n of the DOS for $t'$ = 0 at $\\Gamma=$ 0.07, 0.1 and 0.2 meV for s-wave\n symmetry which reflect the results of Fedro and Koelling. Fig.5b shows the same DOS for d-wave symmetry. In both cases the Fermi energy\n(set as the zero of energy) lies to the left of the van Hove singularity.\nThere is the peaks' asymmetry which is more pronounced at large $\\Gamma$.\n\\section{ Models and Methods}\nIn this paper we use the method for calculation of the DOS presented in [1].\nThe tunneling DOS of a superconductor is determined by the imaginary part\nof the retarded single particle Green's function\n\\begin{equation}\nN(E)=-\\frac{1}{\\pi}Im\\sum_{k}|T_k|^2G^{R}(k,E)\n\\end{equation}\nFor the superconducting state\n\\begin{equation}\nG^{R}(k,E)=\\frac{u_{k}^{2}}{E-E_k+i\\Gamma}+\n\\frac{v_{k}^{2}}{E+E_k+i\\Gamma}\n\\end{equation}\nwhere $u_k^2$ and $v_k^2$ are the usual coherence factors,\n\\begin{eqnarray}\nu_k^2=\\frac{1}{2}(1+\\frac{\\xi_k}{E_k})\\cr\nv_k^2=\\frac{1}{2}(1-\\frac{\\xi_k}{E_k})\n\\end{eqnarray}\nand $\\Gamma$ is the\nquasiparticle lifetime broadening factor. The energy spectrum of\nquasiparticles in the superconducting state is determined by\n\\begin{equation}\nE_k=\\sqrt{|\\Delta(k)|^2+\\xi_k^2}\n\\end{equation}\nwith the effective band structure extracted from ARPES experiments [13]\n\\begin{eqnarray}\n\\xi_k=C_0+0.5C_1[cos(k_xa)+cos(k_ya)]~~~~~~~~~~~~~~~~~~~~\\cr\n~~+C_2cos(k_xa)cos(k_ya)+\n0.5C_3[cos(2k_xa)+cos(2k_ya)]\\cr\n~+0.5C_4\n[cos(2k_xa)cos(k_ya)+cos(k_xa)cos(2k_ya)]\\cr\n+C_5cos(2k_xa)cos(2k_ya)\n\\end{eqnarray}\nHere $\\xi_k$ is measured with respect to the Fermi energy ($\\xi_k$=0), and\nthe phenomenological parameters are (in units of eV) $C_0=0.1305$,\n$C_1=-0.5951$, $C_2=0.1636$, $C_3=-0.0519$, $C_4=-0.1117$, $C_5=0.0510$.\\par\n\\end{multicols}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=2.1in\n\\epsffile{6a.eps}\n\\hspace{0.5cm}\n\\epsfxsize=2.1in\n\\epsffile{7a.eps}\n\\hspace{0.5cm}\n\\epsfxsize=1.6in\n\\epsffile{8.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n\\noindent {\\bf Fig.6}(left) 3D-plot of energy spectrum of normal\nstate according to formula (9).\\\\\n{\\bf Fig.7}(center) 3D-plot of coherence factor $u_k^2$ according to formula (8).\\\\\n{\\bf Fig.8}(right) Fermi surface corresponding to the $\\xi_k=0$ in formula (10). Dark straight line shows the line of directional tunneling, the dashed lines show the angular spread $\\Theta_0$.\n\\begin{multicols}{2}\nFig. 6 shows the three dimensional image of function (10) .\nThere are saddle\npoint in ($\\pi$,0) and flattening out of the energy band at (0,0) which\nlead to the van Hove singularities in the DOS. The three dimensional graph of the coherence factor $u_k^2$\nis shown in Fig. 7.\\par\nSince quasiparticles with momentum perpendicular to the barrier interface\nhave the highest probability of tunneling, the tunneling matrix element $|T_k|^2$ reveals\na need for factors of directionality $D(k)$ and group velocity $v_g(k)$ [1]. The group velocity\nfactor is defined by\n\\begin{equation}\nv_g(k)=|\\vec \\nabla_k\\xi_k.\\hat n|=\n|\\frac{\\partial\\xi_k}{\\partial k_x}cos(\\theta)+\n\\frac{\\partial\\xi_k}{\\partial k_y}sin(\\theta)|\n\\end{equation}\nwhere the unit vector $n$ defines the tunneling direction as shown in Fig. 8,\nwhich is perpendicular to the plane of the junction.\\par\nThe directionality function $D(k)$ is defined by\n\\begin{equation}\nD(k)=exp[-\\frac{k^2-(\\bf k.\\hat n)^2}{( {\\bf k.\\hat n})^2\\Theta_0^2}]\n\\end{equation}\nHere $\\Theta_0$ defines the angular spread of the quasiparticle momentum with none-negligible tunneling probability\nwith respect to n. The tunneling matrix element $|T_k|^2$ is written as\n\\begin{equation}\n|T_k|^2=v_g(k)D(k)\n\\end{equation}\nThe three dimensional graphs of group velocity $v_g(k)$, directionality $D(k)$\nand tunneling matrix element $|T_k|^2$\nfunctions are shown in Fig 9.\n\\section{Results and discussions}\nDifferent factors may lead to the changing of the energy gap $\\Delta _0$ in HTSC.\nIn particular, strong effects are caused by nonmagnetic impurities [14].\nIn superconductors with d-wave symmetry the nonmagnetic impurities\ndestroy the superconductivity very efficiently. Possibility to destroy of Cooper pairs by\nimpurities leads to their finite lifetime.\nIf the state with the quasiparticle is not stationary state, it must attenuate with time\ndue to transitions to other states.\nThe corresponding wave function has the form\n$e^{-i\\xi (p)t/\\hbar-\\Gamma t/\\hbar}$, where $\\Gamma$ is\nproportional to the probability of the transitions to the other states.\nIt may be interpreted as the energy of the quasiparticle\nhas the imaginary addition $-i\\Gamma$. The relation between $\\Gamma$ and\n\\end{multicols}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=2.1in\n\\epsffile{9a.eps}\n\\epsfxsize=2.3in\n\\epsffile{9b.eps}\n\\epsfxsize=2.0in\n\\epsffile{9c.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.9} 3D plot of the group velocity function (a), the directionality function\n(b) and the tunneling matrix element according to formulas (11), (12) and (13),\ncorrespondingly.\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{10as.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{10bs.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-1.78cm}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{10ad.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{10bd.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.10} The changing of DOS with energy gap $\\Delta_0$ for s- (a,b) and d-wave (c,d) symmetry at $\\Gamma$=3 meV (a,c) and $\\Gamma$=9 meV (b,d) without of effects of directionality and group velocity.\n\\begin{multicols}{2}\n\\noindent lifetime of quasiparticle $\\tau_s$\nis $\\Gamma=\\hbar/\\tau_s$. Hence, the impurities lead to changing $\\Delta_0$ and\nwe may do modeling of the\ninfluence of impurities on tunneling conductance by numerical calculation of DOS\nN(E) considering different\nvalues of $\\Delta_0$ in formula (6). Here we present results of calculation N(E) at\n$\\Delta_0=\\alpha\\Delta_{00}$,\nwhere $\\alpha$=0.2, 0.4, 0.6, 0.8, 1 and $\\Delta_{00}$= 46 meV.\\par\nThe peculiarities of the quasiparticle energy spectrum (10) play an essential role in\nexplanation of the\nconductance features. Here, based on the numerical calculation of DOS\nwe consider that\nthe underlying asymmetry of the conductance peaks is primarily due to the features\nof quasiparticles energy\nspectrum. The d-wave gap symmetry simply enhances the degree of the peaks\nasymmetry. The last one is also\nchanged by changing the tunneling direction.\\par\nFig. 10 shows the results of the numerical calculations of the DOS at\n$\\Gamma _0 =3$ meV (a,c) and $\\Gamma _0 =9$ meV (b,d) without effects\nof group velocity and directionality as for s-wave (a,b) and d-wave (c,d) gap\nsymmetry, correspondingly for different values of energy gap $\\Delta_0$.\nWe have decreased the energy gap $\\Delta _0$, starting from $\\Delta _0=46$meV.\nFor clarity we present only three characteristic curves, which corresponds to\n$\\alpha \\Delta _0$ with $\\alpha=$1, 0.6 and 0.2.\nWe exclude the effects of group velocity and directionality to demonstrate that\nthey are not responsible\nfor peaks asymmetry.\\par\nThere is the asymmetry of the quasiparticle peak heights as for s- and d-wave\nsymmetry. So, the origin of\nthe peaks asymmetry is not due to d-wave symmetry of the energy gap of HTSC.\nThere is more flat subgap behavior of DOS in the case of s-wave symmetry in\ncomparing with the d-wave case. The increase of lifetime broadening factor\n$\\Gamma$ leads to the enhance of the peaks' asymmetry. There are van Hove\nsingularities in the DOS at small $\\Gamma$. The increase of $\\Gamma$ leads to\nthe confluence\nof the quasiparticle and VHS peaks and this results to the enhance\nof the DOS peaks asymmetry due to saddle point in energy spectrum (10)\nat ($\\pi$,0).\nAlso note to the asymmetry of the background as for s- and d-wave\ngap symmetry.\\par\nFig. 11 shows the $\\Delta _0$-dependence of DOS taking into account the effects\nof group velocity and directionality at $\\Gamma=3$ meV (a,c) and\n$\\Gamma=9$ meV (b,d) for s-wave (a,b) and d-wave (c,d) gap symmetry.\nAs in [1] we have taken $\\Theta =0.25$ and $\\Theta _0 =0.1$.\\par\nThere is also quasiparticle peaks' asymmetry similar to s- and d-wave cases.\nBut in d-wave case the asymmetry is more stronger than in s-wave.\nThe effects of group velocity and directionality lead to disappear of the VHS\nin DOS.\nThe increase of $\\Gamma$ enhances the quasiparticle peak asymmetry.\nThe most strong effect of energy band structure on the DOS occurs along\n$k_x$-axis due to van Hove singularity at $(\\pi,0)$.\\par\n\\end{multicols}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{11as.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{11bs.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-1.8cm}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{11ad.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{11bd.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.11} The changing of DOS with energy gap $\\Delta_0$ for s- wave\n(a,b) and d-wave (c,d) gap symmetry at $\\Gamma$=3 meV (a,c) and\n$\\Gamma$=9 meV (b,d) with the\n effects of directionality and group velocity.\n\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{12as.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{12bs.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-1.78cm}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{12ad.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{12bd.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.12} Effects of directionality on the DOS as in s-wave (a,b) and d-wave gap symmetry at $\\Gamma$=3 meV (a,c) and $\\Gamma$=9 meV (b,d).\n\n\\begin{multicols}{2}\nFig. 12 has demonstrated this effect. We have presented the DOS at different\n$\\Theta$ at $\\Gamma=3$ meV (a,c) and $\\Gamma=9$ meV (b,d) as for s-wave\n(a,b) and d-wave (c,d) gap symmetry.\nIn the case of s-symmetry the position of the quasiparticle peaks is constant excluding the direction\nalong $k_x$( $\\Theta=0$). We pay attention to the strong peaks' asymmetry\nin this case.\\par\nIn the case of d-wave symmetry we have practically the same behavior around\n$k_x$ direction as for s-wave, but energy gap is changed due to the\n$\\Theta$-dependence of $\\Delta_0$ and correspondingly, the quasiparticle peaks\nare shifted to the zero energy.\\par\n Fig.13 shows the $\\Theta_0$-changing of DOS at $\\Gamma=3$ meV (a,c)\nand $\\Gamma=9$ meV (b,d) as for s-wave (a,b) and d-wave (c,d) gap symmetry.\nThe increase of $\\Theta_0$ means the taking into play (inclusion) the states,\nclose to ($\\pi,0 $). It is reflected as an appearance of the van Hove singularity\nas in case of s-wave and d-wave gap symmetry at small $\\Gamma$. The VHS\nis more pronounced in case of d-wave in comparing with s-wave symmetry. The increase of $\\Gamma$ leads to confluence of the quasiparticle and VHS peaks.\\par\nWe consider that the absence VHS peak on the experimental\n$dI/dV$-characteristics means the enough large lifetime broadening factor\n$\\Gamma$ in that HTSC material.\n\nThe origin of the peaks asymmetry in the tunneling DOS was studied in [1]\nby considering the role of the tunneling matrix element $|T_k|^2$ in the clean\nlimit case ($\\Gamma=0$), where for the calculation N(E) was used the formula (5).\\par\nWe repeat the explanation of the paper [1] because we believe that the following\nconclusion on the origin of the peaks asymmetry must be different.\nAt $E> 0$ (positive bias voltages) the first term of (5) contributes to the N(E) because of\n$\\delta(E_k-E)$.\nIn this case, as can see from Fig. 7 and Fig. 9c and Fig. 14a $|T_k|^2$ selects only a\nrelatively short region of states in k-space in which $u_k^2>0$.\nThese are the states with $\\xi_k>0$ (above the FS).\nFor the majority of states integrated over ( see again Fig.7\nand Fig.9c).\nAt $E< 0$ (negative bias voltages) the second term of (5) contributes to the\nDOS because of $\\delta(E_k+E)$. In this case, as can see from Fig.7 and\nFig. 9c and Fig. 14b, $|T_k|^2$ selects out a large region of k states where $v_k^2 >0$,\nin fact, equal to one. These states are below the Fermi surface, where $\\xi_k<0$.\nThe overall effect then is to have a large negative bias conductance compared\nto the positive one. This is true as for s- and d-wave symmetry. Hence, the\nunderlying asymmetry of the conductance peaks is primarily due to the band\nstructure $\\xi_k$ and d-wave symmetry simply enhances the degree of asymmetry\nof the peaks. So, the peaks' asymmetry existing as for s-wave and d-wave symmetry\nis sensitive to\nband structure $\\xi_k$.\n\\end{multicols}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{13as.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{13bs.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-1.78cm}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=3.0in\n\\epsffile{13ad.eps}\n\\hspace{-0.6cm}\n\\epsfxsize=3.0in\n\\epsffile{13bd.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.13} Numerical calculation of the quasiparticle DOS with a s-wave (a,b)\nand d-wave (c,d)\ngap symmetry at $\\Gamma$=3 meV (a,c)\n and $\\Gamma$=9 meV (b,d)\nfor different spread $\\Theta_0$.\n%\\vspace{-1cm}\n\\begin{figure}\n\\begin{center}\n\\leavevmode\n\\hbox{%\n\\epsfxsize=2.0in\n\\epsffile{15a.eps}\n\\hspace{1cm}\n\\epsfxsize=2.0in\n\\epsffile{15b.eps}}\n\\end{center}\n\\end{figure}\n\\vspace{-0.6cm}\n{\\bf Fig.14} ARPES energy spectrum along $\\Theta=0.25$. The values of\ncoherent factors correspond to $E>0$ (a) and $E<0$ (b).\n\\begin{multicols}{2}\nIn summary, by changing of the energy gap $\\Delta_0$ in HTSC one may\nmodel the influence of\nnonmagnetic impurities on the DOS. We consider that the asymmetry of the\nquasiparticle peaks is due to the\nspecific features of the energy spectrum of HTSC and that the d-wave gap symmetry\nonly enhances\n the peaks' asymmetry.\\par\nThe absence of the VHS peak on the experimental $dI/dV$-charactristics\nmeans the large enough lifetime\nbroadening factor $\\Gamma$ in HTSC.\n\\section{Acknowledgement}\nWe thank Professor Y. Sobouti and Professor M.R. H. Khajehpour for useful discussions.\n\\end{multicols}\n\\begin{multicols}{2}\n\\begin{references}\n\n\\bibitem{} Yusof Z., Zasadzinski J.F., Coffey L., Miyakawa N., Modeling of tunneling\nspectroscopy in the high-Tc superconductors incorporating band structure,\ngap symmetry, group velocity, and tunneling directionality, {\\it Phys.Rev.} {\\bf B 58}\n(1998) pp.514-521.\n\\bibitem{} Schlenga K., Kleiner R., Hechtfischer G., Moessle M., Schmitt S., Mueller\nPaul, Helm Ch., Preis Ch.,Forsthofer F., Keller J., Veith M., Steinbess E.,\nTunneling Spectroscopy with intrinsic Josephson junctions in $BiSrCaCuO$ and\n$TlBaCaCuO$,{\\it Phys.Rev.} {\\bf B 57}(1998) pp.14518-14536.\n\\bibitem{}Shukrinov Yu.M., Nasrulloev Kh., Mirzoaminov Kh., Sarhadov I., Layered\nmodels and tunneling in HTS,{\\it Appl. Superconductivity.} {\\bf 2} (1994) pp.741-745.\n\\bibitem{} Shukrinov Yu.M., Stetsenko A., Nasrulloev Kh., Kohandel M., Tunneling in\nHTS, {\\it IEEE Transaction on Appl. Supercond.} {\\bf 8} (1998) pp.142-145.\n\\bibitem{} Latyshev Yu.I.,Yamashita T., Bulaevskii L.N., Graf M.J.,Balatsky A.V.,\nMaley M.P., Interlayer Transport of Quasiparticles and Cooper pairs in\n$BiSrCaCuO$ Superconductors, Cond-Mat/9903256, 17 Mar 1999, pp.1-4.\n\\bibitem{} Shukrinov Yu., Seidel P., Scherbel J., Quasiparticle current in the\nintrinsic Josephson junctions in $TBCCO$, will be published.\n\\bibitem{}Cucolo A.M., Zero bias conductance peaks in high-Tc superconductors:\nclues and ambiguities of two mutually excluding models, {\\it Physica} {\\bf C 305}\n(1998) pp.85-94.\n\\bibitem{} Ozyuzer L., Yusof Z., Zasadzinski J., Li T., Hinks T., Gray K.E.,\nTunneling Spectroscopy of $Tl_2Ba_2CuO_6$, Cond-mat/9905370, 25 May 1999,pp.1-7.\n\\bibitem{} Y.DeWilde, N.Miyakawa, P.Guptasarma, I.Iavarone, L.Ozyuzer, J.F.Zasadzinski, P.Romano, D.G.Hinks, C.Kendziora, G.W.Crabtree, and K.E.Gray, Unusual Strong-Coupling Effects in the Tunneling Spectroscopy of Optimally Doped and Overdoped $BiSrCaCuO$, {\\it Phys.Rev.Lett.} 80 (1998) pp.153-156.\n\\bibitem{} K.Kouznetsov and L.Coffey, Theory of tunneling and photoemmission spectroscopy for high-temperature superconductors, {\\it Phys.Rev.}{\\bf B, 54} (1996) pp.3617-3621.\n\\bibitem{} J.R.Kirtley and D.J.Scalapino, Inelastic-tunneling model for the linear conductance\nbackground in the high-$T_c$ superconductors, {\\it Phys.Rev.Lett.}, 65 (1990) pp.798.\n\\bibitem{} A.J.Fedro and D.D.Koelling, Interplay between band structure and gap symmetry in a two-dimmensional anisotropic superconductor, {\\it Phys.Rev.}{\\bf B, 47} (1993) pp.14342-14347.\n\\bibitem{} M. R. Norman, M. Randeria, H. Ding, and Campuzano, Phenomenological models for the gap anisotropy of $Bi_2Sr_2CaCu_2O_8$ as measured by angle-resolved photoemission spectroscopy, {\\it Phys. Rev.} {\\bf B 52 }(1995) pp.615-622.\n\\bibitem{} A. A. Abrikosov, On the nature of the order parameter in HTSC\nand influence of impurities, {\\it Physica} {\\bf C}, 244 (1995) 243-255\n\\end{references}\n\\end{multicols}\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n"
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"name": "cond-mat0002077.extracted_bib",
"string": "\\bibitem{} Yusof Z., Zasadzinski J.F., Coffey L., Miyakawa N., Modeling of tunneling\nspectroscopy in the high-Tc superconductors incorporating band structure,\ngap symmetry, group velocity, and tunneling directionality, {\\it Phys.Rev.} {\\bf B 58}\n(1998) pp.514-521.\n\n\\bibitem{} Schlenga K., Kleiner R., Hechtfischer G., Moessle M., Schmitt S., Mueller\nPaul, Helm Ch., Preis Ch.,Forsthofer F., Keller J., Veith M., Steinbess E.,\nTunneling Spectroscopy with intrinsic Josephson junctions in $BiSrCaCuO$ and\n$TlBaCaCuO$,{\\it Phys.Rev.} {\\bf B 57}(1998) pp.14518-14536.\n\n\\bibitem{}Shukrinov Yu.M., Nasrulloev Kh., Mirzoaminov Kh., Sarhadov I., Layered\nmodels and tunneling in HTS,{\\it Appl. Superconductivity.} {\\bf 2} (1994) pp.741-745.\n\n\\bibitem{} Shukrinov Yu.M., Stetsenko A., Nasrulloev Kh., Kohandel M., Tunneling in\nHTS, {\\it IEEE Transaction on Appl. Supercond.} {\\bf 8} (1998) pp.142-145.\n\n\\bibitem{} Latyshev Yu.I.,Yamashita T., Bulaevskii L.N., Graf M.J.,Balatsky A.V.,\nMaley M.P., Interlayer Transport of Quasiparticles and Cooper pairs in\n$BiSrCaCuO$ Superconductors, Cond-Mat/9903256, 17 Mar 1999, pp.1-4.\n\n\\bibitem{} Shukrinov Yu., Seidel P., Scherbel J., Quasiparticle current in the\nintrinsic Josephson junctions in $TBCCO$, will be published.\n\n\\bibitem{}Cucolo A.M., Zero bias conductance peaks in high-Tc superconductors:\nclues and ambiguities of two mutually excluding models, {\\it Physica} {\\bf C 305}\n(1998) pp.85-94.\n\n\\bibitem{} Ozyuzer L., Yusof Z., Zasadzinski J., Li T., Hinks T., Gray K.E.,\nTunneling Spectroscopy of $Tl_2Ba_2CuO_6$, Cond-mat/9905370, 25 May 1999,pp.1-7.\n\n\\bibitem{} Y.DeWilde, N.Miyakawa, P.Guptasarma, I.Iavarone, L.Ozyuzer, J.F.Zasadzinski, P.Romano, D.G.Hinks, C.Kendziora, G.W.Crabtree, and K.E.Gray, Unusual Strong-Coupling Effects in the Tunneling Spectroscopy of Optimally Doped and Overdoped $BiSrCaCuO$, {\\it Phys.Rev.Lett.} 80 (1998) pp.153-156.\n\n\\bibitem{} K.Kouznetsov and L.Coffey, Theory of tunneling and photoemmission spectroscopy for high-temperature superconductors, {\\it Phys.Rev.}{\\bf B, 54} (1996) pp.3617-3621.\n\n\\bibitem{} J.R.Kirtley and D.J.Scalapino, Inelastic-tunneling model for the linear conductance\nbackground in the high-$T_c$ superconductors, {\\it Phys.Rev.Lett.}, 65 (1990) pp.798.\n\n\\bibitem{} A.J.Fedro and D.D.Koelling, Interplay between band structure and gap symmetry in a two-dimmensional anisotropic superconductor, {\\it Phys.Rev.}{\\bf B, 47} (1993) pp.14342-14347.\n\n\\bibitem{} M. R. Norman, M. Randeria, H. Ding, and Campuzano, Phenomenological models for the gap anisotropy of $Bi_2Sr_2CaCu_2O_8$ as measured by angle-resolved photoemission spectroscopy, {\\it Phys. Rev.} {\\bf B 52 }(1995) pp.615-622.\n\n\\bibitem{} A. A. Abrikosov, On the nature of the order parameter in HTSC\nand influence of impurities, {\\it Physica} {\\bf C}, 244 (1995) 243-255\n"
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